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Filozofski vestnik | Volume XLI | Number 1 | 2020 | 249–282 | doi: 10.3986/fv.41.1.10
* Centre National de la Recherche Scientifique (CNRS), France; (2) Institut Franco-Argentin d'Études sur le 
Climat et ses Impacts, UMI 3351 (CNRS-CONICET-UBA-IRD), Buenos Aires, Argentina
Denisse Sciamarella*
A topological reading of Ernesto Laclau
Pitagora: La regola sta al principio.
Eco: Ma trasporti questo atteggiamento nella vita sociale e 
politica. Cosa ne nasce? Una visione aristocratica e conserva-
trice. Non a caso Lei ha dovuto fuggire da Crotone, perché il 
partito democratico vedeva nella Sua scuola un centro di pen-
siero aristocratico e reazionario. Nella vostra fiducia nelle leg-
gi eterne del mondo voi pitagorici non potevate comprendere 
la mutazione, [...] che la realtà nasce anche dal dolore, dalla 
lotta, che l’armonia è un punto d’arrivo, sempre provvisorio, 
ma guiai a considerarla un punto di partenza, definitivo.
Pitagora: Dunque non hai capito.
U. Eco, Pitagora
1
 
Introduction
Ernesto Laclau asserts that the category of ‘relation’ is central to his analysis, 
and that this distinguishes his approach from other contemporary theories. In 
his own words, “Alain Badiou, for instance, sees set theory as the terrain of a 
fundamental ontology. Given the centrality to set theory of the notion of exten-
sionality, however, the category of relation can, at best, play only a marginal 
role.”
2
 The centrality to topology of the notion of relation
3
 suggests that a top-
ological reading of Laclau may be fruitful. This work is an attempt to relate La-
clausian categories and topology from the particular perspective of the theory 
of dynamical systems. Such theory has instability as its central category, but 
in order to study instability, dynamicists resort to topology. The categories of 
1
 Umberto Eco a Pitagora, Le interviste impossibili, Bompiani 1975, available at: http://www.
giutor.com/doc/pitagora/eco-pita.html.
2
 Ernesto Laclau, On Populist Reason, Verso, London and New York 2005, p. 68.
3
 Jean Ladrière, El reto de la racionalidad, La ciencia y tecnologia frente a las culturas. Edi-
ciones Sigueme, Salamanca/UNESCO, Paris 1978, p. 31. 

250
denisse sciamarella
relation and instability are therefore simultaneously central to the marriage be-
tween topology and dynamics.
The methods used to build a topological structure associated with a system’s 
dynamics are developed in the so-called “topological program”.
4
 This work con-
structs an analogical correspondence between concepts of the topological pro-
gram and the main categories in Laclausian theory. The possibility of drawing a 
parallel between such disparate areas of thought relies on the fact that both ap-
proaches constitute efforts to describe structures resulting from an articulatory 
practice. The starting point is, in both cases, a series of disaggregate elements. A 
remark is necessary before we embark on this task. The analogy is not intended 
to reduce theoretico-political categories to a mathematical scheme, or to postu-
late the topological program as a Laclausian matheme. The aim is to outline a 
two-way relationship between the two conceptual fields that leaves neither of 
them unaffected. If the analogy unfolds a productive set of theoretical affinities, 
these may both lead to alternative formulations of some problems in political 
theory, and contribute to deconstructing the nature, conception, and uses of 
mathematical tools.
5
This article is organised as follows. An introduction to the topological program 
is provided in Section 1. Section 2 is devoted to the premises of the analogy. 
Conceptual correspondences are presented in Section 3. The productivity of the 
analogy is discussed in Section 4. 
Section 1. Topology and Dynamical Systems
A dynamical system is a system whose evolution is dictated by rules
6
 which, in 
the more general case and by their mathematical form, combine determinism 
4
 Robert Gilmore and Marc Lefranc, The Topology of Chaos: Alice in Stretch and Squeezeland, 
Wiley, New York 2002. The spelling “program” will be used herein for the sake of consistency. 
5
 Blaise Pascal and Henri Poincaré are some of the precursors in this effort. 
6
 The rule predicts the evolution of the system. More precisely, it predicts the evolution of cer-
tain variables that are assumed to characterise the system. A physical pendulum oscillating 
within a plane, for instance, is a system that can be described using two variables: position 
and velocity. In the example of the pendulum, the rule can be derived from consideration of 
the forces that act upon it. How many variables are needed to adequately describe a prob-
lem? This depends on the problem. Once the variables are chosen, how the problem is to be 
attacked is defined, and this limits what can be said about the system to what can be said 

251
a topological reading of ernesto laclau
with unpredictability. The theory of dynamical systems proposes a set of math-
ematical tools to study the phenomena governed by these kinds of rules (non-
linear rules). Given an evolution rule, initial values are chosen for the variables 
describing the system and the rule is used to compute the values of the variables 
that determine the subsequent states
7
 of the system. For the case of nonline-
ar rules, predictability becomes difficult for one reason: an astonishingly small 
difference in the initial value chosen for a variable makes the system evolve in 
a considerably different way. This is how unpredictability enters the scene with-
out opposing determinism. A future state fails to be predicted not because there 
is no rule governing the system, but because the rule gives rise to radically dif-
ferent results for slightly different initial values. In order to avoid unpredictabil-
ity, infinite precision is needed in the knowledge of the initial state and in the 
application of the rule. But this infinite precision is generally unavailable, not 
only in the case of experimental data, but also for numerical computations. A 
tiny difference that can make the system evolve in remarkably different manners 
is exponentially amplified by the mathematical nature of the rule that governs 
the system, hindering predictability in the long term. The error that is inherent 
to every measure or numerical computation brings “the contingent in the nec-
essary.”
8
 In the field of dynamical systems, this property is known as sensitivity 
to initial conditions. When data are generated with a nonlinear rule, the data 
present a seeming disorder. In other words, the rule governing the data is not 
transparent when the data are examined directly, but is concealed. This is why 
dynamicists talk about unveiling an underlying dynamics from data of this type. 
And this is why it is of interest to learn, for a time series of experimental data 
presenting this seeming disorder, if there exists – or not – an underlying deter-
ministic dynamics. Notice that the finding of an evolution rule for such a system 
will entail an understanding of the mechanisms involved in the behaviour of the 
system. Scientific knowledge is not restricted to prediction. 
about the variables. If the pendulum has a fixed length z and moves within a plane, position 
and velocity are sufficient. Variables are also called state variables or dynamical variables.
7
 A system is in a given state when the ensemble of variables that characterises it adopts a 
value. To correctly define the state of a pendulum with variable length, for instance, values 
should be assigned to three variables: position, velocity, and length. If the length cannot 
change, the length is again a parameter and the state of the system is defined by assigning 
values to two variables: position and velocity.
8
 Ernesto Laclau and Chantal Mouffe, Hegemony and Socialist Strategy, Verso, London and 
New York 2001, p. 114.

252
denisse sciamarella
The topological program devises methods to unveil the type of organisation un-
derlying a data series. The starting point is a time series associated with the 
phenomenon that one is attempting to understand. The rule is not known in ad-
vance, but it is tacitly supposed to exist. In other words, the system is supposed 
to be deterministic. The first step in the analysis consists in representing the 
data in a mathematical space, called phase space, in which time is implicit and 
in which each point stands for a state of the system. Notice that this space does 
not have a predetermined dimension. Its dimension depends on the number of 
significant variables in the problem. Representation in phase space
9
 does not 
focus on when exactly the system visits a certain state, but on whether there are 
privileged states to which trajectories tend to evolve, if there are sets of states 
that are forbidden, and so on. Representation in phase space breaks with the 
synchronic-diachronic dichotomy.
10
When the system evolves from a starting point representing its initial state, it 
visits a sequence of states, leaving a trace in phase space called a trajectory, as 
shown in Fig. 1. A trajectory in phase space is a line that is read in the direction 
in which time evolves. This direction is indicated with an arrow. If the rule that 
governs a system is known, the rule can be used to compute trajectories in phase 
space. Trajectories that obey a certain rule do not circulate freely. Their route is 
dictated by the rule that governs the dynamical system.
9
 Phase space is a geometrical space that enables a representation of the states of the dy-
namical system. Each point in phase space corresponds to a value of the variables that 
characterise the system, i.e. each point in phase space stands for a possible state of the 
system that is defined with those variables. It could also be called ‘state space’. In the case 
of the pendulum, phase space can be constructed using two coordinate axes. One for the 
position of the pendulum (x) and one for the velocity (y). In this case, phase space has two 
dimensions just because two variables are enough to identify a state of the system. If we 
consider the length z as a variable, we would need three coordinate axes (x,y, z). The phase 
space dimension is determined by the number of variables that define a point (a state of 
the system) in phase space. Phase space therefore depends on the number of relevant 
variables of the problem. Let us stress that phase space is a mathematical space that has 
nothing to do with physical space.
10
 Phase space provides neither a photograph of an ensemble of variables, nor a film of suc-
cessively ordered events. If a system recurrently visits a zone in phase space at different 
times, a dense cloud of points will be found in this zone. It is a mathematical space that 
focuses on the representation of dynamical states. 

