(Note: I have changed my view on this topic considerably. Please scroll down or click this link to see a video of a talk I gave on this topic which ends on a more skeptical note.)

It is well-known that Badiou proposes a new connection between mathematics and ontology. In the first place, this is a move internal to the field of philosophy, as his work does not aim to actively contribute theorems and proofs to mathematics proper. Rather, the aim is to show how certain classical ontological/philosophical questions can be approached by examining abstract (but foundational) mathematical theories. This examination remains distinctly philosophical; insights are lifted from the strictly mathematical language and interpreted ontologically. How is this move justified? Perhaps most importantly: How is the relation between mathematics and philosophy understood here? In this short text, I aim to explain how this new assembly of mathematics and ontology is motivated, how it works and why it contributes something to the philosophical field.

Ontology is the doctrine of being. Throughout the history of philosophy, there have been many divergent ontological orientations. Among others, an orientation in ontology is determined by

- The degree of separation from other philosophical themes; is the question of being faced head-on or does it get treated somewhere along the way, nested in a larger scheme?
- The way the question itself is understood; are we merely asking about the different
*species*of being or does the question of being transcend the creation of a catalogue? - The degree of convergence with questions of consciousness and subjectivity; is the concept of being necessarily correlated to a (human) point of subjectivity or thinking entity? Or is it to be treated independently? Why?
- The emphasis on questions of language; is language found to be inadequate to address being, or worse: is the question of being perhaps induced by an erroneous use of language? Or does a privileged language for ontology exist after all? Is it poetic or formal in character?
- The emphasis on abstraction and the formal; is the most general concept of being given by abstracting away from more particular beings or is being rather given in some aspect of experience? Is being a pure concept or an experience of overflowing?
- The influence of intended models; to what extent do other sciences, like biology, physics, or theology predetermine (and simultaneously enable) an approach by supplying an intended model? For example, since being may seem all-encompassing, a theologically-minded philosopher may graft the question of being onto a figure of god (onto-theology). What are the options?

This list is not exhaustive and reminds of certain authors, traditions, and debates more than of others. My point is that a philosophical question is to some extent situated in an orientation. The next step is to explain how Badiou indeed offers an ontological orientation in this sense and how he responds to the above questions.

## The one and the Multiple: Badiousian observations

Locating the beginning of Badiou’s systematic philosophy in the first volume of *Being and Event*, one can describe his orientation to ontology as follows. As always, the formation of a new orientation starts with the collecting of observations, giving us a context. It is from there that a systematic philosophy formulates its opening statements. For Badiou, the central observation is that many of the great ontological efforts ran aground on the pernicious question of the one and the multiple. Part of this may be due to the apparent but misleading simplicity of the matter. Badiou likes to cite Leibniz in this context: “What is not *a *being is not a *being*.”[1] This quote suggests the proximity of the question of being to the one/multiple couple. More precisely, Leibniz seems to say that all possible objects of our domain of discourse (i.e. all beings) are individuals since we refer to them in the singular. On this conception, being in the abstract is thus always *one* in this seemingly trivial way. Being is one since a being is *one* being. This correlation is supported on at least two levels: both consciousness and language seem to depend on the oneness of its objects. Continuing our inquiry, we ask: what about the obvious fact that most (if not all) entities are also wholes of *many* parts – making them multiples as much as we found them to be ones before? Are they multiples of beings which are themselves one? Or can parts of multiples be multiples themselves? Perhaps this is admissible if we reach some simple atoms eventually? And is a *multiple *also always *a* multiple (applying Leibniz’ formula to the multiple-beings)? What about a multiple which encompasses all beings, the final All if you will? Does it exist as a stable point of reference? Lastly, is there an important difference or correlation between the one/multiple-couple and the whole/part-couple?

