Filozofski vestnik Filozofski vestnik THINKING THE INFINITE PENSER L’INFINI Edited by | Sous la direction de David Rabouin, Jana Ndiaye Berankova, Jelica Šumiè Riha Filozofski vestnik ISSN 0353-4510 Uredniški odbor | Editorial Board Matej Ažman, Rok Benèin, Aleš Bunta, Aleš Erjavec, Marina Gržiniæ Mauhler, Boštjan Nedoh, Peter Klepec, Tomaž Mastnak, Rado Riha, Jelica Šumiè Riha, Tadej Troha, Matjaž Vesel, Alenka Zupanèiè Žerdin Mednarodni uredniški svet | International Advisory Board Alain Badiou (Pariz/Paris), Paul Crowther (Galway), Manfred Frank (Tübingen), Axel Honneth (Frankfurt), Martin Jay (Berkeley), John Keane (Sydney), Steven Lukes (New York), Chantal Mouffe (London), Herta Nagl-Docekal (Dunaj/Vienna), Aletta J. Norval (Essex), Oliver Marchart (Dunaj/Vienna), J.G. A. 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Box 306, SI-1001 Ljubljana, Slovenia Tel.: (01) 470 64 65 Phone: +386 (1) 470 64 65 E-pošta: narocanje@zrc-sazu.si E-mail: narocanje@zrc-sazu.si © 2020, ZRC SAZU, Filozofski inštitut | Institute of Philosophy, Založba ZRC Oblikovanje | Design: Barbara Predan Tisk | Printed by: Cicero Begunje Naklada | Print run: 500 Filozofski vestnik Thinking the Infinite Penser l’infini Edited by | Sous la direction de David Rabouin, Jana Ndiaye Berankova, Jelica Šumiè Riha XLI | 2/2020 Izdaja | Issued by ZRC SAZU, Filozofski inštitut Institute of Philosophy Založnik | Published by Založba ZRC Ljubljana 2020 CIP - Kataložni zapis o publikaciji Narodna in univerzitetna knjižnica, Ljubljana 125:51(082) THINKING the infinite = Penser l'infini / edited by David Rabouin, Jana Berankova, Jelica Šumiè Riha ; izdaja ZRC SAZU, Filozofski inštitut = issued by Institute of Philosophy. - Ljubljana : Založba ZRC, 2020. - (Filozofski vestnik, ISSN 0353-4510 ; 2020, 2) ISBN 978-961-05-0509-9 1. Rabouin, David COBISS.SI-ID 45168387 Les éditeurs de ce recueil remercient les organisateurs du colloque « Alain Badiou : Penser l’Infini », qui eut lieu les 11 et 12 avril 2018 a la Galerie Nationale de Prague, pour avoir initié les débats qui sont a l’origine de ce recueil. Ce colloque fut organisé par le Cercle axiomatique de Prague et les éditions Suture avec l’Institut philosophique de l’Académie des sciences de la République Tcheque, l’Université Princeton, l’Institut fran­çais de Prague et l’Université Charles et la Galerie nationale de Prague. Plus de rensei­gnements peuvent etre trouvés sur le site des éditions Suture : https://suturepress.com The editors of this volume would like to thank to the organizers of the conference “Alain Badiou: Thinking the Infinite”, which took place on the 11thand 12th April 2018 in the National Gallery in Prague. This conference initiated many of the debates that are at the origin of this volume. It was organized by the Prague Axiomatic Circle and Suture Press along with the Institute of Philosophy of the Czech Academy of Sciences, Princeton University, the French Institute in Prague, Charles University and the National Gallery in Prague. More information can be found on the website of Suture Press: https://suture­press.com Contents/Sommaire Filozofski vestnik | Volume XLI | Number/Numéro 2 | 2020 7 Jana Ndiaye Berankova, David Rabouin, Jelica Šumiè Riha Préface 15 Alain Badiou Ontologie et mathématiques : Théorie des Ensembles, théorie des Catégories, et théorie des Infinis, dans L'Etre et l'événement, Logiques des mondes et L'Immanence des vérités Le Triangle philosophie – mathématiques – psychanalyse The Triangle of Philosophy – Mathematics – Psychoanalysis 37 Oliver Feltham “One or Many Ontologies? Badiou’s Arguments for His Thesis ‘Mathematics is Ontology’” 57 Nick Nesbitt Bolzano’s Badiou 69 Jelica Šumiè Riha La place de la mathématique : Badiou avec Lacan Le Modele ensembliste en discussion The Set-theoretical Model under Discussion 105 Michael Hauser Badiou and the Ontological Limits of Mathematics 119 Ronald Bolz Mathematics is Ontology? A Critique of Badiou's Ontological Framing of Set Theory 143 Tzuchien Tho Sets, Set Sizes, and Infinity in Badiou's Being and Event Le « Voir » et le « dire »: théorie des ensembles / théorie des categories “Seeing” and “Saying”: Set Theory / Category Theory 181 Charles Alunni Relation-objet et onto-logie, ensembles ou categories. Identité, objet, relation 199 René Guitart L’infini entre deux bouts. Dualités, univers algébriques, esquisses, diagrammes 249 David Rabouin Espace et nombre : deux voies dans l’ontologie ? 285 Norman Madarasz Beyond Recognition: Badiou’s Mathematics of Bodily Incorporation Grands Cardinaux et attributs de l'absolu Large Cardinals and the Attributes of the Absolute 311 Frank Ruda To the End: Exposing the Absolute 341 Jana Ndiaye Berankova The Immanence of Truths and the Absolutely Infinite in Spinoza, Cantor, and Badiou 361 Norma M. Hussey A New Hope for the Symbolic, for the Subject 397 Fernando Zalamea An Elementary Peircean and Category-Theoretic Reading of Being and Event, Logics of Worlds, and The Immanence of Truths 411 Notes on Contributors / Notes sur les auteurs 415 Abstracts / Résumés Kazalo Filozofski vestnik | Letnik XLI | Številka 2 | 2020 7 Jana Ndiaye Berankova, David Rabouin, Jelica Šumiè Riha Predgovor 15 Alain Badiou Ontologija in matematika: teorija množic, teorija kategorij in teorija neskonènosti v Biti in dogodku, Logikah svetov in Imanenci resnic Trikotnik filozofija – matematika – psihoanaliza 37 Oliver Feltham »Ena ali mnoge ontologije? Badioujevi argumenti za njegovo tezo ‘Matematika je ontologija’.« 57 Nick Nesbitt Bolzanov Badiou 69 Jelica Šumiè Riha Mesto matematike: Badiou z Lacanom Model teorije množic v razpravi 105 Michael Hauser Badiou in ontološke meje matematike 119 Ronald Bolz Je matematika ontologija? Kritika Badioujeve ontološke razlage teorije množic 143 Tzuchien Tho Množice, velikosti množice in neskonèno v Badioujevi Biti in dogodku »Videti« in »reèi«: teorija množic/teorija kategorij 181 Charles Alunni Razmerje-objekt in onto-logija, množice ali kategorije. Identite, objekt, razmerje 199 René Guitart Neskonènost med dvema koncema: Dvojnosti, algebrski univerzumi, skice, diagrami 249 David Rabouin Prostor in število: dve poti v ontologiji? 285 Norman Madarasz Onstran pripoznanja: Badioujeva matematika telesne inkorporacije Veliki kardinali in atributi absoluta 311 Frank Ruda Do konca: razkrivanje absoluta 341 Jana Ndiaye Berankova Imanenca resnic in absolutno neskonèno pri Spinozi, Cantorju in Badiouju 361 Norma M. Hussey Novo upanje za simbolno, za subjekt 397 Fernando Zalamea Elementarno peirceovsko in kategorijsko teoretsko branje Biti in dogodka, Logik svetov in Imanence resnic 411 Podatki o avtorjih 415 Povzetki Filozofski vestnik | Volume XLI | Number 2 | 2020 | 7–14 Jana Ndiaye Berankova, David Rabouin, Jelica Šumiè Riha Préface Le rapport que la philosophie d’Alain Badiou entretient aux mathématiques est a la fois intime, simple dans sa formulation et complexe dans ses attendus. Intime, il l’est de nécessité, par la nature meme de ce qu’est a ses yeux la phi-losophie. Pour Badiou, le discours philosophique n’est pas par lui-meme pro-ducteur de vérités. Il se trouve, et s’est toujours trouvé, dans une situation de questionnement et de réflexion par rapport a d’autres discours ou ces vérités se produisent et que Badiou nomme ses « conditions » : l’art, la politique, l’amour et les mathématiques. A l’époque de L’Etre et l’événement, le rapport a cette derniere condition se lais­sait résumer dans une formule simple et restée célebre : « les mathématiques sont l’ontologie » (formule dont on verra qu’elle fut justement questionnée par son créateur au congres de Prague dont ce recueil est issu). Les mathéma­tiques nous livrent la structure ultime de ce qui est : « l’etre en tant qu’etre ». En conséquence, la philosophie doit accompagner cette doctrine de l’etre dans ses évolutions. Elle doit notamment prendre acte de la forme radicalement nou­velle qu’elle a prise avec l’émergence de la théorie des ensembles et le retour de « l’infini actuel » dans l’ouvre de Georg Cantor. C’est vers cette théorie que s’est donc d’abord tourné Badiou, convaincu que la philosophie devait penser cette ontologie débarrassée de la figure séculaire du « Un » et donner libre cours a une pensée du « pur multiple » ou « multiple sans un ». Cette nouvelle articulation du rapport entre mathématiques et philosophie conduisait également a un rapport nouveau a la question de la vérité. De fait, l’axiomatique complexifie le rapport interne des mathématiques a la vérité en montrant non seulement que plusieurs modeles de la théorie des ensembles sont possibles, mais qu’il n’y a pas moyen de décider lequel est le « bon ». Tel fut l’apport décisif de Paul Cohen inventant dans les années 1960 la technique du « forçage » (forcing) qui permettait ainsi de créer des modeles ensemblistes en forçant littéralement certains énoncés a etre vrais. Pour Badiou, cela montrait que l’etre est toujours débordé par des « événements » qui viennent trouer le savoir et ou le Sujet se constitue dans une posture de fidélité a ces trouées évé­nementielles. Ainsi, Badiou rejette l’idée que le concept de « vérité » désigne la correspondance entre un certain discours et une « réalité ». Pour lui, la « vérité » est nécessairement attachée a la production meme de ce réel que recele l’idée d’indécidable. Il s’agit la du socle fondamental des theses exposées dans L’Etre et l’événement. Elles ouvraient a deux questions naturelles : ne pouvait-on envisager que les mathématiques continuent a se transformer d’une maniere comparable au changement radical qui était advenu avec l’émergence de la théorie des en­sembles ? C’est ce que semblait indiquer le développement de la théorie des ca-tégories, née apres-guerre, mais advenue surtout a partir des années 1960 dans les travaux de William Lawvere et Alexandre Grothendieck. Tel était d’ailleurs le ressort d’une possible objection contre le reve d’une « ontologie intrinseque » des mathématiques que fit valoir immédiatement Jean-Toussaint Desanti.1 Les mathématiques changent et si le philosophe y indexe l’ontologie, il devrait ré­viser son systeme a chacun de ces changements. L’autre question naturelle por­tait sur la théorie des ensembles elle-meme. Certes, les mécanismes de forcing inventés par Paul Cohen montraient que certains énoncés comme l’Hypothese du continu ne sont pas décidables dans la théorie axiomatisée par Zermelo et Fraenkel (dite « ZFC » quand on y adjoint l’axiome du choix). Il semble donc vain de considérer que cette théorie a un modele attendu et la vérité ne peut donc etre de l’ordre d’une naive correspondance. Mais ne pourrait-on envisager que ce phénomene ne soit pas intrinseque, mais provienne d’un défaut de cette axiomatique particuliere ? Une autre axiomatique ne pourrait-elle pas régler, au moins pour une part, cette indécidabilité ? Apres tout, certaines théories ma-thématiques puissantes comme la géométrie analytique cartésienne sont, ainsi que l’a démontré Alfred Tarski, décidables. Meme si nous savons depuis Gödel qu’une théorie au moins aussi puissante que l’arithmétique de Peano devra tou­jours etre incomplete, on peut se poser la meme question avec l’arithmétique de Peano et se demander si d’autres axiomatisations ne sont pas préférables ? Ceci donna lieu a tout un programme de recherche qui entendait compléter ZFC par Jean-Toussaint Desanti, « Quelques remarques a propos de l’ontologie intrinseque d’Alain Badiou », Les Temps Modernes 45 (526/1990), pp. 61–71. l’ajout de nouveaux axiomes, revenant a la position de cardinaux tres « grands » (large cardinals), sur l’existence desquels ZFC ne peut pas statuer par elle-meme. Tel fut, sans surprise, le ressort des deux tomes suivants de l’entreprise de L’Etre et l’événement : une confrontation serrée avec la théorie des catégories, sous l’égide de la notion de Topos, que Badiou proposa d’interpréter comme une phé­noménologie appuyée sur le socle ontologique fourni par les ensembles (Lo-giques des mondes) ; puis une analyse fine des grands cardinaux et de la struc­ture progressivement mise au jour grâce a eux de l’univers ensembliste, qui lui permit de repenser l’idée d’absolu (V, horizon inaccessible d’un « ensemble de tous les ensembles ») et de ses attributs (L’Immanence des vérités). Au terme de ce parcours, il apparaît qu’un des fils directeurs de l’ouvre est indé­niablement la question de l’infini – ou peut-etre plus précisément des différentes formes de l’infini. C’est ce theme qui servit de motif au grand colloque organisé a Prague en avril 2018, dont ce recueil est issu. Réunis dans la majestueuse Galerie Nationale, a l’invitation du Cercle Axiomatique de Prague en 2018, mathémati­ciens, philosophes et historiens des mathématiques furent amenés a se pronon­cer sur la pensée de l’infini que propose Alain Badiou et, plus généralement, le rapport que sa philosophie entretient aux mathématiques.2 A cette époque, rap-pelons-le, le troisieme tome, L’Immanence des vérités, n’était pas encore paru, et seules quelques personnes avaient pu en parcourir le texte. Mais c’est une autre annonce qui électrisa la salle ce jour-la : prenant la parole pour conclure le colloque, Badiou expliqua en effet que la these qui avait agité tant de commen­tateurs, selon laquelle les mathématiques sont l’ontologie, était en fait de l’ordre du slogan et devait etre finalement entendue cum grano salis. Nous reprenons le texte d’Alain Badiou en ouverture de ce volume, car il retrace précisément les différentes étapes de son parcours en clarifiant la maniere dont philosophie et mathématiques y ont été articulés. Meme si le slogan « les mathématiques sont 9 l’ontologie » avait le mérite de frapper les esprits, il avait l’inconvénient, précise 2 Le colloque « Alain Badiou : Penser l’infini » eut lieu du 11 au 12 avril 2018 a la Galerie Nationale de Prague (Veletržní Palác). Les intervenants furent : Charles Alunni, Alain Ba-diou, Burhanuddin Baki, Evelyne Barbin, Roland Bolz, Pierre Cartier, Oliver Feltham, René Guitart, Michael Hauser, Norma Hussey, Norman Madarasz, Jana Ndiaye Berankova, Nick Nesbitt, David Rabouin, Frank Ruda, Jelica Šumiè Riha, Tzuchien Tho et Fernando Zala­mea. Cet événement fut organisé en collaboration avec l’Institut philosophique de l’Acadé­mie des Sciences a Prague, l’Université Princeton, les éditions Suture, l’Institut français de Prague et l’Université Charles. Pour plus d’information : https://suturepress.com Badiou, de laisser croire qu’un choix nécessairement philosophique, relatif a l’ontologie, pouvait émerger de la mathématique elle-meme. Il entreprend donc de clarifier ce choix dans ses attendus et ses conséquences philosophiques, tout en le revendiquant comme tel (notamment par rapport a un autre modele, offert par la théorie des catégories). C’est de cette question du choix que part également l’article d’Oliver Feltham dans la section initiale de notre volume dont l’objectif est de placer l’ouvre de Badiou dans un contexte intellectuel plus général. Feltham se concentre sur une décision cruciale : celle d’une ontologie unique, plutôt que d’une pluralité d’ontologies. Il répond aux remarques critiques concernant le projet de Badiou soulignant les risques de la circularité dans le raisonnement qui élit la théo­rie de Zermelo-Fraenkel avec axiome du choix comme modele pour l’ontologie. La modélisation de l’ontologie par ZFC, implique-t-elle de nier la pluralité des modeles ? La réponse de Feltham a cette question est un tour de force théâ­tral évoquant l’utilité d’une certaine « anatomie de l’échec ». Nick Nesbitt, dans sa contribution « Le Bolzano de Badiou » propose de combler une lacune im­portante dans la réception de l’ouvre de ce dernier : le lien entre l’ontologie mathématique et la pensée du mathématicien, logicien et philosophe pragois Bernard Bolzano. Nesbitt lit Bolzano comme un précurseur lointain de Badiou. Il trace les similarités et les divergences entre ces deux penseurs : la critique de l’idéalisme post-Kantien de Bolzano, sa défense de l’infini actuel contre l’infini qualitatif hégélien, sa conception des mathématiques comme langage adéquat de l’ontologie, son réalisme rationaliste et platonicien, son intéret pour l’axio­matisation. Nesbitt remarque également qu’une étude attentive de la pensée de Bolzano pourrait ouvrir la voie a une véritable analyse structuraliste de ce que Marx décrivit comme « forme sociale ». Jelica Šumiè Riha conclut cette série d’articles en comparant les recours d’Alain Badiou et de Jacques Lacan aux ma-thématiques et analysant la triangulation de la philosophie, de la psychanalyse et des mathématiques dans leurs ouvres. Pour Lacan, le matheme est situé a la jonction de la vérité et du savoir. Si les mathématiques représentent pour lui un modele d’acces au réel de la structure, ce réel est saisi au sens de la rencontre d’un point d’impossible a écrire dans les termes de cette structure. Badiou re-marque qu’un tel d’impossible – un réel non-mathématisable – est représentatif d’une position « archi-scientifique » que cherche le psychanalyste. Le philo­sophe, lui, préfere plutôt rester sous condition des vérités qui surgissent dans le domaine des mathématiques. Une autre série d’articles s’attachent ensuite a questionner les décisions philo­sophiques qui sous-tendent au choix de la théorie des ensembles comme can-didat pour exprimer l’ontologie. Deux stratégies sont proposées. La premiere, positive, que met en ouvre Michael Hauser, consiste a préciser la maniere dont fonctionne la « condition » mathématique en spécifiant trois niveaux : celui de la philosophie proprement dite, qui opere a titre de méta-structure ou de mé­ta-ontologie ; celui des mathématiques en tant que le philosophe y opere cer­tains choix ; celui enfin de l’état donné des mathématiques dans leur ensemble. Hauser entreprend alors de montrer que cette structuration est reflétée dans l’ontologie ensembliste a partir du théoreme d’Easton (qui établit en substance que, pour un cardinal infini donné, le cardinal du nombre de ses parties peut etre choisi, sous certaines conditions minimales, parmi n’importe lequel de ses successeurs). L’espace du choix est donc ouvert de l’intérieur du second niveau (la théorie mathématique choisie par le philosophe comme socle ontologique) et commande une forme de circulation entre les trois niveaux. Une seconde stra­tégie d’approche, plus critique, consiste a questionner le choix de la théorie des ensembles comme modele a partir d’autres théories du multiple concurrentes. Une premiere alternative est fournie par la méréologie (c’est-a-dire la théorie des touts et des parties). Roland Bolz argumente que c’est dans cette derniere que Badiou aurait du chercher la théorie du « multiple pur », s’il voulait rester cohérent avec sa relecture de l’histoire de la philosophie. De fait, il est clair que les conceptions de « l’un et du multiple » qui se sont succédées depuis Platon et par rapport auxquelles se situe explicitement Badiou, se sont faites dans le cadre privilégié des rapports tout/partie. Bolz entend alors montrer que l’exi­gence d’une pensée du pur multiple « sans un » est parfaitement réalisée par certaines axiomatisations méréologiques contemporaines. Tzuchien Tho suit une stratégie similaire mais en questionnant plus directement la conceptuali­sation du comptage et de la mesure des infinis. Dans le cadre de la théorie des ensembles, en effet, un ensemble infini est défini par la propriété d’etre équi-potent a une de ses parties propres. En cela, la théorie de Cantor et Dedekind fait rupture avec les conceptions précédentes qui considéraient qu’il s’agissait la d’un paradoxe empechant la possibilité meme d’un infini actuel. Or des théories récentes, dites des « numérosités », ont montré qu’il est tout a fait possible de construire une théorie ensembliste de l’infini dans laquelle on préserve l’ancien axiome euclidien selon lequel « le tout est toujours plus grand que la partie ». Tho entreprend d’explorer les conséquences d’une telle possibilité au regard du choix fait par Badiou. La troisieme section de notre volume explore également une alternative au mo-dele ensembliste, mais fournie cette fois par une orientation que Badiou lui­meme a thématisé comme telle dans Logiques des mondes: la théorie des caté­gories. Comme le rappelle Charles Alunni, la question porte ici sur l’articulation entre la lettre et le diagramme – ou, selon les termes qui guident l’exploration de René Guitart, entre le « dire » et le « voir ». Reprenant le parcours qui a conduit Badiou a faire de la théorie des catégories, et plus particulierement de la théo­rie des topos, le ressort d’une phénoménologie, Alunni met en avant la dimen­sion profondément géométrique et spatialisante qui gouverne cette orientation de pensée. Il y trouve le ressort d’un questionnement sur la maniere dont une chose peut se trouver soustraite au réseau de relations qui la caractérise dans un monde. René Guitart s’installe également au lieu de la tension entre le « voir » et le « dire » pour indiquer comme on peut la faire jouer de l’intérieur de la théo­rie des catégories (plutôt qu’entre théorie des topos et théories des ensembles). Il propose pour cela sa propre construction a partir de la notion d’intégrateur, qui permet de retrouver les structures ensemblistes (notamment les cardinaux infinis) de l’intérieur du dispositif catégorique et de déployer sur cette base une pensée intrinseque du déploiement de l’infini « en personne ». Poursuivant la question de l’articulation du « voir » et du « dire », David Rabouin revisite l’his­toire de la théorie des ensembles elle-meme a partir de la tension entre théma­tisation du nombre et thématisation de l’espace. On y voit, en effet, opérer un double rôle des ensembles selon qu’on les considere comme langage ou comme théorie. Ceci permet de poser la question de l’articulation entre langage et onto-logie, mais aussi d’indiquer comment le couple ensemble/catégories peut etre reproblématisé dans ce cadre. Finalement, Noman Madarasz revient sur le dis-positif de Logiques des mondes et l’évolution qui fait passer du générique a ce que Badiou définit en termes catégoriques comme « corps » d’un sujet de vérité. Retraçant la construction de l’ouvrage, il s’interroge alors sur la maniere dont elle entend se déployer comme « phénoménologie », la ou semble mis en avant un programme d’inspiration plutôt « structuraliste ». La derniere série d’articles de ce recueil porte sur les grands cardinaux et la théo­rie de l’absolu. Elle reprend les éléments nouveaux présentés par Badiou dans L’Immanence des vérités. Frank Ruda inaugure cette section en démontrant que la question a laquelle Badiou essaie de répondre dans cet ouvrage pourrait etre reformulée ainsi : qu’est-ce qui rend une vérité vraiment vraie ? Réponse : son absoluité. L’Immanence des vérités déploie une stratégie argumentative dans la­quelle, comme chez Hegel peut-etre, l’absolu est aupres de nous des le début. La catégorie de l’absolu se trouve au centre du noud borroméen qui noue l’etre, l’apparaître et les vérités – elle comble des lacunes qui apparaissent comme des effets de la construction conceptuelle de L’Etre et l’événement et de Logiques des mondes. L’Immanence des véritésest donc avant tout une affirmation d’une liberté de la pensée, d’un choix entre le constructible et le générique. Le choix et la liberté du mathématicien sont également au centre de l'attention de Jana Ndiaye Berankova qui remarque que le lien entre la philosophie et les ma-thématiques doit etre pensé selon le raisonnement inductif et non pas déductif – a l’origine de ce lien se trouve la décision philosophique qui doit etre ensuite vérifiée par la richesse de ses conséquences (mathématiques ou autres). Ndiaye Berankova présente la hiérarchie des infinis proosée dans L’Immanence des vérités et analyse ce livre a travers la perspective des écrits de Georg Cantor – notamment a travers la notion de « infinitum absolutum » et la distinction entre infini consistant et inconsistant présente dans sa correspondance avec De­dekind. Elle remarque que le fait de décrire l’univers V de toutes les multiplicités pensables comme un « lieu intelligible » est une maniere de contourner le risque que cet absolu retombe dans le domaine de l’infini potentiel. L’Immanence des vérités est avant tout une tentative ambitieuse du renouvellement de la notion spinoziste des attributs de l’absolu. Norma Hussey supplémente cette présentation de L’Immanence des vérités en évoquant les développements les plus récents dans les mathématiques des grands cardinaux, notamment les hypotheses présentées par le mathématicien Hugh W. Woodin. Elle évoque la perspective de l’univers et du multivers en ma-thématiques et remarque que la conjecture « V = L-ultime » de Hugh Woodin n’a pas forcément les memes conséquences que l’hypothese traditionnelle de constructibilité « V = L ». Les concepts mathématiques présentés dans cet article vont certes au-dela de l’appareil mathématique utilisé par Badiou dans L’Im­manence des vérités, mais Hussey reste convaincue qu’ils peuvent etre intégrés dans le projet philosophique badiousien. Enfin, pour conclure, Fernando Zala­mea récapitule la trajectoire conceptuelle des trois principales ouvres d’Alain Badiou. Il fait une remarque provocatrice en imaginant un quatrieme volume fictif de L’Etre et l’événement dédié a la notion des universaux dans la théorie des topoi de Grothendieck et la théorie homotopique des types. Par ailleurs, Zalamea retrace les perspectives et les points d’interrogations en comparant son projet aux catégories de Charles Sanders Peirce. Au terme de ce parcours, ce volume offre ainsi un cheminement a travers l’en­semble de l’ouvre de Badiou, suivant le guide, parfois discretement présent, parfois au tout devant de la scene, de l’infini. Nous espérons qu’il donnera au lecteur l’envie de s’y plonger, ou de s’y replonger. Mais surtout, d’en prolonger le geste dans d’autres pensées vives. Filozofski vestnik | Volume XLI | Number 2 | 2020 | 15–34 | doi: 10.3986/fv.41.2.01 Alain Badiou* Ontologie et mathématiques Théorie des Ensembles, théorie des Catégories, et théorie des Infinis, dans L'Etre et l'événement, Logiques des mondes et L'Immanence des vérités I Comme nombre d'entre vous le savent, pour moi, la philosophie n’existe qu’au­tant qu’existent des procédures de vérité, et elle est sous condition de l’état his-torique de ces procédures. J’ai rangé les vérités dont l’espece humaine s’est, au fil des millénaires de son existence, montré productrice, sous quatre grands genres : les sciences, les arts, la politique et l'amour. La question est alors de savoir comment on interroge, tout au long de l’histoire de la philosophie, le lien entre la philosophie et ses conditions. La difficulté, que je voudrais ici résumer, est qu’il faut s’occuper, dans l’investigation du cor­pus historique de la philosophie tel que nous en héritons, de trois processus distincts. Le premier est de prendre en compte, a tout moment de l’histoire de la philoso­phie,l’état des quatre conditions et de leur impact sur la philosophiedans un lieu déterminé. C’est la vue panoramique, essentiellement historienne. Elle autorise qu’on distingue, plus ou moins efficacement, des époques ou des territoires philosophiques. Ainsi, quand on parle de « philosophie antique », ou de « phi-losophie médiévale », ou encore de « philosophie continentale », opposée a la « philosophie analytique », principalement américaine. Le second processus s’attache au repérage d’un probleme, interne a une condi­tion, et qui modifie tout le rapport antérieur de la philosophie au dispositif com-plet des conditions. C’est évidemment le cas de la mutation mathématique entraînée vers le Ve siecle A.C. par la découverte des longueurs « incommen­surables », qui fait basculer les mathématiques grecques, de l’arithmétique pythagoricienne vers la géométrie d’Eudoxe et Euclide, et la philosophie, de la recherche de l’Harmonie a une théorie des ruptures. On pourrait aussi bien évoquer les effets politiques de la Révolution française, qui contraint la philoso- * Professeur émérite a l'Ecole Normale Supérieure phie allemande, a partir de Fichte, a des remaniements dialectiques fondamen­taux, faisant venir au jour la force créatrice de la négativité. Le troisieme processus réside dans la possibilité qu’une philosophie – donc, d’abord, un philosophe – intervienne du point de la philosophie elle-meme, dans la dynamique d’au moins une des quatre conditions. C’est le processus rétroactif, de la philosophie vers ses conditions. Il n’est pas douteux par exemple que le platonisme ait, a longue portée, influencé la vision sociale de l’amour dans l’époque de sa spiritualisation courtoise, ou que la dialectique hégélienne ait eu une importance constituante pour la politique communiste telle que fon­dée par Marx. Ou encore, on peut suivre a la trace l’influence de la philosophie matérialiste et libertine, dérivée d’Epicure, dans l’ouvre théâtrale de Moliere et de quelques autres. Tout ça pour rappeler que le mot « condition » est distinct du mot « cause ». Il s’agit finalement, avec les arts, les sciences, les politiques, l’amour et la philo­sophie, de cinq processus enchevetrés, si meme il doit rester clair que la phi-losophie occupe la position singuliere de ne pouvoir exister qu’avec les quatre autres, lesquels, eux, peuvent exister par eux-memes. II Quand j’ai diffusé, il y a trente ans, comme on fait en politique d’un mot d’ordre, la formule « l’ontologie, c’est les mathématiques », je ne doutais pas de son succes, mais je n’anticipais pas correctement ses inconvénients. Car, a tout prendre, cette formule a l’avantage d’etre frappante, mais l’inconvénient d’etre approximative. Rapportant de façon en quelque sorte identitaire et brutale un concept typiquement philosophique, celui d’ontologie, a la disposition d’une science particuliere, les mathématiques, la formule ne tient pas assez compte du caractere complexe des relations entre la philosophie et ses conditions. Je vais donc revenir sur la relation entre mathématiques et philosophie a partir de mes considérations introductives. Partons du premier des trois processus que j’ai définis dans mon premier point, a savoir l’histoire globale du quatuor des conditions. Comment ces quatre conditions telles qu’elles se présentaient a moi, en France, il y a mettons cin­quante ans, dans le dernier tiers du XXe siecle, deviennent opératoires dans le champ philosophique ? Quelles inventions, quelles créations, quels problemes, attirent alors mon attention ? 1. Dans le devenir des mathématiques, c’est l’ouvre de Paul Cohen, qui, avec la théorie du forcing et le concept d’ensemble générique remanie la théorie des ensembles, meme par rapport aux inventions géniales de Gödel dans les années trente et quarante. C’est aussi la véritable percée de la théorie des catégories, qui tend a remplacer dans le champ mathématique la notion d’objet par celle de relation. 2. En politique, nous avons le bilan contrasté des vastes mouvements de masse qui ont animé la jeunesse universitaire et la classe ouvriere, presque dans le monde entier, pendant les années soixante et soixante-dix, notamment Mai 68 en France et la Grande Révolution Culturelle Prolétarienne en Chine, bilan que domine finalement l’échec global de ces mouvements, échec qui ac-compagne et commande la faillite des Etats socialistes, Russie et Chine com­prises, ainsi que les leçons que les communistes doivent tirer de cette faillite. 3. Dans les arts, la plus consistante et durable nouveauté est repérable dans les arts plastiques, avec les performances, qui font du corps de l’artiste un élé­ment décisif de son ouvre, et les installations, qui enregistrent la dimension provisoire et locale des structurations spatiales. Dans les deux cas, il s’agit de rendre provisoire tout agencement esthétique, de relativiser l’ouvre d’art dans le temps et dans l’espace, et de mettre ainsi fin a l’idée selon laquelle ladite ouvre aurait une valeur objective et éternelle. 4. En amour, la nouvelle liberté sociale dans le champ de la sexualité, la crise des autorités familiales, l’émancipation des femmes, la légalisation des pro-cédés anticonceptionnels, la promotion d’une vision festive de l’existence, l’autonomisation du simple désir comme un droit revendiqué : tout cela converge vers une précarisation du lien amoureux, voire – avec les « sites de rencontre » – vers une sorte de calcul commercial quant a sa valeur et a son éventuelle mise en ouvre. Cependant, travaille autrement le remaniement par Lacan du point de vue psychanalytique sur l’amour, avec la fameuse formule : « l’etre, c’est l’amour qui vient a y aborder dans la rencontre », laquelle fait de l’amour le lieu possible d’une ontologie du sujet. Regardés dans leur détail, ces matériaux conditionnants conduisent assez na­turellement, dans l’ordre de ce qui se présente comme « philosophie », a deux types de conséquences. D’une part, on trouve un relativisme culturel qui ne laisse plus de place a la notion de vérité universelle, tenue pour impériale et fictive, et qui privilégie la multiplicité des langues et des coutumes, le bariolage planétaire, et aussi les identités multiformes, préférées systématiquement aux grandes constructions a prétention globale. D’autre part, se développent des doctrines qui affirment la supériorité des actions sur les pensées, du mouve­ment pur sur l’organisation, de l’intuition sur l’Idée, de la vie sur les structures, de l’approche locale sur la valeur globale, des multiplicités complexes sur le dualisme dialectique, de l’affirmation sur la négativité, bref, qui reviennent, contre Platon, Descartes, ou Hegel, vers les stoiciens, Hume ou Nietzsche, ce qu’accomplit parfaitement Deleuze. Cependant, c’est a partir des memes matériaux « en situation » que mon désir proprement philosophique discerne, lui, comme tâche essentielle, de recons-truire, contre les deux tendances naturellement dominantes et du reste lar­gement complices, une discursivité spéculative capable d’organiser de façon neuve les questions que les courants dominants écartent de leur devenir, nom­mément celle de l’etre, celle de la vérité, et celle du sujet. Et puisque c’est de l’ontologie que nous sommes partis, commençons par elle. III Si je considere mon travail de pensée sur l’etre en tant qu’etre dans le contexte de l’histoire de cette question, je vois qu’on peut distinguer rien de moins que six possibilités quant a ce qui a finalement été nommé « ontologie ». D’abord deux positions finalement négatives : 1. Le concept d’etre est vide, il n’a aucune signification. C’est le point de vue dominant aujourd’hui, pour les raisons que j’ai dites. C’est depuis toujours la position sceptique ; c’est la position positiviste aussi bien, comme on le voit chez Auguste Comte ; c’est la position explicite des vitalistes, et d’abord de Nietzsche ; mais c’est aussi la position de Wittgenstein et de tout le cou-rant analytique américain. Pour tous ces penseurs, le mot « etre » est en fait une substantivisation illégitime du verbe « etre ceci ou cela », substantivisa­tion qui produit un pur non-sens. 2. Le concept d’etre a un sens, il a une valeur positive. Mais nous ne pouvons pas avoir une connaissance effective de son contenu. La « chose en soi » est située au-dela de nos facultés cognitives. C’est comme on sait la position de Kant, mais finalement c’est la position « historiale » de Heidegger : le nihi­lisme contemporain, lié a la souveraine violence de la technique, fait que nous avons non seulement (comme l’a fait la métaphysique depuis Platon) oublié le vrai sens de l’etre, sa destination, mais que nous avons oublié cet oubli meme. Nous sommes donc devenus totalement étrangers non seule­ment au sens de l’etre, mais meme a la question de ce sens, qui cependant nous constitue historialement. Ensuite quatre positions positives : Elles affirment toutes que le mot « etre » a un sens réel, et que nous pouvons avoir une connaissance vraie, fondée, de ce sens. Mais cette affirmation pre­miere se distribue ensuite en orientations essentiellement distinctes, et meme radicalement opposées. 3. La troisieme position ouvre la voie aux différentes formes du monothéisme : l’etre se donne, de façon explicite et concentrée, sous la forme de l’Un, le Grand Un, ou l’Un comme Infini. C’est largement la position de la métaphy­sique classique, que Heidegger n’a pas tort de définir comme « l’arraisonne­ment de l’Etre par l’Un ». En fait, c’est déja la position d’Aristote, pour qui l’etre est exhibé comme « acte pur » dans la transcendance d’un Dieu. Et le chemin de la donation du sens de l’etre prend chez lui la forme philoso­phique, qui sera longtemps dominante, d’une preuve de l’existence de l’Un-de-l’etre comme tel. Cependant, il y aura aussi le courant mystique, pour lequel l’acces a la transcendance de l’etre est une expérience vitale et non une preuve. Expérience dont le récit est poétique plutôt que logico-mathé­matique : il releve de la condition artistique, comme on le voit chez Saint-Jean de la Croix, et non de la condition scientifique, comme on le voit par exemple chez Malebranche. Mais dans les deux cas, la pensée-vie n’accede a l’etre que dans la forme d’une ascension vers l’Un-infini, forme moderne de l’Un-acte-pur d’Aristote. 4. Dans cette quatrieme orientation, l’etre se donne, non comme transcen-dance de l’Un-Dieu, rationnelle ou extatique, mais comme totalité de soi­meme, incorporant des expressions multiples de soi, toutes immanentes a sa propre unicité. L’envoi de cette orientation est donné par Parménide, qui conclut de l’inexistence du non-etre a l’absoluité-une de l’etre dont tous les existants apparents sont comme des facettes irréelles. L’apogée spéculative de cette vision est évidemment réalisée dans le systeme de Spinoza, ou l’on démontre l’unicité de la Substance (ou Nature), laquelle prodigue a l’inté­rieur d’elle-meme des modes multiples dont tout l’etre dérive de l’Un subs-tantiel. Hegel propose une version dynamique de l’orientation immanente : l’etre, en tant qu’absolu, est identique a son propre devenir multiforme. L’etre est le devenir dialectique de soi-meme, et le Savoir absolu procede a une récapitulation circulaire de ce devenir. 5. Dans cette cinquieme orientation, l’etre n’est donné comme pensable qu’en faisant l’économie de toute transcendance de l’Un, comme de toute totalisa­tion unifiante. L’etre, en effet est pure dispersion multiple, sur fond de vide. Autrement dit : l’Un (le vide) est du côté du non-etre, cependant que l’etre est dissémination atomique de soi-meme. C’est, depuis Démocrite, l’orientation qu’on peut dire matérialiste, en ce qu’elle fait l’économie de tout sens géné­ral de l’etre, au profit de la matérialité des atomes et de leurs combinaisons dans le vide. Epicure et Lucrece s’en réclament. 6. La sixieme orientation, enfin, affirme que la vraie pensée de l’etre ne réside ni dans l’Un transcendant, ni dans l’Un immanent, ni dans la dispersion ato­mique, parce que l’etre n’a pas d’autre etre que la relation et les mouvements qui transforment et lient entre elles les relations. Autrement dit l’etre se com­pose de relations entre relations. C’est la position d’Héraclite, et plus pres de nous de Nietzsche, de Bergson ou de Deleuze. Elle est irriguée aujourd’hui par la mathématique des catégories. En particulier, la catégorie des caté­gories propose une pensée diagrammatique de l’etre comme Relation des relations entre relations. Telle est l’orientation que résume le concept fonda-mental de foncteur, et son organisation systémique en faisceaux. C’est au regard de cette complexité de l’héritage philosophique quant a ce que peut etre une ontologie que, armé de ma vision de l’état contemporain des conditions, j’ai du choisir mon orientation propre. Bien entendu, « choisir » n’est ici que métaphorique : l’orientation s’impose a un sujet-philosophe plus qu’elle n’est tranquillement choisie parmi les six possibilités. Et elle s’est im­posée a raison de ma conviction, venue plus de la politique que des mathéma­tiques, qu’il fallait proposer une ontologie « matérialiste », c’est-a-dire étran­gere a toute transcendance, et qui cependant fasse l’économie du concept in-consistant de « matiere », lequel ne désigne jamais que l’Un caché, et en vérité finalement impensable, de la multiplicité évidente de ce qui est. Et c’est alors, comme déja l’avait fait le jeune Marx, vers la cinquieme orientation que je me suis tourné : l’affirmation que l’etre n’est que multiplicité pure, sans Un, et sans attribut spécifique, de type « matiere » ou « esprit ». Et voici qui est important : C’est seulement de l’intérieur de ce mouvement de pensée que je suis revenu vers la condition mathématique, pour chercher s’il s’y trouvait une structuration, aussi rigoureuse que possible, de ma décision spé­culative. Et je l’ai trouvée dans la théorie des ensembles, parce que j’ai interpré­té cette théorie, singulierement dans son axiomatisation de type ZFC, comme n’étant rien d’autre que l’étude systématique de toutes les formes possibles de multiplicités, sans Un ni qualité particuliere. Je suis alors retourné vers la phi-losophie, muni d’une possible fondation formelle de ma décision ontologique primordiale. Nous avons donc une sorte de cheminement circulaire, impliquant l’histoire de la philosophie quant a la question ontologique, mon etre de sujet-en-philo­sophie, l’état actuel de la condition mathématique, et derechef mon etre-phi­losophe, lequel va s’incorporer a l’histoire de la philosophie quant a la ques­tion ontologique. Cette circularité peut aussi se dire : état des possibles onto-logiques ; prise (par moi) d’une décision philosophique en rapport avec ces possibles ; mouvement rétroactif vers la condition mathématique ; décision philosophico-mathématique d’y trouver une forme adéquate a la décision on-tologique ; investissement de cette deuxieme décision dans la premiere, par la formalisation mathématique du concept de multiples-sans-Un et de ses va­riantes ; incorporation a l’histoire de la philosophie d’une proposition ontolo­gique supposée nouvelle (le livre L'Etre et l'événement). Dans ce mouvement circulaire, il est certes impossible d’examiner séparément mon usage des mathématiques et ma décision philosophique. Mais tout autant d’en tirer l’équation « ontologie = mathématiques ». Parce que l’énoncé de la décision initiale, a savoir « l’etre est multiplicité-sans-Un » n’est d’aucune façon un énoncé mathématique. Et le détour par la théorie moderne des ensembles ne vaut pas preuve de la validité de cet énoncé initial. L’alliance organisée entre mathématiques et philosophie n’est forte que quand on observe ses consé­quences. Et ce n’est qu’assez loin dans ces conséquences qu’on peut réellement apprécier cette portée. Dans les mathématiques, il faut aller au moins a la hau­teur des théoremes de Cohen concernant les sous-ensembles génériques. Et en philosophie, l’ampleur spéculative de mon propos ne se déchiffre que dans la dialectique entre etre et événement, ce qui veut en réalité dire : entre détermi-nation axiomatique et généricité, ou encore : entre les multiples singularisés par des propriétés précises, et les multiples universalisés par leur soustraction a toutes ces propriétés Ce long préambule me permet de revenir, dans des conditions nouvelles, a une question centrale, encore aujourd’hui tres disputée, tres critiquée : quelle est en fin de compte la fonction exacte de la théorie des ensembles dans le discours philosophique qui est le mien ? On peut répondre ainsi a cette question : Le systeme mathématique ZFC pro­pose au philosophe une connaissance scientifique claire et rigoureuse de toutes les formes possibles de la multiplicité pure (sans Un et sans prédicat empirique de type « matiere », « esprit », « atomes », « flux », etc.). Ces formes sont exclu­sivement définies par des éléments anonymes (des « ensembles ») a l’exclusion de quoi que ce soit d’autre, puisque les éléments d’un ensemble sont également des ensembles. Il n’y a pas de définition de ce que c’est qu’un ensemble, ce qui est cohérent avec leur fonction de pures formes de l’etre, constituées de rien d’autre que d’autres formes. La « donation » des formes se fait seulement par des axiomes qui spécifient certaines propriétés relationnelles, nécessaires pour qu’on puisse identifier ce que c’est qu’une telle forme. La relation de base, appelée « appartenance », et notée ., sert, en écrivant par exemple x . y, a inscrire que l’ensemble x est un élément de l’ensemble y. La relation . peut etre considérée comme unique : toute autre relation dans ZFC doit en effet etre définie a partir d’elle, dans le contexte formel de la logique classique. Finale-ment, l’axiomatique fixe les propriétés de la relation . dans un contexte logique déterminé, et permet a partir de la de définir toutes sortes d’autres propriétés des formes ensemblistes du multiple pur, du genre « etre transitif », « etre infi­ni », « etre bien fondé », « etre un ordinal », « etre l’ensemble des parties d’un autre ensemble », « etre une fonction », « etre générique », « etre un cardinal inaccessible », etc. Toutes ces propriétés donnent au philosophe les moyens de se mouvoir conceptuellement, avec une grande souplesse, dans ce qu’il en est des ressources propres de l’etre en tant qu’etre ainsi activé dans l’arene philo­sophique par le stimulant mathématique. On demandera pourquoi il est nécessaire que cette exploration spéculative des ressources ontologiques se meuve – comme Aristote le dit déja dans le livre Gamma de sa Métaphysique – dans le contexte de la logique classique, contexte défini essentiellement par le principe de non-contradiction (on ne peut affirmer a la fois la proposition p et la proposition non-p) et par le principe du tiers exclu (étant donnée une proposition p qui est bien formée, ou bien p est vraie, ou bien il est vrai que non-p, il n’y a pas de troisieme position). La réponse phi-losophique réside en ceci : la plupart des propositions de l’ontologie exigent, comme le dit majestueusement Parménide, d’en passer par le raisonnement par l’absurde. Parménide commence en effet son parcours spéculatif en affirmant qu’il est impossible de montrer directement que seul l’etre est, mais qu’on peut établir cette proposition en démontrant que le non-etre n’est pas. A son école, on constate en effet souvent, en théorie des ensembles, qu’on ne peut démon­trer directement l’existence de telle ou telle forme du multiple pur. Nombre de formes ne peuvent pas avoir de preuve de leur existence qui soit constructive et si possible intuitive. En revanche, on peut parvenir a des résultats de ce genre : « si je nie l’existence de cette forme du multiple, cela entraîne que je dois aussi nier la validité d’une proposition dont j’ai précédemment démontré qu’elle est vraie ». Le raisonnement par l’absurde permet alors de conclure que la forme considérée du multiple existe. On peut – on doit – aussi admettre une regle de tolérance maximale, qu’on peut formuler ainsi : « si cette forme du multiple, par exemple un certain type de multiplicité infinie, peut etre définie clairement dans le langage formel, tant que je n’ai aucune preuve de la négation de son existence, je peux – en fait, je dois – admettre cette existence ». Tout le point est que la relation fondamentale d’appartenance, soit ., est marquée ontologi­quement du sceau de la logique classique. En effet, étant donné un ensemble x et un ensemble y, ou bien x . y, ou bien non-(x . y). Il n’y a pas de troisieme hypothese, et on est donc sous la loi du tiers exclu, caractéristique de la logique classique. Mon énoncé spéculatif sera donc : l’ontologie est classique. Maintenant, je dois aussi montrer que les axiomes de la théorie classique des ensembles, le théorie ZFC, peuvent se prévaloir d’une légitimité philosophique. Je l’ai fait je crois consciencieusement pour la totalité des axiomes du systeme ZFC. Je prendrai ici seulement trois exemples, portant sur les axiomes les plus contestés, y compris par certains philosophes. Premier exemple : je valide, pour des raisons proprement ontologiques, le re-doutable, contre-intuitif et souvent décrié « axiome du choix », qui est une des importantes caractéristiques du systeme ZFC. Cet axiome dit qu’étant donné un ensemble d’ensembles – ce qu’est, rappelons-le, tout ensemble –, il existe tou­jours une fonction qui m’autorise a exhiber un et un seul élément de chacun de ces ensembles, et ce sans exception. Autrement dit, étant donné un ensemble A, avec ses éléments x1, x2, x3…..x …., il existe une fonction F, appelée fonc- n,xn + 1 tion de choix, qui « extrait » de chacun des éléments x1, x2, x3…..x n, xn+1…, un et un seul élément de cet élément. On a en somme F(A) tel que pour tout xn de A, on a un yn . F(A), tel que cet yn est le seul élément de F(A) qui soit un élément de xn. La fonction F « choisit » un élément de chacun des éléments de A. Si bien que F(A) est comme une assemblée nationale de représentants des éléments (depuis Macron, on dit des « territoires ») de A, un élu par élément, F étant en quelque sorte la procédure électorale de désignation de ces représentants. L’axiome du choix ne pose pas de probleme électoral tant qu’on manipule des ensembles finis. Mais dans l’infini, comment définir une fonction qui associe un représentant a chaque élément de l’infinité des éléments de l’ensemble ini­tial ? Le plus souvent, on ne peut pas prouver l’existence d’une procédure bien définie capable d’extraire d’un ensemble infini une telle infinité de représen­tants. L’axiome du choix a été contesté, parce qu’il affirme l’existence d’une pro-cédure qu’on ne parvient pas a construire. En réalité, l’axiome du choix, dans le cas des ensembles infinis, affirme l’existence d’un infini particulier, qui est le résultat du choix simultané d’un élément de chacun des éléments, en nombre infini, de l’ensemble initial. Mais l’existence de cet ensemble ne peut pas, en général, etre prouvée ou construite, et son existence n’est alors garantie, par l’axiome du choix, que comme un principe a priori. J’admets cependant cet axiome pour trois raisons philosophiques : La premiere est ce que j’appelle le principe de maximalité : l’ontologie matéria­liste pose que toute forme du multiple clairement définie doit etre acceptée en tant que possiblement réelle dans un monde, sauf preuve du contraire. Toute restriction de l’existence des formes du multiple est ontologiquement inaccep-table, si ses raisons concernent uniquement la capacité de nos esprits finis a en construire effectivement les éléments. Ce serait la retomber dans l’empirisme relativiste. Notre impuissance a construire une forme de l’etre multiple ne sau­rait etre une bonne raison d’en refuser l’existence. Faute de contre-exemple, l’axiome du choix doit etre tenu pour valide. Il nous présente une multiplicité clairement définie comme « représentative » d’une autre multiplicité, ce qui est en soit intéressant, et s’est avéré pratiquement nécessaire en Analyse moderne. La seconde raison est logique. Par le beau théoreme de Diaconescu, lequel opere dans le contexte de la théorie des catégories, l’axiome du choix impose, comme nous le désirons, que le contexte logique soit classique. La négation de l’axiome du choix ouvrirait donc la possibilité que la logique soit non classique, ce qui est ontologiquement inacceptable. La troisieme raison est plus proche des méta-mathématiques : Gödel a prouvé que si la théorie ZF (sans axiome du choix) est cohérente (sans contradiction interne), alors la théorie ZFC (avec axiome du choix) l’est aussi. L’admission de l’axiome n’introduit donc par elle-meme aucun risque particulier. Exemple du principe de maximalité, garantie du classicisme logique, confor­mité a la cohérence du contexte, l’axiome du choix est un précieux principe de l’ontologie spéculative. Mon deuxieme exemple est le suivant : J’accepte pour une raison philosophique majeure, l’axiome de fondation. Cet axiome dit que tout ensemble possede au moins un élément (ou plusieurs) qui n’a (ou n’ont) aucun élément commun avec l’ensemble initial. Pour ceux qui trouvent les formules plus claires que les dis-cours, on peut ainsi écrire l’axiome de fondation : Pour tout ensemble x : x Il existe au moins un ensemble y : y Tel qu’il est élément de l’ensemble initial : y . x Et tel que si z est un élément de y : z . y Alors z n’est pas élément de l’ensemble initial : z /. x Apres quoi, vous pouvez ponctuer tout ça de façon lisible en une seule for-mule, dont vous notez qu’elle est infiniment plus courte que le meme énoncé en langue maternelle ( x) ( y) [(y . x) et [(z . y) › (z ./ x)]] L’importance exceptionnelle de cet axiome dans le champ philosophique, et déja a vrai dire dans les conditions de vérité politique ou amoureuse, tient a ce qu’il affirme ceci : ontologiquement, l’Autre est présent dans toute Identité. Toute forme multiple, en effet, admet en elle-meme un élément dont la composi­tion propre, l’etre-multiple, lui sont étrangers. On peut aussi dire que l’axiome de fondation affirme l’immanence de la négativité : un point d’etre existe dans toute forme multiple qui n’est pas du domaine de cette forme elle-meme. Il en résulte, dans ma philosophie des vérités, que rien de vrai ne peut etre stricte­ment identitaire.Autrement dit, « vérité » et « universalité » sont inséparables. De l’axiome de fondation, résulte qu’on ne peut jamais avoir l’énoncé réflexif pur x . x. On le démontre sans trop de mal. Cela signifie, philosophiquement, qu’aucune forme multiple ne peut etre élément de la forme multiple qu’elle est. Ce qui est en somme assez évident : la forme, ontologiquement, est un combat contre l’in-forme en tant que non-etre, et ne peut, dans ce combat, s’affirmer comme étant déja elle-meme par et en elle-meme. On peut aussi interpréter l’impossibilité absolue de l’énoncé (x . x) du côté de la théorie du Sujet. Cette impossibilité se dira alors : il n’existe aucune réflexivité qui soit intégrale. Ou encore : tout Cogito est partiel. J’accepte enfin sans aucune restriction l’axiome de l’infini, lequel affirme l’exis­tence d’un ensemble infini (et, par l’effet des autres axiomes, l’existence d’une suite infinie de types d’infinité). Cet axiome revient a dire qu’il existe des formes du multiple qui sont infinies, mais il ne le peut qu’en définissant avec précision un concept de l’infini. Il existe plusieurs façons de proposer une telle définition. Elles sont toutes opératoires et évitent des approches vagues et para-intuitives, du genre « l’infini, c’est ce qui est tres grand ». Les plus communes de ces défi­nitions consistent a définir une opération sur les ensembles, et a indiquer que cette opération peut etre réitérée sans point d’arret. Par exemple, soit un ensemble x, absolument quelconque, et soit l’ensemble dont le seul élément est x, qu’on appelle le singleton de x, et qu’on note {x}. Re-marquez au passage que x est nécessairement différent du singleton {x}, pour la raison suivante : si l’on a x = {x}, il s’ensuit que l’ensemble {x} n’obéit pas a l’axiome de fondation. En effet, d’apres cet axiome, il devrait y avoir un élément du singleton qui n’a aucun élément commun avec le singleton lui-meme. Mais le seul élément du singleton est x. Donc il devrait y avoir un élément de x qui n’est pas un élément du singleton. Si le singleton est égal a x, on aboutit a l’absurdité selon laquelle il existe un élément de x qui n’est pas un élément de x. On a donc toujours, puisque nous assumons l’axiome de fondation : {x} . x. Affirmons dans ces conditions l’existence d’un ensemble Inf tel que, si l’on a x . Inf, alors on a toujours aussi {x} . Inf. Il est clair que ce faisant, on ouvre une réitération sans point d’arret du type : x, {x}, {{x}},…,… réitération sans point d’arret qui est « toute entiere » contenue dans Inf. On conviendra alors que Inf est une forme infinie du multiple. Une propriété absolument fondamentale des ensembles infinis est qu’une par-tie stricte de ces ensembles peut etre aussi grande que l’ensemble lui-meme. L’axiome de la mathématique grecque, « le tout est plus grand que la partie », n’est pas valable pour les formes infinies du multiple. Il est en fait tres simple de voir, sur l’exemple le plus intuitif d’ensemble infini, a savoir l’ensemble des nombres entiers positifs, nos bons vieux nombres « naturels », un, deux, trois, et la suite, que le prétendu axiome « le tout est plus grand que la partie » est faux dans l’infini, comme l’a remarqué avec force Galilée. En effet, il existe par exemple autant de nombres pairs que de nombres tout court. Vous faites tout simplement correspondre a tout nombre son double. A 1 correspond 2, a 2 cor­respond 4, et ainsi de suite, si bien qu’au total infini des nombres entiers, 1, 2,,3,…n, n+1… correspond exactement le total des nombres pairs, 2, 4, 6,…2n, 2(n+1). Et ceci, bien que les nombres pairs soient une partie stricte des nombres entiers. Eh bien, cette partie est aussi grande que le tout. C’est du reste, a mon avis, une raison majeure d’admettre l’axiome de l’infini, que de s’engager dans une étude des formes du multiple qui excede nos intui­tions, généralement limitées au fini. La aussi, en somme, doit valoir le principe de maximalité : de tout ce qui est, sans contradiction formelle, clairement dé­fini, on doit affirmer l’existence, car nos intuitions élémentaires n’ont aucune raison d’etre la mesure de l’etre en tant qu’etre. Cependant, définie sous la condition de cette part des mathématiques qui étu-die les différentes formes du multiple pur, l’ontologie ne crée nullement a elle seule la possibilité d’une connaissance de ce que c’est que la création d’une vérité particuliere dans un monde particulier. Certes, la pensée de ce que sont les formes du multiple, soit de ce que peut etre l’etre de tout ce qui est, est néces­saire. De fait, toute science est mathématisée peu ou prou. Meme Lacan conclut que la psychanalyse a le matheme comme idéal. Cependant, toute théorie des vérités, et du sujet des vérités, armée d’une pensée de l’etre comme tel, doit éga­lement inscrire son propos dans la singularité d’un monde et de ce qu’il propose comme matériaux a la pensée créatrice. La philosophie doit donc tirer de ses conditions, et transformer en concepts, une théorie générale de ce que peut etre, dans un monde singulier, un pro-cessus subjectivé a valeur universelle. Mais d’abord, qu’est-ce qu’un monde ? Bien entendu, un monde est ontologiquement composé d’une multiplicité, do-tée d’une forme définissable, elle-meme composée de multiplicités dont la ma-thématique peut parvenir a penser les formes. Mais quelle est la nature exacte, singuliere, de cette « composition » ? IV Proposer une réponse a cette question est le but de mon livre Logiques des mondes, et je considere qu’il est désormais impossible de séparer L’Etre et l’évé­nement de Logiques des mondes, comme depuis il est devenu impossible de le séparer de L’immanence des vérités. Car on ne peut séparer l’universalité des vérités, établie par L'Etre et l'événement, de leur singularité, pensée dans Lo-giques des mondes, et de leur absoluité, réfléchie dans l’immanence des vérités. Je ne présente ici que quelques caractéristiques tres générales de Logiques des mondes, dans le but de faire comprendre quels sont, dans ce livre, les usages de la condition mathématique. —Le concept central de tout le livre est celui d’identité dans un monde don-né (ou aussi bien, dialectiquement, de différence dans un monde donné). Le concept ontologique d’identité est strictement extensionnel : deux formes du multiple sont différentes si, et seulement si, il existe au moins un élément qui appartient a l’une des formes et pas a l’autre. Si ce n’est pas le cas, elles sont identiques. Le concept « mondain » d’identité, et donc de différence, est au contraire intensionnel, qualitatif, et relatif au monde considéré. Un ensemble x et un ensemble y appartenant au meme monde sont affectés, en tant que paire, d’un degréd’identité variant entre un minimum m (les deux objets sont, dans ce monde, totalement différents) et un maximum M (les deux objets sont, dans ce monde, pratiquement identiques). Les différentes valeurs possibles des degrés d’identité sont tirées d’un objet du monde affec­ té d’une structure d’ordre, qui est le transcendantal du monde en question. — Un ensemble appartenant a un monde, vu sous l’angle de ses degrés d’iden­ tité avec tous les autres éléments du meme monde, est un objet de ce monde. On voit donc qu’etre un ensemble qui est élément de la totalité du monde – monde qui est lui aussi, dans son etre, un ensemble – n’est pas une défini­ tion suffisante de ce que c’est qu’un objet, pour la raison qu’appartenir a un ensemble est une détermination seulement ontologique. Il faut, pour définir l’objectivité, prendre en considération le concept qualitatif et variable de de­ gré différentiel d’identité, qui est en général extremement variable. — Un objet du monde est, bien entendu, mais aussi existe. La présence d’un objet dans un monde déterminé est elle-meme affectée d’un degré, qui est le degré d’identité a lui-meme de l’objet considéré. Ce degré fixe ce qu’est l’existence de l’objet dans un monde. Si l’existence est maximale (le degré d’existence de l’objet est M), l’objet existe dans le monde « absolument ». Si en revanche l’existence est minimale (le degré est m), l’objet n’est pas absent du monde, puisque son etre d’ensemble appartient a l’etre du monde, mais 29 il est un inexistant du monde. Les degrés intermédiaires s’attachent a des existences dans le monde plus ou moins intenses. — La logique globale de tout cet arsenal théorique concernant les objets sup­ pose naturellement la théorie des ensembles (l’etre d’un objet du monde est fixé par son appartenance, en tant qu’ensemble, a la forme de multiple qu’est le monde), mais son cour est bien plutôt dans la théorie formelle des relations qu’est la théorie des catégories. Un objet est en effet existentiel­ lement défini comme tel par ses relations variables avec tous les objets du monde, y compris lui-meme (par la détermination de son degré d’existence dans le monde). — Des considérations plus techniques, tirées des notions fondamentales de la théorie des catégories, permettent de conclurequ’un monde possede globa­lement la structure d’un Topos de Grothendieck. Il y a donc autant de raisons de dire que pour moi « relation » ou « action », ou « existence dans un monde », ou encore « identité et différence », concepts majeurs de tous les vitalismes, relevent de la théorie des catégories, que de dire que mon ontologie est la théorie des ensembles. Mais en réalité, dans les deux cas, un concept philosophique, sans corrélat mathématique fixe, reste au centre de la relation circulaire entre la philosophie et sa condition mathématique. Ce concept est « etre comme multiplicité-sans-Un » dans le second cas, et « appa­raître dans le monde comme intensité d’existence » dans le premier. Dans L'Etre et l'événement, le concept philosophique d’universalité est ontologi­quement supporté par le concept mathématique d’ensemble générique. Un en­semble générique est un sous-ensemble d’un ensemble infini donné, qui ne peut pas etre défini par une propriété commune de ses éléments, propriété disponible dans le répertoire des propriétés définissable dans le monde a l’intérieur duquel on opere. Autrement dit : vous avez un ensemble infini d’un côté, mettons A, de l’autre les propriétés P définissables dans ZFC a l’aide des constantes exis­tantes, des opérateurs logiques, et de la relation .. Un sous-ensemble de A sera « générique » s’il n’est pas défini, en utilisant l’axiome de séparation, par une quelconque des propriétés P disponibles. Donc s’il est bien un sous-ensemble G de A, mais qu’il n’est pas défini, ni définissable, comme « l’ensembles des éléments de A qui ont la propriété P ». On peut finalement dire que la seule propriété de G est d’etre un sous-ensemble de A. En quoi il est un sous-ensemble générique, un « pur » sous-ensemble, non relié comme tel aux propriétés disponibles dans le langage de la théorie. J’ai montré qu’on trouve ici, potentiellement, une formalisation de la distinction, particulierement importante chez Heidegger, entre « savoir » et « vérité » : un savoir est une propriété commune a des choses du monde, propriété exprimée dans le langage dominant. Une vérité est une création « hors savoir », inexpri­mable telle quelle, au moment de sa création, dans le langage dominant (et donc souvent refusée par les partisans acharnés de ce langage). On croise aussi en ce point l’opposition platonicienne entre « opinion » et « connaissance vraie ». La premiere est toujours déja circulante dans les propos partagés, la seconde de­mande un mouvement radical d’épuration, au centre duquel se profile une Idée. On comprend aisément que dans l’ontologie du multiple, le support des opi­nions ou des savoirs communs soit un sous-ensemble de la situation extrait de cette situation par l’axiome de séparation, donc la réunion de tous les multiples qui ont la meme propriété, propriété elle-meme repérée par tous dans le lan-gage dominant. Mais que si création d’une vérité il y a, elle exige d’avoir comme support d’etre un sous-ensemble générique, indifférent au langage dominant. Paul Cohen a découvert, au tout début des années soixante du dernier siecle, une méthode générale pour produire des ensembles génériques, dans le contexte d’un modele ensembliste de la théorie ZFC. C’est pour moi le support enfin découvert, au niveau de son etre pur, de ce que c’est qu’une vérité univer­selle, puisqu’un ensemble générique se tient au-dela de toutes les identités repé­rées dans le monde existant. S’agissant du concept de singularité, son statut est clairement défini, dans le contexte de la théorie des Topoi, sous-partie de la théorie des catégories, par le concept d’existence d’un objet dans un monde. Finalement, l’antique distinction entre « universalité » et « singularité » est élucidée, sous la condition de contextes mathématiques distincts (théorie ZFC, théorie des Topoi), d’abord par la distinction entre etre-multiple et existence-dans-un-monde, ensuite par la distinction plus technique entre « ensemble générique » d’une part et « degré d’existence relationnelle » dans un monde d’autre part. J’ai donc rendu possible, dans mes deux premiers livres de métaphysique contemporaine, d’opposer l’universalité – la généricité – des vérités a la sin-gularité – l’existence, en un monde donné – des opinions. Mais il reste a com-prendre d’ou peut se soutenir que les vérités sont absolues, c’est-a-dire non seu­lement opposées a toute interprétation empiriste, mais encore garanties contre toute construction transcendantale, ce qui veut dire, dotées d’un etre indépen­dant du ou des sujets qui en furent cependant les acteurs, en quelque sorte historiques, dans des mondes déterminés. Disons-le autrement. DansL'Etre et l'événement, je montre comment l’excep­tion universelle d’une vérité peut surgir, événementiellement, sous les especes d’une multiplicité générique. Dans Logiques des mondes, je montre que les vé­rités, dont l’etre est exceptionnel, n’en existent pas moins, comme singularités, comme ouvres marquées de finitude, dans des mondes réellement existants. Ainsi j’ai pu garantir la possibilité ontologique de multiplicités suffisamment distantes du monde ou elles adviennent pour avoir une valeur universelle. Et j’ai pu garantir cependant que l’universalité d’une ouvre de vérité n’exclut nul­lement qu’elle soit le résultat d’opérations particulieres, et que ses matériaux primitifs aient existé dans un monde particulier. Finalement : — Avec le dispositif ontologique de la multiplicité pure, je suis sorti du regne de la transcendance, ou de l’Un, comme unique garantie d’etre du Vrai. — Avec le dispositif de l’universalité générique, je suis sorti de l’empirisme et du relativisme, qui nient l’existence de vérités universelles. — Avec la théorie de l’existence dans un monde de la construction des vérités génériques, et donc de leur singularité, je suis sorti de l’idéalisme, qui tente d’extraire entierement du monde réel la puissance du Vrai pour en faire un processus entierement subjectif. Mais il fallait aussi garantir, contre le relativisme dominant, le point que voici. Le fait que les vérités dépendent, quant a leur surgissement, d’un appareillage événementiel, et que leur etre soit générique, n’interdit nullement qu’une fois ouvrées dans un monde, elles soient absolues en un sens précis. Et ce sens ne se dégage cette fois, en relation avec la condition mathématique, ni des procé­dures du forcing de Cohen, ni des subtilités de la théorie des Topoi, mais d’un autre secteur encore de la mathématique fondamentale, qui est la théorie des infinis, encore en plein essor dans les dernieres décennies. C’est tout l’enjeu de mon troisieme livre, L’Immanence des vérités, achevé comme vous savez en 2018. V Je ne peux ici qu’esquisser la démarche de ce dernier livre, quant a la relation entre la philosophie et sa condition mathématique. L’universalité ontologique ne garantit pas a elle seule l’absoluité des vérités. Le relativiste peut toujours dire que ce n’est que la garantie d’une circulation possible d’une ouvre a support générique d’un monde a un autre, voire l’im­position impériale d’une généricité locale a des mondes culturels disparates. C’est ce que m’objectait, par exemple, mon amie Barbara Cassin : « l’universa­lité est toujours l’universalité de quelqu’un », me disait-elle, avec cette force en quelque sorte naive qui résulte toujours d’une fusion entre empirisme (la pré­tendue universalité comme une chose sensible et culturelle, un fait de langue) et idéalisme (tout existe « pour quelqu’un »). Le livre nouveau lui répond en substance que l’absoluité du vrai est elle-meme garantie par le type d’infinité avec lequel l’ouvre-vraie – qui est toujours, onto-logiquement, un fragment fini d’un devenir générique et donc universel – entre en relation. Le cour de L’Immanence des véritésest d’élucider ce qu’est cette relation immanente entre l’ouvre de vérité et l’infini, relation qui fonde l’abso­luité du vrai. Je m’attends bien entendu a ce que Barbara Cassin, prenant l’exemple des re­ligions, me dise que « l’absoluité est toujours l’absoluité de quelqu’un ». Mais une ouvre de vérité est un existant dans les mondes, et n’a donc rien a voir avec une religion. Une révolution, un amour, un tableau, un théoreme, sont la, sous les yeux de tous. Et tous finissent par y saisir l’invariance de ce qui a valeur universelle, parce que tous participent, dans l’expérience subjective qui les y confronte, a la relation entre l’ouvre finie et l’infini latent de son etre. De la médiation mathématique de ce point, particulierement complexe, je ne peux pas donner ici ne serait-ce qu’une claire idée générale. Disons que la théo­rie contemporaine des infinis autorise qu’on définisse philosophiquement ce que c’est qu’un attribut de l’absolu. L’absolu, quant a lui, pour les mathéma­ticiens,ne peut etre que formel: c’est la collection, mentalement situable (dé­finissable), mais logiquement inconsistante, de toutes les formes possibles du multiple-sans-Un. Notons au passage que les mathématiciens, toujours intuitivement géniaux dans les nominations, ont donné a cet absolu qui, bien que logiquement in-consistant, existe suffisamment pour qu’on en décrive certaines propriétés, le nom de V – lequel peut dire bien des choses, sans doute : Grand Vide, mais aussi bien : Lieu des Vérités. Ce qui « infinitise », et par la-meme absolutise, une ouvre de vérité qui est dans son réel a la fois finie, singuliere (existante) et universelle (générique), c’est son lien médié avec l’absolu ainsi défini. La médiation est assurée par un des attri-buts dont peuvent « participer », – comme c’est le cas dans les intuitions déci­sives de Spinoza sur ce point, quand il définit les « attributs de la Substance » – des ouvres réellement existantes dans des mondes particuliers. Le coup de génie mathématique a été de définir clairement ce que c’est qu’un tel attribut de l’absolu. Le nom mathématique est : « classe transitive de V (l’abso­lu), sur laquelle existe un plongement élémentaire de l’absolu lui-meme ». Je ne peux entrer ici, meme approximativement, dans le sens exact de cette défi­nition, meme si, comme toujours en mathématiques, l’idée sous-jacente est bien plus claire que les calculs qui en sous-tendent la validité. Toujours est-il que cette définition existe. Et qu’en outre, on connaît une condition fondamentale de l’existence d’au moins un attribut de l’absolu : c’est l’existence d’un « tres grand » infini (« large cardinal », disent les anglophones) de type spécial, appe­lé par moi « complet » pour de solides raisons philosophiques. J’ai pu alors montrer, toujours par une circulation serrée entre philosophie et mathématiques, que la définition (mathématique) de ce qu’en philosophie je re-nomme un « attribut de l’absolu », supporte clairement le sens spéculatif que je lui donne. Il ne reste alors au philosophe, quel qu’il soit, qu’a entrer dans les dé-tours difficiles de la construction des infinis, et a y trouver le chemin d’une abso­lutisation des ouvres de vérité, dont j’ai déja démontré, allant et venant entre phi-losophie et mathématiques, qu’elles étaient a la fois singulieres et universelles, et dont il ne restait plus qu’a montrer qu’elles peuvent etre également absolues. Ce que je crois avoir fait. Je l’ai fait a un âge suffisamment avancé pour que cette réussite soit vraiment réconfortante ! Permettez-moi de finir sur cette orgueil­leuse assertion ! Le Triangle philosophie – mathématiques – psychanalyse / The Triangle of Philosophy – Mathematics – Psychoanalysis Filozofski vestnik | Volume XLI | Number 2 | 2020 | 37–55 | doi: 10.3986/fv.41.2.02 Oliver Feltham* “One or Many Ontologies? Badiou’s Arguments for His Thesis ‘Mathematics is Ontology’.” This is not a story of hubris, of a philosopher seeking to make his name by an­nouncing a provocative thesis “mathematics is ontology” and then, decades later, fighting to prevent his name from being unmade by commentators who cannot quite bring themselves to accept that identity as definitive. This is an en-quiry into an argumentative strategy which leads to a crucial question: which initial decisions on ontology lead to there being one or many ontologies? Part of the appeal and strength of Badiou’s philosophy lies in complex way he sets up the tasks of a set-theory based ontology in Being and Event, and then of a category-theory based phenomenology in Logics of Worlds. It is the number of argumentative steps and choices made in these preambles – many of which are re­inforced in the recently published seminars – that turn Badiou’s philosophy into fertile ground not just for contesting interpretations but for the genesis of other philosophies. Indeed, it is the case that Badiou’s philosophy not only spawns or­thodoxies and imitations, like those of Derrida and Deleuze, but also heterodox­ies and rival philosophies. But the challenge for any departure from a putative Badiousian orthodoxy is to retain both the audacity and systematicity of Badiou’s work. Can we, in turn, take steps in a new conceptual construction, and then trace the consequences of each of these steps, in a manner constrained by a previous-ly-existing and consistent discourse, such as ZFC set-theory? Such a task goes be­yond the question of making or unmaking, repeating or forgetting, proper names. 37 In this preliminary enquiry, let us explore Badiou’s initial decisions on ontology. The argument that mathematics is ontology In the first meditation of Being and Event, Badiou sets out its inaugural thesis: “Mathematics is ontology – the science of being qua being”.1One page later he 1 Alain Badiou, Being and Event, trans. Oliver Feltham, Bloomsbury, London 2005, p. 4. * American University of Paris, College International de Philosophie clarifies, “Mathematics writes that which of being itself, is pronounceable in the field of a pure Theory of the multiple”.2 In Meditation One he sets out re­quirements for ontology and then in Meditation Three he proceeds to identify a particular kind of mathematics that satisfies those requirements.3 But this is too simple: there is a puzzle here: what comes first – the argument that determines the requirements for ontology, or the identification of set theory as ontology? One cannot argue that a particular discourse is uniquely suitable for the task of ontology without pronouncing as to the nature of being and thus engaging, at least in a preliminary manner, in ontology yourself. Is there not a problem of circularity here? On the one hand, there are these strong readings of canonical texts in the histo­ry of metaphysics that are supposed to set up the choice of set theory as ontol­ 2 Ibid., p. 5. 3 Ibid., p. 29. In a broad outline this passage appears deceptively simple; however, there are several difficulties that confront the interpreter who pays attention to the details. For instance, there has been much debate over the concepts of inconsistent multiplicity and consistent multiplicity: do they form Badiou’s version of Heidegger’s ontological differ­ence (Hallward, Nancy, Pluth, Feltham vs Brassier); does Badiou have or even need an adequate account of the consistent multiplicity of ‘non-ontological situations’ (Feltham, Besana, Hallward); what does it mean to claim that the count-as-one has no agent (De-santi, Feltham and Clemens)? What is the status of the claim that there must be a redou­bled count-as-one establishing the state of a situation, because being is not presented as chaos, or what is the status of the claim that this second count-as-one has to include the name of the void, which errs in presentation, for otherwise there would be chaos? There are various ways of resolving these difficulties, but in order to develop a coherent inter­pretation of the argument ‘mathematics is ontology’ it seems to me there is a more gener­al puzzle that takes priority, and that is the puzzle of what comes first, the determination of the criteria for ontology or the identification of mathematics and, in particular, ZFC set theory as ontology. See Ray Brassier, “L’Anti-phenomene – présentation et disparaître”, in Ecrits autour de la pensée d’Alain Badiou, ed. B. Besana and O. Feltham, Harmattan, Paris 2007, pp. 55–64 ; Bruno Besana, “Quel multiple ?” and “Replique ; l’événement de l’etre”,Ibid., pp. 23–40, pp. 125–30 ; J-T. Desanti, “Some Remarks on the Intrinsic Ontol­ogy of Alain Badiou” in Think Again: Alain Badiou and the Future of Philosophy, ed. P. Hallward, Continuum, London 2004, pp. 59–66; O. Feltham & J. Clemens, “An Introduc­tion to Alain Badiou’s philosophy”, in A. Badiou, Infinite Thought: Truth and the Return of Philosophy, ed. and trans. O. Feltham & J. Clemens, Continuum, London 2003, pp. 1–38; Peter Hallward, Badiou: a Subject to Truth, University of Minnesota Press, Minneapolis 2003; Jean-Luc Nancy, “Philosophy without conditions”, in Think Again: Alain Badiou and the Future of Philosophy, ed. P. Hallward, pp. 39–49; Ed Pluth, Badiou: a Philosophy of the New, Polity Press, London 2010. ogy, yet at the same time the strong theses and leaps in these readings seem to be either anchored or driven by the prior choice of set theory as ontology. Sam Gillespie wrote “Given that being qua being is given to us exclusively through ontology, it follows that it is very difficult to summon a mathematical ontology to a tribunal of ontology which would tell us whether or not it is a legitimate ontology. Ontology is in no way a set of descriptions of being that preexists its own operations”.4 There are two solutions to this general puzzle. They break the circularity by pos­iting a basic order. — One can argue that it is the philosophical arguments that come first, name­ly a kind of history of being to rival that of Heidegger, and the election of set-theory comes second. Let’s call this the ‘philosophy solution’, or rather, the ‘argument from philosophy’. — Or one can argue that what comes first is the naming of Cantorian set-theory as a truth-procedure within the field of science, which subsequently condi­tions philosophy and generates this sub-discipline of ‘metaontology’. Let’s call this the ‘argument from the condition’. The challenge for these two solutions, the criteria for us choosing one over the other, will be whether it succeeds in chasing down and eliminating the occa­sional appearance of arbitrariness in the argument that explains why mathe­matics is ontology.5 Let’s first examine the ‘argument from philosophy’. 39 4 Sam Gillespie, “L’etre multiple présenté, représenté, rendu vrai”, in Ecrits autour de la pensée d’Alain Badiou, ed. B. Besana & O. Feltham, p. 73. 5 Or do we accept that philosophy is a not an enterprise uniquely motivated by conceptual construction and argumentation; rather, Alain Badiou was seeking to give his own name to being qua being via the law of the father as Quentin Meillassoux provocatively sug­gests in his essay “Décision et indécidabilité de l’événement” in Autour de Logiques des Mondes, ed. O. Feltham, D. Rabouin and L. Lincoln (eds.), Editions des Archives Contem­porains, Paris 2011, pp. 135–6. The argument from philosophy The initial philosophical argument in Being and Event results in the following requirement: being must be thought as inconsistent multiplicity. This claim is set up in four steps: 1) The One is not: The starting point, anchored in the history of philosophy, is the thesis that the One is not, explored by Plato in the last four hypotheses of the Parmenides. 2) There is inconsistent multiplicity: Parmenides’ exploration of the hypothesis “If the One is not, nothing is” leads to the concept of plethora, of a multiple that disseminates itself internally without limit and thus without ever en­countering some ultimate elements or atoms: this is what Badiou calls ‘in­consistent multiplicity’. 3) There is a count-for-one: Nevertheless, Badiou argues, there is some One­ness, an effect of unity and so there must be an operation of unification that distributes inconsistent multiplicity (before) and consistent multiplicity (af­ter its effect). A consistent multiplicity is unified. 4) The nothing is: within consistent multiplicity, inconsistent multiplicity is nothing and as such it subsists in structured presentations as the void. What is ‘void’ within a structured presentation is both the operation of its count-for-one and the material from which all structure is composed (inconsistent multiplicity). Badiou then adds a further, separate, requirement for ontology: ontology must be compatible with contemporary praxes of the subject. The strategy of this argument is to claim that a philosophical confrontation with the impasses of the history of ontology entails these four theses on the one and multiplicity. It is these claims which set up the requirements for ontology. After an examination of various kinds of discourse, it turns out that the only discourse capable of exploring and unfolding the implications of these theses is a particu­lar kind of set theory. We find this ‘argument from philosophy’ in Meditations 1, 2 and 6 of Being and Event, developing readings of Heidegger, Plato and Aris­totle, and we find it massively in the Seminar from 1983 to 1986. There are two versions of this argument, which we shall call the via negativa, and the historial. On the one hand, Badiou will engage in a negative demonstration – the via neg-ativa – arguing that ontologies committed to the being of the One end in ruins. For instance, in the Short Treatise of Transitory Ontology, he rapidly pulls apart Aristotle’s necessary supposition of a global unity, a prime-mover in order to re­solve difficulties in the theory of substance as an impossible union of matter and form.6 In Being and Event he claims that ontology repeatedly falls into an abyss or a labyrinth when it tries to resolve the relationship between the discrete and the continuum, and also when it tries to resolve the relationships between the one and the multiple, or parts and the whole.7 This negative demonstration is a little like Kant’s proof of the systematicity of transcendental philosophy through its resolution of the antinomies of pure reason, antinomies that ruin all other philosophies. The simplest form for this demonstration would be an argument from the absurd where an initial premise ineluctably leads to a contradiction and hence one is obliged to jettison the initial premise (namely the being of the One). The problem, of course, is that despite the variety and breadth of Badiou’s ex­amples of paradox and contradiction in the history of ontology, strictly speaking the demonstration can never be exhaustive because the history of philosophy is still open. Someone – perhaps they are reading this article right now! – might well come along and write a coherent ontology on the basis of the premise that being is One. It is difficult to demonstrate impossibility outside the confines of a simple formal system, in which one options for argument are exhaustible. Hence Badiou’s frequent recourse to another version of the argument from phi­losophy, the historial. For instance, he claims that the being of the One is the fundamental commitment of all onto-theology, adopting Heidegger’s term for his purposes. However, the orientation of thinking that Badiou is carving out holds itself contemporary to – or tributary of – Nietzsche’s declaration that ‘God 41 is dead’, a declaration that mortifies all Gods, those of metaphysics, religion and poetry alike.8 There is no going back on Nietzsche’s epochal declaration. Hence the entire project of onto-theology is closed, and ontology must begin on anoth­ 6 Alain Badiou, Court traité d’ontologie transitoire, Editions du Seuil, Paris 1998, p. 15 ; Alain Badiou, Briefings on Existence: A Short Treatise on Transitory Ontology, trans. N. Madarasz, State University of New York Press, Albany 2006. 7 Badiou, Being and Event, pp. 5, 81, 281. 8 See the “Prologue: God is Dead” in Alain Badiou, Briefings on Existence, pp. 21–32. See also Quentin Mellassoux’s excellent pun, mentioned above, on the proper name ‘A bas l’Un­ er basis than the being of the One. Again in the Seminar, and in other texts, one finds such claims: the thesis “every situation is infinite” is pinned to a reading of Pascal, but it is also situated as a defining thesis of modernity, a thesis that opens up modernity, a historical decision. Badiou often adopts this historial ver­sion of the argument from philosophy when he is interpreting and critiquing Heidegger’s history of being: it is as though he is setting up, by means of his own interpretations of canonical philosophical texts, a rival history of being. Here you can see the historial strategy runs into another problem, one of cir­cularity. If being has its own history – which produces these theses ‘the One is not’, ‘every situation is infinite’ – then what is the original language or discourse in which that history is disclosed? We haven’t yet arrived at set-theory, we are still identifying the preliminary theses which will subsequently justify the elec­tion of set theory as ontology. So both the via negativa and the historial versions of the argument from philos­ophy run into problems. I contend that it is these problems that generate the occasional appearance of a prevalence of choice or of decision in Badiou’s con­struction of his interpretations of philosophy. This is not a matter of failing to convince specialists of the cogency of Badiou’s interpretation – specialists are never convinced, even by each other – but of the overly apparent tactical choic­es in his readings. Badiou, of course, is well aware of the problem of circularity in the justification of an inaugural decision as to the nature of ontology and philosophy. It forms one of the central topics of his analysis of the poem of Parmenides in the epon­ymous seminar of 1985-86.9 In the seminar he appears to borrow an idea from Guy Lardreau concerning the foundation of philosophy, which is to argue that the inaugural decision as to the nature of philosophy is actually taken from the Dieu’ in “Décision et indécidabilité de l’événement”, in Autour de Logiques des Mondes, p. 136. 9 Badiou writes, “If philosophy decided itself from this point and this point alone – thinking at the same time the way of being and the way of nothingness, and instituting a regime of decision –, and if this point was the absolute origin of the existence of philosophy’s dis­cursive apparatus, then one must declare, as I do, that philosophy had been decided well before.” See Alain Badiou, Le séminaire: Parménide, L’etre I – Figure ontologique, Fayard, Paris 2014, p. 54. standpoint of another discourse.10 Badiou then claims that ‘philosophy is under a supplementary condition’, and insists on the heterogeneity of this condition and its encounter with philosophy as productive of decisions. This recalls the supplementary fifth thesis stipulating the requirements for ontology that we mentioned above and then left aside: that ontology must be compatible with contemporary praxes of the subject. Let’s retain these two terms ‘heterogeneity’ and ‘encounter’ – they will guide our conclusion. In his seminar Parmenides Ba-diou thus adopts the argument from the priority of the condition as the solution to the problem of circularity, the problem that affects inaugural decisions that open up a philosophy such as the thesis “mathematics is ontology”. Let’s turn to this argument from the priority of conditions. Argument from the priority of conditions The argument from the priority of conditions for the thesis ‘mathematics is ontology’ runs as follows. Philosophy only occurs historically in the form of a ‘com-possibilization’, that is to say, a naming and theorizing of truth-procedures occurring in four different conditions of art, science, politics and love. A philos­ophy develops a coherent system of reference by constructing its own names for these generic truth procedures, procedures that trace out the consequences of anomalous events occurring in each of these extra-philosophical fields. For instance, Being and Event is an attempt to philosophically name what occurs in the condition of science as a truth procedure faithful to the “Cantor-event”, but it also names what occurs in poetry in Mallarmé’s fidelity to the ‘crisis in verse’, there are brief references to Engels and Mao in the political thinking of the state, and there is an engagement with psychoanalysis as intervention in the condi­tion of love via the exegesis of Lacan’s concept of the subject. 43 As such, it is the philosopher ’s initial fidelity to the Cantor-event that decides that Zermelo-Fraenkel set theory with the Axiom of Choice will determine the nature of ontology, and hence ground these philosophical theses such as “there is no being of the One”. First comes fidelity to the condition, then philosophy. Set theory itself is ontology, and its philosophical naming and theorization is called “metaontology”. Alberto Toscano and Ray Brassier explored this strat­egy in a 2004 article where they claimed that “mathematics is ontology” was 10 Ibid., p. 62. Badiou’s own intervention, his own fidelity to the Cantor-event.11 It is Badiou’s fidelity to Cantorian set-theory that determined his understanding of ontology, and his subsequent disqualification of various philosophical ontologies. At this point another suspicion of insufficient justification emerges. In the de­velopment of the argument of Being and Event it turns out that it is not so much mathematics in general but specifically ZFC set theory that is ontology. Com­mentators have asked why one would choose this particular variant of set theory (although Badiou does give a number of cogent reasons during the construction of Being and Event). Others have asked why choose set theory as a metonymy for mathematics and not another sub-discipline of maths, given that the entirety of mathematics is ontology. This second appearance of insufficient justification is addressed in section four of this article. For the moment let’s note the advantage or virtue of the argu­ment from conditions: it leads to a philosophical exploration of the singularity of set-theory, which allows the identification of three peculiar characteristics that are grasped as demonstrating its vocation for ontology. The first characteristic of Zermelo-Fraenkel set theory is that it has no defined object. There is no explicit definition of a set. As such the theory has no stipulat­ed object. Rather the theory employs one primitive relationship between multi­ples; that of belonging, and this relationship may be used in the ways specified by the axioms. Various kinds of set then emerge from that manipulation. The most striking demonstration of this emergence without definition occurs in the explanation of the axiom of the union set.12 This axiom states that for every set there exists the set of the elements of the elements of that set. This immediately cancels out any substantial distinction between ‘sets’ and ‘elements’, which are clumsy terms from natural language. Indeed there is only one kind of variable in this form of set theory. Every element of a set is itself a set composed of elements which are themselves sets and so on. Set theory can thus consistently write the decomposition of multiples as more multiples at other levels. Through the ax­iom of the power-set, it can also write the composition, the putting together of 11 Ray Brassier and Alberto Toscano, “Postface: Aleatory Rationalism”, in Alain Badiou, Theoretical Writings, Continuum, London 2005, p. 255. 12 Badiou, Being and Event, pp. 63–64. larger multiples through the assemblage of all of the sub-groups of elements of an initial set. So on the basis of an initial multiplicity one can generate other multiplicities through analysis or synthesis, below it and above it so to speak. The series of sets generated continues to such a point that Badiou states with glee: set theory’s “proliferation of infinities” achieves “the complete ruin of any being of the One”.13 It is not a foundational definition but the axioms that structure and render con­sistent the situation that is ZFC set theory. This is an exemplary case for prag­matism: use defines being; that is to say, what counts as being, as multiples of multiples, is only ever whatever is encountered through the facility of writing and manipulating sets and whatever might block that facility, such as Russell’s paradox. As such, there is no unifying or totalizing speculative gaze at being; it is neither seen nor grasped but encountered bit by bit in the scriptural construction of different kinds of sets, in a kind of constrained unfolding of multiples of mul­tiples.14 Hence being-qua-being is not an external object, but insists in a writing. The second peculiar characteristic of set-theory that Badiou seizes upon is that, in Lacanian terminology, it encounters the ‘real’ in its symptoms. In non-Lacan­ian terms, set theory has discovered a number of paradoxes and problems that stymied its efforts at formalization. It is Cantor, for instance, and not Badiou who originally develops the concept of an “inconsistent multiplicity”, precisely in reaction to the discovery of sets that could not be totalized without contradic­tion (the set of all sets that do not belong to themselves). Amongst these para­doxes and problems Badiou briefly mentions the controversy around the Axiom of Choice, he devotes several pages to Russell’s paradox, but the entire last third of the book is devoted to the problem of the continuum and its implications. In the Introduction he announces, “What seemed to me to constitute the essence of the famous “problem of the continuum” was that in it one touched upon an obstacle intrinsic to mathematical thought, in which the very impossibility which founds its domain is said”.15Indeed, in Meditation 27 he declares that the 13 Badiou, Being and Event, p. 273. 14 See my claim in “Translator’s Preface”, in Ibid., p. xxiv. 15 Ibid., p. 5. measure of the excess of the cardinality of an infinite set’s powerset over the cardinality of that infinite set, constitutes the “impasse of being”.16 The third peculiar characteristic of set theory as a kind of discourse is that it historically took decisions on these symptoms. That is to say, mathematicians found resolutions to these paradoxes which in turn opened up new and further domains of formalization. For instance, Zermelo’s Axiom of Separation allowed his version of set theory to avoid Russell’s Paradox (Med.3).17 In Badiou’s exe­gesis, set theory encountered the ontological paradox of the relationship be­tween the continuum and the discrete in its discovery of the undecidability of the excess of the cardinality of a powerset over the cardinality of its original set if the latter is infinite. However, in Paul Cohen’s invention of the procedure of forcing a way is found in each individual procedure to decide on that unde­cidable excess. Badiou calls these resolutions of paradoxes or circumventions of impasses in formalisation “decisions on being”. He claims that in a belated echo of Parmenides they show how “the same, is, both thinking and being”.18 At this point of our exploration we can detect two metaontological consequenc­es of these ‘decisions on being’ carried out by set theory. The first concerns the ontological or set-theoretical inscription of the ‘generic’ multiplicity of truth procedures, and the second the possibility of an alternative ‘history of being’. The first consequence of these ‘decisions on being’ concerns their multiple oc­currence within praxes that follow events, that is to say, within what Badiou calls ‘generic truth procedures’.For instance, when it comes to the set-theoret­ical continuum problem, or, in his metaontological terminology, the ‘impasse of being’, he claims that there are four types of decision on that impasse which constitute the four grand orientations of thought: the transcendental, the gram-marian-constructivist, the indiscernible-generic, and the praxical. 16 Ibid., p. 281. 17 Note that Badiou has been fascinated with these decisions at the points of impossibility within formal systems since his early essay “Subversion Infinitésimale”, in Concept and Form, volume I: Key Texts from the‘Cahiers pour l’Analyse’, ed. Peter Hallward and Knox Peden, Verso, London 2012. 18 Badiou, Briefings on Existence, p. 52. He then goes on to claim that Cohen’s solution to the continuum problem by means of forcing and the generic subset provides an ontological schema for all generic-truth procedures. All generic truth procedures – all praxes that are faithful to an event – thus involve decisions on the impasse of being. So at this point the argument from conditions leads to a multiplication of ‘decisions on being’ within all those generic truth-procedures that a philosophy might recog­nize, and compossibilize in its attempt to be contemporary with recent events in its conditions. Interestingly this consequence does not lead to a reduction of the suspicion of insufficient justification that attended the election of ZFC set theory in particu­lar as ontology. Rather it presents an exacerbationof that insufficiency through its transformation into the radical contingency of events and truth-procedures. In other words, there is insufficient justification within ZFC set theory itself for the continuum hypothesis or for other hypotheses as to the cardinal quantity of the powerset of an infinite set – this is why Badiou terms them ‘decisions on being’. If decisions on being occur within truth-procedures not just in the realm of mathematics but also in politics, art and love, then these moments of ‘in­sufficient justification’ multiply in such practices. However, at the same time, the insufficiency at stake in these decisions within these practices can also be understood as a form of radical contingency in line with that of the event. The second consequence of these set-theoretical ‘decisions on being’ is that via their metaontological interpretation they intervene in the philosophical or his­torical problems of ontology. Throughout Being and Event, Badiou remarks that there are a series of unresolved problems in the history of ontology concerning the one and the multiple, the part and the whole, the finite and the infinite, and the relationship between the discrete and the continuum. It just so happens that Zermelo-Fraenkel set-theory provides a series of new solutions to these problems. There is thus a new ‘history of being’ that is generated by Badiou’s argument from the condition of set theory to philosophy. Precisely, he explicitly claims ‘The history of mathematics [periodized by singular praxes] is the histo­ry of being’. It is by means of this alternative mathematical history of being that Badiou will be able to rival Heidegger, and ground his claims with regard to requirements for a contemporary thinking. The argument from conditions thus joins the argument from philosophy in its commitment to a history of being, but this time the history is grounded in an alternative discourse to philosophy. Thus in this regard at least, the argument from conditions is superior to the argument from philosophy. At this point let’s return to this suspicion of a lack of justification over ZFC set theory alone being elected the metonymy of mathematics as ontology. It is clear that we cannot hope for any absolutely solid philosophical demonstration of the necessity of ZFC alone as ontology to the exclusion of all other mathematical and non-mathematical candidates for the discourse on being qua being. Such a demonstration is impossible because the equation ‘maths is ontology’ does not take place within a formal system. I return to Sam Gillespie’s statement: “Given that being qua being is given to us exclusively through ontology, it follows that it is very difficult to summon a mathematical ontology to a tribunal of ontology which would tell us whether or not it is a legitimate ontology”.19 It is also evi­dent that Badiou’s metaontology is not the only possible philosophical exegesis of what is going on inside ZFC. What is our conclusion concerning the argument from the priority of set-theory as a condition for philosophy? It grounds Badiou’s alternative history of being, which is an advantage; it does not eliminate the appearance of insufficient jus­tification in the initial election of ZFC set theory, which is a disadvantage, and as an ontology compatible with truth procedures, it multiplies ‘decisions on be­ing’, which is neither an advantage or a disadvantage but opens up an enquiry into conception of action entailed by such decisions. Let’s try one more approach, the pragmatist approach: what difference does the election of set theory as ontology make to Badiou’s philosophical project? What does set-theory qua ontology do? In Badiou’s terminology this is the question of the status and role of ‘metaontology’. 19 Gillespie, “L’etre multiple présenté, représenté, rendu vrai”, in Ecrits autour de la pensée d’Alain Badiou, p. 73. The status of metaontology The simple answer to this question is that philosophical metaontology sets out the concepts of a philosophical theory of radical transformation.20It allows one to state, for instance, that “the form-multiple of being is generally infinite”.21 It makes the distinction between presentation and representation, a situation and its state, it generates the concept of eventual sites and the ensuing distinc­tion between natural and historical situations, it anchors the claims that there is no totality of Nature nor of History, etcetera. The choice of ZFC set theory as ontology turns out to be extremely rich and pro­ductive in terms of the wealth of concepts generated for this theory of change. But this theory immediately encounters another problem, that of schematism. Badiou opens Meditation 12 on ‘natural multiples’ with the following claim: Set theory, considered as an adequate thinking of the pure multiple, or of the presentation of presentation, formalizes any situation whatsoever insofar as it reflects the latter’s being as such; that is, the multiple of multiples that makes up any presentation. If, within this framework, one wants to formalize a particular situation, then it is best to consider a set such that its characteristics […] are comparable to that of the structured presentation – the situation – in question.22 Particular situations are thus ‘formalized’ by considering a set as the schema of a situation, hence my term ‘schematism’. This passage has caused much 20 Badiou makes the connection between set-theory ontology and his larger project of de­veloping a theory of radical transformation and hence, in the political sphere, of justice in the following terms in the “Introduction” to Being and Event: “It is the act of trusting 49 [mathematicians] forever with the ‘care of being’ which separates truth from knowledge and opens it to the event. Without any other hope, but it is enough, than that of mathemat­ically inferring justice.” (Ibid., p. 15) Badiou also writes in the introduction “All [the thesis ‘mathematics is ontology’] does is delimit the proper space of philosophy…its function is to introduce specific themes of modern philosophy, particularly – because mathematics is the guardian of being qua being – the problem of what-is-not-being-qua-being.” (Ibid., p. 15) Note that the category of ‘what-is-not-being-qua-being’ refers to the event. Badiou also writes with regard to ontology, “the saying of being would hardly make any sense if one did not immediately draw from the affairs of the City and historical events whatever is neces­sary to provide also for the needs of ‘that-which-is-not-being-qua-being’” (Ibid., p. 282). 21 Ibid., p. 266. 22 Ibid., p. 130. consternation amongst commentators. Tzuchien Tho asked Badiou a question about this very passage in the interview after the English translation of The Concept of Model.23 Schematism is already present in many of the metaontolog­ical statements in the early meditations, especially the claim that set theory is the presentation of presentation; that is to say, the presentation of the inconsist­ent multiplicity of all consistent multiplicities in non-ontological situations.24 The first problem with metaontology as a schematism – one that Ray Brassi­er pointed out very early on – is that it turns set-theory into a referential dis­course, gives it supposed objects, those objects being the implicit structures of non-ontological situations (and so Badiou would be a kind of structuralist apres la lettre).25 One loses the radically immanent and scriptural quality of set-theory ontology that I referred to before. The second problem with schematism, the one that bothers most of the An-glo-Saxon commentators though, with their heritage of empiricism, is how to actually demonstrate that a particular set schematizes a particular situation. I asked Badiou this question in the 1999 interview transcribed in Infinite Thought, and for him it was a non-starter, perhaps, though it is a bit of an excuse, be­cause he is not weighed down by the tradition of empiricism.26 One cannot demonstrate that sets provide the schemas of all non-ontological situations without already making some prior decisions as to the adequation of other discourses to those ‘given situations’, and also having a translation 23 See Tzuchien Tho and Alain Badiou “The Concept of Model, Forty Years Later: An Inter­view with Alain Badiou”, in Alain Badiou, The Concept of Model: an Introduction to the Materialist Epistemology of Mathematics, ed. and trans. L. Fraser & T. Tho, re.press, Mel­bourne 2007, pp. 94–5. 24 The argument for what I call the schematism of set-theory ontology is inextricable from Badiou’s argument for ontology being that unique situation that presents inconsistent mul­tiplicity. On the last page of Meditation One he states: “To accede axiomatically to the pres­entation of their presentation, these consistent multiplicities of particular presentations, once purified of all particularity – thus seized before the count-as-one of the situation in which they are presented – must no longer possess any other consistency than that of their pure multiplicity, that is, their mode of inconsistency within situations” (Ibid., p. 30). See Ray Brassier, “L’Anti-phenomene – présentation et disparaître”, in Ecrits autour de la pensée d’Alain Badiou, pp. 55–64. 26 Badiou, Infinite Thought, p. 178. protocol between those discourses and set-theory, and accepting what is lost in those translations (anti-reductionism objections); indeed this entire objection seems to be caught up in the impossibility of reinventing Carnap’s project in theAufbau. I am English, I have to some degree inherited the burden of empiricism, and so I found it difficult to ignore or dismiss this apparent problem of schematism. One solution I found is laid out in another part of Badiou’s oeuvre, his first book on mathematics as a condition of philosophy, The Concept of Model. My hypothesis is that metaontology is the discourse that results from the mod-elling of philosophical ontology by the syntax of ZFC set theory. I develop this argument in Alain Badiou: Live Theory.27 Here I will summarize its conclusion. The operation of conditioning of philosophy involves the selection of a theoret­ical syntax from the language and names of a generic truth procedure – in this case, ZFC set theory. The second step is to select a semantic field – in this case, philosophy, in particular the history of ontology. A model of the theory is said to be produced if its syntax, and the operations that its syntax permits, can be reproduced without contradiction within that semantic field. Hence, in terms of the production of metaontology, if Badiou can reproduce the syntax and operations of ZFC set theory in the semantics of the history of ontol­ogy without encountering contradiction, then he will have produced a model of ZFC set theory.28 The argument from the priority of conditions must therefore be understood via the operation of modelling. This hypothesis neatly resolves the problem of schematism. Badiou’s metaontology does not provide a theoretical schema 51 27 Oliver Feltham, Alain Badiou: Live Theory, Continuum, London 2008. 