253
a topological reading of ernesto laclau
In order to illustrate what a privileged state in phase space looks like, let us im-
agine a trajectory ending up in a point. This means that the system has visited a 
certain number of states and reached a state from which it no longer moves. This 
state is called a stationary state. If this point attracts trajectories to it, this point 
is a privileged point in the system, to which the system tends to evolve. The point 
is said to be an attractor of the system. An attracting point absorbs trajectories in 
phase space (Fig. 2a). In the same manner, one can imagine repellers, i.e. states 
in phase space from which the system escapes (Fig. 2b). These privileged states 
structure trajectories in phase space. Sometimes there are groups of states that 
play this role. A limit cycle, for instance, is an attracting set of points in the form 
of a closed curve. Systems that converge to a limit cycle are systems that attain a 
stable and periodic (cyclic) behaviour (Fig. 2c).
Fig. 1. Points forming trajectories in phase 
space for sequences of values for variables 
x(t),y(t),z(t) as they evolve in time (t) for two 
different initial states.
Fig. 2. (a) attracting fixed point; (b) repelling 
fixed point; (c) attracting limit cycle.
(b) (a) (c) 

254
denisse sciamarella
When a trajectory falls into an attractor, it is clear that even if the system is 
slightly pushed away from this state it tends to fall back into it. An attractor 
attracts the points that lie in a certain basin of attraction in phase space. Why is 
this description in terms of attractors useful? Dynamical systems in the physical 
world tend to be dissipative, i.e. change tends to cease unless there is a force 
that is permanently acting upon the system. In those systems in which an acting 
force and dissipation coexist, the system tends to evolve (after a transient) to 
some kind of typical behaviour that is illustrated by attractors in phase space. 
Of course, several attractors can coexist in phase space for a given system. The 
evolution of trajectories is in this case conditioned by the basin of attraction in 
which trajectories are born.
Let us suppose that there is only one variable available for measurement. The 
starting point of the analysis is a time series. This variable may not exhaust the 
number of variables that are necessary to define a particular state of the system. 
In such case, there exist mathematical criteria for deciding the minimum di-
mension needed to host the data, and there are recipes to build supplementary 
series from the data. Once the time series have been embedded in phase space, 
clouds of points or states are obtained.
Let us consider the example in Fig. 3. The cloud of points comes from a rep -
resentation of the pressure values recorded by a microphone when a speaker 
pronounces the sound /a/ in the first vowel of the Spanish word ‘casa’. Two sup-
plementary series are generated to achieve an adequate representation in phase 
space, which is three dimensional in this case. The result is shown in Fig. 3b. 
Fig. 3. Example of a time series (a), embedded 
in phase space (b).
(a) (b) 

255
a topological reading of ernesto laclau
The next step is to construct, from this cloud of points, a sort of skeleton or 
model kit that articulates the points, according to the manner in which they are 
organised. This ‘manner’ is not geometrical but topological: what defines the 
skeleton is not the location of the points in the cloud, but the structure organis-
ing the cloud of points. Such a skeleton has been named a template or branched 
manifold. The term branched manifold makes reference to the fact that the skel-
eton usually takes the form of a structure of connecting branches. This structure 
can have isolated pieces that are not connected to the others; it can also be of 
one piece (one connected component). Each piece can contain branches, disks, 
holes, torsions, etc. The template is thus the graphical representation of a series 
of structural properties, which are topological in nature. 
But how can this skeleton be reconstructed from the disperse points represent-
ing the states visited by the system? One of the methods to construct the skel-
eton
11
 uses a procedure that is quite straightforward.
12
 A sort of mask is con-
structed covering the points displayed in phase space. This mask is built piece 
by piece, as shown in Fig. 4. The first step in the method consists in gather -
ing points within cells or patches that will gradually cover the cloud of points. 
Points are grouped in cells provided there is a certain ‘local affinity’ or equiva-
lence, which is defined in geometrical terms.
13
 Scattered points distributed in 
the cloud are arbitrarily chosen as ‘nodal points’ (Fig. 4a), which will nucleate 
nearby points around them. How many points should be nucleated around a 
nodal point to form a cell? As many as the affinity criterion allows to gather. If 
11
 Denisse Sciamarella and G. B. Mindlin, “Topological Structure of Chaotic Flows from Hu-
man Speech Data”, Physical Review Letters 82 (7/1999).
12
 The method of branched manifold analysis through homologies is presented in Denisse 
Sciamarella and G. B. Mindlin, “Unveiling the Topological Structure of Chaotic Flows 
from Data”, Physical Review E 64 (3/2001). Before this work, alternative approaches were 
based on knot theory. These were not applied to the mask (the manifold) where the data 
lie, but to a prior reconstruction of orbits approximating the trajectories described by 
the data. The reconstruction of orbits is difficult if the time series is noisy, as in Fig. 3b. 
Approaches based on knot theory have the additional drawback of being limited to cases 
in which the dimension of phase space is lower than or equal to three, since knots fall 
apart in higher dimensions.
13
 In the method, the affinity criterion is expressed geometrically in the following manner. 
The group of points in a cell must be a good approximation of a hyperplane of dimension 
d in a space of dimension n.

256
denisse sciamarella
the number of points per cell around the nodal points is sufficiently large, cells 
start overlapping, as sketched in Fig. 4b. This favours concatenation between 
cells. The points linking cells are called vertices. Each cell can now be labelled 
using a sequence of vertices, numbered in Fig. 4c. Notice that cells are polygons 
that do not necessarily adopt the form of the points that are gathered to consti-
tute a given cell, but this does not matter, because even if the criterion applied to 
the points to construct the cells is geometrical, a given cell is not a geometrical 
approximation of the points, but a topological approximation of the structure in 
which the points are inscribed. This point will be central to the analogy: cells 
represent groups of points without coinciding with the points, or with the fron-
tier points, or with the trajectories that the points are supposed to make up.
Together, the cells constitute a cell complex. The complex inherits the structure 
of the embedded data regardless of the individual trajectories, which are not 
even reconstructed. The equivalential bond between points operates locally, ig-
noring the trajectory to which the point belongs, and by virtue of a criterion that 
is ultimately arbitrary: the laxer the criterion, the greater the affinity between 
the points and the higher the chances of conforming to a one-component com-
plex. This ‘loss of memory’ of the individual trajectories is not destructive. The 
cell is transparent and makes up a parallel structure, which is constructed to 
express the logic of an organisation related to a particular dynamics. It is clear 
that if the criterion is too rigid or too strict it will hinder the construction of the 
complex. Nodal points are also arbitrary: a different collection of nodal points 
may lead to a different distribution of cells. 
The model kit resists this variability because the aim is to reconstruct the un-
derlying topology. Once the vertices are labelled with numbers, the labels can 
be used to denote paths along the complex. One of the paths of Fig. 4c, for in -
stance, is obtained with the cyclic sequence: 1,2,6,9,19,12,5,1.
14
 A chain is any se-
quence of elements of the cell complex. In this case, the cell complex has 0-cells 
(vertices), 1-cells (lines), and 2-cells (polygons). Sequential chains are important 
14
 Vertices are also called 0-cells, and chains of vertices are called 1-cells. Chains can also be 
formed with sequences of cells, e.g. <<2,3,7,6>,<9,8,7,6>, <10,8,9,11>>, called 2-chains. The 
number in 0-cell or 1-chain denotes the dimension of the element of the complex. A 0-cell 
is a point, while a 1-cell or 1-chain is respectively a segment or sequence of consecutive seg -
ments. A 2-cell or 2-chain is a polygon or sequence of concatenated polygons, respectively.