Many of the classical philosophers identified these problems and tried to grapple with them by articulating concepts and arguments in a chosen natural language (Greek, Latin, German, etc.) enriched by a specialized vocabulary. Some approached the matter in an openly metaphysical or theological way whereas others linked these questions to the structure of consciousness or the investigation of the forms of thought. Certainly, this debate included the early days of analytic philosophy with Russell’s distinction between classes as one and classes as many, where the issue is treated (at least in part) as a problem of formal logic for the first time. It was also clear to all these thinkers that the one/many-couple had its place in ontology. Other ontological concepts were felt to rely on these notions.

From the perspective of Badiou’s ontology, some (if not all) of his predecessors held the prejudice that the notion of oneness has some ontological or conceptual priority over the notion of multiplicity. Put more simply, they believed that the concept of being a multiple was secondary to the concept of being one. It is not hard to identify an influence of monotheism in this: god as the all-encompassing one. But a materialist version of this prejudice exists as well: atomism, the idea that all things are made up of indivisible smallest parts. Here, the atoms are the true stuff of the world and composites are derivative. And a biologically oriented approach to ontology employing the concept of life makes paradigmatic the synthetic unity of the organism over the multiplicity of inanimate matter.[2] Again, something which is one and not multiple takes ontological priority. These prejudgments on the question of the one/multiple couple (nonetheless producing original philosophies in their own right) could thrive because of a lack of clarity on the purely conceptual level. Forget god and atoms: *what is the correct conceptual connection between the notion of oneness and the notion of multiple?*

Enter axiomatic set theory. Certainly, many philosophers had tried to treat the question formally (as opposed to metaphysically). But it is undeniable that the set theory developed by Cantor, Zermelo, and others constitutes a major system overhaul regarding any investigation of the one/multiple-couple.[3] Furthermore, it extends the proper territory of mathematics, which was previously understood as the ‘science of numbers and shapes,’ by making explicit and logically exact the reliance of mathematics on more abstract notions which were formerly treated in philosophy. Put simply, set theory axiomatizes the way human beings talk about sets. Considered from the point of view of natural language, sets are simply those referential entities which are internally multiple. But this axiomatization does not merely recreate formally what was already there in natural language. It becomes a context in which one can investigate what mathematical notions can be elaborated based on the concept of set alone. ZFC with its canonical axioms turns out to be very powerful as a mathematical system. Badiou’s move is to combine the observation of the omnipresence of problems regarding the one/multiple-couple in classical ontology with the observation that set theory offers a clean break with regards to such themes. This opens the door to a renewal of ontology and to an ontological framing of set theory – two sides of the same coin in Badiou’s philosophy.

By way of summary, we can categorize classical approaches to ontology and the question of the one/multiple-couple. We discern four paradigms: 1) A metaphysical treatment, in which we ask whether the stuff of the world is one or multiple (Parmenides, Aristotle); 2) a subjective treatment, where we ask how the epistemic subject cognizes oneness into being (Kant, Husserl); 3) a linguistic treatment, where the one/multiple-couple is analyzed as a grammatical phenomenon, where the above questions become pseudoquestions (Wittgenstein, Carnap). Finally, 4) a purely formal investigation of the conceptual aspects of the problem. This can be done 4a) in a philosophical language (Hegel) and 4b) in a mathematical language (Russell, Badiou).

## Mathematical Ontology as a Research Program

I already remarked that Badiou’s systematic philosophy starts with the observation that a mathematical framing of ontology (and vice versa) is possible. Hence, it is only consequential that his positive proposal is succinctly summarized in the slogan “mathematics *is* ontology.”[4] This statement, which founds a *research program* (in the sense of Imre Lakatos) in ontology,[5] starts from a conviction and an observation. The conviction (1) is that any ontology will stand or fall by the lucidity and rigor of its treatment of the one/multiple-couple. The observation (2) is that axiomatic set theory is exactly what we need: a precise mathematical language which formalizes what is rationally thinkable based on the membership relation *x *∈ *y* (to be read as ‘x belongs to y’) alone. I will return to the specific advantages of the mathematical approach. Finally, we conjecture (3) that in this formal framework, it will be possible to fruitfully give new meaning to a wealth of related classical ontological notions also (e.g. void, infinity, boundary, world, subject, truth). To demonstrate (3) thus becomes the larger task and promise of Badiou’s ontology. Like with any research program, the fruitfulness of the initial convictions (1 and 2) can only be evaluated by considering whether its effects (3) are fruitful in the larger field (ontology).