28 I understand Badiou to be using such an argument when he makes a claim like ‘the whole of the thinking unfolded in Being and Event constitutes the demonstration that mathemat­ics is ontology’. He made this claim orally at a seminar “Ontologie et mathématiques” at the American University of Paris on June 17th 2019. See “Ontology and Politics: an Interview with Alain Badiou”, in Alain Badiou, Infinite Thought: Truth and the Return of Philosophy, Continuum, London 2003, where Badiou says “A large part of L’etre et l’événement tries to explain with the means of mathematics why mathematics is ontology. As a matter of fact it is its task”, p. 184. or model of non-ontological situations. Rather these objects or names – such as ‘non-ontological situation’, ‘natural’ or ‘historical situation’, ‘evental-site’, ‘state’, ‘impasse of being’ – are all elements of a model of a theory. In pragmatic terms, Badiou’s metaontology creates a new universe of objects. When I ana­lyse a situation as a historical situation, my metaontological model enters into competition not with given concrete situations (as a vulgar empiricism would have it, somehow measuring ‘theory’ against ‘reality’), but with established universes created by the models of other theories; that is to say, it enters into an ideological battle (the constant theatrical background of Badiou’s project). So the concept of modelling allows the dismissal of the charge of schematism, but there is one last problem. The solution from modelling does not completely eliminate the suspicion of insufficient justification. Simply put, why model ZFC set theory and not another mathematical theory? The practice of modelling entails a plurality of models Here the solution is evident, and it is exemplified by Badiou’s own practice, turning to category theory in Logics of Worlds to entirely remodel the philosoph­ical discipline of phenomenology, and back to the theory of grand cardinals to rework his system again in TheImmanence of Truths. In other words, there is a positive interpretation of the initial choice of ZFC appearing undemonstrated and thus arbitrary: the practice of modelling can entail a plurality of models. In this way, the insufficient justification is transformed into the contingency of a decision. Here we can follow as our guide the many ‘decisions on being’ that take place in the multitude of generic truth procedures; that is to say, we can accept a proliferation of decisions on being, it doesn’t need to make us anxious. Yet if these decisions on being concern the modelling of ontology itself, then we find the way opened to not one but many ontologies! Now this would be quite an­other strategy to adopt on the basis of the initial argument that the being of the One must be rejected and being thought as inconsistent multiplicity. This would be an approach similar perhaps to what Jean-Toussaint Desanti calls ‘extrinsic ontology’, according to which, when one interprets the phrase ‘being qua be­ing’, one works to embrace the maximum of senses encapsulated in being, along the lines of Aristotle’s initial intuition – that being is spoken in many ways – an intuition that Aristotle, and subsequently many philosophers after him, swiftly sets aside.29 There are actually a few indices or openings to this strategy of many ontologies in Badiou’s own work. When he shows that one of the singular characteristics of set theory ontology is that it does not totalize being, does this not open up the possibility of other kinds of inscription of being? When he argues that on­tology is not a transcendental all-englobing situation but merely one situation amongst others, could it not also be one ontology amidst others? There are two evident objections to this strategy of multiple ontologies. First, it will end up in eclecticism: there are conferences on the ontology of this or that, the ontology of emotions, the ontology of social constructs. Second, the choice of multiple ontologies via this eclecticism finally ends up in nihilism. Barry Smith has attempted to construct a ‘Basic Formal Ontology’ for the benefit of engineering and medical databases not to mention, the military and intelli­gence communities including the FBI.30 One can avoid eclecticism and nihilism by dismissing the idea that every single situation comports its own set of existential commitments which constitute a specific ontology. In my own work on action I show that the ontology opens up not as another variant of the discourse of the university, but as a lived enquiry uniquely when a failure or dysfunction occurs in the reception and consequenc­es of an action. The action itself turns out to be ‘equivocal’ in that it is subject to conflicting attributions of its nature – is it a just or unjust action, has Macron’s government ‘ensured a return of the rule of law in Notre Dame des Landes’, or has he ‘destroyed people’s livelihood’? This equivocity expands to include the intention behind the action, its identifiable consequences and even the identi-53 ty and ‘kind’ of the action’s agent. Contestation occurs over the being of these actions, and each side in such controversy develops its own ontology of these actions. In this manner diverse ontologies multiply in so far as people – not just philosophers – explore the nature of failure and dysfunction and controversy in 29 Jean-Toussaint Desanti, “Some Remarks on the Intrinsic Ontology of Alain Badiou”, in Think Again: Alain Badiou and the Future of Philosophy. 30 See the Wikipedia article on Smith: https://en.wikipedia.org/wiki/Barry_Smith_(academic) the occurrence of actions: I call this approach an ‘anatomy of failure’.31 I would hesitate to term this a ‘democratization’ of ontology as a kind of writing, because it does not concern a majority of people, nor is it based on the circulation of opinions. Anatomies of failure occur when people recognize and explore the undoing of opinions through the ambivalence and equivocity of certain actions. This process undoes received ideas and leads to the modification or displace­ment of dominant ideological positions in a political situation. It is democratic, however, in so far as it requires a process of deliberation between a number of actors from different social and political contexts. The result of people carrying out anatomies of failure, and broadcasting them, is that alongside this ‘stellar’ set-theory ontology which schematizes the struc­tures and inconsistent multiplicities underneath all kinds of existential com­mitment, we would embrace a multiplication of sub-lunar ontologies, and so account for the disjunctions and overlaps in existential commitments, such as the disjunction between Creon and Antigone. The level at which these ontol­ogies would make a difference would be in the diagramming of conflicts over what kinds of action exist. The result of such an exercise would not simply be dialogue across conflict – one can read Rorty or Habermas for that. The result of such an exercise would be the remodelling of these conflicts according to these multiple ontologies. Such remodelling – and this is an argument I develop at length with regard to Hume’s History of England – creates new durations.32 And every new duration entails the emergence of a powerful – because new – meas­ure of the gap between the actual and the ideal. In the gap between the actual and ideal arises the hope of justice. References Badiou, Alain, Being and Event, trans. Oliver Feltham, Continuum Books, London and New York 2005 — Briefings on Existence: A Short Treatise on Transitory Ontology, trans. N. Madarasz, State University of New York Press, Albany 2006 — Court traité d’ontologie transitoire, Editions du Seuil, Paris 1998 31 See Oliver Feltham, Anatomy of Failure: Philosophy and Political Action, Bloomsbury, Lon­don 2013. 32 This argument is substantiated in Oliver Feltham, Destroy and Liberate: Political Action on the Basis of Hume, Rowman and Littlefield, London 2019. — Le séminaire: Parménide, L’etre I – Figure ontologique, Fayard, Paris 2014 — “Subversion Infinitésimale”, in Concept and Form, volume I: Key Texts from the ‘Cahiers pour l’Analyse’, ed. Peter Hallward and Knox Peden, Verso, London 2012 — “Ontology and Politics: an Interview with Alain Badiou”, in Alain Badiou, Infinite Thought: Truth and the Return of Philosophy, Continuum, London 2003 Besana, Bruno, “Quel multiple ?”, in Ecrits autour de la pensée d’Alain Badiou, pp. 23–40 — “Replique ; l’événement de l’etre”, in Ecrits autour de la pensée d’Alain Badiou, pp. 125–30  Brassier, Ray, “L’Anti-phenomene – présentation et disparaître”, in Ecrits autour de la pensée d’Alain Badiou, ed. B. Besana and O. Feltham, Harmattan, pp. 55–64, Paris 2007 Brassier, Ray and Alberto Toscano, “Postface: Aleatory Rationalism”, in Alain Badiou, Theoretical Writings, Continuum, London 2005 Desanti, Jean-Toussain, “Some Remarks on the Intrinsic Ontology of Alain Badiou”, in Think Again: Alain Badiou and the Future of Philosophy, ed. P. Hallward, Continuum, pp. 59–66, London 2004 Feltham, Oliver, Alain Badiou: Live Theory, Continuum, London 2008 — Anatomy of Failure: Philosophy and Political Action, Bloomsbury, London 2013 — Destroy and Liberate: Political Action on the Basis of Hume, Rowman and Littlefield, London 2019 Feltham, Oliver and J. Clemens, “An Introduction to Alain Badiou’s Philosophy”, in Alain Badiou, Infinite Thought: Truth and the Return of Philosophy, ed. and trans. O. Feltham and J. Clemens, pp. 1–38, Continuum, London, 2003 Gillespie, Sam, “L’etre multiple présenté, représenté, rendu vrai”, in Ecrits autour de la pensée d’Alain Badiou Hallward, Peter, Badiou: a Subject to Truth, University of Minnesota Press, Minneapolis 2003 Meillassoux, Quentin, “Décision et indécidabilité de l’événement” in Autour de‘Logiques des Mondes’, ed. O. Feltham, D. Rabouin and L. Lincoln, Editions des Archives Con-temporains, Paris 2011 Nancy, Jean-Luc, “Philosophy without conditions”, in Think Again: Alain Badiou and the future of philosophy, pp. 39–49 Pluth, Ed, Badiou: a Philosophy of the New, Polity Press, London 2010 Wikipedia, “Barry Smith”, available at: https://en.wikipedia.org/wiki/Barry_Smith_(ac­ademic) Tho, Tzuchien and Alain Badiou “The Concept of Model, Forty Years Later: An Interview with Alain Badiou”, in Alain Badiou, The Concept of Model: an Introduction to the Materialist Epistemology of Mathematics, ed. and trans. L. Fraser & T. Tho, re.press, Melbourne 2007 Filozofski vestnik | Volume XLI | Number 2 | 2020 | 57–68 | doi: 10.3986/FV.41.2.03 Nick Nesbitt* Bolzano’s Badiou1 Alain Badiou never mentions the pioneering, long-overlooked Czech-German logician Bernard Bolzano in the three volumes of Being and Event. In fact, his name only appears in passing, to my knowledge, on two occasions in Badiou’s oeuvre: once in Number and Numbers, in a list of the modern founders of the thought of number, and once, in a passing reference to Bolzano’s pioneering for-malisation of the concept of the infinite in Paradoxes of the Infinite, in Badiou’s 1994–95 seminar on Lacan.2 Badiou has, moreover, admitted that his knowledge of Bolzano’s work is in fact limited and largely second hand.3 In what follows, I wish, briefly and in a very preliminary sense, to indicate a few of the ways Bolzano’s thought in fact founds many of the essential categories and critiques developed throughout Badiou’s oeuvre. Badiou’s neglect of Bolzano’s thought is hardly surprising, since the philos­opher’s pioneering and foundational work, in set theory, in the critique of post-Kantian Idealism and intuitionism, in the semantic formalization of math­ematics and logic, in the formal nature of axiomatisation, his precocious artic­ulation of a realist, mathematics-based platonism a century before Albert Laut­man’s “transplatonism,” and in many other fields, remained little acknowledged 57 1 The research and work on this study was supported by the Czech Science Foundation (GAÈR) within the project (GA 19-20319S) “From Bolzano to Badiou.” 2 “Les noms de cette premiere modernité [de la pensée du nombre] ne sont pas Proust et Joyce, ce sont Bolzano, Frege, Cantor, Dedekind, Peano.” Alain Badiou, Le Nombre et les nombres, Editions du Seuil, Paris 1990. p. 24. “Apres que l’infini eut reçu dans la mathéma­tique un statut clair, grâce a Bolzano, Weierstrass et Cantor, il cesse de jouer un rôle dans l’argumentation philosophique.” Alain Badiou, Le Séminaire – Lacan, Fayard, Paris 2013 pp. 256–257. In English: Alain Badiou, Number and Numbers, Robin Mackay, trans., Polity Press, Cambridge 2008. Alain Badiou, Lacan: Antiphilosophy 3, trans. Kenneth Reinhard and Susan Spitzer, Columbia University Press, New York 2018. See also Bernard Bolzano, Paradoxes of the Infinite, Routledge, New York 1950 [1851]. 3 Personal communication, New York, 10.18.17. * Princeton University and even less studied until quite recently.4 As late as 1993, Jacques Bouveresse could still decry this “historical injustice” done to “the most gifted and original adversary of German Idealism.”5 Decades before Frege, Husserl, Cantor, Tarski, and Gödel, Bolzano founded or made possible many of the crucial discoveries of modern analytic philosophy and set theory, innovations for which the former would become famous. Following the prohibition of his publications and his early retirement to the Czech countryside, Bolzano’s discoveries remained over­looked after his death in 1848, and thus the breakthroughs of his major works Paradoxes of the Infinite and Theory of Science were only belatedly recognized by Cantor and famously celebrated by Husserl in the Philosophical Investigations.6 Only in recent years have Bolzano’s contributions to philosophy begun to garner the recognition they deserve. Moreover, Bolzano’s conservative moralistic pro­nouncements have nothing of the political daring of Spinoza or Cavailles and Lautman—with whom he nonetheless shares many theoretical points of agree-ment—nor, to be sure, with Badiou’s many developments of and fidelity to the idea of communism, and it is therefore even less surprising that Badiou should never have engaged with Bolzano’s thought.7 Bolzano’s Badiou That said, Bolzano’s vast and still underexplored body of work announces Badi­ou’s thought in a series of crucial dimensions. Here, I wish only briefly indicate 4 Badiou, Being and Event, p. 12. 5 Jacques Bouveresse, “Préface,” in Jacques Laz, Bolzano, critique de Kant, Vrin, Paris 1993, p. iv. 6 “Bernhard Bolzano’s Wissenschaftslehre, published in 1837, a work which, in its treatment 58 of the logical ‘theory of elements’, far surpasses everything that world-literature has to offer in the way of a systematic sketch of logic.” Edmund Husserl, Philosophical Investiga­tions, Routledge, New York 2001 [1900], p. 68. See Bernard Bolzano, Theory of Science, Oxford University Press, Oxford 2014; Théorie de la science, I-II, Gallimard, Paris 2011. 7 On Bolzano’s life and his (in my judgment) relatively banal and conservative moral, politi­cal, and aesthetic philosophy, which I will not address here, see the biographical informa­tion in Paul Rusnock and Jan Šebestík, Bernard Bolzano: His Life and Work, Oxford Univer­sity Press, Oxford, 2019. It should be noted, however, that Bolzano publicly articulated as radical a critique of Viennese militarism as was perhaps possible in his Austro-Hungarian milieu, and it was this in particular that led to the banning of his publications and his forced early retirement from Charles University. On Badiou’s “Idea of Communism,” see Alain Badiou, The Communist Hypothesis, Verso, New York 2015. some of the most evident examples of this relation, each of which remains to be developed comprehensively: 1. Bolzano’s thought remains the most original and decisive critique of post-Kan­tian Idealism in the first half of the nineteenth century. While Badiou cannot be said to reject Hegelian dialectical modes of thought entirely, and in fact has returned repeatedly to interrogate their modalities, it is arguably Bolzano who initiates a tendency in European philosophy to supplement and complete phil­osophical investigations with apodictic demonstrations formulated in the pre­cise, emphatically un-Hegelian mathematical terms of set-based theory. This mode of philosophical demonstration culminates in Badiou’s methodological apparatus deployed throughout the three volumes of Being and Event. While Theory of Science will reiterate and refine the terms of Bolzano’s initial critique of post-Kantian Idealism, Jacques Laz has shown that Bolzano’s 1810 Beiträge zu einer begründeteren Darstellung der Mathematik [Contributions to an Exposition of Mathematics on a Firmer Basis], written when Bolzano was only twenty-nine, already sets forth the principal propositions of his thought.8 Key among these is his systematic critique of Kantian philosophy, attacked at its root via what Bolzano shows to be the contradictory nature of Kant’s claims for an a priori intuition that would ground the entire project of the Critique of Pure Reason.9 While the extraordinary brevity of the Appendix to Bolzano’s Contributions (“The Kantian Doctrine of the Construction of Concepts by Intuitions”) artic­ulates its powerful critique in a mere eleven dense and methodically parsed paragraphs,10 elsewhere Bolzano decries more generally the “love of imagistic language,” lack of expressive precision, and reliance upon “analogies, paradox­es, and tautologies” dominant in the Schellingian and Hegelian thought of the age.11 While Badiou’s often highly imagistic and even poetic turns of phrase are decidedly unlike Bolzano’s generally dry prose, these natural language excurs-59 es are systematically complemented, in Being and Event, by abstract formaliza­tions that seek to produce purely objective statements on the nature of the vari­ous concepts Badiou develops therein. Bolzano unequivocally condemns what he views as a catastrophic tendency of philosophy, “the essence of [which] con­ 8 Bernard Bolzano, Premiers écrits: Philosophie, logique mathématique, Vrin, Paris 2010. 9 Immanuel Kant, Critique of Pure Reason, trans. Paul Guyer, Allen Wood, Cambridge Uni­ versity Press, Cambridge 1998. 10 See Bolzano, Premiers écrits. 11 Cited at Laz, Bolzano, p. 33. sists in [...] playing with images and passing off the slightest superficial analogy between two objects as an identity.”12 The core of this limitation, Bolzano con­cludes, is that “the thinkers of our age do not feel themselves in the least subject to [...] the rules of logic, notably to the obligation always to state precisely and clearly of what one is speaking, in what sense one takes this or that word, and then to indicate from what reasons one affirms this or that thing.”13 Whether or not one judges this an accurate characterisation of Hegelian negative dialecti­cal thought, Bolzano’s critique proved decisively productive for his invention of what Jean Cavailles would famously call, in his posthumous On Logic and the Theory of Science, a “philosophy of the concept” that Badiou has gone on to develop across the three volumes of Being and Event.14 While Cavailles cel­ebrates, in On Logic, Bolzano’s rigorous attention to the necessary modalities of adequate, apodictic demonstration, he nonetheless critiques the ahistorical nature of these conditions, to offer instead a historically developmental con­cept of adequate demonstration.15Badiou can be said in turn to have taken from Cavailles’ critique a positive notion of ontology in its intrinsic relation to science and to mathematics in particular as the adequate language of being as being.16 2. Bolzano’s thought, from its initial formulation in the 1810 Contributions to the posthumous Paradoxes of the Infinite, anticipated by decades not only Dedekind and Cantor’s definitions of infinite sets, but also Russell’s paradox of the set of all sets, and Frege’s definition of number as a set of concepts with isomorphic extension.17 In 1816 Bolzano constructed a proof that is the first 12 Cited at Laz, Bolzano, pp. 32–33. 13 Cited at Laz, Bolzano, p. 32. 14 Jean Cavailles, Sur la logique et la théorie de la science, Vrin, Paris 2008. Note that begin­ ning with his critique of Fregean logicism in “Meditation 3” of Being and Event, Badiou decisively rejects the notion of logic as a purely syntactic operation: “Logic is not a for­malization, a syntax, a linguistic apparatus. It is a mathematized description of possible mathematical universes, under the generic concept of Topos.” Cited at Peter Hallward, Badiou: A Subject to Truth, University of Minnesota Press, Minneapolis 2003, p. 109. I will return to this point below, in reference to Bolzano’s innovative formalization of axiomatic method. 15 Hourya Benis Sinaceur has argued compellingly that Cavailles’ critique of Bolzano in­dicates a subterranean Hegelianism latent in Cavailles’ thought. Houya Benis Sinaceur, Cavailles, Les Belles Lettres, Paris 2013, pp. 114–116. 16 Thanks to David Rabouin for clarifying this point. 17 Laz, Bolzano, p. 42. See also Jan Šebestík, “La classe universelle et l’auto-appartenance chez Bernard Bolzano,” in Mathematical journal of the seminar, ed.P. Zernos, Athens 1986; strictly conceptual formulation of the concept of continuity, a definition that crucially refuses all dependency upon psychologistic notions of intuition. This in itself constituted a powerful rejection of Kantian Idealism, which had judged the concept of continuity to be irreducible to conceptualization.18 It is therefore all the more surprising that neither Badiou’s 1984-1985 seminar L’infini: Aris-tote, Spinoza, Hegel nor Badiou’s culminating, comprehensive statement on the nature of the infinite and human reason, The Immanence of Truths, contains a single mention of Bolzano’s name.19 Paradoxes of the Infinite directly influenced Dedekind and Cantor, and presents, along with Theory of Science, summaries of Bolzano’s principal insights on mathematics and ontology. The concept of the infinite, Bolzano argues in Paradoxes, applies only to pluralities; as such, an object may be defined as infinite if it bears an attribute that indicates an in­finite plurality. Bolzano furthermore offers a proof of the objective nature of the infinite as concept, from the proposition that there exist infinitely many truths en soi.20 In this manner, Bolzano decisively rejects the Hegelian notion of qual­itative infinity, while also distinguishing his actual concept of the infinite from a mere potentiality (as Cauchy argued), as well as from that of Spinoza, whose concept of the infinite Bolzano understands as the “infinite which is capable of no further increase.”21 Instead, Bolzano argues that various concepts of infinite sets – such as that of all points contained in the circumference of a circle, with­out having to count those elements sequentially, have actual, objective exist­ence independent of their psychological cognition (except in cases in which the set in question includes that subjective cognition).22 Bolzano also argues that there exist infinite sets of differing sizes, since one infinite set may logically be a subset of another, an argument from which, as Rusnock and Šebestík point out, Cantor’s theory of transfinite cardinals begins: “namely, by defining two multitudes (finite or infinite) to be equinumerous if and only if there exists a bi­jection (a one-to-one map) between them.”23 This, however, precisely indicates Rusnock and Šebestík, Bolzano pp. 533–540. On the influence of these thinkers upon Ba­diou’s thought, see Hallward, Badiou: A Subject to Truth, Chapter 9, “Mathematics and Science.” 18 Laz, Bolzano, p. 41; Rusnock and Šebestík, Bolzano, pp. 520–533. 19 Alain Badiou, L’infini: Aristote, Spinoza, Hegel, Fayard, Paris 2016; Alain Badiou, L’imma­nence des vérités, Fayard, Paris 2018. 20 Rusnock and Šebestík, Bolzano, pp. 533–534. 21 Cited at Rusnock and Šebestík, Bolzano, p. 534. 22 Ibid., p. 535. 23 Ibid., p. 536. a “paradox of the infinite,” since this definition contradicts the claim that the whole is greater than the part, and thus of unequal size. Though in the face of this contradiction, Bolzano steps back from the Cantorian conception of transfi­nite numbers – the assertion that, in the case of infinite sets, such a one-to-one mapping demonstrates that two sets may have the same number of elements – it has been suggested that Bolzano’s conception of infinite sets might nonetheless constitute a distinctly non-Cantorian theory of the sizes of infinite sets.24 Petr Vo-pìnka summarizes Bolzano’s contribution to the Cantorian theory of the infinite as a veritable asubjective phenomenology: Bolzano’s explanation of how this or that form of the phenomenon of the infinite is produced is [foundational]. For him, these different forms are produced by dif­ferent structures of the corresponding basic multitudes or, as we would say today, in the relational structures of the corresponding communities of objects. From a given community of objects we abstract its relational structure, which means that we replace properties of its members and their relations (what remains is their pure presence) by sets of objects that have such properties, relations by sets of ordered pairs, triples, etc., of objects that enter into the corresponding relations […] and thus immediately find ourselves in the mathematics of the twentieth cen­tury. […] Bolzano’s instructions thus became the program of the set theory of the twentieth century. [...] No one followed it publicly, and no one appealed to it. Mathematicians did not even know Bolzano’s words and, in spite of this, they obediently accomplished what those words commanded. Thus it is not exactly a program in the true meaning of the word, but rather a prophecy which was ful­filled, because it was founded on a clear and far-sighted vision of things to come.25 That Badiou has never to date engaged with Bolzano’s theory of the infinite is certainly understandable for many of the reasons given above, but this genetic relation in the history of thought unquestionably remains a fruitful path for fu­ture research on Badiou’s thought. 24 This has been argued by Paolo Mancosu, a conclusion Rusnock and Šebestík ultimately find unconvincing. See Rusnock and Šebestík, Bolzano, p. 537. 25 Petr Vopìnka, Vyprávìní o kráse novobarokní matematiky [The Story of the Beauty of Neo-baroque Mathematics], Práh, Prague 2004, p. 212, cited at Rusnock and Šebestík, Bolzano, p. 539. 3. Bolzano, decisively influenced on this count by Leibniz, is arguably the first modern philosopher to clearly define mathematics as the adequate language of ontology in the form of a mathesis universalis based upon predicate logic de­rived from Aristotle’s Posterior Analytic.26 Bolzano argues in the Contributions that philosophy is the science addressed to the question “what things are nec­essarily real,” while mathematics, in contrast, addresses the question “What properties must things necessarily possess to be possible?”27 While philosophy attempts to prove the reality of particular objects a priori and unconditionally, mathematics, in Bolzano’s formulation, constitutes the a priori science of the set of universal laws to which all possible objects are subject.28 Scientific meth­od in general is for Bolzano coterminous with the logical rigor of mathematical method.29 While for Bolzano philosophy seeks to deduce the real existence of things (analogous to Badiou’s project to define an asubjective phenomenal logic in Logics of Worlds), mathematics applies its analysis, Bolzano argues, to the possible existence of all objects as governed by general laws. Bolzano can be said to announce Badiou’s demonstration of the laws governing the phenome­nal appearance of things in Logics of Worlds: mathematics, Bolzano affirmed, develops a general theory of forms, which he defined as “a science that treats of the general laws (forms) to which things must conform in their existence.”30 While for Bolzano this constitutes an ontological affirmation, Badiou will re­ject categorial logic as identical with being as such, to argue instead that while mathematics constitutes the adequate language of what is dicible (sayable) of being, a categorial logic offers the means to conceptualize an asubjective phe­nomenology of worlds. 4. Bolzano inaugurates the modern Platonist rationalist realism that would see its fullest development in Gödel and Lautman, a tendency that Badiou has po­lemically affirmed as crucial to his own thought.31 Bolzano’s Contributions al-63 ready formulates in 1810 a philosophy of objective forms [Formen] and the sys­ 26 On Leibniz’s influence on Bolzano, see Laz, Bolzano critique de Kant, pp. 33–35; and on Bolzano’s reconfiguration and critique of Aristotelean logic, see Laz, Bolzano, pp. 27–30. 27 Cited at Laz, Bolzano critique de Kant, p. 29. 28 Cited at Laz, Bolzano, p. 45. 29 Laz, Bolzano, pp. 46–48. 30 Cited at Rusnock and Šebestík, Bolzano, p. 417. 31 See for example Alain Badiou, Plato’s Republic: A Dialogue in Sixteen Chapters, trans. Su­san Spitzer and Kenneth Reinhard, Columbia University Press, New York 2013. tematic connection of truths that defines its structure. This structure, he argues, follows an objective configuration, independent of subjective intuition and psychological certainty. “In the domain of truth,” Bolzano writes in the Contri­butions, “that is to say in the set of all true judgements, there reigns a certain objective connection, independent of all contingent subjective knowledge that we may develop of it. [...] To present this objective connection of judgments, that is to say, to choose a set of judgments and to order them such that any inferred judgment is mentioned as such, seems to me the true goal of scientific exposi­tion.”32 Such is the method Bolzano declares for a science an sich, one in which the objective connection of true judgements remains strictly independent of any subjective thought or feelings of certainty or doubt. Bolzano’s project, which culminates in the Theory of Science, is nothing less than this demonstration of a coherent methodology, one that would develop for mathematical logic a conceptual clarity and definition independent of all psychologism and reliance upon intuition. 5. Badiou’s rejection of Fregean logical grounding in favour of an axiomatic pres­entation, affirmed in Meditation 3 of Being and Event, marks a central moment in his theoretical intervention: “Axiomatisation,” Badiou writes, “is not an artifice of exposition, but an intrinsic necessity. Being-multiple, if entrusted to natural language and to intuition alone, produces an undivided pseudo-presentation of consistency and inconsistency. [...] Axiomatisation is required such that the multiple, left to the implicitness of its counting rule, be delivered without con­cept, that is, without implying the being-of-the-one.”33 While, as David Rabouin points out, Badiou’s notion of axiomatisation draws upon Hilbert and Bourbaki, one might note that Bolzano already presents in the second section of the Con­tributions the first explicit model of axiomatisation, decisively rejecting Kantian intuitionism.34 There, Bolzano does not proceed via a demonstration of the na­ture of the axiom, which would return precisely to the very logicism axiomatisa­tion seeks to overcome (and for which Badiou takes Frege to task in both Being and Event and, in more detail, in Number and Numbers). The axiom, Bolzano argues in terms that decisively announce those of Badiou, is derived neither 32 Cited at Laz, Bolzano, p. 43. 33 Alain Badiou, Being and Event, trans. Oliver Feltham, Continuum, London 2005, p. 43, translation modified. 34 David Rabouin, personal communication. through an intuition, nor even as a minimally and generally acceptable common notion (as with Marx’s definition of capitalism as the general accumulation of commodities), which would rely on a psychological recognition and agreement, but is instead, he argues, indemonstrable, and objectively so. Bolzano argues that it is precisely and minimally the indemonstrability of an axiom, rather than its essential nature, that can in fact be proven. This minimal proof is merely the verification that allows axioms to found the subsequent propositions subject to apodictic demonstration. “Neither deduction, nor demonstration of the truth of a proposition,” Jacques Laz writes, “the Deductio of an axiom is the expo­sition of its status as principal [statut de principe] in an objective sequence of connections between propositions. It is the operation by which are revealed the propositions that are the principals for other propositions.”35 Objective without being a logical demonstration of the truth of an axiom, the Deductio founds the effective conditions of demonstration, deducing only that a given proposition possesses an axiomatic character, in the sense that it cannot be analytically re­duced into subsidiary components.36 6. Finally, and though this may be a less than obvious claim, it is my contention that Bolzano offers compelling conceptual resources to develop the structuralist analysis of what Marx called “social form”: structuralist analysis, that is to say, in the quite specific sense in which Louis Althusser and Pierre Macherey devel­oped it in Reading Capital.37 Here, Bolzano’s concerted critiques of intuitionism, psychologism, and empiricism, and his concept of propositions in themselves can be said to second and further develop the Spinozist critiques that Althusser, Ranciere, Macherey, and Balibar deployed in their readings of Marx’s Capital.38 If Althusser and Macherey in particular looked back three hundred years prior to Spinoza in order to develop their critiques of Hegel and Hegelian Marxism, it is surely no less plausible to suggest that Bolzano, who as mentioned above 65 developed the single most rigorous critique of Kantian and Hegelian Idealism prior to 1848, might offer compelling theoretical arguments to further develop this anti-Hegelian line of thought. Bolzano argued for an objective semantics governing not subjective, hermeneutic knowledge of objects, but their objec­ 35 Laz, Bolzano, p. 55. 36 Ibid., pp. 52–56. 37 Louis Althusser et al., Reading Capital: The Complete Edition, Verso, New York 2016. 38 See Warren Montag, Althusser and His Contemporaries: Philosophy’s Perpetual War, Duke University Press, Durham 2013. tive properties and relations. He inaugurates, this is to say, the affirmation that Badiou will formalize in 1988 as the governing imperative of Being and Event: that mathematics “writes that which, of being itself, is expressible [dicible].”39 This, Bolzano argues, implies the independent existence of these concepts apart from conscious representation. Their meaning, he argues, is rigorously objec­tive and independent from acts of judgment. In fact, I would willingly push this argument even further, to suggest that Bolzano can rightly be said to formulate crucial theoretical resources in the path leading to the Lacanian theory of the symbolic and real, above all perhaps via his realist, semantic critique of the Kantian thing in itself. As Badiou writes of Lacan’s notion of the real, Lacan is not a critic. To be sure, the real differs from reality, which attaches its regime to knowing. But Lacan immediately says: I don’t mean to say the real is unknowable. I’m not a Kantian. [...] Although the real, as distinct from reality, is exempted from the knowable, which is the essence of reality, the real nevertheless does not end up being the absolute unknowable but is instead exposed to being demonstrated.40 Bolzano’s asubjective order of propositions and representations, in a precise and limited sense analogous to what Lacan will call the symbolic order (in what Badiou calls Lacan’s “hyperstructural axiomatic” phase of the 1950s), Bolzano argues, is eminently knowable through acts of human formalization and judg­ment, in contrast to Bolzano’s anti-Kantian notion of the thing in itself as much as the Lacanian real.41 While this objective order presents things as they are in what Bolzano calls the matter [Stoff] of a semantic, symbolic order, it is for Bolzano (unlike Kant) the (real) object of these representations and discursive judgements that remains inaccessible; the real, as Lacan famously stated, is the impasse of formalisation.42 Or as Laz writes, for Bolzano, “we will never be able to grasp the objects of our representation, but only their [objective] meaning through which we represent them.”43 39 Badiou, Being and Event, p. 5, translation modified. 40 Badiou, Lacan, p. 151. 41 Ibid., p. 237. 42 Badiou, Being and Event, p. 5. 43 Laz, Bolzano, pp. 121–122. To suggest a Bolzanian reading of Badiou along the lines that I am suggesting here is surely no more implausible than was Pierre Macherey’s influential read­ing of Spinoza’s Hegel.44 It is to articulate a transversal relation; unlike that which Macherey articulates, however, in Badiou’s case, there is no obscure dis­avowal on his part of a hidden proximity to Bolzano’s historically prior thought, but rather a complex field of relations and implications that remains to be de­veloped and articulated, an investigation that Badiou himself might be the first to welcome. While I have here tried only to suggest a few of these possible paths of research, it seems to me that Bolzano’s thought is no mere antiquarian moment in a histo­ry of axiomatic philosophies, philosophies that hold mathematics to constitute the adequate language of being. At least as promising, for example, would be to further concretize the anti-Hegelian, objective dimensions of apodictic demon­stration that Althusser, Macherey and Badiou himself have argued govern not only much of their own thought, but above all, the critical projects of Marx and Lacan. Such a project might remain faithful to the imperative that Badiou has ar­gued governs his philosophical project as a whole: “To legitimate the claim that a truth can be absolute, while also a localized construction, [...] eternal, while belonging to the time of this world [... and] a-subjective, while demanding, to be grasped, a subjective incorporation.”45 To place Badiou’s philosophy of being and event in dialogue with that of Bolzano in this manner would imply the ex­ploration of truths developed and demonstrated in suspension, truths articulat­ed between these two figures, in the diffraction of their mutual demonstrations, as both the critique and proof of philosophy itself in the set that constitutes its own historicity. 67 References Althusser, Louis et al., Reading Capital: The Complete Edition, Verso, New York 2016 Alain Badiou, Being and Event, trans. Oliver Feltham, Continuum, London 2005 — Lacan: Antiphilosophy 3, trans. Kenneth Reinhard and Susan Spitzer, Columbia Uni­versity Press, New York 2018 — Le Nombre et les nombres, Editions du Seuil, Paris 1990 — L’immanence des vérités, Fayard, Paris 2018 44 Pierre Macherey, Hegel or Spinoza, University of Minnesota Press, Minneapolis 2011 [1979]. 45 Badiou, L’Immanence des vérités, p. 13. — L’infini: Aristote, Spinoza, Hegel, Fayard, Paris 2016 — Le Séminaire – Lacan, Fayard Paris 2013 — Number and Numbers, trans. Robin Mackay, Polity Press, Cambridge 2008 — Plato’s Republic: A Dialogue in Sixteen Chapters, trans. Susan Spitzer and Kenneth Reinhard, Columbia University Press, New York 2013 — The Communist Hypothesis, Verso, New York 2015 Macherey, Pierre, Hegel or Spinoza, University of Minnesota Press, Minneapolis 2011 [1979] Montag, Warren, Althusser and His Contemporaries: Philosophy’s Perpetual War, Duke University Press, Durham 2013 Benis Sinaceur, Houza, Cavailles, Les Belles Lettres, Paris 2013 Bolzano, Bernard, Paradoxes of the Infinite, Routledge, New York 1950 [1851] — Premiers écrits: Philosophie, logique mathématique, Vrin, Paris 2010 — Theory of Science, Oxford University Press, Oxford 2014 — Théorie de la science, I-II, Gallimard, Paris 2011 Cavailles, Jean, Sur la logique et la théorie de la science, Vrin, Paris 2008 Hallward, Peter, Badiou: A Subject to Truth, University of Minnesota Press, Minneapolis 2003 Husserl, Edmund, Philosophical Investigations, Routledge, New York 2001 [1900] Kant, Immanuel, Critique of Pure Reason, trans.Paul Guyer, Allen Wood, Cambridge Uni­versity Press, Cambridge 1998 Laz, Jacques, Bolzano, critique de Kant, Vrin, Paris 1993 Vopìnka, Petr, Vyprávìní o kráse novobarokní matematiky [The Story of the Beauty of Neo-baroque Mathematics], Práh, Prague 2004 Šebestík, Jan, “La classe universelle et l’auto-appartenance chez Bernard Bolzano,” in Mathematical journal of the seminar, ed.P. Zernos, Athens 1986 Filozofski vestnik | Volume XLI | Number 2 | 2020 | 69–102 | doi: 10.3986/fv.41.2.04 Jelica Šumiè Riha* La place de la mathématique : Badiou avec Lacan1 « La science sans conscience »1 La mathématique peut-elle s’interroger sur elle-meme ? Dans la Critique de la raison pure, Kant fait de la mathématique « l’exemple le plus éclatant d’une rai-son pure qui réussit a s’étendre d’elle-meme sans le secours de l’expérience 2 ». La définition kantienne de la mathématique résonne avec le formalisme de la mathématique moderne dont l’ascese particuliere exclut toute considération sur le contenu. Ainsi, la mathématique pure, selon Bertrand Russell, est « la classe de toutes les propositions de la forme ‘P implique Q’ … ou ni P ni Q ne contiennent d’autres constantes que des constantes logiques 3 ». Selon défini­tion russellienne, la mathématique pure se présente comme un ensemble d’im­plications formelles indépendantes de tout contenu. En forçant un peu le trait, on pourrait dire que la mathématique est « la science qui coincide le plus tota­lement avecsapropreécriture 4». Mais pour le mathématicien d’aujourd’hui, l’enjeu de l’écriture mathématique ouvre une autre problématique, celle qui concerne le rapport du mathématicien avec la mathématique. Vu dans cette perspective, le mathématicien est celui qui efface le plus totalement les marques de la production de son texte, tant les marques du sujet qui le produit que les marques de la société dans laquelle il est produit. C’est meme en ce texte mathématique que l’effacement des marques 69 est le plus explicite. Mais ce ne peut etre la qu’un paradoxe, car le caractere ex­ 1 Cet article est le résultat du programme de recherche P6-0014 «Conditions et problemes de la philosophie contemporaine», financé par l’Agence slovene recherche. 2 Emmanuel Kant, Critique de la raison pure, Ouvres philosophique, tome I : Des premiers écrits a la Critique de la raison pure (1747-1781), Bibliotheque de la Pléiade, Gallimard, Paris 1980, p. 1297. 3 Bertrand Russell, Écrits de logique philosophique, trad. Jean-Michel Roy, PUF, Paris 1989, p. 21. 4 René Lavendhomme, Lieu du sujet. Psychanalyse et mathématique, Seuil, Paris 2001, p. 8. * Institut de philosophie, Centre de recherches scientifiques aupres de l’Académie slovene des sciences et des arts plicite de l’effacement devrait le nier comme effacement et devrait donc conduire a effacer l’effacement5. « Ne serait-il pas possible, » demande le mathématicien René Lavendhomme, « qu’a l’intérieur de l’écriture meme du texte mathématique se fasse jour la re­lation que ce texte entretient avec le sujet »?6 En proposant une autre définition de la mathématique selon laquelle « la mathématique est une science ou l’on ne sait jamais de quoi l’on parle, ni si ce qu’on dit est vrai 7», Russell jette une lu­miere particuliere précisément sur le lien que le mathématicien entretient avec la mathématique. Strictement parlant, les deux définitions russelliennes ne sont pas contradictoires. Si la premiere définition met l’accent sur le formalisme dont la rigueur logique bride la liberté dans le maniement des lettres et des sym-boles, la deuxieme inclut une dimension supplémentaire, celle du dire mathé­matique qui met en relief le rapport des mathématiciens avec la mathématique. Le probleme qui se pose alors est le suivant : qu’est-ce que la mathématique si le mathématicien peut ne pas y croire ? La légereté de la lettre, corrélative au délestage du sens, libere-t-elle le sujet mathématicien a tel point qu’il se targue de ne pas savoir de quoi la mathématique parle, ni d’y croire vraiment ? Dans ce qui suit nous ferons référence aux commentaires respectifs de Badiou et de Lacan suscités par cet étrange énoncé affirmant que, en mathématique, on ne sait jamais de quoi on parle ni si ce que l’on dit est vrai. Badiou et Lacan insistent tous les deux sur un point, a savoir que le mathématicien n’est incon­testablement pas libre de ne pas croire a la mathématique. Ou, pour reprendre les propres termes de Badiou, que « le mathématicien est d’abord celui qui croit ‘dur comme fer’ aux mathématiques8». Quant a l’ignorance mathématique, les réponses que donnent Badiou et Lacan divergent, et cela précisément dans la mesure ou, meme si l’acces a la psychanalyse, comme l’acces a la philosophie, est un certain type d’acces aux mathématiques, il ne s’agit tout de meme pas du meme type d’acces. 5 Ibid., p. 7. 6 Ibid. 7 Bertrand Russell, « Work on the principles of mathematics, » TheInternational Monthly, 4, (1/1901), p. 84. Alain Badiou, « La mathématique est une pensée », Court traité d’ontologie transitoire, Seuil, Paris 1998, p. 43. Selon Badiou, lorsque Russell disait « que les mathématiques sont un discours ou on ne sait pas de quoi on parle, ni si ce qu’on dit est vrai », c’était « sans le croire, bien sur », et d’ailleurs « personne en vérité ne l’a jamais cru, sauf les ignorants, ce que n’était pas Russell 9». Si Badiou ne peut pas croire Russell, c’est parce que, selon lui, les mathématiques sont, au contraire, le seul discours qui « sache » absolument de quoi il parle : l’etre, comme tel, quoique ce savoir n’ait nullement besoin d’etre réfléchi de façon intra-mathéma­tique, puisque l’etre n’est pas un objet, ni n’en prodigue. Et c’est aussi le seul, c’est bien connu, ou l’on ait la garantie intégrale, et le critere, de la vérité de ce qu’on dit, au point que cette vérité est l’unique jamais rencontrée a etre intégra­lement transmissible10. Cependant, Badiou semble etre d’accord avec Russell sur un point tout a fait précis, a savoir : la mathématique ne sait pas et n’a pas besoin de savoir qu’elle fait de l’ontologie. Badiou affirme en effet que la mathématique en tant que dis-cours sur l’etre peut tres bien se passer de ce savoir pour se perpétuer toute seule, presque a l’aveugle. Il revient donc a la philosophie de donner la « di­gnité ontologique11 » a ce que les mathématiciens se contentent de mettre au travail. Mais il y a plus important encore : cette ignorance est en quelque sorte constitutive de la mathématique en tant que science de l’etre, puisque, comme le souligne Badiou, il est de l’essence de l’ontologie de s’effectuer dans la forclusion réflexive de son identité. Pour celui-la meme qui sait que c’est de l’etre-en-tant-qu’etre que pro-cede la vérité des mathématiques, faire des mathématiques – et spécialement des mathématiques inventives – exige que ce savoir ne soit a aucun moment représenté. Car sa représentation, mettant l’etre en position générale d’objet, 71 corrompt aussitôt la nécessité, pour toute effectuation ontologique, d’etre désob­jectivante12. 9 Alain Badiou, L’Etre et l’événement, Seuil, Paris 1988, p. 15. 10 Ibid. 11 Ibid., p. 21. 12 Ibid., p. 17. Apres avoir posé que « ce qui est dicible – et dit – de l’etre en tant qu’etre ne releve d’aucune façon du discours philosophique13 » mais de la théorie des en­sembles, la théorie cantorienne du multiple pur – apres avoir reconnu, comme Russell, que les mathématiciens ne savent pas en tant que mathématiciens, pris dans l’exercice et l’épreuve de leur pratique, que ce qu’il font c’est de pro-noncer ce qui est dicible de l’etre en tant qu’etre – Badiou reconnaît également qu’il y a « les moments ou la mathématique semble convoquée a se penser elle­meme, a dire ce qu’elle est. » Il s’agit des moments singuliers ou la mathéma­tique – « sous la contrainte d’une butée réelle, ou du surgissement nécessaire, dans son champ, d’un point d’impossible » – semble requise, au regard de ses propres buts, de penser sa pensée. La mathématique n’est obligée de revenir sur elle-meme que « sous l’injonction de sa butée intérieure ». Le moment de la torsion de la mathématique sur elle-meme est le moment ou la mathématique est confrontée « a sa dimension décisoire14 ». Sur ce point, Badiou semble etre obligé de distinguer deux modes opératoires de la mathématique identifiée a l’ontologie : il y a d’une part, le mode opéra­toire « normal » de la mathématique ou la mathématique pense l’etre sans pour autant se considérer comme « la pensée de la pensée qu’elle est 15 », et, il y a, d’autre part, les moments de crise ou16 « la mathématique, butant sur un énoncé qui atteste en un point la venue de l’impossible, se retourne sur les décisions quil’orientent17». C’est en s’appropriant un theme kantien, celui de l’orientation dans la pensée, que Badiou se propose de traiter la question de savoir si et dans quelles conditions la mathématique est capable de devenir la pensée de la pen-sée qu’elle est. L’orientation dans la pensée, dans le cas de la mathématique, porte sur la norme que se donne la pensée pour décider de l’existence. L’orien­tation dans la pensée porte, plus exactement, sur « ce que la pensée détermine en elle-meme comme voie d’acces a ce qu’elle déclare exister18». Or, la encore, 13 Ibid., p. 20. 14 Badiou, « La mathématique est une pensée », pp. 48–49. 15 Badiou, « L’événement comme trans-etre », Court traité de l’ontologie transitoire, p. 55. 16 Badiou évoque quelques exemples de ces moments critiques : crise des irrationnels dans la mathématique pythagoriciennne, la crise liée aux « paradoxes » de la théorie des en­sembles, la crise liée aux théoremes de limitation des formalismes, les polémiques sus­citées par le statut de l’axiome du choix, etc. Voir Badiou, « La mathématique est une pensée », pp. 46–48. 17 Ibid., p. 51. 18 Ibid., pp. 49–50. le fait que la mathématique, dans les moments particuliers de crise, rencontre une impossibilité, un « bout de réel » ou se rapporte a sa propre pensée selon son orientation, est quelque chose dont la mathématique elle-meme ne peut pas rendre compte, bien qu’elle le manifeste, comme le dit Badiou. Strictement parler, c’est a la philosophie sous condition des mathématiques, et plus parti­culierement, a la philosophie, qui s’est mise sous condition de l’événementia­lité mathématique, puisqu’elle reconnaît la mathématique « comme un lieu de pensée singulier, dont les événements et procédures doivent etre retracés dans l’acte philosophique19 » que revient la tâche de traiter la question : qu’est-ce qu’une orientation dans la pensée, lorsqu’il s’agit de la mathématique ? Pour cerner plus étroitement le partage des tâches entre la mathématique et la philosophie, il faut donc préciser que c’est parce que la mathématique – sauf dans les rares moments de « crise » – « pense l’etre, mais n’est pas pensée de la pensée qu’elle est », qu’il revient a la philosophie « d’identifier la vocation ontologique de la mathématique ».20 Or, admettre que l’ontologie ne se saisit dans la mathématique que par la décision de la philosophie, c’est délimiter en meme temps l’espace propre a la philosophie, celui de ce-qui-n’est-pas-l’etre-en-tant-qu’etre, a savoir la vérité. Comment alors comprendre le lien entre l’on-tologie et la philosophie, si « le dicible de l’etre est disjoint du dicible de la vé­rité21 » ? Si « toute vérité est postévénementielle22», c’est-a-dire la production d’une nouveauté qui est en tant que telle insituable dans le systeme de savoirs connu, comme le soutient Badiou, et si « les mathématiques ne peuvent penser aucune procédure devérité, puisqu’elles éliminent l’événement23», ce multiple paradoxal qui échappe a l’ontologie, les mathématiques « doivent décider s’il est compatible avec l’ontologie qu’il y a des vérités24 ». Et dans la mesure ou l’etre des vérités est conçu sous la forme de multiples non constructibles ou « génériques », la question que pose Badiou est la suivante : « l’ontologie peut-73 elle produire le concept d’un multiple générique ? 25» C’est le mathématicien, Paul Cohen, qui a fourni la réponse a la question posée par la philosophie, dé­ 19 Alain Badiou, Conditions, Seuil, Paris 1992, p. 158. 20 Badiou, « L’événement comme trans-etre », p. 55. 21 Badiou, L’Etre et l’événement, p. 391. 22 Ibid. 23 Ibid., p. 376. 24 Ibid. 25 Ibid., p. 391. montrant l’existence des multiplicités « génériques », c’est-a-dire non construc­tibles. Ce que Badiou cherche et trouve dans la mathématique, c’est la preuve de « la compatibilité de l’ontologie avec la vérité ». Cette compatibilité implique, pour Badiou, que « l’etre de la vérité, comme multiplicité générique, » peut et doit etre « ontologiquement pensable, meme si une vérité ne l’est pas26». C’est dans ce sens que nous nous proposons de lire la remarque a premiere vue assez obscure de Claude Imbert selon laquelle « Badiou pose que la mathématique cantorienne donne a la philosophie sa position de réel en meme temps que les moyens de le penser 27 ». En revanche, pour Lacan, la nescience qui caractérise la mathématique se­lon Russell touche a ce que Lacan appelle « la frontiere sensible entre vérité et savoir 28». Dans son dialogue « improvisé » avec Russell, au cours de son séminaire « Le savoir du psychanalyste », Lacan clarifie ainsi la définition de Russell : M. Bertrand Russell … a pris soin de dire en ses propres termes [que] la mathéma­tique, c’est tres précisément ce qui s’occupe d’énoncés dont il est impossible de dire s’ils ont une vérité, ni meme s’ils signifient quoi que ce soit. C’est bien une façon un peu poussée de dire que tout le soin précisément qu’il a prodigué a la rigueur de la mise en forme de la déduction mathématique, est quelque chose qui assurément s’adresse a toute autre chose que la vérité, mais a une face qui n’est tout de meme pas sans rapport avec elle, sans ça il n’y aurait pas besoin de l’en séparer d’une façon si appuyée29. C’est d’ailleurs dans ce contexte que Lacan introduit le probleme de l’« incom­préhension » des mathématiques. Cette incompréhension mathématique, pour utiliser son propre terme, permet a Lacan de poser la question suivante : quel ordre de vérité peut-on attendre des mathématiques ? Car l’incompréhension mathématique n’est pas une simple carence contingente mais doit etre « consi­dérée comme un symptôme » et Lacan en avance une raison : 26 Ibid. Claude Imbert, « Ou finit le platonisme ? », Alain Badiou : Penser le multiple, Charles Ra-mond (éd.), L’Harmattan, Paris 2002, p. 357. 28 Jacques Lacan, « Séminaire de Jacques Lacan, Le savoir du psychanalyste, 1971-1972 » (iné­dit), 4 novembre 1971. 29 Ibid., 2 décembre 1971. Les sujets en proie a l’incompréhension mathématique attendent plus de la véri­té que la réduction a ces valeurs qu’on appelle, au moins dans les premiers pas de la mathématique, des valeurs déductives. Les articulations dites démonstra­tives leur paraissent manquer de quelque chose qui est précisément au niveau d’une exigence de vérité30. Cette exigence de vérité est, selon Lacan, a la source de l’incompréhension ma-thématique, en tant qu’elle proviendrait d’un décalage, d’une insatisfaction de­vant une valeur de vérité réduite a la condition de la démonstration. Ceux qui ne comprennent pas les mathématiques auraient voulu qu’elles soient vraies, qu’elles disent quelque chose sur quelque chose, et, comme elles ne disent rien sur quoi que ce soit qui leur est extérieur,31 ils les rejettent en bloc. En d’autres termes, c’est parce qu’on veut du vrai que l’on ne comprend pas les mathéma­tiques. Et, de façon plus ramassée : « Cette embrouille autour de l’incompréhen­sion mathématique » signale le symptôme de « l’amour de la vérité 32». Ce qui paraît etre l’évacuation de la pathétique du vrai laisse un vide, mais, a suivre Lacan : « on aurait tout a fait tort de penser que la mathématique est quelque chose qui, en effet, a réussi a vider tout ce qu’il en est du rapport a la vé­rité de son pathétique 33». Bien loin de réussir a éliminer sa « dimension du pa-thétique », l’histoire de la mathématique – a travers la crise des irrationnelles, l’apparition du calcul infinitésimal, les impasses des procédés de démonstra­tions, mais surtout « la peine, la douleur qu’a engendré, » ce que Lacan appelle « l’effraction cantorienne »34, issue d’une rencontre avec la vérité qui a pu aller chez Cantor jusqu’a la menace de la folie – « de quelque rapports du matheme … 30 Ibid. 75 31 C’est parce que la formalisation mathématique « se fait au contraire du sens, j’allais presque dire a contre-sens. Le ça ne veut rien dire concernant les mathématiques, c’est ce que disent, de notre temps, les philosophes des mathématiques, fussent-ils mathémati­ciens eux-memes, comme Russell ». Jacques Lacan, Le séminaire de Jacques Lacan, Livre XX, Encore, Seuil, Paris 1975, p. 85. 32 Lacan, « Le savoir du psychanalyste », 2 décembre 1971. 33 Ibid. 34 Par exemple, lorsque Cantor établit une bijection entre l’ensemble des points d’un carré dont le côté est l’intervalle (0, 1), et cet intervalle lui-meme, il écrit a Dedekind : « Je le vois, mais ne le crois pas ». C’est son systeme de croyance, plus précisément, son systeme d’inter­prétation, son ordonnancement symbolique, qui se trouve atteint. Cité dans Nathalie Char-raud, Infini et Inconscient. Essai sur Georg Cantor, Anthropos-Economica, Paris 1994, p. 65. avec la dimension de vérité 35 ». C’est exactement ce point qui intéressait La-can, le rapport de la vérité et du savoir, ou l’auteur d’une découverte, comme le montre bien le cas de Cantor, le « drame du savant36», se trouve lui-meme nécessairement impliqué. La clarification de l’incompréhension mathématique importe pour la psychanalyse dans la mesure ou la compréhension implique­rait sa réduction a des articulations non-contradictoires qui ne donnent pas sa place a ce que la vérité doit au désir. En effet, pour Lacan, « la non-contradic­tion ne saurait d’aucune façon suffire a fonder la vérité37». Or, c’est exactement sur cette question du lien entre le désir et la vérité que se produit le bouclage entre la mathématique et la psychanalyse. Mais il n’en est pas moins vrai que, si le désir est porteur d’une valeur de vérité qui excede le savoir mathématique, l’abord de celui-ci ramene au savoir mathématique38. Si Lacan insiste sur la problématique de l’incompréhension, c’est aussi parce que l’incompréhension est genante pour la psychanalyse dans la mesure ou le travail de l’analyste, a la différence de celui du mathématicien, implique que, tout en sachant pratiquer la formalisation en analyse, l’analyste n’ait pas re-noncé a la notion de vérité dont se passe quasiment le mathématicien. Ainsi, quand Lacan introduit le concept de matheme, c’est pour signaler « qu’il n’est absolument pas vrai de parler de matheme comme de quelque chose qui, d’au­cune façon, serait détaché de l’exigence véridique39». Mais pour comprendre le caractere de vérité qui est a l’ouvre en analyse, pour comprendre le caractere de formalisation qui est au travail dans sa littéralité, il faut passer par les tra­vaux qui tiennent la vérité pour une valeur parmi d’autres et qui s’effectuent fondamentalement en mathématiques. Deux modes du réel : le réel en mathématique, le réel en psychanalyse Ainsi, en mathématiques, « on commence par mettre des lettres sans dire abso­lument rien de ce a quoi elles peuvent servir 40», car pour penser il faut formali­ 35 Ibid. Jacques Lacan, « Science et vérité », Écrits, Seuil, Paris 1966, p. 870. 37 Ibid. 38 Sur ce point, nous renvoyons le lecteur au beau livre de Nathalie Charraud, Infini et In- conscient. Essai sur Georg Cantor. 39 Lacan, « Le savoir du psychanalyste », 2 décembre 1971. 40 Ibid., 15 décembre 1971. ser, c’est-a-dire donner une forme littérale a la pensée, travailler sur la littéralité de la pensée. Si d’un côté la notion meme de mathématicité se trouve dépouillée jusqu’a une stricte littéralité pour mieux serrer le réel que vise la psychanalyse, c’est ainsi que nous pouvons décrire la maniere selon laquelle Lacan aborde la mathématique, de l’autre côté, ce qui importe, selon Badiou, dans les pro-jets mathématiques, a commencer par celui de Cantor, c’est d’instaurer « une formalisation intégrale, une théorie générale des univers de la pensée pure … Réduire la mathématique a son acte : la puissance d’univocité du formalisme, la force nue de la lettre et de ses codes ». Ainsi, ce que vise la formalisation de l’acte mathématique, n’est rien d’autre que « le dire du réel mathématique41 ». C’est dans cette perspective que Badiou insiste sur la formalisation comme « une exigence de la pensée, aussi bien pour les mathématiciens que pour les philosophes », car ce qui importe a ses yeux, c’est que « la présentation formelle de la mathématique enveloppe une radicalité fondatrice quant a la nature de son acte 42». Quelle leçon peut-on tirer de cette « passion pour la formalisation » qui caractérise les mathématiques du 20eme siecle ? S’appuyant sur les démons­trations gödeliennes, Badiou met l’accent sur le fait que, meme si « toute dispo­sition formalisante de la pensée laisse un reste », la leçon a tirer de cet « échec » inévitable de la formalisation n’est pas qu’il faut abandonner la formalisation comme voie d’acces au réel, mais, au contraire, qu’il faut prendre le « résidu intraité » par une formalisation non aboutie comme point de départ pour une nouvelle formalisation: C’est bien ce qui sépare la formalisation, comme pensée et projet, d’un simple usage pragmatique des formes. Il faut, sans jamais se décourager, inventer d’autres axiomes, d’autres logiques, d’autres manieres de formaliser. L’essence de la pensée réside toujours dans la puissance des formes43. 77 Quant a Lacan, il est donc bien clair que ce n’est pas tant le savoir mathéma­tique qui importe pour lui que la position du sujet-mathématicien par rapport a un désir inédit, celui de mettre un symbole, une lettre, la ou il y a hors-sens. Finalement, ce qui l’intéresse dans les mathématiques, ce n’est pas la vérité, 41 Alain Badiou, Le siecle, Seuil, Paris 2005, pp. 228–229. 42 Ibid., p. 230. 43 Ibid., pp. 230–231. mais leur puissance de construction, meme au prix de voir la vérité réduite a une valeur parmi d’autres. C’est d’ailleurs en ce sens que Lacan a pu dire, ce qui a premiere vue peut paraître contradictoire, que : « Il n’y a de vérité que ma-thématique, c’est-a-dire écrite44». Car ce qui importe pour Lacan, tout comme pour Badiou d’ailleurs, dans les mathématiques, ce n’est pas seulement leur puissance structurante, celle de donner forme a ses objets, mais la puissance d’arrachement a l’expérience et a l’empiricité qui est attribuée aux mathéma­tiques.45 Ainsi, selon Lacan, le discours mathématique ne peut se fonder sur rien d’autre que « ce langage de pur matheme … le seul a pouvoir s’enseigner : ceci sans recours a quelque expérience, qui d’etre toujours, quoi qu’elle en ait, fondée dans un discours, permet les locutions qui ne visent en dernier ressort rien d’autre qu’a, ce discours, l’établir46 ». Pour Badiou, également, la mathématique s’intéresse « a la dimension la plus formelle, la plus abstraite, la plus universellement presque vide, de l’etre comme tel47 ». Ce que la mathématique rend possible, en tant que « ressource spéculative », c’est ce que Badiou nomme « l’ontologie absolue », plus précisé­ment, « l’existence d’un univers de référence, d’un lieu de la pensée de l’etre en tant qu’etre » qui ne se laisse « décrire, ou penser, qu’a partir d’axiomes, ou de principes, auxquels il correspond. Il n’en existe aucune expérience, ni aucune construction qui dépende d’une expérience. Il est radicalement non 44 Jacques Lacan, « Séminaire de Jacques Lacan XXII, R.S.I. 1972-1973 », (inédit) 11 décembre 1972. Et plus précisément : « La mathématique fait référence a l’écrit, a l’écrit comme tel, et la pensée mathématique, c’est le fait qu’on peut se représenter un écrit. » « Séminaire de Jacques Lacan XXV, Le moment de conclure 1977-1978 » (inédit), 11 décembre 1978. 45 Ainsi, pour Platon, la mathématique est une condition du penser parce qu’elle rompt avec l’opinion. Or, dans la mesure ou il s’agit d’une « rupture contrainte », comme le souligne Badiou, « involontaire, inapparente a elle-meme, et surtout dépourvue de liberté », la mathématique, selon Platon, lu par Badiou, « n’établit pas la pensée dans la souveraine liberté de sa disposition propre ». Et plus précisément encore, définie comme « l’entre-deux de la vérité et de la liberté de la vérité », la mathématique est « la vérité encore captive de la non-liberté que réclame le geste violent de répudiation de l’immédiat. La mathé­matique appartient a la vérité, mais dans une figure contrainte de celle-ci ». Pour capter cette oscillation entre la vérité et la liberté, Badiou propose cette formule saisissante ; « la mathématique est trop violemment vraie pour etre libre, ou elle est trop violemment libre (c’est-a-dire discontinue) pour etre absolument vraie ». Alain Badiou, « Philosophie et mathématique », Conditions, 1992, pp. 168–171. 46 Jacques Lacan, « L’étourdit », Autres écrits, Seuil, Paris 2001, p. 472. Alain Badiou, Éloge des mathématiques, Flammarion, Paris 2015, p. 43. empirique48». La mathématique crée donc des objets sans « s’appliquer » a l’ex­périence. Loin de poser l’existence comme extérieure a la mathématique, Lacan et Badiou la posent comme essentiellement déterminée par la mathématique elle-meme. C’est dans ce sens que Lacan peut affirmer : « Il est clair que ce n’est qu’a partir d’une certaine réflexion sur les mathématiques que l’existence a pris son sens49 ». De meme pour Badiou, la mathématique entendue en tant qu’ « ontologie absolue » obéit au « principe de maximalité » selon lequel « toute entité intellectuelle dont l’existence s’infere sans contradiction des axiomes qui la prescrivent existe par cela meme50 ». Cette notion mathématique d’existence est lisible dans l’articulation de ce que Cantor nommait la formation correcte d’un concept. Dans « Les fondements d’une théorie générale des ensembles », Cantor explique ce que l’on doit en-tendre par liberté de concevoir et de conceptualiser en mathématiques. La mathématique est pleinement libre dans son développement et ne connaît qu’une seule obligation, … ses concepts doivent etre non contradictoires en eux­memes et soutenir d’autre part avec les concepts déja formés antérieurement, déja présents et assurés, des relations fixes, réglées par les définitions. En parti­culier, pour pouvoir introduire de nouveaux nombres, elle est seulement requise d’en donner des définitions leur conférant une précision et le cas échéant une relation aux anciens nombres telles que l’on puisse dans des cas donnés les dis-tinguer les uns des autres de maniere déterminée. Des qu’un nombre satisfait toutes ces conditions, il peut et doit etre considéré comme existant et réel dans la mathématique51. La liberté des mathématiques releve de la conception cantorienne de l’exis­tence mathématique. La difficulté que Cantor confronte est celle de montrer que 79 les nombres transfinis ont bien une existence effective, au meme titre que les nombres finis. En liant existence et non contradiction, Cantor considere qu’il suffit qu’un objet mathématique ne contredise aucun énoncé de la théorie a la­ 48 Alain Badiou, L’immanence des vérités, Fayard, Paris 2018, pp. 36–37. 49 Lacan, « Le savoir du psychanalyste », 1 juin 1972. 50 Badiou, L’immanence des vérités, p. 37. 51 Georg Cantor, « Fondements d’une théorie générale des ensembles », paru en 1883 dans Mathematische Annalen, XXI ; pp. 545–586, trad. J.-C. Milner, Cahiers pour l’analyse, no. 10, p. 48. quelle il appartient pour exister, il n’est donc pas besoin d’une démonstration effective de son existence. Conscient de rompre avec les mathématiques traditionnelles, Cantor insiste sur le fait qu’il ne peut « poursuivre [ses recherches sur les nombres transfinis] qu’en étendant au-dela de ses limites antérieures le concept de nombre entier existant réellement. En vérité, cette extension s’oriente dans une direction ou, a ma connaissance, nul ne l’avait jusqu’a présent cherchée52». Or cette nouvelle exigence de considérer les nouveaux nombres, les transfinis, comme existant réellement, se fonde sur la notion meme de nombre : « La notion de nombre, si développée qu’elle soit ici, porte en soi le principe d’une extension nécessaire en elle-meme et absolument infinie53». Ainsi, pour Cantor, les nombres trans-finis sont aussi « réels » que les entiers finis ou les irrationnels. Car le transfini n’est pas simplement un infiniment grand, un « fini variable », « une grandeur pouvant croître au-dela de toute limite finie54 ». Si le transfini est un infini ac-tuel ou infini proprement dit, c’est parce qu’on peut le fixer de façon mathématique par des nombres, cette pensée s’est imposée a moi logiquement, presque contre ma volonté … L’assomption qu’en dehors de l’absolu, de ce qui ne peut etre atteint par aucune détermination, et du fini, il ne devrait pas exister de modifications qui soient déterminables par des nombres, encore que non-finies, et soient par conséquent ce que j’appelle l’infini propre­ment dit – cette assomption ne me paraît justifiée par rien … Ce que j’affirme et crois avoir démontré par le présent travail ainsi que par mes tentatives anté­rieures, c’est qu’apres le fini, il existe un transfinitum (que l’on pourrait aussi nommer suprafinitum), c’est-a-dire une échelle illimitée de modes déterminés qui par nature ne sont pas finis, mais infinis, et qui cependant peuvent etre pré­cisés, tout comme le fini, par des nombres déterminés, bien définis et distin­guables. Ma conviction est des lors que le domaine des grandeurs définissables n’est pas clos avec des grandeurs finies et que les limites de notre connaissance peuvent etre étendues en conséquence, sans qu’il soit nécessaire pour autant de faire violence a notre nature55. 52 Ibid., p. 35. 53 Georg Cantor, Gesammelte Abhandlungen mathematischen und philosophischen Inhalts, E. Zermelo (éd.), Springer, Berlin 1932, p. 95. 54 Ibid., p. 374. 55 Cantor, « Fondements d’une théorie générale des ensembles », pp. 42–43. Ainsi se trouve réfutée une these traditionnellement opposée a l’infini actuel en mathématiques: la finitude de notre entendement nous empeche de conce­voir l’infini qui se situerait entre le fini et l’absolu (Dieu). La these défendue par Cantor est que si bornée que soient en fait la nature humaine, il y a cependant en elle une tres grande part d’infini, et je vais jusqu’a soutenir que si elle n’était pas elle-meme infinie sous bien des rapports, on ne saurait expliquer la conviction et la cer­titude assurées ou nous nous savons tous unis, touchant l’etre de l’absolu. En particulier, je tiens que l’entendement humain est doué d’une aptitude illimitée a former par progression des classes de nombres entiers, qui soutiennent une relation déterminée avec les modes infinis et en constituent les puissances de degré croissant56. Comme tous les mathématiciens ou presque, Cantor adopte une forme de réa­lisme du nombre. Il fait la distinction entre la réalité intrasubjective et la réalité transsubjective : La premiere réalité, dite « intrasubjective ou immanente » re-leve de notre entendement. Mais il y a une autre réalité des nombres que Cantor appelle « transsubjective ou transcendante » ce qui implique que les nombres ont une réalité hors de l’entendement, puisqu’ils sont « comme expression ou reproduction de processus et de relations existant dans le monde extérieur op-posé a l’intellect57». Dans la mathématique, telle que la conçoit Cantor, la réalité immanente prime sur la réalité transsubjective ou transcendante, cette derniere étant assignée a la métaphysique. Or, le fait que, dans la mathématique, il faut «prendre en considération… uniquement et seulement la réalité immanente », permet d’en tirer « une conséquence importante pour la mathématique58» : le fait qu’elle ne doit s’occuper que de la réalité immanente de ses objets, lui accorde une liberté que les autres sciences ne connaissent pas. Mettant ainsi 81 au centre de sa conception des mathématiques sa liberté de création, Cantor adopte en meme temps l’idée que la mathématique pense a travers la formalisa­tion, mais cela suppose qu’il y a un systeme d’axiomes qui ne porte pas sur des objets qui existent dans la réalité, mais sur des objets que les axiomes-prescrip­tions appellent a l’existence. 56 Ibid., p. 44. 57 Ibid., p. 47. 58 Ibid., p. 48. Badiou, pour sa part, souscrit au réalisme des mathématiciens. En effet, si les mathématiques sont la science du réel, c’est parce que la philosophie, qui se fonde sur l’équation la mathématique = l’ontologie, part de « la supposition qu’il y a dans ce qui existe un niveau de généralité ou d’universalité qui est en quelque sorte immatériel. Il y a des structures qui se retrouvent dans tout ce qui existe. L’étude de ces structures en tant que telles, des possibilités structurales, est précisément l’enjeu des mathématiques59 ». Tout comme les mathémati­ciens, Badiou croit que les objets et structures mathématiques « ‘existent’ en un certain sens ». Mais, a la différence des mathématiciens, le réalisme de Badiou se base sur l’expérience d’une résistance dans le travail mathématique, a savoir « que ‘quelque chose’ résiste lorsqu’on fait des mathématiques, qu’on se frotte a une réalité difficile, rebelle60». C’est a partir de la résistance des « choses » a la pensée qu’il faut saisir le réel en mathématiques. Tout le probleme est d’iden­tifier ce « bout de réel » qui se manifeste seulement la ou ça coince, la ou il y a une impasse, une incompatibilité. Badiou rejoint ainsi Lacan, pour qui, ce n’est pas seulement par les mathéma­tiques que l’on pense le mieux l’existence. L’existence est ce que les mathéma­tiques pensent dans sa pointe, c’est-a-dire dans son impossibilité. Si la notion d’impossible joue le rôle d’un pivot essentiel qui organise les rapports que la psychanalyse entretient avec les mathématiques, c’est parce que, comme le dis-cours psychanalytique, le discours mathématique rencontre des points d’im­possible. Située a l’intersection de la psychanalyse et des mathématiques, la notion d’impossible permet d’ordonner le champ psychanalytique d’une part, et de révéler, d’autre part, les impasses constitutives des mathématiques. L’im­possible n’est donc pas tout a fait étranger aux mathématiques puisqu’il est in-carné dans leur histoire par de grands moments de « crise » ou elles se heurtent contre un réel énigmatique qui correspond a la rencontre d’un obstacle excé­dant le cadre axiomatico-déductif. Mais il faut noter cependant que le réalisme lacanien est bien singulier puisqu’il dépend de sa définition du réel comme impossible. Le réel n’est pas la réali­té, mais ce qui se soustrait a sa représentation. Il est ce qui fait trou dans un discours et le pousse a se déplacer et se renouveler. Et c’est précisément pour Badiou, Éloge des mathématiques, p. 59. 60 Ibid., pp. 72–73. mieux cerner l’impossible dans la psychanalyse meme que Lacan fait appel aux mathématiques. Ce que Lacan cherche dans les mathématiques, c’est une écri­ture, car le réel en jeu dans la psychanalyse ne s’atteint que par les impasses des voies de l’écriture. En choisissant d’emprunter aux mathématiques les ou­tils pour le travail a faire en tant que psychanalyste, convaincu « de la valeur des éléments mathématiques pour faire émerger quelque chose qui concerne vraiment l’expérience de l’analyse 61», ce n’est pas seulement pour rompre avec le discours philosophique que Lacan a recours aux mathématiques. C’est parce que les mathématiques ont affaire au réel, qu’elles sont les seules a penser le réel : « Le Réel dont je parle est absolument inapprochable, sauf par une voie mathématique » et « c’est tres précisément en cela que l’effort logicien doit nous etre un modele, voire un guide 62 ». Comme les mathématiques, la psychanalyse ne trouve pas sa limite a l’exté­rieur d’elle-meme, mais la porte en soi.63 Aborder le réel sous l’angle de la limite nous donne, une fois de plus, l’occasion de préciser un point fondamental des rapports de la psychanalyse avec les mathématiques. Nous nous proposons d’illustrer ce paradoxe de la limite a partir de la notion d’infini. Si Lacan a re-cours aux mathématiques et s’il fait de Cantor un interlocuteur privilégié, c’est aussi pour résoudre l’impasse de Freud, celle qui touche a la fin de l’analyse. En tant qu’expérience de parole, la psychanalyse, de structure, ne connaît aucune fin. Or si, pour Freud, l’analyse est « interminable », sans fin, puisqu’il n’y a rien qui puisse arreter le déchiffrage de l’inconscient, Lacan, pour sa part, propose une notion de la fin « logique » de l’analyse. C’est dans cette perspective que la fin de l’analyse, selon Lacan, met la psychanalyse face a un enjeu majeur : « L’enjeu maintenant est de quoi aidera a sortir le réel-de-la-structure : de ce qui de la langue ne fait pas chiffre, mais signe a déchiffrer64». Faute d’etre muni des outils mathématiques, seuls a meme de localiser le point d’impossible, l’ana-83 lyse ne rencontrera aucun réel. 61 Lacan, « Le savoir du psychanalyste », 1 juin 1972. 62 Ibid., 2 décembre 1971. 63 « Si l’inscription analytique est bien ce que j’en dis, a savoir le début, le noyau-clé de sa mathématique, il y a toutes les chances a ce que ça serve a la meme chose que la mathé­matique. C’est-a-dire que ça porte en soi sa propre limite. » Lacan, « Séminaire XXII », 20 novembre 1973.  64 Jacques Lacan, « Télévision », Autres écrits, p. 536. La découverte cantorienne des transfinis va lui permettre d’élaborer une théo­rie de la fin de l’analyse en y incluant son infinitude. Et c’est d’ailleurs dans ce but qu’il a proposé la fameuse procédure de la « passe » qui marque le passage de l’analysant a l’analyste. La psychanalyse qui vise sa fin a pour ambition de cerner, a la fin de l’analyse, ce qui détermine le sujet a son insu. La rigueur de cette détermination dépend d’une structure qui participe du réel. Ce point de réel qui ne se manifeste qu’a la fin de l’analyse, se fonde sur la construction d’un cadre qui est censé englober tous les dits de l’inconscient du sujet. On voit bien l’analogie du franchissement qu’implique la passe lacanienne et le geste de Cantor. Or dans la psychanalyse, « il est question de l’acces a une formule, un .0, qui est celui de chacun, meme s’il y a un .0, pour tous que Lacan a for-mulé dans les termes ‘Il n’y a pas de rapport sexuel’ 65». Le fait de pouvoir toujours ajouter des dits nouveaux, signale que l’on est dans un infini que Cantor qualifie d’ « improprement dit ». Dans ce contexte, la car-dinalité du symbolique, le terrain propre a la psychanalyse, sera toujours finie. Or l’infini proprement cantorien, l’infini actuel, n’est atteint qu’a partir de la rupture avec une notion de l’infini réduit a l’inaccessible, l’indéfini ou l’illimité, rupture qui a permis a Cantor de construire un infini purement quantitatif : les nombres transfinis. Pour soutenir cette opposition inhérente a la psychanalyse, il faut donc se tourner vers la théorie mathématique de l’infini. On commence par admettre que, dans la psychanalyse, le symbolique est infini, puisqu’il est toujours possible d’ajouter un dit supplémentaire a tous les dits précédents. Si Lacan fait appel a Cantor, c’est pour s’appuyer sur sa découverte des nombres transfinis, car la théorie des transfinis lui permet de penser la façon de dépas­ser le Nom-du-Pere, le rôle pivot de la castration finitiste, qui pour Freud pré­sente un point indépassable pour la psychanalyse, de relativiser cette place qui jusqu’alors avait un caractere absolu. La théorie des nombres transfinis intéresse Lacan dans la mesure ou elle pro­pose une progression de nombres infinis qui ne rencontre aucune butée. De meme que le probleme pour Cantor est celui de ne pas pouvoir sortir du dénom­brable pour atteindre la puissance du continu, le déchiffrage de l’inconscient ne peut atteindre sa fin. Le pas de Cantor, pour faire le saut du dénombrable au continu, consiste a comprendre qu’on ne peut dépasser qu’en limitant au Jacques-Alain Miller, « Vers un signifiant nouveau », Revue de l’ECF, (20/1992), p. 54. préalable. C’est par l’adjonction d’un nouveau signifiant, aleph zéro, un nombre ajouté a la suite infinie de nombres et le nommant, que le fini se saisit. La ma-thématique cantorienne présente l’intéret d’avoir formalisé non seulement la notion de limite, mais encore celle des dépassements successifs de limitations successives. Pour atteindre des cardinaux toujours plus grands, il faut appli­quer ce que Cantor appelle « le principe de limitation ». Une fois le premier pas­sage a la limite accompli, il faut donc un nombre transfini de limitations pour envisager l’ultime saut, impossible a écrire parce qu’il est toujours possible d’en rajouter « un en plus » et donc de le dépasser. Le paradoxe du « plus grand transfini » rend la place de tous les ensembles inconsistante. Or, c’est précisé­ment l’une des possibles fins de l’analyse, a savoir la rencontre avec le manque du dernier signifiant au cour du symbolique, l’incomplétude de l’Autre sym­bolique mais qui ouvre la possibilité de « l’invention d’un savoir dans le réel » que Lacan propose comme « une détermination du réel au meme titre que la science66». L’enjeu pour la psychanalyse est de (se) rendre compte qu’en choi­sissant un modele d’infini on décide ce qu’il faut entendre par la notion d’infini. Il faut toutefois noter que c’est seulement le premier modele de l’infini, celui qui se limite au dénombrable, qui est utilisable en psychanalyse. A cet égard, si Badiou reproche a Lacan de ne pas etre cantorien67, car l’infini pour lui se réduit a l’inaccessibilité68, Lacan ne pourrait que lui donner raison dans la mesure ou, tout comme Freud, il n’admet pas l’infini actuel dans la psychanalyse ; ils sont tous deux foncierement « finitistes ». Ainsi, l’impossible d’écrire est propre aux mathématique tout autant qu’a la psychanalyse. On comprend mieux pourquoi le matheme est « le point pivot » de l’entreprise lacanienne. C’est que la voie du matheme, sous la forme meme ou Lacan la propose, est la seule méthode possible a produire des « petits bouts de réel », comme il le dit lui-meme. Le paradoxe est que l’on ne peut produire 85 des « bouts de réel » que par des artifices, tres exactement, par ceux de l’écri­ture. Au paradoxe de voir le réel dans l’écriture, se joint celui de ne voir appa­ 66 Jacques Lacan, « La note italienne », Autres écrits, p. 310. 67 Selon Badiou, la doctrine lacanienne du sujet est essentiellement finie et il en avance la raison: « Jusque dans la logique de la jouissance, la réelle existence de l’infini actuel l’en­ combre plus qu’elle ne le sert. Il ne convoque l’infini que pour le révoquer. L’infini doit res­ ter cette fiction opératoire, qui pointe l’abîme ou la faille ou le sujet se constitue. » Badiou, « Sujet et infini », Conditions, p. 301. 68 Ibid., p. 300. raître le réel que par l’écriture. Et pour donner une image qui se rapproche le plus de cette formalisation qui ne se supporte que de l’écrit, de cette « réduction aux dimensions de la surface qu’exige l’écrit », Lacan prend une image qu’il trouve dans la nature, ce travail de texte qui sort du ventre de l’araignée, sa toile. Fonction vraiment miraculeuse, a voir, de la surface meme surgissant d’un point opaque de cet étrange etre, se dessiner la trace de ces écrits, ou saisir les limites, les points d’impasse, de sans-issue, qui montrent le réel accédant au symbolique69. Sur ce point la philosophie ne semble pas etre en désaccord avec la psychana­lyse. Pour Badiou, les mathématiques « seraient un mode d’approche du réel, y compris le plus insaisissable70». Mais le réel que viseraient les mathématiques selon Badiou n’est pas tout a fait le meme que celui que visent les mathéma­tiques selon Lacan. Pour ce dernier, « Le réel ne saurait s’inscrire que d’une impasse de la formalisation71 ». Plus précisément, c’est seulement a mettre la formalisation a l’épreuve qu’on rencontre un point d’impossibilité. Cependant il faut souligner également que le formalisme de Lacan est tres particulier par bien des aspects. Le fait de prendre la formalisation mathématique comme mo-dele ne revient pas pour autant a confondre les deux modes du réel : le réel qui est en jeu dans la mathématique et le réel que vise la psychanalyse. La mathé­matique, et la science en général, ont affaire avec un réel qu’elles transforment en écriture. Le réel dont il s’agit ici est censé etre mathématisable – sans reste. La psychanalyse, en revanche, étant affaire de sens, ne donne pas du sens au réel. Il s’agit plutôt de viser le réel dans le sens. La mathématique représente donc pour Lacan le modele d’acces au réel de la structure ou le réel est saisi au sens de rencontre d’un point d’impossible a écrire dans les termes de cette structure. Dans le séminaire L’envers de la psychanalyse, Lacan examine une formulation de l’impossible comme fait de structure, justement : « A poser la formalisation du discours et, a l’intérieur de cette formalisation, a s’accorder a soi-meme quelques regles destinées a la mettre a l’épreuve, se rencontre un 69 Lacan, Encore, pp. 85–86. 70 Badiou, Éloge des mathématiques, p. 59. 71 Lacan, Encore, p. 85. tel élément d’impossibilité72». L’épreuve est prise ici au sens ou elle atteint un certain réel. Le dire de Cantor selon Lacan et Badiou : « un pari sur le réel 73 » C’est dans ce contexte que Badiou critique certains aspects de l’usage des ma-thématiques par Lacan. Nous ne saurions entrer ici dans le détail d’une discus­sion de ses arguments tout a fait recevables de Badiou, qui s’appuient sur une lecture attentive de « L’étourdit » et de Encore. Avant meme de s’interroger sur le bien-fondé de sa lecture et/ou critique de Lacan, surtout sur ces points ou Lacan s’appuie sur les mathématiques, il faut se demander a quoi la lecture de Lacan peut servir dans le projet philosophique construit par Badiou. Certes, il y a chez Lacan la voie du matheme pour situer la question du réel qui semble précieuse a Badiou puisque lui aussi vise a cerner « ce qui se soustrait a la détermina­tion ontologique », « ce qui n’est pas l’etre-en-tant-qu’etre74 » et qui n’est donc pas mathématisable. Car le probleme que Badiou se propose de résoudre est le suivant : « si l’ontologie réelle se dispose comme mathématique en éludant la norme de l’un, il faut aussi, sauf a rétablir globalement cette norme, qu’il y ait un point ou le champ ontologique, donc mathématique, se détotalise, ou reste en impasse. Ce point, je l’ai nommé l’événement75». Ainsi, si la philosophie se sépare d’une partie de soi-meme, celle qui s’interroge sur l’etre comme tel, en assignant cette tâche a la mathématique, c’est aussi pour s’établir comme « théorie générale de l’événement. C’est-a-dire de ce qui se soustrait a la sous-traction ontologique. Ou théorie de l’impossible propre des mathématiques76». Mais pour assigner a la philosophie la tâche de déterminer le trans-etre de l’évé­nement, il faut commencer par admettre que « tout n’est pas mathématisable77 ». La question qui se pose alors est celle de savoir qu’est-ce qui dans l’événement mérite le nom du réel proprement dit, le réel en tant qu’impossible d’écrire. 87 C’est précisément sur ce point que Badiou ne peut pas ne pas rencontrer Lacan dans sa tentative d’aborder le réel ininscriptible a partir de la mathématique. 72 Jacques Lacan, L’envers de la psychanalyse, Seuil, Paris 1991, p. 50. 73 Alain Badiou, Théorie du sujet, Seuil, Paris 1982, p. 290. 74 Badiou, « L’événement comme trans-etre », p. 56. 75 Ibid., pp. 56–57. 76 Ibid., p. 57. 77 Ibid. C’est aussi l’occasion de poser la question suivante : pourquoi faut-il avoir re-cours au matheme en psychanalyse ? Si la psychanalyse, comme tout discours visant un réel, s’articule en mathemes, c’est parce qu’elle butte sur une difficul­té : « Sa difficulté tient a ce que, comme savoir, elle traite d’un savoir, lui aussi littéralisable, mais qui ne se sait pas78». La psychanalyse n’est pas la mathéma­tique bien qu’elle privilégie, tout comme la mathématique, la voie du matheme pour accéder au réel, mais, a la différence de la mathématique, elle vise le rési­du de réel qui échappe a toute formalisation. Pour la psychanalyse ce résidu de réel qui n’est pas mathématisable se résume en la formule qu’ « il n’y a pas de rapport sexuel » : il n’y a qu’impossibilité a écrire le 2 d’un rapport entre deux ensembles sexués. Or c’est précisément ce point d’impossible qui soutient un désir de savoir. Il ne s’agit pas d’un désir de savoir malgré tout, malgré cet im­possible, mais d’un désir reposant sur le fait que la vérité n’est pas toute, qu’elle peut s’agrandir, faisant place a de nouveaux savoirs79. Lacan propose de tenter l’écriture du rapport sexuel, bien qu’il en ait avancé l’impossibilité : Sans essayer ce rapport de l’écriture, pas moyen en effet d’arriver a ce que j’ai, du meme coup que je posais son inex-sistence, proposé comme un but par ou la psychanalyse s’égalerait a la science : a savoir démontrer que ce rapport est impossible a écrire, soit que c’est en cela qu’il n’est pas affirmable mais aussi bien non réfutable : au titre de la vérité. Avec pour conséquence qu’il n’y a pas de vérité que l’on puisse dire toute… . La vérité ne sert a rien qu’a faire la place ou se dénonce ce savoir. Mais ce savoir n’est pas rien. Car ce dont il s’agit, c’est qu’accédant au réel, il le détermine tout aussi bien que le savoir de la science80. Ce qui est visé dans les mathemes lacaniens, ce n’est donc ni de l’ordre du vrai ni de l’ordre du savoir, car le matheme est situé a la jonction de la vérité et du savoir, mais le réel qui se manifeste sous forme de la rencontre, du coinçage, de l’incompatible, du non-rapport. C’est pourquoi ce que vise le matheme, ce n’est pas le réel en tant que tel mais l’impossibilité de dire vrai du réel. C’est la que la maniere dont Lacan utilise les mathématiques est a prendre au sérieux. 78 Lacan, Encore, p. 88. 79 Lacan, « Note italienne, » p. 310. 80 Ibid. La voie du matheme, pour situer la question du réel, rejoint le cour des préoccu­pations de Badiou et, plus particulierement, dans sa tentative pour réintriquer la philosophie et les mathématiques. Il nous faut nous interroger sur le point exact ou les chemins de Lacan et Badiou se séparent tout en se croisant parfois. Dans la perspective de Lacan, le fait de s’inspirer du modele mathématique, ne revient pas pour autant a confondre deux modes du réel : un réel qui fait tra­vailler les mathématiciens, le réel en tant que cause du travail de formalisation, et un réel qui est produit par cette formalisation ou, plus exactement, par son échec. Il faut d’abord souligner que le réel est toujours relatif a un discours, car on ne tombe sur le réel qu’a partir d’un discours. Il en découle que l’émergence d’un nouveau réel a pour conséquence le remaniement du discours mathéma­tique. Cela paraît contredire la philosophie spontanée des mathématiciens, le « réalisme platonicien » qui présuppose un univers mathématique préétabli que les mathématiciens n’ont plus qu’a découvrir. Or, si le réalisme des mathé­maticiens et la production de nouveaux réels ne sont pas incompatibles, c’est parce qu’une fois produit, le nouveau réel est posé comme étant déja la. C’est la leçon qu’on peut tirer de l’invention des transfinis : une fois les nou­veaux nombres, les transfinis, construits par Cantor et introduits dans le champ mathématique, tout se passe comme s’il n’avait fait que les découvrir, puisque ils étaient toujours déja la. Ainsi, ce qui n’est que produit d’une invention, est considéré, apres coup, comme étant déja la avant les démarches de Cantor qui l’ont engendré. Cette inversion de l’invention en découverte est le prix a payer pour le réalisme des mathématiciens. Celui-ci implique cette sorte de rétroac­tion pour garantir la « réalité » de ses objets. C’est que la réalité mathématique s’impose a tous comme étant nécessaire. Pour traiter du réel mathématique, il faut donc prendre comme point de départ le fait qu’un réel se produit comme hors de toute juridiction. Ainsi, pour produire les transfinis, il fallait inventer le chemin pour atteindre l’infini mathématique. La méthode de diagonale de Can­tor était contingente, donc non requise par le discours mathématique existant. Il fallait donc l’inventer, trouver le chemin qui n’était pas donné. La méthode n’était pas nécessitée par le savoir mathématique au moment de son apparition, et pour cause, puisque l’invention du chemin, de la méthode, visait a résoudre une faille dans ce savoir meme. Cependant, une fois l’invention des transfinis admise par le discours mathématique, cette contingence devient nécessaire. On serait donc plutôt d’accord avec Badiou, lorsqu’il avance que le dire mathé­matique, en tant qu’événement, ne se soutient d’aucune garantie de la nécessi­té 81. Pour comprendre cette articulation paradoxale, déja évoquée par Cantor, entre la liberté de création et la contrainte 82 qui caractérise la mathématique, il faut préciser que, bien qu’il soit produit par le discours mathématique, le réel des mathématiques, pour etre effectif, c’est-a-dire capable de renouveler le sa­voir mathématique, doit etre posé comme ex-sistant au discours mathématique, et meme comme étant préalable, hors de ce discours dont il procede cependant. Sur cette question de l’ex-sistence il faut spécifier que n’ex-siste que ce qui se manifeste comme une impossibilité a prendre ici au sens qu’il est impossible que ce réel puisse etre affirmé ou nié par la démonstration mathématique. Ce qui revient a dire que le statut de réel de quelque chose qui se présente comme une entrave ou une difficulté dépend de la démonstration mathématique. On pourrait dire aussi que le réel comme ex-sistence, c’est une vérité démontrée comme impossible a démontrer. C’est précisément a ce qu’il ne puisse s’écrire que l’on reconnaît ce réel en mathématiques. L’exemple notoire qui, pour La-can, illustre bien le réel en mathématiques, c’est l’impossibilité de démontrer la validité de l’hypothese du continu. Au lieu de la démontrer par la méthode diagonale, Cantor a fait émerger l’hypothesee du continu comme le réel-impos­sible des mathématiques. Plus généralement donc, si toute démonstration, une fois validée, implique qu’il est impossible de procéder autrement, l’impossibili­té de la démonstration, c’est-a-dire l’impasse de la formalisation, l’impossibilité a écrire ou a ne pas écrire une formule, en revanche, génere un réel en tant qu’impossible. C’est a ce niveau la que se situe la critique que Badiou adresse a Lacan. Pour aborder le rapport singulier de Lacan aux mathématiques, Badiou prend comme point de départ ce qu’il appelle « la triangulation » qui inclut la psy­chanalyse, la mathématique et la philosophie83. Badiou s’appuie sur un repere tiré de « L’étourdit » ou Lacan met l’accent sur la défaillance de la philosophie lorsqu’il s’agit d’identifier l’essentiel de la mathématique : « Pour etre le langage le plus propice au discours scientifique, la mathématique est la science sans 81 Alain Badiou, Le Séminaire. Lacan. L'antiphilosophie 3, 1994-1995, Fayard, Paris 2013, p. 126. 82 Ibid., p. 129. 83 Ibid., p. 39. conscience dont fait promesse notre bon Rabelais, celle a laquelle un philo­sophe ne peut que rester bouché84». En revanche, ce qui caractérise l’approche de la psychanalyse a la mathématique tient au statut singulier que Lacan attri­bue au matheme : d’etre vidé du sens, le matheme assure la transmission inté­grale d’un savoir. Ainsi, pour Lacan, le discours mathématique, défini comme « langage de pur matheme » est « ce qui est seul a pouvoir s’enseigner : ceci sans recours a quelque expérience85». C’est ce côté que le matheme en psycha­nalyse et le mathématisable aurait en commun. Mais il y a aussi un autre côté du matheme qui va au-dela du mathématisable. Pour la psychanalyse, c’est le réel du « il n’y a pas de rapport sexuel », le réel non-inscriptible, mais autour duquel tout dans le discours psychanalytique s’ordonne. C’est aussi ce réel qui ne peut etre signalé que par les impasses du mathématisable : « C’est en quoi les mathemes dont se formule en impasse le mathématisable, lui-meme a définir comme ce qui du réel s’enseigne, sont de nature a coordonner a cette absence priseauréel86». « L’absence prise au réel » évoquée ici, n’est rien d’autre que le réel propre a la psychanalyse, le « il n’y a pas de rapport sexuel », qu’il faut dis-tinguer du réel inscriptible et donc enseignable par la voie du matheme. C’est ce réel dont l’absence est, comme le dit Badiou, « absence dans aucune mathé­matisation, c’est-a-dire son absence dans l’inscription87 ». Quel serait l’équivalent de cette « absence dans l’inscription », pour les mathé­matiques ? La « réalité » des mathématiques tient au fait que leur cohérence s'impose comme contrainte. Et si l’on peut dire que la formalisation revient a une mise en rapport, le réel survient comme impossibilité de mise en rapport, comme rapport impossible. Par conséquent, le réel en mathématique ne peut émerger que dans les lieux ou cette cohérence est mise en défaut. C’est a ce niveau-la que Lacan cherche le réel en mathématique : le réel dans la forma­lisation meme. Le réel dont il s’agit ici est celui qui « ne saurait s’inscrire que 91 d’une impasse de la formalisation ». Ce qui est important a noter ici, c’est que c’est précisément sur ce point que Badiou avance ses objections a Lacan. La position de Lacan serait, selon Badiou, une position « archiscientifique88 ». A suivre la lecture badiousienne de l’articulation que Lacan propose du réel et du 84 Lacan, « L’étourdit », p. 453. 85 Ibid., p. 472. 86 Ibid., p. 479. 87 Badiou, Lacan, p. 43. 88 Ibid., p. 45. matheme, celle-ci n’est tenable que si l’on n’introduit pas le point d’ « un réel du réel ». La formule : le « réel du réel 89» qu’on ne trouvera nulle part chez La-can, comme Badiou l’admet volontiers, tient une place stratégique dans son ar­gumentation a lui, puisqu’elle lui permet de clarifier comment se positionnent respectivement, selon Lacan, la psychanalyse et la philosophie par rapport a la mathématique. Ainsi, a suivre Badiou, pour situer la place de la psychanalyse par rapport a la mathématique et a la science en général, il lui faut redoubler le réel, le réel propre a la science, le réel dont la science« découvre » les lois et qu’elle inscrit en formules, le réel mathématisable et transmissible par la voie du matheme enseignable ; et le réel propre a la psychanalyse, le réel non-mathématisable qui ne peut s’inscrire que par les impasses de la mathématisation. Ce que cherche Lacan, d’apres Badiou, c’est précisément ce point d’impossible dans la mathématique elle-meme qui permet de cerner le réel non mathématisable. Le matheme, tel que le conçoit Lacan, est archiscientifique et non pas scientifique, selon Badiou, parce qu’il est situé au point du réel de la mathématique. Et Ba-diou de conclure, « il ne peut pas etre mathématique, précisément parce qu’il touche au réel de la mathématique elle-meme90 ». Or, c’est précisément cette distinction entre deux modes du réel qui séparent la psychanalyse de la phi-losophie qui, n’étant pas capable de toucher au réel de la mathématique, reste, comme l’affirme Lacan, « bouchée aux mathématiques ». Le fait d’avoir accordé le statut de pensée a la mathématique, seule a toucher le réel, est en meme temps l’occasion pour Lacan de critiquer la philosophie qui, elle, est incapable de reconnaître, dans la mathématique, la voie d’acces au réel qui s’y trouve, et ne peut pour cette raison meme que d’etre bouchée a la mathématique. Si Badiou rejette en bloc cette idée, c’est parce qu’il défend la these selon la-quelle « s’il y a un lieu de pensée qui est bouché a lui-meme, c’est bien la ma-thématique » et la raison qu’il avance n’est rien d’autre que son ignorance quant a « sa propre portée ontologique ». Cette these permet a Badiou de préciser son principal point de désaccord avec Lacan en maintenant que la philosophie, au lieu d’etre bouchée aux mathématiques, est, au contraire, celle qui, identifiant dans la mathématique « un point de bouchon par rapport a sa propre nature on­ 89 Ibid.p. 44. 90 Ibid., p. 46. tologique », tente depuis Platon « de la déboucher ». A cet égard, le philosophe « n’est pas celui qui est bouché aux mathématiques », comme le soutient Lacan, « mais celui qui tente de les déboucher au regard d’elles-memes91 ». Badiou s’interroge sur « cette dimension radicale de la mathématique92 » que la philosophie est censée manquer ou ne pas saisir. Il trouve la réponse dans « L’étourdit », ou Lacan soutient, a propos de la mathématique, justement, que « le dit se renouvelle de prendre sujet d’un dire plutôt que d’aucune réalité93 ». On trouve ce point névralgique dans ce que Lacan propose comme la dialec­tique du dire et du dit, telle qu’il la développe dans « L’étourdit ». Pour Lacan, c’est lorsque « surgit un dire qui ne va pas toujours jusqu’a pouvoir ex-sister au dit », qu’ « un certain réel peut etre atteint 94». Le surgissement d’un dire dont dépend le dit constitue un événement au sens ou il introduit une scansion, une coupure. Mais pour qu’un dire ait lieu, il faut qu’il se soumette a l’épreuve de ses conséquences, du dit, et c’est seulement a ce moment-la qu’un « certain réel peut etre atteint ». Selon la lecture que Badiou propose de cette dialectique singuliere du dit et du dire, on peut etre d’accord avec Lacan qu’il faut qu’il y ait le dire pour pouvoir transformer, inventer ou renouveler le dit. C’est que la mathématique, en se rap-portant au dire, et non pas a une quelconque réalité, se rapporte a une réalité qui lui est immanente, un point sur lequel a déja insisté Cantor. A suivre Lacan, c’est justement ce point que la philosophie, dans son rapport aux mathéma­tiques, rate forcément, a savoir que le renouvellement du dit s’enracine dans le dire, plutôt que dans le sens comme le prétend la philosophie. En s’efforçant de donner un sens a la vérité, dit Lacan, la philosophie méconnaît la vrai nature de la mathématique qui serait, selon Lacan, mais lu par Badiou, « un dire in-sensé qui se réalise comme dit absolu (intégralement transmissible) 95». Et pour etre 93 encore plus précis, « la valeur paradigmatique de la mathématique est d’etre le modele insurpassable d’une pensée qui n’a aucun sens96». Il est curieux d’ailleurs, constate Badiou, que sur cette question, un antiphilosophe, Lacan, 91 Ibid., pp. 49–50. 92 Ibid., p. 113. 93 Lacan, « L’étourdit », p. 452. 94 Lacan, Encore, p. 25. 95 Badiou, Lacan, p. 116. 96 Ibid., p. 118. puisse rejoindre un philosophe, Platon, a une différence essentielle pres : ce que Lacan identifie comme valeur paradigmatique de la mathématique est pré­cisément ce que Platon lui reproche. Car ce que Platon objecte a la mathématique, c’est de fonctionner a partir d’hy-potheses dont elle ne rend pas compte. Traduit en termes lacaniens, Platon sai­sirait, avant Lacan, que la mathématique s’origine dans « un pur dire », que « la mathématique n’est que sous la garantie d’un dire », que Badiou traduit en « dimension axiomatique de la mathématique97». Vu sous cet angle, le dit, dans la mathématique, « procede intrinsequement d’un dire, puisqu’il faut que quelque chose soit d’abord dit, pour qu’ensuite il y ait « un enchaînement fidele a ce dire premier constituant98 ». Tout en critiquant la mathématique, Platon rejoindrait Lacan, puisque, lui aussi, identifie « le primat du dire » dont décou­lerait la réduction du sens. Si la philosophie est le domaine de la dialectique du sens, la mathématique est le domaine de « la prescription du dire » ou « sous la loi de la prescription du dire99 », qui ne se soutient d’aucune garantie de la nécessité. Tirant la leçon de Descartes, Badiou affirme que la spécificité de la discursivité mathématique se fonde sur « l’événement du dire » et ne peut donc etre qu’événementielle et contingente, mais, d’un autre côté, « ne relevant pas de la réalité, est absolument nécessaire », a prendre au sens d’etre « nécessaire sous l’autorité du dire100 ». Ainsi, d’avoir identifié « la mathématique sous la loi du dire », la philosophie doit reconnaître le paradoxe constitutif de la mathéma­tique, le lien entre la contingence et la contrainte101. La lecture que propose Badiou de la « triangulation » : philosophie, psychana­lyse, mathématique, n’est peut-etre pas aussi claire qu’il le prétend. De fait, il y a plus de points de convergence que de points de divergence entre lui et Lacan que Badiou serait pret a l’admettre. Voyons de plus pres comment fonctionne cette dialectique dans le cas de la mathématique, si la place du dire dans la psychanalyse est « l’analogue dans le discours mathématique102 ». En effet, le discours mathématique, si l’on suit Lacan, doit etre situé a partir de la logique Ibid., p. 122. 98 Ibid. 99 Ibid., p. 124. 100 Ibid., p. 126. 101 Ibid., p. 129. 102 Lacan, « L’étourdit », p. 476. du dire et du dit : « C’est ainsi que le dit ne va pas sans dire. Mais si le dit se pose toujours en vérité, … le dire ne s’y couple que d’y ex-sister, soit de n’etre pas de la dit-mension de la vérité. »103 Si cette logique du dire met en place l’essentiel de la conception lacanienne de la mathématique, c’est aussi parce que, dans le discours de la mathématique, « le dit se renouvelle [constamment] de prendre sujet d’un dire plutôt que d’aucune réalité, quitte, ce dire, a le sommer de la suite proprement logique qu’il implique comme dit 104 ». Pour prendre un exemple, celui d’ailleurs que Lacan lui-meme prend, s’il y a un « événement Cantor », une révolution cantorienne, c’est parce qu’il y a le dire de Cantor ou plutôt l’événement de son dire. Autrement dit, pour Lacan, l’in­vention mathématique procede du dire, le dire de Cantor, d’Euclide,… puisque seul un dire pourra renouveler le dit défaillant. Or ce dire intervient comme une contingence qui ne se fonde d’aucune réalité. Au départ, le dire n’est que la possibilité de son existence et de ce fait meme la suspension de la valeur de vérité du dit. Le dit dépend nécessairement de cette possibilité de l’existence du dire, mais c’est en tant que sa condition que l’acte de dire ex-siste au dit. Étant modale, la logique du dire donne ainsi un nouvel éclairage a la question de l’existence. Pour clarifier ce point, Lacan prend deux exemples, celui du dis-cours psychanalytique et celui du discours mathématique. Concernant le dis-cours psychanalytique, pour qu’il y ait ce discours et sa pratique, l’analyse, il faut restituer « le dire de Freud » qui « s’infere de la logique qui prend de source le dit de l’inconscient 105 ». Autrement dit, c’est parce que Freud l’a découvert que le dit de l’inconscient existe. L’autre exemple exposé par Lacan, celui du discours de la mathématique, est destiné a montrer que le dit « se renouvelle de prendre sujet d’un dire plutôt que d’aucune réalité ». Ce dire est un acte, dans son pouvoir de transformation, apportant la nouveauté d’une formule mathé­matique, d’un chiffrage du réel. On peut assigner au dire de Cantor, c’est-a-dire, 95 a un dire qui a déja fait preuve quant a sa puissance de création d’un nouveau savoir mathématique, le statut de réel, le statut d’ex-sistence au systeme de sa­voir en place. Le dit, en revanche, est ce qui, d’etre posé en vérité, doit etre mis a l’épreuve par la démonstration afin de valider, rétroactivement, le dire 103 Ibid., p. 452. 104 Ibid. 105 Ibid., p. 454. inaugural. On pourrait dire aussi que le dit issu de ce dire est sommé – via la démonstration – d’assurer au dire son statut axiomatique. A cet égard, le « dire mathématique », est « le dire » en tant qu’il réalise un savoir. L’avoir lieu du dire présuppose une rencontre inédite, une nouveauté absolue, d’ou l’idée de Lacan de faire du « dire » un événement en tant que ren­contre avec « un bout de réel ». Ce changement de perspective permet de mettre en place le savoir du réel a partir d’un dire. Pour fixer les termes, on pourrait dire qu’il y a du « dire » au moment ou s’établit un avant et un apres. L’introduc­tion d’un savoir nouveau en tant que rencontre avec « un bout de réel », produit cet avant et cet apres. C’est donc cette rupture ou scansion du temps qui fait du « dire » un événement (mais a condition de ne pas le confondre ni avec le discours mathématique ni avec la démonstration, qui se construisent a partir de ce moment du dire). C’est en ce sens que le « dire-événement » peut conduire a la refonte du champ du savoir dans son ensemble. C’est la, pour reprendre les définitions de Russell, que le mathématicien « ne sait jamais de quoi il parle » parce qu’il confond événement en tant que création pure (en termes badiou­siens on dira : la vérité), et savoir en tant qu’ordonnancement rationnel du dis-cours, « ni si ce qu’on dit est vrai » parce qu’il ne fait pas la différence entre les registres qui ordonnent dire et vrai. On voit bien qu’ici, l’affinité entre langage du mathématicien et inconscient se situe au niveau du rapport a la question de la vérité : « le mathématicien a avec son langage le meme embarras que nous avec l’inconscient, a le traduire de cette pensée qu’il ne sait pas de quoi il parle, fut-ce a l’assurer d’etre vrai (Russell) 106 ». Dans la perspective de cette affinité, Lacan dit qu’un certain réel est atteint quand « surgit un dire qui ne va pas tou­jours jusqu’a pouvoir ex-sister au dit 107 ». Pour conclure Sa lecture attentive, mais critique de l’usage lacanien des mathématiques nous permet de voir sous un autre angle la tentative de Badiou de réintriquer philo­sophie et mathématique. Du fait d’avoir fondé son projet philosophique sur la these de l’identité entre mathématique et ontologie, c’est-a-dire d’avoir affirmé que la philosophie est sous la condition des événements de la mathématique, il s’agit chez Badiou d’un tout autre rattachement a la mathématique. En effet, le 106 Ibid., p. 452–453. 