257
a topological reading of ernesto laclau
because the organisation of the complex is provided by the manner in which 
cells are connected to other cells.
In order to determine the topological properties of the complex, another equiv-
alential relation is needed. This equivalential relation is called homology. It al-
lows the determination of in how many non-redundant ways the complex can 
be visited, so that the holes, the number of bands or handles, and the number of 
enclosed cavities (if there are any) come out. Two paths are equivalent when one 
can be continuously deformed into the other. Unlike the criterion operating to 
constitute connected cells in a complex, the criterion for deciding if two chains 
of the complex can be continuously deformed into each other is neither arbi-
trary nor geometrical: it is equivalential and topological. The equivalential logic 
that enables the construction of the cell complex, with its arbitrary character, 
differs from the equivalential logic that unveils the organisation of chains. It is 
the combination of both that finally leads to the articulatory skeleton organising 
the plurality of sites in phase space.
Once the cell complex is obtained, it can be characterised in a “layered” way 
by answering these questions: How many connected components does it have? 
How many inequivalent loops can we draw on it? Does it enclose cavities? The 
answers, which can be calculated using algebraic topology, are the topological 
Fig. 4. From the cloud of points to the cell 
complex.
(b) (a) (c) 

258
denisse sciamarella
properties of the cell complex. These properties lead to the template through the 
computation of the homology groups.
15
Homology groups code structural information.
16
 If we go back to the example of 
the acoustical pressure, homology groups yield the structure that articulates the 
data of our example. These paths are the three inequivalent loops that articulate 
this structure, and that can be used to design the template. If we label the com-
plex in Fig. 4c as complex K, the result can be written as follows: H
0
(K) =[[<1>]] 
~ Z
1
 (we have a 1-piece structure); H
1
(K)=[[L
1
, L
2
, L
3
 ]] ~ Z
3
 (with three inequiva-
lent loops or branches, the three cycles in Fig. 4c) and H
2
(K) ~ Ø (no enclosed 
cavities). It can also be noted that there is a torsion in cycle L
2
 (T(K) ={<2,3>}). 
The three inequivalent cycles make up two bands or branches, which reconnect 
15
 Homology groups are a series of layered sets that condense the information on how the plu-
rality of points is articulated in the topological structure. The set at level 0 (usually labelled 
H
0
), which condenses information on 0-cells, indicates the number of connected compo-
nents of the structure. The set at level 1 (labelled H
1
) contains the information on the number 
of non-redundant loops that can be constructed using 1-cells. The set at level 2 (labelled 
H
2
) indicates the number of enclosed cavities through the non-redundant paths that can be 
constructed with 2-cells, or polygons. The example of Fig. 4c makes us stop at level 2 because 
the cell-complex does not have 3-cells or polyhedra (the manifold is locally bidimensional). 
16
 Here are a few examples of different topological structures in terms of homology groups. 
The cylinder has one component in H
0
, one element in H
1
, and 0 elements in H
2
. The torus 
has one component (one connected piece) in H
0
, two components (two inequivalent loops) 
in H
1
, and 1 component (one enclosed cavity) in H
2
. 
Fig. 5. Template corresponding to the organi -
sation of the data in Fig. 4c.

259
a topological reading of ernesto laclau
in a central disk, with a torsion in one of the two branches. The corresponding 
branched manifold or template is shown in Fig. 5.
The reason why topological properties are interesting is that data associated 
with the same rule lead to the same topological structure. The template there-
fore shows how the data are organised in phase space under the action of the 
underlying deterministic rule, outlining an “ordre dans le chaos.”
17
 Topology 
classifies manifolds
18
 in a qualitative manner, ignoring deformations that do 
not introduce (or remove) holes or cuts.
19
 What does topology identify in an ob-
ject? Topology identifies disconnected components, holes, enclosed cavities, 
and so on. It does not identify sizes, distances, or continuous deformations.
Going back to the example in Fig. 3, suppose that the speaker is asked to re -
peat the word several times. If the rule governing the pressure time series does 
not change as the speaker repeats the word, the geometrical distribution of the 
cloud of points may change, but the topological structure organising the skel-
eton should still be that of Fig. 5. The template structuring the data is said to 
be an invariant,
20
 a topological invariant. Dynamical systems can therefore be 
classified according to their topological structure, so that unveiling the topo-
logical structure amounts to unveiling the governing rule. The topological pro-
gram is a guide leading from the data to the rule.
A few remarks are necessary. First of all, the mathematical procedure for ob-
taining the template from the data is not unique: several methods have been 
proposed. The method we have outlined above has the particularity of apply-
ing the concept of homologies to the ‘mask’ in which the data lie. Homology is 
17
 Pierre Bergé, Yves Pomeau and Christian Vidal, L’ordre dans le chaos: vers une approche 
déterministe de la turbulence, Hermann, Paris 1988.
18
 A manifold is a mathematical space that on a small enough scale resembles the Euclidean 
space of a specific dimension, called the dimension of the manifold. For instance, the sur-
face of a sphere is a manifold of dimension 2, because on a small enough scale it resembles 
a plane. 
19
 Topologically, a sphere and a plane are different objects. But topology would not distin-
guish a slightly deformed sphere from one that was not deformed. A sphere and a cylinder, 
for instance, are topologically different, because it is impossible to deform one into the 
other, unless two lids are cut from the sphere or glued to the cylinder.
20
 The concept of invariance has been extremely fertile in physics. A classic example of an 
invariance principle is the conservation of energy.

260
denisse sciamarella
an equivalence relation between the elements that constitute the cell complex. 
Secondly, the topological properties obtained in the application of the topolog-
ical program are data dependent. This means, on the one hand, that results are 
not always reliable: they can even be deceptive if the data series is too short, 
for instance, or if it is too noisy. Thirdly, the rule that is attained through the 
topological strategy does not exhaust the possibilities regarding the behaviour 
of the system. The topological program describes a dynamics that is encrypted 
in the data under analysis, but it does not encompass the totality of mecha-
nisms governing the system from a meagre series of data. More precisely, the 
mechanisms that will be unveiled by the topological strategy are those that 
are contained in the data. If the data series is enlarged, the template may be 
altered. Topological changes in the template may have different origins: a var-
iation in the parameters
21
 involved in the rule generating the data, a change in 
the mathematical form of the rule, or the necessity to add variables hitherto 
disregarded. In short, the template is just the manner in which the data un-
der analysis are articulated in a space that has been conveniently defined to 
host the dynamics of the data. The template condenses a relational necessity
22
 
which differs from the natural necessity of the system that it partially repre-
sents. Topology provides an invariant expressing a relational necessity that, in 
the case of deterministic systems, can be used as a guide towards the natural 
necessity expressed by the rule. 
Section 2. The premises of the analogy
Let us examine on what grounds the concepts of topology applied to dynamical 
systems can work as analogues of the main Laclausian categories.
First of all, Laclau and Mouffe maintain that the social order cannot be under-
stood as an underlying principle.
23
 This would represent an obstacle if our aim 
was to apply the topological program to data of socio-political origin, because 
21
 Parameters can be thought of as variables that are supposed to remain constant for the 
system. Parametric studies typically show that the template remains the same for certain 
ranges of parameter values. It is in the jump from one range to another that a change in 
parameter values may create or delete branches in the template. 
22
 Ernesto Laclau and Chantal Mouffe, Hegemony and Socialist Strategy, Verso, London and 
New York 2001, p. 154.
23
 Ibid., p. 96.

261
a topological reading of ernesto laclau
the goal of the program is to unveil an underlying rule. Nothing prevents, in 
fact, the application of the topological program to socio-political data, but un-
dertaking this task implies either that there is or that there can be an underly-
ing principle in the inspected data. Moreover, the choice of the time series for 
the analysis (e.g. the evolution of the human poverty index developed by the 
United Nations in a certain region and time interval) pre-identifies the type of 
underlying principle that is being looked for. But this is not what is being pro-
posed in this work, which is analogical and not mathematical. The connection 
between the topological structure and the mathematical rule is only the last 
step of a method that otherwise deals with the problem of converting a cloud 
of states representing a dynamics into a skeleton. The last step can be taken if 
the system is deterministic, and is successful as long as the elements excluded 
from the analysis remain dynamically irrelevant. In Laclausian terms, the last 
step is suturing,
24
 inasmuch as it transforms a fixation that is partial and open 
into a fixation that is closed and complete. It is a reasonable step for dynami-
cists in search of models, but an undesired leap in post-fundamentalist politi-
cal theory. It can therefore be advanced that the analogy will take up concep-
tual elements of the method leading from the disperse data to the topological 
structure, leaving out the leap from topology to determinism, in the assump-
tion that there is not, and there cannot be, an underlying principle of the social. 
Assuming that there is an underlying principle of the social is condescending 
to the metamorphosis of a hypothesis into a prescribing principle. In order to 
illustrate the dangers of establishing a principle of the socio-political through 
mathematical models inspired by the theory of dynamical systems, the reader 
is referred to other texts.
25
 In short, the topological program is not invoked as an 
analytical tool of data of socio-political origin. Nor is it suggested that the logic 
of the political should be, for some mysterious reason, topological in nature. 
The analogy does not attempt to reduce the gap separating a social problem 
from a deterministic problem.
24
 The concept of suture is taken from psychoanalysis. Hegemonic practices are suturing in 
as much as they attempt to complete the incompletable. According to Laclau and Mouffe, 
such a closure of the social is impossible. 
25
 For a case study on the concept of model in politics, see Amilcar O. Herrera, et al. ¿Catást-
rofe o nueva sociedad? Modelo Mundial Latinoamericano. Centro Internacional de investi-
gaciones para el Desarrollo, Agencia Nacional de Investigación y Desarrollo, Santiago de 
Chile 2004.

