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* American University of Paris, Collège International de Philosophie
Filozofski vestnik | Volume XLI | Number 2 | 2020 | 37–55 | doi: 10.3986/fv.41.2.02
Oliver Feltham*
“One or Many Ontologies? Badiou’s Arguments 
for His Thesis ‘Mathematics is Ontology’. ”
This is not a story of hubris, of a philosopher seeking to make his name by an -
nouncing a provocative thesis “mathematics is ontology” and then, decades 
later, fighting to prevent his name from being unmade by commentators who 
cannot quite bring themselves to accept that identity as definitive. This is an en -
quiry into an argumentative strategy which leads to a crucial question: which 
initial decisions on ontology lead to there being one or many ontologies? 
Part of the appeal and strength of Badiou’s philosophy lies in complex way he 
sets up the tasks of a set-theory based ontology in Being and Event, and then of 
a category-theory based phenomenology in Logics of Worlds. It is the number of 
argumentative steps and choices made in these preambles – many of which are re -
inforced in the recently published seminars – that turn Badiou’s philosophy into 
fertile ground not just for contesting interpretations but for the genesis of other 
philosophies. Indeed, it is the case that Badiou’s philosophy not only spawns or -
thodoxies and imitations, like those of Derrida and Deleuze, but also heterodox -
ies and rival philosophies. But the challenge for any departure from a putative 
Badiousian orthodoxy is to retain both the audacity and systematicity of Badiou’s 
work. Can we, in turn, take steps in a new conceptual construction, and then trace 
the consequences of each of these steps, in a manner constrained by a previous -
ly-existing and consistent discourse, such as ZFC set-theory? Such a task goes be -
yond the question of making or unmaking, repeating or forgetting, proper names. 
In this preliminary enquiry, let us explore Badiou’s initial decisions on ontology. 
The argument that mathematics is ontology 
 
In the first meditation of Being and Event , Badiou sets out its inaugural thesis: 
“Mathematics is ontology – the science of being qua being”.
1
 One page later he 
1
 Alain Badiou, Being and Event, trans. Oliver Feltham, Bloomsbury, London 2005, p. 4.

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oliver feltham
clarifies, “Mathematics writes that which of being itself, is pronounceable in 
the field of a pure Theory of the multiple”.
2
 In Meditation One he sets out re -
quirements for ontology and then in Meditation Three he proceeds to identify a 
particular kind of mathematics that satisfies those requirements.
3
 But this is too 
simple: there is a puzzle here: what comes first – the argument that determines 
the requirements for ontology, or the identification of set theory as ontology? 
One cannot argue that a particular discourse is uniquely suitable for the task 
of ontology without pronouncing as to the nature of being and thus engaging, 
at least in a preliminary manner, in ontology yourself. Is there not a problem of 
circularity here? 
On the one hand, there are these strong readings of canonical texts in the histo -
ry of metaphysics that are supposed to set up the choice of set theory as ontol -
2 Ibid., p. 5.
3 Ibid., p. 29. In a broad outline this passage appears deceptively simple; however, there 
are several difficulties that confront the interpreter who pays attention to the details. For 
instance, there has been much debate over the concepts of inconsistent multiplicity and 
consistent multiplicity: do they form Badiou’s version of Heidegger’s ontological differ -
ence (Hallward, Nancy, Pluth, Feltham vs Brassier); does Badiou have or even need an 
adequate account of the consistent multiplicity of ‘non-ontological situations’ (Feltham, 
Besana, Hallward); what does it mean to claim that the count-as-one has no agent (De -
santi, Feltham and Clemens)? What is the status of the claim that there must be a redou -
bled count-as-one establishing the state of a situation, because being is not presented as 
chaos, or what is the status of the claim that this second count-as-one has to include the 
name of the void, which errs in presentation, for otherwise there would be chaos? There 
are various ways of resolving these difficulties, but in order to develop a coherent inter -
pretation of the argument ‘mathematics is ontology’ it seems to me there is a more gener -
al puzzle that takes priority, and that is the puzzle of what comes first, the determination 
of the criteria for ontology or the identification of mathematics and, in particular, ZFC set 
theory as ontology. See Ray Brassier, “L’Anti-phènomene – présentation et disparaître”, 
in Ecrits autour de la pensée d’Alain Badiou, ed. B. Besana and O. Feltham, Harmattan, 
Paris 2007, pp. 55–64 ; Bruno Besana, “Quel multiple ?” and “Replique ; l’événement de 
l’être”, Ibid., pp. 23–40, pp. 125–30 ; J-T. Desanti, “Some Remarks on the Intrinsic Ontol -
ogy of Alain Badiou” in Think Again: Alain Badiou and the Future of Philosophy , ed. P. 
Hallward, Continuum, London 2004, pp. 59–66; O. Feltham & J. Clemens, “An Introduc -
tion to Alain Badiou’s philosophy”, in A. Badiou, Infinite Thought: Truth and the Return of 
Philosophy, ed. and trans. O. Feltham & J. Clemens, Continuum, London 2003, pp. 1–38; 
Peter Hallward, Badiou: a Subject to Truth, University of Minnesota Press, Minneapolis 
2003; Jean-Luc Nancy, “Philosophy without conditions”, in Think Again: Alain Badiou 
and the Future of Philosophy, ed. P. Hallward, pp. 39–49; Ed Pluth, Badiou: a Philosophy 
of the New, Polity Press, London 2010.  