107 Lacan, Encore, p. 25. rattachement de la philosophie consiste en ce qu’elle accepte de reconnaître la mathématique comme l’instigatrice possible d’une détermination inédite de la philosophie elle-meme. En tant que « condition » de la philosophie, la mathé­matique est considérée ici comme puissance d’une pensée libre et laique dont l’apport est de l’ordre d’une chance de nouveauté donnée. Or, dans cet effort pour relier de nouveau la philosophie et la mathématique, une figure étrange est convoquée : Lacan, qui n’est ni philosophe ni mathématicien, mais psycha­nalyste. Et c’est dans cette tentative de renouer avec la mathématique, que Ba-diou critique l’usage de la mathématique par Lacan. La « triangulation » (mathématique, psychanalyse, philosophie) serait, selon Badiou, une articulation singuliere bâtie par Lacan dans le but de mieux cir-conscrire son recours aux mathématiques. Or, Badiou, lui aussi, a besoin d’une telle « triangulation » pour mettre en lumiere la maniere dont la philosophie, théorie générale des vérités événementielles,abordeles mathématiques iden­tifiées a la pensée de l’etre. Et lui aussi fait référence aux impasses de la forma­lisation mathématique qui attestent que la mathématique, d’avoir touché a une impossibilité qui lui est propre, se fonde sur un réel. Le réel indépassable de la psychanalyse : il n’y a pas de rapport sexuel ; c’est le réel propre a la psychana­lyse en tant que telle, le réel autour duquel tout le « reste » s’ordonne, tout ce qui peut s’inscrire, se mathématiser, formaliser, etre transformé en mathemes. Or, Badiou, n’est-il pas, lui-aussi, obligé de reconnaître qu’il y a un réel ininscrip­tible de la mathématique cantorienne, le « il n’y a pas de rapport » mesurable, numérable entre le dénombrable et le continu, entre l’ensemble des éléments d’un ensemble et l’ensemble de ses parties, pour s’assurer de la compatibilité entre la philosophie en tant que pensée de l’événementialité des vérités et l’on­tologie=mathématiques en tant que pensée de l’etre-multiple ? Car dans son effort de renouer les liens entre la philosophie et les mathéma­tiques, la philosophie est forcée de se battre sur plusieurs fronts : contre les philosophies qui ne reconnaissent pas les mathématiques comme une pensée créative (Heidegger, par exemple) ; contre l’antiphilosophie qui assigne les ma-thématiques a la non-pensée ; et contre l’antiphilosophie singuliere de Lacan qui accorde aux mathématiques le statut de pensée parce que les mathéma­tiques, selon Lacan, sont les seules capables de toucher le réel. Mais il ne leur accorde ce privilege qu’au prix de s’arroger le droit d’identifier le point ou les mathématiques défaillent : le point d’impossible ou le réel non mathémati­sable, ininscriptible en mathemes, et qui ne peut se manifester qu’a travers les impasses de la formalisation. Car, quelle est l’objection principale que Badiou adresse a Lacan ? De prendre la position que Badiou qualifie d’ « archiscienti­fique » vis-a-vis des mathématiques dans le but d’identifier le point d’impos­sible indépassable des mathématiques, le « réel du réel » des mathématiques, comme le dit Badiou, plus précisément, en en mot, de vouloir « dire le vrai du réel » des mathématiques. Sortant ainsi du strict cadre des mathématiques, Lacan se hisse a une position que Badiou qualifie d’ « archiscentifique », une position qui serait capable de circonscrire le réel non-mathématisable dans les mathématiques elles-memes. Mais en dépit de points de convergence que nous avons essayé de circons­crire, la philosophie badiousienne et la psychanalyse lacanienne n’ont pas et ne peuvent pas avoir le meme recours a la mathématique. Pour Lacan, la ma-thématique, tout en faisant partie de ces disciplines dont elle ne peut pas se désintéresser et tout en admettant qu’elle y occupe une place privilégiée, ne constitue pas une de ses conditions, au sens que Badiou donne a cette notion. Il se tourne vers la mathématique pour garder le cap sur le réel, pour cerner le réel propre a la psychanalyse, tentant en meme temps de produire un savoir transmissible a tous via les mathemes. Cependant la psychanalyse ne dépend pas de la mathématique. D’ailleurs l’invention de la psychanalyse par Freud en témoigne puisque, pour établir la psychanalyse, Freud n’a pas besoin des mathématiques. Or, tel n’est pas le cas pour Badiou. La philosophie dépend des événements qui ont eu et auront lieu dans la mathématique parce que c’est a partir des événe­ments que de nouvelles vérités se produisent, que la tâche de la philosophie est de penser. La philosophie n'est donc pas et ne peut pas etre indifférente a ce qui se passe dans la pensé de l’etre en tant qu’etre. Ainsi, comme cela a déja été noté et comme Badiou lui-meme le reconnaît, toute mathématique, identifié a l’on-tologie, n’est pas compatible avec la philosophie des vérités événementielles. En effet, seule une mathématique capable d’identifier et de penser les points d’impossible, son réel, qui se présente sous la forme de ses impasses, apories, difficultés, fournit la place nécessaire pour que les événements aient lieu. On dira que la philosophie sous la condition dépend non seulement des vérités que ses conditions produisent, mais meme plus encore, qu’elle dépend du réel de ses conditions et tout particulierement du réel de la mathématique=ontologie. Badiou lui-meme nous donne une idée de cette dépendance de la philosophie du réel de la mathématique. Citons le passage central de l’Etre et l’événement : C’est depuis ses origines que la philosophie, anticipant la butée cantorienne, a scruté l’abîme qui sépare la discrétion numérique du continu géométrique. Cette abîme n’est autre que celui qui sépare w0, domaine infini dénombrable des nombres finis, de l’ensemble de ses parties p(w0), seul apte a fixer la quantité des points dans l‘espace. …Nous pouvons maintenant dire que c’est l’etre meme, tel que flagrant dans l’impasse de l’ontologie, qui organise l’inexhaustion de sa pensée, des lors que nulle mesure ne se laisse prendre du lien quantitatif entre une situation et son état, entre l’appartenance et l’inclusion. Il y a tout lieu de croire que c’est pour toujours qu’est ouverte dans l’etre cette provocation au concept qu’est le dé-rapport entre présentation et représentation. … Si le réel est l’impossible, le réel de l’etre, soit l’Etre, sera précisément ce que détient l’énigme d’un anonymat de la quantité108. L’errance de l’etre, invoquée par Badiou, est insupportable pour la pensée qui veut, au contraire, que « le désancrage quantitatif de l’etre » cesse, et qui donc ne peut se satisfaire de « la dé-mesure ontologiquement attestée», qui vise pré­cisément a mesurer l’exces, a cerner l’etre la ou « il n’est plus exactement di­cible 109 ». Et Badiou examine attentivement trois stratégies de la pensée, trois orientations dans la pensée ontologique, comme il le dit, pour « parer a l’ex­ces », de dire l’impossible-a-dire de l’ontologie : la voie du constructible, la voie des grands cardinaux et la voie du générique. Si la voie du constructible et la voie des grands cardinaux tentent de maîtriser le réel de l’ontologie, soit « par le bas », c’est-a-dire en réduisant les multiplicités au constructible, soit « par le haut », en formulant des hypotheses d’existences des cardinalités énormes dans le but de prescrire « une disposition hiérarchique ou rien ne saurait er-99 rer110 », l’enjeu de la voie du générique, en revanche, n’est pas de réduire ou de contrôler l’exces, mais de le rejoindre. Pour s’assurer de la compatibilité de la philosophie, théorie générale de l’événementialité des vérités, et de l’ontologie, science de l’etre, il ne suffit pas de seulement choisir la voie générique. Il faut en plus anticiper en quelque sorte que la démonstration de l’hypothese du continu, 108 Badiou, Etre et événement, pp. 311–312. 109 Ibid., p. 312. 110 Ibid., p. 313. postulant que la cardinalité de R (le continu ou l’ensemble des nombres réels) est successeur immédiat de celle de N (dénombrable, ensemble des entiers), ne validera jamais sa vérité. La philosophie, telle que Badiou la conçoit, nécessite une ontologie capable de garantir « pour toujours » qu’il y aurait de nouvelles contingences événementielles dans lesquelles s’originent les vérités, y compris celles de l’ontologie. L’Immanence des vérités semble reprendre et renouveler ce questionnement entamé dans l’Etre et l’événement, un questionnement au cour duquel gît le statut indécidable de l’hypothese du continu. Badiou opte, comme toujours, pour sa fausseté. La théorie des grands cardinaux, situé dans le camp de l’adversaire d’antan, avec le théoreme de Jensen, qui permet de déterminer la puissance d’un ensemble, nommé 0#, semble confirmer le bien-fondé du choix badiousien. N’empeche que l’hypothese que Badiou lui-meme fait sur l’hypo­these du continu, a savoir sur sa fausseté fonciere, ne se fonde que d’un pari : « un pari sur le réel », qui tout comme le réel de la psychanalyse : il n’y a pas de rapport sexuel, un réel indépassable pour la psychanalyse, constitue l’ho­rizon indépassable pour l’ontologie pour qu’elle puisse etre compatible avec la philosophie, pensée des vérités événementielles. Ce que Badiou cherche, c’est un analogue du réel de la psychanalyse, et il le trouve sous une double forme : celle de la fausseté supposée de l’hypothese du continu, d’une part, et, d’autre part, celle de l’existence supposée d’un ensemble 0#. Dans les deux cas, il s’agit de parier sur l’existence des ensembles infinis non constructibles. Le probleme que le « lemme de recouvrement » de Jensen, comme Badiou l’appelle, résout est le suivant : « Il existe un ensemble, nom­mé 0#, zéro diese, tel que, s’il n’existe pas, le recouvrement est toujours possible, et que, s’il existe, le recouvrement de certaines grandes multiplicités est impos­sible111 ». Or, comme pour l’hypothese du continu, l’existence de 0# n’est pas 100 démontrable dans ZFC, l’existence de 0# « releve d’une décision », celle qui pro-clame son existence « au risque meme … que toute l’architecture de l’ontologie formelle s’effondre112 ». Or, ce risque ne peut qu’etre assumé par le choix qui vise la création ou l’émancipation. Finalement, toute pensée créative, écrit Badiou, est obligée de prendre l’existence de 0# comme axiome, pour pouvoir franchir le cadre finitiste de la situation donnée. Quelle est la leçon a tirer de l’examen minutieux de la théorie des grands cardinaux que présente L’Immanencedes 111 Badiou, L’immanence des vérités, p. 445. 112 Ibid., p. 446. vérités. La réponse que propose Badiou est la suivante : parier sur l’existence de0# doit etre considéré comme une arme pour mener a bien la lutte dans des situations concretes. Et pour mener a bien cette lutte, il faut déterminer le seuil, qui est aussi le lieu du combat. Il faut penser l’infini complet – qui nous dit que tout n’est pas soumis a la finitude – mais aussi penser et pra­tiquer0# – qui nous dit ou se joue la sortie du vieux monde. Et cet « ou », infecté qu’il est par les lois du vieux monde et éclairé aussi bien par l’infini nouveau, nul n’en peut économiser les aspects retors, embrouillés, épuisants 113. Ainsi, si la mathématique peut créer une possibilité nouvelle pour la philoso­phie, elle ne le peut que grâce a la contingence de ce que le dire mathématique y produit. La possibilité nouvelle que produit la mathématique n’est donc pas une simple affaire de démonstration, mais implique un acte qui est sans garan-tie. Nous voici, encore une fois, face a la dialectique du dire et du dit, ou, pour Badiou, face a la dialectique de l’acte et de l’ouvre : tout comme pour Lacan, pour qui c’est a travers l’épreuve du dit que s’assure le statut d’un vrai acte du dire créatif, novateur, pour Badiou, « les actes ne sont rien s’ils ne sont eux aussi des ouvres114 ». Références Badiou, Alain,Théorie du sujet, Seuil, Paris 1982 - L’Etre et l’événement, Seuil, Paris 1988 - « Philosophie et mathématique », Conditions,Seuil, Paris 1992 - « Sujet et infini », Conditions - « La mathématique est une pensée », Court traité d’ontologie transitoire, Seuil Paris 1998 101 - Le Siecle, Seuil, Paris 2005 - Le Séminaire. Lacan. L’antiphilosophie 3, 1994–1995, Fayard, Paris 2013 - Éloge des mathématiques, Flammarion, Paris 2015 - L’immanence des vérités,Fayard, Paris 2018 Cantor, Georg, Gesammelte Abhandlungen mathématischen und philosophischen In-halts,E. Zermelo (éd.), Springer, Berin 1932 113 Ibid., p. 462. 114 Ibid. - « Fondements d’une théorie générale des ensembles », trad. J.-C. Milner, Cahiers pour l’analyse(10/1969) Charraud, Natalie,Infini et Inconscient. Essai sur Georg Cantor, Anthropos-Economica, Paris 1994 Imbert, Claude, « Ou finit le platonisme ? », dans Alain Badiou : Penser le multiple, Charles Ramond (éd.), L’Harmattan, Paris 2002 Kant, Emmanuel, Critique de la raison pure, dans Ouvres philosophiques,tome I : Des premiers écrits a la Critique de la raison pure (1747–1781), Bibliotheque de la Pléiade, Gallimard, Paris 1980 Lacan, Jacques, « Science et vérité », dans Écrits, Seuil, Paris 1966 - « Séminaire de Jacques Lacan, Le savoir du psychanalyste, 1971–1972 » (inédit) - « Séminaire de Jacques Lacan, XXII, R.S.I. 1972–1973 », (inédit) - Le Séminaire de Jacques Lacan, Livre XX, Encore, Seuil, Paris 1975 - « Séminaire de Jacques Lacan, XXV, Le moment de conclure 1977–1978 », (inédit) - Le Séminaire de Jacques Lacan, Livre XVII, L’envers de la psychanalyse, Seuil, Paris 1991 - « L’étourdit », dans Autres écrits, Seuil, Paris 2001 - « La note italienne », Autres écrits - « Télévision », Autres écrits Lavendhomme, René, Lieu du sujet. Psychanalyse et mathématique, Seuil, Paris 2001 Miller, Jacques-Alain, « Vers un signifiant nouveau », Revue de l’ECF (20/1992) Russel, Bernard,Écrits de logique philosophique, trad. Jean-Michel Roy, PUF, Paris 1989 - « Work on the principles of mathematics », The International Monthly,4 (1/1901) Le Modele ensembliste en discussion The Set-theoretical Model in Discussion Filozofski vestnik | Volume XLI | Number 2 | 2020 | 105–117 | doi: 10.3986/fv.41.2.05 Michael Hauser* Badiou and the Ontological Limits of Mathematics1 Mathematical axioms are not axioms of general truth. […] But the mathematician argues, from his finite truths, through habit, as if they were of an absolutely general applicability – as the world indeed imagines them to be. E.A. Poe, The Purloined Letter Philosophy and the composition of its mathematical condition In the history of philosophy we can find various ways of philosophy relating to mathematics (Eleats, the Atomists, Plato, Descartes, Leibniz, etc.). In keeping with this tradition, Badiou constituted a new relationship between philosophy and mathematics. In Badiou’s words, “mathematics can tell us something about being as such, without knowing that this is the very meaning of mathematics.”2 This relationship formed a new position of philosophy regarding mathematics, which I will gradually elucidate. Mathematics in the shape of set theory was the science to which Badiou re­lated philosophy in Being and Event, providing the thesis that mathematics is ontology (the science of being). “Insofar as being, qua being, is nothing other than pure multiplicity, it is legitimate to say that ontology, the science of being qua being, is nothing other than mathematics itself.”3 Badiou understood the 105 Zermelo-Fraenkel axiomatisation of set theory to be a truth procedure which formed a “condition” for his philosophy. Badiou’s relating of philosophy to this axiomatisation of set theory, which appeared as ontology, transformed philos­ophy into “metaontology”. Oliver Feltham, in the Preface to Being and Event, 1 This work was supported by the Czech Science Foundation under the grant: “Unity and- Multiplicity in Contemporary Thought”, No. 17-23955S. 2 Alain Badiou, Sometimes, We Are Eternal, Suture Press, Lyon 2019, p. 47. 3 Alain Badiou, Being and Event, trans. Oliver Feltham, Continuum, London and New York 2006, p. xiii. * Institute of Philosophy of the Czech Academy of Sciences, Prague mentioned other axiomatisations of set theory, such as W. V. O. Quine’s, and concluded that these multiple axiomatisations show the contingency of philos­ophy’s conditioning.4 Badiou’s Logics of Worlds was related to a different type of mathematics, i.e. category theory (Samuel Eilenberg and Saunders Mac Lane). The Immanence of Truthsdraws upon the theory of large cardinals. These vari­ous attachments that align philosophy with specific sorts of mathematics seem to be consequences of “axiomatic decisions” which claim no extrinsic justifica­tion for themselves. Many scholars assume that axiomatic decisions create the foundations of Badiou’s philosophy.5 Badiou himself writes that set theory and category theory provide different conditions for philosophy, which instigates “a debate crucial for the construction of the space of philosophy that invariably takes the form of a matrix of ontological choices.”6 Badiou clearly shows that a mathematical condition of philosophy does not amount to an interpretation of the given state of mathematics that is the business of the philosophy of math­ematics – a discipline Badiou rejects. Here, Badiou’s crucial idea is that the space of philosophy is “the construction” resulting from philosophical acts that are “ontological choices”. As Norman Madarasz put it, Badiou performs moves in the field of mathematics and makes a decision concerning the relationship among various segments of mathematics.7 This decision is not an interpretative choice among various accounts of a part of mathematics that appertains to the philosophy of mathematics, nor does it directly intervene in the field of mathe­matics so as to develop its axioms, theorems, and techniques. Badiou emphasis-es that he presents “nothing in mathematics which has not been established.”8 He follows his own philosophical ends and prefers some mathematical axioms and theorems to others. These “ontological choices” are the acts that consist­ently constitute the mathematical space that conditions philosophy. By these 4 Oliver Feltham, “Translator’s Preface”, in Badiou, Being and Event, p. xvii. 5 Zachary Luke Fraser, “New Directions”, in A. J. Bartlett, J. Clemens (eds.), Alain Badiou. Key Concepts. Acumen, Durham 2010, p. 177. 6 Alain Badiou, Mathematics of the Transcendental, trans. A. J. Bartlett and A. Ling, Blooms­bury Publishing, London and New York 2014, p. 16. Category theory is seen to be a com­plementary theory of set theory. While the latter enables us to conceive a being as a pure multiple, the former provides the topological localisation of a being, or of its appearing in a world. 7 Norman Madarasz, “On Alain Badiou’s Treatment of Category Theory in View of a Transi­tory Ontology”, in G. Riera (ed.), Alain Badiou. Philosophy and Its Conditions, State Univer­sity of New York Press, Albany 2005, p. 35. 8 Badiou, Being and Event, p. xiv. choices Badiou composed a consistent mathematical condition of philosophy.9 It means that we divide the mathematical axioms and theorems into two parts. Some are those that we relate to philosophy, the others are those which we leave aside. For this reason, philosophy “constructs” its mathematical condition as a composition of various mathematical fragments. In Manifesto for Philosophy, Badiou defined this role of philosophy as follows: philosophy only composes the given system of its conditions.10 Philosophy composes elements given in the field of mathematics so as to construct a consistent scientific condition of philos­ophy. This philosophical construction consists in selecting axioms, theorems, and techniques such as the axiom of separation, Cohen’s generic set, the tech­nique of forcing, etc. These mathematical elements are composed as the con­ceptual space which philosophy relates itself to. So, we have the selection and composition of the axioms that condition Being and Event. Another selection and composition of mathematical concepts condition Logics of Worlds. Then, there are the ones that create the mathematical condition for The Immanence of Truths. We can see that these conditions are not pre-given but are constructed by philosophy using various mathematical elements. Philosophy, thus, completes a consistent fabric of its condition. Mathematics is a multiple universe of axioms, theorems, techniques, theories, interpretations, and disputes. Due to this complexity, mathematics as such cannot provide philosophy with a consistent “space” that would condition the development of philosophical concepts. The selection of elements from this im­mense complexity of mathematics is decisive for the composition of the mathe­matical condition of philosophy. Needless to say, the composition of the mathe­matical condition of philosophy is an exacting task. Badiou mentioned that he had been working on set theory for almost twenty years.11 9 Badiou noted that he drew on the mathematical material which is dispersed throughout Robert Goldblatt’s book Topos: The Categorial Analysis of Logic and throughout other books. He aimed at interpreting mathematical demonstrations such as the one showing that “every ‘Boolean’ transcendental is expressible in a set-theoretic form, and that, in this sense, there is a kind of legislation of appearing which is isomorphic to that of its being.” Badiou, Mathematics of the Transcendental, p. 183. 10 Alain Badiou, Manifesto for Philosophy, trans. N. Madarasz, SUNY Press, Albany 1999, p. 39. 11 Badiou, Sometimes, We Are Eternal, p. 41. We will now address Badiou’s specific concept of conditioning, which I propose to depict as a three-layered model. Philosophy with its concepts and problems (being, truth, subject, infinity, etc.) forms the top level of this model. The math­ematical condition of philosophy is the second level, i.e. the compositive con­struction produced by philosophy. And finally, the given state of mathematics with its universe of axioms, theorems, lemmas, techniques, etc., represents the third level. Mathematics enables philosophy to construct its own mathematical space, i.e. the second level in the model. So, we have a twofold conditioning. The constructed mathematical space (the second level) conditions philosophy (the first level) and simultaneously the given state of mathematics (the third level) conditions the mathematic space (the second level), providing philosophy with axioms, theorems, etc., to construct it. Following this introduction, I can present two consecutive theses intended to depict the position of Badiou’s philosophy and its dynamic relation to mathe­matics. Philosophical concepts as an excess in the metastructure The first thesis: the mathematical condition is a “situation”, and philosophy acts as its metastructure. Philosophical concepts (event, truth, subject, etc.) appear as a wandering excess from a mathematical point of view. This excess is an effect of the relation between philosophy (metastructure) and its mathematical condition (situation). This excess is described by Easton’s theorem. Žižek maintains that Badiou transposed the real into the discourse of the Master and “what Badiou precludes is the possibility of devising a discourse that has as 108 its structuring principle the unnameable ‘indivisible remainder’ which eludes a discursive grasp.”12 Žižek pays attention to the fact that Badiou did not keep the real as an element of a Lacanian discourse, but overlooked another form of the real in Badiou’s metaontology which comes to light when we unfold the context of set theory. Badiou conceives of the real as “an impasse of formalisa­tion” which is a result of the operation “the count-as-one”, which excludes and Slavoj Žižek, “From Purification to Subtraction: Badiou and the Real”, in P. Hallward, (ed.), Think Again: Alain Badiou and the Future of Philosophy, Continuum, London and New York 2004, p. 177. forgets “pure” multiplicities. This exclusion opens lacunae as symptoms of the real in a situation.13 In keeping with this meaning of the real, Badiou grasps the real in mathematics as being “deprived of sense.” The real is an impasse that makes it impossible to resolve a conflict among interpretations.14 We can, there­in, find a theorem that conveys the real as being inherent in the representative structure. In the remarkable “Meditation twenty-six” in Being and Event, Badiou shows how the void reappears in the state of a situation, which is a metastruc­ture in the sense of representation, even though the state of a situation had en­sured by the operation “the count-as-one” that the void would not be presented. Badiou draws on Easton’s theorem, which states that given an infinite cardinal ., the value of a multiple p(.), i.e. the quantity for the state whose situation is the multiple, can be designated as any superior successor cardinal of ..15 In Ba­diou’s terms, the state of a situation is quantitatively larger than the situation, but it is impossible to determine the quantitative difference between these two instances. The state of a situation is a representation of the presented situation. From Easton’s theorem it follows that representation involves an excess that cannot be counted. We can understand this quantitative excess to be the errant void in the representative structure, the one that corresponds to the real. As Ba-diou stresses, we have an irremovable rift between the representative structure, i.e. the metastructure, and the situation. This rift produces an un-relation: there is no adequate relation between representation and presentation. In Manifesto for Philosophy,Badiou introduces the un-relation as the real. “This relation has the form of a wandering excess: it is known that the parts are more numerous than the members (Cantor’s theorem), but no measure of this ‘more’ can be es­tablished. It is moreover at this real point – wandering excess in the ‘quantita­tive’ infinite – that the great orientations in thought are established.”16 109 My idea is to conceive of Badiou’s philosophy as a representative structure (a metastructure) and its mathematical condition as the presented situation. This interpretation corresponds to Badiou’s understanding of the relationship be­tween philosophy and mathematics: philosophy is metaontology and mathe­ 13 Badiou, Being and Event, p. 54. 14 Alain Badiou, Briefing on Existence. A Short Treatise on Transitory Ontology, State Univer­ sity of New York Press, New York 2006,p. 56. 15 Badiou, Being and Event, p. 279. Cf. Bartlett et al. (eds.), Badiou. Key Concepts, p. 53. 16 Badiou, Manifesto for Philosophy, p. 80. Easton’s theorem refers to Cantor’s. matics is ontology. Then, Badiou’s metaontology can be understood in the light of Easton’s theorem, which enables us to elucidate a “quantitative excess” that arises in philosophy as related to mathematics. This excess explains the fact that Badiou’s philosophy provides concepts without an adequate relation to mathematical axioms and theorems, even if mathematics is its scientific condi­tion. These concepts appear in the frame of philosophy and form elements that “quantitatively” exceed the ones that we find in the mathematical composition that conditions philosophy. The glaring example of this excess is represented by the concept of an event that is not related to a mathematical axiom. Badiou presents it as “the intervening doctrine” because mathematics is the science of being-qua-being (i.e. ontology), while the doctrine of the event, precisely, designates ‘that-which-is-not-being-qua-being’.17For this reason, mathematics cannot offer an axiom of the event. Its concept results from the philosophical in­tervention. The philosophical relation to mathematics is an “un-relation” in this sense. Having an “un-relation” to mathematics as ontology, Badiou’s philoso­phy is not the philosophy of mathematics but a “metaontology” that establishes philosophical concepts without defining their relation to mathematical axioms and theorems. This “meta-position” means that philosophy “represents” mathematics as on­tology. As mentioned above, philosophy composes its condition by selecting some mathematical axioms, theorems, techniques, etc., in order to construct a consistent system thereof that would act as the condition of philosophy. In prin­ciple, this procedure corresponds to the operation of “counting” that creates a consistent situation as a “presented multiple.”18 The mathematical condition of philosophy works as the “situation” composed by philosophical “counting” consisting in selecting axioms, theorems, etc., from the inconsistent multiplicity of mathematical concepts.19 The philosophical “count” simultaneously presents these axioms, theorems, etc., in such a way so as to limit their original mathe­matical complexity.20 This “count” reduces an immense multiplicity of axioms, theorems, and their various formalisations and interpretations. As Badiou put it, 17 Badiou, Being and Event, p. 13. 18 Ibid., p. 24. 19 Understandably, these axioms, theorems, etc., had been “counted” and presented in the frame of mathematics. 20For instance, cf. Badiou’s philosophical presentation of Cantor’s theorem with its original complex formalisation. Ibid., p. 273. a metastructure operates in such a way so as to make a situation consistent.21 Phi­losophy itself correspondingly intervenes in the inconsistent multiplicity of ax­ioms, theorems, lemmas, formalisations, systems, etc., that create the universe of mathematics. By this operation, philosophy composes its consistent condi­tion. Therefore, Badiou’s philosophy acts as the metastructure that composes its mathematical condition as a consistent multiplicity. Using the proposed model of conditioning, philosophy, being on its top level, represents the metastructure, the constructed mathematical space (the condition of philosophy) on the second level forms a consistent multiplicity, i.e. the presented situation, while mathe­matics, on the third level, is an inconsistent multiplicity of mathematical axioms, theorems, formalisations, etc. We can conclude: if Badiou’s philosophy operates as a metastructure, we can apply Easton’s theorem to it. Easton’s theorem shows that Badiou’s philosophy as a metastructure involves the wandering excess of representation that cannot be removed. This excess is removable only on the condition that philosophy would lose its position as a metastructure. In this case, philosophy as metaontology would change into the philosophy of mathematics, which does not create specifically philosophical concepts. This excess of representation means that philosophical concepts in the metastructure do not have a defined relation to mathematical axioms. For instance, the concept of the event conveys ‘that-which-is-not-being-qua-being’ and therefore, it cannot have a defined relation to the mathematical axioms that only cover ontology as a science of being-qua-being. This relation is an “un-re­lation” that arises as philosophy operates as a metastructure. Easton’s theorem helps us to see the meaning of a metastructure regarding philosophy. If philoso­phy ceased to act as a metastructure, it would lose its position as metaontology. Consequently, philosophy could not deploy specifically philosophical concepts such as event, truth, subject, etc. The concepts with an “un-relation” to mathe­matics create philosophical elements in the form of a wandering excess, which is the real in the metastructure. After Badiou established these concepts in the space of set theory, he came to realise that they opened a new philosophical field that could not be covered by set theory. These concepts provoked the problematic of how truths appear in worlds, with many theoretical consequences and resulting issues. This new Badiou, Being and Event, p. 98. field of philosophical investigation could not be deduced from the space of set theory because concepts such as event, truth, and subject, which had opened this new field, have an un-relation to the space of set theory. In Badiou’s words, “if the beginning for Being and Event is without philosophy, the consequences of this beginning are, really, within philosophy. In some sense, the movement of the book is from mathematics to something which has no signification from a mathematical point of view.”22 The philosophical concepts in Being and Event were an excess as seen from the position of set theory. This excess was “wander­ing” regarding the potentially infinite number of issues, concepts, and interpre­tations that “event”, “truth”, and “subject” can generate. Mathematics and the dynamic dialectics of Badiou’s metaontology The second thesis: philosophical concepts (event, truth, subject) as an excess in the metastructure open a new philosophical field that is not covered by the given mathematical condition (set theory). Consequently, Badiou composes a math­ematical condition (category theory) that enables philosophy to elaborate new concepts (singularity, relation, world). And again, these concepts open another philosophical field with “the immanence of truths”, “absoluteness”, and “infinity”, which call for the composition of a new mathematical condition (large cardinals). This move explains the dialectical dynamics of Badiou’s ontology. From the application of Easton’s theorem to Badiou’s philosophy as a metastruc­ture it follows that the relation between metaontology and ontology (mathemat­ics) is an un-relation, i.e. a relation that can never be defined. Easton’s theorem says that the state of a situation (a metastructure) is quantitatively larger than the situation. As stated above, philosophy as a metastructure generates a possi­bly infinite number of new issues, concepts, and interpretations, which shows that philosophy (metaontology) is quantitatively larger than its mathematical condition (ontology). Regarding Easton’s theorem, it is impossible to determine the quantitative difference between these two instances. To wit, philosophy as a metastructure (metaontology) would always be quantitatively larger than its mathematical condition (ontology), even if we composed this scientific condi­tion of philosophy from any elements of mathematics in any number. 22 Badiou, Sometimes, We Are Eternal,p. 52. We saw that the “un-related” philosophical concepts in Being and Event are a wandering excess that opens a new philosophical field with potentially infinite consequences that cannot be covered by means of set theory. In other words, Badiou’s philosophy as deployed in Being and Event contains “more” elements than set theory can provide. These excessive elements show lacunae in the given mathematical condition that was composed from the set theoretical el­ements, i.e. the “situation” on the second level in the model of conditioning. We can explain the dialectical dynamics of Badiou’s ontology along these lines. The lacunae in the given mathematical condition of philosophy provoked the construction of a new one that enabled philosophy as a metastructure to devel­op philosophical possibilities which had emerged in Being and Event. Badiou composed the mathematical condition by adopting some elements of category theory. This composition conditioned the elaboration of philosophical concepts in Logics of Worlds, i.e. singularity, transcendental, object, relation, world, etc. This conditioning is the relation between mathematics and philosophy that can be described as an “un-relation” because philosophy again was a metastructure regarding its mathematical condition composed of elements from category the­ory (situation). These philosophical concepts were an excess from the point of view of category theory. They opened a new possibility of how to put forward other philosophical concepts such as infinity, absoluteness, and idea. It was necessary to prepare the mathematical space composed of fragments of the the­ory of large cardinals. This space conditioned the development of the concepts elaborated in The Immanence of Truths. Potentially, these concepts break new ground in other philosophical domains. Hypothetically, philosophy is step by step going to turn to every field of math­ematics that is available. The point is that we will have a lacuna within the mathematical condition even after philosophy composed it from elements of the last field of mathematics remaining. This happens because every section of mathematics is “smaller” than philosophy as a metastructure in which an excessive new philosophical field appears, and the same holds for the last field available. For instance, the concept of infinity in The Immanence of Truths calls for the application of infinity to various forms of politics and economics. Some types of politics and economics proved to be finite, while other types appear to be infinite. In this light, Badiou addresses Marxism by stating that Marxist economics draws on egalitarian principles and opposes the dominant liberal economy, which represents the oppression of infinity.23 If we were to develop new philosophical concepts in this direction, we would need to compose the scientific condition involving some elements from acceptable “non-mathemat­ical” sciences such as Marxist economics. These are the ones that primarily do not deal with mathematical subject matter (axioms, theorems, hypothesis, etc.), but they apply mathematic formalisation if need be. I argue that, consistently with Badiou’s understanding of philosophy as metaon­tology, a scientific condition of philosophy can be composed from elements stemming from non-mathematical sciences that embrace mathematic formali­sation. These non-mathematical scientific conditions are meant for developing concepts generated by metaontology (Being and Event, Logics of Worlds, The Immanence of Truths), but, principally, they cannot be covered by the mathe­matical domain only. For instance, the development of political and economic notions that were established by an ontological concept of infinity requires the construction of a scientific condition with elements coming from Marxist eco­nomics. Badiou introduced some non-mathematical sciences that operate with mathematic formalisation. These are phonology in linguistics, the foundations of Marxist economics, a part of the anthropological theory of kinship, and a segment of psychoanalysis. These fields are supposed to be exempt from the ideology that largely dominates in the human sciences.24 Due to their complete­ly different subject matter, we can put aside phonology and the anthropological theory of kinship. What remains is “the foundation of Marxist economics” and “a segment of psychoanalysis” (i.e. the Lacanian one), which can be conceived as being consistent with Badiou’s understanding of philosophy. In The Imma­nence of Truths, Badioufully endorses a meaning of Marxist economics and explicitly includes a part of Marxism into his philosophy. Actually, we can see that a part of Marxist economics has formed a condition of Badiou’s philosophy in all its development. Badiou only dismissed its other parts, such as “Marxist politics” tied to the state. Alberto Toscano showed that Badiou retained what he regarded as the core Marxist principles. This was “a minimal Marxism that con­joins the political hypothesis of non-domination with the rational identifica­tion of the sites of subversion, without thereby committing political practice to Alain Badiou, L’Immanence des vérités, Fayard, Paris 2018, p. 81. 24 Badiou, “Afterword: Some Replies to a Demanding Friend”, in Hallward (ed.), Think Again: Alain Badiou and the Future of Philosophy, p. 234. aninstrumental,revolutionary orprogrammaticframework.”25 Thisapproach to Marxism of Badiou provides a telling example of a non-mathematical condi­tion for the development of philosophy. Lacanian psychoanalysis works in a similar way in Badiou’s philosophy. Ba-diou embraced some of the key Lacanian notions and their definitions, for instance the real as “the impasse of formalisation.”26 He elaborated their new meaning within his philosophy, which is another example of how the ontologi­cal concepts of truth, infinity, absoluteness, etc., can instigate the development of notions that are related to a non-mathematical domain. So, we have Badiou’s concept of the unconscious, i.e. “an inner transcendence of consciousness,” which is out of its control regarding the fact that this transcendence is the infin­ity of a truth procedure.27 Badiou speaks about his “vexed, or vexatious, fideli­ty” to Lacan, which is imprinted in Being and Event andLogics of Worlds.28 For Badiou, Lacanian psychoanalysis is the crucial form of anti-philosophy that “aims at an act that it believes is an unconditioned break, a transformation without determination, a groundless leap into the new.”29 Badiou adopted some elements of Lacanian psychoanalysis in order to compose a condition of philos­ophy. As Badiou put it, “a contemporary philosopher, for me, is indeed someone who has the unfaltering courage to work through (traverser sans faiblir; literal­ly ‘to traverse without weakening’) Lacan’s anti-philosophy.”30 Here, we have another example of Badiou’s composition of a non-mathematical condition of philosophy. Marxism, as much as Lacanian psychoanalysis, represents the non-mathematical sciences that are consistent with Badiou’s understanding of philosophy. 115 25 Alberto Toscano, “Marxism Expatriated: Alain Badiou’s Turn”, in J. Bidet and S. Kouve­ lakis (eds.), Critical Companion to Contemporary Marxism, Brill, Leiden and Boston 2008, p. 533. 26 This formulation I quoted above appeared as far back as in Theory of the Subject. Alain Badiou, Theory of Subject, trans. B. Bosteels, Continuum, London and New York 2009, p. 23.Here, Badiou declared his commitment to Lacanian psychoanalysis. 27 Badiou, L’Immanence des vérités, p. 188. 28 Alain Badiou, Lacan. Anti-philosophy 3, trans. K. Reinhardt and S. Spitzer, Columbia Uni­ versity Press, New York 2018, p. xl. 29 Ibid., p. xxiv. 30 Ibid. p. xxv. Conclusion: mathematics is ontology only if philosophy is metaon­tology We determined that Badiou established the relation between philosophy and mathematics that can be understood as the one between a metastructure and a situation. This finding enables us to nuance the statement that mathematics is ontology. René Guitart and other mathematicians understand this statement in such a way that mathematics forms a part of philosophy – sometimes a log­ical part, sometimes an ontological part. Badiou considers this interpretation false, as he stated that mathematics is different from philosophy.31 Mathematics, however, constitutes a condition of philosophy that is a sort of relation. We can interpret this relation as the one between a metastructure and a situation, which shows that mathematics appears as a composed mathematical condition (situa­tion). Metaontological concepts are deployed in philosophy as a metastructure. In keeping with Easton’s theorem, these concepts are the excess that emerges in the relation between a metastructure and a situation. They are different from mathematics precisely as a consequence of this type of relation. They are con­ditioned by mathematics and simultaneously are different from it because they represent an excess from the mathematical point of view. One of the most stunning features of mathematics is that it supplies a theorem indicating the ontological limits of mathematics itself. Poe was not quite right in writing that mathematics credits its axioms and theorems with “an absolutely general applicability” and presents them as a general truth of the world. Mathe­matics provides us with a theorem that limits its general applicability. Mathematics viewed as a multiple universe of all given axioms, theorems, tech­niques, interpretations, and systems (set theory, category theory, etc.) is not on­tology. It is mathematics as the inconsistent multiplicity on the third level in the model of conditioning. Mathematics becomes ontology only if philosophy com­poses its own scientific condition by using various fragments from mathemat­ics. It is mathematics as the consistent multiplicity on the second level in our model. So, mathematics is ontology provided that philosophy as metaontology has changed mathematics into ontology. To wit, it composed its mathematical condition. Philosophy, thus, manifests that mathematics as a multiple universe Badiou, Sometimes, We Are Eternal, p. 52. is not ontology. Philosophy as metaontology shows the ontological limits of mathematics. References Badiou, Alain, “Afterwords: Some Replies to a Demanding Friend”, in Think Again: Alain Badiou and the Future of Philosophy, ed. Peter Hallward, Continuum, London and New York 2004 — Lacan. Anti-philosophy 3, trans. K. Reinhard and S. Spitzer, Columbia University Press, New York 2018 — Being and Event, trans. Oliver Feltham, Continuum Books, London and New York 2005 — Briefing on Existence. A Short Treatise on Transitory Ontology, State University of New York Press, New York 2006 — L’Immanence des vérités, Fayard, Paris 2018 — Manifesto for Philosophy, trans. N. Madarasz, SUNY Press, Albany 1999 — Mathematics of the Transcendental, trans. A. J. Bartlett and A. Ling, Bloomsbury Pub­lishing, London and New York 2014 — Theory of Subject, trans. B. Bosteels, Continuum, London and New York 2009 Feltham, Oliver, “Translator´s Preface”, in Being and Event Fraser Zachary, Luke, “New Directions”, in Alain Badiou. Key Concepts, ed. A. J. Bartlett, J. Clemens, Acumen, Durham 2010 Madarasz, Norman, “On Alain Badiou´s Treatment of Category Theory in View of a Tran­sitory Ontology”, in Alain Badiou. Philosophy and Its Conditions, ed. G. Riera, State University of New York Press, Albany 2005 Toscano, Alberto, “Marxism Expatriated: Alain Badiou´s Turn”, in Critical Companion to Contemporary Marxism, ed. J. Bidet and S. Kouvelakis, Brill, Leiden and Boston 2008 Žižek, Slavoj, “From Purification to Subtraction: Badiou and the Real”, in Think Again: Alain Badiou and the Future of Philosophy, ed. Peter Hallward, Continuum, London and New York 2004 Filozofski vestnik | Volume XLI | Number 2 | 2020 | 119–142 | doi: 10.3986/fv.41.2.06 Roland Bolz* Mathematics is Ontology? A Critique of Badiou’s Ontological Framing of Set Theory The subject of this article is Alain Badiou’s well-known assertion that “mathe­matics is ontology” (henceforth: M = O), a phrase that originates in the open­ing chapters of Being and Event, and which has been an essential orientational maxim for followers of Badiou ever since.1 In this article, I pose two questions: a) How important is the identification of mathematics and ontology for Badiou’s larger philosophical project in Being and Event? And: b) Does Badiou give con­vincing reasons for accepting it? My answers, upon careful deliberation, are a) crucial, and b) no. The key to my counterargument is the idea that set-theoretic multiples (collections) should not be conflated with the types of ‘multiples’ (i.e. parts and wholes, atoms and totalities) that are traditionally within the purview of ontology. In what follows, I aim to complicate Badiou’s idea that the ontolog­ical theme should be moved to the realm of set theory.2 To understand the importance of M = O for an otherwise vast philosophical text, we must recall the design of Being and Event. Its basic chapter structure follows that of a typical introduction to axiomatic set theory (supplemented by exten­sive philosophical commentary). The scope of such an introduction to set theory includes the following subjects: -a motivation and explanation of the axioms of Zermelo-Fraenkel set theory (ZFC) vis-a-vis the naive concept of set/collection; 119 1 Alain Badiou, Being and Event, trans. Oliver Feltham, Continuum Books, London and New York 2005, pp. 4, 15, 19. The role of “mathematics is ontology” for the second and third volume of the Being and Event trilogy is a more complex matter. It is clear that the later volumes build on the first. In this paper, I only consider the argument as it is presented in the first volume, the original Being and Event. 2 However, I do not claim that set theory cannot be used as a conceptual tool in the inves­tigation of ontological matters. In fact, it is widely presupposed as a background theory for ontological investigations that use formal logical theories as languages which ‘cap­ture’ ontological reasoning and commitments. However, I take issue with Badiou’s much stronger claim that set theory is the science of being qua being. * Humboldt University of Berlin -the development of several mathematical concepts in the language of sets – functions, power sets, ordinal numbers, cardinal numbers, infinite sets,etc.; -a presentation of the independence of the continuum hypothesis from the axioms of ZFC (a joint proof of Kurt Gödel and Paul Cohen). However, the philosophical prestige of Being and Event does not come from its mathematical content. There are no new mathematical theorems in the book. Its status as an important work of philosophy comes from the fact that Badiou purports to extract a series of non-trivial ontological insights from the edifice of the mathematical theory of sets. Here, ontology is a field associated with phi­losophers such as Plato, Spinoza, Hegel, Heidegger, and Deleuze. Badiou often invokes set-theoretic results to criticise the views of these thinkers. The intensity with which Badiou reads these two traditions together is more or less unique to him.3 As a sceptical reader, I have significant doubts that they communicate to this degree without lapsing into a somewhat loose exercise in analogy-making. It is critical to see that, prima facie, the mathematical theory of sets and phil­osophical ontology are two only minimally overlapping fields of inquiry, or-ganised around quite different questions. I will return to this matter below. The formula “mathematics is ontology” is what enables Badiou’s ontological her­meneutics of set-theoretic results.4 If the two fields have nothing to do with each other, it will be futile to try to critique some ontological views of, for example, Spinoza using set theory. But if he can convince us of M = O, then set-theoretic results can indeed be read vis-a-vis the ontological tradition, although, need­less to say, different interpretations of the mathematical theorems may ensue. If the identification (or conflation) of the two disciplines is ill-founded, then his exegeses of set-theoretic findings must remain at the level of creative metaphor 3 Perhaps the way Riemannian geometry informs the work of Deleuze and Guattari is some­what analogous. Although Badiou’s project is not nominally associated with philosophical hermeneutics, I submit that in Being and Event he is primarily concerned with a philosophical interpreta­tion of set-theoretic results. (He speaks at length about his “philosophical interpretations” of set theory in the introduction: ibid., p. 19 ff.) This leads me to describe his philosophical practice as importantly hermeneutic (interpretive) in nature. By describing his project as an ontological hermeneutics of set theory, I also wish to foreground that his interpretive strategy is highly idiosyncratic in its preoccupation with ontology. It is also exactly this hermeneutic strategy that I wish to contest here. and analogy.5 This would not undermine his work entirely, but it would change its entire argumentative framing (and the strategies necessary in our reception of his work). Apart from the exposition of set-theoretic concepts and theorems, Being and Event contains many pages of philosophical commentary. One can usefully distinguish between two main functions of these passages. The first is to convince the read­er of the identification of set theory with ontology (the argument for M = O). The second is to develop an ontological interpretation of the set-theoretic results (an ontological reading of set theory). The latter, more voluminous task, depends on the successful completion of the former. This serves to remind us what is at stake in evaluating Badiou’s identification of mathematics and ontology. To see how resolute Badiou is about M = O, we turn to a telling passage from Being and Event. Here, Badiou goes beyond the set theorist Paul Cohen’s ac­count of his discovery and identifies the mathematical concept of an indiscerni­ble part of a multiple (Cohen calls this a generic extension) with the ontological (or philosophical) concept of a truth.6 Let us see how he frames the relationship between the mathematical results and the ontological discussion: It is mathematics which must judge whether it makes any sense to speak of an indiscernible part of any multiple. […] it can decide whether it is compatible with ontology that there be truths. Decided at the level of fact by the entire history of humankind – because there are truths – the question of the being of truth has only been resolved at a de jure level quite recently (in 1963, Cohen’s discovery); without, moreover, the mathematicians – absorbed as they are by the forgetting 121 5 Indeed, I take Badiou’s mathematical passages as extended analogies that help the reader to understand his philosophical points. That said, I also take metaphors and analogies to be indispensable to philosophy (and to human thought more generally) and certainly do not wish to suggest that the use of metaphors discredits a thinker. However, Badiou himself seems to understand his own method as quite distant from analogy. I consider this a misrepresentation of the philosophical work that he does and especially of the justifica­tory structures that underly his main philosophical claims. This makes it hard to engage with him as a theoretical thinker. I am at present working on a book that is concerned with the role of analogy and metaphor in philosophical conceptual labour. 6 Paul Cohen, Set Theory and the Continuum Hypothesis, Dover Publications, Mineola 2008. For Cohen’s own account of his discovery, see Paul Cohen, “The Discovery of Forcing”, Rocky Mountain Journal of Mathematics 32 (4/2002), pp. 1071–1100. of the destiny of their discipline due to the technical necessity of its deployment – knowing how to name what was happening there (a point where the philosophi­cal help I was speaking of comes into play).7 The relationship between mathematics and philosophy, as Badiou sees it, is the following: mathematicians come up with formal innovations that they do not fully interpret. The philosopher is then able to give the right ontological framing to the matter – “the philosophical help.” For Badiou, Cohen’s proof from 1963 resolves a more profound ontological question, without Cohen being aware of it. This has been Badiou’s line ever since Being and Event, although a similar motif already runs through Theory of the Subject.8 What are the philosophical underpinnings for this interpretive practice? Is his conflation of set theory with ontology justified? Favourable commentators have accepted Badiou’s identification of mathemat­ics and ontology, often without even recognising its prima facie implausibility.9 One might explain this by noting that in contemporary philosophy ‘ontology’ describes a whole array of only loosely related approaches and questions.10 It is not my goal to propose an alternative vision of ontology here, only to cast doubt on Badiou’s identification of mathematics (set theory) with ontology. In comparison with his commentators, Badiou seems quite aware of the im­plausibility of his identification of ontology and mathematics. He fully accepts that he must deliver a “proof that mathematics is ontology.”11 But although Ba-diou claims to give such a proof in Being and Event, it is not very clear which 7 Badiou, Being and Event, p. 341. 8 Alain Badiou, Theory of the Subject, trans. Bruno Bosteels, Continuum Books, London and New York 2009, pp. 148–157. 9 See, for example, Steven Corcoran (ed.), The Badiou Dictionary, Edinburgh University Press, Edinburgh 2015; A.J. Bartlett and Justin Clemens (eds.), Alain Badiou: Key Concepts, Acumen Publishing Limited, Durham 2010; Burhanuddin Baki, Badiou’s Being and Event and the Mathematics of Set Theory, Bloomsbury Academic, London and New York 2015. Since Being and Event is built entirely on the proposition that mathematics and ontology can be identified, any interaction with Badiou’s (systematic and philosophical) work al­most presupposes acceptance of it. 10 The field is hard to circumscribe, ‘ontology’ seems to mean something different for most authors, some believe that it does not constitute an actual field of inquiry, etc. 11 Badiou, Being and Event, p. 19. parts of the book pursue this goal. For instance, there is no passage in which he indicates to the reader that he considers the proof complete (presumably it is concluded by the end of Part I). Neither does he pause to state any kind of adequacy criteria therefor – what would make for a successful proof of his equation? Against the mathematical background of much of the material dis­cussed in Being and Event, Badiou’s notion of a philosophical proof (for M = O) is remarkably implicit. So, if one wishes to evaluate the central philosophical claims of Being and Event charitably(especially concerning its mathematical content), one cannot get around formulating adequacy criteria for the identi­fication of mathematics and ontology oneself. What kind of argument would settle this matter to our satisfaction? Theoretical discourses with different statuses flank both sides of the identity M = O. Badiou’s reading of the history of philosophy is that the ontological tradi­tion failed to fulfil its theoretical desiderata. However, mathematics (here in the guise of modern set theory) satisfies these desiderata without knowing it.12 So the identification relies on a) establishing that ontology in all its historical vari­ance ultimately circles around a handful of central theoretical themes and chal­lenges, and on b) showing that ZFC manages to address (resolve?) these themes and puzzles in a more apt way than the classical philosophical programmes did. This description of what Badiou has to show is still sketchy, but it is fair to the spirit of Being and Event.13 To summarise: to be successful, an argument in fa-vour of M = O would have to achieve three things: a) describe the themes and desiderata of ontology anew; b) describe the theme and achievements of set theory anew; c) check that these desiderata and achievements match. I believe that Badiou fails to make a convincing case for M = O on account of his implausible descriptions of both ontology and set theory. The impression that these two discourses align is a consequence of his tendentious characterisation of both. 12 Ibid., pp. 9–11. 13 See, for example, the meditation on Spinoza, ibid., pp. 112–21. Let us zoom in on a) and b). Badiou’s argument is that ontology (qua collec­tive philosophical endeavour) has arrived at a point where ontology recognises the need for a theory of the multiple (a term that I aim to problematise in what follows), but has failed to supply anything theoretically respectable.14 Set the­orists, on the other hand, have managed, he alleges, to create precisely such a theory – a theory of the pure multiple – without fully realising the ontological implications of their creation. As critical readers, all we have to do is check that the word “multiple” means the same thing on both sides of the equation M = O, since Badiou’s principal contention is that despite the tunnel vision of math­ematicians, their work speaks to the ontological theme directly.15 I object that there is too much semantic slippage between the two uses of ‘multiple’ (first referring to the ontological multiple, then referring to the sets from set theory) to accept Badiou’s identification. This issue wreaks havoc on Badiou’s philo­sophical project in Being and Event (but not on set theory as an independent mathematical discipline, of course). In what follows, I will go back and forth between ontology and mathematics in order to spell out how the signifier “mul­tiple” takes on a quite different meaning on both sides. There are two types of problems on the side of ontology. The first is that the onto­logical word ‘multiple’ indeed means something different than in mathematics (the mismatch problem). The second problem is that Badiou’s call to put a figure of the multiple (as opposed to a figure of the one) at the centre of ontology ap­pears ad hoc, despite Badiou’s efforts to imbue the idea with a mood of epochal necessity (the motivation problem).16 14 It is not entirely clear who is taken to hold this view. The mood is one of diagnosing epoch­al philosophical tendencies. 15 In fact, Badiou holds the view that set theory is so fruitful ontologically that all first-order ontological theorising is done by set theorists in the first place. Philosophy’s function is second-order commentary. Ibid., pp. 1–22. An alternative explanation is of course that Badiou started with some analogical inter­pretations of set-theoretic results, only to find a fitting philosophical rubric – ontology – and afterwards to package them so as to appear more attractive to the philosophical read­ership. This view is corroborated by the fact that Theory of the Subject (a book written before Being and Event) already contains an interpretation of Gödel’s and Cohen’s results. However, in the earlier book there is no mention of a systematic ontological role for set theory as a new theory of the pure multiple. Here, Badiou appears content to present his approach to mathematical thought as being more hermeneutical in character. Badiou, Theory of the Subject,p. 148. To evaluate Badiou’s proof, we have to look at his descriptions of the ontological problematic against the backdrop of what we know about set theory and vice versa. In other words, we should study his characterisations of ontological im­passes and then ask ourselves whether mathematical set theory offers a sound solution. The shift that Badiou wants to effect is the following. Without M = O, the rele­vance of Gödel’s and Cohen’s work on the independence of the continuum hy­pothesis is completely intra-mathematical (the local relevance of Gödel-Cohen). If set theory is somehow concerned with “being qua being,” a term coined by Aristotle, on the other hand, this will make their results more universally rele­vant – given some philosophical guidance (the global ‘philosophical’ relevance of Gödel-Cohen). This is the work that the identification of mathematics (set the­ory) and ontology does for Badiou’s philosophical project in Being and Event.17 It is quite natural to question the claim of the global relevance of set theory to themes from classical philosophical ontology. Modern set theory arose as a re­sponse to the increasing use of an intuitive (naive) concept of collection/set in mathematics (sets of objects, sets of numbers, sets of functions, extensions of concepts, etc.). This led mathematicians to investigate infinite sets and the dis­covery of a framework for distinguishing different sizes of infinite sets (Cantor). Of course, classical ontological thinkers had also concerned themselves with in­finity, insofar as they tried to elucidate the foundational concepts of mathemat­ics (e.g. Hegel’s Science of Logic, the section on Quantity).18 But at the foreground 17 See Badiou’s remark that Cohen’s results should become an “intellectual topos at least as famous as Godel’s [sic] famous theorems […]. They resonate well beyond their technical validity.” Badiou, Being and Event, p. 16. 125 18 Georg Wilhelm Friedrich Hegel, The Science of Logic, trans. George di Giovanni, Cam­bridge University Press, Cambridge 2010. With regard to infinity, one should probably sep­arate two aspects. The first concerns the elucidation of different concepts of infinity. In this regard, Cantor and the set theorists are of course important for showing that a certain conceptualisation of infinity (in the context of sets/collections) leads us to accept a typol­ogy of different types or sizes of infinity – a typology that is of considerable mathematical complexity and utility. But this conceptual innovation should not be conflated with met­aphysical or ontological questions that are concerned with the finite or infinite nature of the universe or of any fragment of reality, let alone of being qua being. This second aspect presupposes some concept of infinity and then asks regarding certain entities whether they are finite or infinite. The cultivation of concepts of infinity and the investigation of the ‘size’ of certain ontologically relevant entities are two separate but related intellectual of classical ontology is the elucidation of the concepts of being, entity, object, nature, substance, quality, necessity and possibility, space and time, concept, matter, subject, agency, etc. – concepts that seem presupposed in most, if not all, of our conceptual dealings and behaviours, regardless of the precise mat­ter at hand.19 Insofar as mathematics sometimes develops formal counterparts of these concepts (classically: space and quantity), it offers tools for answering questions posed by philosophers. However, it is of utmost importance to differ­entiate Badiou’s approach from the more typical mingling of philosophy and mathematics. What makes his approach unique is that he approaches the con­cept of being itself via the notion of multiple, which he then identifies with the mathematical notion of set. Returning to set theory, one should keep in mind that the naive concept of set is grounded in our basic experience with collections of objects that are not col­lections themselves. For example, we might consider the collection of chairs in room 101 – a collection that contains basic objects of our everyday experience. It is only after a second step that mathematicians after Cantor started to focus on collections of abstract objects (numbers) and on collections of collections (subsets of the set of natural numbers), which led to the development of a for­mal set theory that does not involve any non-set objects. The most well-known example of such a theory is ZFC. In any case, it is not at all obvious how the concept of set/collection (in whatever stage of formalisation) can be used to elu­cidate any classical ontological category not directly concerned with quantity or extension, such as the concepts of object, being, or substance. We can use set-theoretic tools to model metaphysical categories. Still, we should not iden­tify the domain of discourse of ZFC with the realm of being qua being (without an intricate argument). In contrast, the widely held (and, in my opinion, correct) view is that formal set theories are axiomatisations of the naive concept of collection – nothing more enterprises. It is certainly possible to wholly affirm the conceptual innovations of the set theorists without also having to submit to the much more contentious view that being qua being (or any other metaphysical or physical category!) is infinite. 19 The preference for the label ‘ontology’ over the label ‘metaphysics’ here merely shows the prioritisation of the concept of being in the wider investigation. Also, needless to say, classical ontology also often goes hand in hand with theological themes (onto-theology), something which I have not emphasised in my discussion here. and nothing less (the common sense scope of set theory).20 Whenever we are concerned with talk of collections, extension, and cardinality, set theory can become relevant. But it is certainly not prima facie evident that the very concept of collection is of foundational significance to classical ontology. But this is pre­cisely Badiou’s central claim in Being and Event when he declares that mathe­matics is ontology. How does his argument work? It seems to me that, in Badiou’s view, the central notion of ontology is the no­tion of any being whatsoever. (Other notions that could play a role are those of object and entity, although Badiou avoids these terms. He often uses the Aris­totelian phrase “being qua being”.) I take it that the task of ontology is to char-acterise being qua being adequately. Now, one can undoubtedly pause here to ask whether this even constitutes a sufficiently clearly circumscribed theoret­ical field – a certain scepticism about this project is widespread even among philosophers, myself included. But Badiou is not agnostic on these matters; he brings a surprising and strong desideratum into play at this level: being qua being must not be one (this is captured by the slogan “the one is not”).21 There are two conceptual ingredients to this desideratum. First of all, the concept of parthood (pure being must have several parts, components, constituents, etc.). Second, the ban on unity (pure being must have several parts, pure being is not to be approached through the theme of unity). But as I will show, we should be careful to not conflate different types of part-whole relations here. 20 The naive concept of collection has several origins. On the one hand, instead of consid­ering the objects on the table individually, we can reify this into the set of objects on the table. This spontaneous talk of collections of objects (however concrete or abstract) is one 127 source of intuition regarding the concept of collections. Another, related, background as­pect is the notion of an extension of a concept (Frege). The extension of the concept ‘dog’ is not a spatio-temporal object but a collection of dogs. But neither of these are in an obvi­ous way related to the ontological category of being qua being. Rather, they are related to everyday talk of quantities and extensions. Presumably, our competence with the notion of set/collection builds on our familiarity with the embodied practice of putting several objects inside of a container, allowing us to treat them as one object whilst also retaining the distinctness of the objects inside of the container. The mathematical concept of set is an abstraction of this embodied concept. See George Lakoff and Rafael E. Núnez, Where Mathematics Comes From: How the Embodied Mind Brings Mathematics into Being, Basic Books, New York 2000, pp. 140–54. 21 Badiou, Being and Event,p. 23. My challenge to Badiou is that this desideratum is poorly motivated philosophi­cally (arbitrariness) and that it does not get us to set theory on the side of mathe­matics (poor fit). I will work backward here and start by explaining why “the one is not” may seem to pave the way to a set-theoretic ontology, only to problema­tise this trajectory afterward. In the second step, I will question whether there is an intrinsic ontological motivation for “barring oneness from being” at all. According to Badiou, being must be devoid of all oneness. The practical con­sequence of this claim is that if we can find a theory describing (formalising) entities that are multiple but not one, we may proceed to treat it as our new on­tology, and hence, as a theory with broad philosophical consequences. Here is a key fact about ZFC which contributes to the impression that it is a “theory of the pure multiple”: (ZFC has no ‘atoms’ or urelements).22 The entities of ZFC are pure sets. That is, there are only sets of sets, but no primitive entities that are not sets.23 Hence, the entities of ZFC are ‘multiples’ composed of further ‘multiples’. They do not decompose to an atomic base-level of non-set objects. Let us summarise. On the side of ontology, Badiou identified a demand for a theory of being without oneness; on the side of mathematics, he came across a formal logical theory of multiples that appear not to reduce to atoms (multiples without ‘the one’). It seems like a perfect fit. But there are two open questions: 1) it is unclear what exactly motivates the ontological desire for a multiple without the one; 22 Terminologically, ‘urelements’ is really more accurate here than ‘atoms’. Even in set the­ories that do admit urelements, the relation between sets and urelements is not the one between a spatio-temporal object and its smallest constituent parts. Any talk of atoms in the context of set theory is metaphorical, which is why the more technical terminology of ‘urelements’ is preferable. Badiou seems to make much of the fact that formal set theories (ZFC and its extensions) talk of sets only, without needing to refer to non-set objects. If we take a theory such as ZFC in isolation, it seems to commit us only to a universe of sets, and nothing else. However, this is simply a side effect of the fact that in formal logical theories we consider certain conceptual systems in isolation. Another formal theory, such as Peano Arithmetic (PA), considers the system of natural numbers in isolation. But neither ZFC nor PA should lead us to the idea that “there are only numbers” or that “there are only multiples.” Badiou, Being and Event, p. 61. 2) we also have not sufficiently explored whether or not sets are the type of multiples that can play the role of Badiou’s ontological ‘multiple without the one’. What kind of multiples are sets exactly? It is undoubtedly true that the objects of ZFC, developed out of the need for a more rigorous concept of collection than the quotidian (naive) one, can be char-acterised as ‘multiples’. The word ‘multiple’ itself suggests little more than that the object in question is composed of many. Indeed, most sets have many ele­ments. If they have exactly one element, that element is a set itself, so one level deeper it decomposes into many elements. Some synonyms for ‘set’ are ‘aggre­gate’, ‘collection’, ‘Menge’, and ‘ensemble’. These all emphasise the aspect of plurality and de-emphasise the aspect of unity, that which holds the elements together. This is certainly an important part of the set concept, since nothing much is required in order to keep the elements together to yield a legitimate set.24 This stands in stark contrast to the type of unity that holds together the parts of a spatio-temporal object, for example. But is the set-theoretic notion of a collection the kind of multiple that is adequate for being qua being? Why would it be? Let us proceed by comparing and contrasting what is known about set theory with Badiou’s description of the pure multiple as an ontological figure. One important aspect of set theory, which seems quite alien to ontology as a theory of being qua being, is the following: sets (or ‘collections’) have a distinct­ly combinatorial flavour. In short, the idea is that any combination of elements whatsoever that does not lead to paradoxical situations should yield a legiti­mate set. A simple example helps to demonstrate the combinatorial wealth of set theory. The set X:= {5, {1,2,5}, {{5}}} has three elements: 5, {1,2,5}, and {{5}}. Here, the number 5 occurs three times, on several levels: once as an ele­ment of X, once as an element of an element of X, and once as an element of an 129 element of an element of X. Since these are all viably different combinations, X is a legitimate set. Also, notice that 2 is not an element of X, even though it is an element of an element of X. It is no exaggeration to claim that set theory was created to encompass these types of embedded (‘nested’) membership systems. But this does not seem to be a feature of being qua being on any available inter­pretation of the term. It seems to be a unique and defining feature of sets, which differentiates them from pretty much any other type of entity. 24 Except for the care needed to prevent paradoxical sets. Another formalism needs to be introduced here: mereology. Whereas modern logicians and mathematicians developed set theory to axiomatise the notion of collection, they also developed the theory known as mereology to axiomatise the part-whole relationship. Oddly, mereology is never mentioned by Badiou. Introducing mereology prepares the second step of my argument, in which I aim to show that Badiou’s reading of classical ontological arguments suggests a mereological, not a set-theoretic interpretation. My point will be that in his reading of philosophical authors such as Lucretius and Plato, Badiou problem­atically conflates these quite distinct concepts into a single concept of ‘multi­ple’, which seems to be a mixture of the two. To a large degree, the identification of ontology with set theory can only arise due to the conflation of the collection concept (set theory) with the part-whole relation (mereology). We can familiarise ourselves with the mereological notion of parthood by tak­ing a Volkswagen Golf and its parts as an example.25 The steering wheel is a part of the car. The car is a whole comprised of such parts (a ‘multiple’?). The relationship between a mereological whole and its parts should be kept distinct from the relation between a set and its elements. To see that they are different relations, consider the mereological example of the driver’s seat and its head­rest. The headrest is part of the seat, and the seat is part of the car. By virtue of this, the headrest is a part of the car as well. This parthood relation is tran­sitive.26 Axiomatisations of the mereological parthood relation usually include transitivity either as an axiom or as a direct consequence of other axioms. Com­paring this to the set-theoretic axiomatisation of membership (.), we note that the elementhood relation is not transitive. To see an example of this, consider a flutist named Allen, who is a member of a marching band (viewed as a collection of musicians). The marching band itself is a member of the set of active musi­cal groups in France. But Allen himself is not a member of the set of musical groups. Set-theoretic elementhood is intransitive. Sets and mereological wholes are different types of multiples.27 There are several other essential dissimilarities between the two notions, which is why they yield different axiomatisations. 25 For a more extensive introduction, see Achille Varzi, “Mereology”, Edward N. Zalta (ed.), Stanford Encyclopedia of Philosophy, 2016, https://plato.stanford.edu/archives/win2016/ entries/mereology/. 26 A relation R is transitive if: a b c(Rab Rbc › Rac). 27 That said, a subset of a set is like a mereological part of a set. (Sets have parts analogous to how mereological wholes have parts.) However, this should not allow us to infer the To summarise: mereological axioms, not set-theoretic ones, formalise the mate­rial parthood relation. Soon, we will see that the thinkers that Badiou discusses (aiming to establish the necessity of a set-theoretic ontology) use what look like mereological concepts. Hence, their propositions regarding such matters are best formalised using mereology. It should be pointed out that as a family of formal theories, mereology does not come with such surprising results as the independence of the continuum hy­pothesis. Unlike set theory, mereology is not replete with paradoxical niceties that might invite philosophical interpretation. My suggestion here is not to re­place Badiou’s set-theoretic ontology with a mereological one with similar am­bitions (and then to offer philosophical interpretations of meta-theorems about the mereological theories). Instead, my point is simply that the parthood vocab­ulary used in certain philosophical discussions (and as we will see, in Badiou’s treatment of ancient sources such as Lucretius and Plato) is best formalised us­ing mereological theories.28 Set theories are simply not what is called for when formalising these ideas since they do not concern the extension of sets, but the parthood structure of mereological objects. As we will see, Badiou seems una­ware of this. Now that we have introduced the distinction between collection-multiples and mereology-multiples, we can return to Badiou’s claim that a theory of the pure multiple is needed in contemporary ontology. We must now ask which flavour of formal theory fulfils his ontological desideratum. As stated earlier, Badiou’s ontological demand is that being qua being is a multiple without unity. In other words, being qua being ought to be some sort of multiple whose parts are also 131 inverse statement: that all parthood relations reduce to set theory. This is clearly not the case even at an intuitive level, since we can employ the mereological vocabulary of parts and wholes without treating our objects as collections. For example, nothing obliges me to treat my car as a set just because I want to talk about its parts. I might be able to name many mereological parts of the car. My description may exhaust our knowledge of the car, including its subatomic particles. Set-theoretic elements are not needed on any level. Similarly, I might use a list to describe the components of the car. But this does not commit me to the view that the car itself is list-structured. There is a difference between what is cognitively salient and what is ontologically real. 28 There is considerable literature that applies modern mereological thought to reconstruct ancient doctrines. One example is: Verity Harte, Plato on Parts and Wholes: The Metaphys­ics of Structure, Clarendon Press, Oxford 2002. multiple. We have already seen why Badiou believes that this description fits the set-theoretic framework. However, let us study some of the passages in which he describes what conceptual innovations are needed in ontology, passages where he engages in a reading of classical ontological texts. We will see that a mereo-logical concept of part-whole is more suited. On several occasions Badiou presents his ontology of the pure multiple as a modern realisation of the ontological programmes of ancient thinkers such as Parmenides, Plato, and Lucretius. In a companion essay to Being and Event, entitled The Question of Being Today, Badiou quotes the following lines from Lucretius’ On the Nature of Things: Such is the nature of the place, of the gigantic space: Were it to slide, forever drawn away by time, Lightning would never see distance reduced The whole enormous reservoir of things is open In all directions.29 He describes Lucretius as “the one who directly confronts thought to subtrac­tion from the One, which is none other than inconsistent infinity, that is, what nothing can collect.”30 This requires some critical commentary. First of all, let me repeat that Badiou’s goal in this passage is to establish that ontology re­quires a theory of the multiple without oneness (i.e. set theory). Presumably, he cites Lucretius as an early adherent of (a pre-modern anticipation of) such a doctrine. We must ask two questions here: a) What kind of multiple is Lucretius describing? b) What role does the subtraction of the one play for Lucretius? Turning to the first question: it is quite clear, despite the brevity of the passage, that Lucretius is describing the nature of a spatial whole (“the whole enormous reservoir of things”) – something like the universe or the cosmos. His main claim here seems to be that it is infinite in size (“open in all directions”). How­ever, such a spatial structure should not be mistaken for a set! A spatial whole 29 This is a translation of Lucretius, De Rerum Natura, 1.1002-8. Cited in Alain Badiou, “The Question of Being Today”, in Briefings on Existence: A Short Treatise on Transitory On­tology, trans. Norman Madarasz, SUNY Press, New York 2006, p. 35. Madarasz bases his translation on “the French version used by Badiou.” Footnote 6, p. 174. 30 Ibid. has (mereological) parts, not set elements. So, if one were to capture Lucretius’ theorising up to this point in some kind of formal logical system, it would be a mereological (or mereo-topological) theory, not a set-theoretic one. Set-the­oretic vocabulary may of course come in when we need to discuss how many parts (extension!) a spatial whole has. But in that function it serves to formulate quantitative (or extensional) statements about objects which are themselves not sets, but spatial wholes. So this would lead us away from Badiou’s idea that set theory speaks directly to the ontological category of being qua being. With regard to the second question, concerning the subtraction of the one, we have seen that Lucretius claims that there is no outside border of the universe, that it is open in all directions. This is akin to saying that the universe is not a finite (‘closed’) mereological whole (or unity). Paraphrasing, one could claim that the universe is not ‘one’ in the sense that a finite object is one (if we identify unity with boundedness). So, on this route we may indeed come to the conclu­sion that Lucretius “confronts thought to subtraction from the One,” as Badiou writes, although it remains a tendentious formulation. Most importantly, Lu­cretius’ description is indicative of a mereological vocabulary (parts, wholes, boundaries), not a set-theoretic one. He considers the universe from the per­spective of its mereological structure. It is not clear how to paraphrase Lucre­tius’ specific view in the language of sets, since it is about a spatial whole, not a set. Instead, we should probably read him as claiming that the universe is unlike any cognitively more familiar region of space, insofar as it is unbound­ed.31 His claim is primarily preoccupied with the existence of outer boundaries, and not with questions of extension or numerosity. Although this is certainly a revolutionary position by the standards of Lucretius’ contemporaries, it remains opaque how he can be cited in favour of a connection between being qua being and set theory. 133 Here, I would like to make a final remark as regards understanding Lucretius from a contemporary perspective. As stated, his main concern in the cited pas­sage seems to have been the boundlessness of the physical universe. Spatial 31 This is not to say that mereological or mereo-topological statements cannot be modelled in a set-theoretic context. The point is that the specific “subtraction of the one” that Badiou finds in Lucretius concerns a very specific claim about the boundedness of the universe, which simply does not contribute anything to the much more outlandish claim that the ZFC axioms codify something about being qua being. boundlessness is a variation on the theme of infinity, but not quite the same as the extensional infinity of sets. Since the idea of actually infinite extensions was unknown to the Greeks, we cannot expect Lucretius to have also explored the correlative idea that an unbounded universe (spatial infinite) may have in­finitely many (extensional infinite) parts. However, from a modern point of view, this question comes naturally. How many parts does this unbounded universe have? This type of question mixes mereological and set-theoretic vocabulary. We are now considering the cardinality of a set of parts of a material whole. This is a totally natural situation, which motivates the presence of set-theoretic vocabulary. But notice that this is a set of non-set entities (the parts of space). So, it does not suggest that being qua being is set-theoretically legible. Rather, it merely suggests that when considering questions of cardinality (and combinat­orics), set theory is a natural conceptual tool. Very similar criticisms apply to Badiou’s take on Plato’s Parmenides in medita­tion two of Being and Event. Here he focuses on unity not at the largest level, but at the smallest, atomic level. Badiou goes on to conclude that ZFC set theory, which has no urelements, is a description of the kind of multiple without the one that Plato considers, but fails to elaborate, in the Parmenides. Let us take a look at the passage from Parmenides to see whether the key propositions have a set-theoretic or a mereological character. One of the questions that Plato investigates in the Parmenides is whether we can describe being if we drop the characteristic of unity – the hypothesis “if the one is not.”32 For Plato (or at least for the dramatis personae of the Parmenides), the result is deeply aporetic. The interlocutors in the Parmenides abandon the option that being without unity can be thought at all. But for Badiou, in con­trast, the way the aporia is spelled out by Plato unwittingly indicates a new way forward, which he labels “the multiple without the one.” Although Plato did not see a way to theorise being without the one consistently, Badiou believes that modern set theory delivers precisely such a theory. Here is the passage from the Parmenides that Badiou believes indicates a set-theoretic opportunity: If one took the point of being which seemed to be the smallest, much like a dream within sleep, it would immediately appear multiple instead of its semblance of Plato, Parmenides, 160b-166c, as quoted in Badiou, Being and Event, p. 31. one, and instead of its extreme smallness, it would appear enormous, compared to the dissemination that it is starting from itself.33 This line of thought is an outlandish consequence of the hypothesis of being without the one. More specifically, the idea seems to be that if there are no smallest parts, then every part must contain more parts, ad infinitum. Another way to phrase this is that if there is no fundamental unity to being, then there is no level at which one will encounter undecomposable atoms. The image is one of infinite zoom levels – at each level there are more parts to discover. This view may be hard to fathom for Plato’s interlocutors (and possibly for Plato himself) because they tacitly subscribe to an atomistic picture of spatio-tempo­ral objects. Without there being smallest units, there is nothing the bigger com­posites can be composites of. So they reject this position as aporetic.34 We should now keep two concerns separate. First of all, it is essential, once more, to firmly establish that the type of parthood relation that occurs in the cited passage is mereological. We take some fragment of being (we quite literally ‘take’ some chunk of reality and examine it), then try to find its smallest part, only to discover that it has many parts all over again. This type of language, used in the passage cited above, is indicative of spatial mereological wholes and parts, not of sets, since only spatial wholes have smaller parts.35 By way of con­trast, there is no guarantee that an element of a set is ‘smaller’ than the set itself (if we understand ‘smaller’ in the sense of cardinality – but that is the only no­tion of size available for sets). Although any interpretation of Plato’s Parmenides 33 Plato, Parmenides, 164d, as quoted in Badiou, Being and Event, p. 34. Parmenides address­ 135 ing Aristotle. 34 A similar sentiment is found in Leibniz, in one of his letters to Arnauld: “I maintain that there is no better way to put philosophy back on its feet and turn it into something precise than by focusing on individual substances or complete entities that have genuine unity, their changes all being caused from within themselves; everything else is mere phenome­na, abstractions or relationships. We’ll never find any rule or recipe for making a genuine substance out of many entities by aggregation.” Gottfried Wilhelm Leibniz and Antoine Arnauld, The Correspondence between Leibniz and Arnauld, trans. Jonathan Bennett, 2017, https://www.earlymoderntexts.com/assets/pdfs/leibniz1686a_1.pdf, p. 62. 35 In fact, the Greek word root µ..- (part) occurs over 50 times in the discussion between Parmenides and Aristotle, concerning the different hypotheses about being and the one. ‘Mereology’ takes its name from this Greek root. is bound to be contentious – I certainly do not aim to develop my own here – my goal has been merely to point out that more work is needed to cast the central propositions in set-theoretic terms. It is undoubtedly true that Plato’s interlocutors (Parmenides and Aristotle) con­sider the idea of an atomless reality hard to swallow. However, modern mere-ology teaches us that we can introduce an axiom that states that all objects are atomless into a mereological theory without running into any mathematical in­consistencies: The axiom of atomlessness: Every object has at least one proper part. If an atom is an object without proper parts, then this axiom bars the existence of atoms.36 A mereological theory that includes this axiom would perhaps be a ‘theory of the pure multiple without the one’ in the spirit of the above passage of the Parmenides. From a contemporary mathematical perspective, such theories are known to be consistent.37 However, since this question concerns the make­up of our physical universe, one would have to leave the final word to physicists. But modern atomic physics has long surpassed any sort of simple mereological approach to atoms, shifting its focus instead to the forces at work in keeping matter together. And even if quarks are the smallest (indivisible) particles, it still does not mean that physical objects are sets (unstructured collections) of quarks. Instead, they are some sort of structured constellations of them. The structure of such constellation is what modern physics aims to understand. The mereological aspect plays little to no role. To summarise, if one wanted to pick up the argument where the interlocutors of Plato’s Parmenides leave it, one would do so with mereological ideas (supple­mented by physical theories), not set-theoretic ones. (Set-theoretic cardinality concerns can come in later, but do not imply that the entity in question, a spa­tial fragment of reality, is a set.) So, it is quite unclear how any reading of the Parmenides can lend support to the proposal that set theory is a privileged site 36 ‘Atom’ is here used in the sense of ‘smallest indivisible part’, not in the sense of contempo­rary physics, where atoms have smaller constituents. The open sets of a Euclidean space are a model of an atomless mereology. See Varzi, “Mereology”. of ontological speculation. More specifically, the proposition of excluding unity at the microscopic level of being (the idea explored in the Parmenides) is not at all captured by ZFC set theory, which is not a medium for theorising about the part-whole structure of being at all, but a theory of collections. ZFC, as a theory of sets of sets, has no real purchase on the entities discussed by the ancients. The above objection is more or less independent of the role that unity plays in Badiou’s argument. As I have indicated, the propositions a) that there is no uni­versal whole (Lucretius) and b) that there are no atoms (Plato’s thought exper­iment in Parmenides) are stated in mereological vocabulary. The broader class of mereological theories includes theories that affirm or deny these theses. So, at the outset, mereology (the study of a class of formal theories that isolates the part-whole relation) is neutral with regard to these propositions, and can hence serve as a vehicle for further investigation. Now, regarding the matter of the barring of the motif of unity from ontology: my position is that there is no strong philosophical reason for assuming that being qua being does or does not have the characteristic of unity. Also, as we have seen, Badiou’s framing of the passages from Plato and Lucretius is somewhat tenden­tious, since he frames the specific questions of totality and atomicity as matters pertaining to the broader rubric of unicity. Regardless of these minor framing concerns, one can also ask whether Badiou gives a substantial and independent ontological motivation to bar ‘the one’ from being qua being. Although unity is sometimes more projected than real, it is nonetheless a fundamental (yet not fully understood!) facet of our conceptual (and perceptual) apparatus. Although Badiou claims the opposite, modern set theory can equally be construed as an extension of our cognitive talent at uni­fying heterogeneous substance into a whole (in an extended sense) – set theory as a consequence of the human talent for unification. Returning to Badiou, we have already seen that when it comes to his ancient references, it is not straight­forward why a contemporary ontological programme must avoid the theme of unity, since a close reading of these texts invariably leads to a discussion of issues within the purview of modern physics (atoms, the universe, etc.). Apart from these analyses of ancient philosophical texts, Badiou gives several other reasons to push the pure multiple to the foreground of the ontological agenda. Some of these build on interpretations of other philosophers (e.g. He­gel, Nietzsche, Heidegger, Spinoza), and some of them have a more argumenta­tive character. For example, meditation one of Being and Event concerns itself with establishing the necessity of postulating that being qua being must be pure multiplicity.38 However, upon closer inspection, Badiou gives no independent argument here. Rather, he introduces the maxim “the one is not” as a sort of philosophical axiom. If anything, the indirect argument is that those ontological programmes which made unity a strong characteristic of being qua being failed. However, there are reasons to doubt that Badiou demonstrates this. At best, he establishes that one may attempt the alternative. It appears to me that there is no independent framing of the theme of ontology here, one not designed with the final goal in mind of recognising set theory as a theory of the pure multiple. In subsequent parts of the book he does not dwell on whether M = O has been successfully established. Let us close this examination by looking at one of Badiou’s remarks on Nietzsche. Nietzsche famously declared that God is dead. Insofar as theology sees in the monotheistic God of Christianity a figure of the one, we could inter­pret Nietzsche’s intervention as a warning against the theme of unicity in ontol­ogy. This is what Badiou indeed does in an instructive passage of Number and Numbers, which compresses his proof of M = O to the point of extreme brevity: Modernity is defined by the fact that the One is not (Nietzsche said that ‘God is dead’, but for him the One of Life took the place of the deceased). So, for we mod­erns (or ‘free spirits’), the Multiple-without-One is the last word on being qua be­ing. Now the thought of the pure multiple, of the multiple considered in itself, without consideration of what it is the multiple of (so: without consideration of any object whatsoever), is called: ‘mathematical set theory’. Therefore every ma­jor concept of this theory can be understood as a concept of modern ontology.39 We can undoubtedly embrace Nietzsche’s healthy scepticism towards monothe­ism. But does that get us all the way to the claim that the ZFC axioms codify being qua being? It seems not. As we have seen throughout this paper, the leap 38 Badiou, Being and Event, pp. 23–30. 39 Alain Badiou, Number and Numbers, trans. Robin Mackay, Polity Press, Cambridge 2008, para. 7.16. is enormous. Why do we need to replace the monotheistic God (a figure of the one) with another ontological figure (the pure multiple)? Should set theory take the place of onto-theology? If God does not exist, then it would appear that no specifically theological word on being qua being needs to be spoken at all. And even if we were to accept the necessity of a replacement figure, the subtraction of oneness from being would not get us to the sets of ZFC. It would get us to atomless mereology, a position that might be of interest to some logicians, but not to someone like Badiou, who wants to bridge ontology with the theory of subjectivity. What is most important about the cited passage, however, is the final sentence, its outcome. If the identification succeeds, then Badiou can treat the fruits of set theory (“every major concept”) as direct ontological insights. This indeed seems to be the real justification of “mathematics is ontology”: it serves to disguise the rather loose and analogical nature of Badiou’s use of mathematics. Instead of using mathematical tools to answer questions about the nature of reality, Badi­ou instead interprets mathematical results directly. He presents his idiosyncrat­ic interpretations as rigorous consequences of the work of Cantor, Gödel, Cohen, etc. The complicated intermediate steps have been compressed by way of a di­rect identification. This makes it exceptionally difficult to evaluate his ontology on a strictly philosophical basis, since most, if not all, of his central concepts are elaborated through complex analogies between philosophical themes and the theory of sets. Instead of giving his reader a guide to interpreting his analogies as analogies, he continues to present his work as a rigorous mathematical ontol­ogy. A philosophical project that is in truth experimental and heuristic in nature is presented as a faithful unpacking of the consequences of the innovations of others (Cantor, Gödel, etc.).40 It is this meta-philosophy itself (i.e. an informal model or picture of philosophical activity) to which I object most strongly (inso-139 far as it opens the door to extreme forms of dogmatism). 40 This stylisation of philosophical activity through the heterogeneous field of mathematics is not entirely innocent. It has recently been characterised as a “project of the re-education of philosophy through mathematics,” in François Laruelle, Anti-Badiou: On the Introduc­tion of Maoism into Philosophy, trans. Robin Mackay, Bloomsbury Academic, London and New York 2013, p. vii. Similar complaints have been lodged in Ian Hunter, “Heideggeri-an Mathematics: Badiou’s Being and Event as Spiritual Pedagogy”, Representations 134 (2016), pp. 116–56. Despite their observant remarks, both authors frequently lapse into caricature. A final remark on how I believe we should read Badiou. Despite his great love for mathematics, he is probably most known for philosophical ideas that do not fall within the purview of classical ontology. His central doctrines concerning events and truth procedures are perhaps better understood as falling within the fields of philosophical anthropology or social ontology since they are ultimately concerned with the possibility of certain types of (collective) human action.41 The central question is always what it means to be a human being situated in a socio-historical cultural field of artistic, scientific, political, and amorous in­novations. Perhaps it is time we read Badiou on that level, letting go of the idea that his anthropological claims are mathematical consequences of a deeper in­volvement with ontology (which is how he, at least part of the time, describes his systematic endeavours in the Being and Event trilogy). What if we turned Badiou on his head and finally understood that his ontology is analogical and heuristic in nature, yet ultimately indexed to his avant-garde anthropology, and not the other way around? * * * Postscript At the 2018 conference in Prague where I presented a shorter and less critical version of this paper, Badiou gave a response paper (addressed to all the con­tributors) in which he explicitly addressed the role of his formula “mathematics is ontology.” An edited version of this response is included in this volume.42 Since his talk was rather surprising, I would like to end with a brief response. In his talk, Badiou admitted (“confessed”) that the formula M = O is only “a sort of mediatic formula,” or “a propagandist formula” which is “condemned to a 140 terrible failure.”43 He furthermore claimed that he brought ZFC set theory into ontology because he saw a new tool to continue the ontological orientation of Badiou occasionally describes his work as having an anthropological scope. See Alain Badiou, “Affirmative Dialectics: From Logic to Anthropology”, The International Journal of Badiou Studies 2 (1/2013), pp. 1–13. 42 Alain Badiou, “Ontologie et mathématiques : Théorie des Ensembles, théorie des Catégo­ries, et théorie des Infinis, dans L’Etre et l’événement, Logiques des mondes et L’immanence des vérités”, pp. 15–34. 43 Alain Badiou, “Five Points – Final Speech”, Youtube, 2018, https://www.youtube.com/ watch?v=iWws287P1OU. such thinkers as Lucretius and Democritus in it . But on the other hand, therein he also admitted that he “forced” something onto mathematics, since ZFC is not a theory of the multiple in any available ontological sense of “multiple.” Despite these admissions, it is my view that Badiou falls short of providing a satisfactory alternative narrative with regard to the exact role of mathematics for his systematic philosophical works. In any case, he seems far from recast­ing his interpretations as extended examples of analogical reasoning, which is my present view. I have already discussed why the alleged connection between the ontologies of Lucretius and Democritus, on the one hand, and modern set theory, on the other, is a spurious one. And Badiou admits that the connection is rather loose. However, he continues to describe the study of ZFC and its ex­tensions as the study of “all possible forms of the multiple,” where the concept of multiple is taken to have a direct ontological resonance. Furthermore, he em-phasises that his ontological decision can be appreciated “at the level of the consequences,” e.g. upon consideration of his interpretation of Gödel’s and Co­hen’s theorems. But if the basis of the analogical argument is spurious, how can we accept or appreciate the consequences? The whole project revolves around using mathematical proofs in support of anthropological propositions. It seems that Badiou’s auto-criticism falls short of admitting the creative and analogical nature of his interpretations. References Badiou, Alain, Being and Event, trans. Oliver Feltham, Continuum Books, London and New York 2005 — “Five Points – Final Speech”, Prague Axiomatic Circle, Prague 2018 [Youtube], availa­ble at: https://www.youtube.com/watch?v=iWws287P1OU — Theory of the Subject, trans. Bruno Bosteels, Continuum Books, London and New York 2009 — “The Question of Being Today”, in Briefings on Existence: A Short Treatise on Transi­tory Ontology, trans. Norman Madarasz, SUNY Press, New York 2006 Baki, Burhanuddin, Badiou’s Being and Event and the Mathematics of Set Theory, Blooms­bury Academic, London and New York 2015 Bartlett, A. J. and Justin Clemens (eds.), Alain Badiou: Key Concepts, Acumen Publishing Limited, Durham 2010 Corcoran, Steven (ed.), The Badiou Dictionary, Edinburgh University Press, Edinburgh 2015 Cohen, Paul, Set Theory and the Continuum Hypothesis, Dover Publications, Mineola 2008 — “The Discovery of Forcing”, Rocky Mountain Journal of Mathematics 32 (4/2002), pp. 1071–1100 Leibniz, Gottfried Wilhelm and Antoine Arnauld, The Correspondence between Leibniz and Arnauld, trans. Jonathan Bennett, 2017, available at: https://www.earlymodern­texts.com/assets/pdfs/leibniz1686a_1.pdf George Lakoff and Rafael E. Núnez, Where Mathematics Comes from: How the Embodied Mind Brings Mathematics into Being, Basic Books, New York 2000 Harte, Verity, Plato on Parts and Wholes: The Metaphysics of Structure, Clarendon Press, Oxford 2002 Hegel, Georg Wilhelm Friedrich, The Science of Logic, trans. George di Giovanni, Cam­bridge University Press, Cambridge 2010 Varzi, Achille, “Mereology,”, Stanford Encyclopedia of Philosophy, Edward N. Zalta (ed.), 2016, available at: https://plato.stanford.edu/archives/win2016/entries/mereology/ Filozofski vestnik | Volume XLI | Number 2 | 2020 | 143–178 | doi: 10.3986/fv.41.2.07 Tzuchien Tho* Sets, Set Sizes, and Infinity in Badiou’s Being and Event Introduction In 2006, Lucca Fraser published the article “The Law of the Subject: Alain Ba-diou, Luitzen Brouwer and the Kripkean Analysis of Forcing and the Heyting Calculus.”1 Still one of the best Anglophone commentaries on Badiou’s L’Etre et l’Événement, she argued that Kripke forcing was sufficient to render the theory of the subject in Badiou’s L’Etre et l’événementwithin an intuitionistic frame­work. This pluralization of logical frameworks allowed us some insight into the possibilities of mathematical ontology of which Badiou was not aware. That is, it forced us, and eventually Badiou, to recognize that his commitment to clas­sical logic, where the principle of excluded middle was true, was unnecessary to the claims about the subject within his project. In turn, Badiou agreed and acknowledged this point but argued that this only applied to the subjective di­mensions of mathematical ontology.2 Classical logic was still needed to express the core of the ontological analysis, based on Zermelo-Frankel set theory with the axiom of choice (hereafter ZFC).3 The paper here follows in Fraser’s example. Yet instead of dealing with the the­ory of the subject in L’Etre et l’événement, I turn to the theory of the “count”, the fundamental aspect of Badiou’s theory of Being and beings in L’Etre et l’événe­ment. Further, I am not interested here to challenge the classical logical frame-143 work upon which the work is couched, although it is far from unassailable. What is in question here is instead the relation between the count and multiplicity. In particular, I argue that Cantorian transfinite cardinality, a method of reckoning 1 Zachary Luke Fraser (Lucca Fraser), “The Law of the Subject: Alain Badiou, Luitzen Brou­wer and the Kripkean Analyses of Forcing and the Heyting Calculus”, Cosmos and History: The Journal of Natural and Social Philosophy 2 (1-2/2006), pp. 94–133. 2 Alain Badiou, “New Horizons in Mathematics as a Philosophical Condition: An Interview with Alain Badiou [with Tzuchien Tho]”, Parrhesia 3 (2007), pp. 1–11. 