262
denisse sciamarella
Secondly, the analogy is possible as long as we are ready to sacrifice a number 
of characteristics that are attributed by default to the naturalist paradigm. La-
clau and Mouffe maintain that the type of objectivity constructed by the dis-
course of the natural sciences sets limits that exclude the notions of metaphor, 
analogy, metonymy, or contradiction. The naturalist perspective is therefore 
considered to be incompatible with the theoretical development that follows. 
But is this so? Are all the toolboxes, in the Foucaultian sense of the expression, 
developed by natural scientists or mathematicians unproductive when used 
in an unprescribed manner? This proposal is based on the belief that unpre-
scribed uses are not always infertile, and that thought is not ordered in sealed 
compartments. Blaise Pascal, who promoted the mixture of heterogeneous 
sources, asserted that there can be harmony where we do not expect to find 
it, and there can also be fusion problems in the homogeneous.
26
 Other authors 
have raised and analysed questions concerning the contingency of the a priori, 
the problem of language mediation in the naturalist paradigm, of partialness in 
formal languages, or of mathematics and metaphor.
27
 Using a ‘language’
28
 is a 
practice that can be compared to accepting the perspective of a picture. Taking 
concepts of two ‘languages’ and mixing them, or setting them face-to-face, as 
two mirrors, can produce perspective effects that may appear as strange, but 
that are not necessarily unfruitful, especially if we believe, with Pascal, in the 
possibility of harmony between irreducible perspectives.
Thirdly, let us mention a few points that seem to favour the formulation of the 
analogy. One of these points is the defence put up by Laclau as to the compat-
ibility of a partial order with the absence of an ultimate principle.
29
 The topo-
logical program is, after all, a strategy for searching for an organisation that is 
conditioned by a series of choices relative to the representation of a system in 
phase space. This organisation can be altered, as mentioned before, in a series 
of manners, which have been typified. Another point has to do with the charac-
ter of the program, which is not topological tout court. It is already a marriage 
26
 Maurice Caveing and Évetyne Barbin, “Les Philosophes et les Mathématiques”, Raison 
présente 123 (1997).
27
 Vladimir Tasić, Mathematics and the Roots of Postmodern Thought, Oxford University 
Press, Oxford 2001.
28
 Whether mathematics is or is not a language is a controversial issue. See, for instance Jean 
Petitot-Cocorda, Morphogenèse du Sens, PUF, Paris 1985, p. 19.
29
 Laclau and Mouffe, Hegemony and Socialist Strategy, p. 114.

263
a topological reading of ernesto laclau
between topology and dynamics. The topological structure is a structure con-
structed not in physical space, but in phase space, and as a consequence it is as-
sociated with the deployment of a dynamical behaviour. The categories in La-
clau’s theory aim at a description of fixations that are representative of certain 
processes. Last but not least, there is a series of terminological coincidences 
between Laclausian theory and the topological program that invite this analog-
ical exercise. The concepts of equivalence, chain, and nodal point are central to 
the construction of the template through homologies, and they are also central 
to the Laclausian conception of a popular identity. The concept of regularity in 
dispersion
30
 may redirect to the mechanisms leading to the emergence of top-
ological invariants in nonlinear systems, in which regularity and dispersion 
coexist. Dispersion is present through the sensitivity to the initial conditions, 
which makes nearby initial conditions diverge after a certain time. Regularity 
is present by a certain recurrence in phase space, where motion is eventually 
bounded. These two ‘contradictory’ mechanisms – dispersion and contraction –  
force nearby trajectories reaching maximum separation to begin to approach 
each other again.
31
 The fundamental idea of the topological program is that the 
regularity and dispersion processes, applied in a non-periodic manner, provide 
a sort of partial fixation.
32
 Topology provides the embodiment of this partial 
fixation. Fixation is partial because the topology invariance can be subverted 
by an exterior
33
 constituted by all that is left out of the analysis: the disregarded 
variables, the mechanisms that are not expressed by the data series, etc. 
Fourthly, let us make a few remarks concerning the usage of topological con-
cepts in non-mathematical texts. If Laclausian categories are constructed in 
view of some of the philosophical ideas introduced by Lacan, a legitimate ques-
tion is whether the ‘affinity’ between topology and Laclausian categories is 
somehow ‘inherited’ through Lacan, who does appeal to topology. Our analogy 
30
 Ibid., p. 106.
31
 These opposite tendencies are reconciled by operating in different directions, and this 
explains why there is no chaos (no complex, long-term unpredictable behaviour) in 
two-dimensional phase space. The third dimension is necessary for divergent trajecto-
ries to be squeezed back without self-intersecting. See, for instance, Pierre Bergé, Yves 
Pomeau and Moniques Dubois-Gance, Des rythmes au chaos, Editions Odile Jacob, Paris 
1994, p. 118, or Gilmore Robert and Marc Lefranc, The Topology of Chaos: Alice in Stretch 
and Squeezeland, p. 127.
32
 Laclau and Mouffe, Hegemony and Socialist Strategy, p. 113. 
33
 Ibid., p. 111.

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is not based on Lacanian references to topology. If it were, the analogy would 
be conditioned by pre-established connections, or by the kind of usage that 
Lacan makes of topology.
34
 Lacan borrows from topology a series of specific ob-
jects, such as the torus, the Möbius strip, and the Klein bottle. These objects are 
particular topological structures, and, as such, they are examples of the many 
possible results of the application of the topological program. But the parallel 
drawn in this article is not rooted on the topological objects themselves, but 
on the conceptual elements leading to the construction of a topological organ-
isation. It is on the topological program as a method (and not on its possible 
results) that the analogy is built. Therefore, if there is a Lacanian influence un-
derlying this affinity, it can probably be traced back to more general questions. 
The use of topology in Lacan may be seen as motivated by the inadequacy of 
the opposition interior-exterior to pose certain problems. The irresolvable inte-
riority/exteriority tension is central to Laclau’s arguments, and it can also be 
seen as essential to the preservation/subversion of topological invariance in 
phase space. Let us recall that the topological structure is always threatened 
by an exterior.
35
Before we move on to the terms of the analogy, a last comment may be pertinent 
as to the mathematical character of the topological program. Being mathemat-
ical, the program does not restrict its application to a particular kind of data: 
the origin of the data need not be specified for the program to be applied. But 
if this is so, what exactly delimits the context, the original terrain, of the topo-
logical program? The terrain is delimited by the operating rules underlying the 
practice of the program. Some of these rules are explicit: for instance, the pro-
gram applies to deterministic systems.
36
 To allow for the analogy, the topologi-
34
 Lacan appeals to mathematics as a formal system of writing capable of transmitting inte-
grally, without remainder, a piece of psychoanalytic knowledge [un savoir psychoanalyt-
ique]. Alain Badiou, another author that takes up some of Lacan’s concepts, and whose 
works contain mathematical inscriptions, does not operate in Lacan’s way. He “seeks to 
capture the power of mathematics for the sake of a conceptual development.” Alain Ba-
diou, The Concept of Model: An Introduction to the Materialist Epistemology of Mathemat-
ics, ed. and trans. Zachary Fraser and Tzuchien Tho, re.press, Melbourne 2007, p. xi.
35
 In this sense, it is a mathematical object that could be described, to borrow Rorty’s expres-
sion, as radically contingent. Ernesto Laclau, Emancipation(s), Verso, London and New 
York 2007, p. 118.
36
 The transparency of tacitly operating rules in mathematics has been analysed by several 
authors. It has been held that, if formal language supresses the ambiguity of ordinary lan-

265
a topological reading of ernesto laclau
cal program must be taken out of context at least in this sense. This contextual 
displacement can be compared to the change of focus carried out by Freud from 
the mystery of the form: “the ‘secret’ to be unveiled through analysis is not the 
content hidden by the form (the form of commodities, the form of dreams) but, 
on the contrary, the ‘secret’ of this form itself”.
37
 