The motto ‘mathematics is ontology’ is succinct and therefore somewhat vague. The opening section of *Being and Event *gives it a precise meaning. If one were to unpack the phrase, one could write: ‘the first-order language of set theory with the Zermelo-Fraenkel (+ other optional) axioms is ontology.’[6] But one could be even more precise: ‘the investigation of the consequences of axiomatic choices in the language of ZFC such that these axioms formalize the concept of multiple-formation *is *the investigation of fundamental ontological decisions and consequences.’ An axiom becomes a decision, which can be justified in certain ways, and a theorem becomes a provable consequence of a decision.[7] This also brings us to a central question in Badiou’s work: how are such ontological decisions (axioms) *justified*? In short, this has to be argued and explored for the axioms individually. It is what makes up the substance of his main theoretical works (the three volumes of *Being and Event*). Let us focus on some other relevant features of ZFC for now:

- It is a particular first-order logical language with a single primitive sign ‘∈’. Such languages are extremely well understood (meta-mathematically) and have many desirable properties. The vocabulary is very controlled. Hence, in contrast to natural languages, ZFC is itself a well-circumscribed (meta-)mathematical object. When Badiou says that ‘mathematics is ontology’, he means a very specific and well understood language: ZFC. More strongly, one could say that the knowledge of mathematics in this narrower sense is everything proven from the ZFC axioms. This identification does relevant philosophical work; it gives us a very clear demarcation of ontology. Such a clear demarcation would never be possible if natural language were the medium of our investigation.
- Inside ZFC, one can define mathematical concepts such as numbers, shapes, structures, orders, spaces, functions, etc. In other words, the universe of ZFC already contains a host of mathematical concepts, one merely needs to define them using the terminology of sets and elements. (And thus, ZFC also contains all we need for mathematical science.) This notably includes a treatment of infinite sets (for which we have at least one axiom to consider). The combination of the strength and simplicity of ZFC has made it the standard choice for a foundation of all of mathematics.[8]
- The axioms of ZFC all correspond (to different degrees) to intuitive aspects of sets. One can gauge the naturalness of these axioms by comparing them to our intuitions regarding sets. Hence, ZFC really is a mathematical formalization of the one/multiple-pair.
- This is not to say that some of the axioms do not require both mathematical and ontological discussion. There are highly non-trivial choices to be explored. This is where philosophy enters for Badiou. The central theme is that the basic ZFC axioms do not fully determine a single universe of sets quite yet. The addition of further (independent) axioms has a discernable effect on the final structure of the universe of sets (for Badiou the realm of being qua being). The openness of ZFC for further axioms makes it possible to compare the effects of additional decisions regarding being. For example, a central distinction is the one between a universe which only contains so-called
*constructible*sets and one containing*generic*extensions. In*Being and Event*, Badiou uses this mathematical distinction to clarify the antagonism between important orientations in classical ontology.[9]

