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* Porto Alegre, Brazil
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 math-
ematics 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 subjec -
tive 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 philoso -
phy 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 perspec -
tive 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 Gener -
al 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. 
1
 The version of the General Continuum Hypothesis Badiou applies and reads in Being and 
Event is demonstrated in Meditation 29. Cantor, however, did not define whether the nat -
ural 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 nat -
ural numbers is the smallest infinite collection of elements, while “state” names the next 
infinite set, ω
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.

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norman madarasz
Contemporary French philosophy can be described in part as a sequence of the -
oretical 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 exist -
ence 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 dis -
tinct 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 implica-
tions 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 con-
dition 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 con-
ceptually, a generic, infinite set gathers together the names of processes whose 
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 

287
beyond recognition: badiou’s mathematics of bodily incorporation
French philosophers and social scientists alike since structuralism appeared to 
wash away the face of the subject from modern fields of conceptual inquiry. Ba-
diou’s theory of subject is post-evental. As such, it leads further than previous ar -
guments 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 propos -
al works from the basis of understanding correlations between various logical 
possibilities regarding the appearance of truth. It thus provides a complex un-
derstanding 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 tran-
scendence. 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 

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norman madarasz
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 formu -
la “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 pro -
vides 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 demon-
strate 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 spontanei-
ty, 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 bod -
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 
2 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 espe -
cially, section 7. 
3 Ibid., p. 111.
4 Alain Badiou, Logics of Worlds, trans. Alberto Toscano, Continuum, London 2009, p. 35.

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beyond recognition: badiou’s mathematics of bodily incorporation
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 cate -
gory 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 num-
ber of categories. For the purposes of Badiou’s system, it captures the poten-
tially 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 
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 homeo -
morphism”. Olivia Caramello, Theories, Sites, Toposes. Relating and Studying Mathemati-
cal 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.

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norman madarasz
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 fac -
es 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 dia-
lectic”. 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 pri-
macy 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 
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.

291
beyond recognition: badiou’s mathematics of bodily incorporation
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.”
12
 Such 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 re -
quired 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 philoso -
phy – 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-there-
in-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 struc -
tures. 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. 
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.

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norman madarasz
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 bina-
ry, the intuitionistic and the paraconsistent.
18
 The particular context, or strictly 
speaking “world”, acquires consistency by virtue of being deduced from a spe -
cific 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:
ε
           ___
  ⇒  π
¢
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 indis -
cernibility, 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 af -
fords 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 Badi-
ou, Logics of Worlds, p. 532.
19
 Badiou, Logics of Worlds, p. 53.
20
 Ibid., p. 53.
21
 Ibid., p. 167.

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beyond recognition: badiou’s mathematics of bodily incorporation
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 undecida-
bility 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 par -
tial 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 “frag -
ment 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 ar -
rows 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. 
22
 Ibid., p. 103.

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norman madarasz
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 theo -
ries 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 effica-
cious 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 multiplici-
ty.”
 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 
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 mani-
fests itself, nor indeed was it initially contemplated should its destination be de -
viated and possibly suppressed. None of these immediate shortcomings weak -
23
 Ibid., p. 88.
24
 Ibid., p. 46.
25
 Ibid., p. 27.

295
beyond recognition: badiou’s mathematics of bodily incorporation
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 sub -
ject.”
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 dis -
continuous 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 re -
turns 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-sub -
ject 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 se -
lected 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 perspec -
tive 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 
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.

296
norman madarasz
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 regulat -
ed by a law. By the time of Logics of Worlds, thinking the effective or concrete be -
coming 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 
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. 

297
beyond recognition: badiou’s mathematics of bodily incorporation
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 concep -
tual 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 some -
thing 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 cor -
relations;
- 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 the -
ories, 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.
31
 Yet they can also be considered the reader’s responsibili-
ty, if not deficiency. Once again, Logics of Worlds delegates responsibility to the 
reader to decide upon the points at which truth-valued decisions are determi-
nate for the next steps in keeping the event trace active. 
31
 Cf. Vladimir Tasic, “Badiou’s Logics: Math, Metaphor, and (Almost) Everything”, Journal 
of Huma nistic Mathematics 7 (1/2017), pp. 22–45; or Shiva Rahman, “On Why Mathematics 
Can Not be Ontology”, Axiomathes 29 (2019), pp. 289–296.

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norman madarasz
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 ap -
pearance 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 cor -
relation 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 var -
iation 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” phenomenolo -
gy
33
 aims for a conceptual exposition, while mirroring, even though at a meth-
odological 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 refer -
ences to Lacan in Book VII, to whose work he references the “second body”, 
Badiou explicitly undermines any identification between his “operational phe -
nomenology” 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 perti-
nence. The return to the intentional and to lived experience would hinder a the -
ory 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. 

299
beyond recognition: badiou’s mathematics of bodily incorporation
form, for it recognizes no radical event. Badiou’s phenomenology is a “descrip -
tion 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 phenomenol -
ogy as a whole. He recognizes the risk just as much when evoking how “phe -
nomenology, 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 in-
deed 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 naïve 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 war -
rant, 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 
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, “ À quoi bon l’ontologie ? Les mondes selon Badiou”, 
Critique (719/2007), pp. 279–294.

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norman madarasz
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 con-
structing the theory of appearing, in which even the category of transcendental 
breaks with historical precedent. The broader sense of the philosophical meth-
odology 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 com-
plete 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 
as important as, although it could, in the final analysis, be more important than. 
41
 “I could not be structuralist because structuralism was, in its most extreme form, the aban-
donment of the concept of subject.” This admission, one of Badiou’s most explicit state -
ments on structuralism, was made at the “Conférence d’Athènes: Introduction à L’Être 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.

301
beyond recognition: badiou’s mathematics of bodily incorporation
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 Log -
ics 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 symbol -
ic 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 sym-
bolic 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 be -
come. 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 prima-
ry 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 
from that in the first place.  
As one can see, the interest in infinite sets emerges through the founding prin-
ciple 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.

302
norman madarasz
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 cap -
ture 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 pos -
sibility 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 present -
ed 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 oth-
er 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 
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. 
45
 Alain Badiou, Conditions, trans. S. Corcoran, Continuum, London 2009, [1991], Figure 8.1, 
p. 121.
46
 Ibid., p. 381.
47
 Ibid., p. 46–47

303
beyond recognition: badiou’s mathematics of bodily incorporation
In the mathematical framework, category theory has been especially used to es -
tablish correlations, bridges and equivalences between different areas of math-
ematical work. Although it has been well over a decade since Logics of Worlds 
presented philosophy with this framework, what can be said with relative con-
fidence 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 conceptu -
al 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 
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.

304
norman madarasz
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 sys -
tem 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 broad -
er 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 philoso -
pher 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 fundamen-
tally 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.

305
beyond recognition: badiou’s mathematics of bodily incorporation
sets, functions, and relations. As Badiou’s commitment to irreducible multiplic -
ity relies on a subject-form that is indiscernible, intrinsically non-visible and im-
manently bound to those who decide to take part in its validation through dis -
parate 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 phe -
nomenal dimension merely through logic and not mathematics? We have exam-
ined 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 address
55
, 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 multiplic -
ity 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 
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 En-
sembles, théorie des Catégories, et théorie des Infinis, dans L ’Être et l'événement, Logiques 
des mondes et L 'Immanence des vérités”.

306
norman madarasz
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