Similarly, the analogy is focused not on the mystery behind the form of the data 
in phase space (the governing rule), but on the mystery of this form and its 
construction, on the articulatory mechanisms of a structure representing an 
open dynamics.
38
 The contextual displacement can be virtuous if the interest 
in topological invariants is exclusively guided by cogitation on the concepts 
of articulation and instability on the same relational playground. A certain 
distortion of the original tools is associated with the contextual displacement, 
since these tools are not used to calculate or to unveil an unknown, but to reflect 
upon certain politico-philosophical categories. Such a distortion will affect, 
for instance, the status of some of the elements of the topological program. In 
the original terrain, the template is a concept with a quasi-ontological status 
that stands independently of the method designed to unveil it, while the con-
cepts of cell, chain, nodal point, equivalence, and homology are concepts with 
a methodological status. This distinction will not survive once the contextual 
displacement is operated. Cell, chain, nodal point, equivalence, homology, and 
template will have the status of conceptual tools allowing for an arrangement 
[agencement]
39
 of Laclausian categories.
guage, this is achieved at the price of preserving a particular kind of ambiguity: generality. 
Henri Poincaré asserted that we inevitably operate with certain mathematical prejudices, 
and that many notions (and the notion of identity is an example) are in fact pre-logical 
(they are motivated but not well-founded). These beliefs underlie every rational practice 
and constitute its unstated condition of possibility.
37
 Slavoj Žižek, The Sublime Object of Ideology, Verso, London and New York 2006, p. 1.
38
 Open is used in the sense introduced by H. Poincaré. Objects that are impredicatively 
defined (invoking the reference to a totality) are never immutable or fixed. They are open 
because their identity is not present in them but maintained or supported by a nominative 
operation. 
39
 This expression is borrowed from Alain Badiou, who asserts that the effects of the philo-
sophical text owe their force and duration to the mere arrangement [agencement] of con-
cepts. Alain Badiou, The Concept of Model, p. xi.

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Section 3. Analogical correspondences
 
Discourse
The category of discourse is conceived by Laclau as a complex of elements in 
which relations play a decisive role.
40
 The fact that objectivity and relation are 
synonymous is a principle that may also apply to phase space. Phase space is 
a field of representation that functions as an open relational complex between 
a number of state variables that are defined as pertinent. It fixes, at least pro-
visionally, what will and will not be spoken about. The relations that cannot 
be represented in phase space are a result of an exclusion. The exclusion has 
practical reasons relative to the type of description that is attempted. Objects 
in phase space are therefore “threatened” by all that has been excluded, in an 
exclusion that is constitutive. Phase space is the primary terrain for the constitu-
tion of objects, and it is therefore the analogue of Laclau’s “precarious totalising 
horizon,” the “failed totality” necessary to start speaking about something. 
Discourse is described by Laclau as being “split between the particularity which 
it still is and the more universal signification of which it is the bearer.” Phase 
space can also be thought of in these terms, and these terms lead to the concept 
of hegemony. Hegemony is defined as the operation by which a particularity 
bears a totalising signification. Borrowing this definition, we could say that we 
talk about phase space by virtue of a hegemonical operation. 
The term moment is used to refer to the differential positions insofar as they 
appear articulated within a discourse.
41
 These definitions let us establish two 
initial conceptual pairs for our table of correspondences: the first one is the pair 
discourse-phase space and the second one is moment-state. 
40
 “Discourse is the primary terrain of the constitution of objectivity as such. By discourse, as 
I have attempted to make clear several times, I do not mean something that is essentially 
restricted to the areas of speech and writing, but any complex of elements in which rela-
tions play the constitutive role. This means that elements do not pre-exist the relational 
complex but are constituted through it. Thus ‘relation’ and ‘objectivity’ are synonymous.” 
Ernesto Laclau, On Populist Reason, Verso, London and New York 2005. 
41
 Laclau and Mouffe, Hegemony and Socialist Strategy, p. 105.

267
a topological reading of ernesto laclau
Demand
The minimal unit of analysis in Laclau’s theory is the socio-political demand
42
 
and the scope of his theory is aimed at consideration of how demands are artic-
ulated. In our parallel, let us tentatively associate demand and trajectory. A tra-
jectory in phase space is just the path defined by a series of sequential states. The 
construction of topological objects in phase space is meaningful insofar as the 
topological object holds bundles of trajectories in phase space that are articulat-
ed together. The trajectory can therefore be seen as the minimal unit of analysis 
for topological organisation. Cross-relating the terms of the analogy, a demand 
will be born under the form of a trajectory that initiates a path in phase space. 
What happens to a demand when it is satisfied? “A fulfilled demand ceases to 
be a demand.”
43
 If the pair demand-trajectory is kept, we should be able to de-
cide when a trajectory ceases to be a trajectory. If there exists a point in phase 
space that attracts trajectories that are born in different initial points covering a 
region in phase space, this point ends up absorbing the trajectory. This absorp-
tion makes the trajectory disappear. What happens to a system whose evolution 
is described by a trajectory that is absorbed by a point? It reaches a stationary 
state. We can imagine a group of initial states in phase space. After a transient, 
there is a state (a point) that is chosen as the definite dwelling. This point is 
called an attractor. But attractors are not always points. They can also make up 
closed curves in phase space. In such cases, the stationary state achieved by the 
system is not a single state but a closed sequence of states. 
To proceed with the analogy, let us take a step further and associate the achieve-
ment of a stationary state with the satisfaction of a demand. The transient walk 
of the trajectory before it is absorbed by an attractor is the analogue of the time 
interval during which a demand is still only a request. “If the demand is satis-
fied, that is the end of the matter.”
44
 Otherwise, the trajectories persist in phase 
space without being absorbed. This persistence turns demands into claims and 
trajectories into recurrent trajectories, i.e. the trajectories associated with this 
recurrence and with the impossibility of reaching a stationary state function as 
the analogue of the unsatisfied demands. Let us consider a system for which all 
42
 Laclau, On Populist Reason, p. 224.
43
 Ibid., p. 127.
44
 Ibid., p. 73.

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trajectories find, after a certain transient, their attractor. An example of such 
a system could present several point attractors distributed in space, without 
barriers impeding the encounter between the trajectory and the dwelling that 
hosts it. This situation is the analogue of a society in which “any social need 
should be met differentially; and there would be no basis for creating an inter-
nal frontier.”
45
 But what happens if an attractor transforms into a repeller, thus 
separating a certain group of demands from their satisfaction? This inaugurates 
clearings or voids, delimiting frontiers in phase space. Why? Not because trajec-
tories cannot transit these regions, but because after a certain transient, none of 
them will survive in them. These voids delimit regions in phase space and define 
borderlines or frontiers. Internal frontiers are the first of a series of necessary 
preconditions for the emergence of populism. 
The theory of dynamical systems provides tools to describe bundles of trajecto-
ries distributed in surfaces with branches and holes, trajectories that never be-
come stationary, even if they never abandon a bounded region of phase space. 
This region that attracts trajectories without making them stationary is called a 
strange attractor.
46
 It is an attractor because it is a set towards which trajectories 
evolve over time, but it is ‘strange’ because trajectories evolve in a recurrent but 
non-periodic manner, without ever repeating themselves.
The most famous example of a strange attractor is perhaps the Lorenz attrac-
tor.
47
 The rule that governs the Lorenz system is a nonlinear rule with three state 
45
 Ibid., p. 78.
46
 These attractors are more complex than a point, a surface, or a volume in phase. Their par-
ticular feature is that they form geometrical objects with a non-integer dimension. A point 
has dimension 0, a curve has dimension 1, a surface has dimension 2, but a strange attrac -
tor is a geometrical object that is, for instance, more than a curve but less than a surface. 
47
 The rule that governs the Lorenz system is a rule that can be derived from a simplified 
version of the laws for natural convection in the atmosphere. The Lorenz attractor is a 
paradigmatic example of a system in which the capacity of prediction is hindered. This is 
illustrated with the time series of the plot below.