So Badiou’s ontological orientation is ultimately the choice of a particular logical language (and its axiomatic extensions) as the site for the articulation of ontological consequences (theorems). The meta-discussion of this language, its axioms, etc. then becomes ontology in the philosophical sense, or meta-ontology as Badiou sometimes calls it.[10] This set-up has the advantage that we can clearly demarcate actual sentences of this language (and hence of ontology) from those of other domains of discourse (including meta-ontology). This point is not to be underestimated, since it is what gives a (technically) precise separation of ontological language from other instances of the use of language, philosophical or other. Both ontological decisions (axioms) and ontological consequences (theorems) are simply classes of sentences formulated in ZFC. Hence, a full appreciation of Badiou’s strategy requires some familiarity with ZFC as a particular *proof system* and not just familiarity with the informal set-theoretic *style* of argumentation employed in much of present-day mathematics.[11] Since the choice is for a strong (yet simple) axiom system, it is to be expected that many more advanced ontological themes may be recovered on its basis. (Admittedly, this may be more obvious to readers already familiar with logic and set theory than to unexperienced readers.) Let us now investigate in more detail the corresponding picture of ontology that Badiou paints in the opening section of *Being and Event**.*

## The Impasse of Prior Ontology and ZFC as a Solution

I have already indicated some of the questions which arise around the one/multiple-couple. Badiou spends almost no time with existing positions on the matter and instead opens the book (Meditation I) with a short paragraph regarding the “impasse of ontology.”[12] By this he means pre-mathematical attempts to approach the question of being.[13] This impasse results from a (sometimes implicit) ontological *choice* regarding the one/multiple-couple. To understand the nature of this choice, let us first approach its setting agnostically. Being philosophers, we find ourselves attempting to determine whether there are general concepts which refer to being as such. Given the fact that being is perhaps the most abstract level at which we can grasp objects (of perception and thought alike), it is unsurprising that a great many concepts cannot apply to being *as such*, since they would particularize it. Are there any concepts which are somehow relevant to being as such? An important insight at this stage is that the choice of a predicate which splits the domain into two (into those entities to which the predicate applies and those to which it does not) will not help us. Instead of approaching the question of being with ready-made categories in hand, we should reflect on the very *setting of categorization* itself. Ontology is the science of the setting of grouping and categorization. Or as Badiou calls it, ontology is the “presentation of presentation.”[14] Being as such then simply becomes the amenability to arbitrary groupings and subgroupings. A category or grouping then gathers *as one* a number of instances, objects, individuals, or parts.

But this is only the very first step. It leads us to one possible way to tie these concepts (one, multiple, being) together. This brings us to what Badiou refers to as the impasse of classical ontology. This impasse amounts to either an error or an indecision with regards to the status of the one/multiple-couple in our understanding of the ontological setting. Required is a decisive point of view. At this point we could proceed either by describing the impasse in more detail or by sketching the solution first. As is often the case, the problems of older paradigms are only clear from the point of view of the new. So, we choose to start with the solution and try to answer the question: How does the identification of set theory with ontology give us a resolute point of view on the question of the one/multiple-couple?

In fact, how does set theory present a theoretical position on the one/multiple-couple *at all? *A brief look at the symbols used in set theory tells us that ZFC does not contain (primitive or defined) predicates of the sort ‘x is a set’, ‘x is multiple’, or ‘x is one’. These are not expressions of the language at all. So how can set theory clarify these concepts? Do we not require a link between the concepts we aim to clarify and the linguistic expressions available in the very language we develop for this purpose? The answer is that set theory *implicitly axiomatizes* the predicate ‘x is a set’ (which for Badiou becomes synonymous with ‘x is a pure multiple’). One takes a first-order logic with a single primitive binary predicate ‘∈’ and adds axioms which will ensure that everything in the domain of that language behaves as we expect sets to behave. Since most of these axioms are universally quantified, they apply to everything in the domain our variables range over.[15] If these axioms encode what we expect to be true of sets (and of sets alone), the domain thus effectively becomes a universe of sets. *Everything* in such a universe is a set and hence, ZFC does not give us anything to separate sets from non-sets. Rather, it axiomatizes a rational space in which everything is a set (and which also aims to capture *all *sets) and more importantly, in which the consistent thinking of multiple-being is explored fully. The members of sets are neither atoms nor some primitive given matter, but simply sets themselves. In ZFC, multiples are multiples *of *multiples. Even the elusive empty set is not one in a discernable way. Rather, as Badiou has it, the empty set is a “multiple of nothing.”[16] This is true since ZFC indeed does not distinguish between multiples and ones in any way.