3 Ibid., p. 7. * University of Bristol the measure of the sizes of sets, is not a necessary feature of the coherence of the project of mathematical ontology in L’Etre et l’événement. The implication of this for Badiou’s project is that the “subtraction of the one” also implies a pluralism of the one. That is, the pluralization of how unity is constituted, located, and operational. While subtraction is a rejection of the givenness of the one, the re­jection of a view entrenched in traditional metaphysics, it retains the form of the one as the result of the count. Here the one is not a given but a result. From this count, the set theorical universe is populated by the entities generated by the count. However, these entities need not be measured in the standard Cantorian way and can be subject to different forms of measurement. As will be argued, the implication of the pluralism of the one is that the count-as-one of finite and transfinite sets is indifferent to Cantorian cardinality. This indifference is crucial because the count is responsible for the count of the in­consistent multiple at the very heart of the project. This initial count is one that formally introduces the set (within the discourse of ZFC) as the basic term of Badiou’s mathematical ontology. This renders questions of measure and, in par­ticular, the measure of the infinite orthogonal to this theory. Badiou emphasizes this in his work, noting that the infinite has no intra-mathematical meaning. Badiou’s ontological project is thus not only a subtraction from the metaphysics of the “one” but also a subtraction from the metaphysics of infinity, which is dependent on it. What results instead is a pluralism of these terms based on the more fundamental distinction, as Badiou himself puts it, between concept and existence.4 A pluralism with respect to the unfolding of the ontological concept is thus no challenge to dimension of existence. This orthogonality reveals that Badiou’s ontological project is free to embrace a pluralism about unity and in­finity unanticipated by Badiou and the founders of set theory. This pluralism of the one reveals new contours of theoretical possibility previously unexamined by Badiou and his commentators. Counting and the inscription of the void The central goal of Badiou’s L’Etre et l’événement is to employ the structures provided by set theory (especially ZFC) to analyze existence. What this means is not at all straightforward. Badiou argues neither for a Pythagorean-Platonic ontology in the vulgar sense, where existence reduces to sets, nor mathematical Alain Badiou, Being and Event, trans. O. Feltham, Continuum, London 2005, p. 159. reality as ultimate reality, nor for set theory as a model for existence. Rather, no structure, be it mathematical, or something else, could serve as a direct map of ultimate reality since any ontology must first reckon with the fact that Being is itself unstructured. Any ontology must first force Being to relate to a structure that would be alien to it. Hence, for Badiou, since Being is itself unstructured, in order for ontology to be possible, it must be drawn into structure by the means of the count. Hence, this first step into the possibility of a mathematical ontol­ogy cannot be given by set theory itself. This “drawing in” is operated by “in­scription”, the localization of unstructured being into the structure of set theo­ry. Basically, the unstructured is inscribed within structure by the count of the void, since whatever is not already within the structure must appear as, or be a presentation of, a “nothing”. The basic picture then is that Badiou makes the link between set theory and un­structured Being by counting it (Being) as a set containing the void. The positivi­ty of the void in set theory therefore stands in for what is “underneath” structure but nonetheless localized within the set theoretical discourse. This bridge be­tween unstructured Being and the structures of set theory rejects the two tradi­tional tendencies in ontology. The first tendency is that ontology or metaphysics in general is a rational description of the fundamental structures of being. This presumes that Being has an inherent structure that is knowable and reducible to some basic form. We can set this aside because Badiou maintains the view, shared by a lineage of thinkers since the Platonists that Being is inaccessible to us through the means of the categories we apply to beings or the senses. The second notion is that ontology plays the role of drawing out the immanent but obscure qualities of Being-as-such. Badiou similarly rejects this path of analyz­ing the “deep” allure of the Being by cutting off our relation to the “presence” of Being in favor of reducing it to a count of the void. In short, all that is “above” and “beyond” cannot appear as a “one” (i.e. an entity), therefore it is counted as a “void”. This act, which we may term “subtraction”, produces nothing myste­rious. It is simply the count of the void. This is the null set {} or O, the void after it has been counted, that forms the basis of the arithmetic counting of pure sets. From this O, we count its successor {O}, and its successor’s successor {O, {O}}, and {O, {O}, {O, {O}}}, and so on. The success of this process allows Badiou to provide a key distinction in the mathematical ontology of L’Etre et l’événement. That is, the arborescent branching of the count of the null set (the count of the void), allows us to retroactively designate Being, seen through the lens of con­sistent structure, as “inconsistent”. Hence, instead of subordinating Being to the logic of beings (the unit entity), or searching for a non-rational presence of the Being behind and beyond being, unstructured Being is taken up into ontol­ogy (the realm of consistent multiplicity) as “inconsistent multiplicity”. This is wholly dependent on the existence of a structurally coherent (and well-ordered) consistent multiplicity. Badiou’s act of ontological subtraction may be seen as deflationary to any met­aphysics. It is certainly fair to judge it so from the perspective of the traditional goals of ontology. Yet, it is not deflationary of ontology insofar as the subtraction does not restrict ontology to consistent multiplicity. Instead, the distinction be­tween consistent and inconsistent multiplicity provides the central materials for analysis in L’Etre et l’événement. Ontology, in Badiou’s sense, has therefore the task of unfolding the unexpected relations between consistent and inconsistent multiplicity. More importantly, the fundamental aim of the book, the claim that truths and events exist, is the claim that, because of the incompleteness to any sufficiently strong set theory, there are sets that can be proven to exist but can­not be constructed (non-constructible sets). Hence, the inscription of Being into set theory through the count of the void will entail the unfolding, albeit incom­plete, of the inconsistent multiple within the structure of consistent multiplicity. From the point of the inscription of Being, qua inconsistent multiplicity, the op­eration of the count is crucial in distinguishing between ordered and structured presentation (sets in set theory), and inconsistent multiplicity (the inscribed void). In this way, structured presentation, what Badiou calls “consistent mul­tiplicity” is given by the ordered universe of pure sets. With the inscription of the count, further counts produce a structure of sets that are recursive counts of the inscription. We can illustrate this with the sequence of natural numbers (0,1,2,3, etc.). We can note that this counting structure maps succession order with size. How­ever, this correspondence between order and size will not be so obvious when it comes to counts that go beyond the finite numbers. What is conspicuous here is that Badiou’s ontology in L’Etre et l’événement relies on the count and there­fore ordinality. Cardinality plays a much less important role. Why does this dif­ference matter? It matters insofar as the count cannot be subordinated to the difference between the finite and the non-finite (or infinite). The oneness of the set is indifferent to whether the set is the void qua inconsistent multiple, a defined quantity (such as the keys on a qwerty keyboard), or the uncountably many points on a continuum. Badiou’s use of set theory to anchor mathematical ontology relies fundamentally on ordinality. We shall underline this claim and its stakes further. What is the difference between cardinality and ordinality? Now, the two concepts of cardinality and ordinality are not determined by the axioms of ZFC. The axioms indicate and restrict the kinds of sets that can exist. Cardinality and ordinality are instead structures we use to analyze these sets in terms of “how many” and “which one”, respectively, of a given set. In other words, they measure and count sets, respectively. Cardinality can be determined by matching or mapping (measurement) a set onto another pre-given set (i.e. numbers) or itself while ordinality is determined by ordering. Under normal fini­tary circumstances, cardinality and ordinality correspond neatly to each other, a set of five marbles has a cardinality of five because the five marbles can be mapped to a set of five entities in a pre-given set or to the first five natural num­bers. On the other hand, putting the marbles in order will get us to the “fifth” marble, thereby completing the ordinal count of the marbles.5 Given the set of finite natural numbers, which does not terminate, any arbitrarily large number will be finite and the ordinal that counts the sequence that allows us to arrive at that number will also tell us the size (cardinal) of the set that is given. The situation is slightly different for non-finite sets. For finite cases, we can arrange the natural numbers in order and map them to their subsets (e.g., the squares, cubes and fourths, etc.). This one-to-one map­ping is the structure of cardinality that allows us to see that these sets are the same size. Naturals n 1 2 3 4 5 … Squares n2 1 4 9 16 25 … There are ways to deliberately distinguish cardinality and ordinality in finite sets, exploit­ing the fact that ordinalities are pertinent to ordering while cardinalities are indifferent to order as we shall later examine. Cubes n3 1 8 36 64 125 … Fourths n4 1 16 81 256 625 … This is a principle known at least since Thabit ibn Qurra but better known through Galileo who puts forth this mapping in the first day of the Discours­es and Mathematical Demonstrations Relating to Two New Sciences.6 The point here is that, through mapping, we can clearly see that the measure of these dif­ferent sets of numbers are equivalent even though square, cubes, and fourths are subsets of the set of natural numbers. In other words, they share the same cardinality. Now it is possible to demonstrate the difference between ordinality and cardi­nality in finite sets as well. If we take two subsets of the natural numbers, say, odds and evens, we can make a new set of the union of the two. This maps to the natural numbers and thus allows us to claim equal cardinality. However, since we remain within the context of the finite, if we start with the odds, there will be no way to “reach” the evens. [1, 3, 5, 7, 9 … 2, 4, 6, 8…] Regardless of any assumptions about the infinite, there would be no determi­nate “which one” except for the odds since the ordering does not allow access to the evens in a finite number of steps (i.e. the order means that we would never access the evens since the first ellipsis implies a non-terminating sequence). In other words, we will not know where the number “2” arises in the sequence. This problem accentuates the conceptual difference between ordinality and car­dinality in order to underline why size and order are distinct structures even For a reconstruction of the “alternative” history of the infinite, involving figures likes Ibn Qurra, please see Mancosu’s 2016 essay collection. Paolo Mancosu, “Measuring the size of infinite collections of natural numbers: Was Cantor’s theory of infinite number inevi­table?”, Abstraction and Infinity, Oxford University Press, Oxford 2015, pp. 116–153. This paper was earlier published as Paolo Mancosu, “Measuring the size of infinite collections of natural numbers: Was Cantor’s theory of infinite number inevitable?”, Review of Sym­bolic Logic 2 (4/2009):612-646. For Galileo’s demonstration see Galileo Galilei, Discourses and Mathematical Demonstrations Relating to Two New Sciences, trans. H. Crew and A. de Salvio, Macmillan, New York 1914, pp. 31–37. within the finite cases. In non-finite cases, this difference is accentuated follow­ing a similar kind of reasoning. Without giving its demonstration, let us assume that there are non-finite cardi­ 0 as the size of the finite . nalities and ordinalities. For the cardinal, we designate ordinals collected as a set. This is not a finite number because if it were finite, it would be part of the set being measured. It is non-finite. It is transfinite and infinite in technical and common parlance respectively. However, we hesitate in using the term “infinite” because it is significantly different from traditional notions of infinity as the completion of a non-terminating sequence. By “tradi­tional” I mean here the wavering notions of the infinite as either the “potential” infinite of an unending sequence of successive finite terms, or the “number of numbers”, the “actual” totality of numbers qua termination of the succession of finite terms. Here, the traditional ideas concerning the infinite is indeed a family of ideas and intuitions that runs the spectrum from the negative imagination of the “very large” but only ever finite, to an infinite totality that, at the pain of con­tradiction, must invoke a transcendence over numbers themselves. Variations across this spectrum run the gamut in the history of the infinite since antiquity. The origins of set theory offers an alternative. This alternative breaks with the traditional dialectic of the infinite with the one: either the infinite is not one, and therefore a negative entity, or the infinite is a one, and therefore a totality. As neither a potential infinite nor a totality of finite terms, the modern infinite affirms the non-termination of the count and therefore its non-totality. By the time of the formalization of ZFC, this break with the traditional dialectic of the one and the infinite has already been accomplished by the innovations of Cantor and Dedekind. However, it is clarifying to see its canonical expression in the ZFC axioms. The axiom of infinity in ZFC specifically bars us from this archa­ic attachment of the infinite to the one. The axiom of infinity7 states: The axiom states that there is an infinite set in the sense that there is a set with O as its element, and also the successor of that set which is the union of that set Kenneth Kunen, Set Theory, revised edition, College Publications, London 2011, p. 17. and its subset. Therefore, there is at least one infinite (or non-finite) set in the sense that it collects the set of the null set and all its successors. This expresses infinity neither as termination nor even transcendence of a non-terminating se­quence. Rather it expresses a limit. Regardless of the size of the set offered, there is a set to which it and its successor belongs. While being one of two of the ZFC axioms that assert the existence of a particu­lar kind of set, the other one being the axiom of the void or null set, it does not assert that the infinite set terminates some non-terminating sequence (i.e. there is no “last” finite number). In the ZFC axiom of infinity, there is no “last one” that counts the series for the series to be measured. In fact, the very existence of a “last one” indicates a finite, rather than an infinite set according to this very axiom. Traditionally, the infinite could only be defined against the backdrop of its rela­tionship to the “one”. This dialectic of the one and the infinite pushed the tradi­tional notion to polarize between a potential infinite (an indefinite, non-termi­nating term), and a contradictory notion of a “number of numbers” that counts all the numbers. That is, either a negation of the one-total (hence potential), or the embrace of the one-total (hence contradictory). In this modern ZFC context, this dialectic fails to hold. There is neither a “greatest” finite number that counts all the finite, nor is there a completion of the non-terminating series of finite numbers. That is, it elides both the actual and potential infinity. The cardinal is instead a measure of that non-terminating series of finites. Hence, the nat-0. ural numbers, the squares, cubes, etc., are all measured by the same cardinal. returns the same cardinal, moreover, 0 . Adding a finite number to the cardinal adding squares, cubes, fourths (though there are some shared numbers), each also returns the same cardinal. This feature of the 0 . measuring the cardinal cardinal matches up with intuitions of inexhaustibility traditionally associated with the infinite, where the subtraction of a finite from an infinite returns an infinite, and the addition of an infinite returns an infinite. This was Galileo’s observation in the text related to the principle noted above: Salviati. So far as I see we can only infer the totality of all numbers in infinite, that the number of squares is infinite, and that the number of the roots is infinite; neither is the number of squares less than the totality of all the numbers, nor the latter greater than the former; and finally that the attributes “equal,” “greater,” and “less,” are not applicable to the infinite, but only to finite quantities.8 However, whereas Galileo presented this feature of the infinite as a kind of puz­zlement, Cantor and his faction took it up as a positive property of the infinite. Dedekind, the other founder of modern set theoretically based mathematics (ar­riving at results independently from Cantor), took the Galilean puzzle instead as the positive definition of the infinite. A system S is said to be infinite when it is similar to a proper part of itself; in the contrary case S is said to be a finite system […] My own realm of thoughts, the totality Sof things, which can be objects of my thought is infinite. For if ssignifies an element of S, then the thought s’, that s can be object of my thought, is itself an element of S.9 For Dedekind, the system S was his terminology for a set S, “similar” here means “equal in size”. The infinite set he denotes here is a set where the proper subset is equal in size (ie. “similar”) to itself. In turn, the set that Dedekind gives as an example is the set of his “thoughts”. Since every thought can be thought of, thoughts of thoughts provide a simple way to generate a subset that is equinu­merous to the original set. The thought of a thought is a thought which already belongs to the original set. Hence, any set that expresses this same feature can be considered infinite. What is crucial here is that the finite sets are determined negatively only as those that are contrary to this case. It is worth remarking that the finite here is the negative case of the prior definition of the infinite set. The ground has shifted from a constructive move from the finite to the infinite to that of a prior infinite multiplicity.10 151 Let us designate this shifting of ground by naming this kind of measurement as “Cantor-Dedekind measure”. This can be contrasted to “Euclidian measure” which designates that proper subsets must always be smaller than the origi­ 8 Galileo,Discourses and Mathematical Demonstrations Relating to Two New Sciences, pp. 32–33. 9 Richard Dedekind, Essays on the Theory of Numbers, Dover Publications, Mineola 1963, pp. 63–64. 10 Cf. Alain Badiou, Number and Numbers, trans. R. Mackay, Polity Press, Boston 2008, pp. 38–54. nal set. We recognize that Euclidian measure applies for finite sets. The set of odd numbers in a finite set will be strictly less than the original set (composed of elements other than odd numbers) because it is a proper subset. In turn, as Dedekind argues, the property of infinite sets is to have proper subsets that measure up to the original set.11 Ordering and transfinitude Now, we can shift focus away from measurement and turn to ordering. Ordering or ordinality identifies a very different structure. If given a set, say the empty set, we can always move up in order by counting that set to produce a new set that is a successor of that set. By doing this, we can model the ordering of numbers. Number Von Neumann ordinal 0={} O 1={0} {O} 2={0,1} {O, {O}} 3={0,1,2} {O, {O},{O, {O}}} … … The ellipsis at the bottom of the chart indicates that this constructive process of the ordinal “count” is non-terminating and therefore opens up into the transfi­nite sphere as a well. 11 Cantor argues roughly the same thing in Mitteilungen zur Lehre vom Transfiniten. Georg Cantor, “Mitteilungen zur Lehre vom Transfiniten”, Zeitschrift für Philosophie und philos­ophische Kritik 91 (1887–1888), pp. 81–125, 240–265. Quoted in Mancosu, Abstraction and Infinity, p. 132. This hierarchy produced by the ever-widening elements contained in each suc­cessor. The “V” shape is due to this. Hence, it is proper to ask the measure of each of these sets that identify the ordinality of the set. In this way, measure and order inform each other. For instance, the first number 0 contains no elements while the third number contains three elements constructed from previous counts of the set with no elements. However, the difference here is that ordinal-ity requires a correspondence to a well-ordering of the elements, a property in­dependent from measurement. Measurement, whether in the Cantor-Dedekind sense or in the Euclidian sense, requires no appeal to the “least” element. How­ever, for ordinality to make any sense, it must begin with a least element. In the example above drawn from the natural numbers, this least element is 0, denoted by the empty set. In other words, in order to “arrive” somewhere in the sequence, it must “begin” somewhere. This difference of structure means that the transfinite ordinals, denoted by . n, function in a different way to the . The main difference can be reflected by the properties n . transfinite cardinals of the traditional infinite mentioned earlier. If we add a finite to the infinite, we get back the infinite. This principle holds in the domain of cardinality. Adding something finite to a set with a transfinite measure does not change its measure. However, adding something finite, like 1, to a transfinite ordinal, can indeed make a difference. The point here is rather simple. The first transfinite ordinality is a border between two orders of succession. Whatever comes before it can be subsumed under it, whatever comes after it moves beyond the border. This is simply what follows from the property of well-ordering between orders. Hence, the addition of an element at the start of the sequence is not the same as the addition of an element at the end of the sequence. z+. = . < . + z Though ordinality and cardinality differ in structure, they are not opposed. They are tools that capture different structures in the analysis of sets, especially non-finite sets, which cannot be collapsed into one another. However, insofar as Badiou privileges the count over measure, it is ordinality, not cardinality, that is the guiding structure of his mathematical ontology in L’Etre et l’événement. The count is the count of succession rather than that of measurement. Hence, al­though Badiou accepts the entire machinery of ZFC set theory, the work in L’Etre et l’événement nonetheless selects different aspects of the theory to demonstrate certain ontological claims. What is clearly primary is the structure of order, the feature of ZFC that grants the “count” of the inconsistent multiplicity entry into the status of being an “ontology”. What is secondary is the analysis of their measure, which pertains only to the representations of this primary structure. The key indication of this is the obsolescence of the notion of “infinity” in the text. While we have already hinted at this, we shall turn to make this point more concrete in what follows. The obsolescence of infinity The aim of distinguishing between cardinality and ordinality as structures in the previous section aims at pluralizing the notions of the non-finite (or infinity) that results from the advent of the Cantorian revolution. The goal of this section is to draw out how this implies an obsolescence of the “infinite”. This will in turn highlight the pluralization of the “one” in Badiou’s ontological project in L’Etre et l’événement. What are the ontological implications for Badiou in distinguishing cardinality and ordinality? First, this distinction between measure and order indicates two different kinds of structure. Cardinality is operated by the one-to-one mapping of one set to another or itself. In non-finite cases, a set is equal in measure to at least one of its proper subsets. Second, ordinality requires a least element, the constructive hierarchy of sets admits to no termination and, more impor­tantly, is indifferent to the properties of traditional infinity. Badiou identifies the structure of ordinals (from finite to the transfinite), and not cardinals, as the arborescent domain of ontology. The parallel but independent structure of cardinality is not rejected but taken up in a secondary role. This point is best viewed through what I shall call the obsolescence of infinity at work in L’Etre et l’événement. The argument here is simple at first glance. Cardinality and ordinality deal with infinity in very different ways. A non-finite or transfinite cardinal is one where a proper subset is equal measure to the original set. A non-finite or transfinite ordinal marks the border between the finite and non-finite in a non-terminat­ing sequence. However, the addition of an element from the “left” of the sum returns the same ordinal but an addition to the right does not. This is also true of non-terminating sequences in the finite case. For ordinals, the “count” or the ordering of succession determines the order of transfinitude to which a given set corresponds. The count, as succession, is entirely indifferent to whether it is progressing in the finite or non-finite domain. There are two crucial implications here. First, the count is indifferent to the distinction of the finite and non- or trans- finite. The count, in Badiou’s sense, is not restricted to the countable (denumerable) sets. This means that such a distinction, between the denumerable and indenumerable is only a secondary analysis made upon the count. We shall examine this later on. The second im­plication is that the traditional dialectic between the one and the infinite loses all significance in this context. What we get in return is a theory of the “count” that, through “counting-as-one”, is neither finite nor infinite in the traditional sense. This installs a reckoning of ontological consistency subtracted from both the notion of the one-consistent as finite and the many-inconsistent as infinite. Badiou’s act of “subtraction of the one”, as he calls it, is not only a move to priv­ilege the multiple against “the one” but also to thoroughly withdraw the relation of the one and the infinite. We shall later examine what this obsolescence of the infinite means for infinity. Let us treat the first implication first. The traditional paradoxes of infinity con­cern the one and the many. The infinite arises in cases where the many cannot be reduced to the one or some rule of the one. Hence, traditionally, the infinite cannot be real because it cannot be reduced to a sum of entities (definite parts) that are actual. More importantly, it cannot be a totality except in a non-quan­titative sense. Therefore, appeals to actual infinity from Antiquity until the late modern period tend to be what the Scholastic medievals called “hypercategore­matical”. The hypercategorematical may be some ultimate Being (God) or abso­lute reality that outstrips the discursive resources of the very notion of quantity. Alternatively, to speak of an infinite or infinitesimal quantity in the early mod-155 ern period, theorists like Leibniz used a “syncategorematical” notion of infinity which corresponded to a “fictional” or “manner of speaking” to designate ob­jectivity to the infinite within a restricted domain of discourse (ie. a differential ratio). This allows for the engagement with the structures available through a commitment to the infinite or infinitesimal without a commitment to its reality.12 12 Richard T.W. Arthur has argued for Leibniz’s commitment to the actuality of syncategore­matical infinities. However, the point here is simply that the syncategorematical use of in­finity does not commit us to its actuality. Richard T.W. Arthur, “Leibniz’s Syncategorematic In brief, the primacy of the one is the reason why the infinite cannot be actual. The infinite is neither reducible to “ones” nor can the “ones” form a totality (a whole) in any quantitative sense. The implication to be drawn here is that the irreducibility of the infinite to the one is what forbids its actuality. In turn, the difference between finite and infinite is simply marked by this property of the quantitative reducibility to the one either as part or totality. The limits of this discourse to the reducibility to the one is precisely what is rejected in L’Etre et l’événement. This is of course what is meant by Badiou’s notion of the “subtrac­tion of the one”. However, this means the subtraction of the infinite as well. Since the concept of the infinite is grasped as the transcendence from the do­main of the one, the subtraction of the one implies the obsolescence of the in­finite as well. From this, Badiou’s ontological use of the count is also subtracted from the one and the infinite. The count-as-one is therefore the designation of an entity (i.e. a set) that does not fall into the traditional dialectic of the one and the infinite (many). The further implication is that Badiou’s count does not re­spect the border between the quantities made up of ones (the realm of finitude) and the counts of non-finite multiplicities. This indifference of the count to the finite and non-finite is subject to possible misunderstandings. If the count is understood under the aegis of the traditional relation between the one and the infinite, the distinction between the countable and uncountable (denumerable and indenumerable) quantities would present a challenge to Badiou’s project. If this were the case, the count could not be universally or generically applied across finite and non-finite cases. That is, the count would itself be subject to the dialectic of the one and the infinite. Instead what Badiou introduces with the count is the creation of the border between the consistent and the inconsistent. The count is therefore orthogonal to the distinc­tion between the finite and non-finite (i.e. the transfinite) and produces, rather than be produced by, the distinction between the consistent and inconsistent. Actual Infinite”, in O. Nachtomy, R. Winegar (eds.), Infinity in Early Modern Philosophy, Springer, Cham 2018, pp. 155–179. Richard T.W. Arthur, “Leibniz’s syncategorematic infin­itesimals”, Archive for History of Exact Sciences 67 (5/2013), pp. 553–593. David Rabouin and Richard T.W. Arthur, “Leibniz’s syncategorematic infinitesimals II: their existence, their use and their role in the justification of the differential calculus”, Archive for History of Exact Sciences 74 (5/2020), pp. 401–443. What this implies is that the count is the border between inconsistency and con­sistency, rather than the distinction between the finite and non-finite. This fact indicates that the count privileges ordinality as its primary functional role. Just to be explicit, if, for the sake of argument, we take Badiou’s count to be cardinal measure, we would have to apply some measure to the inconsistent multiple. This is impossible since the inconsistent multiple cannot be measured unless it is first counted-as-one. As such, the inconsistent multiple would have the car­dinality of the void. The void stands as what is “before” any count and there­fore any measure. Insofar as the inconsistent is neither finite nor non-finite, this does not identify any relation between the finite and non-finite.13 The ordinal structure through which the count functions, on the other hand, operates on succession. For the domain of pure sets, the count, outside of the “first” count of the void, operates only by the recursive counting of the prede­cessor. Hence, nothing about the inconsistent multiple has to be assumed for the count to function. As such, the inconsistent multiple can be taken as a void or really any other kind of ground suitable for an ur-set. There is hence no need for the assumption of any distinction between the finitary status of the count it­self. Further, as the count moves up the hierarchy of successors, the count itself does not distinguish between finite and transfinite. That is, while it is true that .0 must be handled in a different way than a finite ordinal (i.e. it does not com­mute), the count itself, the process of succession operates indifferently between the finite and infinite. From this we must assert the fact that the count-as-one does not distinguish between the finite and infinite. This is crucial because the basis of Badiou’s on­tological project in the L’Etre et l’événement does not begin from finite cases that slowly build towards transfinite cases. The count-as-one does not imply finitude 157 or denumerability for either what is counted or what results from the count. Most importantly, it does not imply the difference between denumerability and indenumerability. 13 It should be noted that Badiou has, in L’Immanence des vérités (2018) introduced another approach to inconsistent multiplicity. Here, he uses the Von Neumann universe of sets (V), which is not itself a set but a class, to handle the relation between inconsistent multiplicity and its relationship to sets. The later work certainly implies a more committed relationship to the positivity to the infinite. I reserve my analysis only to the structure worked out in L’Etre et l’événement. Alain Badiou, L’Immanence des vérités, Fayard, Paris 2018. This indifference of the count to denumerability and indenumerability of the one lead us to the second implication here, the obsolescence of the dialectic between the one and the infinite. The traditional dialectic rejects any “actual” or “determinate” infinity because it cannot be made a one-totality. In its tradition­al Pre-Cantorian form, the infinite can only be hypercategorematical, a “one” transcending quantity, or syncategorematical, a stipulated infinity (ie. a fiction) based on a restricted domain. Of course, from this traditional perspective, the post-Cantorian transfinite would not qualify as “infinite”. However, the trans­finite can be considered “infinite” in the sense that the modern view remains the (very useful) distinction between the denumerable and the indenumerable. 0 identify the measure and count of .0 and the ordinal . Insofar as the cardinal the denumerables (respectively), thereby forming the limit between the denu­merable and indenumerable, the transfinites can be considered “infinite” in a meaningful way. This is however gained only by correlating the transfinites with some pre-given (and familiar) sequence of numbers (the rationals vs. the reals) and with geometrical properties (e.g., the continuum). These are legitimate ap­plications of ZFC but correlations of this sort are not intrinsic to it. The axiom of infinity describes a limit through the operator of belonging and the principle of succession. Denumerability and non-denumerability are applications of this construction. The main implication here for the purposes of understanding Badiou’s count-as-one from the axiom of infinity is that the oneness of any set is granted regard­less of its finiteness or non-finiteness. From this, unity and infinity do not form a determinate negation. Transfinitude or the form of non-termination is a robust form of the count-as-one rather than its exception. Given that the Cantor-Dedekind definition of infinity is encoded in a less quan­titative way in the axiom of infinity in ZFC, the analysis of sets moves away from a reliance on number fields as a form of correlation. Hence, the move away from the traditional dialectic between unity and infinity does not take place in the form of a mere rejection of this dialectic but rather as an orthogonal side-step­ping of this relation. Within the framework, the one and the infinite are not opposed. Instead, transfinite orders are successive orders of limits which cor­respond neither to the concept of the one nor the infinite (in the traditional sense). We hence assert the obsolescence of infinity through its replacement by a non-terminable but well-founded hierarchy of successive sets. If we take this view retrospectively back into the history of mathematics, it seems that this obsolescence of infinity does not really come at any cost to math­ematics. The idea of an absolute infinite never had any real mathematical con­tent (even while sustaining a metaphysical and theological importance). The infinite either remained thoroughly potential or had to be, for the most part, fictionalized.14 Instead, where this obsolescence of infinity matters most is in the transformed status of the “one”. It is unfortunate that the Cantorian trans­formation of mathematics is sometimes reduced to a reform of the concept of the infinite and a declaration of its actuality. The “one”, through the count qua succession is formally retained. However, the one no longer plays the role of the distinguishing mark of the finite, and thus no longer separates the finite from and non-finite. Instead it plays the role of marking consistent multiplicity across these traditional distinctions. With this subterranean transformation of the “one”, we make a more thorough reckoning with this guiding concept of the “subtraction of the one”. That is, as Badiou argues elsewhere, the act of subtraction is characterized by the retaining of the positivity of a negation.15 In the case of ontology, the subtraction of the one is not a rejection but rather the retaining of its positive role in ontology through the rejection of its traditional dialectic with the infinite. This notion, in parallel to the obsolescence of the infinite does play an important mathematical role. The most crucial of these is the availability of the transfinite ordinals, the use of unity without totality in treating non-finite terms. What this implies is the pluralization of the notion of the “one”. We shall examine this further on. Cardinality as representation Before turning to the pluralization of the one, we turn briefly to the positive role 159 played by cardinality in L’Etre et l’événement. If, as we have seen, the ontological project in L’Etre et l’événement proceeds through the structure carved out by ordinality, it is also important to indicate what remains of cardinality in the project. As has been emphasized multiple 14 Mancosu points to some exceptions to this mainstream history of the infinite. Cf. Manco­ su, “Measuring the size of infinite collections of natural numbers: Was Cantor’s theory of infinite number inevitable?”, pp. 117–119. 15 See Alain Badiou, “Destruction, Negation, Subtraction – On Pier Paolo Pasolini”, Lacan­ ian Ink, https://www.lacan.com/badpas.htm, accessed Aug. 2020, Los Angeles 2007. times, Badiou does not reject cardinality, but treats it as a secondary structure. In fact, for Badiou the structure of cardinality is appropriate for the analysis of representation. For L’Etre et l’événement, presentation and representation differ in their grafting onto two kinds of operations we can make on sets. Presentation is associated with “belonging” (.). Therefore, whatever is presented in a situation are the elements of the set.16 Representation, on the other hand, is associated with “in­clusion” (either as subset . or proper subset .). This inclusion is an operation that recognizes subsets of the situation (situation qua set). Of course, all the elements of a set can be considered as what “is included” in the set but the subsets also involve all possible combination of those elements. This concept of representation involves mere subsets up to the set of all the subsets of the situation. This is the set of all subsets produced by the operation of inclusion is the powerset. This takes us to Cantor’s theorem. The theorem states that given a set A, its pow-erset P(A) is of a strictly higher cardinality than A. For finite cases, this is obvi­ous. A set of three elements {x, y, z} will have subsets {O}, {x},{y},{z}, {x,y},{x­,z},{y,z}, {x,y,z}. Hence, a set of three elements, a cardinality of 3, will have a powerset made up of 8 sets, hence a cardinality of 8. The basic reckoning here is that the powerset will have a cardinality of 2n, where n is the cardinality of the original set. For non-finite sets, Cantor argued, the same follows. Hence the is therefore also of a strictly greater cardinal. The implication here is that there are cardinalities high­er than the set of all infinite numbers that can be indicated by the application of the powerset operation P(A). Furthermore, for any set, finite or nonfinite, the 160 application of the powerset will render a set of strictly higher cardinality. This feature of the main development in set theory is interpreted by Badiou as the difference between presentation and representation. Badiou refers to the powerset as the “state”. One could find reasons to quarrel about this description Although Badiou credits Lyotard for the terminology of “situation”, he cites Barwise and Perry on treating a set as a situation. Cf. Jon Barwise, “Situations, Sets and the Axiom of Foundation”, Logic Colloquium 1984, J.B. Paris, A.J. Wilkie, G.M. Wilmers (eds.), Elsevier Science Publishers, Amsterdam 1986, pp. 21–36. Cited in Badiou, Being and Event, p. 484. Thanks to Julian Rohrhuber who carefully pointed out this passage recently. 0 . . This powerset of 0. , is 2 0 . powerset of all finite numbers, of the state but we take it up here only as a technical term. Given a situation, there is a state of the situation. This is the difference between a set and its pow-erset. This also lines up with presentation and representation. On this account, Badiou offers two analyses. The first is that given the operation of the power-set, the sequence of cardinal numbers is interminable. The second is that the general application of the powerset results in an outstripping, not only of the quantity of the sets involved in the operation, but the outstripping of the scale of measurement. That is, insofar as one moves upwards in cardinality by the pow-erset of all the sets in that cardinality, each new powerset installs a new scale of quantity (the hierarchy of transfinites) that is irreducible to the “how many” of the lower scale. Here, Badiou notes that, “the natural measuring scale for mul­tiple-presentations is not appropriate for representations. It is not appropriate for them, despite the fact that they are certainly located upon it. The problem is, they are unlocalizable upon it.”17 The point here is that for non-denumerable sets (non-finite cases) cardinality locates lower sets within them but cannot lo­calize them due to the expansion in cardinality. Reinterpreting Galileo’s demon­ . 0 . stration, all sets of natural and rational numbers are of the same cardinality In the same way, any mapping of transfinite cardinalities to their powersets will result in this same “unlocalizability”. The distinctions of a lower dimension are “lost” in the higher dimension. The “state” of a situation, whose constituents are subsets of the situation is therefore always at a higher cardinality than the set of the constituents itself. The resources of cardinality are employed in L’Etre et l’événement. However, they are used in order to address the difference between presentation (governed by belonging), and representation (governed by inclusion). This gap, for Badi­ou highlights what he calls the “impasse of ontology”. This is undergirded by of 0 . Easton’s theorem which roughly states that the cardinality of the powerset 2 0. .0. is arbitrarily greater than 0 . the given powerset of a cardinal . Badiou puts it in the following way: 0 . than 18 Essentially, this means that the cardinality of and greater is any cardinal arbitrarily greater 17 Badiou, Being and Event, p. 278. 18 A few conditions apply here. Crucially, the set in question is regular (ie., obeying the axi­om of regularity/foundation). This theorem roughly says the following: given a cardinal ., which is either .0 or a successor cardinal, it is coherent with the Ideas of the multiple to choose, as the value of -p(.)- ¦ that is, as quantity for the state whose situation is the multiple ¦ any cardinal ., provided that it is superior to . and that it is a successor cardinal.19 Easton’s theorem here is a deepening of the Galilean “paradox”. It turns out that the immeasurability of quantities beyond the finite is a general condition that goes beyond the gap between the denumerable and the indenumerable. This is what Badiou will call the “quasi-total errancy”. The scales of cardinal infinites interplay with an uncontrollable degree of arbitrariness. To what degree is this an impasse of ontology then? Badiou argues that, Consequently, Easton’s theorem establishes the quasi-total errancy of the excess of the state over the situation. It is as though, between the structure in which the immediacy of belonging is delivered, and the metastructure which counts as one the parts and regulates the inclusions, a chasm opens, whose filling in depends solely upon a conceptless choice.20 The “quasi-total errancy” that Badiou refers to here is precisely the role of car­dinality in L’Etre et l’événement. That is, given well-ordered sets and consistent multiplicities beyond the finites, the application of cardinality reveals a field of arbitrariness. In other words, all questions of quantity and measure fall into a state of underdetermination where measure is stipulated (chosen) rather than deduced. From this Badiou offers an interpretation of this arbitrariness of the domain of the infinite consistent with our pluralist argument: Being, as pronounceable, is unfaithful to itself, to the point that it is no longer possible to deduce the value, in infinite extension, of the care put into every pres­entation in the counting as one of its parts. The un-measure of the state causes an errancy in quantity on the part of the very instance from which we expect-ed-precisely-the guarantee and fixity of situations. The operator of the banish­ment of the void: we find it here letting the void reappear at the very jointure be­tween itself (the capture of parts) and the situation. That it is necessary to tolerate Badiou, Being and Event, p. 279. 20 Ibid., p. 280. the almost complete arbitrariness of a choice, that quantity, the very paradigm of objectivity, leads to pure subjectivity; such is what I would willingly call the Cantor-Gödel-Cohen-Easton symptom. Ontology unveils in its impasse a point at which thought-unconscious that it is being itself which convokes it therein-has always had to divide itself.21 The pluralization of the “one” Through examining the difference between cardinality and ordinality, we saw that Badiou’s count-as-one is expressed in a transformed concept of the “one”. It no longer forms part of the dialectic between the one and the infinite, where the infinite, if it is in any sense actual, must be presented as a totality. What results is a pluralization of the “one”. The argument so far has been the following. The Cantor-Dedekind revolution, while traditionally interpreted as the actualization of infinite sums, should in­stead be understood as the obsolescence of infinity. The key reason for this is that the “one” as repetition and totality, within this new domain, ceases to play the role of distinguishing between the finite and the infinite. From a cardinality perspective, although denumerability and non-denumerability sustains the tra­ditional border between the finite and infinite as measure, the “infinite” in this sense fails to correspond to the traditional notions of the infinite. Even if we take the infinite to be non-denumerability, we see the problems of the indefinite re­produced via Easton’s theorem. Hence the “infinite” in this case, or the realm of the “infinite”, fails to be “actual” if we understand this actuality as conditioned, at least, by determination. If we take this question from the perspective of ordi­nality, we do recover the border between denumerability and indenumerability. The transfinite ordinal as a limit ordinal is logically prior to transfinite cardinal-163 ities.22 But here, as we have argued, the structure of ordinality does not require 21 Badiou, Being and Event, p. 280. 22 It should be noted here that Cantor saw ordinality and its well-ordering to be more funda­mental than cardinality. Georg Cantor, “Über unendliche, lineare Punktmannichfaltigkeit-en”, in E. Zermelo (ed.), Gesammelte Abhandlungen mathematischen und philosophischen Inhalts. Mit erläuternden Anmerkungen sowie mit Ergänzungen aus dem Briefwechsel, J. Springer, Berlin 1936; reprinted Olms, Hildesheim 1966, pp. 165–209, p. 169. Although the advent of the cardinal approach to infinity was historically prior, the priority of the ordinal is asserted from a logical perspective. Cf. Hans Niels Jahnke, “Cantor’s cardinal and ordi­ the infinite in order to stipulate a transfinite limit ordinal. In turn the primacy of ordinality in Badiou’s ontological project in L’Etre et l’événement, based on the count-as-one ordinality, renders infinity an obsolete concept for ontology. The claim here is to focus instead on the pluralization of the “one”. The pluralization of the notion of the “one” is simply rendered by the notion of the set. The set is a “one” in that it is counted in the ordinal sense. It therefore matters that this count is not confused with measurement. Sets that are built out of the ZFC axioms are placed within a family of branching sets that fulfils the condition of well-ordering.23 The basic structure that this ordinality analyses is therefore the structure of succession (i.e. for any set, say a pure set, there is a successor set that counts that set). However, this only gets at the structure and not the existence of sets, the notion of a “one” produced by counting (succes­sion). What then accounts for this parallel structure of existence? For the project in L’Etre et l’événement, Badiou asserts throughout that it is the count that produces the one, eschewing the notion of the “one” as a given. Hence it is the count, an act of counting, that introduces the existence at the ba­sis of the analysis. To stave off misunderstanding, mathematical ontology does not go around counting what exists. Instead it is a rational and logically classi­cal analysis of the result of the act of the count, the unfolding relation between inconsistent multiplicity, what is counted as void, and consistent multiplicity, the aggregation of counts following on the count of the void and its successors (ordinals). What constitutes the grounds of this ontological project theorized by the two “seals” [sceaux] that Badiou introduces in Meditation 14 of L’Etre et l’événement. Badiou’s approach pluralizes the “one” treated as the product of the count and thereby concretizes the subtractive aspect of the ontology by revising its traditional basis. The two “existential seals” occur as limits. The first limit is that which seals off inconsistent multiplicity from consistent multiplicity. Hence, this is the count of the void. Whatever is or is in inconsistent multiplicity is excluded from the nal infinities: An epistemological and didactic view”, Educational Studies in Mathematics (48/2001), pp. 175–197. 23 Note that while not all sets per se can be well-ordered, in ZFC, Zermelo’s theorem shows the equivalence between the axiom of choice and the well-ordering theorem. domain of set theoretical entities by inscribing the inconsistent as the void of the consistent structure carved out by sets in their ordinal arrangement. From this, as we have already discussed, sets branch off in ordinals by counting the void. This results in a well-ordered tree of sets grounded in this first seal. Now, the second seal is the limit between the finite ordinals, the first transfinite ordi­nal, and the transfinite orders that follow after the limit as successors. Since the structure of ordinality is constructed by succession, nothing within this struc­ture offers a distinction between the finite and transfinite. The first tranfinite ordinal is stipulated rather than given from the background structure. This bor­der between the finite and transfinite is thus the limit of the finite series and the start of the transfinite series, a second existential seal. We have discussed the first seal previously and thus set it aside. The second existential seal has also been briefly addressed. However, it is important here to underline that although, under a traditional reading, this limit ordinal intro­duces an “actual” infinite in the practice of set theory, what it actually provides, in Badiou’s reading, is a further application of the subtraction of the one. Here, what is crucial is that the transfinite ordinal is not a new “one” that counts the finite series in completeness. Rather, it is an inscription of a consistent multi­plicity beyond finitude. In this sense, the infinite and the transfinite limit do not coincide. As Badiou argues: [W]e have not yet defined infinity. A limit ordinal exists; that much is given. Even so, we cannot make the concept of infinity and that of a limit ordinal coincide; consequently, nor can we identify the concept of finitude with that of a successor ordinal. If . is a limit ordinal, then S(.), its successor, is ‘larger’ than it, since .Î S(.). This finite successor — if we pose the equation successor=finite — would therefore be larger than its infinite predecessor — if we pose that limit = infinite — however, this is unacceptable for thought, and it suppresses the irreversibility of the ‘passage to infinity’. If the decision concerning the infinity of natural being does bear upon the limit ordinal, then the definition supported by this decision is necessarily quite different. A further proof that the real, which is to say the obstacle, of thought is rarely that of finding a correct definition; the latter rather follows from the singular and eccentric point at which it became necessary to wa­ger upon sense, even when its direct link to the initial problem was not apparent. The law of the hazardous detour thereby summons the subject to a strictly incal­culable distance from its object. This is why there is no Method.24 Badiou moves on from this to define infinity according to this limit ordinal. Any set where the limit ordinal belongs is infinite, any set belonging to the limit is finite. The ontological significance here is not captured by this distinction. What this second seal indicates is that it replays the distinction between the inconsist­ent and consistent multiples by treating the finites as the inconsistent (because interminable) multiples. That is, just as nothing within consistent finitude can generate the limit transfinite ordinal, nothing in inconsistent multiplicity can generate the count-as-one of the void. This leads us to the key distinction of our investigation. Here Badiou argues, In the order of existence the finite is primary, since our initial existent is O, from which we draw {O}, S{O}, etc., all of them ‘finite’. However, in the order of the concept, the finite is secondary. It is solely under the retroactive effect of the existence of the limit ordinal .0 that we qualify the sets O, {O}, etc., as finite; otherwise, the latter would have no other attribute than that of being existent one-multiples.25 What Badiou goes on to develop here, at the end of Meditation 14, is the concep­tual effect of the Cantorian revolution. This is the distinction between existence and concept. According to the criterion of the order of existence, there is no relation between the finite and infinite, since, qua ordinals, the sets succeed each other. If {.} is finite, so is its successor S{.}. It is only with the limit ordinal .0 that the limit between the finite and non-finite is marked. Yet, as Badiou ar­gues, this limit is stipulated, and it is only through this stipulation that the finite and infinite can be distinguished. As Badiou argues here in the passage above, what this reveals is not so much a realm beyond the finite, that is, the infinite, but rather a region of the infinite classified as the “finite”. This second seal is therefore a repetition of the first seal. Just as the inconsistent multiple can be recognized as inconsistent only after the first seal which designates consistent multiplicity, finitude can only be recognized (and defined) by the designation of the transfinite limit ordinal. 24 Badiou, Being and Event, p. 157. Ibid., p. 159. Badiou’s argument here reflects onto our earlier discussion of Dedekind’s argu­ment for the non-finite by means of infinite cardinality. In both cases, the finite is treated as a special case for the ordinary “infinitude” of sets. Hence, finitude is only available as a concept with respect to the definition of the infinite. Here, we can quibble with Badiou on the elaboration of this point. Just as the inconsistent multiple is not inconsistent before the forming of consistent multiplicity, the existence of sets and their successors are not finite before the transfinite limit ordinal. It is therefore incorrect to treat the finitude as primary in the order of existence. The crucial difference here is that the count, the structure analyzed by ordinality, is fundamentally indifferent to the distinction between finite and infinite whereas cardinality is fundamentally sensitive to this difference. Badiou here reinforces the subtraction of the one by drawing out the further implication that the one, the counting-as-one (and forming-into-one) as a result of an act of counting rather than a given. The one, the concept through which being must be analyzed, will always be alien to what is analyzed. This holds in the case of the original count (the count of the void) as well as within consistent multiplicity. That is, the one, within the post-Cantorian context, is released from its association with finitude. The forming-into-one of a set is shown to be finite only in specific contexts but generically non-finite. The limit ordinal demon­strates this in Badiou’s interpretation because it shows finitude to itself be a result of a “one” that is formally outside of it and thus capable of “counting” it (qua ordinal) and standing as a limit, a “one” that stands beyond the “ones” that populate finitude. The new “one” within the post-Cantorian context is thus completely trans­formed. Another way of saying the same thing is to reaffirm that the dialectical relationship between the one and the infinite no longer holds. The one is no longer distinguished from the infinite but rather part and parcel of it. Just as the infinite is no longer the “beyond” of finitude, the one is no longer the obstacle of infinity. The reasoning here concerning the one indicates a deep rift between treating ontology by means of ordinality or cardinality. Cardinality relies on the one-to-one mapping between sets (or a set to itself ) and therefore relies on a traditional notion of the one as a pre-given unit-entity. Here, the infinite or the transfinite involves the equal cardinality of wholes and parts, a suspension of the identity between oneness and wholeness (or totality). Hence despite the general coher­ence in set theory between ordinal and cardinal treatments of the transfinite, Badiou’s emphasis on ordinality, via the structure of the count, implies the ac­commodation of different means of reckoning the measure (i.e. cardinality) of any given set. Hence Badiou’s ontological interpretation of ordinality and the limit ordinal implies a pluralization of the one in the sense that the ordinal structure can accommodate different theories of cardinality. In what follows, we will examine an alternative non-Cantorian treatment of car­dinality and revisit the primacy of ordinality in the mathematical ontology of L’Etre et l’événement. What we aim to underline is the pluralization of the “one” as the consequence of the ontological project of L’Etre et l’événement. Numerosity and ordinality If we go back to the traditional problems of infinity and the Cantorian revolu­tion, we find that the hallmark of the new “infinity” (in cardinal terms) was the conceptual break with Euclidian measurement. Recall that for Dedekind, an infinitely sized set is one where at least one subset is equal in size to the original set. Cantor’s transfinite cardinal also has this property. Also recall that this is logically independent of transfinite ordinality, a question that is relevant but not determinant in questions of measurement per se. However, an alterna­tive approach to infinity has always existed in the sub-currents of mathematics and its philosophical expositors even in the Cantorian age. This is an alternative approach that rejects the relinquishing of traditional Euclidian measurement. Hence, to deepen our inquiry into the pluralization of the one, we should exam­ine the recent resurgence of this Euclidian approach. The refusal to relinquish Euclidian part-whole relations is to express a commitment to the notion of the oneness as wholeness. Its friction with the canonical Cantorian view will allow us to grasp what the pluralism of one offers us. In what its proponents call “numerosity theory”, the Euclidian principle, which maintains that the part is always lesser than the whole, is maintained. Although Bolzano, a senior contemporary of Cantor and Dedekind, developed some fea­tures of a concept of the infinite that maintains the Euclidian principle in his 1851 Paradoxes of the Infinite[Paradoxien des Unendlichen], numerosity theory is a distinct recent development.26 This emerged in a series of papers since 2003 by Benci and Di Nasso around “Numerosities of labelled sets: a new way of count­ing.”27 Since 2003, the research program has grown significantly to include sev­eral aspects of mathematics including probability. It has also sparked harsh criticism in the philosophy of mathematics notably by Parker.28 Significantly, it has also been taken up philosophically and historically by Mancosu, moving beyond the narrow domain of non-standard analysis and set theory. Numerosity theory, from Benci and Di Nasso’s 2003 paper, extends the Euclidi-an principle standard from finite cases to infinite cases. Citing from Mancosu’s reconstruction of the paper29, we take Benci and Di Nasso’s theory as operating from the maintenance of three principles: 1. if there is a bijection between A and B then v (A) = v (B) 2. if A . B then v (A) < v (B) 3. If v (A) = v (A’) and v (B) = v (B’) then the corresponding disjoint unions (.) and cartesian products (x) satisfy: v ( A . B) = v(A’ . B’) v (A x B) = v ( A’ x B’) The first is simply the definition of equivalence from bijection, a standard prin­ciple of cardinality. The third is a definition of sums and products in numerosity also standard to sets of this kind. What is distinctive is the second principle that maintains the strictly “lesser than” difference between a set and its proper sub­set. This is what is under contention. 169 26 Bernard Bolzano, Paradoxien des Unendlichen, C. H. Reclam, Leipzig 1851. 27 Vieri Benci and Mauro Di Nasso, “Numerosities of labeled sets: A new way of counting”, Advances in Mathematics, 173(2003), pp. 50–67. Cf. V. Benci, M. Di Nasso, and Marco For-ti, “An Aristotelian notion of size”, AnnalsofPureandAppliedLogic, 143 (1–3/2006), pp. 43–53. 28 Matthew Parker, “Set size and part-whole principle”, The Review of Symbolic Logic, 6 (2013), pp. 589–612. 29 Mancosu, “Measuring the size of infinite collections of natural numbers: Was Cantor’s theory of infinite number inevitable?”, p. 139–145. How does numerosity work? Intuitively, what the theory does is provide labels to sets such that subsets can be indexed (labelled) together in finitary way. Man-cosu uses, as an illustration, the Italian game of Tombola, a version of Bingo. On a master board with numbers 1–90, pegs are placed when a number is drawn from a well at random. If a called number corresponds to a number on one’s own board (an arbitrary rearrangement of 1–90), one pins the number down. The goal is to cover the board with pegs corresponding to the master board. In the middle of the game, how does one check one’s progress in the game? There are three options. First, one could ignore the order of the numbers and simply check if the num­ber of pegs on one’s board corresponds with the number of pegs on the master board. This bijective function would be equivalent to the Cantorian approach to cardinality. Second, one could list the pegs in order such that the numbers 1-90 are listed in order. This corresponds to an ordinal approach. Finally, one could label the pegs by designating the numbers by decades: 1-10, 11-20, 21-30, etc. Each label would be partial, given we are in the middle of the game. Under practical circumstances, this would indeed be an odd way to see how one is pro­gressing in the game of Tombola. Nonetheless, what this does is the breakdown of a potentially linear sequence into incomplete partial sums. The third approach is thus an intuitive image of numerosity. The goal, as Manco­su argues, “is to split a set of objects into boxes each one containing only finitely many objects. The metaphor of putting things in box number 10, 20, and so forth will be captured by the idea of a labelled set. From now on we deal only with countable sets.”30 A potentially confusing point here is the fact that in a game, each of the boxes (ie. labels), will be incomplete as we have not finished the game. The point is that each of these partial sums constitute an approximation of the cardinality marked by each label. The whole sum will be the sequence of these approxima­tions. The formal definition of numerosity will then rely on how one is to operate these sequences of approximation. Here, I will again cite Mancosu’s simplified outline of a “calculus” for numerosities. 30 Ibid., p. 139. Def. 3.1 The sum of two labeled sets A, B is A . B = where lA . lB (x) = lA (x) if x is in A and lB(x) if x is in B. [Caveat: take disjoint unions of A and B only] Def. 3.2 The product of two labeled sets A, B is A . B = where lA . lB (x,y) = max { lA (x) ; lB(y)}. Def. 4 Definition of numerosity A numerosity function for the class L of all countable labeled sets is a map num: L -> N onto a linearly ordered set < N, . > such that the following properties are satisfied31: (1) If #An .#Bn for all n, then num(A) . (B) (2) x>> je signifie : « est égal ultimement a », la succession dans (.( des temps 1 entre les chiffres, s’écrit : 1+ 1+ 1+ ·· · + 1+ ·· · =>>> . chaque somme partielle finie devient toujours plus grande que tout nombre fixé d’avance, et les nombres on n’en finit pas de les dire ; tandis que la succession des intervalles en lesquels (AB(, de longueur 1, est vu Ac2-1, c2-1c2-2, ..., c2-n, c2-(n+1), ..., recouvrent (AB( les longueurs correspondantes donnant l’écriture : 1/2 + 1/4 + 1/8 + ··· + 1/2n + ··· =>>> 1. Pour chaque n on a c2-n = 1 - 2-n. Et quoique . ne figure jamais dans la ligne infinie discrete (.(, sa « représentation » au niveau de (AB(, continu, se voit, c’est-a-dire le point B = c2-. = 1 - 2-. = 1, avec donc l’écriture 1 2-. = = 0. 2. Autour de quoi s’arrangent les paradoxes de Zénon. Si l'on compte les étapes, évidemment il n'y en a pas de derniere, celle ou enfin Achille atteindrait la tortue, et donc, jamais, a aucun instant, Achille ne la rattrape. Entre les deux bouts A et B, se trouve logé l'infini dénombrable des étapes marquées, avec tou­jours des trous. Quelques milliers d'années plus tard, Georg Cantor démontrera qu'en effet on ne peut couvrir toutes les positions possibles lors d'un compte, c'est-a-dire qu'il n'existe pas de surjection de N sur (AB(. 3.3 Pour une histoire du2 Cette dualité du discret et du continu est fondatrice de celle du comptage arith­métique (dire) des nombres, et de la découpe géométrique (voir) des grandeurs, de l’arithmétique et du géométrique, et partant, quand la littéralisation portera l’arithmétique, avant la géométrie, vers l’algebre, fondatrice de la dualité entre l’arithmétique et la géométrie, et partant — de guingois en quelque sorte — entre l’algebre elle-meme et la géométrie. Ainsi donc, apres Descartes, la géométrie prenant l’algebre a son service, la dualité trouve a s’exprimer de façon interne a l’algebre ainsi engrossée. Main­tenant, de nos jours, nous avons la dualité entre corpuscules et ondes, entre le transport et la force, les vecteurs et les covecteurs, les espaces vectoriels et les espaces duaux, etc. L’histoire des mathématiques — semble-t-il — est largement celle des métamorphoses de la dualité. Nous avons aussi la dualité entre les notions de groupes et d’espaces, subsumée par l’unification de ces notions en celle de topos par Alexandre Grothendieck, mise au travail par la topologie et la géométrie algébrique au XXe siecle. Ou encore l'accouplement de ces notions sous la forme de la notion d'espace fibré ou de groupoide topologique, a la façon de Charles Ehresmann. Un mode d’apparaître élémentaire de la dualité est la dualité dite de Boole, entre les algebres de Boole (côté logique ou discours des raisons) et les espaces booléens (côté géométrie ou topologie, discours des lieux), et le fait que l’en­semble P(P(E)) dont chaque élément est une partie de l’ensemble des parties de l’ensembleE, est l’algebre de Boole atomique complete associée a l’ensemble E. Du reste, cette dualité est intimement liée a celle ci-dessus actée par la notion de topos, puisque l’essentiel d’un topos est qu'au plan logique, il comporte un analogue de la construction P. L’idée que j’ai proposée a partir de 1970 est de fonder le concept d’univers (au­trement dit l’idée de lieu de déploiement d’une activité mathématique complete) sur la donnée axiomatique d’une dualité, de cette dualité de Boole spéciale­ment, et plus spécialement encore sur ce qui l’engendre fonctionnellement, a savoir la dualité entre une catégorie d’ensemble EnsUet la catégorie duale EnsopU, déterminée concretement par le cas du foncteur contravariant P* in-troduit par Georg Cantor, du moins sur les objets que sont les ensembles, en définissant P*(E) comme l’ensemble des parties de E ; foncteur dont il s’agissait alors de déterminer une axiomatique, laquelle j’ai nommée d’abord catégorie cantoriennepuis univers algébrique. Le point crucial était que, sur une catégorie donnée l’axiomatique ne soit pas complete, au sens ou la catégorie ne détermine pas uniquement le P covariant, ou le P* contravariant, mais soit complete au sens ou l'opérateur P ou P*et ses transformations permettent de définir tout ce que la logique du premier ordre permet de définir, en quelque sorte « sous condition » deP*, dans la catégorie donnée. Il faut que la production des structures ensemblistes y soit répétable de façons variées. J’y reviens ci-apres, mais pour l’instant, soulignons que, avec O = 0et {O} = 1, on a la « construction » des entiers {O,{O}} = {0,1} = 2, {0,1,2} = 3, etc., n .{n} = n + 1. On a aussi P(E) = 2E, soit l’ensemble des fonctions de E vers 2, et alors un théoreme de Cantor : 2E > E La preuve logique montre que pour une fonction quelconque f : E› P(E), alors f n’est pas surjective, car il n’existe pas de utel que f(u) = D, si on prend D = {x ; x . f (x)} : car alors on ne peut avoir ni u . Dni u . D. On remarque aussi une preuve par récurrence dans le cas fini, que j’aime a don-ner en exemple : on a 2°= 1 > 0, 21= 2 > 1, et, si n > 0, alors 2n+1= 2n + 2n > n + n > n + 1. Ce qui est joli dans cette preuve, c’est qu’elle peut etre étendue en une preuve de ce que pour tout ket pour Nassez grand en fonction de k, on a, pour tout n > N, 2n > nk. Je recommande vivement a celui qui veut bien saisir la premiere pensée de l’infini qui s’incarne dans le principe de récurrence, de s’essayer a bien en écrire une preuve. En sus de l'argument diagonal, on observe, avec Cantor, que si l’on part d’une bijectionR .2N supposée, alors on aura ceci : L’ensemble N ne peut couvrir R, ni couvrir (AB( ci-dessus. Mais l’essentiel est ailleurs, dans la décision douteuse du privilege de l’idéo­logie analytique constructive (de bas en haut) — version noble du constructi­visme finitiste —, qui prétend saisir le géométrique et les grandeurs au titre des nombres et de leurs constructions. Alors qu’il est tout-a-fait envisageable et sain de construire les nombres dans l’apres-coup du donné géométrique, comme dans l’approche motivique moderne en tant qu’algebre de la mise bout a bout des segments ; mais il est vrai qu’alors on est tres pres des vue aristotéliciennes. Aussi faire apparaître l’impossibilité de couvrir Ravec N est un cache-misere subtil, du fait de mettre au rang d’évidence la décision de construire un modele numérique du segment, meme si cela doit échouer, la ou une approche axio­matique réussirait, ce dont le géometre ne se privera pas. Aussi, que l’infini représenté par N soit irrésistiblement dépassé par celui de R ne fait sens qu’au titre d’une montée depuis le rien, ou a défaut depuis l’objet Ø, qui n’est déja plus le rien ; sinon, que se passe-t-il dans la descente depuis l’absolu, ou le prétendu absolu, qui, a mes yeux est représentable par le foncteur P*lui-meme, ou bien P*P*, voire par la construction St des types de structures et le calcul des su-pra-relations ci-apres ébauché? 4.Ensembles, prédicats, parties, structures, supra-relations Il y aurait donc, au départ des ouvrages mathématiques, d’abord des ensembles E, éléments d’un univers U (soit une collection d’ensembles stable par certaines constructions), des ensembles et de la logique mélés — ce dont fait preuve notre expérience pédagogique, d’avoir a enseigner l’un ou l’autre — par usage des prédicats p décrits en formules logiques elles-memes éléments d’ensembles de formules spécifiés — et par la formation de l’ensemble sélectionné par un pré­dicat pdans E : {x. E ; p(x)}. Il y a aussi l’ensemble infini N des entiers naturels (donné par Dieu, paraît-il). Il y a, pour un ensemble E, le produit cartésien E × Eet les opérations binaires E × E › E, telles celles de l’arithmétique + et × quand Eest l’ensemble N des entiers naturels. Et surtout on considere l’ensemble des parties de E, noté P(E) = {X ; X . E}, qui permet de représenter une relation binaire R . A × B comme fonction quel­conque (voir la Figure 1) : R- : A › P(B) : a . Ra = {b . B; (a,b) . R}, voire comme une fonction sup-compatible de P(A) vers P(B) : R- : P(A) › P (B) : X . RX = {b . B; .a . X (a,b) . R}. FIGURE 1 – La fonction R- : A › P(B) associée a une relation R. Et puis il y a les structures. Sur un ensemble donné E on peut considérer une « structure » Sde type donné s, spécifiée en terme d'opérations et de familles de parties, telles que les topologies, les ordres, les groupes etc., ce que l’on no-tera (E, S). Ainsi on pourra désigner par Top(E) l’ensemble des topologies surE, Ord(E) l’ensemble des ordres sur E, Grp(E) l’ensemble des groupes surE, et aussi par Ult(E) l’ensemble des ultrafiltres sur E, par Upl2(E) l’ensemble des couples ou 2-uplets de E, aussi noté E2, Upln(E) l’ensemble des n-uplets deE, aussi noté En, par Reln(E) l’ensemble des relations n-aires surE, soit les parties de En = Upln(E), ce qui est P(En) = P(Upln(E )), par Fonn(E) l’ensemble des fonctions n-aires sur E a valeurs dans E, lesquelles par l’intermédiaires de leurs graphes s’identifient a certainsélémentsdeReln+1(E), soit Fonn(E ) .Reln+1(E ), etc. Pour bien souligner le rôle crucial de P(E), voici quatre propositions nouvelles, quoique faciles, ou l’on pose Pn+1(E) = P(P n(E)). J'indique a partir de maintenant comme « Propo­sitions » ce qu'il lui faut, sous couvert des notations et notions précédentes, essayer de démontrer ou vérifier par lui-meme, ou, a défaut ce sur quoi il lui faudrait se renseigner, et qui sera utilisé ensuite. Proposition 4.1. 1. Par x . {x}, l’ensemble E s’identifie a une partie de P(E), soit, abusivement E . P(E). 2. L’ensemble Upl2(E) = E2 des couples de E s’identifie a une partie de P2(E), soit, abusivement Upl2(E).P2(E), par la construction des couples (a, b) = {{a},{a,b}}. 3. Par récurrence on voit que, pour tout entier n, on a : Upl2n(E).P2n(E). 4. Si 2n-1 < m . 2n, alors l’ensemble Uplm(E)des m-uplets de E s’identifie a une partie de Upl2n(E), par (x1,...,xn) . (x1,...,xn,xn,...,xn), et partant a une partie de P2n (E), soit Uplm(E).P2n(E). Proposition 4.2.1. Les données respectivement d’une topologie ou d’un ultra-filtre, ou d’un ordre, ou d’un groupe, sur un ensemble E sont spécifiables comme un élément de, respectivement, P2(E), P2(E), P3(E)et P5(E), soit : Top(E) .P2(E), Ult(E) .P2(E), Ord(E) .P3(E), Grp(E) .P5(E). Proposition 4.3. Une action fidele . : A × E › E , correspondante a l’application adjointe injectiveA › EE = Fon1(E) : a . (x..(a,x)), s’identifie a une application injective A › P3(E), et donc l’ensemble Act-fid(E)de toutes les actions fideles sur E s’identifie a une partie de P4(E): Act-fid(E). P4(E). Ainsi, chaque « type de structure » usuel possible sur un ensemble E est un élément s(E) de l’un des ensembles Pn+1(E), soit une partie de Pn(E), ou bien une application de but un tel Pn(E), et toute structure Sde ce type sest un élément Sde s(E) : S . s(E) . Pn+1(E). Du coup, des que l’on dispose de P on peut, pour envisager ces structures, se dispenser de parler de logique et de formules logiques, il suffit de penser en terme de spécification par un s(E), sans savoir comment ce s(E) est lui-meme déterminé analytiquement. Au demeurant, nombre de déterminations analy­tiques sont possibles autres que logiques, et notamment nous allons le préciser dans les univers algébriques. On observera aussi que cette mise en scene — que je qualifierai de supra-rela­tionnelle, nous allons tantôt voir pourquoi — unifie les structures topologiques et les structures algébriques, constituant une sorte de mini-version de l’uni­fication que Grothendieck voit dans ses topos, qui comprennent aussi bien les groupes que les espaces, comme je l’ai indiqué plus haut. Miniversion, ou bien version internalisée au topos des ensembles. En tous cas, la dualité al­gebre-géométrie s’y trouve comme mystérieusement dissoute. Mon idée est que c’est dans la production de telles dissolutions de dualités que la mathématique progresse, et que l’ontologie devrait viser ce qui est invariant dans le processus de telles dissolutions, dans l’installation de tels entre-deux. Proposition 4.4. En posant, pour tout ensemble X, . X= {y ; .x (y . x . X)}, et puis en introduisant St(E) = .{Pn(E) ; n .N} = .n.0Pn(E) = {z ; .n . 0, z . Pn(E)}, le monoide E* = .n.0Endes mots sur l’alphabet E s’identifie a une partie de St(E), et si L . E* est un langage sur E on a : L . E*. St(E), et en particulier N = {0}*.St({0}).St(Ø). Alors un langage sur l’alphabet E est une partie de E*, un élément donc de P(E*) et donc de P(St(E)), et une transduction F › P(E*) est alors une relation particuliere de F vers St(E), une fonction . : F › P(St(E)). Ainsi Badiou a-t-il tout a fait raison de mettre l’accent sur les deux constructions P(X) et .X, mais il pourrait améliorer l’économie de sa théorie en considérant ce que nous venons de mettre en évidence, le fait que le « composé » St = .{Pn ; n .N } contient comme parties les codages de toutes les structures utiles, et notam­ment de N — d’ou le choix de la notation St comme l’initiale de « Structure » — puis en remplaçant l’analyse logique par l’usage axiomatique de P et ., et St, et des transformations naturelles structurales entre ces constructions, comme dans la théorie des univers algébriques. On fait d’ailleurs d’une pierre deux coups, puisque les univers algébriques peuvent aussi etre mis a la place des topos, et notamment des .-Ens. Toutefois l’accomplissement de cette mise en place demande de préciser les caracteres fonctoriels de ces constructions, et partant la considération explicite des caté­gories d’ensembles et des catégories de structures ; ou bien, de façon équiva­lente, la détermination des compositions des relations A › P(B), hyper-relations A › P2(B), transducteurs A ›P(B*), voire des supra-relations de A vers B, soit les applications A › St(B). On a évidemment Proposition 4.5.Pour tout entier n, la construction Pn est fonctorielle d’au moins 2n façons, en associant, a toute suite s de n signes +, et -, telle par exemple s = + + - + -- le foncteur donné, dans l’exemple, pour chaque fonction f par Ps(f) =P(P(P*(P(P*(P*(f)))))). A fortiori il y a une infinité de fonctorialités sur St. Mais c’est une autre histoire a venir, tres au-dela aussi de la question des sé­mantiques et langages d’ordres supérieurs qui doit y etre incorporée. 5. Univers algébriques et bornes Pour avancer plus profondément dans l’analyse des structures telles que mises en scenes ci-avant, autour du foncteur P ou du foncteur P*, il faut, puisque je veux éliminer la présentation logique a priori, établir axiomatiquement une description alternative de la spécification. C’est ce qui est fourni avec la no­tion d’univers algébrique et de transformation structurale naturelle, dont voici quelques indications. J’en profite ensuite pour montrer comment l’infini des or-dinaux et des cardinaux peut se récupérer dans ce contexte, avec notamment la notion de borne. 5.1 Univers algébriques Si donc U est un univers et EnsU la catégorie des applications entre ensembles éléments de U, on y dispose — c’est une supposition sur l’univers en question — de la construction P, dite « foncteur parties » covariant P : EnsU› EnsU qui a chaque ensemble Xassocie l’ensemble des parties de X, soit P(X) = {A; A . X}, et a chaque fonction f : X › Yassocie la fonction P(f) : P(X) › P(Y) : A . P(f)(A) = f(A) = {y; .x . A, f(x) = y}, et on dispose aussi du « foncteur parties » contravariant P*: EnsUop › EnsUdon­né par P*(f) : P(Y) › P(X) : B . P*(f)(B) = f-1(B) = {x; f (x) .B}. On dispose encore du foncteur produit cartésien a 2 variables × : EnsU × EnsU › EnsU qui a chaque paire d’ensembles (X,Y) associe X × Y . De plus nous avons quelques transformations génératrices spécifiées, qui sont les deux projections, la diagonale, l'atomisation, le couplage, la négation, la rencontre, l'inclusion: prXX,Y: X × Y › X : (x,y) . x, prYX,Y: X × Y › Y : (x,y) . y, .X : X › X × X : x . (x,x), .X : X › P(X) : x . {x}, .X : X × X › P2(X) : (x,y) . {{x}, {{x,y}}, .X : P(X) › P(X) : U . {x; x . U}, .X : P(X) › P2(X) : U . {V; U . V . O}, .X : P(X) › P2(X) : U . {V; V . U}. Ces données satisfont entre elles un certain systeme d’équations, que nous ne détaillerons pas, qui en font ce que l’on appellera un univers algébrique. Les trois premieres sont que le foncteur P* est adjoint au foncteur P*op, c’est-adire que l’on a une bijection naturelle des fleches r : X › P(Y) vers les fleches s : Y › P(X) déterminée par s = P*(r)..X..X ; puis que. détermine P comme sous-foncteur du foncteur P*opP* =: . ; et puis que pour tout f : X › Y , P(f) est adjoint a gauche a P*(f), pour la structure qui dans le cas ensembliste est la structure d’ordre complet libre sur X de chaque P(X), déduite des transformations génératrices spécifiées. Étant donné un univers algébrique quelconque, soit une catégorie Céquipée de foncteurs et transformations encore notés (abusivement) comme dans le cas en-sembliste ci-avant, en composant entre eux ces foncteurs sur C qui sont donc P, P*, etc., et les transformations spécifiées ., ., etc., on atteint ce que l’on appelle les foncteurs types T, T', etc., et et les transformations naturelles types t : T › T', et, entre eux, les équations structurales (t : T › T', t' : T › T'), et puis les fonc­teurs structuraux de la forme de noyaux s = ker(t, t') = [t = t']. Alors, dans le cas ensembliste, une s-structure sur un ensemble Xest un « élé­ment » de s(X). Les structures ainsi spécifiées sont donc équationnelles sur l’univers. Proposition 5.1.1. En sus de l’exemple initial de Ens avec P etc., indiqué ci- U avant, nous avons en fait une structure d’univers algébrique sur tout topos élé­mentaire, avec P*(X) = .X. 2. Si A est un monoide abélien complet dans Ens , ou bien dans un topos élémen- U taire, alors on détermine une structure d’univers algébrique, avec P(X) = AX. Cela vaut en particulier avec A une algebre de Heyting. Proposition 5.2.Les structures du premier ordre, et les topologies aussi, les structures uniformes, sont équationnelles sur l’univers algébrique ensembliste EnsU, et sont donc spécifiables dans tout univers algébrique, tel ceux indiqués dans la proposition 5.1. Proposition 5.3.Les univers algébriques viennent de l’examen des travaux de topologues, tels Ernest Michael et Gustave Choquet, dans les années 1950, qui équipaient l’ensemble des fermés d’un espace topologique de topologies, pour développer la théorie des relations continues. En sus, on peut développer la théo­rie des espaces topologiques internes a un univers algébrique, en posant, pour toute hyper-relation ou fonction spécifiant les « voisinages » v : X › P2(X), OX = .P(X).X, FX = .P(X).X, Iv= P*(v)OX, Av= P*(v)FX, ce qui définit l’intérieur et l’adhérence associées a v, comme modalités de néces­sité et de possibilité associées a cette fonction de voisinage. 5.2Bornes et opérations logiques Dans un univers algébrique, on appelle borne unsous-foncteurstrictBde P contenantØ, soit tel que O . B . P, et cela revient a la donnée d’une transformation naturelle de P vers P2. Une borne est réguliere si c’est une sous-monade de la monade P. Une transforma­tion naturelle de P vers P*2 est appelée opération logique binaire. Proposition 5.4.Dans le cadre ensembliste les opérations logiques binaires, au sens ci-dessus, correspondent en effet aux connecteurs logiques binaires, soit les fonctions binaires 2× 2› 2. Et les bornes B telles que B(Ø) = Øsont de la forme Bß pourßun cardinal fixé : Bß(X) = {A . X; card(A) . ß}, De plus B est réguliere si et seulement si ß est un cardinal régulier. Ainsi tant la lo-gique connective que le calcul des cardinaux sont déterminables « par en haut », a partir de la seule donnée d’univers algébrique, et basiquement du foncteur P* . 6.Formes, présentations, esquisses 6.1 Formes et co-formes Prenons l’exemple de la forme d’un carré montrée dans la Figure 2. Le carré, a droite dans la figure, est constitué d’instructions de placements u, v, etc., de­puis un site contenant les différentes pieces a assembler, les coins et les côtés, dans le grand rectangle a bords arrondis a gauche, avec, entre ces pieces des relations de cohésion h, g, etc. Le carré est vu comme le recollement ainsi spé­cifié de ses morceaux. Plusgénéralement,laforme d’un objet Xd’une catégorie C sera décrite suivant la Figure 3, par un foncteur ./X : C/X › C : FIGURE 3 – Le foncteur ./X, forme d’un objet Xd’une catégorie C La catégorie C/Xa pour objets les fleches u : Y › Xde but X, de source variable Y, et un morphisme hde uvers v : Z › Xconsiste en une fleche h : Y › Ztelle que v.h = u; le foncteur ./Xassocie Y a u, et h a h. Nous avons, évidemment — c’est la substance du lemme de Yoneda — la récupé-ration de l’objet Xcomme limite inductive de sa forme : colim(./X) = X. Et puis, nous avons la notion de J-forme d’un objet X, ou forme relative a un foncteur J : D › C, suivant la Figure 4. La J-forme de X est le foncteur .J/X := ./X.J/X = J.J*(./X) : D/X › C, ou D/Xa pour objet un couple (Y',u) d’un objet Y' de D et d’une fleche u : J(Y')=Y › X, et pour morphisme de (Y',u) vers (Z',v) une fleche h : Y'› Z' telle que v.J(h) = u. Autrement dit, D/Xest le produit fibré de ./Xet de J. Une présentation d’un objet Xde C est un foncteur p : W › C tel que la limite inductive ou colimite de psoit isomorphe a X, c’est-a-dire que X s’obtienne en recollant les données de W suivant p, soitcolim p . X. Si donc on considere en guise de ple foncteur .J/X : D/X › C, il est, par définition, une présentation de lim(.J/X)que l’on notera brievement XJ, et l’on a un morphisme de comparaison XJ › X,qui est un isomorphisme précisément si .J/Xest une présentation de X, si Xse retrouve en recollant sa J-forme. Nous avons aussi la notion duale de co-forme, liée alors aux limites projectives, ce qui est la forme de X en tant qu’objet de la catégorie duale Cop, et, donc les co-présentations ou foncteur q : V › C tel que la limite projective de qsoit iso­morphe a X, c’est-a-dire que Xs’obtienne en co-recollant les données de V sui­vant q, soitlim q . X. 6.2 Esquisses, modélisation par formes, sans logique Toute théorie peut se spécifier par la donnée de contraintes de formes et de co-formes, c’est-a-dire par une catégorie T sur laquelle des contraintes de formes et co-formes virtuelles sont spécifiées. Une contrainte de forme sur T est un foncteur B : I › T avec un cône inductif . : B › S, soit, pour tout I . I une fleche .I : B(I) › Stelle que, pour tout v : I › J on ait .J.B(v) = .I. Une réalisation dans une catégorie E d’une contrainte (B, .) sur T est un foncteur R : T › E tel que .induise un isomorphisme colim(RB) = RS On définit de meme une contrainte de co-forme, soit un foncteur B : I › T avec un cône projectif . : P › B, avec les conditions .J = B(v)..I, une réalisation étant un Rtel que .induise un isomorphisme RP = lim(RB). Un univers de référence U étant fixé, on appelle U-esquisse ou simplement es-quisse la donnée d’un triplet . = (T, P, I) ou T est une catégorie petite (appar-tenant a l’univers U), P un ensemble petit de contraintes petites de coformes (projectives), I un ensemble petit de contraintes petites de formes (inductives). Une réalisation ou modele de . dans E est un foncteur R : T › E réalisant les contraintes de P et de I. On appelle catégorie des modeles de . relativeal’uni­vers U la catégories des transformations naturelles entre les réalisations de . dans E = EnsU, on la note ModU(.) ou EnsU .. Une catégorie C est esquissable par .si C = ModU(.). Proposition6.1.Toute catégorie de modele d’une théorie du premier ordre est esquissable, c’est-a-dire spécifiable sans logique, par une esquisse indiquant des contraintes de forme et de co-forme. Ainsi sont les groupes, les anneaux, les corps, les faisceaux sur un site, les objets borroméens, etc. 6.3 Diagramme localement libre, groupoide d’ambiguité Considérons un foncteur esquissable M : ModU(.) › ModU(.), soit un foncteur induit par composition avec m : . › ., un morphisme d’es­quisse, en associant a tout R le composé M(R) = Rm. Si .et .ne comportent pas de contrainte inductive (de forme), mais seulement des contraintes projectives (de co-forme), alors il existe un foncteur adjoint a gauche a M, noté L : ModU(.) › ModU(.), déterminant donc chaque L(X) comme objet libre sur X, relativement a U, c’est-a-dire tel que pour tout Y on ait une bijection naturelle entre l’ensemble Hom(LX,Y) et l’ensemble Hom(X,MY) : Hom(LX,Y). Hom(X,MY). On obtient ainsi, comme exemples de structures libres, le monoide libre E* sur un ensemble E, les groupes libres, les algebres de Lindenbaum, le faisceau as-socié a un faisceau, les limites ou colimites d’un foncteur, etc. Voici un exemple, du côté de l'arithmétique élémentaire. Proposition 6.2.Dans le cas des structures libres, il faut souligner la dualité de l’engendrement et du co-engendrement, et le rôle de l’infini en cette affaire, comme on peut l’exhiber par exemple pour l’arithmétique et le pgcd comme suit. Si {a,b}est une partie a 2 éléments du groupe additif Z des entiers relatifs, alors il y a un sous-groupe Gen({a,b})de Z engendré par {a, b}, que l’on atteint de deux façons : — de bas en haut par élaboration de termes Gen({a,b}) = {a,b}. {a,b}. {a,b}... = . n{a,b}n, 23 ou {a,b}n+1est l’union de tous les composés et opposés dans Z d’éléments de {a,b}n — de haut en bas, par conditionnements cumulés, comme intersection des sous-groupesS de Z contenant{a,b}: coGen({a,b}) = .{a,b}.S.Z S. On a alors Gen({a,b}) = coGen({a,b}). Les deux procédés sont en un sens duaux, d’économies opposées. Le deuxieme concretement ne donne qu’une existence de principe, le premier une construc­tion formelle dans Z. Beaucoup de théoremes en mathématiques consistent d’abord a avoir l’existence, puis la construction formelle dans quelque chose (ici dans Z), puis, surtout, une réduction de cette construction. Ici la réduction bien connue de Gen({a,b}) a Gen({a.b}) = Gen{d} vient de ce que en calculant le pgcd d = a.b de aet b, on a, par le théoreme de Bézout, un uet un vtels que ua + vb = d, et l’application bijective [u,v] : Gen({a.b}) › Gen({a,b}) : md . mua + mvb =: [u,v](md). On doit comprendre cette réduction, soit la bijectivité de [u,v], comme un prin­cipe de ré-écriture : dans Z, ce qui s’écrit avec aet bse réécrit avec d. Gen({a,b}) › Gen{d}. Au passage, on aura implicitement utilisé la construction absolument formelle, hors de Z, d’un groupe abélien libre engendré par a et b, soit — en désignant, pour une lettre u, par L(u) .Zle groupe abélien libre engendré par u— le groupe L({a,b}). L(a) . L(b) .Z .Z .Z × Z, avec l’application et le morphisme {a,b} ›Z × Z : a . (1,0); b . (0,1), Z .Z › Z : (x,y) . x + y. On observera que l’infini (éventuel et potentiel) des S, et des ndans les {a,b}n et l’infini (assuré et actuel) de Z.Z, en tant que L(a).L(b) sont entrés en jeu implicitement, si bien que pour toutes ces constructions l’infini est présent, aux deux bouts de la dualité, et dans la construction, qui la surplombe (et la dirige), des structures libres. Proposition 6.3.Si . et . sont des U-esquisses qui comportent aussi bien des contraintes inductives et des contraintes projectives, et si m : . › . est un mor­phisme déterminant le foncteur M : ModU(.) › ModU(.), alors un adjoint L n’existe plus nécessairement. Toutefois il existe encore, pour tout X objet de Mo­dU(.), non plus un objet L(X)mais un diagramme localement libre associé a X, petit— c’est-a-dire tel que .(X)appartienne a l’univers U : L(X) : .(X) › ModU(.), satisfaisant naturellement en Y a : colimI..(X)Hom(L(X)(I),Y). Hom(X,MY). Si, de plus, on procede a l’inversion de tous les éléments de .(X), on obtient un groupoideAmbM(X)que l’on appelle groupoide d’ambiguitéde X relativement a M, mesurant l’équivocité dans les possibilités de relevement universel en objet a la source de M pour l’objet X de son but. Cette proposition 6.3 admet une preuve utilisant le fait que U soit un univers, et dans cet univers les ordinaux et une construction par récurrence transfinie dont on montre qu’elle s’arrete, sous la condition qu’il existe un cardinal ré­gulier majorant les cardinaux des catégories indexant les cônes projectifs et inductifs de.. Mais une fois la preuve faite, avec donc ses outils ensemblistes, on peut décider d’oublier la logique et les ensembles, et repartir de la donnée de ce théoreme comme propriété fondamentale de la catégorie des esquisses. On peut comprendre alors ce théoreme comme une version catégoricienne de Lowenheim-Skolem, pierre de touche du recours a la limitation des infinis dans les constructions mathématiques générales. Comme exemples, en sus des structures libres, nous avons les constructions des groupes de Galois, les constructions des spectres, des catégories des points de topos classifiants, le calcul d’algorithmes par schémas de Herbrand. Et puis, ce théoreme et la notion d’esquisse d’une part, et la théorie du foncteur partie et des structures en celui-ci d'autre part, s’unifient au titre de la construc­tion pour toute catégorie C localement petite de la catégorie localement petite Diag(C) des petits diagrammes D : A › C, ou donc Aest une petite catégorie, et un morphisme de Dvers D' étant un couple (F,f) d’un foncteur F : A › A' et d’une transformation naturelle f : D › D'.F. En effet on a deux choses : D’une part D . L, si un diagramme D a une limite inductive L, est une opération partielle de Diag(C) vers C, etc. Et d’autre part, si C est une catégorie discrete soit un ensemble E, alors Diag(C) = P(E). Mais c’est une autre histoire, a reprendre, avec la théorie des machines et auto­mates que j’exposais jadis autour de Diag. 7. Vers l’infini diagrammatique, sans logique, par deux bouts Au départ de la théorie des ensembles et des structures ensemblistes, deux points essentiels sont en un certain sens duaux : d’un côté la construction P et le théoreme de Cantor, et de l’autre l’algebre de Lindebaum et le théoreme de Lowenheim-Skolem. Les deux peuvent etre axiomatisés, dans un cadre catégorique fonctionnel, voire relationnel, mettant ainsi au second plan la donnée ensembliste soi-disant fondatrice. Cela donne d’un côté la notion d’univers algébrique et une construc­tion axiomatique équationnelle de tous les types de structures ; et, de l’autre côté, le théoreme du diagramme localement libre DLL, et un processus général de développement de tous les modeles libres de tous les types, algébriques ou non, décrit géométriquement par esquisses ou contraintes de formes. Dans les univers algébriques, on peut se dispenser et de logique et d’ordinaux, puisque ces données se retrouvent apres-coup, a partir de la donnée de P, avec la notion toute simple de borne, en quelque sorte en « descendant » depuis P : l’infini descend de P, dans la spécification de ses sous-foncteurs, les bornes. Avec les formes et les esquisses, le théoreme du diagramme localement libre DLL donne une construction et un contrôle de l’infinitude de la taille des struc­tures librement ou localement librement engendrées, en partant donc de l’ana­lyse « formative » ou présentation par forme, seule, sans logique encore. L’infini en jeu monte de la donnée génératrice par découpes et collages de ses pieces, jusqu’a son point de stabilisation dans le diagramme localement libre. Dans la présentation et l'analyse des structures, l’infini s’atteindra donc par deux bouts ainsi duaux, le côté univers algébriques et le côté esquisses : chez l’un plutôt en descendant, a partir de P*, chez l’autre plutôt en montant a partir de Ø. De plus chacun des deux bouts est lui-meme au départ d’une dualité : l’au­to-dualité de P*, pour les univers algébriques, la dualité entre le jeu des formes et celui des co-formes, pour les esquisses, côté DLL. On peut alors rever aux « analogues diagrammatiques » des cardinaux, si, a l'instar de ce qui se passe avec P, on veut les identifier a des bornes diagram-matiques, a savoir a des sous-foncteurs de Diag. Un bel objet de ce type est le sous-foncteur Exa de Diag tel que Exa(X) soit le sous-ensemble de Diag(X) constitué des extensions de Kan absolues au-dessus de X. Partant il y a le projet d’alors re-penser l’infini a partir de Diag construit sur CatU, au lieu de P sur EnsU. Question aussi d'axiomatiser ce Diag — dont j’ai déja donné la propriété universelle qui étend celle de P — l’axiomatiser comme les univers algébriques axiomatisent P, en le lisant comme un grand-transcen­dantal (au sens de Badiou pour ce terme) qui, en plus de la logique, « délivre » toute la théorie de l’infini, comme P donne les cardinaux et les opérations logiques ; mais de plus, Diag « délivrera » les principes de constructions par extensions, comme le Diagramme Localement Libre. On est donc a la recherche de l’infini diagrammatique. 8. Conclusion : Des cardinaux aux structures Pour conclure, voici a propos de l'infini la position que je soutiendrais, lisible dans sa différence ou son rapprochement avec celle d'Alain Badiou que j'ai mar-qués au long du parcours ouvert ci-avant. La pensée de l'infini n'est pas une pensée de l'ontologie a priori, mais une pensée du travail mathématique, au niveau phénoménal de production de ce travail. Ce travail consiste a inventer les formations de l'infini dans la tension entre voir et dire ce que l'on voit ; et ce que l'on voit en mathématiques, ce sont des structures, ou, c'est pareil, les théories. Les ensembles sont utiles pour décrire les structures, et a ce titre sont des entités idéales ou idéalités en théorie des structures, comme les nombres imaginaires en algebre, les points a l'infini en géométrie, etc., sur la base desquelles on sait, par descriptions prédicatives, construire des codages d'autres ensembles et des structures. Mais en fait les structures sont premieres, ce sont les phénomenes que l'intui­tion mathématique entrevoit synthétiquement et leur saisie ensembliste les dé­construit. Il faut donc penser le travail mathématique, et notamment penser l'infini, dans l'alternative ou dialectique entre ensembles et structures. Techni­quement parlant, si l'on prend les ensembles comme formes substantielles ou socles, et les structures comme dynamiques fonctionnelles d'ou des calculs se déploient sur les éléments d'un socle, cela revient a la question d'une dialec­tique entre théorie des ensembles et théorie des catégories, qui ne vont pas l'une sans l'autre, au plan de la compréhension de l'invention mathématicienne. En fait, depuis les paradoxes de Zenon, la mathématique a développé nombre de questions sur l'infini, et diverses théories « résolvantes », parmi lesquelles les méthodes d'Archimede, la mesure des grandeurs, les calculs de séries et limites, la récurrence, les infimes, le calcul différentiel et l'analyse des contacts de courbes, la théorie des cardinaux, les limites dans les catégories. Je consi­dere chacune comme une théorie de l'infini, qui examine l'infini se produisant dans un certain écart entre deux termes, deux bouts, qui sont des instances particulieres de la combinatoire finie et du continu, du dire et du voir. Ce sont des théories de l'infini non pas tant parce qu'elles actualiseraient les infinis comme objets, que parce qu'elles permettent le développement de l'infinitude spécifique d'un calcul. C'est pourquoi il faut, par exemple, penser a l'analyse non standard et aux in-fimes comme d'abord ce que cela permet comme calcul d'infinis. On sait que Badiou s'y intéressa dans les années 60, et c'est par ses écrits que je commen­çais a l'époque a connaître cette théorie. On peut bien penser l'infini en chacune de ces circonstances, d'Archimede a Cantor et apres, particulierement, mais je crois qu'il y faut cependant mettre alors en avant cette question de l'écart que telle ou telle théorie de l'infini vise a peupler, a remplir, comme effet dialectique entre ses bouts. Retournez a Zénon, regardez encore ma photographie « Bivue », ou l'on voit et la question du discret et la question du continu, et leur conflit a résoudre. C'est pourquoi aussi il faut, par exemple, penser a la théorie des cardinaux comme un tel remplissage. C'est bien ce qu'accomplit Badiou aujourd'hui, entre le vide et l'absolu (voire entre le Néant et l'Etre...), au regard de la théorie des grands cardinaux. A juste titre, Badiou utilise basiquement les deux constructions P et ., de la proposition 4.4. ci-avant, ce qui conduit aisément a l'opération que j'y note St, et dont j'établis que pour tout ensemble X, l'ensemble infini St(X) contient toutes les structures utiles sur l'ensemble X. En fait, dans le développement des structures ensemblistes, le foncteur P ou P*, puis maintenant St, jouent un rôle fondamental, que j'ai axiomatisé dans les années 70, d'abord sous le nom de catégorie cantorienne, puis sous celui d'univers algébrique. Les univers algébriques ont trois qualités. D'abord ils sont tres généraux, com-prenant comme cas particuliers aussi bien les topos que les ensembles flous. Ensuite, ils permettent un développement interne équationnel (algébrique) de la théorie générale des structures et théories mathématiques. Enfin ils sont in-trinsequement porteurs de la logique et de la théorie des cardinaux, via ce que j'appelle lesbornes. Ce que j'ai rapidement évoqué ci-avant. Du coup, je peux remplacer la question de l'infini considérée comme celle du remplissage entre le vide et l'absolu, par celle du remplissage entre O et P, voire entre O et St, dans un univers algébrique quelconque. Pour ce remplacement j'ai besoin de considérer au passage le cas d'univers algébrique qu'est la catégorie EnsU, et donc sur elle le P et le St qui y sont construits. Entre O et St il y a chaque type de structure particulier T, spécifié comme sous-construction de St. Alors l'intervalle entre O et un tel T, est sous condition de la dialectique entre EnsU et ModU(T), catégorie des modeles de T dans EnsU. On est donc bien passé de la question du remplissage entre le vide et l'absolu par les cardinaux a celle du remplissage entre ensembles et struc­tures, et par les structures. Et a fortiori, avec les univers algébriques abstraits, a celle plus générale du remplissage entre O et St. Ce mouvement de pensée accompli techniquement ici, des cardinaux aux struc­tures, pour y penser l'infini, est probablement hors du systeme de Badiou qui n'indique pas que pour lui les ensembles peuplent, comme objets ou phéno­menes mathématiques, un monde EnsU. Mais sur ce point il faudrait interroger son retour aux ensembles apres son excursion vers les topos, avec, en plus, une relecture de ses idées sur les modeles. Admet-il de penser l'ontologie ensem­bliste avec, au départ, la donnée d'un modele U d'univers ? Et si oui, admet-il, toujours dans la perspective ontologique, de passer de la vue de U a celle de la catégorie EnsU? Ce n'est pas impossible, vu ce qu'il dit sur Logiques des mondes et son déplacement « vers l'aval », que je rapporte en section 1. Mais mon im­pression est que cela va contre la pente naturelle qu'il a pris en déployant son analyse de l'aventure des grands cardinaux, et cela, qui plus est, sous condi­tion expresse de la logique. Dans les univers algébriques, on n'a donc pas besoin d'utiliser a priori un trans-cendantal, tout vient de P et donc la logique peut etre bien plus indéterminée que la logique intuitioniste, mais surtout il n'y a pas lieu de s'en servir pour construire, tout découle des équations structurales. En fait, les structures elles­memes sont décidées, dans l'univers en question, par la donnée St. Mais il y a un autre moyen, en quelque sorte dual de la spécification a priori via P ou St, pour éviter la logique, dans la spécification constructive, cette fois, des structures. C'est l'approche des structures par esquisses, contraintes de formes et co-formes. J'expliquais sommairement ici ce qu'il en est, avec le théoreme centraldu diagramme localement libre(DLL), lequel est sous conditions d'exis­tence de cardinaux réguliers. On peut alors a nouveau considérer la dialectique entre entre EnsU et ModU(T), soit maintenant entre EnsU et ModU(.), avec . une esquisse. Ultérieurement, on peut unifier univers algébriques et esquisses dans la théo­rie des machines, et la considération du foncteur « diagrammes » Diag sur la catégorie CATU des petits diagrammes sur des catégories localement petites. On recommencera la théorie des bornes, comme étant maintenant celles des sous-foncteurs de Diag, contemplant ce que l'on peut appeler l'infini diagram-matique, entre ses deux bouts O et Diag. 9. Epilogue : ou sont les structures, les bouts, l'infini ? Dans la version ensembliste pure, de Badiou, il n'y a pas de structure, seule­ment les ensembles, il y a, comme bouts, les cardinaux, et l'infini est ce qui se loge entre deux cardinaux, par exemple entre X et P(X), voire entre le O et l'ab­solu. Alors la pensée de l'infini est sous condition de l'ontologie (ensembliste). Dans ma proposition, il y a d'abord les structures, qui sont les pensées de ce que l'on perçoit en mathématiques, soit les phénomenes mathématiques, au cours du travail mathématique (et non pas en interprétant les théories et théoremes pour leur donner du sens « réel » dans le monde). Ces structures sont, dans un premier temps, décrites a partir des ensembles, au-dessus des ensembles. On Les obtient comme éléments des ensembles St(E), ou bien comme objets de catégories ModU(T) ou bien ModU(.). La premiere donation est descendante, a partir de St, ou du moins de P, la seconde est montante et constructive, via des constructions de structures libres ou bien de DLL. Ces deux donations sont en quelque sorte duales, constituent deux bouts « méthodologiques », entre lesquelles on loge l'infinité de l'activité mathématicienne. Étant données deux structures, prises comme bouts, on doit construire l'infinité des comparaisons entre elles, comme c'était déja le cas avec la structure des entiers N et celle du continu [0,1], avec les paradoxes de Zenon. Cette infinité a elle-meme une struc­ture, et pas seulement un cardinal, etc. Dans un second temps, grace au lemme de Yoneda, et a la considération de la forme et de la co-forme, chaque structure, et notamment telle qu'ensembliste­ment déterminée, apparaît tout simplement comme un objet d'une catégorie. Et bien sur la structure de catégorie n'échappe pas a cette approche, et est donc un objet de la catégorie CatU des petites catégories ou de CATU des catégories localement petites, relativement a un modele U de la théorie des ensembles. Alors on peut renverser la perspective, et, partant d'une catégorie abstraite C, par exemple d'un univers algébrique, ou plus particulierement d'un topos, on peut décider de remplacer le monde des ensembles EnsU, par la catégorie C, et les objects de C, pris alors comme structures premieres. On peut faire cela par exemple avec C = TopU, la catégorie des espaces topologiques, ou bien avec C = CatU. Si par exemple on considere comme bouts EnsU et ModU(T), il y a comme infini celui constitué des foncteurs de l'une vers l'autre, et formant, avec les transfor­mations naturelles, une catégorie, qui en constitue la structure. De meme avec ModU(.) et ModU(.). De meme avec deux catégories quelconques C et B. Il y a donc la catégorie BC des foncteurs et transformations naturelles de C vers B. La théorie des ensembles permet, via les cardinaux, de démontrer des théo­remes comme celui du DLL. Toutefois, un théoreme de ce type peut etre pris comme postulat, et alors remplacer l'usage des cardinaux et partant des en­sembles. La théorie des ensembles permet de construire le foncteur « structures » St ou du moins P. En réalité, la théorie des ensemble demande, en plus du langage ensembliste, une logique et, exprimés dans cette logique, des postu-lats, comme l'axiome du choix. Dans le point de vue ici développé, on peut faire l'impasse sur la théorie des ensembles et sur la logique. Il reste la théorie du re-collement des diagrammes, d'ou l'on exprime les structures qui doivent interve­nir dans les problemes et leurs solutions. Au lieu de former des ensembles par des spécifications prédicatives, on spécifie des structures par des contraintes de formes. On peut donc oublier les nécessités ensemblistes laborieusement lo-giques, et, comme déja souligné, utiliser le foncteur Diag, plutôt que P. Ainsi, dans l'oubli de leurs premieres déterminations ensemblistes, les struc­tures, les bouts, les infinis entre ces bouts, sont tous, tout simplement, des objets de catégories, et des catégories eux-memes. En particulier l'infini entre deux catégories C et B est la catégorie BC. Dans ce cadre « phénoménologique » la pensée de l'infini devient donc l'examen des structures BC, avec d'un côté les classes d'isomorphies ou d'équivalences de catégories, et, de l'autre, comme suggéré déja, les sous-foncteurs de Diag. Ainsi se constituent deux nouveaux bouts « méthodologiques » entre lesquelles on loge l'infinité de l'activité ma-thématicienne. Références Badiou, Alain,La République de Platon, Fayard, Paris 2012 — Le Séminaire. Parménide. L’etre 1 — Figure ontologique. 1985-1986, Fayard, Paris 2014 — L’Etre et l’événement, Seuil, Paris 1988 — L’immanence des vérités.L’Etre et l’événement, 3, Fayard, Paris 2018 — Logiques des mondes. L’Etre et l’événement, 2, Seuil, Paris 2006 Dehornoy, Patrick, La théorie des ensembles : introduction a une théorie de l'infini et des grands cardinaux, Calvage et Mounet, Paris 2017 Foreman, Matt et Akihiro Kanamori (dirs.), Handbook of Set Theory, Springer, Berlin et Heidelberg 2010 Goldblatt, Robert, Topoi: The Categorical Analysis of Logic, Elsevier, Amsterdam 1984 Jech, Thomas, Set Theory, Springer, Berlin Heidelberg 1997 Kanamori, Akihiro, The Higher Infinite, Springer, Berlin Heidelberg 1994 Lacan, Jacques, «Acte de Fondation » [1964], repris dans Autres écrits, Seuil, Paris 2001 Guitart, René, textes disponibles sur : http ://rene.guitart.pagesperso-orange.fr. Rabouin, David, Oliver Feltham et Lissa Lincoln (dirs.), Autour de « Logiques des mondes » d’Alain Badiou, Éditions des archives contemporaines, Paris 2011 Filozofski vestnik | Volume XLI | Number 2 | 2020 | 249–284 | doi: 10.3986/fv.41.2.10 David Rabouin* Espace et nombre : deux voies dans l’ontologie ? Introduction Cet article poursuit un dialogue engagé a la sortie de Logiques des mondes a partir de trois grandes lignes de questionnement : 1. La premiere, la plus immé­diate, est le sens qu’il convient de donner au célebre slogan « mathématiques = ontologie ». C’est autre chose, en effet, d’avancer que les « mathématiques sont l’ontologie », comme l’avait promu l’Etre et l’événement explicitement1, et de dire que la théorie des ensembles seule est l’ontologie (comme l’avance Lo-giques des mondes, ainsi que d’autres textes contemporains). Il semble qu’il y ait en ce point une inflexion importante du systeme, au demeurant non thé­matisée comme telle ; la théorie des ensemble est-elle une maniere d’exprimer l’ontologie, c’est-a-dire les mathématiques, ou est-elle l’ontologie elle-meme ? 2. Ceci conduit a une interrogation plus large sur le rapport, en mathématiques, entre expression et ontologie, ou « langage » et « etre ». Ici, je voudrais indiquer que, contrairement a ce que l’on pourrait croire, il y a souvent une ambiguité entre l’un et l’autre non seulement chez Badiou, mais plus généralement dans les discussions de philosophie des mathématiques. Si cette distinction est per-tinente — et j’essayerai de montrer pourquoi elle doit l’etre —, alors on ne peut pas conclure trop vite du fait que les mathématiques ont adopté une expression unifiée grâce au langage ensembliste au fait que la forme de l’etre qu’elles expri­ment est de nature ensembliste (que l’etre est « multiple pur » dans le vocabu-249 laire de Badiou) ; 3. Enfin, je voudrais creuser le fait que le langage ensembliste a justement donné lieu a la thématisation de deux orientations que l’on pourrait tout aussi bien qualifier d’« ontologiques » (dans un sens différent, donc, de celui que lui donne Badiou) ; la premiere met en avant le nombre, tandis que l’autre met en avant l’espace (plus tard nommé « topologique »). Que l’on dis­pose d’un langage apte a les décrire de maniere homogene ne préjuge pas alors de ce que nous ayons affaire a un seul domaine d’objets. Je voudrais montrer 1 Alain Badiou, L’Etre et l’événement, Seuil, Paris 1988, p. 10. * (CNRS), the research group SPHERE (UMR 7219, CNRS – Université de Paris) que cette tension traverse les mathématiques contemporaines, et par voie de conséquence, la pensée d’Alain Badiou plus qu’il ne veut l’admettre (notam­ment au titre de ce qu’il nomme « onto-logique »). Elle est d’ailleurs au cour de différentes tentatives proposées en mathématiques pour parvenir a des formes plus satisfaisantes d’unification que celle procurée par les seuls « ensembles ». 