269
a topological reading of ernesto laclau
variables: phase space is therefore three dimensional. Trajectories computed 
with this rule generate an object that is shown in Fig. 6a, with trajectories circu -
lating in a sort of ribbon or strip, making up a locally bidimensional manifold. 
The strip circumscribes the trajectories but does not coincide with them. It is a 
sort of holder or container, which may be represented graphically with the mod-
 The plot shows two-time series for one of the three state variables of the Lorenz system. 
These time series are obtained using the Lorenz rule. Both time series start at time zero, 
with x values that are almost identical. The difference between both x values amounts to 
0.000001. During the first time period (which lasts about 15 seconds), the evolution of the 
state variable x coincides for both time series and we can only see a single red curve. This 
poses no problem for prediction. But for a certain time the series that were coincident 
become drastically different, as if they had nothing in common. A prediction beyond the 
first 15 seconds with this rule will therefore fail. The rule is known but the rule does not 
suffice to predict the long-term behaviour of the system, due to sensitivity to the initial 
conditions. This kind of behaviour, illustrated here by the Lorenz system, is called chaotic. 
There is no consensus on a formal definition of deterministic chaos, but the term ‘chaotic’ 
is used to refer to deterministic phenomena described in terms of variables that vary in a 
particularly irregular fashion in time and space. It is known that a chaotic behaviour is 
constructed through the action of two basic (and opposed) mechanisms in phase space: 
stretching and squeezing. Stretching is connected with the property of sensitivity to the 
initial conditions. Squeezing is associated with the fact that variables adopt values within 
a limited range of values in phase space. For instance, in meteorology, pressure and tem-
perature are variables that adopt values within a certain range in the atmosphere. When 
both mechanisms are simultaneously present, diverging trajectories in phase space end 
up approaching each other in the long run. An imperfect regularity results from the com-
bined action of these two opposed mechanisms of dispersion and contraction, producing 
the ‘mille feuille’ object of Fig. 6a. 
Fig. 6. (a) Trajectories in phase space 
generated with the Lorenz system. (b-c) 
Two alternative but equivalent forms of 
representing the model kit of the strip within 
which trajectories circulate.
(b) (a) (c) 

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el kit shown in Figs. 6b and 6c. Both model kits are topologically equivalent. The 
first version resembles the attractor in Fig. 6a, but the strip can be continuously 
deformed until it coincides with the second version.
48
 The model kit in either (b) 
or (c) is the template of the Lorenz system. 
The first important point to retain from this example is that trajectories that 
coexist with a strange attractor are distributed in phase space, forming bun-
dles that can, however, be confined or inscribed in the structure that articulates 
them. The second important point is that trajectories trapped in a strange attrac-
tor move in a disorderly manner that does not appear to be generated by a rule, 
even if this is the case. But the structure in which the trajectories move can be of 
help to unveil the rule, when it is not known in advance.
The trajectories coexisting with a strange attractor can be conceived as unstable 
orbits that never close upon themselves to form a periodic orbit. We can think of 
a strange attractor as a set that hosts trajectories without absorbing them. Why 
does it host them? Because there is an object in phase space with well-defined 
topological features that contains them. Why does it not absorb them? Because 
stationarity is never reached.
Laclau remarks that the lack of fulfilment of a demand can take the form of 
a downright rejection – as in the case of a repeller expelling a trajectory from 
its zone of influence – but it can also take the form of an unstable equilibri-
um between satisfaction and rejection.
49
 This intermediate situation between 
satisfaction and rejection is comparable to the particular situation encountered 
by a trajectory that coexists with a strange attractor. These trajectories change 
permanently but within certain limits, and these limits preserve a mathematical 
form that is well described by topology. In the field of dynamical systems, we 
say that there is topological invariance and we associate this invariance with 
the dynamical properties of the rule governing the system. The strange attrac-
tor maintains its topology unless the system suffers radical changes (through a 
parameter variation, through the incorporation of dynamical ingredients that 
were absent, or through the inclusion of previously excluded variables). Within 
48
 Recall that the transformation process of one version into the other only forbids discontin-
uous actions such as cutting or welding.
49
 Ibid., p. 120.

271
a topological reading of ernesto laclau
the apparent disorder of trajectories that never settle to a stationary state, there 
subsists an order of a higher level: the level of the topological structure organ-
ising the trajectories. 
But how can we jump from the level at which there is disorder – the level of the 
trajectories – to the level at which there is order – the level of the template? By 
establishing an equivalential relation at the level of the trajectories. The estab-
lishment of equivalential relations between demands is the second precondi-
tion for the emergence of populism. 
What kind of equivalential relations are introduced in the topological program? 
In the first place, a local equivalence is practiced on nearby states, enabling the 
grouping of stretches of trajectories that admit a local
50
 common inscription, 
provided certain criteria are met.
51
 What comes out of the establishment of this 
local equivalential relation? A number of organised cells representing stretches 
of grouped trajectories, constructed around nodal points. This local equivalence 
leads from the trajectories to the template.
The Laclausian equivalential chain between demands presents trends that are 
similar to the equivalential bond constituting a cell in the cell complex. The 
more extended the equivalential relation, the laxer the criterion that is applied 
to constitute it. The arbitrary nature of this criterion has an analogue in the 
emptiness of the signifier:
52
 the laxer the criterion, the emptier the signifier. 
Another trend that admits an analogical translation is the ontological function 
overriding the ontic content.
53
 The fact that nearby trajectories have nothing in 
common when they are followed out of the region of interest does not interfere 
with the inscription of their common stretch around a nodal point to form a 
cell. The ontological need to be unified under a signifier (or to be grouped un-
der a criterion having an arbitrary common reference) is stronger than the ontic 
difference between demands (or mutually diverging trajectories). The method 
50
 The equivalence between the states in the cell is local because their continuation out of 
the cell can be completely different, but this does not matter.
51
 This local inscription can be seen as the analogue of a “levelling instinct”, Ibid., p. 102.
52
 “A signifier like ‘workers’, for instance, can, in certain discursive configurations, exhaust 
itself in a particularistic, sectional meaning; while in other discourses – the Peronist 
would be an example – it can become the name par excellence of the ‘people’.” Ibid., p. 87 .
53
 Ibid., p. 87.

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that we are describing does not even attempt a reconstruction of the trajectories 
from the disperse points corresponding to the available data. And this is not 
attempted because the identification of the trajectories does not matter to the 
reconstruction of the topological structure. The focus is on how disperse states 
are articulated in phase space, independently of the trajectories to which these 
states might belong. 
The demands that enter into an equivalential relation with other demands are 
called popular demands. The construction of the cell complex in the topologi-
cal program provides an image of the Laclausian equivalential process. Popular 
demands have their analogue in trajectories admitting an inscription within a 
cell complex. The analogy is even respectful of the fact that “equivalence does 
not attempt to eliminate differences.”
54
 The cell does not replace the trajectories, 
it does not coincide with them, nor does it make the trajectories coalesce into 
a unique trajectory; the cell is an object that can be imagined as overlying or 
supporting different trajectories when certain criteria are met. 
The topological program admits a second type of equivalential relation. It is an 
equivalence relation that operates upon the cell chains that can be defined along 
the cell complex, i.e. at the level of the structure supporting the initially dis-
perse states and contributing to the determination of the topological properties 
of such structure. This equivalence relation between chains is termed homology 
and it leads to the recognition of the number of connected components, of in-
equivalent loops, of enclosed cavities, etc. This stage of the topological program 
can be seen as the third and last precondition for the emergence of populism: 
the unification of the various demands into a stable system of signification.
55
Institutionalism
Let us turn back to the case in which phase space is populated with non-strange 
attractors. These attractors play the role of institutional forms satisfying isolat-
ed (democratic) demands. And isolated trajectories (democratic demands) are 
all alike in the sense that all of them lie in the basin of attraction of an attractor 
that will, sooner or later, capture them. This ‘homogeneous’ scenario can be 
considered the analogue of the institutionalist totality.
54
 Ibid., p. 79.
55
 Ibid., p. 74.

273
a topological reading of ernesto laclau
Populism 
Moving away from this scenario, let us imagine that repellers inaugurate interior 
frontiers and that the homogeneity mentioned above is broken. Laclau speaks 
of a fractured space, of a division into camps. But how is this division created? 
Who creates it? How is space effectively fractured? Does the break admit rep-
resentation in phase space? If not, how can it be read in terms of the analogy?
The borders of the manifold in which trajectories coexisting with a strange at-
tractor move are frontiers between zones of phase space where there are tra-
jectories and zones of space which are empty. However, these empty spaces 
are not representative of the action that empties them. In the case of a deter-
ministic system, the “power” that “creates” these voids is legislated by the rule 
governing the system. These voids may be created in phase space by changing 
the mathematical rule that governs the system. But who produces the chang-
es in the rule? The natural answer is “the system.” But the terms “rule” and 
“system” are often used interchangeably. A distinction can, however, be made 
between the system generating the data and the rule that models it. The system 
can unexpectedly exhibit a behaviour that is not prescribed by the rule that was 
supposed to describe it. This makes the model that is expressed by the rule in-
adequate and calls for a re-adjustment of the rule. The system therefore exceeds 
the representation of its behaviour in phase space, and constitutes ‘something 
out there’ that may force the modification of the representation that is initially 
chosen as appropriate. This is what creates the fracture in the original context. 
But what about the fracture as a political category? The dynamics of the system 
has borders delimiting voids in phase space, but if we ask about the action be-
hind the void, our relational playground (phase space) remains voiceless. Let us 
consider an example: let us imagine that we describe a physical system in which 
two elastic masses collide periodically against each other under the action of 
certain forces. Let us suppose that we choose one of these masses and that we 
represent the velocity and position of this mass in phase space. A collision with 
the other mass may be modelled using a nonlinear term in the mathematical 
rule (a saturation term). This nonlinear term produces the effect of stopping the 
mass that we are looking at (the mass that plays the role of our system). Our 
mass cannot visit certain areas of phase space due to the action of this satu-
ration term. But this forbidden region in phase space inaugurated by the sat-
uration term in the rule does not tell us if the mass is drastically stopped in its 
motion at a certain position because it is colliding against another mass, or if 