This already allows us to formulate how Badiou has ZFC contribute to the question of the one/multiple-couple. Read as an (implicit) contribution to an age-old philosophical debate, set theory says: take ‘x is a multiple’ as that which is axiomatized implicitly in a theory of pure multiple-formation and stop giving ontological weight or significance to the concept ‘x is one’. And indeed, Badiou summarizes this in another slogan: ‘the one is not’. Badiou also gives an overview of different philosophical notions of oneness, some of which are acceptable on his view and others which are delegitimized by the ideational intervention of set theory.[17] His claim is furthermore that only mathematical ontology is up to the task of properly differentiating these notions. The notion of oneness only survives as the *effect of a count*: a set is that which is counted-as-one (or counted-for-one, depending on the translation). To say ‘x is’, that ‘x is *a* being’, amounts to saying ‘this multiple has been counted as one’. This lets us ask the question of oneness without basing it on the assumption of prior existing primitive parts, since every one is now the one of a multiple, the result of a count. The notion of oneness does *not *survive as something which exists ontologically prior to multiple-composition, simply since the discourse of set theory does not require it as a primitive notion! Internally, set theory does not distinguish between derivative multiples and original atoms from which the former would be built. As stated earlier, it is effectively the axiomatization of the composition of multiples from multiples, which functions without using any initial or final one-elements anywhere.

From here we can identify the impasse of prior ontology more easily. Regarding the question of the one/multiple-couple, it can now be seen to have chosen the wrong option, namely to have prioritized the one over the multiple – to have taken the one as an aspect of primitive existence and not as an effect. Its axiom has thus been ‘the one *is’*. The error or indecision on this issue is of course an effect of working outside of a formal context, which is why Badiou often refers to Cantor’s contributions to mathematics as an event with which philosophy must come to terms – with which it can reorganize ontology. This is not to say that philosophers before Cantor made some unforgivable error; it is rather that philosophers *after *Cantor are given the unique task to rethink these ontological themes in a philosophical setting now properly conditioned by the Cantor-event, i.e. the invention of axiomatic set theory. The earlier impasse amounted to not seeing the strictly formal possibility to make multiple-being the organizing concept, leading one to instead treat the one/multiple-couple as a dialectical pair of sorts (e.g. Hegel) or worse, to declare one-being ontologically prior to multiple-being (atomism, onto-theology, etc.).

Earlier thinkers also thought about the connections between simpler ontological concepts such as the one/multiple-couple and more complicated ones like space, number, infinity, etc. But with a basic grasp of set theory, it becomes painfully obvious that it is really set theory which allows the proper logical transition from the primitive axiomatization of multiple-being to the treatment of higher ontological concepts. This is attested by the mathematical success of set theory, above all. Given strong enough axioms, these (derivative) concepts can simply be defined and explored inside ZFC. The concept of pure multiple allows passage to more advanced ontological themes without hindrance. The rest of ontology thus becomes a part of the meta-ontological investigation of the consequences of axiomatic decisions regarding the pure multiple. The impasse is thus not only the ungrounded privileging of the one, it is also the unawareness of the resources afforded by a proper axiomatization of the concept of multiple alone.