39
“one or many ontologies? badiou’s arguments for his thesis ‘mathematics...
ogy, yet at the same time the strong theses and leaps in these readings seem to 
be either anchored or driven by the prior choice of set theory as ontology. Sam 
Gillespie wrote “Given that being qua being is given to us exclusively through 
ontology, it follows that it is very difficult to summon a mathematical ontology 
to a tribunal of ontology which would tell us whether or not it is a legitimate 
ontology. Ontology is in no way a set of descriptions of being that preexists its 
own operations”.
4
There are two solutions to this general puzzle. They break the circularity by pos -
iting a basic order. 
— One can argue that it is the philosophical arguments that come first, name -
ly a kind of history of being to rival that of Heidegger, and the election of 
set-theory comes second. Let’s call this the ‘philosophy solution’, or rather, 
the ‘argument from philosophy’. 
— Or one can argue that what comes first is the naming of Cantorian set-theory 
as a truth-procedure within the field of science, which subsequently condi-
tions philosophy and generates this sub-discipline of ‘metaontology’. Let’s 
call this the ‘argument from the condition’. 
 
The challenge for these two solutions, the criteria for us choosing one over the 
other, will be whether it succeeds in chasing down and eliminating the occa-
sional appearance of arbitrariness in the argument that explains why mathe -
matics is ontology.
5
 
Let’s first examine the ‘argument from philosophy’. 
4 Sam Gillespie, “L’être multiple présenté, représenté, rendu vrai”, in Ecrits autour de la 
pensée d’ Alain Badiou, ed. B. Besana & O. Feltham, p. 73.
5 Or do we accept that philosophy is a not an enterprise uniquely motivated by conceptual 
construction and argumentation; rather, Alain Badiou was seeking to give his own name 
to being qua being via the law of the father as Quentin Meillassoux provocatively sug -
gests in his essay “Décision et indécidabilité de l’événement” in Autour de Logiques des 
Mondes, ed. O. Feltham, D. Rabouin and L. Lincoln (eds.), Editions des Archives Contem-
porains, Paris 2011, pp. 135–6.

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oliver feltham
The argument from philosophy  
The initial philosophical argument in Being and Event results in the following 
requirement: being must be thought as inconsistent multiplicity. This claim is 
set up in four steps:
1) The One is not: The starting point, anchored in the history of philosophy, is 
the thesis that the One is not, explored by Plato in the last four hypotheses 
of the Parmenides. 
2) There is inconsistent multiplicity: Parmenides’ exploration of the hypothesis 
“If the One is not, nothing is” leads to the concept of plethora, of a multiple 
that disseminates itself internally without limit and thus without ever en-
countering some ultimate elements or atoms: this is what Badiou calls ‘in-
consistent multiplicity’. 
3) There is a count-for-one: Nevertheless, Badiou argues, there is some One -
ness, an effect of unity and so there must be an operation of unification that 
distributes inconsistent multiplicity (before) and consistent multiplicity (af -
ter its effect). A consistent multiplicity is unified. 
4) The nothing is: within consistent multiplicity, inconsistent multiplicity is 
nothing and as such it subsists in structured presentations as the void. What 
is ‘void’ within a structured presentation is both the operation of its count-
for-one and the material from which all structure is composed (inconsistent 
multiplicity). 
Badiou then adds a further, separate, requirement for ontology: ontology must 
be compatible with contemporary praxes of the subject. 
The strategy of this argument is to claim that a philosophical confrontation with 
the impasses of the history of ontology entails these four theses on the one and 
multiplicity. It is these claims which set up the requirements for ontology. After 
an examination of various kinds of discourse, it turns out that the only discourse 
capable of exploring and unfolding the implications of these theses is a particu -
lar kind of set theory. We find this ‘argument from philosophy’ in Meditations 
1, 2 and 6 of Being and Event, developing readings of Heidegger, Plato and Aris -
totle, and we find it massively in the Seminar from 1983 to 1986. There are two 
versions of this argument, which we shall call the via negativa, and the historial. 
 