1. Quel est le sens de l’énoncé « les mathématiques sont l’ontologie » ? 1.1. « Ontologie » et devenir historique des mathématiques Accordons pour le moment que « les mathématiques », toutes les mathéma­tiques, soient « l’ontologie ». On devrait alors dire que les Élémentsd’Euclide sont un traité d’ontologie au meme titre que « la formidable Introduction a l’analyse en 9 volumes, de Jean Dieudonné »2. Mais une telle affirmation n’est pas sans poser de difficulté. Il ne s’agira pas alors seulement de pointer les em-barras qui peuvent naître des lors que le philosophe délegue au mathématicien le soin d’etre seul prescripteur en matiere d’ontologie. Ces difficultés, Alain Ba-diou a d’ailleurs fini par les reconnaître et cela l’a conduit a moduler un énoncé brandi d’abord sans autre qualificatif en énoncé de nature plutôt stratégique. Mais meme si l’on concede que l’énoncé initial est de nature stratégique, meme si l’on concede qu’il signe des choix qui peuvent rester irréductiblement philo­sophiques, il n’en accompagnera pas moins une certaine vue sur les mathéma­tiques qui en fait un discours sur « l’etre en tant qu’etre ». Dans l’affirmation précédente, on n’entend pas notamment qu’Euclide et Dieudonné aient formu­lé des ontologies et encore moins que ces ontologies puissent etre différentes. Le cour du slogan, qu’il soit stratégique ou non, est qu’ils nous offrent l’un et l’autre un certain rapport a « l’ontologie », parce que l’un et l’autre décrivent « ce qui est ». Or ce face-a-face avec l’etre semble soustraire les mathématiques, au moins pour une part, a leur devenir historique. Plus exactement, elle semble rejeter ce devenir du côté des modalités de la seule expression. Il faut que les mathéma­tiques d’Euclide, quoiqu’exprimée dans une terminologie qui leur est propre, nous parlent d’entités que nous reconnaissons, d’une maniere ou d’une autre, comme les memes que celles dont parle Dieudonné (au moins pour les parties de leurs discours qui se recouvrent, par exemple ce qui a trait aux nombres et Ibid., p. 20. aux grandeurs en général). Il ne s’agit pas la d’une vue particulierement origi­nale face a cette science singuliere, a la fois déployée dans une histoire et pour-tant, comme l’avançait déja Cavailles, « négatrice d’histoire ». De fait, le devenir semble pouvoir y etre ressaisi apres coup comme déploiement d’une nécessité purement conceptuelle qui en efface les scories contingentes. Les mathémati­ciens ne cessent de relire leur passé en agissant comme si certains théoremes étaient les « memes » et qu’ils portaient sur « les memes » objets. Sous ce point de vue, c’est bien la maniere d’exprimer ces objets qui seule varie. Peut-etre est-ce la, d’ailleurs, une des origines de la croyance aux concepts comme pures « idées » éternelles et immuables, dont les mathématiques ont toujours servi de support exemplaire. Mais que cette croyance soit tres largement partagée ne la rend pas moins questionnable. Supposons, en effet, que les mathématiques d’Euclide se rapportent, non moins que les nôtres, a « l’etre en tant qu’etre ». Une chose paraît acquise a qui les lit : elles ne sont certes pas formulées dans un langage qui porterait sur des objets du type « ensembles ». Les objets dont elles traitent se nomment plutôt « nombres » et « grandeurs », et meme, plus souvent encore, « triangle », « cercle », « droite », « rectangle », etc. (mais le livre V des Éléments nous apprend que toutes ces formes géométriques sont des « grandeurs »)3. L’affaire n’est pas sans importance, parce que le fait que les mathématiques grecques classiques — le constat se transfere aisément, en effet, a Archimede aussi bien qu’a Apollonius — s’expriment selondeux grands domaines d’objets commande leur organisation théorique. Ainsi, il n’y a chez Euclide aucune démonstration qui circule entre les livres géométriques (I-VI pour la géométrie plane) et les livres arithmétiques (VII-IX), alors meme que de nombreux résultats se répondent d’un domaine a l’autre. Non moins surprenant est le fait qu’il y ait alors besoin de deux théo­ries des proportions pour opérer avec ces objets (exposées respectivement aux 251 livre V et au livre VII des Éléments), alors que leurs propriétés sont pourtant identiques quand on les applique a l’un ou l’autre type d’entités (pensons au fait que les rapports entrant dans une proportion peuvent etre « renversés » ou, comme disaient les Grecs, « alternés »). C’était d’ailleurs la un des exemples favoris d’Aristote pour faire valoir que l’etre se dit « en plusieurs sens », qui correspondent a autant de « genres d’etre » incommunicables : pour Euclide comme pour Aristote, meme si certains énoncés valent en apparence de toutes 3 Euclide, Les Éléments, trad. fr. Bernard Vitrac, PUF, Paris 1990-2001. les entités, on ne pourra jamais prouver le géométrique avec l’arithmétique, et réciproquement, parce que l’un et l’autre visent des « genres d’etre » distincts4. Si nous rétorquons alors que ce n’est la qu’un phénomene de surface et qu’Eu­clide se meut dans la meme « ontologie » que nous, qui n’est rien d’autre que « l’ontologie » éternelle, dont il reviendrait simplement au philosophe d’explici­ter les attendus, alors nous devons du meme souffle admettre plusieurs theses problématiques. Tout d’abord, cela veut dire qu’une théorie mathématique peut relever d’un langage qui se rapporte en apparence a des domaines d’objets, tandis qu’elle se rapporte en fait a d’autres. Ensuite, cela conduit a faire des mathématiques une discipline soumise a des formes de progres qui ne seraient pas seulement a trouver du côté des résultats, mais des modalités d’expression de cette réalité sous-jacente — certaines se révélant « meilleures » que d’autres. Ainsi on dira que le langage euclidien porte « en apparence » sur des nombres et des grandeurs, mais que « en réalité » il traite déja de multiples purs. On devra également tenir que les embarras dans lesquels il se meut a sacrifier a son langage de surface doivent etre imputés, en derniere analyse, a des mala-dresses d’expression dont nous serions heureusement sortis. Mais entre autres difficultés subséquentes s’avance alors le fait que ces theses valent de droit de toute théorie mathématique et donc, en particulier, de la théo­rie des ensembles (et, plus généralement, de toute théorie en vigueur qui préten­dra livrer l’expression la « meilleure » des entités mathématiques). Si elles sont vraies, il faut donc admettre déja qu’il n’y a rien dans la théorie des ensembles, prise en elle-meme, qui permettent de savoir si elle parle vraiment d’ensembles ou si c’est la simplement une maniere de parler d’autre chose — exactement comme le mathématicien contemporain prétend que les mathématiques d’Eu­clide semblent parler de « nombres » et de « grandeurs », mais que tous deux relevent d’une essence commune (celle qu’exprimeraient les « ensembles »). En outre, il n’y a rien dans la théorie des ensembles qui en fera le dernier mot de la recherche de la meilleure expression a laquelle sera porté, par sa nature meme, le développement des mathématiques5. 4 Voir par exemple Analytiques Seconds, I, 7, 75 a 38–b 6 (Aristote, Seconds Analytiques. Or-ganon IV, trad. fr. P. Pellegrin, GF, Paris 2005, p. 103). 5 On n’aura d’ailleurs pas de peine a trouver des mathématiciens arguant du fait que les mathématiques, meme quand elles parlent d’ensembles, ne portent pas sur des ensembles et que c’est la une indication qu’il ne saurait s’agir de la meilleure expression des enti­tés mathématiques. Je donnerai des exemples dans la derniere section, mais notons des Mais remarquons également que si les theses que je viens d’évoquer sont fausses, s’il n’y a pas d’écart entre expression et ontologie, la situation n’en sera pas moins problématique : car on se retrouvera alors avec des types d’enti­tés différentes d’une expression a l’autre, et notamment d’une période a l’autre, d’une culture a l’autre, d’un auteur a l’autre, d’un auteur a lui-meme, etc. La question se posera, par exemple, de savoir non seulement si Euclide se meut dans la meme « ontologie » que Dieudonné, mais si Leibniz se meut dans la meme « ontologie » que Newton, Lagrange que Euler ou Brouwer que Hilbert, voire que Heyting – et meme, a termes, si le jeune Newton (le virtuose des mé­thodes symboliques qui découvre le développement du binôme) se meut dans la meme ontologie que le vieux Newton (celui qui défend la primauté de la géo­métrie synthétique a l’ancienne)6, etc. etc. On ne pourra meme pas s’en sortir alors avec une solution de type « aristotélicienne » qui entendrait par « onto-logie » une description des grands « genres de l’etre », car ces genres n’auront plus rien de « grands ». Ils se démultiplieront avec les formes d’expression que l’historien nous apprend a discerner et dont le nombre ne cesse de croître a me-sure que s’affinent nos descriptions. 1.2.Langage et ontologie La difficulté soulevée dans la section précédente touche a un probleme philoso­phique plus général et plus profond, sur lequel il peut etre utile de s’arreter brie­vement. Il s’est trouvé naturellement associé a une conception du langage ou le formalisme logique élaboré pour ce que l’on nomme aujourd’hui « la logique du premier ordre » a prétendu fournir un premier modele formel. Le mécanisme de la « référence » s’y trouvait alors idéalement représenté par le rapport entre une syntaxe supposément « vide », vue comme pur jeu de symboles contrôlé par des regles, et une sémantique donnée par une « interprétation » de ces symboles. Du fait que les mathématiques sont déclarées une science « formelle » — mais a nouveau veut-on dire par la toutes les mathématiques a travers l’histoire ? — et du fait qu’on peut effectivement exprimer la plupart des structures mathé­matiques a l’aide du formalisme que je viens de décrire, on croit alors que ce a présent que c’était la un des motifs des objections de Desanti contre ce qu’il appelait le reve d’une « ontologie intrinseque » des mathématiques, cf. Jean-Toussaint Desanti, « Quelques remarques a propos de l’ontologie intrinseque d’Alain Badiou », Les Temps Modernes 45 (526/1990), pp. 61–71. Voir Niccolo Guicciardini, Isaac Newton on Mathematical Certainty and Method, MIT Press, Cambridge, Massachusetts 2009. modele du rapport entre syntaxe et sémantique, meme s’il a rapidement montré ses limites dans la formalisation du langage naturel, s’y trouve exemplairement validé7. Or derriere cette évidence se cache un probleme désormais classique que des auteurs comme Hilary Putnam ont mis en avant a propos du fonctionnement de la « référence ». L’intéret particulier de l’approche de Putnam, meme si elle prend son départ aux antipodes de la pensée de Badiou, est qu’elle appuie en partie son argument sur le point aveugle de l’historicité des sciences, dont je suis également parti8. C’est pourquoi je la rappelle ici. Supposons donc que le langage scientifique puisse etre idéalement représenté dans un langage formel « transparent » (par exemple celui de la logique du premier ordre) au sein du­quel il nous serait possible de décrire directement les objets a l’aide de ce que Russell appelait des « descriptions définies »9. Une entité, quelle qu’elle soit, devrait alors etre caractérisable par une collection de telles formules qui consti­tueront sa « description complete »10. Dans ce cadre, la seule forme d’historicité que l’on semble pouvoir attribuer aux sciences, au-dela des améliorations liées a la seule expression (invisibles du point de vue du langage formel), est celle de la correction des erreurs et d’un progres cumulatif : une description peut s’avérer fausse, au sens ou elle ne correspond finalement a « rien » (comme le « phlogistique ») ; mais il peut également arriver qu’elle soit considérée a un moment comme complete, alors qu’elle n’était que partielle (comme lorsque l’on pensait, par exemple, que toute « fonction continue » devait également avoir une tangente en chaque point). Dans le premier cas, on remplacera une descrip­ 7 Badiou a d’ailleurs consacré a ces questions une de ses premieres interventions philoso­phiques en défendant un point de vue tres critique sur la tendance a exporter le concept de modele des mathématiques aux autres sciences (pour ne rien dire du langage !), cf. Le concept de modele, Maspéro, Paris 1969. Mais il n’en a pas moins conservé en mathéma­tiques une entente assez stricte du fonctionnement du mécanisme de la référence. 8Voir, par exemple, en traduction française : Hilary Putnam, « Langage et réalité [1975] », dans Textes clés de philosophie des sciences, Vol. 2, dirs. S. Laugier et P. Wagner, Vrin, Paris 2004, pp. 61–104 et « Explication et référence », De Vienne a Cambridge, dir. dans P. Jacob, pp. 337–365, Gallimard, Paris 1980. 9 Bertrand Russell, « On denoting », Mind 14 (56/1905), pp. 479–493; trad. fr. Jean-Michel Roy, Écrits de logique philosophique, PUF, Paris 1989, pp. 203–218. 10 On peut toujours mettre bout a bout toutes ces descriptions a l’aide de conjonctions et obtenir ainsi un seul énoncé définitionnel que j’appelle « complet ». tion par une autre ; dans le second, on completera une description existante en la précisant. Or l’histoire des sciences modernes met a mal une telle vue en procurant un tres grand nombre d’exemples ou la connaissance a progressé non pas en rem-plaçant ou en complétant, mais en niant des descriptions antérieures du meme objet. Les noms de Bachelard et de Canguilhem, dont Badiou hérite, viennent ici a l’esprit de tout lecteur français. Un exemple que propose Putnam et que nous pouvons suivre, est celui des descriptions de l’électron dans les théories de Bohr-Rutherdord et de Schrödinger11. Le point clef sur lequel Putnam veut attirer l’attention est le suivant : plusieurs « descriptions définies » données dans ces deux modeles apparaissent comme incompatibles. Dans le premier, notamment, l’électron est une entité qui a une position et une vitesse détermi­nées, mais pas dans le second. Nous nous trouvons alors face a un dilemme : ou bien nous considérons qu’il s’agit de différentes tentatives pour approcher théoriquement le meme objet ; c’est la tendance naturelle a laquelle est porté l’historien des sciences quand il distingue différentes conceptions de « l’électron » a travers l’histoire. Mais le prix a payer sera alors de concéder que nous ne nous référons donc pas aux objets par l’intermédiaire des descriptions (qu’elles soient completes ou incom­pletes). Le probleme de la description initiale de Bohr, en effet, n’est pas qu’elle est incomplete, mais qu’elle est inadéquate et persiste a penser l’électron sur le modele d’un objet de la mécanique classique auquel on devrait ajouter des propriétés particulieres pour le transformer en objet quantique. Ou bien nous devons accepter que nous avons affaire a des entités de types différents, disons 255 11 Dans le premier modele, l’électron est conçu comme tournant autour du noyau de l’atome a la maniere d’un satellite autour d’une planete (mais sur une orbite qui serait circulaire). L’idée est alors d’intégrer a un tel modele les spécificités issues des découvertes liées a la quantification, c’est-a-dire le fait que les électrons restent sur des orbites stationnaires correspondant a des niveaux d’énergie déterminés et qu’ils peuvent néanmoins passer d’un niveau a l’autre par émission ou absorption d’un certain quantum d’énergie. Mais ce modele souffre du nombre d’hypotheses ad hoc qu’il nécessite. Il laissa donc rapidement la place, sous l’impulsion de Bohr lui-meme, a un modele de nature tres différente, pro-babiliste. Dans cette nouvelle description, dont Schrödinger fut le maître d’ouvre, on ne parle plus d’orbite, mais d’orbitale et il ne s’agit plus d’isoler des régions physiques stricto sensu, mais de concevoir plutôt des « nuages de probabilité » dans lesquels la notion de trajectoire n’a plus de sens clair. l’électronBohr etl’électronSchödinger, que ne relient entre elles qu’une homonymie de surface forcée par le déroulement des affaires humaines. Dans ce cas, nous pourrons dire, par exemple, que l’on a cru que l’une d’entre elles existait alors qu’elle n’existait pas. Le probleme ne serait donc pas que nous entretenions des croyances fausses sur « l’électron » (ce qui nous ramenerait a la possibilité d’une référence hors description adéquate), mais que notre langage de surface nous faisait croire a une stabilité de la référence la ou en fait, nous nous réfé­rions a deux entités différentes : l’électronSchödinger et l’électronBohr — ce dernier s’étant avéré aussi peu existant que d’autres entités fantastiques qui peuplent l’histoire des sciences : « l’humeur noire », « l’éther » ou le « phlogistique ». La difficulté est alors que l’histoire de la connaissance humaine se dispersera aussitôt en une multitude incontrôlable d’entités produites par notre langage au travers des siecles et en perpétuel devenir (l’« électron » auquel nous croyons aujourd’hui n’est déja plus exactement l’électronSchödinger). Le monde rassurant des idées se révelera alors ni plus ni moins bariolé et fluctuant que celui du devenir sensible. Meme si Putnam lui-meme n’a guere utilisé son modele pour traiter des mathé­matiques12, le probleme qu’il souleve s’y transfere aisément et rejoint alors notre interrogation initiale. En mathématiques, non moins qu’ailleurs, on trouvera a travers l’histoire des descriptions incompatibles de certains objets. Nous y avons fait allusion en proposant de comparer Euclide et Dieudonné dans leurs traitements respectifs des opérations entre nombres et entre grandeurs. Pour prendre un exemple plus élémentaire, il est bien connu que les Grecs anciens ne considéraient pas l’unité comme un nombre (et encore moins le zéro, dont ils n’avaient pas idée). Nous pourrions croire qu’il s’agit alors simplement de « compléter » leur systeme numérique par des entités qu’ils auraient simple-ment « oubliées ». Mais c’est la une vue tres simpliste de l’histoire des mathé­matiques, sur laquelle les historiens ont attiré l’attention depuis longtemps. Son objectif était plutôt de défendre l’idée que les mécanismes de la référence ne passent pas par des descriptions — ce qui paraît d’abord une these assez curieuse lorsqu’on l’ap­plique aux mathématiques. Il a néanmoins, a plusieurs reprises, confirmé qu’a ses yeux, la vérité mathématique avait un statut quasi expérimental qui la rapprochait plus qu’on ne le croit ordinairement des sciences naturelles, cf. « What is mathematical truth ? », dans Hilary Putnam, Mathematics, Matter and Method. Philosophical papers, vol. 1, Cambridge University Press, Cambridge 1975, pp. 60–78. Comme le faisait déja remarquer Simon Stevin des la fin du XVIeme siecle13, la conception grecque entraînait, en effet, des conséquences « infortunées » dans leur maniere de penser le parallele entre nombres et grandeurs, dont les conséquences furent longues a etre perçues. En particulier, il leur était natu-rel de mettre face a face le point, « principe des grandeurs », et l’unité, « prin­cipe des nombres » (alors, rappelait Stevin, que c’est le zéro qu’il aurait fallu mettre a cette place pour assurer un parallele cohérent). Comme je l’ai indiqué précédemment, cette conceptionstructure la mathématique grecque et nourrit l’idée qu’il y a, en mathématiques, deux domaines d’objets incommunicables, que seules des analogies (au demeurant mal fondées) peuvent unir. L’évolution du concept de « nombre » au temps de l’algebre symbolique (Viete, Stevin, puis Descartes), qui elle-meme prenait la suite d’une longue tradition initiée avec les débuts de l’algebre arabe, ne consista donc pas seulement a compléter la conception du nombre : elle en changea aussi radicalement la nature (de sorte qu’il fut progressivement a meme de recouvrir l’entiereté du domaine ancien des « grandeurs »)14. C’est tout le probleme qui se pose quand on avance qu’Eu­clide et Dieudonné se réferent a l’occasion aux « memes » objets (par exemple a l’objet « nombres entiers »). On pourrait prendre bien d’autres exemples de ce phénomene tres répandu. Ainsi, pour rester sur des cas élémentaires, lorsque Blaise Pascal aborde les « sections coniques » d’un point de vue que nous dirions « projectif » et y integre le point (comme un cas de conique qu’on dirait aujourd’hui « dégénérée »), il fait bien plus que compléter la classification d’Apollonius. Il change profondément la nature des « coniques » en les voyant non comme des sections du cône, mais comme des projections du cercle. Or certaines propriétés du point ne sont pas compatibles avec celles des coniques conçues comme courbes. L’idée meme de « courbe » a d’ailleurs fortement évolué au cours du temps. Ainsi, de meme qu’il 257 13 Simon Stevin, « Arithmétique (1585) », dans The Principal Works of Simon Stevin, éd. E. Crone, E.J. Dijksterhuis, R.J. Forbes et al., 6 vol., t. 2 B, N. V. Swets & Zeitlinger, Amsterdam 1955–1966, p. 498. 14 Sur cette évolution, on peut renvoyer le lecteur philosophe au livre classique de Jacob Klein, « Die griechische Logistik und die Entstehung der Algebra(1934/1936) », dans Quellen und Studien zur Geschichte der Mathematik, Astronomie und Physik, Abteilung B: Studien. Band 3, Erstes Heft, Springer, Berlin 1934, pp. 18–105 und Zweites Heft, Berlin 1936, pp. 122–235). Notons au passage que les solutions de Viete et de Stevin relevent d’ailleurs de deux « ontologies » différentes pour la résolution de ce probleme. aurait été inconcevable a un ancien qu’une courbe (« dégénérée ») soit un point, de meme il aurait été incompréhensible a un auteur du XVIIeme siecle qu’une courbe (« monstrueuse ») puisse emplir entierement une surface. Or c’est ce que peut la courbe de Peano qui emplit le carré. L’idée de « surface » a évidemment connu le meme sort. Pour s’approcher plus pres de nous, un célebre théoreme de Nash, dit « de plongement », établit par exemple que toute surface orientée et fermée peut etre plongée, tout en préservant les distances, dans une boule de l’espace euclidien ordinaire de taille arbitrairement petite. Mais ceci n’aurait eu aucun sens il y a encore un siecle : comment imaginer que je puisse déformer continument la terre pour la faire entrer dans une balle de ping pong tout en préservant les distances? Ceci a pourtant été a la base de l’élaboration récente d’objets baptisés « fractales lisses »15. Or les « fractales » avaient précisément été conçues originellement au titre des objets « non lisses » (éventuellement continus, mais pas différentiables), etc. etc. A chaque fois, un meme type d’ob-jet est pourvu a différentes périodes de propriétés incompatibles, par lesquelles semble se manifester le progres de notre connaissance a son sujet. Les exemples sont nombreux de telles avancées ou il ne s’agit pas tant d’étendre un domaine que de nier certaines propriétés qu’on croyait appartenir a son « es­sence » (au sens le plus neutre qu’on puisse imaginer, disons a son « ce qu’il est »). Dans chaque cas, c’est l’écart entre expression et ontologie qui se mani­feste : aucun langage, aussi formel qu’il soit, n’est intégralement transparent a l’etre et, tel est le fond de l’argument de Putnam, il faut que nous puissions nous référer aux entités existantes indépendamment de l’expression par les­quelles nous tentons de les décrire. Ce mécanisme est le ressort de tout progres scientifique, en mathématiques comme ailleurs. Il est le soutien de son devenir a travers le temps et l’espace. Mais il interdit du meme coup de faire de telle ou 258 telle expression (dans le cas de Badiou, l’expression « ensembliste ») le dernier mot de « l’ontologie ». 15 Pour une présentation accessible a tous, voir Vincent Borrelli, « Gnash, un tore plat ! » — Images des Mathématiques, CNRS, 2012 (https://images.math.cnrs.fr/Gnash-un-tore-plat. html, consulté le 15 mars 2020). 2. Quel est le sens de la révolution cantorienne sous laquelle se place Alain Badiou ? 2.1. Langage et ontologie dans l’histoire de la théorie des ensembles Les remarques qui précedent sur le rapport entre langage et ontologie per-mettent de soulever une difficulté générale sur le rôle que sont censé avoir joué les « ensembles » dans l’histoire des mathématiques et, par voie de consé­quence, le rôle qu’ils peuvent jouer dans une interprétation philosophique comme celle que propose Badiou. De fait, l’émergence de ce paradigme chez des auteurs comme Cantor et Dedekind se fait sous l’égide d’une théorie que l’on décrit rétrospectivement comme « naive » et qui produit ses effets les plus remarquables avant d’avoir été axiomatisée. Il y a donc un raccourci a dire que les ensembles ont procuré « un langage universel pour toutes les branches des mathématiques », comme l’avance a juste titre L’Etre et l’événement16et de faire comme si on parlait alors de la théorie,au sens axiomatique du terme (théorie dite aujourd’hui « ZFC », pour « Zermelo-Fraenkel avec axiome du Choix »). Le développement du langage et celui de la théorie (du moins sous sa forme axio­matique) correspondent, en effet, a deux moments différents de l’histoire. Or, nous apprennent les historiens, le succes de l’un fut relativement indépendant de l’autre. On peut ici rappeler la mise en garde de José Ferreirós, auteur d’une histoire de la théorie des ensembles qui pointe une importante distinction a faire entre « la théorie des ensembles comme branche autonome des mathé­matiques — comme lorsque l’on parle de théorie des ensembles transfinis ou de théorie des ensembles abstraite — et la théorie des ensembles comme outil de base ou langage pour les mathématiques : l’approche ou le langage ensem­bliste»17. Et Ferreiros de rappeler : « comme indiqué précédemment, la théorie des ensembles abstraite vint a l’existence apres que l’approche ensembliste eut commencé a se développer, et non le contraire »18. 259 Ainsi les ensembles forment d’abord un langage qui est introduit, par Cantor et Dedekind, pour traiter de problemes mathématiques particuliers, en l’oc­currence des questions de convergence des séries trigonométriques et des 16 Badiou, L’Etre et l’événement, p. 49. 17 José Ferreirós, Labyrinth of thought. A history of set theory and its role in modern mathemat­ics, Birkhäuser, Basel-Boston-Berlin 1999, p. xix (ma traduction). 18 Ibid. problemes de divisibilité dans les systemes de nombres. C’est dans le premier domaine que Cantor obtient ses résultats les plus marquants des le début des années 1870, notamment son grand théoreme d’unicité qui établit que deux sé­ries trigonométriques qui ont la meme limite simple ont les memes coefficients. A la meme époque, Dedekind élabore les premieres versions de sa « théorie des idéaux », sous-structure de ce que nous appelons aujourd’hui un « anneau » et qui permet d’y préserver dans un cadre général la possibilité d’une décomposi­tion unique en éléments premiers (sous réserve de redéfinir dans ce cadre ce a quoi correspondent les « nombres premiers »). Si le second contexte forme les prémices du développement de l’« algebre mo-derne », le premier voit apparaître chez Cantor un certain nombre de concepts relatifs aux « ensembles de points », c’est-a-dire a ce que nous désignerions au­jourd’hui sous le nom de « topologie générale » (comme le concept d’ensemble « parfait », « dérivé », « dense », etc.). C’est un des domaines ou la théorie va connaître ses premiers succes aupres des contemporains, bien plus que dans le domaine de l’arithmétique de l’infini, ou elle suscite au mieux l’indifférence, au pire la défiance. On le voit tres bien en regardant sa réception en France aupres d’auteurs comme Lebesgue, Borel ou Baire (avant l’élaboration d’une théorie axiomatique par Zermelo en 1908) — artisans de la théorie moderne de la mesure et parfois qualifiés de « semi-intuitionnistes ». De fait, ces auteurs, pourtant grands défenseurs du langage mis au point par Cantor, n’en refusent pas moins l’utilité de la théorie abstraite que ce dernier cherche ensuite a déve-lopper sous la forme d’une arithmétique transfinie. A propos de Baire, Hélene Gispert rappelle qu’« il adopte une démarche radicalement différente de celle de Cantor qui cherche a dégager sa généralisation du concept de nombre de ses premieres considérations sur les ensembles de points et ne privilégie d’aucune 260 façon l’infini dénombrable dans son étude du transfini »19. Elle cite a ce propos une lettre a Borel de 1905 ou Baire évoque la « corvée assommante»que repré­sente l’article qu’il doit écrire sur les ensembles pour la version française de l’Encyclopédie des sciencesmathématiques : En écrivant mes Leçons sur les fonctions discontinues, j’avais les coudées franches, j’exposais de la maniere qui me paraissait la plus claire les théories Hélene Gispert, « La théorie des ensembles en France avant la crise de 1905 : Baire, Borel, Lebesgue... et tous les autres », Revue d’histoire des mathématiques 1 (1/1995), p. 58. dont j’avais l’intention de me servir. Je ne suis plus ici dans les memes condi­tions, je n’ai plus le droit de faire dévier la pensée de G. Cantor. Il me faut bon gré, mal gré, parler de l’addition, de la multiplication des types ordinaux, etc., choses dont je ne connais pas la moindre application. Je ne peux tout de meme pas en inventer [Lettres, p. 83. C’est moi qui souligne]20. On aurait tort de croire qu’il s’agit la de quelques résistances rapidement sur­montées et que la théorie axiomatique aurait justement balayées. Aujourd’hui encore, c’est la meme réponse qu’on recevrait de la part de nombre de mathéma­ticiens. Je cite, par exemple, ce que dit a ce sujet Yves André dans ses Leçons de mathématiques contemporaines — dont un des chapitres est significativement intitulé : « (Non-)influence de la Théorie des ensembles sur les Mathématiques ». Que retiennent les Mathématiques de tous ces travaux sur les multiplicités infi­nies ? La réponse est double, et tres tranchée. En ce qui concerne l’usage et le langage (élémentaire) des ensembles, a la ma-niere de Dedekind disons, ils ont envahi toutes les Mathématiques. L’usage des ensembles a ouvert la voie a la Topologie générale, a la Théorie de la mesure, et a l’Analyse fonctionnelle (ou l’on traite d’ensembles de fonctions comme s’il s’agissait de points d’un espace). D’autre part, le langage des ensembles, sous l’impulsion de Bourbaki notamment, a beaucoup contribué a la précision du lan-gage mathématique en général. […] En revanche, en ce qui concerne les travaux de Cantor sur la combinatoire transfinie, l’axiomatique ensembliste et tous les développements ultérieurs de la Théorie des ensembles, les Mathématiques (hors Théorie des ensembles et Lo-gique) n’en retiennent quasi-rien21. 261 On voit donc que les historiens, comme les acteurs de l’époque et comme ceux de notre temps, s’accordent tous sur l’importance qu’il peut y avoir a distinguer langage et ontologie ensembliste (au sens de ce que porte la théorie formelle axiomatisée). Si le triomphe de la théorie des ensembles comme langage est in­ 20 Cité par Gispert, ibid. L’édition des lettres est : Lettres de René Baire a Émile Borel, Cahiers du séminaire d’histoire des mathématiques 11 (1990), pp. 33–120. 21 Yves André, Leçons de Mathématiques contemporaines a l’IRCAM, IRCAM, France 2009, p. 104 (archives ouvertes. https://cel.archives-ouvertes.fr/cel-01359200/document (consulté le 25 janvier 2020). C’est moi qui souligne. déniable et si l’on peut tout a fait soutenir que ce langage a permis de mieux ex-primer certains aspects de la réalité mathématique, cela ne nous dit encore rien sur cette réalité elle-meme — surtout s’il s’agit de voir dans la théorie axioma­tique l’expression d’une telle ontologie du « multiple pur » et de son reglement22. Un point sur lequel je voudrais insister est que cette situation n’est nullement spécifique a la Théorie des ensembles. A de tres nombreuses reprises dans l’histoire des mathématiques, on assiste a la mise au point d’un langage qui s’avere fécond et qui va etre adopté par un grand nombre de mathématiciens, alors meme que ces mathématiciens sont en désaccord sur le type d’entités qui est associé a ce langage (comme on vient de le voir avec l’exemple de Baire et de Cantor). Ceci rend la tâche particulierement ardue pour le philosophe qui pense qu’il lui revient d’expliciter le discours « ontologique » sous-jacent. J’ai déja cité a ce propos le langage euclidien des proportions, contesté des l’Antiquité tar-dive par ceux qui estimaient que nombres et grandeurs pouvaient tout a fait « communiquer » (mouvement qui fut continué et amplifié dans les mathéma­tiques arabes), mais on peut aussi penser au langage algébrique cartésien et aux débats qui s’ensuivirent pour savoir si les courbes décrites par Descartes étaient, comme il le prétendait, les seules entités dignes d’etre reçues dans la « géométrie », au calcul différentiel leibnizien et aux débats qui déchirerent ses premiers défenseurs pour savoir si l’algorithme nécessitait d’accepter ou non d’authentiques entités infinitésimales, au langage des développements en série et a la question de savoir si toute « fonction » est exprimable sous cette forme, au langage des « epsilon/delta » et aux querelles sur l’existence d’entités ne satisfaisant pas l’axiome d’Archimede, jusqu’aux débats agitant au début du XXeme siecle les topologues pour savoir si l’approche par les « ensembles de points » était la meilleure pour capturer la notion d’espace ou si, au contraire, elle nous faisait perdre l’essentiel (par contraste avec la topologie qu’on appelait alors « combinatoire » et qu’on appelle aujourd’hui « algébrique »)23. A chaque 22 Je n’ai pas la place de développer ce point ici, mais il est non moins remarquable que le développement de la théorie dite « descriptive » des ensembles se soit fait en grande partie par une réflexion tres poussée, jusque dans ses attendus théologiques, sur les rapports entre langage et etre — et plus précisément encore sur la question de la nomination. Je renvoie sur ce point a l’étude de Jean-Michel Kantor et Loren Graham, Au nom de l’infini, Éditions Belin, Paris, 2010. Sur les réticences que certains topologues ont pu avoir a accepter que les notions spatiales soient adéquatement capturées dans le langage ensembliste, voir M. Bélanger et J.-P. Mar­quis, « Menger and Nöbeling on pointless topology », Logic and Logical Philosophy22, (2/2013), pp. 145–165. fois, un langage triomphe (souvent assez rapidement s’il permet la résolution de problemes latents) tandis que les débats font rage sur le type d’entités auquel il nous engage « vraiment ». Meme si l’on conçoit l’« ontologie » proprement dite comme un méta-discours qui aurait a trancher dans ces débats, le probleme majeur est qu’il ne s’agira nullement de trancher seulement sur le dernier venu et qu’il faudra prendre position sur tous les débats advenus dans l’histoire des mathématiques. Un point clef qui se dégage des considérations précédentes est qu’il ne semble donc pas qu’on puisse conclure de l’adoption d’un langage a la donnée conco­mitante d’une ontologie et que ceci vaut de la théorie des ensembles comme de la plupart des théories antérieures. L’histoire des mathématiques nous offre de nombreux exemples ou l’on voit un discours se rapportant en apparence a un do-maine d’objets etre soumis a des discussion sur la nature du domaine qu’il vise réellement— exactement comme on a vu les physiciens se quereller sur la nature exacte de ce qu’est un « électron ». Mais au-dela du cas particulier des contro-verses, il arrive bien plus souvent encore — c’était notre constat de départ — que telle ou telle reformulation postérieure fasse apparaître la distinction entre différents domaines d’objets comme simple « effet de langage ». Ceci est meme inhérent a la lecture rétrospective que les mathématiciens portent sur leur dis­cipline et au fait qu’il leur faut alors tenir que si tel ou tel pan des mathéma­tiques a été formulé dans telle ou telle terminologie, il parlait en fait « toujours déja » des memes objets que ceux que nous reconnaissons aujourd’hui (sans quoi, remarquons-le, il ne sera pas possible de dire que tel ou tel théoreme a réellement été démontré dans une période antérieure, faute de porter sur les memes choses). Pis, comme j’ai essayé de l’indiquer, il semble que toute posi­tion, comme celle de Badiou, qui voudrait a la fois s’appuyer sur une théorie en vigueur pour délivrer un sens « ontologique » profond et tenir qu’une telle théo­rie n’a pas simplement une portée régionale, mais capture tous les sens d’etre déployés par « les mathématiques » a travers leur histoire, soit dans l’obligation de s’appuyer sur un tel écart entre langage et ontologie. Le probleme, j’y insiste, est doncinterne a la perspective développée par Badiou. 2.2. De quoi parle-t-on lorsqu’on manipule des « ensembles » ? Dans le cas de la théorie des ensembles, on pourrait rendre les considérations qui précedent plus précises, comme l’évoque Yves André dans le passage de la citation précédente que j’ai omis : Au reste, la plupart des ensembles considérés par les mathématiciens sont des ensembles définis « en compréhension », pour lesquels l’appartenance veut dire, concretement, satisfaire une certaine propriété explicite ; a ce niveau basique, les ensembles n’offrent guere qu’un langage « réaliste » un peu plus commode que le maniement logique des propriétés elles-memes24. Ainsi certains auteurs comme Stewart Shapiro considerent que les mathé­matiques ensemblistes parlent, en réalité, d’objets qui ne sont pas des « en­sembles », mais des « structures » et qui doivent etre décrites dans un cadre qui n’est pas a proprement parler ZFC (formulé au premier ordre avec des schémas d’axiomes), mais la logique du second ordre25. Dans ce cas, « l’objet » propre de la théorie que nous interprétons comme « ensembles » doit plutôt etre ressai­si, ainsi que le rappelle Yves André, comme une maniere commode de parler d’autre chose (des propriétés et des relations). Les mathématiques, comme cela a été soutenu a de nombreuses reprises dans l’histoire, ne serait qu’une science des relations26. Un autre exemple qui vient immédiatement a l’esprit étant donnée l’orientation générale de l’ouvre de Badiou est l’alternative proposée a la théorie des en­sembles par la « théorie des catégories », qui elle aussi met fortement en avant le caractere relationnel des entités mathématiques. Sous ce point de vue, il peut etre utile de rappeler tout d’abord que le rapport Théorie des ensembles/Théo­rie des Catégories n’est pas simplement ici, comme l’estiment trop souvent les philosophes, de querelle sur la prétention a « fonder » les mathématiques. Il peut également toucher l’interprétationdes memes objets, en l’occurrence des 24 Yves André, Leçons de Mathématiques contemporaines a l’IRCAM, p. 104 25 Stewart Shapiro, Foundations without Foundationalism: A Case for Second-Order Logic, Oxford University Press, Oxford 1991. 26 De fait, c’est la une lecture assez répandue des mathématiques « structurale » et il paraît effectivement difficile d’identifier une structure comme celle de « groupe », par exemple, a un objet du type « ensemble », meme si tout groupe peut etre interprété comme un en­semble. Cela tient au fait que nous voulons précisément que cette structure puisse valoir d’ensembles de natures tres différentes, notamment en ce qui concerne leur cardinalité (ensembles finis ou infinis, groupes discrets ou continus). En identifiant la structure a l’en­semble, nous commettrions donc un abus de langage qui permet de se rapporter a toutes ces interprétations possibles dans un langage qui ne porterait plus sur des relations, mais sur des objets – ou, comme le dit Yves André, nous adoptons un langage « réaliste » com­mode pour parler « en fait » de systemes de propriétés. « ensembles ». Il existe notamment des versions purement catégoriques de ZFC, comme la « théorie algébrique des ensembles » (TAE) développée par Joyal et Moierdijk, ainsi que par Awodey, qui montrent que l’on peut se situer dans un meme cadre fondationnel (au sens ou il s’agit dans les deux cas d’exprimer ZFC) et se rapporter en apparence aux memes entités (les « ensembles » au sens de ZFC) selon deux interprétations (on voudrait dire « ontologies ») différentes. Comme l’indique Brice Halimi, la TAE « incarne une combinaison heureuse de théorie des ensembles et de théorie des catégories ». Et de préciser : D’un point de vue ensembliste, une application est un ensemble : il n’y a pas de fleche. Les seules fleches correspondent aux aretes du graphe d’appartenance propre a l’univers ensembliste d’arriere-plan [..]. Le point de vue de la théorie algébrique des ensembles consiste, au contraire, a mettre en vedette les fleches, a l’aide du « foncteur de codomaine » — ce qui est une perspective typique de théorie des catégories. La théorie des catégories fibrées permet ainsi de ressaisir la théorie des ensembles tout en la transformant en une théorie fondée sur les fleches plutôt que sur les objets27. Ainsi la théorie algébrique des ensembles peut etre décrite comme « la greffe de la théorie des catégories a ZFC, ce qui est bien plus fécond que la rivalité habi­tuelle entre théorie des ensembles et théorie des catégories »28. Cette remarque s’inscrit dans une réflexion plus vaste sur le changement de point de vue opéré avec les catégories. Ainsi la théorie des topos — puisque c’est un aspect de la théorie des catégories auquel s’intéresse particulierement Badiou — peut etre vue, elle aussi, comme une « théorie des ensembles », mais locale29. Ici ce n’est plus au sens de ZFC, mais au sens ou l’on extrait un noyau 265 27 Brice Halimi, « Sets and Descent », dans Objectivity, Realism and Proof, dirs. A. Sereni & F. Boccuni, pp. 123–142, Springer, Basel 2016. Ma traduction et mes italiques. 28 Ibid. 29 Un topos est une catégorie, c’est-a-dire une collection des fleches et d’objets, munie de propriétés additionnelles concernant l’existence de certaines fleches associées a la collec­tion initiale. Une maniere condensée de l’exprimer est de dire que cette catégorie possede toutes les limites et colimites finies, les exponentielles, ainsi qu’un objet distingué appelé « classificateur de sous-objets » (qui, comme son nom l’indique, permet de capturer au moyen de diagrammes l’idée qu’un objet est un « sous-objet » d’un autre). Les topos qui intéressent particulierement Badiou sont les « topos de Grothendieck », un cas particulier de la définition générale précédente. Ils peuvent etre définis comme des faisceaux d’en­ opératoire a toute « théorie des ensembles », y compris des versions plus faibles que l’axiomatique qui a fini par s’imposer au début du XXeme siecle. C’est le point de vue qui a été développé notamment par J. Bell dans son livre Toposes and local set theories30 (qui prolonge le point de vue initial de Lawvere sur l’idée qu’un topos exemplifie l’idée d’ensembles variables). Dans un exposé précédent, j’avais essayé d’indiquer pourquoi cette idée d’une mathématique « locale » (par opposition aux mathématiques « absolues »)31 peut etre féconde et comment elle court-circuite ce qui peut apparaître chez Badiou comme un raccourci, a savoir l’alternative ou il cherche souvent a en-fermer son interlocuteur : « soit l’absolu, soit le relativisme ». De fait, la théorie des ensemblelocale permet de donner un sens parfaitement bien déterminé a la notion de vérité locale et elle donne lieu non a un relativisme, mais a une théo­rie de la relativité (mathématique)32. Bien plus, l’idée de vérité « locale » paraît présupposée par la notion de vérité « absolue » plutôt que le contraire : dans le systeme de Badiou, appuyé tres fortement sur l’idée de Cohen que l’on peut « forcer » certaines vérités dans des modeles de l’univers ensembliste, le point de départ est précisément que l’ensemble des vérités ne peut pas etre déployé devant nous une fois pour toutes et a priori33. Une vérité doit apparaître dans un ou plusieurs mondes (c’est tout le sens du projet de Logiques des mondes d’ex­pliciter les modalités de cette apparition). Son « absoluité » (au sens que donne Badiou aux vérités « éternelles ») est alors simplement postulée a partir du fait de sa réactivation possible dans d’autres mondes, voire dans tout autre monde (c’est ce que Badiou présente comme une relecture de la doctrine cartésienne de la « création des vérités éternelles »)34. sembles sur un site (une généralisation des espaces topologiques). Sous ce point de vue, un topos est une maniere de voir des ensembles qui varient d’une région a l’autre d’un 266 30 espace et se recollent selon certaines regles de compatibilité. J. L. Bell, Toposes and local set theories: An introduction, Oxford Logic, Guides: 14, Claren­ don Press, Oxford 1988. 31 Bell lui-meme en a développé le programme philosophique dans « From Absolute to Local Mathematics », Synthese 69 (3/1986), pp. 409–426. Voir mon analyse dans : « Tous en­ semble ? Sur le rapport d’Alain Badiou aux mathématiques » dans Autour d’Alain Badiou, dirs. F. Tarby et I. Vodoz, pp. 81–102, Germina, Paris 2011. 32 Le parallele avec la théorie de la relativité est au cour de l’article de John Bell cité dans la note précédente. Voir également, René Guitart, « Caractere global et caractere local de la vérité », conférence donnée a la Lysimaque, 23 septembre 1990 (disponible sur le site de l’auteur : http://rene.guitart.pagesperso-orange.fr/preprints.html), en particulier p. 8 pour la référence a la théorie de la relativité et l’opposition relativité/relativisme. 33 C’est justement le point de départ de Bell dans l’article cité note précédente. Je ne reviendrai pas sur ces discussions autour de l’approche « locale » des mathématiques ici et me concentrerai sur un autre point : la maniere dont elle modifie le rapport entre multiplicité spatiale et multiplicité numérique. De fait, l’idée d’une théorie des ensembles « locales », et, plus simplement encore, le nom meme de topos, suppose une forme de revanche du spatial dans la com-préhension du « multiple pur », qui réactive la tension entre nombre et espace dans le paradigme ensembliste. 3. Espace et nombre, a nouveau Le fait que les catégories, et en particulier la notion de topos, ait pu redistribuer les rapports entre nombre et espace, discret et continu, arithmétique et géo­métrie, etc., a été pointé par de nombreux auteurs, a commencer par l’inven­teur de cette derniere notion, Alexandre Grothendieck. Je cite un passage bien connu de Récoltes et Semailles : C’est le theme du topos qui est ce « lit » ou viennent s’épouser la géométrie et l’algebre, la topologie et l’arithmétique, la logique mathématique et la théorie des catégories, le monde du continu et celui des structures « discontinues » ou « discretes ». Il est ce que j’ai conçu de plus vaste, pour saisir avec finesse, par un meme langage riche en résonnances géométriques, une « essence » commune a des situations des plus éloignées les unes des autres35. 267 34 Les différents sens de l’absoluité, dans lesquels je ne peux entrer ici, sont au cour du troisieme tome de l’Etre et l’événement intitulé L’immanence des vérités. Celui qui nous venons d’évoquer (qui se marque du fait qu’un événement se transcrit dans une ouvre dotée du maximum d’existence dans un monde) n’est qu’un aspect d’une absoluité qui s’atteste plus précisément du rapport qu’une ouvre de vérité entretient avec « l’absolu », c’est-a-dire « l’ensemble de tous les ensembles » – ou plus précisément avec un « attribut de l’absolu », dont l’ouvre témoigne par la structure intriquée des infinis qu’elle implique (voyez notamment le chap. VII de l’Immanence des vérités, Fayard, Paris, 2018). 35 Alexander Grothendieck, Récoltes et Semailles, p. 59. Ce texte est encore inédit, mais il est aisé de s’en procurer une version numérique sur divers sites internet (par exemple ici : https://webusers.imj-prg.fr/~leila.schneps/grothendieckcircle/recoltesetc.php). On trouve des déclarations tres similaires chez Mac Lane et Moerdijk, mais avec une référence additionnelle a Cohen qui n’est pas sans intéret en contexte ba­diousien: Un aspect frappant de la théorie des topos est qu’elle unifie deux domaines ma-thématiques en apparence completement distincts : la topologie algébrique et la géométrie algébrique, d’un côté, la logique et la théorie des ensembles de l’autre. De fait, un topos peut etre considéré a la fois comme un « espace généralisé » et comme un « univers ensembliste généralisé ». Ces différents aspects ont émergé indépendamment vers 1963 : avec A. Grothendieck et sa reformulation de la théorie des faisceaux en géométrie algébrique, avec William F. Lawvere dans sa recherche d’une axiomatisation de la catégorie des ensembles et de celle d’ensembles « va­riables », et avec Paul Cohen dans son usage du forcing pour construire des nou­veaux modeles de la théorie des ensembles de Zermelo-Fraenkel36. Le point sur lequel j’aimerais insister est l’évocation par Grothendieck d’une « essence commune » a des situations qu’il dit « les plus éloignées les unes des autres » – ou, chez Mac Lane et Moerdijk de l’unification de « deux do-maines mathématiques en apparence completement distincts ». Nous avons la un exemple typique de la maniere dont on réinterprete un langage de surface en disant qu’en fait il ne parlait pas de ce qu’on croyait, qu’il y a derriere une « es­sence » cachée qui permet de saisir l’unité des « genres » d’etre antérieurement distingués37. Bien plus, ce geste n’est pas sans affinité avec celui qui reprochait aux mathématiques grecques la scission maintenue entre discret et continu. Mais dans ce cas, ce n’est plus sur la mathématique ancienne que porte cette reformulation : elle portesur la mathématique ensembliste elle-meme. Grothen­dieck, pas plus que Lawvere, et tant d’autres mathématiciens aupres eux, ne 268 croient que ZFC soit la meilleure maniere d’exprimer des entités sous-jacentes a ce que le langage séparait encore le long de la frontiere entre arithmétique et géométrie (soit exactementle meme argument que celui que pouvait faire 36 Saunders Mac Lane et Ieke Moerdijk, Sheaves in Geometry and Logic. A First introduction in topos theory, Springer-Verlag, New York 1992, Prologue p. 1. Ma traduction. Ce vocabulaire de « l’essence cachée » est bien plus répandu en mathématiques qu’on ne pourrait le croire. Mark Wilson en a fait le ressort d’une analyse philosophique du dis-cours mathématiques dans son article « Frege: The Royal Road From Geometry », Nous 26 (2/1992), pp. 149 –180. valoir un mathématicien « ensembliste » en se rapportant aux mathématiques anciennes !). C’est en ce point, bien plus que dans le fait que la théorie des catégories nous livre en apparence une ontologie des relations plutôt que des objets, des mor­phismes plutôt que des éléments, que l’on devrait, me semble-t-il, faire porter la confrontation. Avant meme de prétendre qu’elle nous livre une nouvelle vi­sion de l’« ontologie » (qu’en sait-on si l’on s’en tient a son seul vocabulaire descriptif ?), nous devons dire que la théorie des catégories nous montre, une nouvelle fois, que ce que nous avions pris pour principe de l’élucidation onto-logique ultime ne l’était pas et peut etre reformulée a son tour dans une théorie dont l’objet apparent est différent (qui elle-meme se trouvera reformulée dans d’autres langages qui modifieront éventuellement l’interprétation de la réalité visée, etc., etc. Nous en verrons un autre exemple sous peu)38. Dans la derniere partie de cet article, je voudrais développer plusieurs re-marques relatives a ce constat. Tout d’abord, l’unification invoquée en lien avec l’émergence destopos nous conduit a relire le développement de la théorie des ensembles elle-meme. De fait, la tension entre nombre (multiplicité numérique) et espace (topologique) pourrait y etre constitutive (a la maniere dont la refor­mulation ensembliste elle-meme nous avait permis de voir que la coupure entre nombre et grandeur était constitutive de la rationalité mathématique grecque classique). La question centrale évoquée par Grothendieck et Mac Lane n’est pas de savoir si on décrit les objets mathématiques en termes d’éléments plutôt qu’en termes de fleches – elle est de savoir si on a manqué quelque chose de l’unité des objets mathématiques quand on les a décrit dans les termes d’une 269 38 Remarquons que l’idée qu’une science est soumise a des progres constants dans l’ex­pression de la réalité qu’elle vise laisse ouverte plusieurs options métaphysiques. On peut y voir, bien sur, un témoignage de l’existence d’une réalité immuable qui résiste a ces expressions, qui en forment autant d’approximations. Une telle position conduit assez naturellement a une forme de platonisme, dit parfois « naif », que l’on retrouve chez nombre de mathématiciens au travail. Mais on peut aussi considérer que la position d’objet est immanente a l’expression elle-meme et que sa « réalité » n’est pas a trouver dans une entité indépendante et séparée, mais dans le mécanisme de l’expression ou de la visée elle-meme – une position qu’ont pu tenir nombre d’interlocuteurs de Badiou sous des formes tres différentes (Desanti, Deleuze, certains relativistes d’inspiration wit-tgensteinienne ou latourienne). théorie comme ZFC39. Et leur réponse est sans ambiguité : on manque quelque chose de l’essence commune du nombre et de l’espace parce que les ensembles n’ont justement pas réussi a vraiment unifier ces deux types d’entités (c’est-a-dire autrement que dans un langage que la formulation catégorique aura per-mis de dépasser). Mais cela signifie notamment que l’ontologie sous-jacente a la théorie des ensembles (quelle qu’elle soit) n’est précisément pas exprimée adéquatement par ce langage. On peut certes contester cette affirmation. Mais il paraît difficile de le faire, comme semble y etre contraint Badiou, en tenant exactement la meme position a propos de la mathématique antérieure (qui ex-primerait « en réalité » des multiples purs). Un autre point particulierement intéressant est que la tension entre multiplicité numérique et multiplicité spatiale, que la notion de topos est censée permettre de dépasser, rejoint une ligne de démarcation entre deux orientations philoso­phiques. On peut l’associer a deux propositions divergentes pour une « ontolo­gie » du multiple pur telles que thématisées par Deleuze et Badiou. Badiou a lui-meme mis en scene cette opposition a plusieurs reprises et fait valoir ce qu’il considere comme la faiblesse d’une position s’appuyant sur des intuitions spa-tiales originaires. Son argument est précisément que les modeles spatiaux, ty­piquement les modeles différentiels dont raffole Deleuze, apparaissent comme un cas tres particulier de ce qu’on peut exprimer dans un cadre ensembliste40. De fait, tant qu’on reste confiné a un langage ensembliste, ce constat semble s’imposer. J’ai déja eu l’occasion d’indiquer qu’il y a, cependant, une réponse possible a l’objection et que cette réponse consiste non pas a défendre la pri­mauté du continu et du différentiel, comme l’ont fait nombre de deleuziens, mais a s’installer dans la tension entre discret et continu – une version du pla­tonisme qu’a illustrée remarquablement la perspective d’Albert Lautman dont 270 se réclame justement Deleuze (tandis que Badiou serait plutôt, a mon sens, du côté de l’héritage de Jean Cavailles). Je voudrais revenir brievement sur cette réponse dans le cadre que j’ai retracé ci-dessus ou c’est le langage ensembliste 39 On peut d’ailleurs remarquer, comme y a insisté Bernard Vitrac dans sa traduction des Éléments, qu’Euclide dispose déja d’un langage « commun » qui lui permet de décrire les nombres et les grandeurs (par exemple a partir de la notion de « multiplicité »/pléthos, mais aussi des rapports tout/parties, ajouter/soustraire, etc.). 40 Voir, outre son Deleuze : La clameur de l’etre (Hachette Littératures, Paris 1997), l’article donné par Alain Badiou pour le numéro spécial « Badiou/Deleuze » de la revue Futur An-térieur (43, avril 1998) sous le titre : « Un, multiple, multiplicité(s) ». lui-meme qui s’avere porter cette tension. Enfin, je voudrais indiquer en conclu­sion que le développement récent des mathématiques accompagne, comme on pouvait s’y attendre un nouveau changement de paradigme dont la philosophie n’a pas encore pris toute la mesure et qui revisite a son tour la reformulation catégorique elle-meme (au sens de Grothendieck et Mac Lane), pour la ressaisir a un niveau d’unification « supérieur ». Or, comme j’y insisterai en conclusion, la référence spatiale y tient une place encore plus prépondérante que dans le cadre toposique. Sur le premier point, je passerai vite puisque j’y ai déja fait allusion en rappe­lant les travaux de Cantor et leur réception, mais il est important de voir que le langage ensembliste se prete immédiatement a deux formes de thématisations différentes, que l’on peut dire « topologique » et « ordinale »41. Quand on suit la préhistoire de la Théorie des ensembles, on ne peut qu’etre frappé de consta­ter que cette thématisation est déja présente chez Riemann dans un passage célebre sur les mannigfaltichkeiten. Or ce passage se trouve etre également le point de départ de Deleuze des le Bergsonisme (1966) : Les concepts de grandeur ne sont possibles que la ou il existe un concept général qui permette différents modes de détermination. Suivant qu’il est, ou non, pos­sible de passer de l’un de ces modes de détermination a un autre, d’une maniere continue, ils forment une multiplicitécontinue ou une multiplicité discrete. […] Une partie d’une multiplicité, séparée du reste par une marque ou par une li-mite, s’appelle un quantum. La comparaison des quanta au point de vue de la quantité, s’effectue, pour les grandeurs discretes, au moyen du dénombrement ; pour les grandeurs continues, au moyen de la mesure. La mesure consiste dans une superposition de grandeurs a comparer ; il faut donc, pour mesurer, avoir un moyen de transporter la grandeur qui sert d’étalon de mesure pour les autres. 271 Si ce moyen manque, on ne pourra alors comparer entre elles deux grandeurs, que si l’une d’elles est une partie de l’autre, et encore, dans ce cas, ne pourra-t-on décider que la question du plus grand ou du plus petit, et non celle du rapport numérique. Les recherches auxquelles un tel cas peut donner lieu forment une 41 En fait, on devrait dire trois en ajoutant la thématisation algébrique dont Dedekind s’em­parait au meme moment. Cela permettrait, par ailleurs, de rappeler le souvenir des trois « structures meres » bourbakistes (algébrique, topologique et d’ordre). Mais je laisserai cette troisieme thématisation de côté parce que sa jonction avec la seconde a été immé­diate et s’est plus facilement opérée. branche générale de la théorie des grandeurs, indépendante des déterminations métriques, et dans laquelle elles ne sont pas considérées comme existant indé­pendamment de la position, ni comme exprimables au moyen d’une unité, mais comme des régions dans une multiplicité42. Le point clef est que ces différentes thématisations ont conduit, entre autres choses, a des axiomatiques distinctes comme celle de l’arithmétique transfinie cantorienne d’un côté, formalisée par Zermelo et ses successeurs (qui conduit a « ZFC »), et celle de la topologie ensembliste formalisée initialement par Hausdorff et Weyl de l’autre (qui conduit a la définition en termes de régions ou de « voisinages » des variétés, puis plus généralement des « espaces topolo­giques »). Le fait que la premiere axiomatise en général le langage des ensembles dans le­quel se trouve expriméela seconde (comme on le voit chez Hausdorff, mais pas chez Weyl) peut d’abord donner l’impression qu’on a affaire a une unification ontologique. Mais cette impression doit etre nuancée par deux remarques im­portantes. La premiere est que les différentes propriétés des espaces topolo­giques qui vont alors émerger comme centrales (connexité, compacité, conti­nuité, etc.) ne paraissent pas analytiquement dérivables, pour reprendre le vocabulaire kantien, du concept de multiplicité pure – a la différence de ce qui se passe pour les concepts numériques. Autant il est aisé de dériver du concept pur de « multiple » ou d’« ensemble » l’idée de zéro et de successeur, puis d’édi­fier progressivement sur cette base tous les systemes de nombres (a l’aide d’un certain nombre de fonctions et de relations), ainsi que leurs structures opéra­toires, autant il paraît difficile d’en tirer l’idée qu’un espace est, par exemple, « d’un seul tenant » (« simplement connexe »). Pour l’expliquer, prenons, a nouveau, un exemple élémentaire : peut-on dire que les entiers naturels forment un ensemble « d’un seul tenant »? Peut-on dé-river cette propriété de l’analyse conceptuelle de cette multiplicité ? La réponse va dépendre des régions que nous allons considérer. Car le multiple « entiers naturels » ne porte avec soi aucune indication des « régions » qu’on peut y discerner. Si, par exemple, l’ensemble lui-meme est la seule région considérée 42 Bernhard Riemann, « Sur les hypotheses qui servent de fondement a la géométrie », Ouvres mathématiques, trad. Laugel, Gauthier-Villars, Paris 1898, p. 282. (avec la région « vide »), il va sans dire qu’il est d’un seul tenant ! C’est ce qu’on appelle aujourd’hui la « topologie grossiere » (de fait, tout ensemble muni de la topologie grossiere est simplement connexe). Si, en revanche, nous consi­dérons l’ensemble des entiers comme « héritant » de la topologie usuelle de la droite numérique réelle dans laquelle nous avons l’habitude de le voir plongé (les « régions » sont donc les points a coordonnées entieres sur la droite), alors l’ensemble des entiers naturels sera cette fois … totalement déconnecté ! Il sera meme le prototype d’espace discret. Cette situation crée des phénomenes intri­gants pour ceux qui peinent a distinguer ces thématisations et dont la valeur pédagogique est bien connue. Ainsi, on apprend aux débutants qu’un ensemble comme le « triadique » de Cantor peut avoir la puissance du continu (selon la thématisation numérique), alors qu’il est pourtant discontinu (selon la théma­tisation topologique)43. On leur apprend également a munir la droite réelle de topologies différentes de la topologie usuelle, par exemple d’une topologie qui la rende « non séparée », c’est-a-dire dans laquelle la condition partes extra partes n’est pas satisfaite – alors meme que la droite réelle apparaît comme le cas le plus simple, a une dimension, de l’espace « ordinaire », caractérisé par le fait que l’on peut y découper des parties extérieures les unes aux autres (partes extra partes). Il est d’ailleurs possible de munir la droite réelle d’une topolo­gie qui la rende « discrete », par un procédé symétrique a celui par lequel on avait rendu les entiers « d’un seul tenant » : il suffit de prendre tous les points comme « régions » de notre découpage. Le simple fait que l’on puisse ainsi mu-nir un « multiple » de différentes topologies, aux propriétés incompatibles entre elles, montre bien qu’il ne saurait s’agir d’une dérivation « analytique » au sens kantien (les différentes propriétés considérées ne sont pas dérivables du seul concept du multiple considéré). La seconde remarque, plus historique, tient au constat suivant : l’idée que les 273 espaces topologiques soient de nature ensembliste fut immédiatement l’objet de doute de la part de certains mathématiciens, souvent issus de la tradition de la topologie algébrique, qui considéraient que la structure topologique propre­ment dite était indépendante de cette expression (comme ils en faisaient eux­memes l’expérience en analysant ces espaces sans les concevoir comme des en­ 43 C’est la raison pour laquelle Cantor objecta justement a Dedekind que la propriété de com- plétude (numérique) ne pouvait suffire a définir la continuité géométrique et qu’il fallait également y adjoindre des propriétés de connexité. sembles de points). J’y ai déja fait allusion au titre des controverses qui agiterent les peres fondateurs : nombre d’entre eux considéraient, en effet, la machinerie cantorienne des ordinaux transfinis comme dénuée d’utilité et marquant plu­tôt le fait que la théorie du multiple pur, prise pour elle-meme, nous égarait dans des abstractions sans signification (le meme reproche qu’on fit par la suite a la théorie des catégories prise pour elle-meme et qualifiée alors d’abstract nonsense). Ce mouvement fut notamment a l’origine de l’idée d’une topologie qui se ferait « sans points » (pointless topology)44. Or cette remarque historique n’est pas anodine dans notre développement puisque les « mondes » que consi­dere Alain Badiou dans Logiques des mondessont adossés a des structures qui généralisent celle d’espace topologique et que l’on peut justement exprimer par la structure algébrique sous-jacente de leurs régions (« l’algebre de leurs ouverts »). Dans le cas d’espece, elles forment la structure logico-algébrique d’« algebre de Heyting complete »45. Mais cette structure est précisément celle qui s’est révélée etre un objet privilégié de… la topologie sans points (c’est-a-dire qu’elle peut etre exprimée entierement sans évoquer la notion d’ensemble). C’est d’ailleurs un des problemes théoriques majeurs de Logiques des Mondes que le dispositif ne force nullement le caractere ensembliste des structures décrites et que cette contrainte doive donc etre ajoutée par un postulat ad hoc (que Badiou appelle « postulat du matérialisme »)46. Il est tres intéressant que Logiques des mondeschoisissedélibérémentune pré­sentation logique et ensembliste du cadre dans lequel sont censé varier les en­sembles lorsqu’ils « apparaissent » et n’introduise l’approche topologique de ce meme cadre que dans un second temps, au chapitre III, tout en confessant que cette derniere est néanmoins « plus fondamentale »47. Mais pourquoi préciser qu’elle est « plus fondamentale » ? Une étude fine du dispositif montre qu’il ne s’agit pas la d’une facilité de formule. De fait, on a besoin de cette présentation spatiale pour comprendre la structure de monde non pas seulement comme va­riation de l’apparaître par rapport a une grille d’évaluation, mais comme recol­ 44 Voir l’article de Bélanger et Marquis, « Menger and Nöbeling on pointless topology ». Pour une présentation générale du formalisme de Logiques des mondes, je me permets de renvoyer a : « Objet, relation, transcendantal. Une introduction au formalisme de Logiques des mondes » dans Autour de « Logiques des mondes », dirs. D. Rabouin, O. Feltham et L. Lincoln, Editions des Archives contemporaines, Paris 2011. 46 Badiou, Logiques des mondes, p. 264. 47 Ibid., p. 267. lement de ces informations, c’est-a-dire, en vocabulaire mathématique, comme « faisceau ». Cet aspect occupe l’essentiel de la partie III consacrée a ce que Badiou désigne par l’expression curieuse (dans son systeme) d’« onto-logique ». C’est ce qui permet de construire tout « monde » comme « topos de Grothen­dieck », c’est-a-dire comme catégorie de faisceaux d’ensembles (sur un site)48. Le point clef est alors le suivant : pour construire un « monde » au sens de Ba-diou, il ne suffit pas d’avoir une grille de valeurs a partir de laquelle évaluer les variations des existants (un « transcendantal » dans le vocabulaire de Logiques des mondes), encore faut-il aussi que ces variations soient cohérentes entre elles et, pour cela, qu’elles satisfassent a des conditions de recollement. Ici s’ouvrent de nombreuses questions que je vais essayer de formuler brieve­ment en guise d’ouverture a cette étude et en restant le plus possible, comme auparavant, a l’intérieur du systeme d’Alain Badiou : 1. Tout d’abord, il devient alors particulierement clair que la construction des « mondes », comme structure d’apparaître de l’etre, s’est faite par un certain nombre de choix philosophiques qui déborde le simple choix d’un fondement ensembliste et opere parmi les objets mathématiques considérés comme pertinents. En particulier, on n’a pas choisi n’importe quel topos, mais un topos qui se laisse exprimer sous une certaine formespatiale (qu’on dit « localique »). Mais comment justifier un tel choix si d’autres topos appa­raissent non pas dans la théorie abstraite des catégories, a titre de purs pos­sibles non réalisés, mais dans des situations mathématiques réalisées ?49 Si « les mathématiques sont l’ontologie », en effet, il n’y a aucun moyen de ba-layer ces apparitions (qui ne correspondent pourtant pas a la définition d’un « monde ») comme n’étant pas « réelles »50. Autant on peut comprendre que la théorie des catégories abstraite puisse etre décrite comme une logique 275 48 Un site est une catégorie équipée d’une topologie de Grothendieck. Il fournit une géné­ralisation de la notion d’espace topologique opérée a partir de l’idée centrale de « re-couvrement ». Pour une description technique, mais accessible, voir Antti Veilahti, « Alain Badiou’s mistake. Two postulates of dialectic materialism », arXiv: 1301.1203, p. 19 (https://arxiv.org/abs/1301.1203, consulté le 15 mars 2020). 49 Pour les détails techniques, voyer l’article d’Antti Veilahti cité dans la note précédente. 50 Si j’étudie, par exemple, les actions d’un groupe discret G, les G-ensembles forment un topos BG (dit « topos classifiant » de G) qui n’a aucune raison d’etre localique. Je remercie Mathieu Anel pour m’avoir indiqué cet exemple lors d’une de nos discussions sur les topos. générale des mondes possibles tant elle s’aventure souvent dans des terres ou la mathématique « ordinaire » ne semble pas pénétrer51, autant on ne comprend pas tres bien que certains de ces possibles abstraits apparaissent dans les mathématiques ordinaires, lieu meme de « l’ontologie », sans satis­faire aux réquisits de ce qu’est un « monde ». Par contraste, on voit que les contraintes spatiales sont impensées dans ce modele, alors meme que ce sont elles qui forcent, en derniere instance, la structure de ce qui est accep­table ou non (par le philosophe). 2. D’ou une deuxieme remarque importante : comment justifier que « l’onto­logique », ou structure de l’apparaître, coincide précisément pour Badiou avec un processus de spatialisation ? C’est un probleme qui hante la phi-losophie depuis Platon au titre de la chôra, mais qui n’en est pas moins un point aveugle de tout platonisme, celui de Badiou comme celui de son maître. L’espace vient suturer l’écart entre l’etre et l’apparaître en situant la manifestation de l’etre quelque part. Et c’est le meme probleme qui re-surgit jusqu’a Kant dans le fait que la donation intuitive doit se faire via la forme-espace, alors meme que la dérivation de cette prétendue « nécessité » mobilise déja toute une entente préalable de la spatialité, a commencer par celle qui permet de distinguer entre un sens « interne » et un sens « ex-terne ». La « forme-espace » ne peut advenir que sur la donnée préalable d’une extériorité qui ne peut pourtant advenir dans aucune « forme-es-pace » (puisqu’elle est antérieure a sa possibilité meme). Derriere se terre le probleme, dans un vocabulaire que reprend Badiou lui-meme des l’époque de l’Etre et l’événement, du passage de l’etre a l’« etre-la ». Dit autrement : pourquoi la forme générale de la manifestation doit-elle s’opérer par une forme de spatialisation (le « la » de l’etre-la), alors meme que la doctrine gé­nérale de la manifestation relegue l’espace a n’etre qu’une forme particuliere de ce qui se manifeste ? Ces remarques, qu’on pourrait développer bien plus avant, me conduisent aux deux aspects auxquels je souhaitais parvenir : tout d’abord, on voit que si les mathématiques elles-memes évoluent vers une conception ou la spatiali­té n’a pas un statut dérivé par rapport a une ontologie fondamentale, mais au contraireconstitutif – alors la difficulté va se trouver aggravée. Or c’est bien ce a Alain Badiou, Court traité d’ontologie transitoire, Seuil, Paris 1998, p. 198. quoi a conduit le développement libre (hors des choix philosophiques forcés par Badiou) de la théorie des topos et la question qui s’est ouverte progressivement de formes d’unité de niveau « supérieur » entre nombre et espace, arithmétique et géométrie. Cette idée a notamment donné lieu a l’émergence d’une prise en compte du caractere irréductiblement spatial de toutes les relations mathéma­tiques, a commencer par les fleches du catégoricien.J’y reviendraibrievement en conclusion de cette étude, mais on peut noter des a présent que le pas est considérable par rapport a une premiere approche ou certaines catégories (ty­piquement les « topos ») étaient censées capturer l’essence de la spatialité (par différence avec d’autres). L’autre aspect est évidemment lié a la querelle Badiou-Deleuze. Comme je l’ai rappelé, Badiou a objecté a Deleuze que son entente des multiplicités était li­mitée a cause de son attachement a des intuitions originaires spatialisantes. C’est la meme critique que pourrait faire Kant a quelqu’un qui prétendrait que l’espace est une structure de la pensée elle-meme (et non seulement de l’intui­tion sensible) et qui se verrait objecter que le domaine de la pensée est bien plus vaste que ce qui s’en exprime via la forme-espace. Mais, comme l’avait déja objecté le mathématicien Johann Heinrich Lambert52, l’entente de l’espace a laquelle s’adosse cette réponse repose sur le refoulement préalable de tout un régime de spatialité attaché a la structure de l’etre en tant que se manifes­tant53. On cherchera alors a rabaisser cette intervention originaire du spatial au rang de simple métaphore (c’est la stratégie que suit Badiou dans Logiques des mondes en présentant d’abord les structures spatiales de l’apparaître dans leur expression logique et algébrique et en ne recourant au vocabulaire topologique que dans un second temps, comme une maniere imagée de les exprimer). Mais reste alors a expliciter le sens littéralqui rendrait la métaphore inopérante et, en particulier, lorsque l’on parvient a ce sens « plus fondamental » de la spatiali-277 sation ou elle ne sert pas seulement a exprimer les variations, mais a en assurer la cohérence. 52 Emmanuel Kant, Correspondance, Vrin, Paris 1991, p. 79. 53 C’est la meme réponse que pourrait faire un spinoziste, comme j’y ai insisté dans d’autres études (Vivre Ici, PUF, Paris 2010) : si l’espace et la pensée sont deux attributs distincts de l’etre, cela ne signifie pas qu’on puisse concevoir un exces de l’un sur l’autre. Il n’y a rien de la pensée qui ne soit exprimable dans la spatialité et réciproquement : c’est tout le sens du mal-nommé « parallélisme » spinoziste. Conclusion En guise de conclusion, je voudrais indiquer quelques prolongements possibles a mes questions initiales. Tout d’abord, si l’on considere que les ensembles forment d’abord un langage pour décrire l’etre mathématique, alors on constate aisément qu’un des avan­tages de la théorie des catégories est précisément, indépendamment de toute question fondationnelle, d’offrir un nouveau langage dans lequel la référence a des « ensembles » est préservée, mais étendue a des situations nouvelles. Ici une question naturelle est donc : y a-t-il des situations mathématiques que les catégories permettent d’exprimer mieux que ce que faisait le vocabulaire ensem­bliste ? La réponse est indéniablement oui. Tel fut meme le principal moteur du développement de cette théorie (exactement de la meme façon qu’un grand succes de la théorie des ensembles fut de pouvoir exprimer des situations qui étaient inaccessibles au langage de la « grandeur »). Le lieu prototypique de déploiement de ce langage a été, et est toujours, la topologie algébrique. Pour prendre l’exemple le plus célebre, il paraît tres difficile d’exprimer ce qu’est une homologieen général ou une homotopie en général («homotopie supérieure») sans passer par ce langage – meme si, bien entendu, on peut toujours trouver des situations ensemblistes ou l’on pourra exprimer ces différentes notions dans des contextes particuliers (puisque c’est dans ces contextes qu’elles ont émergé). Ceci conduit a une seconde remarque : la topologie contemporaine, et tout parti­culierement la théorie de l’homotopie, offre ici un triple défi dont il faut prendre acte et qui reste a penser par la philosophie. Tout d’abord, ses constituants de base, les « types d’homotopie » ne se laissent pas bien exprimer dans un cadre extensionnel. Ce point peut etre rendu précis au moyen d’un théoreme qui montre que la catégorie associée ne peut pas etre rendue « concrete » (c’est-a-dire qu’elle ne peut pas etre plongée « fidelement » dans celle des ensembles)54. Comme l’a indiqué Jean-Pierre Marquis, un des rares philosophes a s’etre pen­ 54 On dit qu’un foncteur entre deux catégories est « fidele » si l’application qui associe les morphismes de la premiere a leurs images dans la seconde est injective. Plus intuitive-ment (mais aussi moins précisément), la seconde catégorie représente « fidelement » la premiere, car elle n’identifie pas des morphismes distincts de la premiere. Le théoreme mentionné est du a Peter Freyd (1969) dans le cas de la catégorie homotopique htop (géné­ralisé depuis aux catégories de modeles) et il indique donc que la catégorie des ensembles ne représente pas fidelement les types d’homotopie. ché sur ces questions, nous semblons alors quitter les rives de la mathématique « extensionnelle »55. Or, comme il le rappelle également, certains mathémati­ciens considerent, non sans arguments, que les « types d’homotopie » sont aux formes spatiales ce que les nombres premiers (d’ailleurs non moins mystérieux) sont aux nombres entiers : des sortes de « composants ultimes » qui en reglent les structures fondamentales. Sous ce point de vue, le fait que l’on ait pu expri-mer les nombres entiers ou l’espace au moyen du langage ensembliste n’atteint pas pleinement a la nature « profonde » de ces objets, parce qu’il n’a pas prise sur leurs « composants ultimes ». Ensuite, la perspective homotopique s’est développée jusqu’a pouvoir se pré­senter comme un véritable changement de paradigme permettant d’exprimer des situations généralisant ce qu’exprimaient l’égalité extensionnelle et les isomorphismes des catégories « simples »56. Nous ne sommes donc déja plus au moment ou les catégories viendraient contester la prééminence du langage ensembliste, mais au moment suivant ou ce sont ces memes catégories qui peuvent désormais etre ressaisies a un niveau « supérieur ». Le phénomene tout a fait remarquable, au regard des développements qui précedent, est que ce point de vue « supérieur » se fasse précisément par une accentuation du ca-ractere spatial conféré aux entités fondamentales : les fleches des catégories simples étaient, en effet, réglées par un principe d’identification donnée par les isomorphismes, mais vues comme objets spatiaux, elles peuvent également etre vues comme des « chemins » entre les objets, que l’on peut ou non déformer les uns dans les autres. Dans ce cas, c’est la notion d’homotopie qui fournit le bon critere d’identification, la notion d’isomorphisme apparaissant comme une ma-niere d’écraser (ou de « tronquer ») la richesse des diverses possibilités données pour identifier des chemins entre eux. 55 Jean-Pierre Marquis, « Mathematical forms and forms of mathematics: leaving the shores of extensional mathematics », Synthese 190 (2013) pp. 2141–2164. 56 L’expression est de Bertrand Toën : « Very briefly, the expression homotopical mathemat­ics reflects a shift of paradigm in which the relation of equality relation is weakened to that of homotopy », phrase qu’accompagne la note suivante : « It is is very similar to the shift of paradigm that has appeared with the introduction of category theory, for which being equal has been replaced by being naturally isomorphic » (Bertrand Toën, « Derived algebraic geometry », EMS Survey in Mathematical Sciences 1 (2014), pp. 153–240. Je cite a ce propos un des tenants de ce « changement de paradigme » en géo­métrie, qui va jusqu’a proposer de substituer au nom de « mathématique », trop associé aujourd’hui au paradigme structural ensembliste, celui de « mathéma­tiques homotopiques » (dont il faut bien comprendre qu’elles ne constitueraient donc pas une forme particuliere des « mathématiques », mais au contraire une extension)57 : Tout au long de ce travail, j’ai aussi essayé de montrer que les résultats de ce mémoire ne sont pas du tout indépendants les uns des autres et qu’ils appar­tiennent tous au domaine de la mathématique homotopique. Les mathématiques sont fondées sur la théorie des ensembles et la notion de structure (au sens de Bourbaki), tandis que dans les mathématiques homotopiques, les ensembles sont remplacés par les types d’homotopie et les structures se trouvent alors en-richies sur la théorie homotopique des espaces […]. La philosophie générale (qui est probablement assez ancienne et, je suppose, remonte a Bordman, Dwyer, Kan, Quillen, Thomason, Waldhausen, Vogt, …) semble etre qu’une grande par-tie des mathématiques possede des extensions intéressantes et utiles dans le contexte des mathématiques homotopiques.58 Ceci permet finalement de revenir mieux informé au débat que j’ai évoqué entre l’approche de Badiou et celle de Deleuze sur les « multiplicités ». De ce qui précede, on peut, en effet, conclure qu’il y a deux façons assez différentes d’entendre le doublet multiplicité discrete/multiplicité continue. La premiere consiste a y voir le motif d’un choix philosophique en termes de « fondements ». Il s’agirait alors de choisir laquelle de ces deux voies a la priorité sur l’autre. C’est ainsi que s’y rapporte Badiou en faisant valoir que le langage ensembliste porte bien plus de possibilités que ce que sa prise dans certains modeles géomé­ 280 triques pourrait laisser croire. Il objecte alors a Deleuze de s’etre lié les mains en ne se plaçant pas immédiatement au degré le plus grand de généralité. A l’opposé, on pourrait rappeler les déclarations de René Thom sur la primauté irréductible du continu, et celles de son principal défenseur en philosophie des mathématiques : Jean Petitot. Pour bien comprendre cet énoncé, il faut bien garder a l’esprit que les catégories homoto-piques ne sont pas « concretes » (c’est-a-dire qu’elles ne se laissent pas représenter fidele­ment par des structures sur des ensembles). 58 Bertrand Toën,Homotopical and Higher Categorical Structures in Algebraic Geometry, arX­iv:math/0312262, consulté le 16 mars 2020, ma traduction. Alain Badiou et Jean Petitot considerent tous les deux – l’un pour le critiquer et l’autre pour le défendre – que ce sont les positions de Deleuze lui-meme. A mon sens, tel n’est pas le cas. Deleuze ne choisit pas un des modeles comme « meil­leur » que l’autre, mais part de la dualité elle-meme des formes du multiple, telle que l’évoquait Riemann. C’est meme la une des bizarreries de son interprétation de Bergson, puisqu’il reverse la coupure entre deux types de multiplicités a l’in­térieur des mathématiques – alors que Bergson cherchait plutôt a y fonder la différence irréductible entre une approche mathématisante du qualitatif (plus tard de la « durée ») et un régime de multiplicités « intensives » qui échapperait de jure a la mathématisation. Il y a la un trait constant de l’ouvre de Deleuze auquel on n’a pas preté suffisamment attention : a chaque fois qu’il pose un modele continu (et meme souvent différentiel), il l’accompagne d’un autre mo-dele discontinu censé exprimerla meme réalité59. Ceci s’accorde avec sa concep­tion « lautmanienne » des mathématiques et une ontologie qui serait celle des idées-problemes, et non des idées-propositions60. Par rapport aux questions qui nous ont intéressés dans cette étude, il s’agit d’une maniere radicalement dif­férente d’investir le rapport entre expression et ontologie en prenant acte de la labilité de la référence et en reversant l’ontologie proprement dite dans le mécanisme meme d’expression (comme articulation d’un plan d’expression et d’un plan de contenu). C’est ce que Deleuze lui-meme a appelé une logique non de l’etre, mais du sens. Sa grande these sur Spinoza avait d’ailleurs déja indiqué comment cette logique peut s’exprimer métaphysiquement en reversant toute la part de l’expression dans l’ontologie elle-meme61. En ce point, la position de Deleuze paraît a meme d’affronter les difficultés que nous avons soulevées pour commencer. Plutôt que de vouloir résorber l’écart entre langage et etre, elle s’y installe pour identifier l’ontologie au mécanisme de l’expression et en faire une véritable « logique du 281 sens ». Plutôt que de mettre espace et pensée face a face, elle s’installe dans leur parallélisme et leur articulation mouvante, toujours reconfigurée. Plutôt que de rigidifier l’espace selon une norme absolue qui l’assignerait au rang d’objet 59 Ainsi du pli de René Thom et des fractales de Mandelbrot, de la variété de Riemann et du tapis de Sierpinski, etc., cf. David Rabouin, « Un calcul différentiel des idées ? Note sur le rapport de Deleuze aux mathématiques », Revue Europe 996 (avril 2012), numéro spécial Deleuze, sous la direction de E. Grossmann et P. Zaoui, pp. 140–153. 60 Gilles Deleuze, Différence et répétition, PUF, Paris 1968, chap. IV. 61 Gilles Deleuze, Spinoza et le probleme de l’expression, Les Éditions de Minuit, Paris 1968. pour la pensée, elle se donne les moyens de s’ouvrir aux multiples formes dis-cordantes de sa variabilité locale et de son expressivité (constitutive de ce qu’est « penser »). La tension entre espace et nombre n’y apparaît plus alors comme un probleme a résoudre, mais comme un probleme sous la condition duquel nous pensons. Que ce probleme résiste, a n’en plus finir, a toutes les tentatives de le réduire, y compris en mathématiques, paraît d’ailleurs le meilleur témoignage de la fécondité de ces vues. Références André, Yves, Leçons de Mathématiques contemporaines a l’IRCAM, IRCAM, France 2009, archives ouvertes: https://cel.archives-ouvertes.fr/cel-01359200/document (consulté le 25 janvier 2020) Aristote,Seconds Analytiques. Organon IV, trad. fr. P. Pellegrin, GF, Paris 2005 Badiou, Alain,Court traité d’ontologie transitoire, Seuil, Paris 1998 — Deleuze : La clameur de l’etre, Hachette Littératures, Paris 1997 — Le concept de modele, Maspéro, Paris 1969 — L’Etre et l’Événement, Seuil, Paris 1988 — l’Immanence des vérités, Fayard, Paris 2018 — « Un, multiple, multiplicité(s) », Futur Antérieur 43 (avril 1998) Bélanger, M. et J.-P. Marquis, « Menger and Nöbeling on pointless topology », Logic and Logical Philosophy 22 (2/2013), pp. 145–165. Bell, J. L., Toposes and local set theories: An introduction, Oxford Logic, Guides: 14, Clarendon Press, Oxford 1988 — « From Absolute to Local Mathematics », Synthese 69 (3/1986), pp. 409–426 — « Tous ensemble ? Sur le rapport d’Alain Badiou aux mathématiques », dans Autour d’Alain Badiou, dirs. F. Tarby et I. Vodoz, pp. 81–102, Germina, Paris 2011 Baire, René/Borel, Emile, Lettres de René Baire a Émile Borel, Cahiers du séminaire d’his­toire des mathématiques 11 (1990), pp. 33–120 Borrelli, Vincent, « Gnash, un tore plat ! », Images des Mathématiques, CNRS, 2012, dis-ponible a: https://images.math.cnrs.fr/Gnash-un-tore-plat.html(consulté le 15 mars 2020) Deleuze, Gilles,Différence et répétition, PUF, Paris 1968 — Spinoza et le probleme de l’expression, Les Éditions de Minuit, Paris 1968 Desanti, Jean-Toussaint, « Quelques remarques a propos de l’ontologie intrinseque d’Alain Badiou », Les Temps Modernes 45 (526/1990), pp. 61–71 Euclide,LesÉléments, trad. fr. Bernard Vitrac, PUF, Paris, 1990-2001 Rabouin, David, « Objet, relation, transcendantal. Une introduction au formalisme de Logiques des mondes » dans Autour de « Logiques des mondes », dirs. D. Rabouin, O. Feltham et L. Lincoln, Editions des Archives contemporaines, Paris 2011 Ferreirós,José,Labyrinth of Thought: A History of Set Theory and Its Role in Modern Mathematics, Birkhäuser, Basel-Boston-Berlin 1999 Gispert, Hélene, « La théorie des ensembles en France avant la crise de 1905 : Baire, Borel, Lebesgue... et tous les autres », Revue d’histoire des mathématiques 1 (1/1995), pp. 39–81 Grothendieck, Alexander,Récoltes et Semailles, disponible a: https://webusers.imj-prg. fr/~leila.schneps/grothendieckcircle/recoltesetc.php Guitart, René, « Caractere global et caractere local de la vérité », conférence donnée a la Lysimaque, 23 septembre 1990, disponible sur le site de l’auteur : http://rene.guitart. pagesperso-orange.fr/preprints.html Halimi, Brice, « Sets and Descent », dans Objectivity, Realism and Proof, dirs. A. Sereni & F. Boccuni, pp. 123–142, Springer, Basel 2016 Kant, Emmanuel, Correspondance, Vrin, Paris 1991 Kantor, Jean-Michel et Loren Graham, Au nom de l’infini, Éditions Belin, Paris 2010 Klein, Jacob, « Die griechische Logistik und die Entstehung der Algebra (1934/1936) », dans Quellen und Studien zur Geschichte der Mathematik, Astronomie und Physik, Abteilung B: Studien. Band 3, Erstes Heft, pp. 18–105, Springer, Berlin 1934 — Quellen und Studien zur Geschichte der Mathematik, Astronomie und Physik, Zweites Heft, pp. 122–235, Springer, Berlin 1936 Mac Lane, Saunders et Ieke Moerdijk, Sheaves in Geometry and Logic. A First introduc­tion in topos theory, Springer-Verlag, New York 1992 Marquis, Jean-Pierre, « Mathematical forms and forms of mathematics: leaving the shores of extensional mathematics », Synthese 190 (2013) pp. 2141–2164 Niccolo Guicciardini, Isaac Newton on Mathematical Certainty and Method, MIT Press, Cambridge, Massachusetts 2009 Putnam, Hilary, « Langage et réalité [1975] », dans Textesclésdephilosophiedessciences, Vol. 2, dirs. S. Laugier et P. Wagner, pp. 61–104, Vrin, Paris 2004 — « Explication et référence » dans De Vienne a Cambridge, dir. P. Jacob, Gallimard, Paris 1980, pp. 337–365 — « What is Mathematical Truth? », dans Hilary Putnam, Mathematics, Matter and Method. Philosophical Papers, vol. 1, pp. 60–78, Cambridge University Press, Cam­bridge 1975 Rabouin, David, « Un calcul différentiel des idées ? Note sur le rapport de Deleuze aux mathématiques »,Revue Europe 996 (avril 2012), pp.140–153 Roy, Jean-Michel, Écrits de logique philosophique, PUF, Paris 1989, pp. 203–218 Riemann, B., « Sur les hypotheses qui servent de fondement a la géométrie », Ouvres mathématiques, trad. Laugel, Gauthier-Villars, Paris 1898 Russell, Bertrand, « On denoting », Mind 14 (56/1905), pp. 479–493 Stewart Shapiro, Foundations without Foundationalism: A Case for Second-Order Logic, Oxford University Press, Oxford 1991 Stevin, Simon, Arithmétique, (1585) », dans The Principal Works of Simon Stevin, dirs. E. Crone, E.J. Dijksterhuis, R.J. Forbes et al., 6 vol., t. 2 B, N. V. Swets & Zeitlinger, Amsterdam 1955–1966 Toën, Bertrand, « Derived algebraic geometry », EMS Survey in Mathematical Sciences 1 (2014), pp. 153–240 — Homotopical and Higher Categorical Structures in Algebraic Geometry, arXiv:­math/0312262 (consulté le 16 mars 2020) Veilahti, Antti, « Alain Badiou’s Mistake. Two Postulates of Dialectic Materialism », arXiv: 1301.1203, displonible a: https://arxiv.org/abs/1301.1203 (consulté le 15 mars 2020) Wilson, Mark, « Frege: The Royal Road From Geometry », Nous 26 (2/1992), pp. 149–180. Filozofski vestnik | Volume XLI | Number 2 | 2020 | 285–308 | doi: 10.3986/fv.41.2.11 Norman Madarasz* Beyond Recognition: Badiou’s Mathematics of Bodily Incorporation By placing Alain Badiou’s philosophy under the sign of thinking infinity, the splendor, originality and radicality of the way he relates philosophy and math­ematics shines forth. The scope of this relation is such that it leads directly into what thought displays as its promise and strength, that is, novel subjec­tive change. The proposal for a radical restructuring of ontology put forth by Badiou does not emerge from a need to integrate the infinite as such (l’infini) into philosophical thought per se. This had already been achieved in philoso­phy previously, but mainly from a transcendent perspective. Instead, Badiou seeks to ground ontology according to infinity, here understood technically in terms of infinite sets and conceptually as pure multiplicity. From the perspec­tive of mathematical logic, the existence of such sets is not a new finding. It has been a topic central to set theory ever since Georg Cantor postulated the Gener­al Continuum Hypothesis.1 Subsequently, Kurt Gödel showed that the negation of Cantor’s Hypothesis regarding the existence of an intermediary infinite set between the natural and the rational numbers could not be proved, so that in the end there would exist but one infinite of a larger size than the set of natural numbers. The argument set forth in Being and Event bases its claim on a far more radical prospect, which is that the nonexistence of a set of an intermediate order could not be proved, thus making the Hypothesis independent from the axiomatized set theory (ZFC). In virtue of this “independence”, there might very well be many other orders of infinite sets. 285 1 The version of the General Continuum Hypothesis Badiou applies and reads in Being and Event is demonstrated inMeditation 29. Cantor, however, did not define whether the nat­ural numbers were a set, nor even that the power set axiom involves such a set. E. Zermelo is who defined the natural numbers as a set. As far as Badiou is concerned, the set of nat­ural numbers is the smallest infinite collection of elements, while “state” names the next infinite set, .1. For any “state of the situation”, there corresponds to it an excess in regard to the smallest infinite. Alain Badiou, Being and Event, trans. Oliver Feltham, Continuum, London 2008, pp. 25ff. * Porto Alegre, Brazil Contemporary French philosophy can be described in part as a sequence of the­oretical proposals stemming from a concept of multiplicity that is irreducible to a single totalizing unit, say the One. Arguably, there has not been any proposal as broad ranging as Badiou’s regarding the irreducibility of multiple infinites. The argument set up in Being and Event makes a claim on how the state of exist­ence we inhabit, Badiou calls it the “state of the situation”, can be represented in set-theoretic terms. This state can be framed as a gathering together of dis­tinct elements and parts bounded by a single infinite, to which the elements either belong or are included as parts regardless of how many such elements the state of the situation might actually have. This description changes little when considering the very localized fields of the state of the situation in which truths, instead of opinions, beliefs or lies, are produced. Known as the “conditions” according to the lexicon Badiou composes for what is otherwise presented as a philosophical system, these fields do not exceed the boundaries of the state of the situation so long as a truth does not come to be produced within them. Being and Event explores the thesis that the principle of radical change in any condition of the state of the situation can be recognized through the implica­tions of the mathematical proof allowing for infinity to exceed the boundaries of a set whose elements can be counted. The measure of radical change goes by a continual creation of truth in procedures specific to each condition. Truth is thus held to refer back to an event that has ruptured the order of a condition. As such, the ontology sets out to show how the fact that radical change occurs in a con­dition can be both proved and understood by the set-theoretic theorem whereby a generic, indiscernible set, that is, one whose elements are not yet known, may be demonstrated as a real extension to the state of the situation. Thought con­ceptually, a generic, infinite set gathers together the names of processes whose 286 achievement will have been to produce a radically new truth in a condition. The limits to which this truth can expand are those of the condition itself. Philosophically speaking, this is a radical conclusion, though it takes getting to the later Meditations of Being and Event to grasp the considerable consequences Badiou’s application has for conceptual thought. In Meditation 16, for example, he introduces the subtle paradox of the notion of event. In Meditation 35, he outlines the formal traits of a theory of subject molded upon the implications an event has for changing the terms by which truth is worked out in a condition. The Meditation reads like a list of the most polemical conclusions reached by French philosophers and social scientists alike since structuralism appeared to wash away the face of the subject from modern fields of conceptual inquiry. Ba­diou’s theory of subject is post-evental. As such, it leads further than previous ar­guments and models into the dissolution of the ‘humanist’ subject. The upshot of such exploration is that the evental subject is materially indiscernible. Such is the requirement for this subject to maintain its formal structure intact prior to acquiring materiality. As soon as this process is engaged, it works against the inertial pressures produced from within a condition. These pressures can be expected to force a reduction upon the subjective process in order for it to return to the conformity of a substance-like entity. In the following discussion, we seek to retrace the challenge to what lay ahead for Badiou’s articulation of the subject. On the one hand, the requirements for thinking infinity as seen from within mathematics, by either philosophers of mathematics or those working mathematicians receptive to the philosophical problems their field fosters, point to the limitations of set theory to think infinite relations in real terms. On the other, the proposal for radical subjective change required a novel concept of a “second body”, or “bodies of truth”, by which to situate objects, change and worlds as they become part of the process by which a truth appears. Logics of Worlds joins these two problems in another striking proposal by which to relate philosophy and mathematics. This broad work on the process of appearing pursues the importance of thinking infinity. Its propos­al works from the basis of understanding correlations between various logical possibilities regarding the appearance of truth. It thus provides a complex un­derstanding of what bodies of truth can, and must, achieve were radical truth to have a chance to be created in a world – or, indeed, as a world. From the generic subject to the body of truth Philosophically speaking, the challenge for Badiou after Being and Event was to show what the structure of the subject is as it forces itself into existing in a condition in an unbounded process. In works such as Conditions and Ethics, he demonstrated how radical practice cannot be mediated by a category of tran­scendence. Moreover, as it is determined by a chance event, Badiou holds that the subject is not the result of a dialectical sublation. This is a key feature of an ontology deemed to be intrinsic, as the structure of subjectivation is always localized within a condition.2 This is a crucial point to his argument on and around the event in Being and Event, especially when illustrated from within the political condition by the figure of the activist, that “patient watchman of the void instructed by the event.”3 It also lies at the subtle core of his argument regarding how the mathematics of set theory can instruct philosophy as to the formal mode by which to build a general science of being qua being. The formu­la “mathematics is ontology” thus warrants that the former provide a complete model for the latter, insofar as set theory is a theory of infinity. But it also pro­vides a model for a set, the generic set, that is the extension of the set-theoretic universe as such. Neither of the operations by which sets are built and theorems derived is dialectical in nature – though they are demonstrably deductive. Thus, while the set-theoretic universe in Being and Event may be that of infinites of different sizes and of irreducible multiplicities, what it is not is that of bodies and appearance. In Logics of Worlds, Badiou applies category theory to demon­strate the logical dynamic of appearing. Yet he also suspends the problem as to whether category theory is more adequate as an ontology. We have little choice but to follow his argument. A pressing concern arises with the risk of having truth reduced and eliminated in a hegemonic world that would tend to reduce bodies to a general social physics, in which they act merely as communicational poles, regardless of how expressive or emotional such communication might become. What bodies can do is provide a subject with freedom, but that requires truth. According to Badiou’s conviction, freedom is not linked to a body’s spontanei­ty, nor to its intersubjective relations, but to how it literally incorporates truth. From the outset of Logics of Worlds, he declares that “the most significant stake of Logics of Worlds is without a doubt that of producing a new definition of bod­288 ies, understood as bodies-of truth, or subjectivizable bodies.”4 Subsequent to a series of critical objections, mainly regarding the claim about set-theory being While the term “intrinsic” has come to identify Badiou’s ontological proposal as a result of Jean-Toussaint Desanti’s critique of his ontology in Les Temps modernes, the notion of intrinsic is used throughout Being and Event to describe the non-referential nature of sets and the indiscernibility of the generic. For example, Alain Badiou, Being and Event, trans. Oliver Feltham, Continuum Books, London and New York 2005, 34 as a whole, and espe­cially, section 7. 3 Ibid., p. 111. 4 Alain Badiou, Logics of Worlds, trans. Alberto Toscano, Continuum, London 2009, p. 35. mathematics, to which we shall not return here, the coherence of the ontology as a building block for a broader system comes to depend on an adjacent claim, one warranted analytically by a structure in which logical forms are correlated. In his response to his critics, Badiou argues that such a task is captured by cate­gory theory and especially by Alexandre Grothendieck’s notion of topos.5 According to Grothendieck, a topos is a “metamorphosis of the notion of space”, the promise of which is to renew traditional topological spaces and geometry.6 Topos alternates in his terminology precisely with multiplicity. Grothendieck stresses that a topos can be expressed as a category, provided the structure be one of a sheave category, being that topos is like the “envelop” or “habitat” of the new geometry.7 Furthermore, his newly crafted notion of topos is unifying as it provides “a common geometric intuition for topology, algebraic geometry and arithmetic.”8 Topos can be thus understood as involving an infinite num­ber of categories. For the purposes of Badiou’s system, it captures the poten­tially infinite number of correlations established between the possible modes by which to exist in givens worlds as they are structured according to distinct logical forms. From the philosophical perspective, topos theory serves Badiou as a response to the complexities faced by a generic subject as it appears as a worldly existence structured by truth-producing procedure. Topos captures a world in which truth is an exception through which to build a site of existence, of being-there. In this regard, it is understandable how Badiou considers topos 5 According to Olivia Caramello, “the introduction of the concept of Grothendieck topos stemmed from the observation that many important properties of topological spaces, such as compactness and connectedness, admit reformulations as categorically invariant 289 properties of the associated categories of sheaves of sets; moreover, if a topological space X is sufficiently well-behaved (technically speaking, sober, cf. Remark 1.1.28), it can be recovered from the associated category Sh(X), as well as from the frame O(X), up to homeo­morphism”. Olivia Caramello, Theories, Sites, Toposes. Relating and Studying Mathemati­cal Theories through Topos-Theoretic Bridges, Oxford University Press, Oxford UK 2018, p. 12. While Badiou does not systematically distinguish between Grothendieck topos and F.W. Lawvere’s elementary topos, his reference is to the former alone (although he would regularly reference Lawvere’s category theory in his 1990’s Doctoral seminars.) 6 Alexandre Grothendieck, Récoltes et Semailles, Université des Sciences et Techniques du Languedoc, Montpellier, 1985-1987, p. 46 ft. 27, 56, 273 and 355. Translations are my own. 7 Ibid., p. 58. 8 Ibid., p. 78. as “among the most remarkable mathematical structures that saw the light in the 1950s and 1960s. ”9 From the outset of his follow-up to his ontological proposal, Logics of Worlds fac­es off two idealizations of the body determining a world. The book now provides a name for the state of the situation: “democratic materialism”. Characteristic of its belief system is a naturalized form of thinking by which objective existence is conceded to two broad paradigms: languages and bodies. As in Being and Event, the pivotal concept by which such thought is radically contested are truths. And the fact of truths sets the stage for a way of thinking termed a “materialist dia­lectic”. From the perspective of truth, the latter structures another perspective on how bodies and languages ought to be thought. Provided it be a vehicle for the radically new, irreducible, generic subject, a truth can be sustained insofar as a “second” body understood to be the creative and practical basis for conveying truth into appearance. To set up this claim, mathematics serves the aims of the system more than it did even in Being and Event. However, it is as a general theory of logic that mathematics is used to map the preconditions for the emergence of a body of truth. In this framework, logic signifies “purely and simply the cohesion of appearing.”10 We may now unpack the radical terms according to which a body of truth appears. The conceptual construction of the body is arranged in two phases. First, in the introduction to Logics of Worlds, Badiou presents the argument by which the current hegemonic system is that of bodies and languages. Plurality and pri­macy of bodies is set up as one of the bastions of the current order. At the same time, a process of subjective transformation of the body as a vehicle for truth 290 takes hold of a fundamental place in the emergence of new subjective forms in which languages must also be structured accordingly. As such, “the norm of life is, quite naturally, that the genealogy of languages be adequate to the power of bodies”11. That which is opposed to this norm is the exceptional nature of truths. Truths reach appearance thanks to a body. The program is thus set out in Logics of Worlds: “Given that a subjectivizable body is a new body, this problem 9 Badiou, Logics of Worlds, p. 295. 10 Ibid., p. 100. 11 Ibid., p. 35. requires that one know what the ‘appearance’ of a body means, and therefore, more generally, that one elucidate what appearing, and therefore objectivity, may be.”12Such a task invokes perception as well as thought, which is a strategy deviating from the ontological field explored in Being and Event. In an allusion to Jacques Lacan’s Seminar XX, the new program stipulates that truths “are re­quired to appear bodily [en-corps] and to do so over again [encore].”13 The notion of truth Badiou had reinstated as central – and crucial to philoso­phy – thus confronts another crucial criterion. The definition of body refers to “that which, bearing a subjective form, confers upon a truth, in a world, the phenomenal status of its objectivity.”14 From a philosophical perspective, the aforementioned “fundamental thesis” stipulates that just as ontology refers to the thought of being qua being by mathematics, “so appearing, or being-there­in-a-world, is thought by logic.”15 Badiou establishes a claim according to which the dynamic by which a body of truth evokes the relational exteriority of distinct worlds can be mapped according to a Grothendieck topos, which he seeks to organize in a full exposition of its relational variations and categorical struc­tures. What affects the truth physically is regulated by the scale of appearance as defined according to degrees or “intensities” of existence. Truths continue to respond to events whose discharge stems in this new context from generic non-existence, labelled the “inexistent” in Logics of Worlds. An event is defined as a process by which an inexistent achieves maximal intensity of existence. The theoretical program thus depends on proving how specific worlds show a capacity for sublating the inexistent of an object.16 As such, Logics of Worlds provides a theory of object without a subject. This sudden emergence is what traces an unhindered process of subjectivation singular to a generic subject whose hold in a world is only ensured by a body. 291 The notion of worlds and object are drafted as “a logical theory, wholly alien to any doctrine of representation or reference.”17 Insofar as this theory is a coher­ 12 Ibid. 13 In fact, Badiou evokes the program in a confessional tone, admitting how the problem relat­ ing body to truth “was the problem whose breadth I was yet unable to gauge”. Ibid., p. 46. 14 Ibid., p. 36. 15 Ibid. 16 Ibid., Book V on dialectical sublation. 17 Ibid., p. 37. ent one, Logics of Worlds can be acknowledged for having achieved a common space in which to correlate three irreducible logical forms: the classical or bina­ry, the intuitionistic and the paraconsistent.18 The particular context, or strictly speaking “world”, acquires consistency by virtue of being deduced from a spe­cific logical form. What provides a basis for analyzing the coexistence of these worlds is the Topos category. Logics of Worlds is replete with historical examples and formal modes by which the generic subject appears in a world. The positive analysis of the body of truth is limited to only one among four possible subjective forms. The faithful subject, as illustrated by its matheme, is the only one to align a category of body of truth to the trace of an event.19 The following is the synoptic image of this form: . ___ . . c The implication of an alignment between an event trace in a world and the body of truths responding to it is what underwrites the possibility of a new present. Yet the fundamental point is how a “divided (and new) body becomes, under the bar, something like the active unconscious of a trace of the event.”20 The latter points to the new framework in which what does not appear does not exist per se. Inasmuch as it is unprecedented, the new is thought from a point of indis­cernibility, which evokes in this book the aforementioned inexistent Each world is thus set as a field or territory of variable scales of appearing. A transcendental is the mechanism introduced to refer to what regulates the phenomenal diversity of appearing, or of being-there in that world. As Badiou refers to the transcendental as a “complete Heyting algebra” or “locale”21, it af­fords the synchronic iteration warranting the emergence of any world lodging a truth procedure. Completeness in this context refers specifically to the infinite 18 Marios Constantinou and Norman Madarasz, “Being and Spatialization: an interview with Alain Badiou”, Environment and Planning D: Society and Space 27(2009), p. 791. Also Badi­ou, Logics of Worlds, p. 532. 