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this periodical stop is caused for another reason. The reason is beyond what 
can be expressed in our representation frame. In terms of the analogy, we can 
say that the concrete actions that are responsible for the existence of unsatis-
fied demands do not constitute part of the partial totality in which the demands 
move. Emptiness is the only visible effect in the relational playground to which 
meaning is restricted. This allows us to say, with Laclau, that the fracture we are 
dealing with is a purely negative object. “The void points to the absent fullness 
of the community.”
56
 The holes or fractures in phase space are also a negative 
representation of a positive action exerted ‘out there’. These actions are repre-
sented in phase space as regions expelling trajectories, regions where stability 
or long-term survival becomes impossible. But if there is objectivity only within 
a certain representation frame, what happens with objects that are constituted 
in different frames? Is there a possible encounter between such objects? This 
point will be taken up when we discuss the category of heterogeneity. 
Let us move onto the second precondition for the emergence of populism. The 
transition towards the populist identity consists in the establishment of an 
equivalential bond between demands. In the parallel we are developing, this 
equivalential bond takes place with the establishment of a local equivalence 
relation that enables the inscription of temporally bounded demands into a cell. 
As expected, the field of the trajectories that are inscribed into a cell is not a 
“neutral terrain.” The “uneasy alternation between cold and warm” demands 
in Laclau
57
 is the analogue of the contradictory processes of dispersion and con-
traction leading to a topological invariance.
The inscription within an equivalential chain “gives the demand a corporeality 
which it would not otherwise have. It ceases to be a fleeting, transient occur-
rence, and becomes part of […] a discursive/institutional ensemble which en-
sures its long-term survival.”
58
 We have seen how this inscription leads to the 
template, the skeleton articulating originally disperse states. The template ex-
presses the long-term survival of the trajectories inscribed in the complex. This 
long-term survival is expressed in terms of topological invariance. But the par-
56
 Ibid.., p. 170.
57
 Ibid., p. 89.
58
 Ibid.

275
a topological reading of ernesto laclau
ticular topology of the template represents something on its own: a particular 
class of articulation, a particular class of dynamics.
59
This makes the template hypostatic in the following sense. Once constructed, 
the template stands by itself and does not coincide with the time series that was 
used to unveil it. The template expresses a property of the system; it represents 
what survives in the long term if the system is well described by the data. In a 
way, the time series data is like Wittgenstein’s ladder, which can be dropped af-
ter climbing. The template overrides the particularistic trajectories and produc-
es meaning
60
 by itself. This hypostasis realises the third and final precondition 
for the emergence of populism. 
Empty signifier
Lacan holds that the concepts of signifier and articulation are consubstantial, 
that all that is presented as articulated is the signifier, and at the same time, that 
the signifier secretes a remnant which it cannot signify. In this sense, every ar-
ticulation is a failed articulation. We have already discussed in what way phase 
space is a failed totality. We have already mentioned that phase space exists 
when we deliberately choose some variables that are considered significative 
and that will therefore “construct meaning.”
61
 This is in line with the manner 
in which Laclau defines the term system: a system is a failed totality that con-
stitutes a horizon.
62
 The excluded categories cancel their differences through 
an equivalence that holds between all that is excluded. The system can signify 
itself as a totality at the price of obliterating (or putting between brackets) the 
differential nature of what is left out. But all that is obliterated or ignored at a 
certain time scale or period may not be insignificant at every time scale or peri-
59
 “The ‘people’ does not act as a clearing-house for the individual demands, for it is trans-
formed in most cases into a hypostasis which starts to have demands of its own.” Ibid., p. 89.
60
 This meaning is more general than the particularistic trajectories, but it is still particular: 
it is a particular kind of articulation, with a certain topology.
61
 Phase space is the analogue of the “zero point of signification which is nevertheless the 
precondition for any signifying process.” Ibid., p. 72.
62
 “The systematicity of the system is a direct result of the exclusionary Iimit, it is only that 
exclusion that grounds the system as such.” Laclau, Emancipation(s), p. 38. 
 “ Any system of signification is structured around an empty place resulting from the impos-
sibility of producing an object which, none the less, is required by the systematicity of the 
system.” Laclau, On Populist Reason, p. 40.

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denisse sciamarella
od. If “totality now becomes the name of a horizon and no longer of a ground,”
63
 
phase space is also a horizon rather than a ground and is therefore something of 
the order of an empty signifier.
64
Let us focus on the cell complex. In the departure context, the cell complex is 
a methodological element that is built to unveil a topological structure. The 
popular identity is the analogue of the template. But to determine the topology 
of the template, the moment of the cell complex construction is critical. The 
method depends on a performative criterion; different criteria will produce dif-
ferent templates and the success of the program relies on a good choice of this 
criterion. We have mentioned that the constitution of cells and equivalential 
chains relies on certain arbitrary criteria, by virtue of which the range of the 
equivalential process is decided. If the topology of a cell complex depends on a 
criterion, the method may lead to false rules. This ambiguity is problematic in 
the original context (since the rule is the scope of the topological program), but 
in the context of political theory this ambiguity is desirable. In the topological 
program, the cell complex is methodological, while in the analogy the complex 
is a constitutive element, with a role that is determinant for cementing the artic-
ulation. In other words, while in the original terrain there is a correct template 
that the method may or may not adequately capture, in the analogy the template 
does not pre-exist the complex but is legitimately constituted through it.
65
 
The cell complex mediates between the clouds of disperse elements and the 
possibility of constructing the template. The analogy is better tuned if this me-
diation is seen as a nominal operation. The name operates as a pure signifier, 
as a structuring element without proper content. The performative operation by 
which the cell brings about the unity of disperse elements is done in the name 
of a criterion which cannot be deduced, it has to be defined. The unity of the 
template is a retroactive effect of its constitution through the criterion leading to 
the cell complex. This criterion is a relation between universality and particular-
ity, between a particular content in phase space (points distributed in a certain 
manner) and the universal function embodied by the criterion. Arbitrariness is 
involved both when phase space is constructed and in the passage from the dis-
63
 Ibid., p. 103.
64
 Ibid., p. 71.
65
 Ibid., p. 68.

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a topological reading of ernesto laclau
perse points to the topology that organises them. The structuring dimension (of 
signification) and the arbitrary (or affective) dimension
66
 converge in the con-
struction of the template and in the construction of the popular identity.
Antagonism, dislocation, and heterogeneity
According to Laclau, demands may receive the structural pressure of rival he-
gemonical projects. In the analogy, this occurs for instance when a bifurcation 
is created in a template so that a single branch opens up into two branches. 
Trajectories close to the bifurcation will receive the ‘structural pressure’ of rival 
branches. 
In the original terrain, topological changes in a structure can originate in three 
increasingly radical ways: through variation of the parameters of the rule, by a 
change in the form of the rule, or by a change in the definition of phase space. 
Dynamicists call events that do not involve a redefinition of phase space pere-
stroikas. Phase space reformulation is necessary if the change is manifested, 
for instance, through the inclusion of previously excluded variables. This dis-
tinction between the different manners that may originate structural instability 
in a template has an almost direct translation into the Laclausian concepts of 
antagonism, dislocation, and heterogeneity. 
Antagonism and dislocation are perestroikas: they do not involve a reformula-
tion of the discoursive inscription. Frontiers play the role of antagonistic rela-
tions, and its possibilities of displacement and modification are affine to dislo-
cation. The concept of social heterogeneity, instead, has a type of “exteriority” 
with respect to the field of representation that is analogous to the most radical 
change, implying phase space reformulation.
67
 
The notion of exteriority, however, remains problematic. In the paragraph de-
voted to populism, it was mentioned that frontiers and voids in phase space 
have an apparential exteriority. An antagonism may induce the bifurcation of a 
branch into two branches. But the exterior whose correlate is the void between 
the separated branches operates inside the scheme through the mobility of the 
frontiers of the topological structure. These voids structuring the template intro-
66
 Ibid., p. 117.
67
 Ibid., p. 148.