41
“one or many ontologies? badiou’s arguments for his thesis ‘mathematics...
On the one hand, Badiou will engage in a negative demonstration – the via neg -
ativa – arguing that ontologies committed to the being of the One end in ruins. 
For instance, in the Short Treatise of Transitory Ontology, he rapidly pulls apart 
Aristotle’s necessary supposition of a global unity, a prime-mover in order to re -
solve difficulties in the theory of substance as an impossible union of matter and 
form.
6
 In Being and Event he claims that ontology repeatedly falls into an abyss 
or a labyrinth when it tries to resolve the relationship between the discrete and 
the continuum, and also when it tries to resolve the relationships between the 
one and the multiple, or parts and the whole.
7
 This negative demonstration is a 
little like Kant’s proof of the systematicity of transcendental philosophy through 
its resolution of the antinomies of pure reason, antinomies that ruin all other 
philosophies. The simplest form for this demonstration would be an argument 
from the absurd where an initial premise ineluctably leads to a contradiction and 
hence one is obliged to jettison the initial premise (namely the being of the One). 
The problem, of course, is that despite the variety and breadth of Badiou’s ex -
amples of paradox and contradiction in the history of ontology, strictly speaking 
the demonstration can never be exhaustive because the history of philosophy is 
still open. Someone – perhaps they are reading this article right now! – might 
well come along and write a coherent ontology on the basis of the premise that 
being is One. It is difficult to demonstrate impossibility outside the confines of a 
simple formal system, in which one options for argument are exhaustible. 
Hence Badiou’s frequent recourse to another version of the argument from phi-
losophy, the historial. For instance, he claims that the being of the One is the 
fundamental commitment of all onto-theology, adopting Heidegger’s term for 
his purposes. However, the orientation of thinking that Badiou is carving out 
holds itself contemporary to – or tributary of – Nietzsche’s declaration that ‘God 
is dead’, a declaration that mortifies all Gods, those of metaphysics, religion and 
poetry alike.
8
 There is no going back on Nietzsche’s epochal declaration. Hence 
the entire project of onto-theology is closed, and ontology must begin on anoth-
6 Alain Badiou, Court traité d’ontologie transitoire, Editions du Seuil, Paris 1998, p. 15 ; Alain 
Badiou, Briefings on Existence: A Short Treatise on Transitory Ontology, trans. N. Madarasz, 
State University of New York Press, Albany 2006. 
7
 Badiou, Being and Event, pp. 5, 81, 281.
8 See the “Prologue: God is Dead” in Alain Badiou, Briefings on Existence, pp. 21–32. See also 
Quentin Mellassoux’s excellent pun, mentioned above, on the proper name ‘A bas l’Un-

42
oliver feltham
er basis than the being of the One. Again in the Seminar, and in other texts, one 
finds such claims: the thesis “every situation is infinite” is pinned to a reading 
of Pascal, but it is also situated as a defining thesis of modernity, a thesis that 
opens up modernity, a historical decision. Badiou often adopts this historial ver-
sion of the argument from philosophy when he is interpreting and critiquing 
Heidegger’s history of being: it is as though he is setting up, by means of his own 
interpretations of canonical philosophical texts, a rival history of being. 
Here you can see the historial strategy runs into another problem, one of cir -
cularity. If being has its own history – which produces these theses ‘the One is 
not’, ‘every situation is infinite’ – then what is the original language or discourse 
in which that history is disclosed? We haven’t yet arrived at set-theory, we are 
still identifying the preliminary theses which will subsequently justify the elec -
tion of set theory as ontology. 
So both the via negativa and the historial versions of the argument from philos -
ophy run into problems. I contend that it is these problems that generate the 
occasional appearance of a prevalence of choice or of decision in Badiou’s con-
struction of his interpretations of philosophy. This is not a matter of failing to 
convince specialists of the cogency of Badiou’s interpretation – specialists are 
never convinced, even by each other – but of the overly apparent tactical choic -
es in his readings. 
Badiou, of course, is well aware of the problem of circularity in the justification 
of an inaugural decision as to the nature of ontology and philosophy. It forms 
one of the central topics of his analysis of the poem of Parmenides in the epon-
ymous seminar of 1985-86.
9
 In the seminar he appears to borrow an idea from 
Guy Lardreau concerning the foundation of philosophy, which is to argue that 
the inaugural decision as to the nature of philosophy is actually taken from the 
Dieu’ in “Décision et indécidabilité de l’événement”, in Autour de Logiques des Mondes, 
p. 136.
9 Badiou writes, “If philosophy decided itself from this point and this point alone – thinking 
at the same time the way of being and the way of nothingness, and instituting a regime of 
decision –, and if this point was the absolute origin of the existence of philosophy’s dis -
cursive apparatus, then one must declare, as I do, that philosophy had been decided well 
before.” See Alain Badiou, Le séminaire: Parménide, L’être I – Figure ontologique, Fayard, 
Paris 2014, p. 54.

43
“one or many ontologies? badiou’s arguments for his thesis ‘mathematics...
standpoint of another discourse.
10
 Badiou then claims that ‘philosophy is under 
a supplementary condition’, and insists on the heterogeneity of this condition 
and its encounter with philosophy as productive of decisions. This recalls the 
supplementary fifth thesis stipulating the requirements for ontology that we 
mentioned above and then left aside: that ontology must be compatible with 
contemporary praxes of the subject. Let’s retain these two terms ‘heterogeneity’ 
and ‘encounter’ – they will guide our conclusion. In his seminar Parmenides Ba -
diou thus adopts the argument from the priority of the condition as the solution 
to the problem of circularity, the problem that affects inaugural decisions that 
open up a philosophy such as the thesis “mathematics is ontology”. Let’s turn 
to this argument from the priority of conditions. 
Argument from the priority of conditions 
The argument from the priority of conditions for the thesis ‘mathematics is 
ontology’ runs as follows. Philosophy only occurs historically in the form of a 
‘com-possibilization’, that is to say, a naming and theorizing of truth-procedures 
occurring in four different conditions of art, science, politics and love. A philos -
ophy develops a coherent system of reference by constructing its own names 
for these generic truth procedures, procedures that trace out the consequences 
of anomalous events occurring in each of these extra-philosophical fields. For 
instance, Being and Event is an attempt to philosophically name what occurs in 
the condition of science as a truth procedure faithful to the “Cantor-event”, but 
it also names what occurs in poetry in Mallarmé’s fidelity to the ‘crisis in verse’, 
there are brief references to Engels and Mao in the political thinking of the state, 
and there is an engagement with psychoanalysis as intervention in the condi-
tion of love via the exegesis of Lacan’s concept of the subject. 
As such, it is the philosopher’s initial fidelity to the Cantor-event that decides 
that Zermelo-Fraenkel set theory with the Axiom of Choice will determine the 
nature of ontology, and hence ground these philosophical theses such as “there 
is no being of the One”. First comes fidelity to the condition, then philosophy. 
Set theory itself is ontology, and its philosophical naming and theorization is 
called “metaontology”. Alberto Toscano and Ray Brassier explored this strat -
egy in a 2004 article where they claimed that “mathematics is ontology” was 
10
 Ibid., p. 62. 