19 Badiou, Logics of Worlds, p. 53. 20 Ibid., p. 53. 21 Ibid., p. 167. possibilities of correlate worlds. The correlation that intelligibly connects the transcendental to a worldly multiplicity is also borrowed from Grothendieck’s arsenal of tools, a “sheaf ”. Although it is “conceptually required for appearing to be bound”, sheaves preserve infinity at the level of postulating worlds.22 Whereas in Being and Event the generic subject was bound to the undecida­bility of an event, in Logics of Worlds unfolds a sequence of immanent acts of non-conscious recognition, the practice of which proceeds to incarnate a par­tial but expansive body. In the diagram above, the split marking the body ‘c’ associates a process of recognition stripped of dependence or reference to past experience. This recurrence and the formal identity of this process implies that truth ultimately refers to a constant continuous plane. The process of producing the new present is initially emphasized as proceeding discontinuously “frag­ment by fragment”. As the book proceeds, this mode of producing fragments becomes one of deciding upon points, the basic insight of which stems from tracing a curve on a Euclidean plane. The body is built from successive, albeit distinct, relations in which a point is linked to previous ones in a categorical composition that is coherent (in categorical terms, is associative and composite) and provides an insightful clarity as to relations (in categorical terms, the ar­rows and points) constituting it. The new perspective on truth production refers to the way in which incorporation as a body of truths is mapped according to a methodological approach having the same structure as category theory. This resulting body is neither passive in any biological sense of the term, nor is it a monistic entity, in which consciousness would be reduced to the brain, for example. As the reader is led into an immanent perspective through which the body broadens the grounds of its certainty, Badiou introduces a general, non-descript “efficacious organ” as a generic operator, internal to the category of body, from which decisions regarding the true are made. With no remote view on the structure’s totality, the efficacious organ is an operation by which truth attribution builds upon points. As the points gather into a coherent category, they come to shape and objects that, in turn, respond to the circulation of truth in a specific world. Ibid., p. 103. In the end, the body of truth is distinguished from the hegemonic body form by an efficacious structure that works punctually, point by point, in “an order of affects which authorises the continuation of the process.”23 Contrary to theo­ries that tend to lend cognizing capacities mainly to the brain, consciousness is not a recognizable category regarding this procedure. In terms of a referential connection, Badiou does not provide one as regards any naturalized theories of body. This break summarizes a philosophical commitment extended to the reader as it also underscores the need for a general theory of structural as well as mathematical analyses. Without them, the reader risks reducing the effica­cious organ to previous notions, theories or fields, be they psychobiological or social constructionist. Furthermore, Logics of Worlds invites the reader to develop a novel perspective on immanence regarding the assumptions behind dialectical thinking as if to probe the very inner dynamic of sublation. In Badiou’s view, it is clear “that the dialectical thinking of a singular subject presupposes the knowledge of what an efficacious body is, and of what a logical and material excess with regard to the bodies-languages system might be.”24 Its magnitude depends on the amount of points linked in a process that is nothing but that of how “a subjective form is ‘carried’, in the phenomenon of this world, by an organized material multiplici­ty.” 25 Whereas the theory of subject is essentially formal, or formalist, the theory of body is material, thus requiring a correlation of logical theories and spatial compositions able to trace its phenomenal appearance. Being and Event was about multiplicities and not about bodies, because its stake was not to enter into the descriptions of the historical processes of rupture and construction, which is how the body emerges in the first place. The subject lies 294 beyond recognition for it required sustaining its indiscernibility for as long as possible in the likelihood of coming to be manifested in a condition. For all the descriptions of generic subject forms in the conditions, the system at that stage in its development did not present the means by which a faithful subject mani­fests itself, nor indeed was it initially contemplated should its destination be de­viated and possibly suppressed. None of these immediate shortcomings weak­ 23 Ibid., p. 88. 24 Ibid., p. 46. 25 Ibid., p. 27. ened the conviction over the real capability a subject has in overthrowing the order by which a condition is secured. As Badiou explains, “that is why we can present the figures of the subject right away, without yet possessing the means to think the effective or concrete becoming of a historically determinate sub­ject.”26 To achieve the latter in a theoretical setting, the methodology requires a descriptive stance. Now, this means that Badiou’s argument is not primarily a historical argument, not even ex post factum. Rather, it preserves deductive inference as fundamental to mathematical reason.27 But that is the extent of the argument, regardless of how detailed and impressive it is set out in Being and Event. Meaning that when we return to the conditions, the subject is not mathematical in material content anymore than is philosophy reduced to mathematics. The task of the second body, that is, the variations on “subjectivated bodies”, is otherwise more subtle. It presents a threshold beyond which history as a dis­continuous regularity is required to speak. Still, the body is nothing without the description the process of history itself provides to it as a context in which it re­turns the favor and literally makes history. The process of non-visible linking of the points of its appearing is what guarantees indiscernibility to the body-sub­ject in the same gesture as what ensures its amplification. No matter how certain readers of Logics of Worlds attempt to steer his program, infinity is still central to the perspective adopted. The Grothendieck topos is se­lected due to its power to make Cantorian set theory correlate with finitist areas of mathematics. The body conceptualized by Badiou can thus aim for a perspec­tive on its projected totality. Recapitulating, Badiou asserts “a limpid abstract formula: a post-evental body is constituted by all the elements of the site which 295 invest the totality of their existence in their identity to the trace of the event. Or, to employ a militant metaphor; the body is the set of everything that the trace of 26 Ibid., p. 47. 27 The power of inference is the entire sense by which Badiou understands mathematization in a philosophical framework. As he puts it, “‘Mathematizable’ means submitted to the literal power of inferences, and therefore entirely indifferent to naturalness as well as to the multiplicity of languages”. Ibid., p. 74. Cf. Badiou, Being and Event, Meditation 24, “Deduction as an Operator of Ontological Fidelity”, especially section 4, on the “Triple Determination of Deductive Fidelity”, pp. 252–255. the event mobilizes.”28 As the support to the event trace, a composition surges forth with an inferential capacity to create a new present, one responding to the truth set to build upon the event. Such is the sense to the claim that “of the subject, there is but a theory”.29 This subject incarnated is not a thing, nor object. It repels idealist postulations as it does ideological ones seeking to falsify it. As mentioned before, the subject’s basic properties, as formulated from the ontology, are set out in Being and Event, Meditation 35. Its terms break with the model of substance, though nor is it the void through which an event becomes manifest. Subject does not refer either to experience, let alone to individualized lived experience. Its conceptual situation is that of being in excess to a condition. Albeit rare, its occurrence is not regulat­ed by a law. By the time of Logics of Worlds, thinking the effective or concrete be­coming of a historically determinate subject “requires a description of the body that functions as its support.”30 The philosophical methodology in Logics of Worlds and its claims As opposed to Being and Event, the argumentative framework of Logics of Worlds takes shape according to two distinct levels. In addition to the formalism in which category and topos theory provide a general logic, Badiou introduces a recast phenomenology centered on rearticulating the notion of material be-ing-in-the-world. In this section, we examine the phenomenological model he applies to conceptually analyze the radical correlations formalized by the cat-egory-theoretic as well as topos-theoretic structures. We question whether the boundedness typical of worlds, as expressed by the notion of horizon specific to phenomenology, might not undermine the radicality of Badiou’s philosophical 296 system. Namely, we submit that Badiou’s methodology is clearly structuralist, and as such requires a clear break with phenomenology for reasons of clarity and coherence. For Logics of Worlds justifies, at least indirectly, the unexplored radical extensions of structuralism once the event and infinity are adjoined to its conceptual arsenal. Phenomenology has never been able to think infinity without returning to a theological commitment. 28 Ibid., p. 467. 29 Ibid., p. 47. 30 Ibid., p. 49. The conclusive statements on the body of truth occur in Book VII of Logics of Worlds, in which one finds the recapitulation of the complete argument, the demonstration, proof and historical illustrations. While attentive to the process by which this structure appears, its simplified model suggests a form of concep­tual circulation not present in the phenomenological context. The structure can be simplified according to five conditions as follows: -An existing world, in which one comes to live, teaming with problems old and new; -The coming to be of a site: from within this site, an event trace is received as a break in a world, not as an extension; -Although a trace is a break, it does have elements, and proves to be some­thing more. A body builds from elements that further integrate points from within a not entirely discernible world. As such, the body is not reducible to a trace; -Part of this body is the efficacious organ, whose elements decide upon the points allowing for its amplification, which proceeds by means of new cor­relations; -Subsequent to these correlations, new concepts envelop the efficacious part itself, a process in and through which a “new organ” is formed for the body. While the articulation of such a model is not unfamiliar to thinkers of sexuation and critics of the gender binary, Badiou’s effort in this book challenges such the­ories, at the very least, to ensure that incorporation is filtered by an event trace that breaks radically with the current world. We would like to assume, at least for the purposes of the current state of the situation, the coherence of Badiou’s bid. To be sure, metaphoric projections are possible, as some critics of Badiou’s philosophy contend.31Yet they can also be considered the reader’s responsibili­ty, if not deficiency. Once again, Logics of Worlds delegates responsibility to the reader to decide upon the points at which truth-valued decisions are determi­nate for the next steps in keeping the event trace active. Cf. Vladimir Tasic, “Badiou’s Logics: Math, Metaphor, and (Almost) Everything”, Journal of Humanistic Mathematics 7 (1/2017), pp. 22–45; or Shiva Rahman, “On Why Mathematics Can Not be Ontology”, Axiomathes 29 (2019), pp. 289–296. In his chapter on “Classical Worlds” (Section V), Badiou clearly expresses his commitment to following what category theory affords in terms of mapping some worlds according to a non-binary logic. But his main focus commits to those worlds that are overwhelmingly standard, despite how the measure of ap­pearance presented by the transcendental works as a continuous grid. Through a purely visual example of the mixture of the sun’s color from a given angle against the ivy grown on a house’s wall, Badiou shows how claiming a specific color for the ivy leads to an undecidable proposition as it depends on a cor­relation with the specific aspects of a light source. As a result, “The world of the house in the autumn evening is not classical, in the sense that it validates neither double negation nor the excluded middle.”32 The occurrence of this var­iation in logical form should not distract from the primacy of standard logic. In the final analysis, standard logic may prove to be the surest way at preserving truth as it appears in a world. The calculated, “objective”, “minimal” or indeed “operational” phenomenolo­gy33 aims for a conceptual exposition, while mirroring, even though at a meth­odological distance, the formal proof. Indeed, Badiou will at times say as much, especially when stating that “without a doubt, due to the extreme rigorousness of the chains of reasoning, the formal exposition is here often more illuminating than the didactic phenomenology that precedes it.”34 Through frequent refer­ences to Lacan in Book VII, to whose work he references the “second body”, Badiou explicitly undermines any identification between his “operational phe­nomenology” and the historical methodology recognizable by that name.35 The resort to this former terminology does not cease to provoke doubt given that later Badiou explicitly rejects the kind of phenomenology that merges with the analysis of consciousness.36 If the phenomenological approach is constitutive here, then it is as from a cut. Furthermore, it is precisely the cut that raises questions as to its overall perti­nence. The return to the intentional and to lived experience would hinder a the­ory of the new, as the transcendental ego is of no relation to the generic subject 32 Ibid., p. 184. 33 Ibid., p. 41, p. 128 and p. 103, respectively. 34 Ibid., p. 197. 35 Ibid., p. 48, p. 478 and p. 480. 36 Ibid., p. 48. form, for it recognizes no radical event. Badiou’s phenomenology is a “descrip­tion without a subject”, although we might hasten to add it is one stabilized by what he terms “eternal truths”. As such, it is not clear whether Badiou manages altogether to elude the theological directionality, if not finality, of phenomenol­ogy as a whole. He recognizes the risk just as much when evoking how “phe­nomenology, in its German variant, is indisputably haunted by religion.”37 His critique of Paul Ricoeur stands as a critique of post-Husserlian phenomenology as a whole: “indestructible latency of a Christian subject at the very heart of the text.”38 To go beyond the theory of subject by which phenomenology has come to be recognized, Badiou switches the register from perception to affect: “it is in­deed by its affect that the human animal recognizes that it participates, through its incorporated body, in some subject of truth, we will say, with Lacan, that ‘it is as incorporated that the structure makes affect’.”39 By reducing the conceptual stratum of the analytic of appearance to a formal level, not even the naive sense of prelinguistic comprehension can maintain its reference to phenomenology. For category theory shows the complexity of this level explicitly in its general theory of relations. The question remains as to whether phenomenology can explain, let alone war­rant, the indiscernible. Here we encounter the controlling instance by which an inexistent, after being propelled into a maximum intensity of appearing, keeps from being immediately drawn back to the hegemony of bodies. In fact, without efficacious organs deciding on the fragments and points that break strictly with the phallic order, there is neither guarantee, nor really directionality in the type of body that is desirable – if, that is, one can agree on whether the phallic order extends across all conditions. At any rate, at no moment has the entire tradition of phenomenology shown itself to be apt at carrying out this rupture. Although P. Maniglier and D. Rabouin in an early discussion of Logics of Worlds emphasize 299 that there is a “structuralism” (their quotes) in the book, they do not consider the phenomenology as potentially undermining the formal topos theory.40 We could conclude that the inexistent is radically independent even from Heideg­ 37 Ibid., p. 516. 38 Ibid. 39 Ibid., p. 480. 40 Patrice Maniglier and David Rabouin, “A quoi bon l’ontologie ? Les mondes selon Badiou”, Critique (719/2007), pp. 279–294. ger’s ontological difference and Ereignis concept, if, that is, Badiou considers Heidegger as an exemplar of phenomenology, which is not clear. In sum, what is kept of phenomenology is the displaced methodology for con­structing the theory of appearing, in which even the category of transcendental breaks with historical precedent. The broader sense of the philosophical meth­odology here espoused seems more structuralist than phenomenological. To be sure, structuralism breaks with the theological baggage at risk with most if not all phenomenologies. Rather surprisingly, though, Badiou prefers to sideline structuralism, or at least consider it part of a past philosophical epoch.41 And he insists: “When we say that the description of these operations makes up a com­plete phenomenology, we mean that the transcendental determination of the minimum, the conjunction and the envelope (or synthesis) provides everything that is needed for being-there to consist as a world.”42 One could add, though, that to consist in a world, the logic behind Logics of Worlds requires relational transparency and complexity, instead of descriptive fidelity. By contrast, philosophy caught within either subject-object or bodies-languages paradigms – democratic materialism – seems very far from understanding the sequence by which thought processes associate and correlate. The proto-para-consistent logical framework of Martin Heidegger’s Being and Time might be an exception, but French phenomenology has largely maintained its speculations bound to a single world – even when integrating the event as a key concept. Thus far, I have been arguing that the question of preserving the theory of subject in Logics of Worlds, whether it be upheld by a phenomenological or structuralist perspective, is at least as important as how mathematics captures infinity. I say 300 as important as, although it could, in the final analysis, be more important than. “I could not be structuralist because structuralism was, in its most extreme form, the aban­donment of the concept of subject.” This admission, one of Badiou’s most explicit state­ments on structuralism, was made at the “Conférence d’Athenes: Introduction a L’Etre et l’événement et Logiques des mondes”, given at the National Technical University of Athens, on January 30, 2008. The transcription was made by R. Lopinska, available at: http://www. entretemps.asso.fr/Badiou/Athenes.htm. The conference is available online at: http:// www.youtube.com/watch?v=CfngSuXwW0g&feature=PlayList&p=BBC9321FB9E98E0 3&playnext_from=PL&index=0&playnext=1, (last access: July 20, 2019) 42 Badiou, Logics of Worlds, p. 103. To follow my point (which in all truth is meant to be Badiou’s), mathematics is strictly defined as ontology. Naturally, mathematics is not as important in Log­ics of Worlds, since the latter is not about ontology, at least not primarily. Now, a number of philosophers of mathematics simply disagree with how Badiou separates category theory from mathematics.43 Contesting Badiou on this point might not be as prudent as it might first seem. Take the case of symbolic logic, which in its syllogistic form is as old as Aristotle. The modernization of symbol­ic logic occurs through Boolean algebra, the articulation of which was initially developed within mathematics, the refinements of which continue through the turn of the century with the axiomatization of set theory. Consider, though, how nowadays a mathematician would have an easier time finding a course in sym­bolic logic given in a philosophy program instead of in mathematics. If category theory is indeed a general theory of logic, no matter how unknown it still is to philosophy programs, it is not impossible to imagine it increasingly becoming more theoretically investigated in philosophy programs, just like logic has be­come. Depending on who is doing the talking, it already is. The question then becomes the following: what does category theory present to Badiou if its purpose is not primarily to compensate for what is too polemical in his argument on “mathematics = ontology”? In extremely schematic form, by considering its minimal tools and definition of identity as a relation; its weak axioms, with no existential quantifier; the visibility offered to the fundamental relational activity behind the constitution of categories, as demonstrated by the “pullback” and “central object” (or subject classifier) 44; and of course with the very notion of the category of sets, infinity is no longer understood at the prima­ry level of existence. Infinity becomes a possibility of correlations and the proof aims at showing its construction to be coherent from within a material form, not of proving its existence as a set. The post-evental generic extension is not far off 301 from that in the first place. As one can see, the interest in infinite sets emerges through the founding prin­ciple of set theory regarding membership or belonging, which is immanent to any set and thus not considered a relation. If Badiou establishes his ontology as 43 Cf. footnote 51. 44 Alain Badiou, Mathematics of the Transcendental, trans. and ed. A.J. Bartlett and Alex Ling. Bloomsbury Press, London 2014. illustrative of the state of the situation, it is true that membership, the axiom of separation, axiom of foundation, power set axiom and Cartesian product cap­ture the intrinsic form of the state of the situation. But when ontology is taken in its special sense, as describing what is singular to the generic set, which implies bracketing the axiom of foundation, then it can be argued that category theory better maps a theory of generic subjectivity, as it especially deals with the pos­sibility of such radical change. Logical possibility might just be enough, given that the rest is up to us anyway. Regarding Being and Event, the Gamma Diagram (Schéma Gamma) first present­ed in 1991 succinctly portrays the categories and steps involved in the forming of a generic subject.45 The Diagram also makes visible the threats to fidelity a subject encounters at the level of its being. Though Badiou deliberately indexes the name “subject” to this structure, in virtue of which its material effects are made visible, it could just as well be an object, meaning that it is undecidable as to its evental form. In other words, the subject is not human per se, possibly being a set of works, theories, organizations or irreducible twos, depending on whether the event ruptures the condition of art, science, the political or love, respectively. In Logics of Worlds, the event takes the following typology of forms: “in politics, Revolution; in love, erotic liberation; in the arts, performance; and in the sciences, the epistemological break. In philosophy, we can detect it in Wittgenstein (‘The world is all that is the case’) as well as in Heidegger (being as coming-to-be, Ereignis).”46 We can understand the evental-subject also in terms of directionality and destination, both of which nonetheless need to transfer the event trace, or situational rupturing, to a new configuration, which is none oth­er than the body of truths. The full definition of subject in Logics of Worlds is the following. Subject is “that which imposes the legibility of a unified orientation 302 onto the multiplicity of bodies. The body is a composite element of the world; the subject is what fixes in the body the secret of the effects it produces.”47 There is no temporal measure of the duration of the process, nor an indication of the tipping point when achievement lies on the horizon. Alain Badiou, Conditions, trans. S. Corcoran, Continuum, London 2009, [1991], Figure 8.1, p. 121. 46 Ibid., p. 381. 47 Ibid., p. 46–47 In the mathematical framework, category theory has been especially used to es­tablish correlations, bridges and equivalences between different areas of math­ematical work. Although it has been well over a decade since Logics of Worlds presented philosophy with this framework, what can be said with relative con­fidence is that the relation between this work and the ontology is concentrated in at least one concept instead of the entire undertaking. In his discussion with F Tarby in Philosophy and the Event, Badiou states: “In order to throw light on truths’ relation to bodies and languages, I use a notion that is the equivalent of forcing in Being and Event, namely the concept of compatibility. A truth body is composed, in fact, of elements that are compatible in both a technical and elementary sense: they let themselves be dominated by a common element.”48 And that element is the truth itself. In the dictionary of concepts offered as an appendix to Logics of Worlds, compatibility is defined as follows: “Two elements of the support-set of an object are compatible if the ‘common’ of their existence is the same thing as the measure of their identity.”49 Following the observations made in McLarty’s review of Transcendental Mathematics, Badiou’s choice of the term identity instead of isomorphism could be seen as problematic from a categorical perspective, leaving doubt as to whether Badiou is not transposing the primacy of his intrinsic ontology to a thought that is only relational.50 In the final analysis, this observation could lead us to explore an underused conceptu­al dimension available only to category theory. By contrast, to hold a point is the experimental practice espoused by Badiou always from the double perspective that choice involves. “To hold a point means to hold this instance in the face of the world. Or, to have the subjective (that is, corporeal and formal) wherewithal to submit the situation to the decisional pressure of the Two.”51 The theory of points is that from which radical change is enabled since it filters the degrees of the transcendental. The theory of points is 303 also what filters the onto-logic back into the ontology. 48 Alain Badiou with Fabien Tarby, Philosophy and the Event, trans. L. Burchill, Polity Press, Cambridge 2013 [2010]), p. 116. 49 Badiou, Logics of Worlds, p. 579. 50 McLarty’s reading is also voiced by Veilahti. Cf. Colin McLarty, “Review of Alain Badiou, Mathematics of the Transcendental, A. J. Bartlett and Alex Ling (trs.), Bloomsbury, 2014”, Notre Dame Philosophical Review (2014) and Antti Veilahti, “Alain Badiou’s Mistake – Two Postulates of Dialectic Materialism”, Math arXiv:1301.1203v22013, 2013. 51 Badiou, Logics of Worlds, p. 590. With regard to the qualified communism of the second body, as demonstrated in Book VII, it is important to emphasize the growing nuances in the system that have allowed it to break with, at least the classical understanding of, binary es-sentialism, seen as early as in the final statement of Manifesto for philosophy.52 There, the Mallarmean Constellation, anticipating the diversification of the sys­tem occurring in Logics of Worlds a decade later, had been summoned to evoke the new poetic significance of the event.53 Therein a subject-thought is said to be disposed “point by point” to evoke a communism of the multiple. Taking hold of the divisible body is the stated task of the second body if and only if it reaches the experience of the “transhuman body”.54 It is difficult not to see in this form the inscription of pure multiplicity itself. Final remarks on the oeuvre Several critics of Badiou’s system have accused the argumentative strategy in Being and Event as lagging behind debates in mathematics regarding the broad­er power of topos theory to analyze the nature of ontology in comparison to set theory. An aspect of these critiques has to do with the problem of the existence of infinite sets, a view aimed at showing they do not exist. While the philoso­pher of mathematics has found in Logics of Worlds a formal proposal adequate to the fact that set theoretic worlds can be analysed by category theory, many philosophers have been more receptive to the work for reasons that Being and Event does not accept. The emergence of radical generic subjectivation, they claim, would not be as rare as previously surmised. Nevertheless, when Logics of Worlds is read as a new and indeed complex theory of subjectivizable body, the evidence of plurality is not forthcoming. The reader is faced with an even more subtle, and demanding, radical philosophy grounded in formalism. Badiou has brilliantly shown how mathematics and philosophy still have much to share. As to the question of whether the destiny of mathematics is fundamen­tally bound to the technologies philosophy takes to be an affront to its ethical principles, it does not sufficiently justify restricting further inquiry into number, 52 Alain Badiou, Manifesto for philosophy, ed. and trans. Norman Madarasz, State University of New Press, Albany 1999, p. 109. 53 Alain Badiou, Briefings on Existence: A Short Treaty on Transitory Ontology, trans. and ed. Norman Madarasz, State University of New York Press, Albany, NY 2006 p. 168. 54 Badiou, Logics of Worlds, p. 481. sets, functions, and relations. As Badiou’s commitment to irreducible multiplic­ity relies on a subject-form that is indiscernible, intrinsically non-visible and im­manently bound to those who decide to take part in its validation through dis­parate practices, it recognizes set theory as its common ontological inscription. Logics of Worlds was a project aimed in its preliminary stages at satisfying two important critical questions. First, if ontology is mathematics, and set theory is used as the prototype for mathematics, as in ZFC and Paul Cohen’s generic extension of the set-theoretic universe, how does category theory, with its richer mathematical possibilities, stand in relation to ontology? Second, if ontology is the field of the objectless subject, how do we justify the overhauling of the phe­nomenal dimension merely through logic and not mathematics? We have exam­ined both of these questions. In this sequence, we can add another thought: is the excess warranted by the Grothendieck topos on worlds, objects, change and bodies a capture of the set theoretic universe or does it merely capture a world consequent to its axioms, theorems and operational possibilities? In his 2018 Prague keynote address55, Badiou loosened his equation of ontology and mathematics to let the former rest solely upon set theory. However, he did not relinquish the axiomatic structure of the ontology in favor of the broader relational possibilities of category theory. Regarding his defiance, is the risk we face with Being and Event’s ontological framework in support of infinity, that is, of the existence of infinite sets, a new metaphysics, as some mathematicians would contend? As long as the formalism is preserved, the concept of multiplic­ity shown to be irreducible and the generic process stipulates an index of in-discernibility, the answer would seem to be no. By contrast, if the category and topos theories in Logics of Worlds turn out to give material support to the generic process regardless of Being and Event’s ontological inquiry and refutations, it 305 will at least have given philosophers a choice of priorities regarding the creation of an ontology. In that regard, we can all appreciate the beauty and assurance infinite sets ultimately provide to a theory of multiplicity, regardless of the relief found in the range of correlations offered by topos theory in its own capacity at preserving truth from the perspective of the infinity of relations. 55 Cf.Badiou’s contribution to this volume: “Ontologie et mathématiques Théorie des En­sembles, théorie des Catégories, et théorie des Infinis, dans L’Etre et l'événement, Logiques des mondes et L'Immanence des vérités”. References Badiou, Alain, Being and Event, trans. Oliver Feltham, Continuum Books, London and New York 2005 — Briefings on Existence. A Short Treatise on Transitory Ontology, trans. ed. Norman Ma-daras, State University of New York Press, Albany, NY 2006 — Conditions, trans. Steven Corcoran, Continuum, London 2009- Le Concept de Modele. Introduction a une Épistémologie Matérialiste des Mathématiques, Fayard, Paris 1969 — Court traité d’ontologie transitoire, Seuil, Paris 1998 — Deleuze, “la clamour de l’Etre”, Hachettes, Paris 1997 — Entretien avec Peter Engelman, Quel communisme?, Bayard, Paris 2015 — Ethics. An Essay on the Understanding of Evil, trans. and intro. Peter Hallward, Verso, New York 2001 — La Tétralogie d’Ahmed, Actes Sud, Arles 2010 — L’Etre et l’événement, Seuil, Paris 1988 — L’Éthique. Essai sur la conscience du mal, Hatier, Paris 1993 — L’Immanence des vérités. L’Etre et l’événement 3, Fayard/Ouvertures, Paris 2018 — Le Nombre et les nombres, Gallimard, Paris 1990 — Logiques des mondes. L’Etre et l’événement 2, Éditions du Seuil, Paris 2006 — Manifeste pour la philosophie, Paris, Seuil 1989 — Manifesto for philosophy, followed by two essays, ed. and trans. Norman Madarasz, State University of New Press, Albany, NY 1999 — “Mathematics and philosophy”, in Virtual Mathematics, ed. Simon Duffy, pp. 12-30, Clinamen Press, New York 2006 — Mathematics of the Transcendental, trans. and ed. A.J. Bartlett and Alex Ling. Blooms­bury Press, London 2014 — Para uma nova teoria do sujeito, Editora Relume Dumará, Sao Paulo 1994 —Second Manifeste pour la philosophie, Fayard/Ouvertures, Paris 2009 — Théorie du Sujet, Paris, Seuil 1982 Badiou, Alain and F. Tarby, Philosophy and the Event, trans. L. Burchill, Polity Press, Cambridge, UK 2013 Badiou Alain and Gilles Haéri, Éloge des mathématiques, Flammarion, Paris 2015 Balibar, Étienne, “Le Structuralisme : une Destitution du sujet?”, Revue de métaphysique et de morale 45 (1/2005), pp. 5–22. Burhanuddin, Baki, Badiou’s Being and Event and the Mathematics of Set Theory, Blooms­bury, London 2015 Barwise, Jon and Lawrence Moss, Vicious Circles: on the Mathematics of Non-Wellfounded Phenomena, CSLI, Stanford, CA 1996 Cantor, Georg, “Foundations of a General Theory of Manifolds: A Mathematico-Philo­sophical Investigation into the Theory of the Infinite” [1883], trans. William Ewald, in From Kant to Hilbert: A Source Book in the Foundations of Mathematics, Volume II., ed. Ewald, Oxford University Press, Oxford 1996 — Contributions to the Founding of the Theory of Transfinite Numbers, trans. Philip Jour­dain, Dover, London 1915 Carmello, Olivia, Theories, Sites, Toposes. Relating and Studying Mathematical Theories through Topos-Theoretic Bridges, Oxford University Press, Oxford 2018 Cohen, P. J., “The Independence of the Continuum Hypothesis”, Proceedings of the Na­tional Academy of Sciences of the United States of America 50 (6/1963), pp. 1143–1148 — Proceedings of the National Academy of Sciences of the United States of America 51 (1/1964), pp. 105–110. Constantinou Marios and Norman Madarasz, “Being and Spatialization: an interview with Alain Badiou”, Environment and Planning D: Society and Space 27 (2009), pp. 783–795 Desanti, Jean-Toussaint, “Quelques remarques a propos de l’ontologie intrinseque d’Alain Badiou”, Les Temps Modernes (526/1990), pp. 61–71 Fraser, Zachary, “The Law of the Subject: Alain Badiou, Luitzen Brouwer and the Krip­kean Analyses of Forcing and the Heyting Calculus”, The Journal of Natural and So­cial Philosophy 2 (1-2/2006), pp. 94–133 Gödel, Kurt, “What is Cantor’s Continuum Problem?”, in Kurt Gödel: CollectedWorks, vol II., ed. Solomon Feferman, et. al., pp. 1938–1974, Oxford University Press, New York 1995 — “The Consistency of the Axiom of Choice and the Generalized Continuum Hypoth­esis”, Proceedings of the national academy of sciences of the U.S.A. 24 (1938), pp. 556–557 Grattan-Guinness, I., The Search for Mathematical Roots. (1870–1940): Logic, Set Theories and Foundations of Mathematics from Cantor through Russell to Gödel, Princeton Uni­versity Press, Princeton, NJ 2000 Grothendieck, Alexandre, Récoltes et Semailles, Université des Sciences et Techniques du Languedoc, Montpellier 1985–1987 Hallward, Peter and Knox Peden (eds.), Concept and Form, Verso, New York 2012 Heidegger, Martin, Being and Time, trans. John Macquarrie and Edward Robinson, Harp­er and Row, New York 1962 Heller, Michael and W. Hugh Woodin, Infinity: New Research Frontiers, Cambridge Uni­versity Press, New York 2011 Hunter, Ian, “Heideggerian Mathematics: Badiou’s Being and Event as Spiritual Peda-gogy”,Representations 134 (Spring 2016), pp. 116–156 Jech, T., Set Theory, Springer-Verlag, Berlin 2003 Lacan, Jacques, Séminaire livre XX: Encore (1972–1973), ed. Jacques-Alain Miller, Editions du Seuil, Paris 1999 — Autres écrits, Seuil, Paris 2001 Lautman, Albert, Les mathématiques, les idées et le réel physique, Librairie Philosophique, J. Vrin, Paris 2006 Lejewski, Czeslaw, “Ontology and Logic”, in Philosophy of Logic, ed. Stephen Korner, pp. 1–63, Basil Blackwell, Oxford 1976 Madarasz, Norman, O Realismo Estruturalista: sobre o imanente, o intrínseco e o inato, Filosofia e interdisciplinaridade, Editora fi, Brasil 2015 — O Múltiplo sem Um. Uma apresentaçao do sistema filosófico de Alain Badiou, Editora Ideias e letras,Sao Paulo 2011 — “On Alain Badiou’s treatment of Category Theory in view of a Transitory Ontology”, in Alain Badiou: Philosophy and Its Conditions, p. 23–43, Suny Series, Albany, NY 2005 Malicki, Maciej, “Mathemes and Mathematics. On the Main Concepts of the Philosophy of Alain Badiou”, ArXiv, May 2014, available at:  2014arXiv1406.0059M Maniglier, Patrice et Rabouin, David, “A quoi bon l’ontologie ? Les mondes selon Badi­ou », Critique (719/2007), pp. 279–294 Marquis, Jean-Pierre, “Category Theory”, The Stanford Encyclopedia of Philosophy, ed. Edward N. Zalta [forthcoming] Mclarty, Colin, “Como Grothendieck simplificou a geometria algébrica”, trans. Norman R. Madarasz, Revista Veritas 61 (2/2016), pp. 276–294 — “Review of Alain Badiou, Mathematics of the Transcendental, A. J. Bartlett and Alex Ling (trs.), Bloomsbury, 2014”, Notre Dame Philosophical Review (2014), available at: https://ndpr.nd.edu/news/mathematics-of-the-transcendental/ — “Review of Simon Duffy, ed., Virtual Mathematics”, Australasian Journal of Philoso­phy 86 (2008), pp. 332–36 — “The Uses and Abuses of the History of Topos Theory”, British Journal for the Philoso­ phy of Science, 41 (1990), pp. 351–75 Putnam, Hilary, Philosophy of Logic, Routledge Revivals, London 2011 Rahman, Shiva, “On Why Mathematics Can Not be Ontology”, Axiomathes 29 (2019), pp. 289–296 Tasic, Vladimir, “Badiou’s Logics: Math, Metaphor, and (Almost) Everything”, Journal of Humanistic Mathematics 7 (1/2017), pp. 22–45 Veilahti Antti, “Alain Badiou’s Mistake. Two Postulates of Dialectic Materialism”, Math arXiv:1301.1203v22013 (2013) Watkin, William, Badiou and Indifferent Being. A Critical Introduction to Being and Event, Bloomsbury, London 2017 Grands cardinaux et attributs de l’absolu Large Cardinals and the Attributes of the Absolute Filozofski vestnik | Volume XLI | Number 2 | 2020 | 311–340 | doi: 10.3986/fv.41.2.12 Frank Ruda* To the End: Exposing the Absolute “Toute infinité requiert une errance.”1 “The task is indeed to demonstrate what the absolute is. But this demonstration cannot be either a determining or an external reflection by virtue of which determinations of the absolute would result, but is rather the exposition of the absolute.”2 “Notez que je suis absolument immanentiste.”3 Introduction The Immanence of Truths is the vineyard in which all the labour of the reader of the first two Being and Event volumes finally, and one might dare to say, ab­solutely pays off. And – as in the famous Jesus parable – those who start with the last volume will receive just as much as those who started years and years ago. Everyone will have received the same currency, notably orientation – and especially a reader of Badiou’s last systematic volume is enabled to see what has any real value – and this means “absolute value.”4 For this reason alone, time does not matter much for the currency that the absolute provides us, as it is that “which in time exceeds time.”5 Yet, the peculiar place where this absolute value 311 system is formulated is difficult to locate. It is close by,6 yet and at the same time it does not exist in any standard sense of the term. The place of the absolute is 1 Alain Badiou, L’Immanence des vérités. L’Etre et l’événement, 3, Fayard, Paris,2018, p. 236. 2 G. W. F. Hegel, Science of Logic, Cambridge University Press, Cambridge 2010, p. 466. 3 Alain Badiou, “L’etre en nombres”, in Entretien 1. 1981–1996, Nous, Paris 2011, p. 177. 4 Badiou, L’Immanence des vérités, p. 417. 5 Ibid., p. 176. 6 Cf. Frank Ruda, “Tovariši absoluta, ali: Kje je ‘aupres de nous?’”, in Problemi, (3–4/2019), pp. 133–150. * University of Dundee like an invisible, inexistent, yet under certain conditions universally accessible Fort Knox for the people, that is: for everyone. In what follows I will take a field trip into this inexistent yet rigid fort of the ab­solute. To do so, I will: 1. elucidate the specific perspective of Being and Event, 3 against the background of the two previous volumes; and 2. I will turn to the immanent but invisible dialectic of the general floorplan of the absolute fort that Badiou calls absolute ontology and offer a reading thereof. This will lead to a discussion (albeit limited) of the place “where all possible forms of” in­finite “multiples”7 are located, and will show in what sense it provides us with a means to “measure”8 a truth’s inner infinity, i.e. the absoluteness of a truth. The article will end by only pointing in the direction of what deserves further elaboration in the future, namely the conceptual requirement needed to con­ceive, metaphorically speaking, of the unfolding of the inner inconsistency of the (concept of the) absolute. How to form a (philosophical) concept (the ab­solute) which is barely still a concept due to the “absolute powderiness”9 that follows from the “intrinsic diversity of [the] types of infinity”10 that inhabit it? That the absolute, “the absolute referent […] in-exists”11 and nonetheless pro­vides an – in this sense fictional12 – orientational measure can be understood, as will be shown, against the background of an impossible yet necessary (i.e. forced) totalisation of the domain of the absolute. The article will signal that at the abyssal ground as well as at the impossible end of the absolute there is a paradoxical experience of freedom. 7 Badiou, L’Immanence des vérités, p. 245. 8 Ibid., p. 333. Previously, it seemed that what Badiou called “nature,” the ordered sequence of ordinals, could provide such a measure. “Nature, even infinite, is absolute,” as he stat­ed. Alain Badiou, Being and Event, Continuum, London / New York, 2006, p. 362. Now, the set of ordinals only provides us with an internal model, an attribute of the absolute, as there are not only ordinals in the latter. Cf. Badiou, L’Immanence des vérités, pp. 371ff. 9 Badiou, “L’entretien de Bruxelles”, in Entretien, p. 107. 10 Badiou, L’Immanence des vérités, p. 374. 11 Ibid., p. 417. 12 What precisely this kind of fiction is will become clearer in what follows. But it should be clear that it is not meant to be a mere construct of language or discourse. Cf. ibid., p. 30. Badiou once linked it to the idea that a truth is always related to what it will have been (and due to this specific temporality has a fictitious being). Cf. Alain Badiou, “Philosophie & Politique”, in Entretiens, p. 132. This is just the roadmap or, if the reader prefers, the activity plan for the envis­aged field trip. I take it that such a roadmap or activity plan (or both) can help when one seeks to exit from whatever cave one comes from (for example, one’s preconceptions of what the infinite is, or what one believes Badiou is doing – anything that can actually work as a cave, and hold us hostage). It is mainly supposed to enable one to not lose the way on the way (out).13 From the being of truths to their appearing… TheImmanence of Truth is not a book that addresses the universality of truth nor its singularity but its absoluteness.14 It therefore does not deal with the concept of the generic (set) that elucidates a truth’s universality nor with that of the in­existent of a world from which its concrete singularity can be accounted for. It deals with the inner link between truth and the absolute and thereby with that of truth and the infinite.15 The genuine perspective of The Immanence of Truth can best be established by depicting why such an absolute point of view is at all sys­tematically necessary.16 This is especially significant since it is, which for the av­erage professional philosopher will sound irritating, a(n impossible) “view from nowhere.”17 But one should recall that “absoluteness”18 already played an obvi­ous role in Being and Event, 1 and therefore Badiou’s entire philosophical project seems to depend on how pertinent the perspective of Vol. 3 will prove to be. 13 Badiou claims that The Immanence of Truths depicts the cave (of finitude), leads us out of it (so that we encounter the blinding brightness of the different sizes of infinities in the absolute, i.e. a heaven of ideas). It then makes us return to the cave, so that we are able to make a difference (between waste [déchet] and work [oeuvre]). Badiou not only depicts the constitution of the (ontological) cave – which surprisingly disappears if we have a dif­ferent orientation – but a philosopher is also he or she who, by confronting us with the 313 absolute, drags us out of it: because one should remember that the prisoners do not want to leave voluntarily. 14 Badiou, L’Immanence des vérités, p. 275. Absoluteness also necessitates addressing the dialectic between the finite and the infinite. Cf. Alain Badiou, Le fini et l’infini, Bayard Culture, Paris, 2010. 15 This is already the gesture of Cantor, whose “ontological thesis is evidently that inconsis­tency […] orientates thought towards the Infinite as supreme-being, or absolute.” Cf. Alain Badiou, Being and Event, Bloomsbury, London / New York, 2013, p. 42. 16 Badiou himself raised this question: “But what is the general necessity for a third book?” Cf. Alain Badiou, “Conférence de Ljubljana”, in Filozofski Vestnik, 32 (2/2011), p. 7. 17 Thomas Nagel, A View from Nowhere, Oxford University Press, Oxford 1986. 18 Cf. appendix 5 in Badiou, Being and Event, pp. 456ff. What thus is the systematic need for a third volume? If it is systematically nec­essary, this necessity must derive from the specific gaps left by the first two in-stalments of Being and Event. These gaps do not turn their respective enterprises into flawed or failed ones. The point is not that they lack something that could have been done better. Rather, their respective perspectives, when taken togeth­er, enable us to see that something is missing (from their points of view), which only appears to be missing because of their points of view. Only after Being and Event and Logics of Worlds are we enabled to see that something is missing. So, what they brought to the fore was a systematic lack: a lack that only exists because they did what they did. After they established and achieved what they set out to do, it became evident that they only determined in a negative manner what necessarily also needs a positive elaboration. What is it that we only get a negative rendering of? Both books depicted only negatively what it means for a truth to emerge in a historically specific situated world as a set of consequences of an event. They did not positively deliver a demonstration of why and in what sense a thusly produced truth can actually be said to be (eternally) true. Badiou always argued that “each truth is, in a sense, absolute,”19 but the first two books did not tell us too much about the inherent conception of the abso­luteness of truths. Rather, they did account for it in oppositional terms. This is a systematic implication of their perspective, since they did not positively begin from (within) the (immanence of the) absolute or infinite and then derive an account, for example, of the finite from it. They all argued for a need to think the infinite, yet did not conceive of it from within the immanence of a truth’s inner unfolding, even if they relied on what specific generic processes made it possi­ble to think.20 But Badiou had to reverse the perspective: a(n almost Platonic) turn was needed.21 To understand the precise coordinates of this turn, one may ask again: Why did the first two volumes of the Being and Event series not de­ 19 Alain Badiou and Maria Kakogianni, Entretien Platonicien, Lignes, Paris 2014, p. 52. 20 In some sense, my rendering is imprecise and rather heuristic. Because already for Being and Event, 1 it is clear that it is the infinite that “carries or bears the finite, a truth that car­ries a subject.” Alain Badiou, “Beyond Formalization. An Interview”, in Angelaki. Journal of the Theoretical Humanities, Vol. 8, No. 2, August 2003, p. 132. And equipped with an adequate concept of infinity, we can see that “the infinite, only the infinite testifies for the truths.” Badiou, L’Immanence des vérités, p. 419. 21 On this motive, cf. Frank Ruda, “Marx in the Cave”, in Slavoj Žižek, Frank Ruda, Agon Hamza, Reading Marx, Polity, London 2018, pp. 62–100. liver everything one needed to conceive of the (non-)relation between its main terms, being and event? Was this not what they set out to do? An answer to this question can be sketched out if we recall the previous books’ achievements in slightly greater detail. Being and Event articulated Badiou’s meta-ontological concept of being – meta-ontological because, mind you, phi­losophy is “not […] ontology.”22 It did so by identifying mathematics, i.e. axi­omatised set-theory with the discourse of ontology. In the same book, Badiou conceptualised how such a philosophical meta-ontology allows the conception of an event, i.e. of ‘something’ which is not really some thing and which there­fore is not what an ontology can possibly account for. It cannot do so, because if an event really is an event, it is not derivable from being (ontologically for-malised) qua being.23 This yields an important consequence for the overall sys­tematic perspective of this volume and the whole project: events can happen even within ontology – as discourse on being. Ontology – but not being! – has a history, because its very concept of being can be transformed.24 The concept of the event and that which is directly linked to it, i.e. the concept of truth, was therefore determined in the first volume of Being and Event in a primarily negative or at least supplementary way. It was the “and”, or as Badiou himself phrased it in almost deconstructive language, it was conceived as what “supplements”25 being. In Being and Event, we start from being and get to that 22 Badiou, Being and Event, p. 3. This is the case, since “ontology […] exists as a separate and exact discipline,” i.e. mathematics. 23 For this, ibid., pp. 184–190. If it were derivable from a given concept of being, it could be described by recourse to “the encyclopaedia of the situation” (ibid., pp. 327ff.) and hence could be constructed by recourse to the already existing language (which conceptually 315 means there could never be any evental supplement to being, everything would already be latently there and hence the book series could have been called simply Being). Ibid., pp. 265–327. This problem will acquire a new dramatic form in The Immanence of Truths. 24 Being and Event philosophically deploys the conceptual consequences of the “Cohen-event,” as something that happened – contingently – in and to ontology. Ibid., p. 16. Yet if what ‘substantiates’ an event is what will have followed from it, there are therefore also necessarily ontological truths: This is why “there is a history of mathematical truths.” Alain Badiou, “Dix-neuf réponses a beaucoup plus d’objections”, in Entretiens, 1: 1981– 1996, Nous, Paris 2011, p. 67. 25 This is why any generic procedure – that “practice” which produces the consequences of an event and thus makes the event into what it will have been – that is inaugurated by an event can be described as an “undecidable supplement” (Badiou, Being and Event, p. 355), which is not being. That-which-“is-not-being-qua-being”26 proved to be as im­portant as that-which-is (being). Therefore, in a dialectical move, we need to say something about the being of that which is not being-qua-being and thereby we say something about (the being of) truth. Its primary characterisation was negative in the sense that it was doubly derived from the meta-ontological per­spective on being: when a truth has originated in what is not-being-qua-being (an event), the being of this truth is – as also holds true for everything that is – also a type of multiple (even though a specific, generic one). Being and Event proposed “a complex thinking of the being of truths.”27 The “impossible possibility”28 of truths arises from what is not (being qua be­ing) – and it is not, because it happens. After determining what is constitutive of truths, one is able to determine them as that which also and necessarily transforms that which is (even being qua being). So, we move from being to that which is not-being to that which results from the latter and transforms the for­mer. Truths thus bring temporalisation (and history) to being – being, this was Badiou’s anti-romantic (anti-Heideggerian) wager,29 has no history of its own. A historicising truth is therefore always a non-derivable, aleatory supplement to being. History is aleatory and alethic time. Being and Event presents a number of concepts for grasping post-evental truths: the event, elucidating its “origin,” the “generic” which determined the being of a truth, the concept of “fidelity” (de­scribing its practical aspect) and the concept of “forcing” that determines “truth as truth,”30 i.e. the being of truth in general. Being and Event says something about being and it elaborates what Badiou once called the subjective process, but it does not give us all the conceptual tools needed to think subjectivisation which is itself sustained by a further, additional, or supplementary supplement, namely “the fiction of a theory” (ibid., p. 246), the “supplementary hypothesis” (ibid., p. 249) of assuming the totalisation and closure of a generic procedure (as in saying: “I will always love you” after falling in love due to encountering someone new). This is a simplified rendering of forcing. 26 Ibid., 15. 27 Badiou, “Dix-neuf réponses”, p. 65. 28 Alain Badiou, Can Politics Be Thought?, Duke University Press, Durham / London 2019, p. 97. 29 The project is a “de-romanticization of the infinite.” Cf. Badiou, “L’etre en nombres”, p. 189. 30 Badiou, “Dix-neuf réponses”, p. 65. proper.31 Even though the concept of forcing – allowing one to conceive of the functioning of the “subject-language,”32 i.e. of the interiority of an engaged sub­jective perspective and the immanent representation of what a subject believes it is doing – and Badiou’s overall philosophy can already here be convincingly described as “genericism,”33 the event and all related concepts were elucidated against the backdrop of being. Logics of Worlds, the second volume of Being and Event, changes the perspective. It does not deal with being as such but with (formalising) all the possible ways in which (something of) being can appear. Everything that is can appear when it is localised in a determined logical space, in a world. Logics of Worlds thus pre­sents a meta-phenomenology where phenomenology, the logic of appearance proper, is related to mathematical algebra and topology.34 Different from tradi­tional phenomenology, the meta-phenomenological perspective elides all refer­ences to an already given subject of (potential) experience. The book elaborated a non-subjective theory of the transcendental,35 of that which governs the laws of appearance of a singular world. All appearances of being are worldly and thus historical.36And even though being appears, appearance has its own logics – roughly put: logic is to appearance what mathematics is to being. Its exposition leads to an account of the multiplicity of different worlds and the different forms 31 For this distinction, which Badiou himself uses (cf. Badiou, “Conférence”, p. 15): Alain Badiou, Theory of the Subject, Continuum, London/New York 2009, pp. 241–275. 32 Badiou, Being and Event, p. 400f. 33 Badiou, “Philosophie & Politique”, in Entretiens, p. 136. 34 Alain Badiou, Logics of Worlds, Being and Event, 2, Continuum, London / New York 2009, 317 p. 234. 35 Ibid., pp. 97–188. 36 A reader of Logics of Worlds might be surprised that within the logics of appearance cer­ tain crucial concepts of the meta-ontological account seem to have disappeared (or been forgotten). But this can mostly be easily explained: What happened to the distinction be­tween nature and history that was crucial for Being and Event? The explanation is obvious: whatever appears appears according to transcendentally governed laws; and even if the most stable form of multiplicities (nature) appears, it appears in a particular setting. Their ontological structure will not be transformed by their appearance (there can be factual changes to a natural world through evolution, for example – and factual changes are not events, which means that there are still no events in nature). But nature meta-ontological­ly, as physis, is obviously quite different from phenomenal nature. of appearance within the frame of an “objective phenomenology.”37 This objec­tive meta-phenomenology, to state the obvious, is not meta-ontology: it deals with a realm of its own in which (something of) being appears in the form of bodies that are accompanied by their respective languages or descriptions.38 This is the backdrop against which Logics of Worlds seeks to elucidate the ap­pearance of what constitutes a truth and the appearance thereof. We thus get in this work also a theory of all the possible ways in which change can appear, the most intense being an event (pun intended). And we also get an account of the ways in which a post-evental truth-practice can appear39and of the appearance of a subject, of a subjective body – since true change that appears also appears as a body.40 The appearance of an event is related – note the structural repetition – to that which in appearance does not appear, to that which appears so mini­mally that it appears not to appear, to what appears to “inexist”41 in a world. An event effectively allows for a change in the laws of appearance, a change of thetranscendental.42It makes exist that which inexisted by presenting a mo­mentary suspension of the transcendental (which is therefore an appearance of disappearance) that consequently effectuates a transformation of its laws. Logics of Worlds accounts for this transformation by offering an account of what it means for a truth to appear within a concrete and particular world. Badiou thus accounted for the appearance of a truth by recourse to that which is not (because it happens, even to and in appearance). 37 Badiou, Logics of Worlds, pp. 277–280. Surprisingly enough, Thomas Nagel used the same term to describe his own, quite different project. Cf. Thomas Nagel, “What Is It Like to Be a Bat?”, in The Philosophical Review, 83, (4/1974), p. 449. 38 Badiou, Logics of Worlds, pp. 191–296. On the meta-ontological level, this is referred to as the “language of the situation” (Badiou, Being and Event, p. 288f). 39 Meta-ontologically speaking, all elements of an evental situation can only refer in a posi­tive or negative way to the event (when examined); meta-phenomenologically speaking, worlds know more variety, which is why Badiou here also must say something about the reactive and obscure subject (when elaborating what a truth-practice looks like). 40 Technically speaking: it is that whose identity with an event is maximal. Cf. Alain Badiou, Le séminaire – Images du temps présent: 2001–2004, Fayard, Paris 2014, p. 422. Badiou, Logics of Worlds, pp. 321–324 and pp. 376–378. There is only one inexistent in a world and it is therefore the point on which the whole consistency of the world hinges and which singularises it. 42Not all transformations affect the transcendental of a world, but an event does. For a cata­logue of the different types of change, ibid., p. 374. A truth appears when something appears, as a consequence of an event, which did not appear (and thus did not seem to exist or be possible) before. This con­version appears in the form of a new body that is engaged in a practice of phe­nomenal self-organisation or auto-transcendentalisation (a classical name for it is freedom). To account for how the product of an event within a world is able to suspend the transcendental and to itself take charge of the laws of appear­ance, Badiou develops a theory of the subjectivisable body. All that appears in a world are bodies and languages, except when truths appear. They appear as a different type of body, as truth-bodies. They are not simply finite, but inscribe something else into the apparently closed universe of a world. Logics of Worlds demonstrates that a subject is not only structured like a language, but also like a dialectic,43 because within it infinity (truth) and finitude (every body is fi­nite) are materially woven together in a truth-body that appears in a particular world. A subjectivised truth-body inscribes the infinite into a world – because it takes decisions what path to take, what type of consequences to explore, with whom to ally itself, etc., and these decisions are crucial not only for the body’s continued existence, i.e. the “bodily [en-corps]”44 materialisation of a truth’s self-organisation, but also for the transformation of a world. These decisions are what Logics of Worlds refers to as points and they are how the infinite is “fil­tered by the form of the two,”45 i.e. by the form of a decision (of how to continue) into the world (again: we are here talking about freedom).46 Being and Event, 2 says something about (the) appearance (of being) and it elaborates conceptual tools to account for the appearance of subjective truth-bodies. But it does not give us an account of what subjectivation looks like from the perspective of an individual who incorporates him- or herself into an active truth-body – surpris­ingly for many of its readers, forcing does not play a very explicit role in it. Even though the complex dialectic of the infinite and the finite is clearly at the centre of Badiou’s account of the truth-body, its constitution and transformation, the 319 43 Cf. Alain Badiou, Le séminaire – L’un: Descartes, Platon, Kant (1983-1984), Fayard, Paris 2016, p. 141. 44 Badiou, Logics of Worlds, p. 46. 45 A point is always the infinite treated in the form of the two. Cf. Badiou, Images du temps présent, p. 384. 46 It should be clear that inscribing the infinite into the finite by forcing it into the form of the two of an either-or choice means that the infinite (of any truth) is closely linked to free­dom. This is an implication that comes fully to the fore in The Immanence of Truths. I will return to this point (pun again intended) below. event and all related concepts, especially that of truth, were elucidated against the backdrop and within the frame of the logics of appearance. … to their immanence Two books, two accounts of eventally originating truths, none of which would put truth first. With the third and final volume, the turn must thus be made: truth first! Before we can see what this entails, it is worth pointing out a pe­culiar problem that arose with and from the publication of Badiou’s Logics of Worlds. The problem can be described in the following simple way: What is the relation between Vols. 1 and 2 of Being and Event? Obviously, one deals with being, the other with appearance, and both deal with the respective relation to eventally originating truths. But is there anything that necessitated the appear­ance of being? If so, one would have to explain the origin of appearance outside of appearance (notably in being). This runs counter to the concept of being that Being and Event elaborates. Nothing in being drives being to appear. For in it­self being is static. Therefore, the appearance of being cannot be derived from the concept of being. But why then is there appearance – is this a supplement to being (and what kind)? Is this just contingent? This set of questions is further complicated by the fact that ontology (as mathe­matics) clearly appears and hence its world – since “ontology is a situation”47 – follows specific, transcendental, laws48 that govern what can appear and what cannot appear within it.49 Here, it is again important to recall that ontology is 47 Badiou, Being and Event, p. 27. 48 Badiou,Logics of Worlds, pp. 183ff. This can mean that there is not first being and “then” also appearance, but that it is as plausible to speak of a primacy of appearing, since it is only within the appearing discourse of ontology, mathematics, that we – if we attempt to identify what it is doing from a meta-ontological position – formulate a concept of being. In this sense, Badiou systematically avoids a two-layer model of ontology and phenome­nology. 49 Ibid., pp. 183–188. The following should be noted: 1. that ontology as such does not have a concept of the event – the event is a meta-ontological, i.e. philosophical concept. Which means “the event” appears as inexistent in the classical world of ontology. 2. It is here important to note that mathematics comes first (as ontology) and logics comes second (as phenomenology). Badiou himself worked in the 1960s within the framework of a “mathematicised logics,” a “logical materialism” in which “the difference between a mathematical theory and a logical theory was itself thought from the interiority of logics.” This obviously changed in the later development of his position. Alain Badiou, a condition and that the whole perspective of Being and Event provides a me­ta-ontological, i.e. philosophical, re-formulation of what Badiou identifies as an eventally arisen truth (which transformed ontology). The key to understand­ing in what way the meta-ontological account of being relates to the meta-phe­nomenological one of appearing thus lies, what else could be expected, in the account of eventally emerging truths. This does not – and cannot – mean that there is a subject mediating the two distinct spheres of ontology and phenome­nology. It rather means that from within a certain post-evental practice we be­come able to make and formalise the difference between being and appearance. This, therefore, does not necessarily mean that the appearance of being is an event and that such eventuality must be inscribed into our concept of being. Appearing as such is as unevental as being, even if the formal conceptualis­ations thereof are. But this also implies that the concept of being is not formu­lated within an objective ontology. It is itself part of a subjectivised truth-body that auto-dictates and regulates its own laws of appearance to itself. “Being” appears in ontology (as discourse) as it does in (philosophical) meta-ontology. This is to say, 1. there is no path that leads from being to appearance: the con­cept of being as formalized after the Cantor and Cohen event in ontology is not generative of appearance. But these events changed something about the (tran­scendental regulation of the) appearance of ontology. There is, thus, 2. a path that leads from appearance to being, under the condition that there is appear­ance. The relation between being and appearance is a one-way street. Being and appearance look very different from the perspective of (meta-)ontology (as being is then just an immobile multiple of nothing and appearance is a form in which it is counted-as-one) and from that of the logics of appearance (where being is what appears and appearance is the variety in which it appears). In a conceptual sense, this means that there is no (reciprocal or deductive) relation between being and appearing. Thereby, the appearance of being is non-deriv- “Préface de la nouvelle édition”, in Le concept de modele. Introduction a une épistémolo­gie matérialiste des mathématiques, Fayard, Paris 2007, p. 30. able(nothingnecessitates it) and hence contingent.50 Yet, when there is appear­ing, what appears is being (even when formalised within ontology).51 If there is no derivable relation between being and appearing, does this mean that – this may almost sound reminiscent of Kantian problems52 – the third vol­ume is supposed to provide the systematic consistency with – and hence rela­tion to – the previous two unrelated accounts? The project of The Immanence of Truths is not to conceive of truths from the standpoint of either being or ap­pearing, but to begin with truth (even though, generally speaking, the structure of the book first deals with the ideology of finitude and then moves on to de­pict its infinite outside and thereby reverses the constitutional relation between the infinite and the finite, so that the infinite finally comes first – we thus get a repetition in miniature of Badiou’s overall path in his last philosophical book). The third volume does not simply fill a gap left open. But it contributes to the ways in which we will have to re-read the previous two volumes. The third vol­ume makes it unambitiously clear that primacy will have been given to the in­finite (and thereby to truths) all along. To elucidate this claim, one could argue that the three volumes are linked with one another as the circles are within a Borromean knot: we have being (in its meta-ontological account), appearance (within its meta-phenomenological account), and the two respective accounts of the evental emergence of truth. Finally, we have an immanent account of truth, a conception of truth on and in its own terms. 50 Does this force us to assume that appearance is therefore necessarily constituted by that which is not-being-qua-being and hence the existence of appearance is itself the result of an event? If this were the case, what kind of event would we be talking about (since it seems rather difficult to account for it in terms of one of the conditions – note: it cannot be an event in ontology, because ontology is the “discourse on being qua being” (Badiou, Being and Event, p. 7), but this cannot mean that whatever takes place in this discourse could actually generate the appearance of being (where there was none before). 51 One may see Badiou’s claim that “[p]roblems of connection and continuity” – between Be­ing and Event1 and 2 – “do remain, namely between ‘generic procedure’ and ‘intra-worldly consequences of the existence of an inexistent’” as pointing also to the problem I address above. Thus, I have formulated a possible solution and follow Badiou’s remark that he “leave[s] them for another time, or for others to solve.” Badiou, Logics of Worlds, p. 39. 52 For a far more convincing reading, cf. Rado Riha, Kant in Lacan’scher Absicht: Die koperni­kanische Wende und das Reale, Turia+Kant, Vienna 2018. It is important to note that when­ever there is appearing, what appears is being, which means that we are not separated from being through appearing. But to speak of truth in and on its own terms, we have to introduce the concept of truth again, we have to say something about the truthness of truth. Badiou does this by means of the concept of the absolute, which as he claims was there­fore “with us”53 all along – in the sense from Being and Event, 1. We thus get the following schema, a Borromean knot with the (inexistent – I will return to this) absolute at its core: Being A Truth Appearing A = Absolute This model suggests, apparently, that each of the domains has an overlap with the others and hence a unique perspective on how to conceive of them.54 Only 53 Badiou, L’Immanence des vérités, p. 415 and cf. also Ruda, “Tovariši”. Here it is also in­teresting to note that with regard to the work Badiou did with(in) his seminar, he moves from: a) the logical theory of models (1980–1983) – already precipitated by his 1969 Con­cept of the Model; to b) (mathematical) set theory (1983–1989) – Being and Event is pub­lished in 1988; and then moves quickly through c) the (mathematical) theory of numbers 323 (1989–1990) – Number and Numbers is the manifestation of this period; addresses d) the (mathematical) theory of categories, topos (1990–1996); and e) the philosophical theory of worlds (1996–2001); before finally dealing with f ) the ideology of finitude (2001–2004); g) how to conceive of an exit therefrom (2004–2012); and ultimately h) deals with the im­manence of truth (2012–2015). 54 To my mind, one could give a similar rendering of Hegel’s philosophical system, with Na­ture, Spirit, and Logic as circles and, again, the absolute located at their intersection. I offered an interpretation of Hegel’s Logic along the lines of The Immanence of Truths in Frank Ruda, “Hegel’s Immanence of Truths”, in Badiou and German Philosophy, ed. by Jan Völker, Bloomsbury, London 2019, pp. 51–69. It might be interesting to discuss if one may read the above rendering as pointing out that the absolute is nothing but its own unfold­ing, its exposition. through their entanglement are we able to see that the absolute is always close-by. The three books form a Borromean knot of being, appearance, and truth: there is being, a being of truth, a being of appearance and an appearance of be­ing, and an appearance of truth; there are truths of the presentations of being, of specific worlds – and, as we will see: the truth of an account of being if formu­lated in a condition of philosophy can lead us to formulate an absolute ontology (I will get to this). In order to keep these rings together, the absolute is needed – and it only in-exists in this entanglement; it becomes conceivable in the mar-gin(s) at the centre. It is the knot’s off-side, out-side, out-of-sight, out-of-space, out-of-place, its hors-lieu.55 One needs being, appearance, and truth to expose it. Another way of putting this is the following: ontology is a condition and hence the site of potential truth procedures. Therefore, the determination of the being of a truth can itself be regarded as the (knowledge-) product of a truth-proce­dure. The absolute is what can be said to determine the truthness of truth. We can understand how to conceive of it by determining the consequences and im­plications of the truth procedure that allows for a new thought of the concept of being, by allowing for a new conception of infinity, and of its appearance as part of the truth-body that is unfolded within ontology in the aftermath of the (Cantor-event and) Cohen-event. This brings with it also a new conception of truth. We get an absolutely immanent account of the truthfulness of truth, of the absoluteness of truth if we take into account that ontology is itself a condition where something happened that philosophy has to incorporate. From this me­ta-ontological perspective, a truly novel ontological account of the being of in­finity and thereby of truth must be read as a result of a truth procedure that has the peculiar features that it – qua result – also provides an immanent “measure” of its own truth-value. An ontological truth procedure produces a thinking of infinity that through the intervention of philosophy can be comprehended as providing an evaluation of a truth’s truthness. This does not lead us to the discovery of the one and only true truth. The abso­lute is not the truth. Rather, it is because of a truth-unfolding in (and as) ontolo­gy that we can forcefully imagine an absolute – and hence no longer historical – ontology. An ontology that allows us to verify that the truth we are engaged in – Badiou, Theory of the Subject, pp. 3–21. This is why Badiou will indicate that the absolute is not a multiple and not a set. sharing its ontological consequences – is actually true. Thereby it provides an immanent “measure” of itself and of all other potential truth procedures. A truth occurs in ontology in such a way that it creates a fiction of knowledge that al­lows us to determine the absoluteness, i.e. the truthness of the truth that created this very fiction. Badiou always insisted that “the potency of a truth depends on the hypothetical forcing. It consists in saying: ‘If we suppose the generic infinity to a truth to be completed, then such or such piece of knowledge must be imper­atively transformed.’”56 An absolute ontology must be an ontology whose very constitution is impact­ed and effectuated by a truth procedure, yet which is only properly constituted with the hypothetical formulation – i.e. with the necessary even though impos­sible and thus fictitious forced totalisation – of all possible effects of this very truth procedure. We create such fictitious totalisation in ontology and thereby we articulate an absolute measure of truthness, which implies that we formu­late a complete, absolute, ontology. It is precisely in this sense that we surpass the “structural” and “historico-existential”57 determinations of truth and start to determine its proper truth character. In its unfolding, we bend the process back onto itself and totalise it.58 Yet this totalisation is necessarily fictitious and, even though necessary, at the same time bears the mark of its own impossibility due to the fact that the consequences of a truth procedure are never just com­pleted – otherwise, at least in an immediate understanding, a truth would be finite or limited.59 Even though a truth procedure absolutises60 itself in ontology and thereby generates the concept of the absoluteness of truth, we must “think a truth as at the same time grasped in its immanence to the absolute and at its distance to this grasp.”61 If we were to forget that the totalisation is a hypo­thetical fiction bridging the impossible, we would reduce truth to knowledge or make truths into one truth (and end up either suspending truth altogether or 325 turn philosophy into religion). An absolute ontology is a necessary and impos­sible project of a philosophy that identifies ontology with mathematics, mathe­ 56 Alain Badiou, “The Ethics of Truth: Construction and Potency”, in Pli 12 (2001), p. 252. 57 Badiou, L’Immanence des vérités, p. 20. 58 This sounds highly compatible with certain contemporary readings of Hegel. 59 The solution, as I will only be able to indicate briefly at the end, will lie in splitting the concept of the end into two. One divides into two: even at and in the end. 60 Badiou, L’Immanence des vérités, pp. 415ff. 61 Ibid., p. 402.a matics with a specific thinking infinity (without one), and infinity (without one) as what allows for conceiving of truth: we must have a measure (of truths), we cannot have a measure, we will have a measure. Impossible, yet necessary: forcing absolute ontology The task is clear. The Immanence of Truths attempts to formulate what makes a truth truly, really true, so that it is ultimately not relative to the situation or world whose truth it is or in which it appears as truth. Even though truths are formed within (and with the material of) a world or situation, they must also be independent of it. Obviously, this demands a non-relative (non-situation specif­ic, non-world specific), i.e. an absolute, take on truth. It necessitates an absolute ontology (of truth), which provides us with a kind of measure or orientational guideline for what is true (and for what might be truer than something else). If a truth is not relative, there must be a measure. And this measure cannot simply be timely and revisable, but it must be absolute. That is to say: to have a concept of truth one needs a concept of absoluteness,62 because if one overemphasises the idea that a truth is always situated and is a truth of this or that situation, one gives up the idea of truth and endorses that everything is externally deter­mined and relative (to a situation or world and this was what remained in part unresolved by the first two volumes of Being and Event). What is at stake here thus also immanently links the concept of truth and that of freedom. If there were no such concept that allows one to immanently determine what truths are, they would also not be transmissible and therefore would not be as universal as they conceptually must be. One thus needs a specific kind of dialectical take on truths’ truth and this is only offered by an immanent perspective. Walter Benjamin once remarked that “the inner [das Innere] of history is reserved for the dialectical gaze,”63 and if truths are what really makes history in all relevant senses of the term, the same holds for Badiou’s project. 62 This even holds for the idea that there is no truth – since it, trivially, claims to be abso­lutely valid (and hence true). It is thus not that it is difficult to say something about truth (we do this all the time – and Hegel’s Phenomenology shows us all possible ways in which we can be wrong about what we take to be true and how we attempt to avoid correcting ourselves), but it is difficult to confront what is true about truth. It is difficult because here we confront the infinite. 63 Walter Benjamin, “Ein Jakobiner von heute. Zu Werner Hegemanns ‚Das steinerne Ber­lin‘”, in Gesammelte Schriften, Vol. 3, Suhrkamp, Frankfurt am Main 1991, p. 263. This necessitates that one creates the forced and forceful fiction of a totalisation, of an absolutisation of an active truth procedure in the condition of ontology. The result of this forcing is the idea of an absolute ontology. In The Immanence of Truths, V is the name of the result of such an operation of forcing. Yet, it is of importance to qualify that V is more than what makes a merely fictitious64 whole of a truth. Since the operation of forcing – i.e. a constitutive element of any truth procedure whatsoever65 – is what enables truths to have an effect on knowledge. Forcing compels new terms into knowledge and in the present case it forces a paradoxical fiction of a knowledge of truth into knowledge.66 It is here that we must emphasise the fictitious qualification of the forced concept of an absolute ontology: that it is fictitious means that we do not reduce and dissipate truth to and in knowledge. But we operate as if there were a knowledge of truth, a real science fiction of truth in the form of an absolute ontology. To reduce truth to knowledge is, on the other hand, what the constructability hypothesis (short­hand for: Gödel – “who constitutes the latent force of the ideology of finitude”67 – since the publication of the first volume of Being and Event) is ultimately about. In its third volume Badiou must make a strong case why the universe of absolute ontology, with the absolute as the place where all possible forms of being are placed (this idea being the result of the operation of forcing), must be different from the constructible universe. Otherwise, we would violate what Being and Event stated about the being of truths, namely their generic nature, bringing together elements that have nothing else in common than having been brought together. The danger lies in seeing in the result of ontological forcing an absolute validity 64 “Fiction” here does not mean that we are dealing with an illusory or merely fake being (of language or metaphysics only – cf. Badiou, L’Immanence des vérités, p. 22). Rather it means that truth (in its totality), in Lacan’s phrasing, has the structure of a (necessary, yet 327 impossible) fiction. 65 This is already the argument of Vol. 1. Cf. Badiou, Being and Event, pp. 391–440. 66 In precisely this sense, Badiou speaks of “an – almost true – knowledge of what can be, grasped in its proper advent, the almost absolute being of truths.” Badiou, L’Immanence des vérités, p. 393. 67 Ibid., 246. “Gödel” stands for a specific kind of problem – since he does not simply defend a formalist interpretation of mathematics, wherein mathematics is understood as a (lan­guage) game, but is a “convinced Platonist” (ibid., p. 67). There is thus a struggle within the Platonic conception of mathematics – one ( philosophical Platonism) divides into two (constructivism and genericism). That is to say: Gödel also absolutises, but the outcome is quite different, namely he stands for “the possibility of an integral absolutisation of finitude.” Ibid., p. 257. of the constructible universe – or in Badiou’s rendering, in assuming “V = L”68, which describes “the victory of what is (constructible) over what is not” and is ultimately “‘realist’ propaganda.”69 Against the assumption that “all that is, is constructible,” which is the foundational axiom of “the ontology of finitude,”70 it is crucial not to misconceive the forced totalisation of an immanently unto-talisable ontological truth procedure in a finitising way. This is what is at stake with the concept of absolute ontology. Forced ontological totalisation must be non-constructible. It can neither lead to “das Seiende im Ganzen,” to “beings as a whole”71 in Heidegger’s sense – as there is no existing whole in set theory – nor can and should we assume that by means of forcing we would find a structural paradigm that enables us to imagine how the absolute – of an absolute ontology – actually exists. This is the funda­mental problem with Cantor’s continuum’s hypothesis, since it assumes that the relation between different sizes of infinity (more adequately: between the size of the infinite set of natural numbers and the size of the larger infinite set of real numbers) could be constructed by applying the successive ordering structure of the set of natural numbers. Then their relation could be described by recourse to the structure of natural numbers.72 The problem with this, as with “whole numbers,” which “have become the alpha and omega of each evaluation when it comes to ideas […] or of the value of historical experience,”73 is that this finitis-es infinity by assuming there is a constructible structure. The domain of infinity is thereby turned into a domain where there is more of the same (finitude). One therefore must create an impossible yet necessary forced totalisation of what is non-totalisable, i.e. one must create an absolute ontology within which one also forcefully wards off the temptation of constructability. 68 Ibid., p. 247. “L” stands for the constructible universe and V for the formalisation of “the Vacuum, the great void, but also the Truths.” Ibid., p. 40. 69 Ibid., p. 247. 70 Ibid., p. 441. 71 Martin Heidegger, What is Metaphysics? Original Version ed. by Dieter Thomä, in Philoso­ phy Today, 62, (3/2018), p. 738. 72 One problem with this is that it does not allow us to see that when we attempt to say some­thing about infinity, we actually do not know what we are talking about. This becomes even more problematic for Badiou when we use the concept of “infinity” without knowing anything about what happened in mathematics in the last fifty years. Cf. Alain Badiou, In Praise of Mathematics, Polity Press, London 2016, p. 43f. 73 Badiou, L’Immanence des vérités, p. 444. If this cannot be done, our attempts are wasted; if we can, there might be an oeuvre. The task is to first provide an outline of such a forced absolute ontology and then to delineate the conditions under which it remains non-constructible. And this is what The Immanence of Truths does, inter alia, by positing that “V is not, itself, a set” and that it has the “capacity to welcome infinitely many infinite forms of possible-being.” Even though it presupposes “a kind of ultra-infinity,”74 this should not lead us to the assumption that an absolute ontology implies an ontology of the absolute. The absolute is the place of all possible forms of be­ing, yet this place does not exist: it is and must be inconsistent.75 It inexists and hence forcing an absolut(ised) ontology – i.e. “the possible anticipation of a real phenomenon”76 – does not ontologise the absolute.77 The absolute is rather the real of truth(s). Dialectics of axioms Almost at its beginning, The Immanence of Truths formulates an absolute ontol­ogy – and it argues that set theory fulfils all criteria to be such an ontology. An absolute ontology is “the existence of a universe of reference, a place of think­ing being qua being”78 that has four defining features. The first of its features is that “it is immobile.” This means that being as such does not have a history – which is itself, more or less obviously, an implication of the term “absolute on­tology.” Being does not have a history because being does not move. It provides the background against which one can conceive of movement and history, but neither is it an internal attribute of being as such. For an absolute ontology, being is unmoved – and this implies that even the mathematical formalisation of movement itself is articulated somewhere, where there is no movement: “the 329 74 Ibid., p. 370. Also instructive here is W. Hugh Woodin, “The Realm of the Infinite”, in Infin­ ity: New Research Frontiers, ed. by Michael Heller and W. Hugh Woodin, Cambridge Uni­ versity Press, New York 2011, pp. 89–118. 75 In this sense, this operation of ontological forcing remains faithful to Cantor, who “does not step back from associating the absolute [absoluité] and inconsistency.” Badiou, Being and Event, p. 41. 76 Badiou, L’Immanence des vérités, p. 41. 77 Badiou therefore repeatedly insists that V is not a multiple (not a set) and hence is truly inconsistent within the universe in which it is formulated. 78 Badiou, L’Immanence des vérités, p. 36. mathematical equation that formalizes the thinking of movement has no specif­ic location, except, in fact its mathematical absoluteness.”79 Being is, secondly, intelligible in all its dimensions on the basis of nothing. This implies that we do not have to refer to any kind of substance (or positive proper­ties), nor to a subject to conceive of being qua being.80 The only thing we need is no-thing. We need (substanceless) nothing – a nothing without nothingness – to unfold this ontology consistently – which obviously is always an advantage in comparison to needing something. Being is composed of nothing and nothing does not move: there is nothing in the grand void which composes being.81 This means that being is not composed i.e. it is “non-atomic.”82 Thirdly, being can only be thought by means of the “axioms to which it corresponds.”83 This allows us to distinguish an absolute ontology from all kinds of phenomenology (being cannot be experienced) and all types of empiricism – being is not empirical (but con­structed by axioms on the basis of nothing). This is crucial, since thinking being is independent of the fact whether it is or is not. In an absolute ontology, thinking and being are the same when we are thinking being. The final and fourth feature of the absolute ontology is that it obeys a principle of maximality, namely that “any intellectual entity whose existence can be inferred without contradiction from the axioms that prescribe it, exists by that very fact.”84 What is rational is real, what is real is rational. That is to say, any “thing”, any being, that can be derived from the axioms of the absolute ontology alone is taken to exist because it can be inferred without contradiction. The four principles relate to one another dialectically. There is a dialectic of the axioms of absolute ontology:85 79 Badiou, In Praise of Mathematics, p. 75. 80 Badiou gives the following example: “Take a revolutionary movement, an uprising that will become historic, such as the storming of the Bastille, let’s say. Considered in terms of its pure political value, as a symbol, a reference point, an absolute beginning of a process, this event cannot be broken down into separate units. It’s not the result of an addition of factors.” Ibid., p. 75f. 81 Here it is interesting to consider the beginning of Hegel’s Logic which Badiou commented on in similar terms in Alain Badiou, The Rational Kernel of the Hegelian Dialectic, re.press, Melbourne 2011; cf. also Rebecca Comay and Frank Ruda, The Dash – The Other Side of Absolute Knowing, MIT Press, Cambridge, 2018, pp. 87-106. 82 Badiou, L’Immanence des vérités. p. 36. 83 Ibid., p. 37. 84 Ibid. 85 In what follows I recapitulate a public discussion with Alain Badiou (and Jan Völker) on the concept of absolute ontology. Cf. the recording at: https://vimeo.com/28417395. Beginning with and within an immobile place (i.e. “V”), which is entirely static, the thought of being does not introduce any kind of movement into it (1st fea­ture). How must this place be conceived to be really immobile? There must be no-thing in it, there must be a pure nothing-ness “in” it (2nd feature). The place where we think being is absolutely empty and hence entirely immobile. Only such nothing(-ness, even though the substantialising postfix is misleading) is immobile, since every-thing, certainly any (concrete) thing moves and is in trans­formation all the time. From the first principle – being is entirely immobile – we are thus necessarily led to assume that an absolute ontology commences in (and with) nothing(-ness). This forces us to consider how we are able to think this nothingness – since it obviously can neither be empirically experienced nor can we simply deduce it from a given preceding concept. There is one possi­bility of how to think it as such, notably by recourse to principles or axioms (3rd feature). Axioms represent a set of theoretical affirmations needed to realise and comply with the first two features of an absolute ontology. The affirmative positing of axioms is thus a consequence of the 2nd feature, which is a dialectical consequence of the 1st feature. The principle of maximality can now also be regarded as a dialectical consequence of the 3rd feature, because if you think on the basis of axioms, you must admit all of their consequences to really think in accordance with them. This is to say, the decision that being is immobile is what leads us to the set of features by means of which an absolute ontology is unfold­ed. Set theory fulfils all these criteria and thereby becomes the paradigmatic candidate for an absolute ontology. As a consequence of a truth procedure taking place in ontology, we can force an absolute ontology that we then meta-ontologically identify with the very dis­course on being that we (already had) identified as ontology.86 The results of ontological forcing are also registered in a meta-ontological manner and affect 331 the conception philosophy has of its ontological condition. It affects meta-ontol­ogy. The formulated absolute ontology debunks – different from Spinoza, who for Badiou was the first to claim that “a consequent materialism demands an 86 I named this operation – an operation which can happen under the specific historic condi­tion that the very existence of conditions, i.e. of truths as such, is threatened – philosophi­cal forcing and have shown in what way it is linked to what psychoanalysis calls “working through”, in Frank Ruda, For Badiou. Idealism without Idealism, Northwestern University Press, Evanston 2015, pp. 127ff. absolute referent”87 – all substantial, i.e. unifying and one-ifying, properties of substance. It is important that V does not exist and incoheres, since thereby it also avoids introducing substantial attributes of an already given subject (and eliminates all substantial links between the two, too). It is important that such non-substantial absolute ontology is absolute, but not objective. It is not an ob­jective ontology because it does not give a description of any kind of object nor any objective account. But this does not make it relative or relativist. Yet, here we encounter a peculiar almost Kantian question (slightly resonant of debates around his table of categories): is this set of axioms of absolute ontology com­plete or not? If it were not complete, it may be accidentally incomplete – and then nothing would change with regard to the status of this ontology – or it may be systemically incomplete – and this would change its absolute character. To avoid jeopardising the absoluteness of ontology, Badiou insists that the set of axioms can be completely determined, even though it has not yet been entirely discovered. Truths “in” the absolute: forced freedom and necessary totalisation Thus far, I have reconstructed Badiou’s claim that ontology is a condition of philosophy and that this implies that there can be truth procedures taking place within it. The specificity of this type of truth procedure lies in the fact that it un­folds the potential of scientific discoveries within the field of mathematics and that this unfolding – in its retroactive and creative dimension88 – leads to a the-orisation of different types of infinity, which in turn can itself be read as offering a conceptualisation of the truthness of truths. This finally allows us to apply its conceptualisation of truths as part of the unfolding of a truth procedure taking place in ontology onto the very concrete procedure that formulates this concept of truth. There is a torsion of the ontological truth procedure. This means that the very truth-conceptualisation which it allows by means of theorising differ­ent sizes of infinity applies to itself and thereby it allows for an absolute – in the sense of being fully detached from everything but itself – and a fully immanent “measuring” of its own truth (i.e. infinity) by means of its own creation. We get a true theory of truths – which evaluates truth solely from within its own imma­ 87 Badiou, L’Immanence des vérités, p. 37. 88 Ibid., p. 588ff. nent practical unfolding. This theory of truth is practically self-absolutising.89 The further we force this immanent and absolute self-measurement, the more we are forced to formulate the idea of an absolute ontology90 – something that philosophy articulates so that this idea will prove compossible with contempo­rary creations in other conditions of philosophy, allowing us, for example, to philosophically identify a truth in politics. The self-absolutisation here chimes with philosophy’s “desire for truth.”91 What has been missing from the argument thus far is how the dialectically inter­connected axioms of absolute ontology relate to the conceptualisation of the dif­ferent sizes of infinities. This move, in all its demanding mathematical intricacy, leads Badiou to undertake in The Immanence of Truths a veritable and majestic renewal of the dialectics of quantity and quality, a dialectics of quantity and quality exploding in all directions – creating the absolute place “V”, which can be imagined to be structured similar to the letter that names it.92 This move into the mathematical discussion of infinity is crucial since only with this step do we complete the torsion, the peculiar circle, and only thereby see how we can force mathematics into the status of an absolute ontology (even though the latter is a meta-ontological concept). We only get a proper absolute ontology that provides us with a real conceptualisation of truths if we explore the domain of infinity, the powderiness of V. This is why two – at least – further clarifications are need­ed concerning the previous remarks: First, an absolute ontology must reject the Continuum Hypothesis – as it would impose a structural model for the relations between different sizes of infinity and thereby the realm of infinities would be made constructible and finite.93 This 333 89 Badiou identifies this move that happened “in the last twenty years” with the work of Woodin. Cf. Badiou, L’Immanence des vérités, p. 68f. For reasons that would explode the length of the present article, I cannot elaborate why this move is justified and yet cannot be repeated, such that after establishing an absolute ontology it is truly absolute. The rea­ son for this is elaborated in Badiou, L’Immanence des vérités, p. 495ff. I will return to this elsewhere. 90 Which is the reason why the absolute must be in a strict identity with itself: ibid., p. 451f. 91 Badiou, “L’entretien de Bruxelles”, p. 91. 92 Cf. schema 2 in Badiou, L’Immanence des vérités, p. 688f. One can read this as Badiou’s take on the famous “parallelism” argument in Spinoza. 93 Which is why Badiou takes it to be false, even though it is compatible with the axioms of pure set theory, cf. Badiou, L’Immanencedes vérités, p. 310. must mean that we simply cannot derive and deduce the infinite from finite suc­cession. To overcome the finitising constructivist orientation, we must disentan­gle thinking the infinite from “the false infinity of hopeless successions.”94 This false succession is false because it has three problematic implications: 1. we have a succession that follows one steady model whereby any infinite set is fol­lowed by a greater infinite set and the relation between the two is always of the same kind (which is constructible in advance). But the infinite can only be prop­erly thought if we unbind it from any link to the one and do not problematically generalise the move by means of which we can get from one infinity to a larger one (otherwise we would in advance unify, make consistent, and count as one all these transitions). There is neither a largest one nor is there any one-struc­ture by means of which we could construct all relations between infinite sets. 2. this succession is false because with each new and larger infinite set, it would happen that we simply finitise the previous set. That is why we are tempted to encounter only potential infinity in the realm of natural numbers, at least if our concept of infinity emerges from the idea of stable succession. 3. It is false be­cause it is a paradigm of what Hegel called bad infinity, since it continues end­lessly – and this internal endlessness is precisely the ruse of potential infinity. In order to formalise actual truth, there must be actual infinity. We would not get an immanent measurement of truth if we simply followed an infinite succession of infinities nor an absolute ontology. Second clarification: it is here that a choice needs to be made: “a crucial choice,”95 which Badiou identifies with the “fictitious opposition Cohen / Gödel.”96 Either everything is constructible and hence finite (Gödel), a position that does not necessitate any choice at all, or there is a choice, a choice for which we have no criteria other than what we can anticipate will follow from opting for one or the other side. A choice which in this sense is determined by the fictitious anticipation of what it entails and which is constructed from a meta-ontolog­ical perspective. Philosophy here becomes an imitator and mimics a feature of the forcing-operation happening in its condition(s). If there is such a choice, 94 Alain Badiou, “Toward a New Thinking of the Absolute”, in Crisis and Critique, Special Issue, 1, (2/2014), p. 21. 95 Badiou, L’Immanence des vérités, p. 257. 96 Ibid., p. 266. philosophical forcing constructs a fundamental fictitious incompatibility:97 two sides that are so infinitely different that each amounts to an entirely different orientation with a whole range of implications (from consequences for the ex­istence or inexistence of absolute truths, to what kind of ontology one defends and political positions one finds defensible, say if one’s ontology is compatible with present regimes of domination or not, etc.). The infinite gap that we en­counter between the finitising orientation of constructability and its finitised, only potential infinite, and the actual infinite of genericism with its “infinite potentiality,”98 on the other side, forces us to choose. We here, through a forced philosophical fiction, surprisingly experience infinity, namely the infinity that separates the two sides of the choice. We experience the infinite when we iden­tify the fictitious choice as a real choice, as one that forces us to be free and we cannot not choose. But we only encounter it if we construct the choice as choice and see that we must choose; otherwise, we have already chosen (constructabil-ity).99And if we identify it as choice, we also have already chosen, since then we see that the constructible orientation absolutises de-absolutisation, i.e. it abso­lutises finitude, and this contradiction cannot but be a reactive (or obscure) de-fence mechanism against some truth taking place. If something truly happened in ontology with Cohen, its anticipated consequences absolutise ontology and this forces philosophy to create the fiction of a choice, which has already been made. With an ontological truth procedure there originates the fiction of an ab­solute ontology and philosophy is itself forced to force a fiction. Philosophy is meta-forced, forced to force the fiction of a choice whose only criterion lies in its anticipated consequences for ontology – which adds another layer of forcing. When we identify the choice as a choice, the choice has immediately been tak­en, it never existed as a matter of deliberation. This is the experience of freedom that The Immanence of Truthstransmits. A freedom of and in thought, the free­ 335 97 That it is a forced philosophical fiction becomes manifest in the fact that this choice does not seem to be a necessary choice at all for the mathematician – but for the philosophi­ cally informed, to put it neutrally, mathematician or for the mathematically informed phi­ losopher it cannot but be one (for Badiou). 98 Ibid., p. 223. 99 Max Horkheimer once remarked in a radio conversation that Sartre believed that people no longer decide. He added critically, and in line with the above, that the true problem is rather that they do decide but are not aware of it. Opting for Gödel is a spontaneous ideol­ ogy that one barely notices as such. dom of a forced choice that we do not (as individuals) arbitrarily decide (accord­ing to our preferences), but that is decided within the immanence of thought. But it is important to note that both sides are equally valid: both are compatible with set theoretical mathematics,100 even though its ontological status changes quite drastically if we opt for the one or for the other. This choice confronts us with freedom as we have no objective reasons to choose the one or the other; there is no normative grounding whatsoever apart from the anticipation of con­sequences and if we see what is at stake with the choice – if we understand the choice as choice – ultimately there is no choice, as there is only one option to choose when we anticipate what the consequences will have been. Between equal rights, forcing decides. As any choice, this choice is a real choice because it is structurally undecidable and this also means that it is already decided when we identify the choice as choice. What we have at our disposal is only the an­ticipation of what results from choosing either side. This anticipation is what Badiou calls “idea.”101 The philosophical thinking of what was thought in its ontological condition manifests in the fictitious construction of a choice. This fiction of choice is force­ful, since as long as one starts to see it, one can no longer choose to escape it and therewith one has already chosen – philosophy reminds us of what we were unaware we knew before. It is a choice of one’s way of thinking that may man­ifest a “non-constructible subjectivity.”102 By being forced to choose one expe­riences freedom, since one experiences something of the infinite in identifying the infinite abyss that separates thinking under the guidance of an idea from the ideology of finitude. Freedom in thought exists only in the fictitious realm of a forced choice that has consequences in and for everything. This fiction provides us not only with a genuine experience of freedom in thinking – or in spirit, as one could say with Hegel, or more precisely: with a genuine experience of free­ 100 Which is why, as Badiou points out, mathematicians as such would not come across this choice, since both positions are compatible with the delineated set theoretical universe. 101 Badiou, L’Immanence des vérités, p. 265. This is the treatment of a point – in the terminol­ogy of Logics of Worlds – I elaborated above. It should be clear why this shatters the pre­dominant conceptions of freedom. I tried to make an argument pointing in this direction in Frank Ruda, Abolishing Freedom. A Plea for a Contemporary Use of Fatalism, Nebraska University Press, Lincoln 2016. 102 Badiou, L’Immanence des vérités, p. 265. dom in absolute spirit. If there are real differences at stake in this constitutive choice – and the subsequent development of the mathematics of different types of infinity will essentially develop these in different ways – this proves that there are some things one can only do within the realm of thought (and not in the same way in concrete life103). But thereby we may get a “new idea of human action,”104 of thought and of truth. This forced freedom touches the absolute – through the infinite gap separating the sides. In the absolute spirit of meta-on­tology’s constitutive choice there is an experience of freedom and of truth, since it is an experience of infinity105 – an experience that is transmissible under the condition that the fiction of this choice is transmissible. The status of an absolute ontology is thus linked to an experience of freedom. It therefore does not come as a surprise that Badiou endorses that the absolute ontology (set theory) must also entail the axiom of choice. This axiom is only relevant for infinite sets and indicates that from any given infinite set, a set can be extracted that is composed of an element of each set that is an element of the original set – yet it does not offer any means to construct it. The axiom of choice indicates that there is a choice but it leaves open the criteria and framework of this very choice. When relating this back to the four features of an absolute ontology, this implies that the latter is fundamentally based on an essentially indeterminate choice: a choice that does not have any objective or subjective or concrete conditions, a choice that is determined by nothing, a pure (inde­terminate) choice, a lawless choice. “It validates a choice whose norm does not pre-exist, a representation whose Law is unknown.”106 The axiom of choice is an inner-mathematical dialectical consequence of the pure and lawless choice be­tween constructability and genericism – on which the very inexistent existence of an absolute ontology relies – which takes axiomatic form. It repeats some­thing of the immediate choice (conditioned by the torsion of ontology), yet in 337 a different form. It is – in Hegelian terminology – a repetition of the immediate 103 But the former fundamentally changes our effective – ontological, etc. – take on the lat­ter. It means that we get a different concept of what thinking (and freedom) is. And we get it in thought and this can change everything. This was what Hegel argued for when he emphasised the non-practical nature of philosophy (in absolute spirit). And should not an absolute ontology be part of absolute spirit? 