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denisse sciamarella
duce an element that functions as the internal negation of the pseudo-totality 
constituted by phase space.
68
 
Immanence and transcendence are interwoven in the portrait developed by the 
analogy. The void opened by the repeller has an apparent exteriority: it can be 
seen as the correlate of the “intrusion” of something external that operates in 
phase space causing changes in the topological structure.
69
 These voids are mo-
bile, since the borders of the branches move to allow for a bifurcation. Laclau 
uses the category of “floating signifier” to refer to the logic of the displacements 
of the frontiers. The empty signifier (represented in the analogy by the process 
allowing for the nucleation of points in cells) takes for granted the presence of a 
stable frontier, while the floating signifier does not.
70
 The distinction is only an-
alytical, since both empty and floating signifiers are complementary categories 
that participate in the construction of the popular identity. Empty signifiers are 
involved in the reference to a partial fixation, while floating signifiers
71
 allow for 
the mutability of these partial fixations. Similarly, the organisation of holes and 
borders fix a topological structure, and perestroikas allow for the mutability of 
this organisation.
Section 4. Conclusions
Ernesto Laclau has rescued the term populism from its pejorative condition. The 
theory of dynamical systems and topology are fields that have rescued the value 
of ‘the qualitative’, making it compatible with mathematical tools that seemed 
restricted to quantification. This work restores the value of ‘the analogical’ to 
apply tools borrowed from the naturalist paradigm to categories of philosophi-
cal order. 
 
68
 “The multiple presence of the heterogeneous in the structuration of the popular camp is 
that the latter has an internal complexity which resists any kind of dialectical homogeni-
zation. Heterogeneity inhabits the very heart of a homogeneous space.” Ibid., p. 152.
69
 This intrusion is the analogue of the “heterogeneous other” that is dialectically irretrieva-
ble in the emergence of an antagonistic frontier. Ibid., p. 172.
70
 Ibid., p. 133.
71
 “The ‘floating’ dimension becomes most visible in periods of organic crisis, when the sym-
bolic system needs to be radically recast.” Ibid., p. 132.

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a topological reading of ernesto laclau
‘The political’, seen as an undecidable game between the floating and the emp-
ty, has an analogue in ‘the topological’ in the structuration of a dynamics.
72
 The 
conjunction between topology and dynamics has an exponent in the so-called 
topological program. This work proposes an analogomorphism between the 
concepts of the topological program and the categories that Laclau defines to 
articulate the dissatisfaction of demands and the construction of the popular 
identity. The term analogomorphism is coined here to refer to an ensemble of 
relationships between terms of two disparate fields, in which the relationships 
have a heuristic function, and in which the contextualised role of the terms in 
the original fields can be denatured or transformed by the “mise en relation.”
73
 
In this analogomorphism, the topological program is invoked as a toolbox in the 
Foucaultian sense of the term. The contextual displacement is hermeneutical, 
the key being the identification of a trajectory with a demand, and the identifi-
cation of a stationary state with the satisfaction of a demand. The analogy is not 
based on borrowing mathematical inscriptions or symbols, but on the common 
articulation of an ensemble of concepts. This differs from the usage of mathe-
matics implemented by other authors, such as Jacques Lacan or Alain Badiou.
In the original context, the mathematical tools of the topological program are 
used to construct a skeleton that articulates trajectories in phase space. Here 
it is argued that the concept of phase space involves the category of hegemony, 
and that the topological skeleton can be related to the category of “people.” 
The template has a structure that is topological in nature; it is defined in terms 
of the number of holes, enclosed cavities, branches, etc. Substantial (qualita-
tive) changes in the system’s behaviour affect the topology of the template: en-
closed cavities are modified, branches bifurcate, holes appear or close, etc. The 
concepts involved in the program can be divided into three groups:
· those involved in the representation of the dispersion of elements available 
before the construction of the template;
· those concerned with the manner in which the template is constructed from 
the dispersion of elements;
72
 Ibid., p. 153.
73
 The connections between the terms need not be exhaustive or bijective.

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denisse sciamarella
· those necessary to describe the different possibilities regarding the mutation 
of the topology of the template. 
The concepts in the first group admit a correspondence with the categories of 
discourse, demand, difference, and institutionalism. Those in the second group 
are related to the categories of frontier, empty signifier, equivalence, and pop-
ulism. The concepts in the third group function as analogues for antagonism, 
dislocation, and heterogeneity. 
The template, constructed over a series of methodological steps, is the analogue 
of the popular camp. Its corporeality is translated in terms of topological invar-
iance. The methodological steps play a central role in the analogy. The concept 
of template in the analogomorphism is denatured in that it no longer condenses 
a dynamics that must be unveiled, but a dynamics that is built provided the 
inscription criteria that cement the articulation operate conveniently. The arbi-
trariness of these criteria is also essential to the analogy. We could assert that 
there is a constitutive link between the template-people and the trajectories-de-
mands that articulate it. This link expresses both its strengths and weaknesses. 
The template is a partial fixation, an open invariant that can be subverted, with 
subversion conditions that take the form of an alteration of frontiers or of a rad-
ical reformulation of the discursive space. 
The analogomorphism can be used as a guide to gain insight into several ques-
tions that are not discussed in this article. One of them is the distinction be-
tween equivalence and homology, i.e. between the equivalential bond at the 
level of the cell and the equivalential bonds at the level of the complex, in terms 
of political theory. 
Naturally, the type of analogical work proposed in this article need not be re-
stricted to the topological program. Nothing prevents invoking other concepts 
of the naturalist paradigm with a similar strategy. For instance, the concept of 
coherence, used in physics to denote certain correlation properties between mag-
nitudes (waves, atoms, magnetic dipoles, etc.), could be proposed, in principle, 
as a concept with some relationship to the category of hegemony. However, when 
a physical system attains a coherent state of some kind, the order is directly intro-
duced at the level of the differences. A collective action achieved by a set of indi-
viduals could be read as a coherent action in the physical sense of the term, but 

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a topological reading of ernesto laclau
in this case the coherence, the convergence in the arrangement, the order in the 
action, is attained by cancelling differences. The equivalential logic that builds 
the topological structure expressed in the template is an order of a different kind. 
Order and disorder coexist without disturbing each other. Moreover, not only do 
they coexist peacefully, but they support each other. Let us remark that if we 
force an order upon the trajectories-demands, disorder disappears and we fall 
back into the institutionalist limit: trajectories become stationary and demands 
encounter satisfaction. The analogy with the template is appropriate because the 
template is not a physical force that orders trajectories into a structure, but rather 
a sort of containment frame. The template hosts trajectories without dissolving 
their differences. Differences are sheltered by virtue of an arbitrary criterion that 
we have related to the affective dimension. Topological invariance, the “ordre 
dans le chaos,” is a product of the coexistence of non-coincident levels: the level 
of the trajectories or demands, and the level of the template or the people. At the 
level of the trajectories, differences and disorder persist. The dynamical order 
that is expressed by the template is not directly or ostensibly present at the level 
of the trajectories and in this sense it is more discursive than phenomenological, 
as much as the template is ‘more mathematical’ than ‘physical’. The notion of 
coherence is therefore more akin to the notion of a general consent, while the 
notion of equivalence in the topological program allows for an order without 
convergence, without consensus, without coincidentia.
74
‘The popular’, ‘the qualitative’, and the ‘analogical’ have a pejorative charge 
that this text has attempted to overcome by showing the fertility of an exercise 
that is not limited to a mere parallelism, since it opens up a transfer process 
of notions, methods, and images. The procedure naturally raises the question 
of the justification of such an affinity. It also highlights the importance of not 
reducing an analogomorphism to an isomorphism.
75
 Every analogy is ultimately 
dialogical and the construction of the analogy unfolded in this work implies 
a pre- and re-interpretative work, both of the dynamico-topological concepts 
as well as of Laclausian categories. Analogomorphisms can be put to work in 
74
 Perhaps the difficulty in harmonising categories such as multitude in Toni Negri and he-
gemony in Ernesto Laclau can be traced back to the difference between the concepts of 
coherence and equivalence, in physics and mathematics, respectively.
75
 Some of the flaws of Alain Badiou’s effort to build an ontology inspired by set theory could 
be the result of forcing an isomorphism there where an analagomorphism would be more 
natural. 

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denisse sciamarella
political theory to complexify certain political notions, to pose some theoretical 
questions diversely, or to increase the precision of the manner in which alterna-
tive theories proceed. On the other hand, mathematical or physical concepts can 
profit from analogomorphisms in a genre that can be seen as a deconstructive 
reflection of scientific practices that are often engaged without consideration of 
the latent notions with which they operate, those notions that Henri Poincaré 
once termed mathematical prejudices.
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