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oliver feltham
Badiou’s own intervention, his own fidelity to the Cantor-event.
11
 It is Badiou’s 
fidelity to Cantorian set-theory that determined his understanding of ontology, 
and his subsequent disqualification of various philosophical ontologies. 
 
At this point another suspicion of insufficient justification emerges. In the de -
velopment of the argument of Being and Event it turns out that it is not so much 
mathematics in general but specifically ZFC set theory that is ontology. Com-
mentators have asked why one would choose this particular variant of set theory 
(although Badiou does give a number of cogent reasons during the construction 
of Being and Event). Others have asked why choose set theory as a metonymy for 
mathematics and not another sub-discipline of maths, given that the entirety of 
mathematics is ontology. 
This second appearance of insufficient justification is addressed in section four 
of this article. For the moment let’s note the advantage or virtue of the argu -
ment from conditions: it leads to a philosophical exploration of the singularity 
of set-theory, which allows the identification of three peculiar characteristics 
that are grasped as demonstrating its vocation for ontology.
The first characteristic of Zermelo-Fraenkel set theory is that it has no defined 
object. There is no explicit definition of a set. As such the theory has no stipulat -
ed object. Rather the theory employs one primitive relationship between multi-
ples; that of belonging, and this relationship may be used in the ways specified 
by the axioms. Various kinds of set then emerge from that manipulation. The 
most striking demonstration of this emergence without definition occurs in the 
explanation of the axiom of the union set.
12
 This axiom states that for every set 
there exists the set of the elements of the elements of that set. This immediately 
cancels out any substantial distinction between ‘sets’ and ‘elements’, which are 
clumsy terms from natural language. Indeed there is only one kind of variable in 
this form of set theory. Every element of a set is itself a set composed of elements 
which are themselves sets and so on. Set theory can thus consistently write the 
decomposition of multiples as more multiples at other levels. Through the ax -
iom of the power-set, it can also write the composition, the putting together of 
11
 Ray Brassier and Alberto Toscano, “Postface: Aleatory Rationalism”, in Alain Badiou, 
Theoretical Writings, Continuum, London 2005, p. 255. 
12
 Badiou, Being and Event, pp. 63–64.

45
“one or many ontologies? badiou’s arguments for his thesis ‘mathematics...
larger multiples through the assemblage of all of the sub-groups of elements of 
an initial set. So on the basis of an initial multiplicity one can generate other 
multiplicities through analysis or synthesis, below it and above it so to speak. 
The series of sets generated continues to such a point that Badiou states with 
glee: set theory’s “proliferation of infinities” achieves “the complete ruin of any 
being of the One”.
13
 
It is not a foundational definition but the axioms that structure and render con-
sistent the situation that is ZFC set theory. This is an exemplary case for prag -
matism: use defines being; that is to say, what counts as being, as multiples of 
multiples, is only ever whatever is encountered through the facility of writing 
and manipulating sets and whatever might block that facility, such as Russell’s 
paradox. As such, there is no unifying or totalizing speculative gaze at being; it is 
neither seen nor grasped but encountered bit by bit in the scriptural construction 
of different kinds of sets, in a kind of constrained unfolding of multiples of mul -
tiples.
14
 Hence being-qua-being is not an external object, but insists in a writing.
The second peculiar characteristic of set-theory that Badiou seizes upon is that, 
in Lacanian terminology, it encounters the ‘real’ in its symptoms. In non-Lacan-
ian terms, set theory has discovered a number of paradoxes and problems that 
stymied its efforts at formalization. It is Cantor, for instance, and not Badiou 
who originally develops the concept of an “inconsistent multiplicity”, precisely 
in reaction to the discovery of sets that could not be totalized without contradic -
tion (the set of all sets that do not belong to themselves). Amongst these para-
doxes and problems Badiou briefly mentions the controversy around the Axiom 
of Choice, he devotes several pages to Russell’s paradox, but the entire last third 
of the book is devoted to the problem of the continuum and its implications. In 
the Introduction he announces, “What seemed to me to constitute the essence 
of the famous “problem of the continuum” was that in it one touched upon 
an obstacle intrinsic to mathematical thought, in which the very impossibility 
which founds its domain is said”.
15
 Indeed, in Meditation 27 he declares that the 
13
 Badiou, Being and Event, p. 273.
14
 See my claim in “Translator’s Preface”, in Ibid., p. xxiv.
15
 Ibid., p. 5.