104 Jean-Paul Sartre, Search for a Method, Knopf, New York 1963, p. 45. 105 Which again proves the old point that truth can only be accessed in a one-sided manner. 106 Badiou, “Toward”, p. 20. as result, but, and this is crucial, now the pure choice repeats not at the ground and foundation of absolute ontology but in the literal heights of its develop­ment, namely in the theory of different sizes of infinity. An absolute ontology is framed by freedom at its ground – the forced choice to unfold the consequence of an event – and peak, the meta-forcical impact that the forced self-absolutis­ing of ontology has for the very concept of meta-ontology. A freedom that is not a given unless we identify the choice as a choice and are thus forced to make it actual, because we have already done so. (Too) brief addendum on the end of infinity Based on a given infinite set, one can easily construct a new sort of infinity by means of what is called orthogonal operation.107 To do so, we measure it and determine its size, i.e. the number of its elements. This holds if the set is finite or infinite. If we call “Hans” the given infinite set – provided that we have man­aged to measure it – we can attempt to determine a set that is so incompara­bly larger than “Hans” that there are at least “Hans” many infinite sets that are smaller than it. So, we have a set, let us call it “Super-Hans” in comparison to which “Hans” is small, since there are “Hans” many infinite sets that are all smaller than “Super-Hans”. “Super-Hans” comprises “Hans” times “Hans”­sized sets. It is “Hans” Hans-times. If “Hans” is infinite, “Super-Hans” is clearly super-infinite. The problem that occurs within this type of construction of in­finite sets is that retroactively the previous infinite set is finitised. “Super-Hans” finitises “Hans”. If this were the case, we would only in appearance be dealing with infinite sets – until they are superseded and then they are finite again. We would thus end up again with a bad infinite and with a potential invalidation of everything we held to be true (if this is what we are talking about when we talk about the infinite). In The Immanence of Truths, Badiou mobilises an astounding theorem that was developed in 1971 by Kenneth Kunen. It elaborates a meta-ontological precon­dition for conceiving of actual infinity (and hence truths). To cut this down vi­olently: it proves that “numerical openness is not the law of infinity” and that there is no endless succession. The only way we encounter actual and true in­ 107 I am taking this elaboration from Alain Badiou, “Toward a New Thinking of the Absolute” (manuscript). All subsequent quotes are from this paper. finity therefore is, so to speak, by ending it somewhere. There is unwilling fini­tisation – in the idea of potentially infinite and eternal succession – and there is a different type of finitisation. The introduction of a limit, a region where the endless “process of the resorption of successive infinities” comes to an end. A bad infinite and – emphasising its Platonic undertones – a good infinite. There is no actual infinity without this peculiar type of end. Once the end of history was proclaimed as a precondition for an adequate concept of history, which im­plies that the resurgence of history will necessarily end and revolutionise what we conceive as history, and the end of art was proclaimed to be preparation for the emergence of a new art that will thoroughly shake what we previously deemed possible to be art. This was always different from the endless blabber about the end of (political history in democracy or art in art-theory, and of) phi­losophy, which Badiou has been heroically combatting for decades now. In the end, what the thinking of actual infinity ontologically and meta-ontologically necessitates is an emancipatory theory of the end. Only then can an actual infin­ity be thought that measures up to the infinity of the forced freedom at its origin. It might not be accidental that The Immanence of Truthsrepresents the end of Badiou’s philosophical enterprise. 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Feltham, Bloomsbury, London / New York 2013 — Le séminaire – Images du temps présent: 2001–2004, Fayard,Paris 2014 — “Toward a New Thinking of the Absolute”, in Crisis and Critique, Special Issue, 1 (2/2014), pp. 19–24 — In Praise of Mathematics, trans. S. Spitzer, Polity Press, London 2016 — Le séminaire – L’un: Descartes, Platon, Kant (1983–1984), Fayard, Paris 2016 — L’Immanence des vérités. L’etre et l’événement, 3, Fayard, Paris 2018 — Can Politics Be Thought?, trans. B. Bosteels, Duke University Press, Durham / London 2019 Benjamin, Walter, “Ein Jakobiner von heute. Zu Werner Hegemanns ‚Das steinerne Berlin’”, in Gesammelte Schriften, Vol. 3, Suhrkamp, Frankfurt am Main 1991, pp. 260–265 Comay, Rebecca and Ruda, Frank, The Dash – The Other Side of Absolute Knowing, MIT Press, Cambridge 2018 Hegel, G. W. F., Science of Logic, trans. G. Di Giovanni, Cambridge University Press, Cambridge 2010 Nagel, Thomas, “What Is It Like to Be a Bat?”, in The Philosophical Review, 83 (4/1974), pp. 435–450 — A View from Nowhere, Oxford University Press, Oxford 1986 Riha, Rado, Kant in Lacan’scher Absicht: Die kopernikanische Wende und das Reale, Turia+Kant, Vienna 2018 Ruda, Frank, For Badiou. Idealism without Idealism, Northwestern University Press, Evanston — Abolishing Freedom. A Plea for a Contemporary Use of Fatalism, Nebraska University Press, Lincoln 2016 — “Hegel’s Immanence of Truths”, in Badiou and German Philosophy, ed. by Jan Völker, London, Bloomsbury, 2019, pp. 51–69 — “Tovariši absoluta, ali: Kje je “aupres de nous?””, in Problemi, (3–4/2019), pp. 133–150 Sartre, Jean-Paul, Search for a Method, trans. H.E. Barnes, Knopf, New York 1963 Woodin, W. Hugh, “The Realm of the Infinite”, in Infinity: New Research Frontiers, ed. by Michael Heller and W. Hugh Woodin, Cambridge University Press, New York 2011, pp. 89–118 Žižek, Slavoj, Ruda, Frank, Hamza, Agon, Reading Marx, Polity Press, London 2018 Filozofski vestnik | Volume XLI | Number 2 | 2020 | 341–360 | doi: 10.3986/fv.41.2.13 Jana Ndiaye Berankova* The Immanence of Truths and the Absolutely Infinite in Spinoza, Cantor, and Badiou The relationship between mathematics and ontology and the slogan “mathe­matics is ontology” has been a constant topic of investigation among think­ers who follow Alain Badiou’s philosophical project. In 2018, at the conference Thinking the Infinite at the National Gallery in Prague, Badiou was supposed to give a talk proving the logical necessity of mathematical ontology. Howev­er, instead of providing the final proof that some members of the public were asking for, he delivered a different talk reworked at the last minute. He elicited great perplexity in the room by retracting the slogan “mathematics is ontology” for being nothing but an advertising simplification of a necessarily complex problem.1 He observed that this clear and easily memorable slogan falls into the sphere of the doxa and cannot exhaust the nuanced relationship between the two fields. My intention in the present article will be to provide an interpre­tation of these puzzling remarks along with an exegesis of the mathematical concepts that appear in the recently published third volume of Being and Event: The Immanence of Truths. In this third volume, Badiou relies on the mathemat­ical theory of large cardinals, which addresses sets and classes so large that they can almost approximate the entire universe of numericity. He proposes a renewed theory of the absolute “substance” and of the attributes of the abso­lute. As abstract and complex as such thinking might seem, it builds upon prob­lems which appeared – although in their naive and schematic form – already in the mathematico-philosophical considerations of Georg Cantor. By reading Badiou’s and Cantor’s texts closely together, we might be able to comprehend the articulation of philosophy and mathematics and the role played by philo­sophical choices. Alain Badiou, “Ontologie et mathématiques : Théorie des Ensembles, théorie des Catégo­ries, et théorie des Infinis, dans L’Etre et l’événement, Logiques des mondes et L’imma­nence des vérités”. * Columbia University Mathematical ontology and its consequences Did Badiou truly disavow the relationship between ontology and mathematics, or more broadly, between philosophy and mathematics? My conviction is that the response to this question might not be scandalous at all: “the discourse of Prague” constitutes nothing but a clarification of the hierarchy between these two fields. Badiou posits that “the thinking of being is a thinking on different forms of the multiple.”2 He departs from the philosophical statement “being is the multiplicity-without-the-one.”3 This starting point is not a mathematical statement. It would thus be misleading to say that “ontology is mathematics” for the latter expression would imply that we can derive knowledge of being di­rectly from mathematics without passing through philosophical decisions and that mathematical inventions must have a direct and immediate effect on our understanding of Being. “Being is the multiplicity-without-the-one”4 is nothing but an initial philosophical decision; it is the philosopher’s choice: the phi­losopher chooses this position over another one by comparing their possible consequences. In other words, the decision situated at the beginning of any philosophical system is an initial “wager”, a “working hypothesis”, perhaps something like a philosophical “axiom”. Such an “axiom” can be legitimated only by the richness of its consequences. The dialogue between mathematics and philosophy can be portrayed as follows: “a philosopher outlines a certain idea, and then uses mathematics, wherein this idea can be verified, and in the end returns to philosophy.”5 It is the “very construction of the [philosophical] system [that] proves the statement. This means that this initial statement is a posteriori proven and vindicated thanks to the scope of its consequences. But we never have demonstrative certainty which would resemble mathematical formalism.”6 In the present volume, Badiou states similarly that “the alliance organized between mathematics and philosophy becomes strong only when we observe its consequences.”7 In other words, mathematics is but philosophy’s methodological tool for verifying its initial philosophical hypotheses. The phi­ 2 Interview with Alain Badiou on The Immanence of Truths, Paris, 13 February 2018. 3 See p. 22 in the present volume. 4 See p. 30 in the present volume. 5 Jana Beránková, “Communism is a New Idea, Interview with Alain Badiou by Jana Beránk­ ová”, Contradictions 2 (2/2018), p. 118. 6 Ibid., 118. 7 See p. 22 in the present volume. losopher needs mathematics in order to build a method, and it is in this sense that a great part of ontology remains “mathematical”, and yet we cannot affirm that philosophical statements are deduced directly from mathematics. To the philosopher, mathematics is but a modality to develop his or her thought; the origin and the finality of thinking remains philosophical. In fact, ontology is grounded in philosophy, not in science. In Prague, Badiou merely clarified the hierarchy between these two fields by alluding to the ineluctable arbitrariness of philosophical decisions; in no manner did he reject the relationship between ontology and mathematics. Instead of “mathematics is ontology,” it might be more exact to say that “ontology uses mathematics as a methodological instru­ment for creating a possibility of our understanding of being.” Mathematics is a mere condition of philosophy along with politics, art, and love. The relationship between philosophy and mathematics is not that of “suture”: philosophy is not sutured to anything; it highlights the existence of truth procedures in its condi­tions (science qua mathematics being one of them). Note that in the above cited passages, the relationship between mathematics and philosophy is portrayed on the basis of inductive, not deductive, reason­ing. Here, the induction generates the possibility of building up a philosoph­ical system from mathematical grounds. As long as mathematics assists phi­losophers by successfully elucidating otherwise unsolvable problems, its use in ontology is legitimate. Such reasoning sounds surprisingly Gödelian, for it can call to mind the concluding paragraph of Kurt Gödel’s 1947 article “What is Cantor’s Continuum Problem?”: “There might exist axioms so abundant in their verifiable consequences, shedding so much light upon a whole field, and yielding such powerful methods for solving problems (and even solving them constructively, as far as that is possible) that, no matter whether or not they are intrinsically necessary, they would have to be accepted at least in the same 343 sense as any well-established physical theory.”8 Badiou – who valued Gödel’s article for the emphasis it placed on the role that mathematical intuition and axiomatic decisions play in solving mathematical problems9 – remarked that “Every thought – and therefore, mathematics – sets off decisions (intuitions) 8 Kurt Gödel, “What is Cantor’s Continuum Problem?”, in Collected Works, Vol. II, Publica­tions 1938–1974, Oxford University Press, Oxford 1990, p. 261. 9 See Badiou’s comments on Gödel’s article in Briefings on Existence: A Short Treatise on Transitory Ontology, State University of New York Press, Albany 2006, p. 92. from the standpoint of the undecidable (of nondeductible inference).”10 Ontolo­gy’s use of mathematics is nothing but an initial decision that can be justified a posteriori by the solutions it provides to otherwise unsolvable problems. Critics of Badiou’s mathematical ontology (some of whom are included in this volume) tend to see as deduction what in reality is inductive reasoning. The criticism usually follows this trajectory: If set theory is ontology (i.e. a rational discourse on being qua being defined as an inconsistent multiplicity without the one), could some other mathematical theory (e.g. category theory, mereol-ogy)11 be ontology too? Why should we privilege set theory, an obscure field of knowledge that seems to be but a burden to many working mathematicians? These questions are legitimate but also tautological. For had we departed from a different philosophical axiom, e.g. that “being is a pure multiplicity without the one,” a different mathematical or scientific theory might be much more suitable to our thinking. It is very likely possible to build ontology on a different basis than a set-theoretical basis. Any true criticism cannot dispense with the long and laborious task of system building. The usefulness of a given mathemati­cal theory for philosophical thought can only be seen after we have sketched and exhausted its consequences in the domain of philosophy. Thinking moves from ontology to mathematics and not vice versa. A correct reasoning cannot be “there is theory y in mathematics, therefore being must be x,” but rather: “being is x, therefore we should use mathematical theory y to develop our reflection.” And even this second reasoning can be vindicated only a posteriori depending on the fruitfulness of its consequences. Philosophy cannot have the certitude of a mathematical formalism; it is consistent but necessarily incomplete. It begins with an axiom and not with a totalising origin that would close its field of the possible. Philosophical axioms, the statements in which the long trajectory of 344 thought originates, can be verified only a posteriori, after accomplishing a cer­tain philosophical journey. Only by elaborating a different and similarly com­plex philosophical system grounded in a competing statement that “ontology is y” will it be possible to measure the richness of its consequences with those of Badiou’s set-theoretical ontology. Philosophical “working hypotheses” do not 10 Alain Badiou, Briefings on Existence: A Short Treatise on Transitory Ontology, State Univer­sity of New York Press, Albany 2006, p. 95. 11 See Roland Bolz’s article in the present volume, "Mathematics is Ontology? A Critique of Badiou's Ontological Framing of Set Theory". act immediately but with delay. Thus, the question “Why is ontology not y?” constitutes a philosophical suspicion, not a true criticism. It can become a se­riously voiced criticism only when – accompanied by the effect of time – it has gone through a patient and laborious procedure of system-building. Badiou’s portrayal of the relationship between philosophy and mathematics reminds us of Georg Cantor’s affirmation of the freedom of mathematics in the 1883 Foundations of a General Theory of Manifolds, a text that was reprinted in 1969, in the 10th and last issue of Cahiers pour l’Analyse on “La Formalisation”, which was coedited by Badiou.12 According to Cantor, in the introduction of new numbers the mathematician is “only obliged to give definitions of them which will bestow such a determinacy and, in certain circumstances, such a relation to the older numbers that they can in any given instance be precisely distin­guished. As soon as a number satisfies all these conditions, it can and must be regarded in mathematics as existent and real.”13According to Cantor, math­ematical concepts appear as causa sui; they contain in themselves an almost functionalist corrective according to which if a concept “is fruitless or unsuited to its purpose, then that appears very soon through its uselessness, and it will be abandoned for its lack of success.”14 Like the above-mentioned philosophi­cal decisions, mathematical concepts are legitimated by their consequences, by inductive reasoning or by reductio ad absurdum. However, it would be absurd to pretend that philosophy can “accumulate” knowledge in the same way as mathematics does and that philosophical statements do not contradict the work of their predecessors. Philosophers’ desire for scientificity makes them confront new conjectures with the work of previous masters (in Badiou’s case, such con­frontation appears namely in his seminars examining the work of thinkers such as Plato, Leibniz, Spinoza, and Kant). And yet, a rigour equivalent to mathemat­ical formalism can be never attained by philosophy. 345 12 See Georg Cantor, “Fondements d’une théorie générale des ensembles”, Cahiers pour l’Analyse (10/1969), pp. 35–52. 13 Georg Cantor, “Foundations of a General Theory of Manifolds: A Mathematico-Philosoph­ical Investigation into the Theory of the Infinite”, in From Kant to Hilbert: A Sourcebook in the Foundations of Mathematics, Vol. II, ed. William Ewald, Clarendon Press, Oxford 2005, p. 896. 14 Ibid., 896. Infinitum Absolutum: the actual or potential infinite? Much of the confusion about the relationship between mathematics and philos­ophy might have been spurred by the fact that Badiou began his trajectory refer­ring to problems proper to the philosophy of mathematics without practicing a “philosophy of mathematics”. Such a detour through this field and mathematics’ own “crisis of foundations” had the function of creating space for a new realist orientation of thought and of “break[ing] with the linguistic turn that has seized philosophy.”15 Badiou rejected the Aristotelian and Leibnizian algorithmic and constructive vision of mathematics and inscribed his own philosophical project into the genealogy of Plato and Spinoza. In some cases, he had to redefine terms commonly used in the philosophy of mathematics. For instance, he proposed that instead of the definition of mathematical Platonism put forward by Benac­eraff and Putnam, i.e. the belief that “mathematical objects are independent of our minds and, unlike physical objects, do not interact with our bodies to cause alterations in our brains that lead ultimately to knowledge of them,”16 Platonism should be understood as “the recognition of mathematics as a thought that is intransitive to sensible and linguistic experience, and dependent on a decision that makes space for the undecidable, while assuming that everything consist­ent exists.”17 Badiou also frequently referred to the crisis of the foundations in mathematics as a moment in which mathematics was “compelled to think its thought as the immanent multiplicity of its own unity.”18 However, he rejected all three major issues of this crisis formulated in the first half of the 20th century (i.e. Frege’s and Russell’s logicism, Hilbert’s approach to conceiving of mathematics as a complete and consistent formal system, and Brouwer’s intuitionism recog­nising only the existence of denumerable sets). He renounced the very delinea­tion of the concepts and questions that these three orientations imply. Badiou 346 criticised what he saw as philosophy’s linguistic turn and the “algorithmic or constructivist finitism”19 in mathematics. For a constructivist, “whatever is not distinguishable by a well-made language is not”20 and the language builds a 15 Badiou, Briefings on Existence: A Short Treatise on Transitory Ontology, p. 111. 16 Paul Benaceraff, Hilary Putnam (eds.), Philosophy of Mathematics, Cambridge University Press, Cambridge 1983, p. 30. 17 Badiou, Briefings on Existence: A Short Treatise on Transitory Ontology, p. 91. 18 Ibid., 54. 19 Ibid., 104. 20 Alain Badiou, Being and Event, Continuum, New York 2006, p. 283. proximity between the presentation and the representation. To these orienta­tions of thought, Badiou opposed a philosophical doctrine in which “number is a form of being”21 and not a linguistic convention. If being qua being is an inconsistent multiplicity without the one, sets and numbers must be more than empty words. Badiou’s three major works are three different takes on this relation between mathematics and philosophy. Being and Event, by focusing on ontology, is grounded in the key notions of set theory and Paul Cohen’s forcing. Logics of Worlds uses category theory in order to explain existence, or what there is. The Immanence of Truths creates a bridge between these two books by mobilising the theory of large cardinals.22 Each of these three books is cantered on a differ­ent notion: Being and Event on that of universality (the book posits the existence of universal and infinite truth procedures), Logics of Words on singularity (it questions how these truths appear in a given world), and finally, The Imma­nence of Truths on the notion of the absolute. By referring to cardinal numbers so large that they can approximate the entire universe of numericity, in the third volume Badiou tries to answer the question: “How do truths relate to the abso­lute?” According to Badiou, “neither universality, nor singularity have a con­stant explicit relation with infinity.”23 The category of the absolute is necessary to connect these two categories. Without the notion of the absolute, a given set might be generic in one world and not in another one. What seems universal and infinite in our world could be merely our own, localised, and culturally-de­termined universalism. To a certain extent, Badiou invokes the absolute as a response to the criticism coming from the side of Anglo-Saxon empiricism and postcolonial studies. For he admits that “without grounding universality in the absolute, it becomes merely empirical.”24 Thus, he describes the goal of The Im­ 347 manence of Truths as “to examine what in the constitution of a truth makes it touch the absolute in such a manner that universality can be created and af­firmed in singularity.”25 21 Alain Badiou, Number and Numbers, Polity, Cambridge 2008, p. 25. 22 Among the rare bibliographical references, Badiou cites Thomas Jech’s Set Theory, Akihiro Kanamori’s book on the Higher Infinite, and the work of Hugh Woodin. 23 Alain Badiou, L’Immanence des vérités, Fayard, Paris 2018, p. 515. [My translation.] 24 Interview with Alain Badiou on The Immanence of Truths, Paris 13 February 2018. 25 Ibid., Paris, 13 February 2018. In The Immanence of Truths, the category of the absolute appears marked by the capital letter V; V denotes the absolute class or the place of all thinkable forms of multiplicities. The absolute class V is not a mathematical object – a place where we can find all possible forms of the multiple cannot itself be a form of the multiple. V is an operator; it is not a mathematical object but “that from which we abstract mathematical objects.”26 The absolute class V is the universe of thought; it is that from which we have to depart in order to think multiplicity. In order to elucidate some of the paradoxes of the absolute class V, it might be useful to return to the prehistory of this concept in Georg Cantor’s philosoph-ico-mathematical essays and correspondence. For although Cantor’s set theory was later described as “naive,” some of his considerations manifest a striking similarity to the problems that Badiou faced. From Aristotle’s failed attempt to solve Zeno’s paradoxes until Cantor, the majority of Western philosophers had been contesting the existence of the actual infinite and had seen the potential infinite as the only imaginable infinite that could resolve these paradoxes. It was only with Georg Cantor’s invention of set theory that the category of the actual infinite (the infinite as actually present, as “what there is”) was brought back into the forefront and mathematicians and philosophers began to consider the infinite as a mathematical object. Cantor famously distinguished between car­dinal and ordinal numbers, by naming the set of all natural numbers by Greek and Hebrew letters such as w (the ordinality of this set) or .0 (its cardinality). Thanks to his discovery of the power set theorem, he was able to surpass the elementary infinite w and build an entire hierarchy of the uncountable actual forms of infinity (beginning with the first uncountable infinite ordinal w1. In his 1887–88 text Mitteilungen zur Lehre vom Transfinitem, Georg Cantor de­ 348 fined the potential infinite as “an indeterminate, always finite, variable magni­tude taking values which become either as small as we please or larger than any arbitrary finite bound.”27He described the actual infinity as a “constant quan­tum which is larger than any finite magnitude of the same kind.”28 How were the potential and the actual infinity articulated in Cantor’s work? Cantor men­ 26 Ibid. Georg Cantor, “Mitteilungen zur Lehre vom Transfinitem”, in Gesammelte Abhandlungen Matematischen and Philosophischen Inhalts, Springer, Berlin 1932, p. 401. [My translation.] 28 Ibid., p. 401. [My translation.] tions that “a variable magnitude x successively taking the different finite whole number values 1,2,3…v… represents a potential infinite, while the set (v) of all whole finite numbers, conceptually determined by a full conceptual law, offers the simplest example of an actual infinite quantum.”29 In Mitteilungen, Cantor divided the infinite into the following categories: · the potential infinite (i.e. the series of natural numbers 1, 2, 3…). Cantor re­ marked that this potential infinity is “improper” and it might even be better not to call it “infinite” at all;30 · the transfinite or increasable [vermehrbares] actual infinite. Cantor gives w, w + 1, w + 2… among examples thereof; · the absolute infinite or the true non-increasable [unvermehrbares] infinite. Cantor remarks regarding this form of infinity that “it is not possible to add to or to take away anything from its size and on the quantitative level it should be seen as the absolute maximum.”31 Cantor states clearly that while the transfinite can be manipulated by mathema­ticians, the absolute infinity “eludes mathematical determination”32; it is not a mathematical object. More specifically, Cantor was convinced that the category of the absolute belongs to theology, and not to mathematics. The absolute inex­ists from the point of view of mathematics. Transfinities are forms of intermedi­ate infinities that are actual without being absolute and are located between the potentiality of the improper infinite and the absolute. In his letter to Wundt of 5 October 1883, Cantor characterised the absolute as “what cannot be enlarged or perfected and is analogous to the ‘absolute’ in metaphysics. My proper infinite, or if you’d like, transfinite numbers w, w+1 are not ‘absolute’ because – although they are not finite – they can be increased. The absolute cannot be increased at all and therefore it is inaccessible to us.”33Elsewhere, Cantor famously declared that “the absolute can only be acknowledged [anerkannt] but never known [erkannt] – 349 and not even approximately known.”34 We can subtract transfinite sets from the absolute, but as large as these sets might be, we will never even approximate 29 Ibid., p. 409. [My translation.] 30 Ibid., p. 404. [My translation.] 31 Ibid., p. 405. [My translation.] 32 Ibid. [My translation.] 33 Georg Cantor, Briefe, Springer, Berlin 1991, p. 139. [My translation.] 34 Georg Cantor, “Foundations of a General Theory of Manifolds: A Mathematico-Philosoph­ ical Investigation into the Theory of the Infinite”, p. 916. the whole universe of numericity. For this absolute universe of numbers is what Cantor described in his late correspondence as an “inconsistent multiplicity”. It is not always very obvious to see the correspondences between the actual and potential infinite and the various categories put forward by Cantor (improper infinite, transfinite, proper infinite). Ignacio Jané divides Cantor’s work into two periods: “between the writing of Grundlagen and the appearance of Beiträge, Cantor conceived the absolute infinite as actually existing (although not as an object of mathematics, while after Beiträge (from 1897 on) he viewed the abso­lute infinite as existing only potentially.”35 Jané remarks that the potentiality of the absolute infinite (which corresponds to what Badiou describes as the “ab­solute class V”) guaranteed for Cantor the actual existence of the intermediate transfinities. In a letter to David Hilbert of 2 October 1897, Cantor emphasised that “the ‘transfinite’ coincides with what has since antiquity been called the ‘actual infinite’,”36 while sets such as the “set of all alephs” are absolutely in­finite and cannot be thought of as existing together. Thus, it might seem that in Cantor’s work the potential infinity did not disappear, it was merely transposed to a higher level. In the letter to Richard Dedekind of 3 August 1899, Cantor delineated the dis­tinction between consistent and inconsistent multiplicities. An inconsistent or “absolutely infinite” multiplicity is that in which “the assumption of ‘being-to­gether’ of all its elements would lead to a contradiction; thus, it is impossible to conceive this multiplicity as a unity, as a ‘finished thing’.”37 The “epitome of all that is thinkable”38 is an inconsistent multiplicity. In consistent multiplicities, “the totality of the elements of the multiplicity can be thought without contra­diction as ‘being-together’ so that it is possible to conceive it as ‘a thing’.”39 Con­350 sistent multiplicities can also be called sets. Following this distinction, Cantor 35 Ignacio Jané, “The Role of the Absolute Infinite in Cantor’s Conception of Set”, Erkenntnis 41 (3/1995), p. 383. 36 Georg Cantor, “Letter to David Hilbert, 2 October 1897”, in From Kant to Hilbert: A Source-book in the Foundations of Mathematics, Vol. II, ed. William Ewald, Clarendon Press, Ox­ford 2005, p. 928. 37 Georg Cantor, Briefe, p. 407. [My translation.] “Finished thing” is my translation of “ein fertiges Ding” in German. 38 Ibid., 407. 39 Ibid., 407. asked if the collection of all that is thinkable is a consistent multiplicity or an in­consistent one. Cantor indexed the system of all numbers, of all that is thinkable by the letter W. He remarked that such a system forms the following increasing sequence: 0,1,2,3… w0, w0 +1,…. g, and so on… Then, he answered his question by asserting that W cannot be a consistent mul­tiplicity (a set of all sets), because if W were consistent, for every well-ordered set there would have to be a number d that would be bigger than all numbers contained in W and that would be its successor. This d would be bigger than everything that is in W, which is a contradiction since W is defined as the set of all sets. For this reason, W must necessarily be an inconsistent multiplicity and there cannot be a set of all sets.40 Following such a trajectory of reasoning, we might get the impression that Cantor’s thought began with potential infinities (the series of natural numbers 1, 2, 3…), then passed through an architecture of actual infinities (w1, w2, ...), in order to finally fall back into the potentiality of W. The absolute place and the hierarchy of the infinite Badiou’s description of the whole universe of numericity as universe or place in line with recent mathematics might be his manner of avoiding the previously mentioned paradox of Cantor: the paradox in which W can be interpreted as a potential infinity that has merely been pushed to the higher level. Badiou remarks that the absolute class V is similar to what Plato describes as “the intel­ligible realm”41 [le lieu intelligible], which means the “non-representable place within which all representation is deployed.”42 V is “the place of all that can validate the propositions about multiplicities as such.”43 It is “the place where 351 the formal possibility of all existent multiplicities is thought, and which can be reduced to neither language nor the power of nothingness.”44 V recalls Lei­bniz’s “being of the possible”45 [l’etre du possible] and Spinoza’s “substance,” 40 Ibid., 408. 41 Badiou, L’Immanence des vérités, p. 42. [My translation.] 42 Interview with Alain Badiou on The Immanence of Truths, Paris, 13 February 2018. 43 Badiou, L’Immanence des vérités, p. 40. [My translation.] 44 Ibid., p. 41. [My translation.] 45 Ibid., p. 41. [My translation.] with the only difference being that V cannot be identified to the intellect of God. In the absolute class V, being and thinking are identical. By describing V as “the absolute ontological place,”46 Badiou avoids collating V with the potential infinity of Cantor’s W. For if Cantor defined the actual infinity as a “constant quantum which is larger than any finite magnitude of the same kind,”47V might not be a quantum, but it is certainly constant and larger than any finite magni­tude. In other words, the “absolute” means that V is a maximum, there is no Vv. A “place” can be conceived as being-together, thus, using this term to describe V as “place” enables its potentialisation to be avoided. The absolute class V is stratified, it has the structure of an unattainable hori­zon; we can create higher and higher approximations of this horizon but we can never reach it. In mathematics, such a universe is generally represented as a triangle standing on its top. There is the horizon V, and the approximations of this horizon: its sub-classes. If we consider, for instance, a class of all ordinals, which as such belongs to the absolute class V, we see that the notion of class is based on a connection of intentional and extensional characteristics: a class is defined by its attributes (being an ordinal) and it also has an extensional rela­tionship with other subclasses (the class of all ordinals belongs to V). However, to say that one class belongs to another class is nothing but a metaphor used by mathematicians because, strictly speaking, the relation of belonging exists only between sets. Understanding these approximations of the absolute and the closeness to the absolute was a major task of Badiou in The Immanence of Truths. In the mentioned book, Badiou distinguishes four different kinds of the finite: · the accessible finite (a new set can be invented only with the resources of the existing situation); · the divisible finite (the decomposition of a large infinite set into smaller sets); · the limited finite [le fini borné] (preferring totality to openness, the particular to the universal); and · the finite that negates all absoluteness. 46 Ibid., p. 42. [My translation.] 47 Cantor, “Mitteilungen zur Lehre vom Transfinitem”, p. 401. [My translation.] These four kinds of the finite are symmetrically juxtaposed with four different species of infinity organised in ascending order (from the smallest to the largest infinite). Each higher order of the infinite “finitises” the previous one. a) Inaccessible infinity or infinity via transcendence This elementary form of infinity is close to the potential infinite and is relatively small; “transcendent” means merely that we cannot gain access to it and that it can be defined through negation. In theology, the God transcendent to the human world might constitute an example of such an inaccessible infinity. In mathematical language, Badiou formalises the infinite via transcendence as the strongly inaccessible cardinal. A strongly inaccessible cardinal k 1) is superior to w; 2) is a limit cardinal; 3) is regular = it is bigger than the union of all cardinals smaller than k; 4) is bigger than the cardinality of the power set of any cardinal that is smaller than this strongly inaccessible cardinal. Badiou remarks that, for a long time, a strongly inaccessible cardinal has been seen as the limit of set theory. Had this cardinal k existed, we could find its corresponding class VKand the latter would become the model of all axioms of ZFC. However, according to Kurt Gödel’s second incompleteness theorem, given a formal system containing basic arithmetic, it is impossible to prove its consist­ency from within that system. Therefore, the existence of inaccessible infinity cannot be proved from within the axioms of ZFC. Inaccessible infinity evokes Blaise Pascal’s wager: we can only wager that God qua inaccessible infinity ex­ists, without ever being able to possess a final proof of the existence thereof. b) The infinity defined by its indivisibility In The Immanence of Truths, Badiou compares this infinity to the division of God into a trinity in Christian theology and the related question of how it is possible that such a division does not diminish God’s power. If we attempt to cut the “in­divisible” infinity into very small parts, these parts will be always able to form a subset that will be of the same cardinality as the entire infinite set. To explain this kind of infinity, Badiou uses the Ramsey theorem. Frank P. Ramsey studied the first countable infinite w and the possibility of dividing it by two. He realised that if this set is divided by two, it will always be possible to form a subset H that will belong to the same half of this divided set and will have same cardinality as the whole set (before division). The Ramsey cardinal, devised by Paul Erdõs and András Hajnal and named af­ter Frank P. Ramsey, transposed the latter’s discovery into the sphere of the un­countable infinite. According to Erdõs and Hajnal, if we divide this uncountable infinite set by any number, it will always be possible to find a manner of classi­fying the parts into which it was divided and transform them into a subset of the initial set, a subset that is homogenous to the partition. Saying that this subset is homogeneous means that it has the same cardinality as the initially divided infinite set. Thus, the Ramsey cardinal evokes the idea of an infinite set that is so compact and dense and whose elements are in such proximity to each other that even if we cut it into small parts, a large infinite residual set will always escape our cutting. Badiou mobilises the above-mentioned concepts in order to reflect on how emancipatory political movements could avoid capitalism’s oppressive tenden­cy to divide them into smaller parts. By helping to sow the division into any genuine emergent political movement, the dominant regime is able to preserve its sovereignty. The division is an operator of finitude. In contrast, any truth procedure will engender an infinity equal at least to the Ramsey cardinal. Any emancipatory political movement, any true “event”, relates to the emergence of new forms of infinite truth procedures. c) The infinity of big parts This form of infinity invokes the question of what it means that something is close to the absolute, or that it is “almost” absolute. Are there any classes inte­rior and inferior to V that can still express the absolute V? Can the absolute be approximated in any manner? Is there anything like bigness in itself? On a formal level, Badiou attempts to respond to this question by using the difficult concept of the non-principal k-complete ultrafilter. A filter on a set is a mathematical apparatus helping us to distinguish small parts from the large ones; it works like a sieve that only catches large parts, while the small ones pass through. If we have a set E, we call a filter on E a set F composed of parts of E, which has the following properties: 1) It does not contain the empty set; the empty set must be “small”. 2) It contains the set E: the largest part. 3) If parts A and B belong to the filter, the filter also contains their intersection. In other words, if both A and B are large parts, their intersection will be large. 4) If the filter contains part A and A is included in part B, it must contain part B, because B is bigger than A. In order to transform this filter into an ultrafilter, a property of exhaustivity has to be added: 5) If we consider any part of E, it either belongs to the ultrafilter or its comple­mentary part (i.e. its negation) belongs to the ultrafilter. Thus, the ultrafilter exhausts everything that there is: either one element belongs to it or its op­posite does. A non-principal ultrafilter means: 6) that this ultrafilter does not contain any singleton (a set containing only one element). Badiou is particularly interested in this property for it enables him to disconnect the infinite from the one. And finally, k-complete 7) is a mathematical procedure of ultrafiltering constructing a huge cardinal k that has this non-principal k-complete ultrafilter on itself. This mathemati­cally complex section of the book translates into philosophical language as follows: the ultrafiltering engenders an infinite set so large that it exceeds the previous two forms of infinities and constitutes a testimony of the ex­istence of the absolute class V. This set is “almost absolute”; it is a proof of the existence of the absolute class V. The statement “truths are absolute” is equivalent to the affirmation that the infinity of a truth is so large that it can attest to the existence of the absolute class V. d) The infinity defined by its proximity to the absolute This infinity corresponds to Badiou’s theory of the attributes of the absolute and is elucidated in the most Spinozist part of the book. In this section, Badiou empha-sises that the relationship between the absolute and one of its attributes implies the existence of a very large infinite set – a complete cardinal – that becomes wit­ness to the existence of the absolute class V. Using Mostowski lemma and Jerzy Lós theorems, Badiou explains the mathematical concept of the elementary em­bedding, which plays a similar structural role in the book as forcing does in Being and Event. In simple words, elementary embedding is a mathematical procedure entailing taking a transitive sub-class of V – which we can call here M (e.g. a class of all ordinals because ordinals are defined by their transitivity) – and transform­ing it into a model of V. In this manner, the absolute class V will be embedded in one of its attributes, in the transitive sub-class M. There will be a relation j between M and V. However, this relation j will not be that of identity: V will remain different from the attribute M and contain sets that we cannot find in M. V is “embedded” in this class somewhat like an edifice is embedded in its concrete foundations. The attributes of the absolute and the philosopher’s choice In Plato’s vocabulary, we could say that this fourth kind of infinity participates in the absolute class V. If V is equivalent to Spinoza’s notion of substance, the fourth infinity equals Spinoza’s notion of the attribute of the absolute: the ab­solute expresses itself through its attributes but nevertheless remains separated from them. Or as Badiou remarks, “the expressive capacity of the absolute is intelligible for us only through the mediation of attributes.”48 Saying that we con­sider “all the sets” is a weak characteristic; it gives us a feeble understanding of the absolute. However, using an attribute such as the “class of all ordinals” can provide at least some grasp of the absolute because we possess a definition of an ordinal: we know that an ordinal is a set that is transitive and well-ordered by .. Thus, by approaching the absolute through its attributes we are able to gain a certain and limited knowledge of the absolute. We can never entirely know the absolute as such, but we can at least approximate it through the use of attributes. It would be interesting to compare Georg Cantor’s own references to Spinoza to those of Badiou. For instance, in Foundations of a General Theory of Manifolds, Cantor averred that “an especially difficult point in Spinoza’s system is the re­lationship of the infinite modes to the infinite one; it remains unexplained how and under what circumstances the finite can maintain its independence with re­spect to the infinite, or the infinite with respect to still higher infinities.”49 Cantor alluded here to “Proposition 22” of Spinoza’s Ethics, according to which “what­ever follows from some attribute of God insofar as it is modified by a modification 48 Interview with Alain Badiou on The Immanence of Truths, Paris, 13 February 2018. 49 Cantor, “Foundations of a General Theory of Manifolds: A Mathematico-Philosophical In­vestigation into the Theory of the Infinite”, p. 892. which, through the same attribute, exists necessarily and is infinite, must also ex­ist necessarily and be infinite.”50 In contrast, in “Proposition 28,” Spinoza postu­lated that “any thing which is finite and has a determinate existence, can neither exist nor be determined to produce an effect unless it is determined to exist and produced an effect by another cause, which is also finite and has a determinate existence.”51Thus, it seems that the infinite and the finite exist in Spinoza in two independent chains of causes and effects which do not interact with each other: on the one hand, the infinite attributes of the substance, on the other hand, the finite modes of our existence. To bridge these two chains, Spinoza proposed the concept of the “infinite mode,” and yet he did not adequately explain how the infinite mode can engender the finite. The lack of connection between the finite and the infinite puzzled Cantor, as well as Badiou, who in his 1984-85 seminar remarked that in Spinoza’s work “the finite produces the finite and the infinite produces the infinite. This is an enigmatic point.”52 Badiou’s concept of “oeuvre”, presented in the concluding chapters of The Im­manence of Truths, constitutes an endeavour to disentangle the unsolved prob­lem of Spinoza. Badiou posits that it is necessary to “postulate that the working of a truth is subjectively structured in tension within a play of various distinct infinities, and the result is certainly a finite oeuvre, or in other words, its abso­luteness is related to the fact that this finitude conquers the status of an oeuvre instead of being a simple waste [déchet] of the infinite. It is a finite result that reaches the level of its infinite causality because it inscribes itself into an attribute of the absolute.”53 If truths, in Badiou’s work, correspond to Spinoza’s attributes of the substance, and the very procedures of these truths to infinite modes, the oeuvre is the finite mode engendered by the infinite. The oeuvre is a paradoxical finite fragment of reality that in spite of its finitude remains indexed to the ab­solute. “All oeuvre of truth is finite, singular, universal, and absolute,”54 states 357 Badiou. The indexation of the finite oeuvre by infinity makes it participate in 50 Benedictus de Spinoza, A Spinoza Reader: The Ethics and Other Works, Princeton Univer­ sity Press, Princeton 1994, p. 101. 51 Ibid., p. 103. 52 Alain Badiou, L’infini, Aristote, Spinoza, Hegel, Le Séminaire 1984-1985, Fayard, Paris 2016, p. 170. 53 Badiou, L’Immanence des vérités, p. 393. [My translation.] 54 Ibid., p. 512. [My translation.] the absolute. The oeuvre of the absolute is a manifestation of the connection between universality and singularity. In The Immanence of Truths, the concept of the oeuvre is juxtaposed with that of a “waste” [déchet]. If the oeuvre is indexed to the absolute, the idiosyncratic property of the “waste” is that it can be covered by constructible sets, that it is nothing but “a mode of existence of multiples which have no other figure than to remain under the law of the world in which they appear.”55 The Immanence of Truths completes Badiou’s critique of the linguistic orientation of philosophy by identifying finitude with the property of constructibility. A given set is finite, or “constructible” if all its elements can be defined by a given language. In the constructivist orientation of thought, the mathematical real is subjugated to lan­guage. Badiou remarks that “any set can become a material for covering and thus become finite if it has as its sole elements other multiplicities that constitut­ed definable parts in an already pre-existing and finite set. Such finite set will be said to be ‘constructible’.”56 Constructivism operates through the logic of “cover­ing” [le recouvrement]: a new potentially infinite multiple is “covered” by already existing multiples. An emerging large infinity is rendered finite by being covered by a multiplicity of finite sets. The inconsistent multiplicity is transformed into a multiplicity of consistent ones. In mathematical terminology, the hypothesis that the word “set” signifies a finite or constructible set has been generally marked by the capital letter L. V = L denotes the hypothesis of a constructible universe, the idea that there are no actual infinities and that the only things that exist are fi­nite, constructible sets. If V = L were true, Georg Cantor’s Continuum Hypothesis (there is no set that is greater than the set of all integers and smaller than the set of all real numbers) would be necessarily and logically valid. Badiou admits, in Chapter 19 of The Immanence of Truths that from the mathe­matical point of view, both options of accepting and rejecting constructivism (V = L) are possible: on the one hand, Kurt Gödel proved that it is possible to preserve the consistency of ZFC set theory by adding to it the axiom of constructibility; on the other hand, Paul Cohen, in the 1960s, invented the technique of forcing, thus showing that it was possible to admit the existence of non-constructible generic multiplicities. Both options are equally admissible for working mathematicians. Ibid., p. 511. [My translation.] 56 Ibid., p. 240. [My translation.] Thus, he is left with nothing but a “crucial choice” between the constructible or truly infinite and a generic orientation of thought. He decides in favour of Cohen against Gödel. The affirmation that “there is the infinite”57 is nothing but a purely philosophical act, in which The Immanence of Truths originates. Such an act can be justified only a posteriori, by the abundance of its consequences. Badiou decides as a philosopher, and not as a mathematician, that there must be some­thing unnamable or indiscernible that cannot be described by the language of the dominant order. Once again, he subordinates mathematics to philosophy. The task of the philosopher might be to cut the Gordian knot that mathemat­ics cannot untie. Such cutting operates in Badiou’s thinking by finding in Spi­noza resources to think an actual non-denumerable infinity and turning them against the ideas of Leibniz, by privileging the realist orientation of the math-eme over the supremacy of language. Maintaining philosophy in close relation to its mathematical condition is necessary for freeing thinking from its capture by the linguistic turn. Mathematics is but Badiou’s shield against the reduction of thought to the constructible; it is in this sense that the three volumes of Being and Event could also be renamed “Three Critiques of the Constructible”. Philos­ophy is not sutured to anything – not even to mathematics – it is free and bound only by its consistency while operating in a necessarily incomplete field. References Badiou, Alain, Being and Event, Continuum, New York 2006 — Briefings on Existence: A Short Treatise on Transitory Ontology, State University of New York Press, Albany 2006 — L’Immanence des vérités, Fayard, Paris 2018 L’infini, Aristote, Spinoza, Hegel, Le Séminaire 1984–1985, Fayard, Paris 2016 — Number and Numbers, Polity, Cambridge 2008 359 — Interview with Alain Badiou on The Immanence of Truths, Paris, 13 February 2018 — Benaceraff, Paul, Hilary Putnam (eds.), Philosophy of Mathematics, Cambridge Uni­versity Press, Cambridge 1983 Beránková, Jana, “Communism is a New Idea, Interview with Alain Badiou by Jana Beránková”, Contradictions 2 (2/2018), pp. 117–132 Cantor, Georg, “Fondements d’une théorie générale des ensembles”, Cahiers pour l’Ana­lyse (10/1969), pp. 35–52. — “Foundations of a General Theory of Manifolds: A Mathematico-Philosophical Inves­tigation into the Theory of the Infinite”, in From Kant to Hilbert: A Sourcebook in the Foundations of Mathematics, Vol. II, ed. William Ewald, Clarendon Press, Oxford 2005 57 Ibid., p. 265. [My translation.] — “Mitteilungen zur Lehre vom Transfinitem”, in Gesammelte Abhandlungen Matema­tischen and Philosophischen Inhalts, Springer, Berlin 1932 Gödel, Kurt “What is Cantor’s Continuum Problem?” in Collected Works, Vol. II, Publica­tions 1938–1974, Oxford University Press, Oxford 1990 Jane, Ignacio, “The Role of the Absolute Infinite in Cantor’s Conception of Set”, Erkennt­nis 41 (3/1995) Spinoza, Benedictus de, A Spinoza Reader: The Ethics and Other Works, Princeton Uni­versity Press, Princeton 1994 Filozofski vestnik | Volume XLI | Number 2 | 2020 | 361–395 | doi: 10.3986/fv.41.2.14 Norma M. Hussey A New Hope for the Symbolic, for the Subject Symbolization and ontology Tradition is the old world of castes, nobilities, religious obligation, … local my­thology, the submission of women, the father’s absolute power over his children, and the official division between a small group of rulers and a condemned mass of toilers, in which the differences inherent to human life were regulated and sym­bolized in a hierarchical form. The most important binaries, like old and young, men and women, … commoners and nobles, town and countryside – were all ad­dressed (in language, in mythologies, in ideologies, and in the established reli­gious models) by recourse to ordered structures setting everyone’s place in a set of overlapping hierarchical systems.1 From the philosophical writings of Alain Badiou (illuminated by the mathemat­ical ontology) it is recognized here that the hierarchical symbolization of ‘tradi­tion’ found its guarantee, at the abstract level, in the supposition that beyond the multiple, the one is. In other words, the ontological basis for the hierarchical form of the traditional symbolic assumes the being of the one, and relies on it for the “normative power”2 of its deployment. As such, it is a symbolic based on finitude, which means that (in the mathematical ontology) it “resides within the elementary immanence of … finite ordinals.”3 This is the case whether the one be finite or infinite. For instance, “Christian monotheism, despite its designa-361 tion of God as infinite, does not immediately and radically rupture with Greek finitism” – it can be based on a thinking in which being as such “remains essen­tially finite;” the assumption of an infinite one does not yet break with the finite, 1 Alain Badiou, “True and False Contradictions of the Crisis”, trans. David Broder, Verso blog, May 29, 2015. Emphasis added. 2 Alain Badiou, Theoretical Writings, trans. Ray Brassier and Alberto Toscano, Bloomsbury Academic, London and New York 2014, p. 44. 3 Alain Badiou, Being and Event, trans. Oliver Feltham, Continuum, London and New York 2007, p. 160. because it is still “hierarchically representable as beyond [or unknowable], yet deducible from,” the finite world.4 According to Badiou, the onto-theological device – trying to save truth – pays the price of the absolute transcendence of the one, thereby subordinating finite multiplicities to the formal authority of the One-infinity that has often been called God.5 Modernity is effectively a break with tradition. In barely three centuries, this break with the world of tradition has swept aside forms of organization that had lasted for millennia. When this break takes a bourgeois, capitalist form, it opens up a gigantic crisis of humanity’s symbolic organization. It does not in fact propose any active new symbolization, but only the brutal, independent play of the economy: the neutral, a-symbolic reign of what Marx called “the icy water of egotistical calculation.”6 It could be said that the breakdown of the traditional symbolic corresponds to modern ontological confusion owing to the death of God,7 i.e. the infinite One. Modernity – in a world structured by globalized capitalism – has revealed that being is essentially multiple; it has exposed the one as pure semblance, and therefore ruined the supposition on which the hierarchical symbolic depends. In his Manifesto for Philosophy, Badiou writes that The only thing we can and must welcome within Capital [is that]: it exposes the pure multiple as the foundation of presentation; it denounces every effect of One as a simple, precarious configuration … To think over and above Capital … we must still have as a departure point what it has revealed: Being is essentially multiple.8 Symbolization and subjective orientation Rejection of the traditional hierarchy and its oppressions exposes the being of the one as un-credible. But freedom from hierarchy, from subordination to the 4 Ibid, p. 142. 5 Badiou, as reported by Jana Berankova in “The Immanence of Truths: the Absolute be­tween the Singular and the universal”, Presentation at the International Conference “Thinking the Infinite”, April 11, Prague 2018. 6 Badiou, “True and False Contradictions of the Crisis”. Emphasis added. 7 Badiou, Theoretical Writings, pp. 27–30. 8 Alain Badiou, Manifesto for Philosophy, trans. Norman Madarasz, SUNY Press, Albany, New York 1996, pp. 56–57. normative power of the one, without symbolic replacement, “creates a subjective crisis … one of whose most remarkable aspects is the extreme and growing diffi­culty that young people in particular face finding themselves a place in this new world.”9 Disorientation arises with the breakdown of symbolization, i.e. with the lack of a common understanding of the symbolic organization. Perhaps this is why the theme of the subject has been a constant for modernity. Faced with the crisis of humanity’s symbolic organization (and the associat­ed crisis of general subjective disorientation) some would have us believe that there is no crisis, and that there is nothing better than this liberal ‘democratic’ model of freedoms weighed down by the neutrality of market calculations. This is Western capitalism’s a-symbolic vision, which creates monstrous inequalities and pathogenic upheavals; under the auspices of ‘freedom’ in the a-symbolic reign, exploitation replaces the oppressions of hierarchy, and often inspires the reactive desire to return to the traditional hierarchical symbolization.10 With respect to the break with tradition then, the subjective orientations of mo­dernity range from the embrace of this liberal a-symbolic model of freedoms (especially by the rich), to the reactive desire to return to the traditional sym­bolization (especially by the ex-privileged), and in the wider context of general disorientation, from imaginary utopianism to hopeless nihilism. But there is an­other subjectivity which, recognizing the non-being of the one and the unavoid­able break with tradition, is convinced that a non-hierarchical symbolization is possible and that it must be invented, i.e. “an egalitarian symbolization that restructures differences – [recognized of equal subjective right] – based on a total sharing of resources.”11 363 A new symbolic requires a coherent ontological referent From reading Badiou, it is my understanding that a symbolic (defined as a uni­versal structure encompassing the entire field of human action and existence)12 can only be based (contrary, it would seem, to Lacan) on a credible thinking of being as such, i.e. a stable and consistent ontology. The normative power of an 9 Badiou, “True and False Contradictions of the Crisis”. 10 Ibid. 11 Ibid. 12 Encyclopedia.com, Symbolic, the (Lacan), available at: https://www.encyclopedia. com>psychology>symbolic-lacan assumed one (finite or infinite) sustained the hierarchical symbolic for millen­nia, but such an assumption is no longer credible – it has been exposed as pure semblance, and being is revealed as essentially multiple. If all is multiple, is a symbolic even possible? That is, a symbolic which recognizes the non-being of the one and therefore cannot rely on the deployment of its normative power, as such a non-hierarchical and therefore egalitarian symbolic? It would at least require a coherent, i.e. stable and consistent, ontology of multiplicity. An ontology of multiplicity The thesis of Alain Badiou which equates ontology with mathematics13is under­scored by the philosophical decision that being, or what we can think under the name of ‘being,’ is composed of pure multiplicities, and that “the one is not,” be it the one of an Idea (Plato), or a God, or Presence, or any-one; “the one exists solely as operation.”14 Since the axiomatic system of mathematical set theory delivers the multiple without implying the being of the one,15 Badiou adopts it as the thinking of being as such (ontology). Set theory is the area of modern mathematics devoted to the study of infinity; mathematics thinks (infinite) mul­tiplicities in the form of the notion of a (infinite) set.16 It is based on a formal axiom system called ZFC (Zermelo-Fraenkel system with the axiom of choice). Quantitative infinity is not deducible from finitude, but is authorized solely by the axioms of mathematical set theory.17 It is by recognition of the mathematical existence of quantitatively distinct infinite multiples, that the ruin of the being of the infinite one is achieved.18 This thorough-going rationalization of actual infinities (as opposed to a transcendent one) teaches us that there is no reason to confine thinking within the ambit of finitude because we can have access to a rational, secular thinking of infinity.19 Mathematical ontology (i.e. set theory) is the proposal of an infinite ontology which is, finally, a radical rupture with the 364 finitism of tradition, and in which “the finite is qualified as a region of being, a minor form of the latter’s presence.”20 13 Badiou, Being and Event, p. 4. 14 Ibid, p. 23. 15 Ibid, p. 43. 16 Ibid, p. 145. 17 Ibid, p. 148. 18 Ibid, p. 273. 19 Badiou, Theoretical Writings, p. 19. 20 Badiou, Being and Event, p. 159. Badiou, in fact, proposes the universe of set theory – conventionally denoted by V – as an absolute ontological referent, where V can be said “to formalize … the place … of everything that can be constructed by axioms.”21 He argues that “the absoluteness of the thinking of the pure multiple, in the atheistic takeover from monotheism … is feasible.”22 But is it a coherent theory? In Being and Event(1988),23Badiou described the then prevailing situation of ontology (set theory), with all of its intricacies and its metaontological implica­tions. In particular, he highlighted the impasse concentrated in the continuum problem which, as Paul Cohen established in 1963, is undecidable from the con­temporary ZFC axiomatic. Since then, the formal axiom system itself has been in crisis – which challenges the very conception of mathematical infinity,24 to­gether with the credibility of set theory as a consistent ontology. It doesn’t auger well for the invention of an egalitarian symbolic which can only be based on an ontology of the multiple, if that ontology is inconsistent or uncredible. The pre­vailing incoherence of the mathematical ontology underscores a contemporary deficit of symbolization which, in turn, yields confusion and conflict in terms of subjective orientation. But something momentous is happening right now in set theory, in response to the recent surprise discovery now encapsulated in the universality theorem (see later); a new axiom, conjectured as yet, promises to resolve the crisis of its for-malization,25 and achieve the pass of the impasse. The consequent realization of a coherent set theory – a stable and consistent ontology of multiplicity – would suggest that an egalitarian symbolization is possible, i.e. a modern symbolic which receives its guarantee solely, but transparently, in its mathematical con­sistency. Indeed, “mathematics is the only discourse in which one has a com-365 21 Acheronta Movebo (Badiou), “Towards a New Thinking of the Absolute”, Crisis and Cri­ tique 1 (2/2014), p. 12. Badiou was one of the contributors to this journal as part of the anonymous collective Acheronta Movebo. 22 Ibid., p. 21. 23 The original edition of Being and Event, i.e. Badiou, L’Etre et L’Événement, Éditions du Seuil, Paris, was published in January 1988. 24 W. Hugh Woodin, “Generalizing Gödel’s Constructible Universe: Ultimate L”, Presentation at the IMS Graduate Summer School in Logic, National University Singapore, June 2018. 25 W. Hugh Woodin, “In Search of Ultimate-L, the 19th Midrasha Mathematicae Lectures”, The Bulletin of Symbolic Logic 23 (1/2017), p. 2. (DOI:10.1017/bsl.2016.34). plete guarantee and a criterion of the truth of what one says, to the point that this truth is unique inasmuch as it is the only one ever to have been encountered which is fully transmissible.”26 This remarkable development in set theory, led by W. Hugh Woodin,27 offers hope for the pursuit of a new symbolic in a modern age of actual infinity, and a consequent resolution of the general subjective disorientation which arises from our current crisis of symbolization. The premise here, then, is that the process for the articulation of a coherent and transmissible ontology on which to ground an egalitarian symbolic has already begun; it originated with the Cantorian inventions of set theory and quantitative infinity, but it is still undergoing development as the contemporary set theoretic axiom system (ZFC) is not yet stable/complete. Set theory – historical development The saga surrounding the development of set theory over the past 150 years – its splits, its advances, its dramatic crises – reflects the extreme difficulty of in­vention of an ontology of multiplicity which is not in the least intuitive as it depends on complex mathematical reasoning and innovation for the thinking and comparison of distinct quantitative infinities. Mathematical infinity is a counter-intuitive concept, even at its simplest level, for instance “the infinite set of even numbers {2,4,6,…} can be placed in a one-to-one correspondence with all counting numbers {1,2,3,…}, indicating that there are just as many evens as there are odds-and-evens,”28 therefore we cannot rely on intuition, but require the patient and painstaking work of mathematical rigor, in order to understand 366 it. The development of set theory also seems (almost uncannily) to have mir­rored the general underlying situation in the social and political world over the same period (see below). 26 Badiou, Being and Event, p. 9. Emphasis added. W. Hugh Woodin is the mathematical set theorist leading the remarkable developments in set theory referenced throughout this article, at Harvard University since 2014, and before that at the University of California, Berkeley. 28 Natalie Wolchover, “To Settle Infinity Dispute, a New Law of Logic”, quantamagazine.org, November 26, 2013 (reprinted on ScientificAmerican.com, December 3, 2013). The nine axioms of ZFC were laboriously developed around the turn of the 20th century, thereby putting Cantor’s still partly intuitive theory of sets on a secure axiomatic (or non-intuitive) footing.29 They accord with the iterative conception of sets as cumulative hierarchies, where sets are constructed by a combination of axiomatically prescribed procedures in successive steps from an assumed foundational empty set O, and the sets accumulate – hence ‘cumulative.’ The universe of sets V (as formalized by ZFC) is calibrated by the ordinals through the definition of the cumulative hierarchy of sets,30 in which the successor step is obtained by the power-set (i.e. the set of all subsets) of the preceding set. According to Badiou, the ordinals – as the ontological schema of natural multi­plicity – “say that nothingness is a form of natural being [the empty set O being an ordinal, and therefore a natural multiple], and that the infinite, far from be­ing retained in the One of a God, is omnipresent in nature, and beyond that, in every situation-being.”31 Constructibility and conservatism Set theory was tentatively concluded with a tenth axiom, i.e. Kurt Gödel’s axi­om of constructibility (1938), which asserts that the universe of sets V is the con­structible universe L, i.e. V = L. Gödel’s definition of L arises from restricting the successor step in the definition of the cumulative hierarchy to the constructible power-set (also called the definable power-set), which is the set of all subsets of the preceding set such that each subset is logically definable (in the set struc­ture) from parameters in the set.32 29 Peter Hallward, “On the development of Transfinite Set Theory”, in Peter Hallward, Ba­ 367 diou: A Subject To Truth, University of Minnesota Press, Minnesota 2003, p. 337. He also notes that Cantor’s own position was anti-axiomatic. 30 The cumulative hierarchy of sets V, is the collection of sets V indexed by the class of or­. dinal numbers. For each ordinal ., a set V. is defined by induction on . as follows: 1. V0 = O, 2. (Successor step) V.+1 = P(V.), the power-set of V., 3. (Limit step) If . is a limit ordinal, then V = U {V|ß<.}, the union of all the V-stages so far. .ß 31 Alain Badiou, Number and Numbers, Polity Press, Cambridge UK and Malden USA 2008, p. 84. 32 W. Hugh Woodin, “The Transfinite Universe”, in Kurt Gödel and the Foundations of Math­ ematics. Horizons of Truth, ed. M. Baaz, C. H. Papadimitriou, H. W. Putnam, D. S. Scott, and C. L. Harper Jr., Cambridge University Press, Cambridge 2011, pp. 449–471. Gödel’s axiom “does provide a clear conception of the universe of sets V,”33 but as a minimal statement about the universe of sets (because if the set is infinite and the axiom of choice holds, then the definable power-set is never the set of all subsets of the set)34 it was never fully embraced as a true axiom for set theory.35 In addition, the constructible universe cannot accommodate the large cardinals in the higher infinite which, although unprovable from ZFC, are now considered to be genuine.36 (Since the existence of such a set cannot be proven on the basis of ZFC, it has to form the object of a new axiom. What is then at stake is an axiom of infinity, stronger than that of ZFC which guarantees only the existence of a limit ordinal,37 i.e. a large cardinal axiom stating that there exists a cardinal (or perhaps many of them) with some specified large cardinal property. Such axi­oms appear to occur in strict linear order by consistency strength, thus forming the large cardinal hierarchy. A measurable cardinal is a relatively ‘small’ large cardinal, i.e. it is low in the hierarchy.38) By Scott’s theorem (1961), there are no measurable cardinals in the constructible universe and therefore no stronger large cardinals either.39 As such, V = L “severely limits the nature of infinity.”40 It could be said that Gödel’s constructible universe represents a first attempt at an abstract referent for an egalitarian symbolic, in that it does yield a model for ZFC, i.e. a stable and consistent framework (albeit restricted) for an ontology of multiplicity. It is a universe for all ordinals (i.e. natural multiplicities) indicating that “the infinite is omnipresent in nature,” but beyond that (i.e. with respect to non-natural multiplicities) it has nothing to say. As such it may be consid­ered as a natural ontology, but it doesn’t account for every situation-being. Its mathematical experimentation coincided with the period of social and political conservatism of the 1940’s and ‘50’s, which is not to suggest that Gödel was re­ 33 Ibid. 34 W. Hugh Woodin, “Strong Axioms of Infinity and the Search for V”, in Proceedings of the International Congress of Mathematicians, Hyderabad, India 2010. 35Woodin writes: “We know that Gödel [himself] rejected the axiom V = L”, in “The Transfi­ nite Universe”. 36 Woodin, “Generalizing Gödel’s Constructible Universe: Ultimate L”. 37 Badiou, Being and Event, p. 513. 38 Colin J. Rittberg, “How Woodin Changed his Mind: New Thoughts on the Continuum Hy­ pothesis”, Archive for History of Exact Sciences 69 (2/2015), pp. 125–151. 39 Ibid. 40 Woodin, as quoted by Wolchover in “To Settle Infinity Dispute, a New Law of Logic”. sponsible for that period of conservatism but, perhaps, that his work explains the abstract thinking behind what was going on at the time. In The Immanence of Truths,41 Alain Badiou claims that we (still) live in a con­structible universe without being aware that this is the case.42 He argues that the constructible universe largely equals … finitude,43 i.e. a modern finitude where the finite is defined (here) as a set that can be ‘covered’ by a constructible set with the same cardinality as the former.44 Badiou asserts that this modern form of finitude is one of ‘movement and innovation,’ as opposed to the classical finitude of ‘stability and repetition;’45 nevertheless it represents a return to the figure of oppression, to the normalization of the one, and a closure of thought to the realm of actual infinity. Such ‘finitude’ is precisely what Badiou rejects. Independence and dissatisfaction The axiom system (ZFC) itself has been in crisis since 1963 when Paul Cohen demonstrated that the most basic problem of set theory – that of Cantor’s con­tinuum hypothesis – was not solvable on the basis of the ZFC axioms; it can neither be refuted nor proven from the axioms of set theory, it is an independent statement.46 This result established the impossibility, within ZFC, of determining the type of multiplicity of the continuum (which corresponds to the power-set of the very first infinity, i.e. all the subsets constituted from whole numbers); set theory could not determine “the measure of the set of parts of an infinite set,” or in Badiou’s metaontological terms, “the excess of the power of the state with respect to that of its situation.”47 Easton’s theorem (1970) extended this result 41 Alain Badiou, L’Immanence des Vérités, Fayard, Paris 2018. 42 Terence Blake, “My Path Through Badiou’s The Immanence of Truths”, available at: https:// 369 terenceblake.wordpress.com/2018/10/06/my-path-through-badious-the-immanence-of-truths. 43 Jana Berankova, Presentation at the Graduate Workshop with Alain Badiou onThe Imma­nence of Truths, November 13, Columbia University, New York 2019. 44 Jana Berankova, “The attributes of the absolute and Alain Badiou’s response to Spinoza”, in Sometimes We Are Eternal, ed. J. N. Berankova and N. Hussey, Suture Press, Lyon 2019, pp. 174–175. 45 Terence Blake, “Badiou’s The Immanence of Truths: Introduction (sketch)”, available at: https://terenceblake.wordpress.com/2017/01/19/ badious-the-immanence-of-truths-intro­duction-sket/ 46 Woodin, “Generalizing Gödel’s Constructible Universe: Ultimate L”. 47 Badiou, Being and Event, p. 275. and established the “quasi-total errancy of the excess of the state over the situ­ation,” in that it is consistent with ZFC that the power-set could be any cardinal as immense as you like, provided that it is a successor.48 Cohen’s method of forcing ushered in a new age of ‘freedom’ from the restricted formalization assumed by the constructible universe; it exposed the formal inde­pendence of not just the continuum hypothesis, but also the axiom of constructi­bility (Gödel’s constructible universe), from the set-theoretical framework49 – it inherently implied the non-obligation of conservatism. In the immediate term, however, the focus was on the unquantifiable excess of the infinite power-set it­self, rather than on the implication of crisis for ZFC, owing to such independent statements. The recognition of the “exorbitant excess,” the “innumerable injus­tice,” implied by the almost-total un-measure of the state with respect to the sit-uation,50 seems to underlie the general dissatisfaction prevailing in the 1960’s & ‘70’s and articulated against the state – whether in the domain of mathematics itself, or in the domains of art, or love, or politics. Incoherence and disorientation But today it’s a question of credibility with respect to the capacity of set theory to deliver a coherent framework for infinite multiplicities. The intractability of the independence phenomenon due to Cohen’s method of forcing, still unresolved more than fifty years later, challenges the foundational issues of truth in set theory (ZFC) itself, and raises the question of whether the continuum hypothesis has a truth value at all, or even whether one can “coherently talk of the realm of the infinite.”51 Hugh Woodin adds that the modern significance of Scott’s theorem – in light of significant advances in large cardinal theory since then – establishes that the axiom of constructibility is false, i.e. that V . L, but he ac­370 knowledges that this claim is not universally accepted.52By Jensen’s dichotomy theorem,53 both hypotheses (i.e. V = L, and V . L) are completely valid in con­ 48 Ibid, p. 280. 49 Woodin, “The Transfinite Universe”. 50 Badiou, Being and Event, p. 282. 51 W. Hugh Woodin, “The Realm of the Infinite”, in Infinity. New Research Frontiers, ed. M. Heller and W. H. Woodin Cambridge University Press, Cambridge 2011, pp. 89–118. 52 Woodin, “The Transfinite Universe”. 53 The Jensen dichotomy theorem shows that V must be either very close to L or very far from L. Refer to Woodin, “In Search of Ultimate-L, the 19th Midrasha Mathematicae Lectures”, p. 2. temporary mathematics, which contributes to the ambiguity of set theory. The prevailing incoherence of the frame of reference for thinking the infinite (or the crisis of formalization) coincides with contemporary widespread disorientation as manifested in dissatisfaction with society, mental ill-health, troubled youth, popular frustration, a feeling of obscure disorder, lack of orientation/stability. Platonism, pluralism, and the skeptic Set theory’s contemporary indeterminacy undermines its candidacy as an abso­lute referent. As such, two major contradictory orientations arise; the universe view (or Platonism), and the multiverse view (or pluralism), together with a third: the reactionary orientation of the skeptic. The universe view (or Platonism) holds the conviction that there is a unique absolute background concept of set, realizable in the corresponding absolute set-theoretic universe, V, in which every assertion has a definite truth value, i.e. that quantitative infinity is rationally thinkable in a coherent framework. Ques­tions such as the continuum hypothesis and others have definitive final answers, and it is considered that the pervasive independence phenomenon is due to an insufficiency of the theory in finding the truth, rather than about truth itself – because the independence of a statement from ZFC tells little about whether it holds or not in the universe.54 With regard to the continuum hypothesis, Gödel wrote that “it must be either true or false … and its undecidability from the axi­oms as known today can only mean that these axioms do not contain a complete description of reality.”55 For Hugh Woodin, also Platonist, this undecidability is “perhaps tolerable on a temporary basis during a period of axiomatic discovery but it certainly cannot be the permanent state of affairs.”56 He also considers that, based on set theoretical work (his own and others), “there is the very real 371 possibility that we can find a new optimal axiom (from structural and philo­sophical considerations) that prevents us from undecidability of some impor­tant properties of set. In which case, we will have returned, against all the odds or reasonable expectation, to the view of truth for set theory that was present 54 Joel David Hampkins, “The Set-Theoretic Multiverse: a Natural Context for Set Theory”, (MR2857736 (2012h:03002)), based on a talk given by the author at the Philosophy of Math-ematicsConference (New York University, April 2009). 55 As quoted by Wolchover in “To Settle Infinity Dispute, a New Law of Logic”. 56 Woodin, “Strong Axioms of Infinity and the Search for V”. at the time when the investigation of set theory began.”57 Alain Badiou writes: “By saying that set theory constitutes an absolute reference, I am assuming that there exists a system of axioms, incompletely discovered as yet, which defines the universe V … and defines it alone. In other words, no important, significant, useful property of sets will remain undecidable once we have been able to fully identify the axioms.”58 The opposing multiverse view (or pluralism) considers that there are many dis­tinct concepts of set, each realized in their corresponding universes which, in turn, exhibit diverse truths. Its adherents argue that we may prefer some uni­verses to others, and that there is no obligation to consider them all as some­how equal. Here, the continuum problem is already settled, having a different answer in different universes, and it is considered that the Platonist’s ‘dream’ solution of a new axiom for set theory by which the continuum hypothesis is decided, is impossible. This view does not necessarily undermine the claim that set theory serves as an ontological foundation for mathematics, but rather is directed against the claim that there is an absolute frame of reference in which set-theoretic truths are immutable.59Absolute, of course, means not relative, and for Badiou, the current of thought corresponding to this vision is relativism. He writes: 60 Regarding the possibility that V could be an ontological referent, the most impor­tant group of opponents is made up of those who have given up on any referent at all and claim that a truth is never anything but relative or local … This current of thought, suited like no other to representative democracy and cultural relativism, is the prevailing one today … [E]ven those who acknowledge a sort of practical value in it … have objected that the statement “there are only opinions” must be absolutely true, otherwise something other than opinions might well exist, and an absolute truth would therefore exist. 57 W. Hugh Woodin, “The Axiom of Determinacy, Forcing Axioms, and the Nonstationary Ideal”, in De Gruyter Series in Logic and its Applications, ed. W.A. Hodges, R. Jensen, S. Lemp, M. Magidor, 2nd ed., De Gruyter, Berlin 1999. 58 Badiou, “Towards a New Thinking of the Absolute”, p. 16. 59 Hampkins, “The Set-Theoretic Multiverse: a Natural Context for Set Theory”. 60 Badiou, “Towards a New Thinking of the Absolute”, p. 13. The multiverse view – which ultimately corresponds to an a-symbolic vision since a symbolic requires an absolute referent – is dominant today so that diso­rientation reigns as we are cast adrift in an a-symbolic world. The skeptic refers to the meta-mathematical position which asserts that the only meaningful existence assertions are those that can be verified in the physical universe – which certainly excludes set theory (the mathematics of infinite sets).61 According to Woodin, it is a position which denies any genuine meaning to a conception of uncountable sets and considers that set theory and its exten­sions do not reflect any mathematical reality – they are merely a reflection of the mathematician. 62 For the skeptic, the continuum hypothesis is neither true nor false, and consistency claims (e.g. for large cardinal axioms) can never be meaningfully made, because the entire concept of the universe of sets is a com­plete fiction; as such, this view is simply a rejection of the infinite altogether.63 For example, taking a view originally espoused by Aristotle, Stephen Simpson64 argues that actual infinity doesn’t really exist and so it should not so readily be assumed to exist in the mathematical universe. As reported by Natalie Wolcho­ver, he leads an effort to wean mathematics off actual infinity, by showing that the vast majority of theorems can be proved using only the notion of potential infinity. “But potential infinity is almost forgotten now,” he says. “In the ZFC set theory mindset, people tend not to even remember that distinction. They just think infinity means actual infinity and that’s all there is to it.”65 The current of thought corresponding to the skeptical vision responds to the crisis of formalization (ontology’s contemporary indeterminacy) with a rejec­tion of infinity altogether, and proposes a return to finitude, i.e. to the one of hierarchy, or a traditional symbolic. One might consider that such skepticism in mathematics is analogous with the political orientation which is very obviously 373 61 Andrés Eduardo Caicedo, reviewing Woodin’s “The Realm of the Infinite”, submitted to Mathematical Reviews/MathSciNet, MR 2767235, June 26, 2012. 62 Woodin, “The Transfinite Universe”. 63 Woodin, “Strong Axioms of Infinity and the Search for V”. 64 Stephen Simpson is a mathematician and logician at Pennsylvania State University, USA. 65 Wolchover, “To Settle Infinity Dispute, a New Law of Logic”. gaining traction in the contemporary world, as manifested by popular support (reflecting obscure revolt)66 of Trump, Orban, Erdogan, Modi, etc.67 Contemporary assessment of set theory (ontology) Set theorists (today) are broadly drawn from three camps, which pursue the de­velopment and refinement of set theory from different angles: the doctrine of constructibility (originating in Gödel’s constructible universe, and pursuing an inner model program for the incremental understanding of large cardinals),68 the doctrine of genericity (originating in Cohen’s theory of generic extensions, and pursuing a forcing program for expanding the width of the universe of sets),69 and the doctrine of large cardinals, where the hierarchy of large cardi­nals, while unmoored from ZFC, is increasingly emerging as an intrinsic, fun­damental conception within set theory.70 As described in Being and Event, the orientations in thought corresponding to these three doctrines, hold mutually antagonistic positions, especially with respect to the question of the state.71 But in recent times perhaps, the ‘universists’ among them find more common cause in recognition of their shared conviction (albeit differing approaches) of the pos­sibility of an absolute framework in which set-theoretic truths are immutable, and they continue its pursuit, often even against reasonable expectation. Since the turn of the 21st century, adherents of the universe view have insisted on a rational discussion and investigation based on a contemporary assessment of the facts and developments in set theory to date. For instance, they recognize that Cohen’s result (1963) and its mathematical descendants have severely challenged any hope for a concise conception of the universe of sets, since it is not provided by the axioms ZFC – as has been well-documented in the (more than) 50 years since.72 In fact Cohen’s method of forcing has been vastly developed and many 66 Badiou, “Reflections on the Recent Election”, Verso blog, November 15, 2016, which is a transcript of his Presentation at the University of California, November 9, Los Angeles 2016. 67 These are the current (2020) democratically elected presidents of the USA, Hungary, Tur­ key, India, respectively – all with characteristics of proto-fascism. 68 Woodin, “In Search of Ultimate-L, the 19th Midrasha Mathematicae Lectures”. 69 Wolchover, “To Settle Infinity Dispute, a New Law of Logic”. 70 Woodin, “The Transfinite Universe”. 71 Badiou, Being and Event, chap. 27. 72 Woodin, “Strong Axioms of Infinity and the Search for V”. questions have been shown to be unsolvable; it’s not only about the continuum hypothesis which is just a statement about an infinitesimal fragment of V.73 Also, large cardinal axioms are not provable from ZFC by Gödel’s 2nd incomplete­ness theorem.74 A large cardinal axiom is a statement that a ‘very large’ infinite set with certain properties exists; these cardinals carry names such as, for ex­ample, inaccessible, weakly compact, Ramsey, measurable, strong, Woodin, superstrong, strongly compact, supercompact, extendible, huge, .-huge, Rein­hardt. “With the definition of huge cardinals, we approach the point of rupture presented by inconsistency,”75 but we cannot determine that point from ZFC. All of this challenges the very conception of mathematical infinity; there is a temptation to yield, and to accept the skeptical assessment that the concep­tion of the universe of sets is incoherent.76The skeptic wants to reject actual infinity for the sake of certainty (ontological coherence); this represents an ex­treme conservatism, to which adherents of the constructible universe are also vulnerable in the face of increasing recognition, among set theorists, of ‘large’ large cardinals without any formal consistency.77 Yet Woodin, for example, still insists that “there are statements (predictions) which we expect to be false (or true), but there does not seem to be a coherent argument that would explain ‘why’ unless we adopt as meaningful the conception of a non-physical realm, a view where it makes sense to talk of the existence of at least some infinite objects.”78 For Badiou, in any case, it is meaningless to propound the theory of finitude since a rational, ramified thinking of infinity exists, i.e. large cardinal theory – which is the modern theory of infinity.79Recall that the theory of fini­tude (for him) extends beyond the classical form (embraced by the skeptic) to the modern form of finitude which is the constructible universe L. 73 Woodin, “Generalizing Gödel’s Constructible Universe: Ultimate L”. 74 Ibid. 75 Badiou, Being and Event, p. 448. Badiou quoting set-theorist Thomas Jech. 76 Woodin, “Generalizing Gödel’s Constructible Universe: Ultimate L”. 77 The constructible universe cannot accommodate even a measurable cardinal – which is a relatively ‘small’ large cardinal in the hierarchy. 78 Woodin, “The Realm of the Infinite”. 79 Badiou, “Towards a New Thinking of the Absolute”, p. 14. Over the last few decades it has become commonplace to doubt that an ultimate completion of set theory can be articulated, or that a natural completion will settlethecontinuumproblem.80 The greater pressure then, is to conform to the dominant view that there can be no certainty with regard to infinity (relativ­ism), that the arbitrariness is precisely what is so exciting about it (liberalism), and in any case the pursuit for a coherent conception of V goes against reason­able expectation. Forcing axioms Meanwhile, forcing axioms (from the doctrine of genericity) continue to extend the frontiers of mathematics – the universe of sets is extended to form a new, wider universe; a forcing axiom called ‘Martin’s maximum’ (after the mathema­tician Donald Martin) discovered in the 1980’s, extends the universe as far as it can go. Forcing axioms fill some problematic holes in everyday mathematics; work over the past few years (including new-found uses of Martin’s maximum) by Stevo Todorcevic81 and others shows that they bestow many mathematical structures with nice properties that make them easier to use and understand. They also solve the continuum hypothesis – deeming it false by adding a new size of infinity; in the extended universe created by forcing, there is a larger class of real numbers than in the original universe defined by ZFC, which means the real numbers of ZFC constitute a smaller infinite set than the full continuum. If the point is to find the most fruitful seeds of mathematical discovery (as op­posed to the grains of truth that yield the most pristine mathematical universe), then forcing axioms emerge as a possible contender for addition to ZFC. Never­theless, according to Todorcevic, “it is difficult to justify Martin’s maximum as an axiom from a philosophical point of view; it can only be justified in terms of the influence it has had on the rest of mathematics.” Woodin considers that it is a rich theory for understanding structures in classical mathematics but “it’s not clear how it would lead to a better understanding of infinity,” which (for him) is what set theory is about.82 80 Abstract for “The Joint Quest for Absolute Infinity and the Continuum – From Cantor to Woodin”, Series of (three) lectures by Hugh Woodin at the University of Turku, May 9-11, Finland 2019. 81 Stevo Todorcevic is a mathematician at the University of Toronto, and the French National Center for Scientific Research in Paris. 82 Paragraph paraphrased from Wolchover, “To Settle Infinity Dispute, a New Law of Logic”. Large cardinal hierarchy As already mentioned, large cardinal axioms are not provable from ZFC. Nor is it known whether the existence of such sets is even consistent with ZFC, with one exception; a basic theorem proved by Kunen (1971)83 shows that the exist­ence of a Reinhardt cardinal is inconsistent with ZFC. The proof relies on the axiom of choice, and it is not known if the existence of a Reinhardt is inconsist­ent with ZF (i.e. ZFC minus the axiom of choice).84 But large cardinal axioms can be well-ordered by their consistency strength, thus yielding the large cardinal hierarchy.85 Also, they can decide some of the problems which are unsolvable from ZFC alone, for example, the existence of a measurable cardinal implies that the universe of sets is larger than Gödel’s constructible universe.86 (This is because the existence of a measurable cardinal implies the existence of Silver’s 0# (zero sharp),87 where0# is the set of true formulae about indiscernibles and order-indiscernibles in the constructible universe L. The existence of 0# howev­er, like that of a measurable cardinal, is unprovable from ZFC).88 But they cannot decide the continuum hypothesis. (This is because the contin­uum hypothesis can be made true or false by forcings (Cohen) which are small in the sense that they only act on the lower (infinite) levels of the cumulative hierarchy, but these small kinds of forcings do not act on the level of a measura­ble cardinal or beyond, therefore the existence of a measurable (or any stronger large cardinal) simply has no bearing on the truth value of the continuum hy­pothesis).89 83For the actual statement of Kunen’s theorem, see Akihiro Kanamori, “The Higher Infinite”, Perspectives in Mathematical Logic, Springer-Verlag, Berlin 1994. This is Ref [5] in Woodin, “The Transfinite Universe”. 84 Rittberg, “How Woodin Changed his Mind: New Thoughts on the Continuum Hypothesis”. 377 85 One theory has a higher consistency strength than another, if the consistency of the former implies the consistency of the latter. The order of consistency strength in the large cardinal hierarchy is not necessarily the same as the order of the size of the smallest witness to a large cardinal axiom. For example, the existence of a huge cardinal is much stronger, in terms of consistency strength, than the existence of a supercompact cardinal, but assum­ing both exist, the first huge is smaller than the first supercompact. 86 Rittberg, “How Woodin Changed his Mind: New Thoughts on the Continuum Hypothesis”. 87 0# (zero sharp) was first introduced in Silver’s 1966 thesis, which was later published as: Jack H. Silver, “Some Applications of Model Theory in Set Theory”, Annals of Pure and Ap­plied Logic 3 (1/1971), pp. 45–110. 88 Woodin, “In Search of Ultimate-L”, p. 3. 89 Rittberg, “How Woodin Changed his Mind: New Thoughts on the Continuum Hypothesis”. Yet the development of set theory after Cohen has led to the realization that formally unsolvable problems have degrees of unsolvability which can be ex­actly calibrated by large cardinal axioms, which has proven to be useful. As a consequence of this calibration, it has been discovered that in many cases very different lines of investigation have led to problems whose degree of unsolva­bility is the same. Thus the hierarchy of large cardinals emerges as an intrinsic, fundamental conception within set theory.90 An interesting example in this context is that large cardinal axioms do not exist in the universe of number theory, yet their existence generates new truths of that universe; more precisely, assuming the large cardinals to be true, one can infer as true specific statements of number theory which arguably cannot other­wise be proven. How then can the number theorist account for these truths? The skeptic of course will object here that all one is asserting about large cardinal axioms is formal consistency, and that such consistency claims cannot be mean­ingfully made since the concept of the universe of sets is so ‘incomplete’ that it is a complete fiction.91 For so long as large cardinal axioms remain independent set-theoretical statements, this skeptical position cannot be refuted. Exploring the universe from below (the local argument) ‘Universists’ however point to the increasingly stable consequences of the large cardinal hierarchy as evidence that we are on the right track towards final an­swers to set-theoretical questions.92 Large cardinal axioms not only fit together, but all extensions of set theory ‘considered in practice’ seem to be equiconsist­ent with large cardinal axioms; their arithmetic consequences are compatible, and (eventually) so are their projective consequences, thus providing, for exam­ple, a coherent theory of the reals (i.e. of V.+1). This is illustrated by the axiom of determinacy (an assertion about infinite games and pivotal to current research in set theory) that turns out to be equiconsistent with the assertion that there are infinitely many Woodin cardinals.93 The axiom of determinacy (with extrinsic justifications) and ‘there exist Woodin cardinals’ (with intrinsic justifications) are proved to coincide. “This sort of convergence of conceptually distinct do­ 90 Woodin, “The Transfinite Universe”. 91 Ibid. 92 Hampkins, “The Set-Theoretic Multiverse: a Natural Context for Set Theory”. 93 Caicedo, reviewing Woodin’s “The Realm of the Infinite”. mains is striking and unlikely to be an accident,” as Peter Koellner94 wrote in 2006.95Yet despite these successes, with the amount of mathematical effort and development which was required – which yielded the correct axioms for a cer­tain part of the universe of sets, i.e. V.+1 (in which the real numbers appear), and noting that this is just an infinitesimal fragment of the universe of sets V, then the prospects for understanding V itself to this same degree (or even just V.+2 in which all sets of reals appear, and which would reveal whether the continuum hypothesis is true) seem daunting.96 Probing the ‘whole’ universe of sets (the global argument) Taking another track, the successes of the inner model program (from the doc­trine of constructibility) were encouraging, and led to the realization that the large cardinal hierarchy is a very ‘robust’ notion.97 The inner model program is the detailed study of large cardinal axioms, by seeking generalizations of the definition of the axiom of constructibility (Gödel) which are compatible with these axioms (such as the axioms for measurable cardinals and beyond),98 i.e. by enlarging the constructible universe (denoted by L) to accommodate targeted large cardinal axioms. The axiom of constructibility, i.e. V = L, severely limits the nature of infinity, but the constructible universe L, does have the benefit of being pristine and analyzable (unlike the cumulative hierarchy of sets V), and enlargements of L, being L-like, offer the same advantage. Inner models, i.e. enlargements of L, are constructed using both constructible elements and extender elements as defining parameters, where an extender is derived from the ‘elementary embedding’ which is closely linked to the large cardinal under consideration.99 Large cardinals can be defined in terms of elementary embed­dings, i.e. an embedding of V itself into some inner model M,100 and extenders are non-trivial fragments of the elementary embedding.101 More precisely, one 379 94 Peter Koellner is a philosophy professor specializing in mathematical logic at Harvard Uni­ versity. 95 As quoted by Zhaokuan Hao, (Fudan University, China) in “Gödel’s Program and Ultimate L”, Presentation at the National University of Singapore, September 2017. 96 Woodin, “Strong Axioms of Infinity and the Search for V”. 97 Woodin, “The Transfinite Universe”. 98 Ibid. 99 Rittberg, “How Woodin Changed his Mind: New Thoughts on the Continuum Hypothesis”. 100 Where M is a transitive model of ZFC containing all the ordinals, and an elementary em­ bedding is a function (not the identity) between models which preserves truth. Ibid. 101 Woodin, “The Transfinite Universe”. has to require that the large cardinals of the inner models which constitute the solution have some form of ancestry in the large cardinals of V. As such, the extender models are constructed as refinements of V, which preserve enough extenders from V to witness that the large cardinal axiom holds.102 By the year 2000, the inner model program had been unconditionally extended to the level of Woodin cardinals – which are limits of strong cardinals, and con­ditionally extended to the level of superstrong cardinals.103 But the incremental nature of the program seems to be an absolutely fundamental aspect: each new construction of an enlargement meeting the challenge of a specific large cardi­nal axiom comes with a theorem that no stronger large cardinal axiom can hold in that model. Since it’s very unlikely that there could ever be a strongest large cardinal axiom, this methodology seems unlikely by its very nature to yield an ultimate enlargement of the constructible universe, i.e. to ever succeed in pro­viding the requisite axiom for clarifying the conception of the universe of sets.104 The pursuits for a new axiom for set theory from below (analyzing the set-theo­retic universe fragment by fragment) looked daunting, and from above (a form of extension of ZFC based on the whole universe) seemed hopeless.105 Then something happened… Circa 2010 the situation changed dramatically, with the “remarkable and com­pletely unexpected” discovery within the inner model program, that at a specif­ic, critical stage in the hierarchy of large cardinal axioms, there is a breakaway point: at the level of one supercompact cardinal, the inner model construction must yield an enlargement which is compatible with all stronger large cardinal axioms, i.e. it must yield an ultimate enlargement of the inner model L.106 The 380 requirement for enough suitable extenders from V to witness that the supercom-pact cardinal holds, necessarily must generate ‘phantom’ witnesses (extenders) for all large cardinals stronger in the hierarchy.107 This is now formulated in the 102 Ibid. 103 Woodin, “In Search of Ultimate-L”, p. 1. 104 Woodin, “Strong Axioms of Infinity and the Search for V”. 105 Rittberg, “How Woodin Changed his Mind”. 106 Woodin, “Strong Axioms of Infinity and the Search for V”. 107 Woodin, “The Transfinite Universe”. ‘universality theorem.’108 If such an inner model could be found, then it would accommodate all large cardinals (that are consistent with ZFC), which amounts to a paradigm shift in the whole conception of inner models.109 Is this not a mathematical event? The entire framework for the inner model program was transformed from a pro­gram for the incremental understanding of large cardinals into a program for perhaps understanding V itself, and it led to the isolation of a candidate for an axiom that V is an ultimate version of L. W. Hugh Woodin is on the brink (in a decadal sense) of demonstrating a transcendent version of Gödel’s axiom of constructibility for the universe of sets; but where Gödel’s axiom is minimal (i.e. the mimimum possible universe of sets containing all ordinals), Woodin’s axi­om is a maximal statement about the nature of the universe of sets. Everything now hinges on the fate of the ‘ultimate-L’ conjecture which, if true, provides an absolutely compelling case that V = ultimate-L.110 Strategic-extender models The various approaches to inner model theory have to be fundamentally al­tered to provide the solution to the inner model problem for one supercompact cardinal.111 The ultimate-L program introduces a new class of model for V – a different, previously unknown class of extender model called the strategic-ex­tender models, which can no longer be layered (as in the non-strategic case);112 based on the strategic-extender models, one can formulate the axiom, V = ulti­mate-L, without referring to the detailed fine-structure theory of these models, or even using the definition of the structures yielding the levels of these models. The point (here) is that, in the context of a proper class of Woodin cardinals, there are naturally defined approximations to ultimate-L and the collection is rich enough to make a definition of the axiom, V = ultimate-L, possible without 381 specifying the detailed level-by-level definition of ultimate-L. Woodin notes that 108 Woodin, “Generalizing Gödel’s Constructible Universe: Ultimate L”. 109 Rittberg, “How Woodin Changed his Mind”. 110 Woodin, “Beyond the Age of Independence by Forcing,” Presentation at the Chinese Math­ ematical Logic Conference”, May 20, 2017. 111 Woodin, “In Search of Ultimate-L”, p. 1. 112 Ibid, p. 93. this situation is analogous to being able to formulate the axiom, V = L, without specifying the definition of L.113 Strategic-extender models were developed by generalizing on the earlier suc­cess in understanding V.+1 and the projective sets – which provided a good the­ory of the reals (assuming the existence of Woodin cardinals). They make use of the class of all hereditarily ordinal definable (HOD) sets, i.e. the class containing all those sets which are definable from ordinal parameters.114 Originally defined by Gödel,115 Woodin gives an equivalent reformulation of the class HOD which highlights it as some sort of merge of the definitions of the cumulative hierarchy V and of the constructible universe L. For the power-set in L, the subset is logi­cally definable from parameters in the set, whereas in HOD, the subset is defin­able without parameters. He also notes here that there is a remarkable theorem of Vopenka which connects HOD and Cohen’s method of forcing,116 and which illustrates why Cohen’s method is so central in set theory (and for reasons other than simply establishing independence results).117 Ultimate-L program The program to prove the ultimate-L conjecture is a 4-stage project. Stage 1 has been completed (~2013) and it survived its first critical test; this was the con­struction of an L-like weak extender model for the finite levels of supercompact­ness. Then, “in an unexpected twist,” Woodin reported, “the remaining three stages collapsed into a single final stage, and subsequent analysis has provided the key clues as to how to proceed.”118 With many evidences leaning toward a positive answer for the ultimate-L conjecture, Woodin is quite confident of the outcome. However, as he wrote in March 2017, “given the series of unexpected events to date on this subject, an abundance of caution seems prudent here.”119 113 Ibid, pp. 94–95. 114 Woodin, “The Transfinite Universe”. 115 More precisely, the definition of the class HOD originates in remarks of Gödel at the Princ­ eton Bicentennial Conference in December, 1946. According to Woodin, the first detailed reference appears to be Ref [8] in Ibid. Theorem (Vopenka). For each set G . Ord, if G . HOD, then HODG is a Cohen extension of HOD. 117 Woodin, “Strong Axioms of Infinity”. 118 Woodin, “Beyond the Age of Independence by Forcing”. 119 Woodin, “In Search of Ultimate-L”, p. 3. As of May 2017 there was a draft of a proof and, according to Woodin (at that time), “things look promising.”120 In 2019, Woodin said that the ultimate-L conjecture reduces the entire post-Co­hen debate on set theoretic truth to a single question which must have an an­swer – true or false, i.e. it cannot be meaningless. Either the conjecture is true, in which case the axiom, V = ultimate-L, is very likely the key missing axiom for V. Or it is false, in which case the program to understand V, by generalizing the (local) success in understanding V.+1 and the projective sets, fails.121 The axiom V = Ultimate-L (conjectured) ·Large cardinal axioms can (and therefore do) exist in the universe given by ultimate-L. This is the key new feature which sets it apart from all previous generalizations of L. Ultimate L is mathematically analyzable – in this sense it is L-like – and yields a much deeper understanding of the large cardinal axioms; it identifies more precisely the transition for large cardinal axioms from the possible to the impossible (i.e. the demarcation of consistent and inconsistent sets), and pro­vides a framework for a continuing evolution in the understanding of this tran­sition.122 One obtains a new generation of inconsistency results for the large car­dinal hierarchy (in the setting where the axiom of choice fails) which includes a mild strengthening of Reinhardt cardinals, and also Berkeley cardinals.123 · The conjectured axiom affirms the ZFC axiomatic, and in particular the axi­om of choice. 