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oliver feltham
measure of the excess of the cardinality of an infinite set’s powerset over the 
cardinality of that infinite set, constitutes the “impasse of being ”.
16
The third peculiar characteristic of set theory as a kind of discourse is that it 
historically took decisions on these symptoms. That is to say, mathematicians 
found resolutions to these paradoxes which in turn opened up new and further 
domains of formalization. For instance, Zermelo’s Axiom of Separation allowed 
his version of set theory to avoid Russell’s Paradox (Med.3).
17
 In Badiou’s exe -
gesis, set theory encountered the ontological paradox of the relationship be -
tween the continuum and the discrete in its discovery of the undecidability of 
the excess of the cardinality of a powerset over the cardinality of its original 
set if the latter is infinite. However, in Paul Cohen’s invention of the procedure 
of forcing a way is found in each individual procedure to decide on that unde -
cidable excess. Badiou calls these resolutions of paradoxes or circumventions 
of impasses in formalisation “decisions on being ”. He claims that in a belated 
echo of Parmenides they show how “the same, is, both thinking and being”.
18
At this point of our exploration we can detect two metaontological consequenc -
es of these ‘decisions on being’ carried out by set theory. The first concerns the 
ontological or set-theoretical inscription of the ‘generic’ multiplicity of truth 
procedures, and the second the possibility of an alternative ‘history of being’. 
The first consequence of these ‘decisions on being’ concerns their multiple oc -
currence within praxes that follow events, that is to say, within what Badiou 
calls ‘generic truth procedures’. For instance, when it comes to the set-theoret -
ical continuum problem, or, in his metaontological terminology, the ‘impasse 
of being’, he claims that there are four types of decision on that impasse which 
constitute the four grand orientations of thought: the transcendental, the gram -
marian-constructivist, the indiscernible-generic, and the praxical. 
16
 Ibid., p. 281.
17
 Note that Badiou has been fascinated with these decisions at the points of impossibility 
within formal systems since his early essay “Subversion Infinitésimale”, in Concept and 
Form, volume I: Key Texts from the ‘Cahiers pour l’Analyse’, ed. Peter Hallward and Knox 
Peden, Verso, London 2012.
18
 Badiou, Briefings on Existence, p. 52.

47
“one or many ontologies? badiou’s arguments for his thesis ‘mathematics...
He then goes on to claim that Cohen’s solution to the continuum problem by 
means of forcing and the generic subset provides an ontological schema for 
all generic-truth procedures. All generic truth procedures – all praxes that are 
faithful to an event – thus involve decisions on the impasse of being. So at this 
point the argument from conditions leads to a multiplication of ‘decisions on 
being’ within all those generic truth-procedures that a philosophy might recog -
nize, and compossibilize in its attempt to be contemporary with recent events 
in its conditions. 
Interestingly this consequence does not lead to a reduction of the suspicion of 
insufficient justification that attended the election of ZFC set theory in particu -
lar as ontology. Rather it presents an exacerbation of that insufficiency through 
its transformation into the radical contingency of events and truth-procedures. 
In other words, there is insufficient justification within ZFC set theory itself for 
the continuum hypothesis or for other hypotheses as to the cardinal quantity 
of the powerset of an infinite set – this is why Badiou terms them ‘decisions on 
being’. If decisions on being occur within truth-procedures not just in the realm 
of mathematics but also in politics, art and love, then these moments of ‘in -
sufficient justification’ multiply in such practices. However, at the same time, 
the insufficiency at stake in these decisions within these practices can also be 
understood as a form of radical contingency in line with that of the event.
The second consequence of these set-theoretical ‘decisions on being’ is that via 
their metaontological interpretation they intervene in the philosophical or his -
torical problems of ontology. Throughout Being and Event , Badiou remarks that 
there are a series of unresolved problems in the history of ontology concerning 
the one and the multiple, the part and the whole, the finite and the infinite, and 
the relationship between the discrete and the continuum. It just so happens 
that Zermelo-Fraenkel set-theory provides a series of new solutions to these 
problems. There is thus a new ‘history of being’ that is generated by Badiou’s 
argument from the condition of set theory to philosophy. Precisely, he explicitly 
claims ‘The history of mathematics [periodized by singular praxes] is the histo -
ry of being’. It is by means of this alternative mathematical history of being that 
Badiou will be able to rival Heidegger, and ground his claims with regard to 
requirements for a contemporary thinking. The argument from conditions thus 
joins the argument from philosophy in its commitment to a history of being, 
but this time the history is grounded in an alternative discourse to philosophy. 

48
oliver feltham
Thus in this regard at least, the argument from conditions is superior to the 
argument from philosophy. 
At this point let’s return to this suspicion of a lack of justification over ZFC set 
theory alone being elected the metonymy of mathematics as ontology. It is clear 
that we cannot hope for any absolutely solid philosophical demonstration of 
the necessity of ZFC alone as ontology to the exclusion of all other mathematical 
and non-mathematical candidates for the discourse on being qua being. Such a 
demonstration is impossible because the equation ‘maths is ontology’ does not 
take place within a formal system. I return to Sam Gillespie’s statement: “Given 
that being qua being is given to us exclusively through ontology, it follows that 
it is very difficult to summon a mathematical ontology to a tribunal of ontology 
which would tell us whether or not it is a legitimate ontology”.
19
 It is also evi -
dent that Badiou’s metaontology is not the only possible philosophical exegesis 
of what is going on inside ZFC. 
What is our conclusion concerning the argument from the priority of set-theory 
as a condition for philosophy? It grounds Badiou’s alternative history of being, 
which is an advantage; it does not eliminate the appearance of insufficient jus -
tification in the initial election of ZFC set theory, which is a disadvantage, and 
as an ontology compatible with truth procedures, it multiplies ‘decisions on be -
ing’, which is neither an advantage or a disadvantage but opens up an enquiry 
into conception of action entailed by such decisions. 
Let’s try one more approach, the pragmatist approach: what difference does the 
election of set theory as ontology make to Badiou’s philosophical project? What 
does set-theory qua ontology do? In Badiou’s terminology this is the question of 
the status and role of ‘metaontology’. 
19
 Gillespie, “L’être multiple présenté, représenté, rendu vrai”, in Ecrits autour de la pensée 
d’ Alain Badiou, p. 73.

49
“one or many ontologies? badiou’s arguments for his thesis ‘mathematics...
The status of metaontology
The simple answer to this question is that philosophical metaontology sets out 
the concepts of a philosophical theory of radical transformation.
20
 It allows one 
to state, for instance, that “the form-multiple of being is generally infinite”.
21
 
It makes the distinction between presentation and representation, a situation 
and its state, it generates the concept of eventual sites and the ensuing distinc -
tion between natural and historical situations, it anchors the claims that there 
is no totality of Nature nor of History, etcetera. 
The choice of ZFC set theory as ontology turns out to be extremely rich and pro -
ductive in terms of the wealth of concepts generated for this theory of change. 
But this theory immediately encounters another problem, that of schematism. 
Badiou opens Meditation 12 on ‘natural multiples’ with the following claim:  
Set theory, considered as an adequate thinking of the pure multiple, or of the 
presentation of presentation, formalizes any situation whatsoever insofar as it 
reflects the latter’s being as such; that is, the multiple of multiples that makes up 
any presentation. If, within this framework, one wants to formalize a particular 
situation, then it is best to consider a set such that its characteristics […] are 
comparable to that of the structured presentation – the situation – in question.
22
Particular situations are thus ‘formalized’ by considering a set as the schema 
of a situation, hence my term ‘schematism’. This passage has caused much 
20
 Badiou makes the connection between set-theory ontology and his larger project of de -
veloping a theory of radical transformation and hence, in the political sphere, of justice 
in the following terms in the “Introduction” to Being and Event: “It is the act of trusting 
[mathematicians] forever with the ‘care of being’ which separates truth from knowledge 
and opens it to the event. Without any other hope, but it is enough, than that of mathemat -
ically inferring justice.” (Ibid., p. 15) Badiou also writes in the introduction “All [the thesis 
‘mathematics is ontology’] does is delimit the proper space of philosophy…its function is 
to introduce specific themes of modern philosophy, particularly – because mathematics is 
the guardian of being qua being – the problem of what-is-not-being-qua-being.” (Ibid., p. 
15) Note that the category of ‘what-is-not-being-qua-being’ refers to the event. Badiou also 
writes with regard to ontology, “the saying of being would hardly make any sense if one did 
not immediately draw from the affairs of the City and historical events whatever is neces -
sary to provide also for the needs of ‘that-which-is-not-being-qua-being’” (Ibid., p. 282).
21
 Ibid., p. 266.
22
 Ibid., p. 130. 

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oliver feltham
consternation amongst commentators. Tzuchien Tho asked Badiou a question 
about this very passage in the interview after the English translation of The 
Concept of Model.
23
 Schematism is already present in many of the metaontolog -
ical statements in the early meditations, especially the claim that set theory is 
the presentation of presentation; that is to say, the presentation of the inconsist -
ent multiplicity of all consistent multiplicities in non-ontological situations.
24
 
The first problem with metaontology as a schematism – one that Ray Brassi -
er pointed out very early on – is that it turns set-theory into a referential dis -
course, gives it supposed objects, those objects being the implicit structures 
of non-ontological situations (and so Badiou would be a kind of structuralist 
après la lettre).
25
 One loses the radically immanent and scriptural quality of 
set-theory ontology that I referred to before. 
The second problem with schematism, the one that bothers most of the An -
glo-Saxon commentators though, with their heritage of empiricism, is how to 
actually demonstrate that a particular set schematizes a particular situation. I 
asked Badiou this question in the 1999 interview transcribed in Infinite Thought , 
and for him it was a non-starter, perhaps, though it is a bit of an excuse, be -
cause he is not weighed down by the tradition of empiricism.
26
 