383 It recognizes all large cardinal axioms (both known, and those yet to be discov­ered) consistent with ZFC, but will eliminate essentially all the large cardinal ax­ioms known to contradict the axiom of choice.124 For example, the existence of 120 Woodin, “Beyond the Age of Independence by Forcing”. 121 W. Hugh Woodin, “The Continuum Hypothesis”, Presentation at the University of Münster (WWU), Germany, June 20, 2019. 122 Woodin, “The Transfinite Universe”. 123 Woodin, “In Search of Ultimate-L”, p. 3. 124 Ibid. weak Reinhardt cardinals – which was already proven to contradict ‘choice’ by the well-known Kunen inconsistency (1971)125 – is actually refuted here.126 More fundamentally, it reveals deep connections between large cardinal axioms and proving instances of the axiom of choice.127 One also obtains, in the context of the axiom of choice, that what seem like natural generalizations of axioms of defin­able determinacy, are false if sufficient large cardinals are assumed to exist.128 · The continuum hypothesis is decided. Surprisingly in fact, it holds,129 which runs counter to the intuitive expectations of notable Platonists, from Gödel, to Woodin himself, to Badiou. The type of mul­tiplicity of the power-set is decided and with a minimum of excess; the contin­uum is the second infinity. In fact the axiom renders Cohen’s method of forcing completely useless for establishing independence from the resulting conception of the universe of sets;130 it settles all questions about ‘small’ sets which have been shown to be independent by that method. This is because the axiom V = ultimate-L (or V = L, for that matter) strongly couples the width of the universe of sets V, to its height. Woodin explains that V has two dimensions, width and height; the height is a large cardinal issue, and the width is – how many reals do you have, i.e. how many sets of reals. V = ultimate-L binds the width to the height; if you change the width, you cannot recover the axiom without changing the height. Cohen’s method changes the width, but preserves the height of V, and that’s why it is useless – it cannot affect the axiom.131 Woodin further conjectures that this axiom is true, i.e. that it will eventually be validated on the basis of accepted and compelling principles of infinity, exactly as the axiom of projective determinacy was validated,132 and the truth of which 384 only became evident after a great deal of work, i.e. mathematical results proven 125 Kenneth Kunen, “Elementary embeddings and infinite infinitary combinatorics”, J. Sym­ bolic Logic 36 (1971), pp. 407–413. This is Ref [7] in Woodin, “The Transfinite Universe”. 126 Woodin, “The Realm of the Infinite”. 127 Woodin, “Strong Axioms of Infinity and the Search for V”. 128 Woodin, “In Search of Ultimate-L”, p. 3. 129 Woodin, “Generalizing Gödel’s Constructible Universe: Ultimate L”. 130 Ibid. 131 Woodin, “The Continuum Hypothesis Set Theory”, Presentation at the California Institute of Technology(Caltech), February 23, 2019. 132 Woodin, “Strong Axioms of Infinity and the Search for V”. by the set theoretic community. This demonstrated (for him) that the discov­ery of mathematical truth is not a purely formal endeavour.133 Using the phrase “truth beyond our formal reach,” Woodin maintains that there’s a component in the evolution of our understanding of mathematics which is not formal, that there is mathematical knowledge which is not entirely based in proofs.134 Some meta-ontological implications Set theory’s contemporary indeterminacy – which renders the impossibility of regarding it as an absolute referent – stems from the independence of large cardinal axioms, and the undecidability of the continuum hypothesis. The non-intuitive nature of the axiom of choice, which is “without any known pro-cedure,”135 doesn’t help either.136 There are many implications raised not just by the fact, but also by the manner of the resolution of these issues by the axiom V = ultimate-L, if indeed it is demonstrated to be true… . The conjectured axiom, V = ultimate-L proposes to complete the system of axioms which define the universe V; it offers a coherent conception of the universe of sets, presenting an unambiguous concept of the transfinite uni­verse. Addressing the questions of independence with respect to large cardi­nal axioms and the continuum hypothesis, it resolves the contemporary cri­sis of formalization. The resulting conception of V is maximal and univocal, i.e. not in the least indeterminate or relative; it thereby affirms the universe (or non-pluralist) orientation, and facilitates the adoption of set theory as an absolute ontological referent. . The skeptical assertion that V is meaningless (i.e. a fiction) is undermined by the articulation of a coherent and ultimately complete (but not closed) 385 set theory. V = ultimate-L provides a framework for understanding and ana­ 133 Rittberg, “How Woodin Changed his Mind”. 134 Woodin, “The Continuum Hypothesis Set Theory”. 135 Badiou, Being and Event, p. 228. 136 The axiom of choice has long been subjected to criticism. It is a purely existential axiom in that it asserts the existence of a function of choice without providing any means of con­structing it; it is impossible to specify any rule that might guide an infinite set of arbitrary choices. Yet it has been proved that it does not introduce any contradiction. Moreover, the axiom of choice is required to establish that every set can be well-ordered. lyzing large cardinal axioms as true axioms about the universe of sets, and no longer as independent set-theoretical statements. Large cardinal axioms exist in the universe given by V = ultimate-L, which allows the number theo­rist, for example, to account for new truths generated in the universe of num­ber theory by their existence. Such examples serve to ground a refutation of the skeptic. . The universe given by V = ultimate-L – unlike the constructible universe (giv­en by V = L) – recognizes all large cardinals (both known, and those yet to be discovered) consistent with ZFC, but refuses an incoherent “prodigality of being.”137 The impossibility, heretofore, of determining consistency in large cardinals is surpassed, so that an axiomatic realized in the ultimate-L model transcends the fear of incoherence, the tension of uncertainty. As such, it offers a coherent way out of the constructible universe. . V = ultimate-L, affirms the axiom of choice. In fact ‘choice’ emerges as a crucial axiom for determining formal consistency in large cardinal axioms, which therefore are no longer independent from the resulting conception of sets. For Badiou, the axiom of choice provides the precise concept of the be­ing (as opposed to the act) of subjective intervention; he associates it with a ‘principle of infinite liberty.’138 But in the contemporary world, freedom is generally associated with a principle of independence. Perhaps freedom will yet come to be re-defined (positively) as the expression of choice within a co­herent yet unrestricted axiomatic, rather than (negatively) as the exemption (independence) from any formalization. . Woodin’s axiom affirms the truth of the continuum hypothesis. Woodin origi­nally set out to falsify the continuum hypothesis; between 1999 and 2004, he presented three very similar yet different arguments against the continuum hypothesis, in each case, exploring the universe from below.139 But in the course of his endeavour to find the true axioms for V.+2 – in which the contin­uum hypothesis would have an answer – he was obliged to start a program in inner model theory which led to the completely unexpected discovery of 137 Badiou, Being and Event, p. 283. 138 Hallward, “On the Development of Transfinite Set Theory”, p. 339. 139 Rittberg, “How Woodin Changed his Mind”. the breakaway point in the hierarchy of large cardinal axioms, and the sub­sequent isolation of an axiom that V = ultimate-L. The truth of the continu­um hypothesis is something demonstrated by this axiom, but the resulting model was not the original expectation or motivating goal. Alain Badiou’s objection to the continuum hypothesis as true (ever since he was working on what was to become Being and Event) stems from a consideration that adopting it leads to a restriction of set theory’s axiomatic powers, and is therefore opposed to the principle of maximality. Yet, as he says, there were rea­sons for hesitating, since adopting it would introduce no contradiction, and the mere negation of the hypothesis doesn’t tell us what type of infinity the conti­nuum really is either.140 Perhaps Badiou’s objection to the truth of the conti­nuum hypothesis stems more particularly from an objection to the restrictions (minimality) of the constructible universe – which is the only situation (here­tofore) which has demonstrated that truth. But Woodin’s axiom is a maximal axiom for V; it accommodates all genuine large cardinal axioms, including that of a measurable cardinal which therefore implies the existence of Silver’s 0#and brings the doctrine of indiscernibles (generic) into the resulting conception of V. The key point here is that ultimate-L is not at all L, that in fact the inner model approach has to be altered – in a fundamental way – to provide the solution for one supercompact cardinal; ultimate-L is not simply an L enlarged to accom­modate large cardinal axioms, but (having developed from a coherent theory of the reals) an L re-cast such that the resulting model can accommodate the reals which are overlooked (indiscernible) in L. The normalization of the state – its minimum of excess (represented by the truth of the continuum hypothesis)141 – which was primitively achieved in the con­structible universe, is accomplished here in the context of an axiomatic which is 387 capable of challenging a much greater indeterminacy, that of consistency with respect to large cardinals; genuine large cardinals are admitted due to their (de­monstrable) formal consistency and false large cardinals are refuted. It seems then that the state proves to be unduly excessive only in the absence of a coher­ent formal framework; beyond the narrow constructible universe (which can’t admit even a measurable cardinal) and before an ultimate universe (which ad­ 140 Badiou, “Towards a New Thinking of the Absolute”, pp. 17–18. 141 Badiou, Being and Event, p. 296. mits all large cardinals consistent with ZFC), the specter of independent large cardinal axioms coincides with an excess of the state which is immeasurable. Once the formal framework has the capacity to identify and accommodate all genuine large cardinals, the state seems to settle into normalcy and lose its (po­tentially) exorbitant power of oppression. . The axiom completely negates the ramifications of Cohen’s method, leading Woodin to present a vista for set theory “beyond the age of independence by forcing.”142 Badiou’s maxim that “there are only bodies and languages, except that there are truths,” is based on the existence of the pure theory of the multiple as an absolute ontological referent. In this respect, he writes “it is that theory, and not the general principle of the existence of truths, that it is a matter of defending here.”143 Set theory as an absolute referent implies a system of axioms fully iden­tified such that no important, significant, useful property of sets will remain undecidable; any such theory has to “banish the specter of undecidability as demonstrated by Cohen’s method of forcing.”144 It is noteworthy here that Alain Badiou recently remarked that he now accepts that his definition of universality – as genericity (due to Cohen) – in Being and Event is purely negative.145 He added that “genericity signifies, exclusively, that universality is not reducible to properties in the situation,” and that maybe it is impossible to solve the question of universality with a negative definition of this kind. He said that universality does not just consist of negative genericity, but also of something which has a form of relationship to an affirmative concept of absoluteness which he proposes in The Immanence of Truths.146 Badiou also 388 notes that ‘elementary embedding’ somehow plays a similar role to ‘forcing,’147 in the sense of as a truth procedure. He says that the absolute class V is embed­ 142 Woodin, “Beyond the Age of Independence by Forcing”. 143 Badiou, “Towards a New Thinking of the Absolute”, p. 13. 144 Woodin, “Strong Axioms of Infinity”. 145 Badiou, Presentation at the Graduate Workshop on Being and Event, Columbia University, New York, October 24, 2017, subsequently published in Sometimes We Are Eternal, p. 58. 146 Badiou, referring to the English translation (yet unpublished) of L’Immanence des Vérités. 147 As reported by Berankova, in “The Immanence of Truths: the Absolute between the Singu­ lar and the Universal”. ded in one of its ‘attributes’ (Spinozist notion), i.e. in the transitive subclass M, which is thereby transformed into a model of V. There is a relation between V and M but this is not the identity, i.e. V will remain different from the attribute M and contain sets that we cannot find in M. Thus, to say that a ‘truth’ is abso­lute does not mean that a ‘truth procedure’ is absolute; rather, the absolute is embedded in its attribute. Ultimate-L – with strategic-extender models and the class HOD – seems to em­brace these truth procedural conditions. The subset in HOD is definable without parameters, and HOD is connected to Cohen’s method of forcing (Vopenka), so it seems that forcings are somehow already inscribed in ultimate-L giving a wid­ening of V. But this widening is strongly coupled with a heightening of V to the level of a supercompact cardinal and (thus) beyond, yielding an inner model which is very ‘close’ to V. Forcing, i.e. the Cohen symptom of a sort of legal insubordination to a restricted (conservative) formalization – which is effective and necessary to demonstrate the insufficiency of ZFC – does not itself propose a solution to the crisis (of ZFC) which it initiated, but the indiscernibles which it indicates must be incorporat­ed in any such solution. It only becomes redundant as an intervention when the formalization is sufficient. This is also the case with the ‘attributes of the abso­lute,’ i.e. large cardinals, which are indicated by the procedure of elementary embedding. Alain Badiou has not written about ultimate-L, but then he generally does not write about what is only conjecture in set theory – he waits for results and their establishment among the set theoretical community. Woodin’s axiom however seems to present the possibility of a new framework for infinite thought which 389 is universally consistent in the sense indicated by Badiou; it completes a system of axioms which define a model which is very ‘close’ to V with the capacity to ac­commodate all of the reals, together with all large, demonstrably consistent ‘at­tributes of the absolute’ (large cardinal axioms). It seems to embrace a principle of maximality,148 and it affirms a ‘principle of infinite liberty’ (axiom of choice). 148 This is in the sense that “any intellectual entity, whose existence can be inferred without contradiction from the axioms that prescribe it, exists by that very fact,” Badiou, “Towards a New Theory of the Absolute”, p. 11. Implications for the symbolic, for the subject The argument presented in this paper is that humanity’s symbolic organization, or a symbolic, requires a credible ontological referent. The breakdown of the hi­erarchical symbolic (of tradition) accompanied the ruin of the ontological sup­position – of the being of the one – on which it relied. The modern revelation of the pure multiplicity of being as such cannot be undone. A new symbolic (if such is possible) has to recognize the non-being of the one, which means that it can only be non-hierarchical, which is to say, egalitarian. An egalitarian symbolic therefore requires a credible ontology of pure (infinite) multiplicity as referent; mathematical set theory is proposed (Badiou). In Being and Event, Badiou wrote that metaontology “serves as an unconscious framework for every orientation in thought.”149 In this respect, the commonplace doubt that an ultimate completion of set theory can be articulated, corresponds to a general conviction of impossibility – with respect to an egalitarian symbolic – which has dominated since the 1980’s; such a conviction equates a symbolic solely with hierarchy, and resigns itself to (or lauds) the reign of the a-symbolic, (unless it attempts a retrogression to hierarchy). But set theory – with the axiom V = ultimate-L (conjectured) – now affords the real possibility of a coherent, ultimately complete, and therefore credible, ontol­ogy of multiplicity. As such it proposes the possibility of an egalitarian symbolic. This is not to suggest that an abstract formalization in itself constitutes a con­crete new symbolic, but a coherent ontology of multiplicity is an a priori require­ment, i.e. a condition of possibility. The articulation – even as conjecture – of a coherent, rational thinking of infinite multiplicities challenges the dominant 390 conviction of impossibility with respect to an egalitarian symbolic;150 it re-opens the question of a symbolic for an age of multiplicity. It implies that the idea of an egalitarian symbolic may be more than just a utopian dream. 149 Badiou, Being and Event, p. 282. 150 It is of course still quite possible that ultimate-L will fail, and that an optimal axiom (or otherwise concluding axioms) for V will eventually take a form other than that of an inner model. Nevertheless, the significance of the supercompact cardinal and the universality theorem cannot be ignored. This can modify, or even flip, a subjectivity; a wishful orientation (unfounded desire), or even a hopeless one, has new reason to hope (desire combined with expectation). A condition of possibility (as opposed to impossibility) gives a subjective motivation to act (rather than hopelessly react) in the concrete world. Ontology may indicate how to proceed: such is Badiou’s endeavor, with the adoption of forcing (in Being and Event) and elementary embedding (in L’Im­manence des Vérités) as the being of truth procedures. It is by such “arduous and protracted procedures” that “the new pulls itself away from the old.”151 But the advance from doubt, or beyond conviction, to reasonable expectation that we can have access to a ‘place’ of thinking where such truth procedures can be decided, gives impetus to the subject to pursue them in the objective world, whenever chance arises, or the ‘event’ happens. More speculatively (and this is an impressionistic response to accounts of the remarkable developments in set theory by Hugh Woodin and colleagues), ontol­ogy might even suggest that recent phenomena such as the generation of mass testimony (‘target witnesses’) by activists in, for example, the ‘Me Too’ or the ‘Black Lives Matter’ movements, to challenge the inherent male/white/power bias in the existing legal, social and economic systems, will have effect. Or that the advocation of ‘choice’ by activists, for instance for same-sex marriage or abortion, is valid. More tantalizingly, the deep awareness of the requirement for social and economic justice by contemporary (genuine) climate activists,152 suggests the inadvertent generation of non-targeted or ‘phantom’ witnesses. All of these orientations have their correspondences in (speculative) ontology in the present moment. Moreover, the development of the axiom V = ultimate-L demonstrates how the unexpected discovery (encapsulated here in the universality theorem) can 391 change everything. It harnesses and combines the efforts of the ‘universists’ in the three camps of set theory, i.e. the inner model program, the generic and the large cardinal doctrines, using the resources of all three but privileging none. Those aspects of mathematical truth discovered separately in each of the three 151 Zachary Luke Frazer in the introduction to Badiou, The Concept of Model, re.press, Mel­ bourne 2007, p. xvi. 152 This is captured by Naomi Klein in “This Changes Everything: Capitalism vs. the Climate” (Simon and Schuster, New York 2014) – again, an intriguing title. doctrines are addressed and deployed, which suggests that in other domains perhaps those of a shared conviction, but opposing methods, might engage with each other’s work. The subject and reversal of exile Before devising the mathematical ontology, Alain Badiou wrote a theory of the subject (1982).153 While he subsequently revised his thinking,154 some of those teachings have resonance here; in particular, his comparison of the subjective figures in Greek tragedy according to Sophocles (in Antigone), and to Aeschylus (in the Oresteia): 155 In Antigone the theme is ‘reversal’ by which the way of the new is barred; it is not governed by any new right. The crucial point is the retrogression towards the origin. This is native reversal, reversal from the formal (what is learned) to the originary native form. It leaves no way out for the subject (Antigone) – wandering below the unthinkable – other than death. This subjective figure must always pre­vail in times of decadence and disarray. Such a subjective figure seems appropriate for our times; it captures the gener­al disorientation which prevails under the triumphant reign of the a-symbolic, with no prospect for any new right (a symbolic in an age of multiplicity) but only the retrogression to finitude by which the way of the new – actual infinity – is barred. This leaves no way out for the subject other than nihilism. In the Oresteia the theme is the ‘rupture’ that allows for the advent of the new; it is a new coherence instituted by the interruption of the repetitive series that made up the whole previous social order. The key point is the interruption of the power of the origin. Orestes (the subject) demands a discussion based on facts; he stands firm and does not give in to the murderous seduction of the Erinyes. There is no return to order, and the subject is prepared to sustain exile, exile-without-re­turn. Reversal of exile will not take place without the re-composition of a different Badiou, Théorie du Sujet, Éditions du Seuil, Paris 1982. 154 Badiou, Logics of Worlds, especially the “Formal Theory of the Subject”, pp. 43–89. 155 Badiou, Theory of the Subject, Bloomsbury Academic, London and New York 2013, pp. 161–168. order – which is not simple. Neither will it take place without the structural an­chorage of the native – indeed it is from the materialistic impasse of the latter that the practical existence of the former proceeds. Here the subjective figure seems to correspond to the universe orientation (or faithful subject) with respect to the situation of set theory; it captures the in­terruption (the Cohen event) of the repetitive series (V = L) that made up the previous natural order. There is no return to order, and the subject is prepared to sustain exile without return (set theory’s indeterminacy). After more than 50 years of roaming (sustained exile) – refusing the native rever­sal of the skeptic, and overcoming the exile-without-return of the generic (due to Cohen) – set theory now affords the possibility, not of a return to order, but the recomposition of a different order (V = ultimate-L). It is only the insistence of the interruption (forcing/generic) which, in the end, allows for the advent of a new right capable of completely recomposing the whole logic of the decision. Risking the hyperbole of understatement, this is not simple. Moreover, the re­versal of exile will not take place without the structural anchorage of the native (“the anchoring of the ordinals in being”156); ultimate-L, as a new, consistent universe capable of accommodating all cardinals, is still structurally anchored in L which is the minimum possible universe of sets containing all ordinals, but with L re-shaped, re-composed. Indeed it is from the materialistic impasse of the latter (i.e. the constructible power-set is never the set of all subsets of the set), that the practical existence of the former proceeds (by generalizing the suc­cess in understanding the reals, i.e. V.+1). Finally – faithful to the conviction that “even though we have to return, and it is this return that makes the subject [set theory] – there can arise an enlightened overcoming of what no longer entails any return.”157 393 References Acheronta, Movebo, “Towards a New Thinking of the Absolute”, Crisis and Critique 1 (2/2014) Badiou, Alain, Being and Event, trans. Oliver Feltham, Continuum, London and New York 2007 156 Badiou, Number and Numbers, p. 83. 157 Badiou, Theory of the Subject, p. 168. — L’Immanence des Vérités, Fayard, Paris 2018 — Logics of Worlds, trans. Alberto Toscano, Continuum, London and New York 2009 — Manifesto for Philosophy, trans. Norman Madarasz, SUNY Press, Albany, New York 1996 — Number and Numbers, Polity Press, Cambridge UK and Malden USA 2008 — Presentation at the Graduate Workshop on Being and Event, October 24, Columbia University, New York 2017 — “Reflections on the Recent Election”, Verso blog, November 15, 2016 — Theoretical Writings, trans. Ray Brassier and Alberto Toscano, Bloomsbury Academ­ic, London and New York 2014 — The Concept of Model, trans. Zachary Luke Fraser and Tzuchien Tho, Re.press, Mel­bourne 2007 — Théorie du Sujet, Éditions du Seuil, Paris 1982 — Theory of the Subject, trans. Bruno Bosteels, Bloomsbury Academic, London and New York 2013 — “True and False Contradictions of the Crisis”, Verso blog, May 29, 2015 Berankova, Jana Ndiaye, “The Immanence of Truths: the Absolute between the Singular and the Universal”, Presentation at the Conference “Thinking the Infinite”, April 11, Prague 2018 — “The Attributes of the Absolute and Alain Badiou’s Response to Spinoza”, in Some­times, We Are Eternal, Suture Press, Lyon 2019 — Presentation at the Graduate Workshop with Alain Badiou on The Immanence of Truths, November 13, Columbia University, New York 2019 Blake, Terence, “Badiou’s The Immanence of Truths: Introduction (sketch)”, available at: https://terenceblake.wordpress.com/2017/01/19/badious-the-immanence-of-truths­introduction-sket — “My Path Through Badiou’s The Immanence of Truths”, available at: https://terence blake.wordpress.com/2018/10/06/my-path-through-badious-the-immanence-of-truths 394 Caicedo, Andrés Eduardo, “Review of Woodin’s ‘The Realm of the Infinite’”, submitted to Mathematical Reviews/MathSciNet, MR 2767235, June 26, 2012 Cheng, Eugenia, Beyond Infinity, Profile Books, London 2018 Hallward, Peter, Badiou: A Subject to Truth, University of Minnesota Press, Minnesota 2003 Hampkins, Joel David, “The Set-Theoretic Multiverse: a Natural Context for Set Theory”, (MR2857736 (2012h:03002)) Hao, Zhaokuan, “Gödel’s Program and Ultimate L”, Presentation at the National Univer­sity of Singapore, September 2017 Kanamori, Akihiro, “The Higher Infinite”, in Perspectives in Mathematical Logic, Spring-er-Verlag, Berlin 1994 Klein, Naomi, This Changes Everything: Capitalism vs. the Climate, Simon and Schuster, New York 2014 Kunen, Kenneth, “Elementary embeddings and infinite infinitary combinatorics”, Jour­nal of Symbolic Logic 36 (1971) Rittberg, Colin J, “How Woodin Changed his Mind: New Thoughts on the Continuum Hy­pothesis”, Archive for History of Exact Sciences 69 (2016) Silver, Jack H, “Some Applications of Model Theory in Set Theory”, Annals of Pure and Applied Logic 3 (1/1971) Encyclopedia.com, Symbolic, the (Lacan), available at: https://www.encyclopedia. com>psychology>symbolic-lacan Wolchover, Natalie, “To Settle Infinity Dispute, a New Law of Logic”, Quanta Magazine, November 26, 2013 (reprinted on ScientificAmerican.com, December 3, 2013) Woodin, W. Hugh, “Beyond the Age of Independence by Forcing”, Presentation at the Chinese Mathematical Logic Conference, May 20, 2017 — “In Search of Ultimate-L, the 19th Midrasha Mathematicae Lectures”, The Bulletin of Symbolic Logic 23 (1/2017) (DOI:10.1017/bsl.2016.34) — “Strong Axioms of Infinity and the Search for V”, in Proceedings of the International Congress of Mathematicians. Hyderabad, India 2010 — “The Axiom of Determinacy, Forcing Axioms, and the Nonstationary Ideal”, in De Gruyter Series in Logic and its Applications, 2nd edition, De Gruyter, Berlin 1999 — “The Continuum Hypothesis”, Presentation at the University of Münster (WWU), June 20, Germany 2019 — “The Continuum Hypothesis Set Theory”, Presentation at the California Institute of Technology (Caltech), February 23, 2019 — “The Realm of the Infinite”, in Infinity. New Research Frontiers, Cambridge University Press, Cambridge 2011 — “The Transfinite Universe”, in Kurt Gödel and the Foundations of Mathematics. Hori­zons of Truth, Cambridge University Press, Cambridge 2011 Filozofski vestnik | Volume XLI | Number 2 | 2020 | 397–409 | doi: 10.3986/fv.41.2.15 Fernando Zalamea* An Elementary Peircean and Category-Theoretic Reading of Being and Event, Logics of Worlds and The Immanence of Truths Badiou’s magnum opus, i.e. his trilogy Being and Event1, Logics of Worlds2, and The Immanence of Truths3, with all its dubious statements, deviations, and con­troversial formulas, but also with all its courage, finesse, and dedicated hard work, represents the greatest effort by a philosopher in the last thirty years to ex­tensively confront the power of contemporary mathematical invention (e.g. forc­ing, toposes, and large cardinals). The enterprise has not passed unnoticed, but has mainly attracted the attention of some strong detractors, as well as a small community of worshippers. Both extremes are unjust as regards this amazing work that now spans three decades and which should be hailed as a unique con­tribution to knowledge, and we will try here to present a simple, hopefully bal­anced, view of the trilogy. In our reading, we will use the force of Peirce’s logical architecture and of some high conceptual instances of Category Theory. In Sec­tion 1, we explore some basic methodological connections, from a very intuitive dialectical perspective between the three shutters of the trilogy. In Section 2, the dialectics are specified through Peirce’s three phenomenological categories and some adjunctions in Category Theory. Finally, in Section 3, we present some open problems related to our simplified, elementary understanding of Badiou’s work. Some dialectical forces in the trilogy The main polarities in the foundations of mathematics in the 20th century are 397 Set Theory and Category Theory. On one hand, Set Theory proposes to under­stand a mathematical object (a “set”) through its analytical decomposition: looking at the interior, the elements comprise a set. Some of the axioms of Set Theory are not simple, since they pretend to capture the structural evolution of 1 Alain Badiou, L’Etre et l’événement, Seuil, Paris 1988, translated into English by Oliver Feltham: Being and Event, Continuum, London 2008. 2 Alain Badiou, Logiques des mondes, Seuil, Paris 2006, translated into English by Alberto Toscano: Logics of Worlds, Continuum, London 2009. 3 Alain Badiou, L’Immanence des vérités, Fayard, Paris 2018. * Departamento de Matemáticas, Universidad Nacional de Colombia infinities (e.g.the Replacement Axiom, the Foundation Axiom). The size of the continuum (represented in Set Theory as the set of real numbers) even goes be­yond the usual axioms, as Cohen showed in 1963 with the invention of forcing. BeingandEventexplores the uses of Set Theory and forcing to propose a static ontology of mathematics, to be extended to general thought. On the other hand, Category Theory proposes to understand a mathematical object (e.g. “identi­ty”, “representable functor”) through its synthetic composition: looking at the exterior, the ambient relations (“aura”) comprise an identity. The axioms of Category Theory are trivial, but extremely powerful: from practically nothing, the theory captures the mathematical behaviour of apparently distant regions (logic, topology, algebra, arithmetic, differential geometry, etc.). Going further, Grothendieck’s Topos Theory (1962) axiomatises many features of set universes inter-spread with geometrical objects. Logics of Worlds explores the applica­tions of Category Theory and toposes to offer a dynamicontologyof mathemat­ics, to be extended to general thought. The singular of the first monograph contrasts with the plural of the second one: the Absolute searched for in Being and Event is contrasted with the Relative sought in Logics of Worlds. Going beyond the Relative, The Immanence of Truths comes back to the One, and deciphers some approximations of an Absolute Uni­versal, through the many layers of large cardinals. Badiou’s commitment is again unique: going painstakingly through the very technical contributions of Cantor, Cohen, Gödel, Jech, Jensen, Kanamori, Kunen, Mostowski, Ramsey, Scott, and Woodin, the many contrasting features of large cardinals are explained to the lay­man, often with a penetrating poetic language, and are related to politics, culture, and essentially to humanity at large. To be complete (but we will not, of course, re­quire such a commitment from Badiou, who has already done more than enough 398 for generations to come!), a fourth shutter (some sort of “Universels relatifs”) of the tetralogy should study the universals inscribed in Topos Theory, i.e. classifier toposes and geometric logics.4 In this way, two singulars (L’Etre, L’Immanence) and two plurals (Logiques, Universels) would produce a beautiful counterpoint weaving of the contemporary mathematical foundations. This “fourth shutter” could also be extended to the promising new perspectives offered by Homotopy See, for example, Olivia Caramello, Theories, Sites, Toposes, Oxford University Press, Ox­ford 2018. Type Theory5, where some strong connections between geometry and computa­tion reveal a structural universality of which Badiou may be very fond. Three fundamental ideas provide the backbone of Cohen’s forcing: (1C) at the base, the use of a countablestandardmodelof the Zermelo-Fraenkel axioms for Set Theory; (2C) at the approximation level, the use of an order on which forcing conditions are inscribed; (3C) at the ideal level (or point at infinity), the con­struction of a generic filter that offers some completeness conditions to provide independence proofs. Many passages ofBeing and Event are related to situa­tions (1C)-(3C)6: (1C) “[…] the demonstration that every truth is necessarily infinite” (p. 328). “A truth (if it exists) must be an infinite part of the situation, because for every finite part one can always say that it has already been discerned and classified by knowledge” (p. 333). “we shall install ourselves in a multiple which is fixed once and for all, a mul­tiple which is very rich in properties (it ‘reflects’ a significant part of general ontology) yet very poor in quantity (it is denumerable)” (p. 356). “This multiple will be both the basic material for the construction of the indis­cernible (whose elements will be extracted from it), and the place of its intelligi­bility” (p. 357). (2C) “the study of local or finite forms of a procedure of fidelity” (p. 327). “the minimal gesture of fidelity: the observation of a connection (or non-con­nection)” (p. 329). “A faithful procedure has as its infinite horizon being-in-truth” (p. 339). 399 “Using a transparent algebra, we will note x(+) the fact that the multiple x is recognized as being connected to the name of the event, and x(-) that it is recog­nized as non-connected. […] We will term enquiry any finite set of such minimal reports. […] It is the enquiry which lies behind the resemblance of the procedure of fidelity to a knowledge” (pp. 330–331). 5 Homotopy Type Theory, Univalent Foundations Project, Princeton 2018. 6 The citation numbers refer to pages in the English translation: Alain Badiou, Being and Event. Emphases in bold in the quotes are mine. “The concept of order is central here, because it permits us to distinguish multi­ples which are ‘richer’ in sense than others; even if, in terms of belonging, they are all elements of the supposed indiscernible” (p. 362). “a condition is useless [...] if it does not tolerate any aleatory progress in the conditioning” (p. 364). “Order, compatibility and choice must, in all cases, structure every set of con­ditions” (p. 364). (3C) “We find ourselves here at the threshold of a decisive advance, in which the con­cept of the ‘generic’ […] will be defined and articulated in such a manner that it will found the very being of any truth.” (p. 327) “We shall therefore say: a truth is the infinite positive total — the gathering togeth­er of x(+)‘s — of a procedure of fidelity which, for each and every determinant of the encyclopaedia, contains at least one enquiry which avoids it. Such a procedure will be said to be generic” (p. 338). “The generic is the being-multiple of a truth” (p. 338). “Thus, any veracity in the extension will allow itself to be conditioned in the situation. [...] an inhabitant is in the position of a subject of truth: she forces veracity at the point of the indiscernible” (p. 411). The use of genericity for philosophy and culture, outside its original technical en­vironment, is one of the greatest contributions of Badiou’s trilogy. In fact, generic processes (love, art, science, politics), offer a complete passage of the categories of Being (multiple, void, nature, infinity) and Event (ultra-one, intervention, fi­delity). 400 On the other hand, one can also find three fundamental concepts in Groth­endieck’s toposes: (1G) at the base, the Grothendieck topologyJ in an adequate category C and its associated site(C,J); (2G) at the approximation level, a col­lection of sheaves with its gluing properties; (3G) at the ideal level (or point at infinity), the emerging complete structure of the topos with its limits, expo­nentials, and classifier object. Many passages of Logics of Worlds are related to situations (1G)-(3G)7: The citation numbers refer to pages in the English translation: Alain Badiou, Logics of Worlds. (1G) “What we are attempting here is a calculated phenomenology. […] Hence a style of formalization that is both more geometrical and more calculating, at the boundary between a topology of localizations and an algebra of forms of order” (pp. 38–39). “The axiomatic deployment of what a place (or a power of localization) is con­sists in finding the principles of interiority” (p. 411). “a site is a multiple which happens to behave in the world in the same way with regard to itself as it does with regard to its elements, so that it is the ontological support of its own appearance” (p. 363). “Logic of the site” and the four forms of change (pp. 369–380). (2G) “it is possible to take even further the thinking of the logico-ontological, of the chiasmus between the mathematics of being and the logic of appearing. But one then needs to equip oneself with a more topological intuition and to treat the degrees of the transcendental as operators of the localization of multiple-beings. […] To use the technical language of contemporary mathematics, this correlation is a sheaf” (p. 197). “the logical identity of appearing is elucidated particularly well by the theory of complete Heyting algebras” (p. 389). “The subjectivation of the new body will acquire the creative form of a constant broadening of structural correlations, of the ‘visibility’ of one structure in an­other. In particular, ‘reading’ algebraic structures in topological structures will become the key to contemporary mathematics. With the concept of sheaf, which synthesizes this type of correlation and serves as its general organ, there un­doubtedly begins the history of a new body, for which Grothendieck arguably played around 1950 the same role that Galois played around 1830” (p. 475). “(…) let’s hold on to the notion, which we have seen at work in both mathematics and poetry, that the sequence world—points—site—body—efficacious part–or­gan is indeed the generic form of what makes it possible for there to be such things as truths. This authorizes the materialist dialectic to contend that beyond bodies and languages, there is the real life of some subjects” (p. 475). (3G) “the idea which has received the name of ‘Grothendieck topos’ and the related idea of a sheaf shine in the sky of pure thought” (p. 540). “The mathematics of appearing consists in detecting, beneath the qualitative disorder of worlds, the logic that holds together the differences of existence and intensity” (p. 39). “The infinite of worlds is what saves us from every finite disgrace. […] We over­come all this when we seize hold of the discontinuous variety of worlds and the interlacing of objects under the constantly variable regimes of their appearanc­es” (p. 514). Here, the use of a materialist dialectics provides a crossing of the diverse logics of the world, where a “universality of truths” emerges beyond individuals and communities (pp. 5–6). Finally, three other main ideas govern the behaviour of large cardinals: (1L) the back-and-forth between absoluteness and localisations; (2L) the back-and-forth betweengenericity (ultrapower) and existence (power); (3L) the back-and-forth between “A-subjectivity” (the index of absoluteness) and subjectivity (art, science, love, politics: “works of Truth”). Many passages of The Immanence of Truths are now related to situations (1L)-(3L)8: (1L) “Absolute place V” (p. 61). negative access to the inaccessible (p. 303). going beyond the generic (p. 275). elementary embedding (p. 296). “point at infinity” (p. 109). (2L) genericity = ultrapower (p. 423). “transgressions” (p. 307). “maximalizations” (p. 339). “partitions” (p. 318). filtering (p. 335). 8 The citation numbers (for both quotes and glosses) refer to pages in Alain Badiou, L’Imma­nence des vérités. (3L) A-subjective = index of absoluteness (p. 514). art: power of form (p. 544). “science: the power of the letter” (p. 579). “love: the stage of the Two” (p. 611). politics: the stage of the Others (p. 639). Here, by the multiplication and ultralimitation procedures of large cardinals, we can see how the singular, set-theoretic, category of Truth present in Being and Event, is extended to the plural, but again set-theoretic, categories of Truth presentin TheImmanence ofTruths. The capital, singular, letter in the term “Immanence”, pointing to the Absolute (the global contradiction 0=1), enters in counterpoint with the lower case, plural, letter in “vérités” (the local real-isations at each large cardinal layer). The dialectics of the One and the Many, the contradiction and the models, the forms of incompleteness and complete­ness, are very nicely explored by Badiou, in order to glue the complexity of mathematics with the complexity of Life and the World. In fact, one can sense that, beyond the set-theoretic intrinsic and natural flavour of large cardinals, a sheaf-theoretic reading is also implicit, first, along cuts and gluings, follow­ing increasingly high consistency results (from inaccessibles to superhuges, through Jónsson, Rowbottom, Ramsey, Woodin, Vopenka, “from the top to the bottom (...) something descriptive and baroque” (p. 281), and second, along the fibres of a universal sheaf, where each large cardinal may be seen as a locali­sation of the Absolute, and each family of large cardinals can be understood through an ultrapower of the Absolute. An elementary Peircean and category-theoretic reading of the trilogy 403 When faced with contemporary thought, we cannot escape a certain transitory ontology9 that at first, terminologically speaking, seems self-contradictory. Nev­ertheless, although the Greek ontotetes sends us, through Latin translations, to a supposedly atemporal “entity” or to an “essence” that ontology would study, there is no reason, besides tradition, to believe that those entities or es­sences should be absolute (in a set-theoretic sense, as in BeingandEvent) and 9 Alain Badiou, Briefings on Existence. A Short Treatise on Transitory Ontology, trans. Nor­man Madarasz, SUNY, Albany 2006. not asymptotic, governed by partial gluings in a correlative evolution between the World and knowledge (in a category-theoretic sense, as in Logiques des mondes). Going beyond the dichotomy,bimodality, in the sense of Jean Petitot10, that is, a dynamic (topological) movement in both physical and morphologi­cal-structural space, is related to such a state of things, where “things” have to be replaced in fact by “processes”. Both prefixes (trans-, bi-) offer a suitable ground to understand the wanderings of contemporary thought. Peirce had already imagined (or discovered, according to our variable ontologi­cal commitment) a wonderful phenomenological tool (Peirce 1886) that helps to unravel the multilayered geometry of the (trans-, bi-)situation11.Phanerosco­py, or the study of the phaneron, i.e. the complete collective spectrum present to the mind, includes the doctrine of Peirce’s cenopythagorean categories (with “ceno-” coming from the Greek kaíno, “fresh”), which observe the universal modes (or “tints”) occurring in phenomena. Peirce’s three categories are vague, general, and indeterminate, and can be found simultaneouslyin every phenom­enon. They are interlaced in several levels, but can be prescised (distinguished, separated, detached) following recursive layers of interpretations, in progres­sively increasingly determined contexts. A dialectics between the One and the Many, the universal and the particular, the continuous and the discrete, the general and the concrete, is multilayered along a dense variety of theoretical and experimental fibres.12 Peirce’sFirstnessdetects the immediate, the spontaneous, whatever is inde­pendent of any conception or reference to something else. Secondness is the category of facts, mutual opposition, existence, actuality, material fight, action and reaction in a given world. Thirdness proposes a mediation beyond clashes, 404 a third place where the “one” and the “other” enter into a dialogue, the catego­ry of sense, representation, synthesis. As Peirce reckons: By the Third, I understand the medium which has its being or peculiarity in con­necting the more absolute first and second. The end is second, the means third. 10 Jean Petitot, Per un nuovo illuminismo, Bompiani, Milano 2009. 11 Charles Sanders Peirce, “One, Two, Three: An Evolutionist Speculation”, in Writings, Indi­ana University Press, Vol. 5, pp. 300–301. 12 For a mathematical introduction to Peirce’s architecture, see Fernando Zalamea, Peirce’s Logic of Continuity, Docent Press, Boston 2012. A fork in the road is third, it supposes three ways. (...) The first and second are hard, absolute, and discrete, like yes and no; the perfect third is plastic, relative, and continuous. Every process, and whatever is continuous, involves thirdness. (...) Action is second, but conduct third. Law as an active force is second, but order and legislation third. Sympathy, flesh and blood, that by which I feel my neighbor’s feelings, contains thirdness. Every kind of sign, representative, or deputy, everything which for any purpose stands instead of something else, whatever is helpful, or mediates between a man and his wish, is a Third.13 Peirce’s vague categories can be “tinctured” with key-words: (1) Firstness: immediacy, first impression, freshness, sensation, unary predicate, monad, chance, possibility; (2) Secondness: action-reaction, effect, resistance, binary relation, dyad, fact, actuality; (3) Thirdness: mediation, order, law, continui­ty, knowledge, ternary relation, triad, generality, necessity. The three Peircean categories interweave recursively andproduce anested hierarchy ofinterpreta­tive modulations. Dynamic cognition yields progressiveprecision through pro-gressiveprescision. Both surgery and gluing form part of a ubiquitous topology of comprehension. Intelligence grows with the definition of ever more contexts of interpretation, and the association of finer and finer cenopythagorean tinc­tures inside each context. On the other hand, Category Theory offers a multitude of tools to understand the (trans-, bi-) imperative. The mathematical theory of categories axiomatis-es areas of mathematical practice, in accordance with the structural similarities of the objects in question and with the modes of transmission of information between these objects (here, categories are close to methodological and philo­sophical approximations, sensitive to problems of transference, as in the case of Peircean pragmaticism). As opposed to set theory, where objects are analysed 405 internally as aggregates of elements, the mathematical theory of categories studies objects by way of their external synthetic behaviour, due to the rela­tions of the object with its environment. The objects are like black boxes, which cannot be analysed and broken into smaller interior sub-boxes, and which can be understood only by way of their actions and reactions with the surrounding medium. The modes of knowledge are then essentially relational: the ways in 13 Peirce, “One, Two, Three: An Evolutionist Speculation”, p. 300. which the information transmitters behave in the environment constitute the mathematical weaving in which thought grows. A morphism is universal with respect to a given property if its behaviour with respect to similar morphisms in the category possesses certain uniquely iden­tifying characteristics which distinguish it within the categorical framework. The basic notions of Category Theory related to universality – those of free ob­ject and adjointness – respond to deep problems related to the search for rela­tive archetypes and relative dialectics. In fact, after Gödel, the turn in math­ematics toward problems of relative consistency (thus, overcoming chimerical longings for absolute foundations) resulted in an explosion of diversity and dif­ferentiation in axiomatic mathematical theories, beyond a certain threshold of complexity. Within the resulting multiplicity, in the broad, variable spectrum of the areas of mathematics, Category Theory managed to find some patterns ofuniversalitywhich facilitated processes of local unfolding and also the tran­scendence of concrete particulars. For instance, in a category, a free object is able to project itself into any object whatsoever taken from a sufficiently wide subclass of the category: it is thus a sort of primordial sign, embodied in all re­lated contexts of interpretation. Hence, relative universals arise beyond relative localisations; these have given a new technical impetus to the classical notions of universality. Although it is no longer possible to presume that we are in a supposed absolute, nor to believe in uniform, stable concepts regarding space and time, Category Theory has reshaped the notion of universality, making it suitable for a series of relative transferences of the universal/free/generic, in which transition is allowed, and in which at the same time it is possible to find remarkableinvariants beyond it.14 Thus, Category Theory explores the structure of certain generals in a way 406 similar to that of Peirce’s late scholastic realism. Indeed, categorical thinking contemplates a dialectics between universal definitions in abstract categories (generic morphisms) and realisations of those universal definitions in concrete categories (structured set classes); moreover, within abstract categories, there may perfectly well be morphisms that are real universals, while at the same For elaborations of these remarks, see Fernando Zalamea, “A Category-Theoretic Reading of Peirce’s System: Pragmaticism, Continuity, and the Existential Graphs”, in New Essays on Peirce’s Mathematical Philosophy, ed. Matthew Moore, Open Court, Chicago 2010, pp. 203–233. time, not being existent (that is to say, they are not embodied in concrete cate­gories: think, for example, about an initial object, readily definable in abstract categories, but which is not realised in the category of infinite sets, in which initial objects do not exist). In the range of pure possibilities, the pragmatistic maxim has to deal with the idea of universal concepts, logically correct, but which could possibly not turn out to be embodied in bounded contexts of exist­ence (such as, in the case of the three Peircean categories: real universals that may not always adequately be realised in concrete existents within the bound­ed contexts). The mathematical theory of categories illuminates this kind of sit­uation with a high degree of precision. Category Theory has actually managed to effect the technical construction of a variety of entities, seemingly as elusive as those real universals with no existence, thanks to a very interesting dialec­tical process between the domains of actual mathematical practice (computa­tional, algebraic, or differential structures, for example) and the possibility of abstract, universal definitions, still not realised in that practice. For instance, following current tendencies in universal algebra and Abstract Model theory [Category Theory has been able to define truly general notions of logic and of relative truth universals, as suitable invariants of given classes of logics. Using Peirce’s Triadicity and Category Theory, Badiou’s trilogy BeingandEvent (BB),Logics of Worlds (LW), and The Immanence of Truths (IT), acquire a simple first adjunction first embedding (sets – categories) (completion) 2 (LW) 3 (IT) second adjunction second embedding (classifier topos – large cardinals) (?) Figure 1: A triadic, category-theoretic reading of Badiou’s trilogy As a (Peircean) First, (BB) proposes a monadic, fresh, intuitive entrance to in­finity and the Absolute. As a Second, (LW) acts and reacts against the first naive approximation, and multiplies the range of the World. As a Third, (IT) mediates between the Absolute (inconsistency 0=1) and the Relative (large cardinals con­sistency layers). Beyond that, since (IT) can be seen as a completion of (BB), one would expect that a fourth shutter of an extended tetralogy would act as a com­pletion for (LW). The dialectical forces at stake now become pretty clear: on one hand, both sets and categories become fundamental for a thorough understand­ing of the polarities One-Many, Absolute-Relative, and both of their projections on culture (“works of Truth”) enrich the transits of reason and heart; on the other hand, the search for a multilayered geometry of knowledge is embodied in triadic ramifications, iterations, embeddings, projections, and adjunctions. Some open problems Badiou’s work is full of questions and suggestions; perhaps what one may ap­preciate the most in a philosophical reflection. The density of the concepts stud­ied, and the correspondingrichness of the perspectives, allow many wander­ings. Below we summarise five of those queries: (i)The study of the pendulum sets – categories, along many of the perspectives presented in the Prague Conference (2018): philosophical openness (Alunni), dualities and dialectics (Guitart), universe L (Berankova), inconsistent multi­plicities (Tho), multiverses (Hussey), bordering metaontology (Baki), historical relativity (Barbin), locality and genericity (Cartier), combinatorial time flow (Sumic), Riemann multiplicities (Rabouin), subtraction gestures (Hauser), the logic of True Life (Nesbitt), textual mathematical operations (Bolz), via negativa (Feltham), configuration excesses (Madarasz), etc. (ii) The construction of a general mediation between Set Theory (analytics) and Category Theory (synthetics), along a third foundational perspective (“horotics”), with axioms for a theory of frontiers and limits, capturing what is common to neighbourhood coverings (analytics) and Grothendieck topologies (synthetics). (iii) The category-theoretic completion of Logics of Worlds, in correlation with the set-theoretic completion of Being and Event provided by The Immanence of Truths, and the study of the many geometrical features of the situation (Figure 1 above). (iv) A comparison of Badiou’s Absolute in The Immanence of Truths, with Floren­sky’s Antinomy as the true foundation of mathematics15: the back-and-forth be­tween contradiction and consistency provides a complex and dense multilayered understanding of mathematics, and its projections onto the World. (v) A comparison of Badiou’s poetical understanding of the infinite with Borges’s many renderings in El Aleph (1949): transcendence, resistance to any division, the power of internal determinations, the intimate relation to the Absolute.16 The development of some of these threads would enhance our understanding of mathematics, the philosophy of mathematics, general philosophy, and cultural studies at large. Our community should be sincerely grateful to Alain Badiou for opening up such possibilities. References Badiou, Alain,L’Etre et l’événement, Seuil, Paris 1988 — Court traité d’ontologie transitoire, Seuil, Paris 1998 — Logiques de mondes, Seuil, Paris 2006 — L’Immanence des vérités, Seuil, Paris 2018 Caramello, Olivia, Theories, Sites, Toposes, Oxford University Press, Oxford 2018 Florensky, Pavel, The Pillard and Ground of the Truth, Princeton University Press, Prince­ ton 2004 HoTT, Various Authors, Homotopy Type Theory, Univalent Foundations Project, Prince­ton 2018 Peirce, Charles Sanders, “One, Two, Three: an evolutionist speculation [1886]”, in Writ­ings: A Chronological Edition, vol. 5., 300–301, Indiana University Press, Blooming­ton 1993 Petitot, Jean, Per un nuovo illuminismo, Bompiani, Milano 2009 409 Zalamea, Fernando, “A Category-Theoretic Reading of Peirce’s System: Pragmaticism, Continuity, and the Existential Graphs”, in New Essays on Peirce’s Mathematical Phi­losophy, ed. Matthew Moore, 203–233, Open Court, Chicago 2010 — Peirce’s Logic of Continuity, Docent Press, Boston 2012 15 Pavel Florensky, The Pillard and Ground of the Truth, Princeton University Press, Princeton 2004. 16 Badiou, L’Immanence des vérités, pp. 282–284. Notes on Contributors Alunni, Charles founded and directed (for twenty-five years) the research centre “Lab­oratoire disciplinaire – Pensée des sciences” at the École Normale Supérieure in Paris. For more than forty years he taught at the Scuola normale superiori di Pisa. He was a visiting professor (Gastdozent) at the Ruhr University in Bochum. He studied with Jacques Derri­da and Eugenio Garin in Paris and Pisa, respectively. He has translated many Italian phi­losophers and legal theorists into French. Recently, he founded the book series “Pensée des sciences”, published by the Hermann publishing house, of which he currently serves as the director. His recent publications include Spectres de Bachelard. Gaston Bachelard et l’école surrationaliste (Paris, Hermann, 2018). Badiou, Alain is a French philosopher and playwright. He is a professor emeritus and former chair of the Department of Philosophy of the École Normale Supérieure in Paris and one of the founding members of the Faculty of Philosophy of the Université Paris VIII. His major works include the three volumes Being and Event, Logics of Worlds, and the recently published The Immanence of Truths. His philosophical oeuvre connects con­tinental and analytical philosophical traditions along with his reflections on set theory and contemporary mathematics. Berankova Ndiaye, Jana is a PhD candidate at Columbia University’s Graduate School of Architecture, Planning and Preservation (GSAPP) and a former student of École Nor-male Supérieure in Paris and École des Hautes Études en Sciences Sociales. Her research interests include the links between continental philosophy and architecture theory, so­cial movements of 1968, and Central European architecture. She is the co-editor of Alain Badiou: Sometimes, We Are Eternal (2019) and Revolutions for the Future: May ’68 and the Prague Spring. She is the founder of the publishing house Suture Press (www.suture­press.com) and member of the Prague Axiomatic Circle. Bolz, Roland is a PhD candidate at Humboldt University of Berlin. He is currently pre­paring his thesis on the role of analogy-making and metaphor in philosophical theories of concepts. Some of his other research interests include political philosophy, aesthetics, philosophy of language, and humour theory. Feltham, Oliver teaches philosophy at the American University of Paris. He works in the area of modern political philosophy, critical theory and psychoanalysis. He trans­lated Alain Badiou’s Being and Event for Continuum Books in 2005, wrote a monograph on Badiou, and more recently Anatomy of Failure: Philosophy and Political Action with notes on contrubutors Bloomsbury in 2013 and Destroy and Liberate: Political Action on the basis of David Hume with Rowman and Littlefield International in 2019. He has recently completed a full-length play Twig. Guitart, René born in 1947 in Paris, taught mathematics at the Université de Picardie, from 1968 to 1970, then at the University Paris 7 Denis Diderot from 1970 to 2012. He also worked at Essilor Company between 1985 and 1992, manufacturing progressive lenses with spline functions. He was programme director at Ciph (College International de Phi-losophie) from 1990 to 1996, leading a seminar on category theory, psychoanalysis and philosophy. His research in mathematics has mainly focused on category and structure theory, theory of theories, logic, and homological algebra, fields in which he proposed, in particular, algebraic universes, the locally free diagram theorem, Borromean logic, and exact squares. His other contributions are in the history of science of the 19th cen­tury, especially regarding physics and geometry (with Monge, Lamé, Poincaré), in epis­temology and questions of teaching, in philosophy on Nietzsche, and Bachelard, and in psychoanalysis concerning Lacan’s work. His articles (about 130) are available on his personal website. Amongst “the latest, the following should be mentioned: “Bachelard et la pulsation mathématique”, Revue de synthese, t.136, 6th series, Nos 1-2, 2015, “Note sur deux problemes”, Revue de synth.se, t.136, 6th series, no. 1-2, 2015, “Nietzsche face a l’expérience mathématique”, in E. Barbin and J.P. Cléro (eds.), Les mathématiques et l’expérience : ce qu’en ont dit les philosophes et les mathématiciens, Hermann, Paris, 2015. He has also published two books: La pulsation mathématique (rigueur et ambiguité, la na­ture de l’activité mathématique, ce qu’il faut enseigner), L’Harmattan, Paris, 1999, and Ev­idence et étrangeté (Mathématiques, psychanalyse, Descartes et Freud), PUF, Paris, 2000. Hauser, Michael is a researcher at the Philosophical Institute of the Czech Academy of Sciences and professor at the Faculty of Pedagogy of Charles University. He is the founder of the civic association Socialist Circle (SOK), which develops socially critical thinking. He is the author of Cesty z Postmodernismu (2012), Prolegomena k filosofii souèasnosti (2008), and Adorno: modernita a negativita (2005). He has also published a book of inter­views with Slavoj Žižek, Humanismus nestaèí (2008). Hussey, M. Norma, PhD. I have published a number of scientific papers (in optical fiber device research), and worked in the telecommunications industry and in telecom­munications regulation. Giving myself the term ‘lay-person’ here – since I’m neither a philosopher nor a mathematician and (currently) not affiliated with a university/insti­tution – I’ve been inspired by the philosophy of Alain Badiou since my first encounter (2000) with his works; Being and Event in particular, and the Infinity Lectures which he delivered at the European Graduate School in 2011. I have since attended graduate work­shops with Badiou, given a short presentation at the conference Thinking the Infinite, and co-edited Sometimes, We Are Eternal with Jana Ndiaye Berankova. Madarasz, Norman is the author of four books, among which, O Realismo Estruturalis­ta: sobre o intrínseco, o imanente e o inato (Editora Fi, 2016). He is also co-editor of six col­lected volumes as well as translator/editor of three books by Alain Badiou. As Research Professor in the Graduate Programs of Philosophy and of Literature and Linguistics at the Pontifical Catholic University of Rio Grande do Sul State (PUC-RS) in Porto Alegre, Brazil, he lectures on and conducts research in the area of contemporary French philosophy (ontology, structuralism, biolinguistics and political philosophy), contemporary French and North American literature and Gender Studies. He is also contributing editor of the periodical, Veritas, having edited five supplements on ontology and continental philos­ophy of science. Nesbitt, Nick is a Professor at Princeton University. He is the author most recently of Caribbean Critique: Caribbean Critical Theory from Toussaint (Liverpool, 2013) and editor of The Concept in Crisis: Reading Capital Today (Duke, 2017). His next book project is en­titled Slavery, Capitalism, and Social Form: From Marx to Caribbean Critique, forthcoming fall 2021 (U Virginia Press). Rabouin, David is a Senior Research Fellow (Directeur de Recherche) at the French National Centre for Scientific Research (CNRS), in the research group SPHERE (UMR 7219, CNRS – Université de Paris). His main interest is in the history of philosophy and mathe­matics in early Modern Times, with special focus on Descartes and Leibniz. He is the au­thor of Mathesis universalis. L’idée de « mathématique universelle » d’Aristote a Descartes, Paris, P.U.F., coll. « Épiméthée », 2009 and, in collaboration with the ‘Mathesis’ Group, of Leibniz. Ecrits sur la mathématique universelle, Paris, Vrin, 2018. With K. Chemla and R. Chorlay, he co-edited the The Oxford Handbook of Generality in Mathematics and the Sciences (Oxford University Press, 2017). He also has a strong interest in Contemporary French Philosophy and has edited (with Oliver Feltham and Lissa Lincoln) : Autour de « Logiques des Mondes », Editions des Archives contemporaines, Paris 2011. His latest publication, co-edited with Emmylou Haffner, is : L’épistémologie du dedans. Mélanges en l’honneur d’Hourya Benis-Sinaceur, Garnier, Paris 2021. Ruda, Frank is Senior Lecturer in Philosophy at the University of Dundee. His book pub­lications include For Badiou: Idealism without Idealism. Northwestern University Press, Evanston 2015; Abolishing Freedom: A Plea For a Contemporary Use of Fatalism. Univer­sity of Nebraska Press, Lincoln 2016 and The Dash – The Other Side of Absolute Knowing (with Rebecca Comay). MIT Press, Cambridge 2018. notes on contrubutors notes on contrubutors Šumiè Riha, Jelica is Professor of Philosophy at the Postgraduate School of Research Centre of the Slovenian Academy of Sciences and Arts and Senior Research Fellow at the Institute of Philosophy, Research Centre of the Slovenian Academy of Sciences and Arts. She was visiting professor at the University of Essex, University Paris 8, Universidad de Buenos Aires and University of Sao Paulo. In 2000-2002 she conducted a seminar “Le pour tous face au réel” at the College international de philosophie in Paris (together with Rado Riha). She has published a number of philosophical works, including Politik der Wahrheit (with Alain Badiou, Jacques Ranciere and Rado Riha), Turia+Kant, Vienna 1997, Universel, Singulier, Sujet (with Alain Badiou et al), Kimé, Paris 2000; Mutations of Ethics (Založba ZRC, 2002); Veènost in spreminjanje. Filozofija v brezsvetnem èasu (Eternity and Change. Philosophy in the Worldless Time, Založba ZRC, Ljubljana 2012; A politíca e a psicanálise: do nao-todo ao para todos, Lume editor, Sao Paulo 2019. Currently she is working on a forthcoming volume entitled Volonté et Désir (Harmattan). Tho, Tzuchien is a philosopher and historian of science. He is currently lecturer at the University of Bristol. Previously, he has been affiliated with the Jan van Eyck Academie in Maastricht (NL), the École Normale Supérieure in Paris (Rue D’Ulm), the Max Planck Institute for the History of Science (Berlin), Berlin-Brandenburg Academy of Sciences, the Institute for Research in the Humanities (University of Bucharest) and the University of Milan. He is currently working on a research project on causality in 18th century phys­ics, focusing on the development of analytical mechanics. He is also currently working on issues related to Badiou’s mathematical ontology, the philosophy of algebra, Leibniz reception in the 20th century, and the critique of contextualism as historical methodology. Zalamea, Fernando (Bogotá, 1959) is Professor of Mathematics at Universidad Na-cional de Colombia. After his Ph.D. in category theory and recursion theory (University of Massachusetts, 1990, under Ernest Manes), Zalamea has been working in alterna­tive logics, Peirce and Lautman studies, and the philosophy of modern (1830-1950) and contemporary mathematics (1950-today). His recent Grothendieck. Una guía a la obra matemática y filosófica (2019) is the first complete guide to Grothendieck’s work. A pro­lific essayist, he is the author of twenty books around cultural studies, philosophy, and mathematics. His book Synthetic Philosophy of Contemporary Mathematics (Bogotá 2008, translated into English 2012 and French 2018) seems to have had some impact. Zalamea has obtained some of the most prestigious essay awards in the Hispanic world: Siglo XXI (Mexico 2012), Jovellanos (Spain 2004), Gil-Albert (Spain 2004), Kostakowsky (Mexico 2001), Andrés Bello (Colombia 2000). He has been included as one of 100 Global Minds. The Most Daring Cross-Disciplinary Thinkers in the World (Roads 2015). Web: https://unal. academia.edu/FernandoZalamea. Mail: fernandozalamea@gmail.com Povzetki | Abstracts Alain Badiou Ontology and Mathematics: Set Theory, Category Theory, and the Theory of Infinites in Being and Event, Logics of Worlds and The Immanence of Truths Key words: mathematics and philosophy, ontology, set theory, category theory, large cardinals, the absolute, Badiou This paper examines the relationship between philosophy and its conditions. The affir­mation “mathematics is ontology”, which I posited thirty years ago, has certain incon­veniences. In this article, I present six varying possibilities for ontology. My own philo­sophical decision was to proclaim that being is a pure multiplicity, without the One and without any specific attribute such as “matter” or “spirit”. This movement of thought brought me to study the mathematical condition of philosophy and to search for a rigor­ous structuration of my speculative decision within the field of mathematics. However, my initial postulate that “Being is the multiplicity without the One” is not a mathemati­cal but a philosophical statement. This paper concludes with a presentation of the rela­tionship between mathematics and philosophy in Being and Event, Logics of Worlds, and The Immanence of Truths. Alain Badiou Ontologija in matematika: teorija množic, teorija kategorij in teorija neskonènosti v Biti in dogodku, Logikah svetov in Imanenci resnic Kljuène besede: matematika in filozofija, ontologija, teorija množic, teorija kategorij, veliki kardinali, absolut, Badiou Prièujoèi prispevek preiskuje razmerje med filozofijo in njenimi pogoji. Trditev: »Mate­matika je ontologija,« ki sem jo postavil pred tridesetimi leti, je trèila na nekaj težav. V prispevku predstavim šest možnosti za ontologijo. Moja lastna filozofska odloèitev je bila razglasiti bit kot èisto mnoštvenost brez Enega in brez posebnega atributa, kakršna sta »materija« ali »duh«. Ta miselna pot me je napeljala na preuèevanje matematiènega pogoja filozofije in raziskovanje strukturacij za mojo spekulativno odloèitev znotraj polja matematike. Vendar pa moj zaèetni postulat: »Bit je mnoštvo brez Enega«, ni matema­tièna trdetev, temveè filozofska. Prièujoèi èlanek sklenem s predstavivijo razmerja med matematiko in filozofijo v Biti in dogodku, Logikah svetov in Imanenci resnic. Oliver Feltham “One or Many Ontologies? Badiou’s Arguments for His Theisis ‘Mathematics is Ontology’.” Key words: ontology, meta-ontology, argument from the conditions, argument from philosophy, modelling, schematism This article explores rival interpretations of Badiou’s strategy behind the claim ‘mathe­matics is ontology’, from his construction of an alternative history of being to that of Hei­degger to his exposure of the radical contingency of the ‘decisions on being’ carried out by transformative practices in the four conditions of philosophy: art, politics, love and science. The goal of this exploration is to open up the possibility of another strategy that responds to Badiou’s initial intuition – that being is multiple – by embracing the writing of multiple ontologies in the sphere of action. Oliver Feltham »Ena ali mnoge ontolgije? Badioujevi argumenti za njegovo tezo ‘Matematika je ontolgija’.« Kljuène besede: ontologija, meta-ontologija, argument, izpeljan iz pogojev, argument, izpeljan iz filozofije, modeliranje, shematizem Prièujoèi èlanek raziskuje med sabo tekmujoèe interpretacije Badioujeve strategije, kot jo lahko razberemo iz trditve, »matematika je ontologija«, pri èemer se opira na njegovo konstrukcijo alternative Heideggerjevi zgodovini biti kot tudi na razkritje radikalne kon­tingentnosti »odloèitev glede biti«, kot jih je mogoèe zaznati v transformacijskih praksah v štirih pogojih filozofije: umetnosti, politiki, ljubezni in znanosti. Namen te preiskave je odpreti možnosti za drugaèno strategijo, ki ustreza Badioujevi izvorni intuiciji, da je namreè bit mnoštvo, in sicer tako, da prenesemo pisanje mnoštva ontologij na podroèje delovanja. Nick Nesbitt Bolzano’s Badiou Key words: Bolzano, Badiou, theory of science, demonstration, Being and Event This article raises a series of points of confluence between Badiou’s philosophy and that of Bernard Bolzano, whom Badiou has identified as a historical predecessor but never di­rectly engaged. These points include their respective critiques of Kant and Hegel, as well as their various concepts of sets, platonist realism, axiomatisation, the infinite, adequate demonstration, structure, and mathematics as the adequate language of being. Nick Nesbitt Bolzanov Badiou Kljuène besede: Bolzano, Badiou, teorija znanosti, dokaz, Bit in dogodek Prièujoèi èlanek zariše vrsto stiènih toèk med Badioujevo filozofijo in filozofijo Bernarda Bolzana, ki ga je Badiou prepoznal kot zgodovinskega predhodnika, vendar se z njim nikoli ni neposredno ukvarjal. Te stiène toèke vkljuèujejo njuni kritiki Kanta in Hegla kot tudi njuni razlèini pojmovanji množice, platnoistiènega realizma, aksiomatizacije, neskonènega, adekvatnega dokaza, strukture in matematike kot ustreznega jezika biti. Jelica Šumiè Riha The Place of Mathematics: Badiou with Lacan Key words: mathematics, philosophy, psychoanalysis, the real, formalisation, Cantor, Lacan, Badiou The paper attempts to give an account of two different ways of relating to mathematics: Lacan’s and Badiou’s. Its starting point is Badiou’s and Lacan’s interpretations of Rus­sell’s infamous definition of mathematics, according to which mathematics is a discourse in which no one knows what one is talking about, nor whether what one is saying is true. While for Badiou, the ignorance that is supposed to characterise mathematics according to Russell only concerns the role philosophy assigns to it, namely, it being identified with the science of being qua being, for Lacan the ignorance constitutive of mathematics should rather be seen as a symptom resulting from the reduction of truth to a mere truth value. In discussing the detected divergences in these two readings, the paper shows how the access to psychoanalysis as well as the access to philosophy coincides with a certain access to mathematics, although we are not dealing with the same type of access. Jelica Šumiè Riha Mesto matematike: Badiou z Lacanom Kljuène besede: matematika, filozofija, psihoanaliza, realno, formalizacija, Badiou, Cantor, Lacan Namen prièujoèega prispevka je pojasniti dva naèina navezovanja na matematiko: Laca­nov in Badioujev. Avtorica prispevka pri tem izhaja iz Badioujeve in Lacanove interpre­tacije slovite Russellove definicije matematike, po kateri naj bi bila matematika diskurz, pri katerem nihèe ne ve, o èem govori, niti ali je to, kar govori, resnièno. Medtem ko za Badiouja ta nevednost, ki naj bi bila znaèilna za matematiko, kot jo razume Russell, za­deva zgolj vlogo, ki jo filozofija pripisuje matematiki, da je namreè znanost o biti kot biti, pa Lacan nevednost, ki naj bi bila konstitutivna a matematiko, pojasnjuje kot simptom, ki izvira iz statusa resnice v matematiki, tj. njene redukcije na resniènostno vrednost. Pri pojasnjevanju ugotovljenih divergenc med Badioujevim in Lacanovim razumevanjem matematike èlanek izhaja iz teze, da vstop v psihoanalizo tako kot tudi vstop v filozofijo sovpada z vstopom v matematiko, a hkrati vztraja, da ne gre za isto modaliteto vstopa v matematiko. Michael Hauser Badiou and the Ontological Limits of Mathematics Key words: metaontology, the real, ontology, Badiou, Easton’s theorem I propose to depict the relationship between Badiou’s philosophy and mathematics as a three-layered model. Philosophy as metaontology creates a metastructure, mathemat­ics as ontology in the form of a condition of philosophy constitutes its situation, and mathematics as a multiple universe of all given axioms, theorems, techniques, inter­pretations, and systems (set theory, category theory, etc.) is an inconsistent multiplicity. So, we can interpret the relationship between philosophy and mathematics as the one between a metastructure and a situation. By using Easton’s theorem, we come to realise that philosophical concepts in the metastructure “quantitatively” exceed the elements that belong to mathematics as ontology. Therefore, philosophy as metaontology shows the limits of mathematics as ontology. Michael Hauser Badiou in ontološke meje matematike Kljuène besede: metaontologija, realno, ontologija, Badiou, Eastonov teorem Avtor obravnava razmerje med Badioujevo filozofijo in matematiko kot trostopenjski model. Filozofija kot metaontologija ustvari metastrukturo, matematika kot ontologija v vlogi pogoja filozofije pa ustvari njeno situacijo, matematika kot mnoštveni univerzum vseh danih aksiomov, teoremov, tehnik, interpretacij in sistemov (teorija množic, teorija kategorij itn.) pa tvori nekonsistentno mnoštvo. Na tej podlagi je mogoèe razmerje med filozofijo in matematiko interpretirati kot razmerje med metastrukturo in situacijo. S pomoèjo Eastonovega teorema pridemo do sklepa, da filozofski koncepti na ravni me-tastrukture »kvantitativno« presegajo elemente, ki pripadajo matematiki kot ontologiji. Filozofija kot metaontologija zato lahko pokaže meje matematike kot ontologije. Roland Bolz Mathematics is Ontology? A Critique of Badiou’s Ontological Framing of Set Theory Key words: Badiou, set theory, ontology, mereology, multiple, Lucretius, Plato This article develops a criticism of Alain Badiou’s assertion that “mathematics is on­tology.” I argue that despite appearances to the contrary, Badiou’s case for bringing set theory and ontology together is problematic. To arrive at this judgment, I explore how a case for the identification of mathematics and ontology could work. In short, ontology would have to be characterised to make it evident that set theory can contribute to it fundamentally. This is indeed how Badiou proceeds in Being and Event. I review his descriptions of the ontological problematic at some length here, only to argue that set theory is a poor fit. Although philosophers working on questions of being were certainly occupied with matters of oneness and the part-whole relationship, I argue that Badi­ou’s discussion of philosophical sources points towards a mereological treatment, not a set-theoretic one. Finally, I suggest that Badiou’s philosophical interpretations of key set-theoretic results are better understood as some sort of analogising between mathe­matics, ontology, and philosophical anthropology. Roland Bolz Je matematika ontologija? Kritika Badioujeve ontološke razlage teorije množic Kljuène besede: Badiou, teorija množic, ontologija, mereologija, mnoštvo, Lukrecij, Platon Avtor èlanka kritizira trditev Alaina Badiouja, da je “matematika ontologija”, pri èemer pokaže, da je kljub videzu o nasprotnem Badioujevo prizadevanje povezati teorijo mno­žic in ontologijo problematièno. Zato da bi utemeljil to sodbo, avtor raziskuje, kako bi lahko delovala identifikacija matematike z ontologijo. Za to bi bilo treba ontologijo opre­deliti na naèin, da bi bilo nedvomno, kako je lahko teorija množic temeljni prispevek k ontologiji. To je dejansko naèin, kako ravna Badiou v Biti in dogodku. Avtor prièujoèega besedila podrobno analizira Badioujeve opise ontološke problematike in na njeni pod-lagi pokaže, da teorija množic ni skladna z vlogo, ki ji jo je Badiou namenil. Èeprav so se filozofi, ki so se ukvarjali z vprašanjem biti, nedvomno ukvarjali tudi z vprašanjem enosti in razmerja del-celota, avtor prispevka zatrjuje, da vodi Badioujeva razprava o filozofskih virih prej k mereološkemu pristopu kot k pristopu teorije množic. V sklepu zato avtor zatrdi, da je treba Badioujeve interpretacije nekaterih kljuènih rezultatov te­orije množic razumeti kot nekakšno analogijo med matematiko, ontologijo in filozofsko antropologijo. Tzuchien Tho Sets, Set Sizes, and Infinity in Badiou’s Being and Event Key words: mathematical ontology, ordinality, cardinality, transfinite sum, limit ordinal, subtractive ontology, numerosity This paper argues that Cantorian transfinite cardinality is not a necessary assumption for the ontological claims in Badiou’s L’Etreetl’Événement(Vol. 1). The necessary struc­ture for Badiou’s mathematical ontology in this work was only the ordinality of sets. The method for reckoning the sizes of sets was only assumed to follow the standard Can-torian measure. In the face of different and compelling forms of measuring non-finite sets (following Benci and Di Nasso, and Mancosu), it is argued that Badiou’s project can indeed accommodate this pluralism of measurement. In turn, this plurality of measure­ment implies that Badiou’s insistence on the “subtraction of the one”, the move to affirm the unconditioned being of the “inconsistent multiple”, results in the virtuality of the one, a pluralism of counting that further complicates the relationship between the one and the multiple in the post-Cantorian era. Tzuchien Tho Množice, velikosti množic in neskonènost v Badioujevi Biti in dogodku Kljuène besede: matematièna ontologija, ordinalnost, kardinalnost, transfinitna vsota, mejni ordinal, subtraktivna ontologija, numeroznost Avtor prispevka zagovarja tezo, da Cantorjeva transfinitna kardinalnost ni nujna pred­postavka za ontološke trditve iz Badioujeve L’Etre et l’Événement (zv. 1). Nujna podlaga za Badioujevo matematièno ontologijo v tem delu je bila zgolj ordinalnost množic. Glede metode za ugotavljanje velikosti množic pa je bila zgolj domneva, da sledijo standardni Cantorjevi meri. Upoštevajoè razliène preprièljive oblike merjenja nefinitnih množic (kot so jih razvili Benci in Di Nasso ter Mancosu), zagovarjamo stališèe, da Badioujev projekt ni neskladen s tem pluralizmom merjenja. Ravno nasprotno, trdimo, da ta plu­ralizem merjenja implicira, da Badioujevo vztrajanje na »odtegnitvi enega«, nujno za zatrditev brezpogojne biti »nekonsistentnega mnoštva«, vodi v virtualnost enega, plu­ralizem štetja, s èimer se še bolj zaplete razmerje med enim in mnoštvom v pocantorje­vski dobi. Charles Alunni Relation-Object and Onto-logy, Sets, or Categories. Identity, Object, Relation Key words: set theory, category theory, ontology, diagram, duality, Plato, Aristotle This article puts into perspective Badiou’s differing approach to set theory and the theory of categories and shows how the distinctions and protocols specific to these fields affect his philosophical thought. It also questions the evolution of Badiou’s approach to these two mathematical theories. The article concludes with questions and examinations re­lated to some of the key points of Badiou’s philosophical orientation that he has not yet addressed. Charles Alunni Razmerje-objekt in onto-logija, množice ali kategorije. Identiteta, objekt, razmerje Kljuène besede: teorija množic, teorija kategorij, ontologija, diagram, dvojnost, Platon, Aristotel Prièujoèi prispevek postavi v doloèeno perspektivo Badioujev selektivni pristop k teoriji množic in teoriji kategorij in pokaže, kako razloèevanja in protokoli, ki so znaèilni za ti polji, zadevajo njegovo filozofsko misel. Tu gre tudi za vprašanje evolucije Badioujevega pristopa k tema dvema matematiènima teorijama. Èlanek sklene z vprašanji in raziska­vami, ki so povezane z nekaterimi kljuènimi toèkami Badioujeve filozofske usmeritve, s katerimi se doslej ni ukvarjal. René Guitart Infinity between Two Ends. Dualities, Algebraic Universes, Sketches, Diagrams Key words: algebraic universe, being, Cantor, category, diagram, duality, form, ontology, phenomenology, power set, limit, shape, sketch, structuralism, Zeno. The article affixes a resolutely structuralist view to Alain Badiou’s proposals on the infi­nite, around the theory of sets. Structuralism is not what is often criticized, to administer mathematical theories, imitating rather more or less philosophical problems. It is rather an attitude in mathematical thinking proper, consisting in solving mathematical prob­lems by structuring data, despite the questions as to foundation. It is the mathematical theory of categories that supports this attitude, thus focusing on the functioning of math­ematical work. From this perspective, the thought of infinity will be grasped as that of mathematical work itself, which consists in the deployment of dualities, where it begins the question of the discrete and the continuous, Zeno’s paradoxes. It is, in our opinion, in the interval of each duality -“between two ends”, as our title states -that infinity is at work. This is confronted with the idea that mathematics produces theories of infinity, infinitesimal calculus or set theory, which is also true. But these theories only give us a grasp of the question of infinity if we put ourselves into them, if we practice them; then it is indeed mathematical activity itself that represents infinity, which presents it to thought. We show that tools such as algebraic universes, sketches, and diagrams, allow, on the one hand, to dispense with the “calculations” together with cardinals and ordinals, and on the other hand, to describe at leisure the structures and their manipulations thereof, the indefinite work of pasting or glueing data, work that constitutes an object the actual infin­ity of which the theory of structures is a calculation. Through these technical details it is therefore proposed that Badiou envisages ontology by returning to the phenomenology of his “logic of worlds”, by shifting the question of Being towards the worlds where truths are produced, and hence where the subsequent question of infinity arises. René Guitart Neskoènost med dvema koncema. Dvojnosti, algebrski univerzumi, skice, diagrami Kluène besede: algebrski univerzum, bit, Cantor, kategorija, diagram, dvojnost, forma, ontologija, fenomenologija, kardinalnost, meja, oblika, skica, strukturalizem, Zenon Avtor tega prispevka prepozna v Badioujevi obravnavi neskonènega na podlagi teorije množic odloèno strukturalistièen pristop. Strukturalizem ni tisto, kar pogosto kritizira­jo, se pravi, naèin, kako uporabiti matematiène teorije na bolj ali manj filozofske probl-me. Prej gre za stališèe v samem matematiènem mišljenju, kar je razvidno iz reševanja matematiènih problemov s strukturiranimi podatki, in to ne oziraje se na vprašanja ute­meljitve. Tako stališèe podpira matematièna teorija kategorij, kar omogoèa osredinjenje na funkcioniranje matematiènega dela. Na podlagi take perspektive je mogoèe mišlje­nje neskonènosti dojeti kot mišljenje samega matematiènega dela, ki sestoji iz razgri­njanja dvojnosti, zaèenši z vprašanjem razloèenega in kontinuiranega ter Zenonovimi paradoksi. Avtor pri tem zagovarja stališèe, da je v intervalu vsake dvojnosti – »med dvema koncema«, kot je zapisano v našem naslovu – mogoèe videti neskonènost na delu. Temu stališèu je mogoèe nasproti postaviti predstavo, da matematika proizvaja teorije neskonènosti, naj gre za infinitezimalni raèun ali teorijo množic, kar je seveda res. Toda te teorije nam zgolj omogoèajo razumeti vprašanje neskonènosti, èe se vanje umestimo, èe jih prakticiramo. Edino v tem primeru je dejansko matematièna dejavnost tista, ki reprezentira neskonènost, kot jo nato prezentira mišljenju. V tem prispevku bi radi pokazali, da orodja, kot so algebrski univerzumi, skice, diagrami omogoèajo, da se ognemo »raèunanju«, ki je vedno povezano s kardinali in ordinali, oziroma da opišemo strukture in rokovanje z njimi, neskonèno in nedoloèno delo lepljenja podatkov, delo, ki na ravni objektov konstituirajo aktualno neskonènost, pri èemer je teorija struktur kalkulacija te neskonènosit. Na podlagi teh tehniènih podrobnost avtor na koncu pre­dlaga Badiouju, da ontologijo obravnava tako, da se vrne k fenomenologiji njegovih »logik svetov«, s èimer bi se vprašanje biti premestilo na podroèje svetov, kjer so resnice proizvedene, se pravi tam, kjer se postavlja vprašanje neskonènosti. David Rabouin Space and Number: Two Ways in Ontology? Key words: Badiou ontology, Set theory, Category theory, homotopy theory, number, space, history of mathematics In this paper, I pursue a dialogue initiated with the publication of Logiques des mondes on the basis of three main lines of questioning: 1. The first, most immediate one, is the meaning that should be given to the famous motto “mathematics = ontology”. Indeed, it is a different statement to claim that “mathematics is ontology”, as was promoted explicitely by Being and the event, and to say that set theory alone is ontology (as ad­vanced by Logiquesdes mondes, as well as other contemporary texts). It seems that there is at this point an important inflection of the system, not thematized as such; is set theory a way of expressing ontology, i.e. mathematics, or is it ontology itself? 2. This leads to a broader questioning of the relationship, in mathematics, between expression and ontology, or “language” and “being”. Here I would like to point out that, contrary to what one might think, there is often an ambiguity between these two aspects not only in Badiou, but more generally in discussions of the philosophy of mathematics. If this distinction is relevant - and I will try to show why it should be - then one cannot conclude too quickly from the fact that mathematics has adopted a unified expression thanks to the language of set theory to the fact that the form of being it expresses is set-theoretic (or “pure multiple” in Badiou’s terminology); 3. Finally, I would like to delve into the fact that the set-theoretic language has precisely given rise to the thematization of two orientations which could be just as well coined “ontological” (but in a different sense, therefore, from that given to it by Badiou); the first is anchored in the concept of number, while the other is anchored in the concept space (later called “topological”). The fact that we have a language capable of describing them in a homogeneous fashion does not entail that we are dealing with a single domain of objects. I would like to show that this tension runs through contemporary mathematics, and consequently through Alain Badiou’s thinking more than he wants to admit. In fact, it is at the basis of various attempts proposed in mathematics to arrive at more satisfactory forms of unification than that provided by “sets” alone. David Rabouin Prostor in število: dve poti v ontologiji? Kljuène besede: Badiou, ontologija, teorija množic, teorija kategorij, teorija homotopij, število, prostor, zgodovina matematike V prièujoèem prispevku avtor nadaljuje dialog, ki se je zaèel z objavo Logik svetov, izha­jajoè iz treh glavnih linij spraševanja: 1. Prva in najbolj neposredna zadeva pomen, ki ga je treba dati slovitemu geslu »matematika=ontologija«. In res, nekaj drugega je trditi, da je »matematika ontologija«, kot je bilo izrecno razglašeno v Biti in dogodku, in na tej podlagi zatrditi, da je edino teorija množic ontologija (kot je zatrjeno v Logikah svetov in drugih soèasnih tekstih). Zdi se, da gre na tej toèki za pomemben premik v sistemu, ki pa kot tak ni tematiziran; ali je teorija množic naèin izražanja ontologije, tj. matematike, ali pa je teorija množic že kar sama ontologija? 2. To vodi k širšemu preiskovanju razmerja znotraj matematike med izražanjem in ontologijo, oziroma med »jezikom« in »bitjo«. V nasprotju s tem, kar bi lahko mislili, avtor opozarja, da imamo tu pogosto opravka z am-bigviteto teh dveh vidikov, in to ne zgolj pri Badiouju, temveè, splošneje, v diskusijah, ki potekajo znotraj filozofije matematike. Èe je to razlikovanje pomembno, in avtorjev namen je pokazati, zakaj bi moralo biti, potem bi lahko – na podlagi dejstva, da je mate-matika privzela enoten izraz, zahvaljujoè jeziku teorije množc, prehitro sklepali, da forma biti, ki jo izraža, dejansko je forma teorije množic (oziroma »èisto mnoštvo«, èe upora­bimo Badioujevo terminologijo). 3. V sklepnem delu se avtor loti vprašanja jezika teorije množic, kar je omogoèilo tematizacijo dveh usmeritev, ki bi ju ravno tako lahko poime­novali »ontološka« (vendar v drugem pomenu, kot je temu izrazu dal Badiou); prva se opira na pojem števila, druga pa na pojem prostora (slednjega so pozneje preimenovali v »topološki«). Dejstvo, da imamo na voljo jezik, ki ju je zmožen opisati na homogen naèin, še ne pomeni, da imamo opraviti z enim samim podroèjem objektov. Avtorjev namen je pokazati, da je ta napetost navzoèa v sodobni matematiki in zato tudi v Badioujevem mišljenju, in to veliko bolj, kot je sam pripravljen prizanati. Avtor zagovarja stališèe, da je na podlagi razliènih prizadevanj v matematiki dejansko mogoèe priti do bolj zadovoljivih oblik poenotenja, kot je tisto, ki so ga ponudile zgolj »množice«. Norman Madarasz Beyond Recognition: Badiou’s Mathematics of Bodily Incorporation Key words: ontology, objective phenomenology, body of truth, category theory, topos theory In Being and Event, Alain Badiou disconnects the infinite from the One and the Abso­lute, thus recasting the basis from which to craft a new theory of generic subject, the existence of which is demonstrated through set theory. In Logics of Worlds, Badiou turns his attention to the modes by which this subject appears in a world. It does so by being incorporated as a subjectivizable body, a body of truth. As opposed to Being and Event, the demonstration of this argument takes shape according to two distinct levels, that of a “calculated phenomenology” and that of a formalism in which category theory provides a general logic, the combination of which delineates an “onto-logic”. In this essay, we trace Badiou’s derivation of the notion of body of truth and evaluate the innovative phe­nomenological methodology applied to explain its association with a world. Norman Madarasz Onstran pripoznanja: Badioujeva matematika telesne inkorporacije Kljuène besede: ontologija, objektivna fenomenologija, telo resnice, teorija kategorij, teorija toposov V Biti in dogodku Badiou razveže neskonèno, Eno ter absolut, s èimer ustvari podlago za razdelavo nove teorije generiènega subjekta, èigar eksistenco je mogoèe dokazati s pomoèjo teorije množic. V Logikah svetov pa se Badiou ukvarja z naèini, kako se ta su­bjekt pojavlja v svetu, in sicer prek inkorporacije v subjektivizabilno telo, telo resnice. V nasprotju z Bitjo in dogodkom dokazovanje tega argumenta poteka na dveh loèenih rav­neh, na ravni »kalkulirane fenomenologije« in na ravni formalizma, za katerega teorija kategorij ponuja splošno logiko, povezava obeh ravni pa omogoèi zaris »onto-logike«. V prièujoèem besedilu sledimo Badioujevi izpeljavi pojma telesa resnice in osvetlimo upo­rabljeno inovativno fenomenološko metodologijo, s pomoèjo katere je pojasnjena vez med telesom resnice in svetom. Frank Ruda To the End: Exposing the Absolute Key words: absolute, appearance, Badiou, Being, forcing, freedom, truth(s) This article reconstructs Badiou’s oeuvre from the vantage point of the last volume of his Being and Eventtrilogy. It determines the perspective and task of The Immanence of Truths and demonstrates in what way an “absolute ontology” is the result of an act of forcing that serves as the measure for a truth’s truthness. This allows to argue that Ba­diou’s is and always was a philosophy of freedom. Frank Ruda Do konca: razkrivanje absoluta Kljuène besede: absolut, videz, Badiou, bit, izsiljenje, svoboda, resnic(e) Prièujoèi prispevek rekonstruira celotno Badioujevo delo, izhajajoè iz zadnjega dela tri­logije Biti in dogodka. Opredeli perspektivo in nalogo Imanence resnic in pokaže, kako je lahko »absolutna ontologija« rezultat dejanja izsiljenja, ki je hkrati mera za resniènost resnice. Na tej podlagi lahko nato zatrdimo, da je Badioujeva filozofija, kar je vedno tudi bila, filozofija svobode. Jana Ndiaye Berankova The Immanence of Truths and the Absolutely Infinite in Spinoza, Cantor, and Badiou Key words: Badiou, Cantor, the absolute, The Immanence of Truths, large cardinals, Spinoza, Being and Event, philosophy and mathematics The following article compares the notion of the absolute in the work of Georg Cantor and in Alain Badiou’s third volume of Being and Event: The Immanence of Truths and proposes an interpretation of mathematical concepts used in the book. By describing the absolute as a universe or a place in line with the mathematical theory of large cardinals, Badiou avoided some of the paradoxes related to Cantor’s notion of the “absolutely infi­nite” or the set of all that is thinkable in mathematics W: namely the idea that W would be a potential infinity. The article provides an elucidation of the putative criticism of the statement “mathematics is ontology” which Badiou presented at the conference Thinking the Infinite in Prague. It emphasizes the role that philosophical decision plays in the con­struction of Badiou’s system of mathematical ontology and portrays the relationship be­tween philosophy and mathematics on the basis of an inductive not deductive reasoning. Jana Ndiaye Berankova Imanenca resnic in absolutno neskonèno pri Spinozi, Cantorju in Badiouju Kljuène besede: Badiou, Cantor, absolut, Imanenca resnic, veliki kardinali, Spinoza, Bit in dogodek, filozofija in matematika Prièujoèi èlanek primerja pojem absoluta v delu Georga Cantorja in v tretjem zvezku Biti in dogodka: Imanenci resnic Alaina Badiouja in ponudi interpretacijo matematiènih kon­ceptov, ki so uporabljeni v tem delu. S tem ko Badiou opiše absolut kot univerzum ali me-sto, v skladu z matematièno teorijo velikih kardinalov, se izogne nekaterim paradoksom, ki so bili povezani s Cantorjevim pojmom »absolutnega neskonènega« oziroma mnoštva vsega, kar je mislivo v matematiki, W: predstavo, da je lahko W potencialna neskonènost. Èlanek pojasni Badioujevo domnevno samokritiko trditve »matematika je ontologija«, kot jo je Badiou predstavil na konferenci Thinking the Infinite v Pragi. Èlanek poudari vlogo, ki jo ima filozofska odloèitev pri konstruiranju Badioujevega sistema matematiène ontologije in predstavi razmerje med filozofijo in matematiko na podlagi induktivnega in ne deduktivnega sklepanja. Norma M. Hussey A New Hope for the Symbolic, for the Subject Key words: ontology, metaontology, symbolization, subjectivity, disorientation, mathematical infinity, set theory, Platonism, pluralism, large cardinal theory, inner model, ultimate-L, universality theorem, supercompact cardinal This paper is perhaps an impressionistic response to accounts of the extraordinary set-theoretical activity being undertaken by W. Hugh Woodin (mathematician) and col­leagues in the present moment, in the context of the mathematical ontology proposed and elaborated by Alain Badiou (philosopher). The argument presented is that the pre­vailing and sustained incoherence of the mathematical ontology (i.e. set theory) under­scores a contemporary deficit of humanity’s symbolic organization which, in turn, yields confusion and conflict in terms of subjective orientation. But a new axiom (conjectured as yet) promises to realize a coherent set theory, i.e. stable, consistent and complete. This remarkable (and completely unexpected) development offers hope for the pursuit of a modern (i.e. non-hierarchical) symbolic, and a consequent resolution of the general subjective disorientation. Norma M. Hussey Novo upanje za simbolno, za subjekta Kljuène besede: ontologija, metaontologija, simbolizacija, subjektivnost, dezorientacija, matematièno neskonèno, teorija množic, platonizem, pluralizem, teorija velikih kardinalov, notranji model, poslednji-L, teorem univerzalnost, superkompakten kardinal Prièujoèi prispevek je morda impresionistièni odziv na izjemno dejavnost, ki smo ji prièa na podroèju teorije množic, zahvaljujoè predvsem delu (matematika) W. Hugha Woodina in sodelavcev v navezavi na problem matematiène ontologije, kot jo je ponudil in razdelal (filozof) Alain Badiou. Argument, ki ga želimo predstaviti je, da prevladujoèa in vztrajna nekoherentnost matematiène ontologije (tj. teorije množic) izpostavi in pojasni današnji manko na podroèju simbolne organizacije èloveštva, ki vodi v zmedo in konflikt med su­bjektivnimi orientacijami. Vendar pa novi aksiom (ki je ta hip na ravni hipoteze oziroma predpostavke) obljublja izdelavo koherentne teorije množic, ki bo stabilna, konsistentna in popolna. Ta izjemni in popolnoma neprièakovani razvoj je podlaga za upanje za obli­kovanje sodobnega (tj. nehierarhiènega) simbolnega in posledièno za razrešitev prevla­dujoèe subjektivne dezorientacije. Fernando Zalamea An Elementary Peircean and Category-Theoretic Reading of Being and Event, Logics of Worlds, and The Immanence of Truths Key words: mathematics, philosophy, set theory, category theory, large cardinals, Peirce The article presents a reading of Badiou’s trilogy, L’Etre et l’événement (1988), Logiques des mondes (2006), and L’Immanence des vérités (2018), and points out the mathematical connections with the works of Cohen, Grothendieck, and large cardinal specialists. A synthetic rendering of these connections is first offered, following precise passages in Badiou’s work, then a category-theoretic and Peircean perspective is explored in order to specify the many dialectics in the trilogy, and, finally, some open problems are proposed. Fernando Zalamea Elementarno peircejevsko in kategorijsko teoretsko branje Biti in do-godka, Logik svetov in Imanence resnic Kljuène besede: matematika, filozofija, teorija množic, teorija kartegorij, veliki kardinali, Peirce V prièujoèem èlanku predstavimo braje Badioujeve trilogije L’Etre et l’événement (1988), Logiques des mondes (2006), in L’Immanence des vérités (2018), pri èemer izpostavimo matematiène povezave z deli Cohena, Grothendiecka in specialistov za velike kardinale. Ponudimo sintetièni pogled na te povezave, pri èemer sledimo natanènim prehodom v Bdioujevem delu, zato da bi se potem obrnili h kategorijsko teoretski in peircejevski per-spektivi, ki jo obravnavamo, zato da bi natanèneje opredelili veè dialektik, ki so dejavne v trilogiji, na koncu pa opozorimo na nekaj odprtih problemov. Obvestilo avtorjem Prispevki so lahko v slovenskem, angleškem, franco­skem ali nemškem jeziku. Uredništvo ne sprejema prispevkov, ki so bili že obja­vljeni ali istoèasno poslani v objavo drugam. Prispevki naj bodo pisani na IBM kompatibilnem ra-èunalniku (v programu Microsoft Word). Priložen naj bo izvleèek (v slovenšèini in anglešèini), ki povzema glavne poudarke v dolžini do 150 besed in do 5 kljuènih besed (v slovenšèini in anglešèini). Za oddajo prispevkov prosimo sledite navodilom: http://ojs.zrc-sazu.si/filozofski-vestnik/information/ authors. Prispevki naj ne presegajo obsega ene in pol avtorske pole (tj. 45.000 znakov s presledki) vkljuèno z vsemi opombami. Zaželeno je, da so prispevki razdeljeni na razdelke in opremljeni z mednaslovi. V besedilu dosledno uporabljajte dvojne narekovaje (npr. pri nava­janju naslovov èlankov, citiranih besedah ali stavkih, tehniènih in posebnih izrazih), razen pri citatih znotraj citatov. Naslove knjig, periodike in tuje besede (npr. a priori, epoché, élan vital, Umwelt, itn.) je treba pisati ležeèe. Opombe in reference se tiskajo kot opombe pod èrto. V besedilu naj bodo opombe oznaèene z dvignjenimi indeksi. Citiranje naj sledi spodnjemu zgledu: 1. Gilles-Gaston Granger, Pour la connaissance philo­sophique, Odile Jacob, Pariz 1988, str. 57. 2. Cf. Charles Taylor, “Rationality”, v: M. Hollis, S. Lukes (ur.), Rationality and Relativism, Basil Blackwell, Oxford 1983, str. 87–105. 3. Granger, str. 31. 4. Ibid., str. 49. 5. Friedrich Rapp, “Observational Data and Scientific Progress”, Studies in History and Philosophy of Science, Oxford, 11 (2/1980), str. 153. 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To submitt manuscript please follow instructions: http://ojs.zrc-sazu.si/filozofski-vestnik/information/ authors. A brief biographical note indicating the author’s in­stitutional affiliation(s), works published and central subject of professional interest should also be enclosed. Manuscripts should not exceed 8,000 words (45,000 characters with spaces) including notes. Papers should be sectioned with clearly marked subheadings. Use double quotation marks throughout the text (e.g. for titles of articles, quoted words or phrases, technical terms), except for quotes within quotes. Titles of books and periodicals, and foreign words (e.g. a priori, epoché, élan vital, Umwelt, etc.) should be in italics. Note numbers should be referred to in the text by means of superscripts. Citations should be presented as follows: 1. Gilles-Gaston Granger, Pour la connaissance philo­sophique, Odile Jacob, Paris 1988, p. 123. 2. Cf. Charles Taylor, “Rationality”, in: M. Hollis, S. Lukes (Eds.), Rationality and Relativism, Basil Blackwell, Oxford 1983, pp. 87–105. 3. Granger, p. 31. 4. Ibid., p. 49. 5. Friedrich Rapp, “Observational Data and Scientific Progress”, Studies in History and Philosophy of Science, Oxford, 11 (2/1980), p. 153. The author-date system is also acceptable with a text reference reading. References in the text are then made as follows: (author’s last name, date: page(s) or sec­tion). Detailed bibliographical information should be given in a separate alphabetical list at the end of the manuscript. Articles will be externaly peer-reviewed. Proofs will be sent to authors. They should be corrected and returned to the Editor as soon as possible. Altera­tions other than corrections of typographical errors will not be accepted. Filozofski vestnik ISSN 0353-4510 Programska zasnova Filozofski vestnik (ISSN 0353-4510) je glasilo Filozofskega inštituta Znanstveno­raziskovalnega centra Slovenske akademije znanosti in umetnosti. Filozofski vestnik je znanstveni èasopis za filozofijo z interdisciplinarno in mednarodno usmeritvijo in je forum za diskusijo o širokem spektru vprašanj s podroèja sod-obne filozofije, etike, estetike, poli tiène, pravne filozofije, filozofije jezika, filozo­fije zgodovine in zgodovine politiène misli, epistemologije in filozofije znanosti, zgodovine filozofije in teoretske psihoanalize. Odprt je za razliène filozofske usme­ritve, stile in šole ter spodbuja teoretski dialog med njimi. Letno izidejo tri številke. Druga številka je posveèena temi, ki jo doloèi uredniški odbor. Prispevki so objavljeni v angleškem, francoskem in nemškem jeziku s pov­zetki v angleškem in slovenskem jeziku. Filozofski vestnik je vkljuèen v: Arts & Humanities Citation Index, Current Con­tents / Arts & Humanities, EBSCO, DOAJ, IBZ (Internationale Bibliographie der Zeitschriften), The Philosopher's Index, Répertoire bibliographique de philosop-hie, Scopus in Sociological Abstracts. Izid revije je finanèno podprla Javna agencija za raziskovalno dejavnost Repu­blike Slovenije. Filozofski vestnik je ustanovila Slovenska akademija znanosti in umetnosti. Aims and Scope Filozofski vestnik (ISSN 0353-4510) is edited and published by the Institute of Phi­losophy of the Scientific Research Centre of the Slovenian Academy of Sciences and Arts. Filozofski vestnik is a philosophy journal with an interdisciplinary character. It provides a forum for discussion on a wide range of issues in contemporary polit­ical philosophy, history of philosophy, history of political thought, philosophy of law, social philosophy, epistemology, philosophy of science, cultural critique, ethics, and aesthetics. The journal is open to different philosophical orientations, styles and schools, and welcomes theoretical dialogue among them. Three issues of the journal are published annually. The second issue is a special issue that brings together articles by experts on a topic chosen by the Editorial Board. Articles are published in English, French, or German, with abstracts in Slove­nian and English. Filozofski vestnik is indexed/abstracted in the Arts & Humanities Citation Index; Current Contents / Arts & Humanities; DOAJ; EBSCO; IBZ (Internationale Bibli­ographie der Zeitschriften); The Philosopher's Index; Répertoire bibliographique de philosophie; Scopus; and Sociological Abstracts. Filozofski vestnik is published with the support of the Slovenian Research Agency. Filozofski vestnik was founded by the Slovenian Academy of Sciences and Arts. Alain Badiou,Ontologie et mathématiques : Théorie des Ensembles, théorie des Catégories, et théorie des Infinis, dans L'Etre et l'événement, Logiques des mondes et L'Immanence des vérités Le Triangle philosophie – mathématiques – psychanalyse The Triangle of Philosophy – Mathematics – Psychoanalysis Oliver Feltham,“One or Many Ontologies? Badiou’s Arguments for His Thesis ‘Mathematics is Ontology’” Nick Nesbitt,Bolzano’s Badiou Jelica Šumiè Riha,La place de la mathématique : Badiou avec Lacan Le Modele ensembliste en discussion The Set-theoretical Model under Discussion Michael Hauser,Badiou and the Ontological Limits of Mathematics Ronald Bolz,Mathematics is Ontology? A Critique of Badiou's Ontological Framing of Set Theory Tzuchien Tho,Sets, Set Sizes, and Infinity in Badiou's Being and Event Le « Voir » et le « dire »: théorie des ensembles / théorie des categories “Seeing” and “Saying”: Set Theory / Category Theory Charles Alunni,Relation-objet et onto-logie, ensembles ou categories. Identité, objet, relation René Guitart,L’infini entre deux bouts. Dualités, univers algébriques, esquisses, diagrammes David Rabouin,Espace et nombre : deux voies dans l’ontologie ? Norman Madarasz,Beyond Recognition: Badiou’s Mathematics of Bodily Incorporation Grands Cardinaux et attributs de l'absolu Large Cardinals and the Attributes of the Absolute Frank Ruda,To the End: Exposing the Absolute Jana Ndiaye Berankova,The Immanence of Truths and the Absolutely Infinite in Spinoza, Cantor, and Badiou Norma Hussey,A New Hope for the Symbolic, for the Subject Fernando Zalamea,An Elementary Peircean and Category-Theoretic Reading of Being and Event, Logics of Worlds, and The Immanence of Truths