One cannot demonstrate that sets provide the schemas of all non-ontological 
situations without already making some prior decisions as to the adequation 
of other discourses to those ‘given situations’, and also having a translation 
23
 See Tzuchien Tho and Alain Badiou “The Concept of Model, Forty Years Later: An Inter -
view with Alain Badiou”, in Alain Badiou, The Concept of Model: an Introduction to the 
Materialist Epistemology of Mathematics, ed. and trans. L. Fraser & T. Tho, re.press, Mel -
bourne 2007, pp. 94–5.
24
 The argument for what I call the schematism of set-theory ontology is inextricable from 
Badiou’s argument for ontology being that unique situation that presents inconsistent mul -
tiplicity. On the last page of Meditation One he states: “To accede axiomatically to the pres -
entation of their presentation, these consistent multiplicities of particular presentations, 
once purified of all particularity – thus seized before the count-as-one of the situation in 
which they are presented – must no longer possess any other consistency than that of their 
pure multiplicity, that is, their mode of inconsistency within situations” (Ibid., p. 30).
25
 See Ray Brassier, “L’ Anti-phènomene – présentation et disparaître”, in Ecrits autour de la 
pensée d’ Alain Badiou, pp. 55–64.
26
 Badiou, Infinite Thought, p. 178.

51
“one or many ontologies? badiou’s arguments for his thesis ‘mathematics...
protocol between those discourses and set-theory, and accepting what is lost in 
those translations (anti-reductionism objections); indeed this entire objection 
seems to be caught up in the impossibility of reinventing Carnap’s project in 
the Aufbau. 
I am English, I have to some degree inherited the burden of empiricism, and so I 
found it difficult to ignore or dismiss this apparent problem of schematism. One 
solution I found is laid out in another part of Badiou’s oeuvre, his first book on 
mathematics as a condition of philosophy, The Concept of Model . 
My hypothesis is that metaontology is the discourse that results from the mod -
elling of philosophical ontology by the syntax of ZFC set theory. I develop this 
argument in Alain Badiou: Live Theory .
27
 Here I will summarize its conclusion. 
The operation of conditioning of philosophy involves the selection of a theoret -
ical syntax from the language and names of a generic truth procedure – in this 
case, ZFC set theory. The second step is to select a semantic field – in this case, 
philosophy, in particular the history of ontology. A model of the theory is said 
to be produced if its syntax, and the operations that its syntax permits, can be 
reproduced without contradiction within that semantic field. 
Hence, in terms of the production of metaontology, if Badiou can reproduce the 
syntax and operations of ZFC set theory in the semantics of the history of ontol -
ogy without encountering contradiction, then he will have produced a model of 
ZFC set theory.
28
 
The argument from the priority of conditions must therefore be understood 
via the operation of modelling. This hypothesis neatly resolves the problem 
of schematism. Badiou’s metaontology does not provide a theoretical schema 
27
 Oliver Feltham, Alain Badiou: Live Theory, Continuum, London 2008.
28
 I understand Badiou to be using such an argument when he makes a claim like ‘the whole 
of the thinking unfolded in Being and Event constitutes the demonstration that mathemat -
ics is ontology’. He made this claim orally at a seminar “Ontologie et mathématiques” at 
the American University of Paris on June 17
th
 2019. See “Ontology and Politics: an Interview 
with Alain Badiou”, in Alain Badiou, Infinite Thought: Truth and the Return of Philosophy, 
Continuum, London 2003, where Badiou says “A large part of L’être et l’événement tries to 
explain with the means of mathematics why mathematics is ontology. As a matter of fact it 
is its task”, p. 184.

52
oliver feltham
or model of non-ontological situations. Rather these objects or names – such 
as ‘non-ontological situation’, ‘natural’ or ‘historical situation’, ‘evental-site’, 
‘state’, ‘impasse of being’ – are all elements of a model of a theory. In pragmatic 
terms, Badiou’s metaontology creates a new universe of objects. When I ana -
lyse a situation as a historical situation, my metaontological model enters into 
competition not with given concrete situations (as a vulgar empiricism would 
have it, somehow measuring ‘theory’ against ‘reality’), but with established 
universes created by the models of other theories; that is to say, it enters into 
an ideological battle (the constant theatrical background of Badiou’s project). 
So the concept of modelling allows the dismissal of the charge of schematism, 
but there is one last problem. The solution from modelling does not completely 
eliminate the suspicion of insufficient justification. Simply put, why model ZFC 
set theory and not another mathematical theory?
The practice of modelling entails a plurality of models 
Here the solution is evident, and it is exemplified by Badiou’s own practice, 
turning to category theory in Logics of Worlds to entirely remodel the philosoph -
ical discipline of phenomenology, and back to the theory of grand cardinals to 
rework his system again in The Immanence of Truths. In other words, there is 
a positive interpretation of the initial choice of ZFC appearing undemonstrated 
and thus arbitrary: the practice of modelling can entail a plurality of models. 
In this way, the insufficient justification is transformed into the contingency of 
a decision. Here we can follow as our guide the many ‘decisions on being’ that 
take place in the multitude of generic truth procedures; that is to say, we can 
accept a proliferation of decisions on being, it doesn’t need to make us anxious. 
Yet if these decisions on being concern the modelling of ontology itself , then we 
find the way opened to not one but many ontologies! Now this would be quite an -
other strategy to adopt on the basis of the initial argument that the being of the 
One must be rejected and being thought as inconsistent multiplicity. This would 
be an approach similar perhaps to what Jean-Toussaint Desanti calls ‘extrinsic 
ontology’, according to which, when one interprets the phrase ‘being qua be -
ing’, one works to embrace the maximum of senses encapsulated in being, along 
the lines of Aristotle’s initial intuition – that being is spoken in many ways –  

53
“one or many ontologies? badiou’s arguments for his thesis ‘mathematics...
an intuition that Aristotle, and subsequently many philosophers after him, 
swiftly sets aside.
29
There are actually a few indices or openings to this strategy of many ontologies 
in Badiou’s own work. When he shows that one of the singular characteristics 
of set theory ontology is that it does not totalize being, does this not open up 
the possibility of other kinds of inscription of being? When he argues that on-
tology is not a transcendental all-englobing situation but merely one situation 
amongst others, could it not also be one ontology amidst others?  
There are two evident objections to this strategy of multiple ontologies. First, 
it will end up in eclecticism: there are conferences on the ontology of this or 
that, the ontology of emotions, the ontology of social constructs. Second, the 
choice of multiple ontologies via this eclecticism finally ends up in nihilism. 
Barry Smith has attempted to construct a ‘Basic Formal Ontology’ for the benefit 
of engineering and medical databases not to mention, the military and intelli-
gence communities including the FBI.
30
One can avoid eclecticism and nihilism by dismissing the idea that every single 
situation comports its own set of existential commitments which constitute a 
specific ontology. In my own work on action I show that the ontology opens up 
not as another variant of the discourse of the university, but as a lived enquiry 
uniquely when a failure or dysfunction occurs in the reception and consequenc -
es of an action. The action itself turns out to be ‘equivocal’ in that it is subject to 
conflicting attributions of its nature – is it a just or unjust action, has Macron’s 
government ‘ensured a return of the rule of law in Notre Dame des Landes’, or 
has he ‘destroyed people’s livelihood’? This equivocity expands to include the 
intention behind the action, its identifiable consequences and even the identi-
ty and ‘kind’ of the action’s agent. Contestation occurs over the being of these 
actions, and each side in such controversy develops its own ontology of these 
actions. In this manner diverse ontologies multiply in so far as people – not just 
philosophers – explore the nature of failure and dysfunction and controversy in 
29
 Jean-Toussaint Desanti, “Some Remarks on the Intrinsic Ontology of Alain Badiou”, in 
Think Again: Alain Badiou and the Future of Philosophy.
30
 See the Wikipedia article on Smith: https://en.wikipedia.org/wiki/Barry_Smith_(academic)

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oliver feltham
the occurrence of actions: I call this approach an ‘anatomy of failure’.
31
 I would 
hesitate to term this a ‘democratization’ of ontology as a kind of writing, because 
it does not concern a majority of people, nor is it based on the circulation of 
opinions. Anatomies of failure occur when people recognize and explore the 
undoing of opinions through the ambivalence and equivocity of certain actions. 
This process undoes received ideas and leads to the modification or displace -
ment of dominant ideological positions in a political situation. It is democratic, 
however, in so far as it requires a process of deliberation between a number of 
actors from different social and political contexts.
The result of people carrying out anatomies of failure, and broadcasting them, 
is that alongside this ‘stellar’ set-theory ontology which schematizes the struc -
tures and inconsistent multiplicities underneath all kinds of existential com-
mitment, we would embrace a multiplication of sub-lunar ontologies, and so 
account for the disjunctions and overlaps in existential commitments, such as 
the disjunction between Creon and Antigone. The level at which these ontol -
ogies would make a difference would be in the diagramming of conflicts over 
what kinds of action exist. The result of such an exercise would not simply be 
dialogue across conflict – one can read Rorty or Habermas for that. The result of 
such an exercise would be the remodelling of these conflicts according to these 
multiple ontologies. Such remodelling – and this is an argument I develop at 
length with regard to Hume’s History of England – creates new durations.
32
 And 
every new duration entails the emergence of a powerful – because new – meas -
ure of the gap between the actual and the ideal. In the gap between the actual 
and ideal arises the hope of justice.
References 
Badiou, Alain, Being and Event, trans. Oliver Feltham, Continuum Books, London and 
New York 2005
— Briefings on Existence: A Short Treatise on Transitory Ontology, trans. N. Madarasz, 
State University of New York Press, Albany 2006
— Court traité d’ontologie transitoire, Editions du Seuil, Paris 1998
31
 See Oliver Feltham, Anatomy of Failure: Philosophy and Political Action, Bloomsbury, Lon-
don 2013.
32
 This argument is substantiated in Oliver Feltham, Destroy and Liberate: Political Action on 
the Basis of Hume, Rowman and Littlefield, London 2019.

55
“one or many ontologies? badiou’s arguments for his thesis ‘mathematics...
— Le séminaire: Parménide, L ’être I – Figure ontologique, Fayard, Paris 2014
— “Subversion Infinitésimale”, in Concept and Form, volume I: Key Texts from 
the ‘Cahiers pour l’ Analyse’, ed. Peter Hallward and Knox Peden, Verso, London 2012
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