## Closing Remarks

We can summarize the above by succinctly stating Badiou’s orientation to ontology. So far, we have merely started to sketch the ontological problems at work and how set theory can be of help. The understanding of being qua being that Badiou extracts by working through this new approach is rather detailed and involves several arguments which are better left for another essay. As stated earlier, although Badiou claims that mathematics is ontology, this does not mean that there is no meta-ontological theorizing to be done. However, what has become clear is the unique role that mathematics plays in this undertaking. For Badiou, set theorists from Cantor onwards have invented a solid formal setting in which we can explore the consequences of axiomatic decisions regarding the notion ‘x is a pure multiple’. Since ontological questions are so caught-up with questions regarding multiple-composition, it is perhaps only natural for a philosopher to consider set theory as a *conditio sine qua non* for the investigation of (meta-)ontology.[18] Inasmuch as Badiou argues that the notion ‘x is a pure multiple’ can be taken as *the *single most important ontological notion, ZFC then becomes exactly that: a very clearly circumscribed ontology. So, the commonplace view that ZFC is merely a mathematical tool is too modest and denies something of its ontological importance. For philosophy, the use of a mathematical language also implies the possibility to disconnect ontology proper from a) the theme of consciousness, and b) the theme of the poetic power of language, two orientations which were dominant at least in continental philosophy before Badiou’s interventions. Instead, we condition our ontological thinking on the work done by set theorists, thinking with them as they propose new axioms and explore their consequences.[19]

A more complicated matter is perhaps how to situate Badiou vis-à-vis analytic philosophy. Some indications can be made here based on the above. For Badiou, mathematics is never one *explanandum *among others. Analytic philosophers, with some exceptions, usually treated mathematics as just that: a linguistic practice with certain ontological commitments which needed philosophical justification. Hence, analysts practiced philosophy *of *mathematics. But for Badiou, mathematics is never something that needs philosophical grounding of any kind. Rather, mathematics is a condition *for* thought, a truth procedure.[20] Mathematics is and always has been “the science of being *qua *being, […] the science of everything that is, grasped at its absolutely formal level […].”[21] Working out the details of these very interesting differences between the analytic school and Badiou’s proposal should also be left for another essay. Even though Badiou does not always do the work of making connections, there is no reason why one could not present his position as a pertinent (and critical) reaction to 20^{th} century analytic philosophy of mathematics.

I understand that for most readers and non-readers of Badiou my clarifications leave unexplained how Badiou finds passage from such mathematical or formal themes to those he is perhaps best known for: his theory of the event, his avant-gardist anthropology, his notion of a truth procedure, and his theories of art, politics, and love. Since for him, the mathematical work is what gives the ultimate philosophical support for his other theories, I thought it would be only natural to dwell on these starting declarations at some length. More so since most commentators passionately go to these themes whilst only giving a rather fuzzy account of his ontology. To be continued.

(Here is a video of a talk I gave more recently on the same topic which ends with a more skeptical assessment.)

## Notes

[1] Leibniz, cited in Alain Badiou, *Being and Event*, trans. Oliver Feltham (London & New York: Continuum Books, 2005), 23.

[2] Vitalism is a repeatedly engaged enemy for Badiou. E.g. “One, Multiple, Multiplicities,” chapter 6 of: Alain Badiou, *Theoretical Writings*, ed. and trans. Ray Brassier and Alberto Toscano (London & New York: Continuum, 2004), 67–80.

[3] Or as we will see, it constitutes a perspective shift. It transposes the proper relation between the concepts of being one and being multiple.

[4] Badiou, *Being and Event*, 4.

[5] For Lakatos, a scientific paradigm is characterized by 1) a *hard core* consisting of ideas and convictions which researchers will strive not to give up unless absolutely necessary and 2) *auxiliary* *hypotheses *which are more expendable. Of course, the analogy has its limits because Badiou’s mathematical ontology is not an paradigm for natural science but a philosophical program, the nature of which is to be clarified here. Imre Lakatos, “Criticism and the Methodology of Scientific Research Programmes,” *Proceedings of the Aristotelian Society* 69 (1969): 149–86.

[6] Among others, this becomes clear from the “technical note,” Badiou, *Being and Event*, 49–51. I will refer to set theory as ZFC occasionally.

[7] In the context of a particular logical language (like ZF), an axiom is a sentence simply assumed to be true at the outset. That is, it is not proven inside the system but is used to prove theorems. If one gives support for an axiom, the reasoning is external to the formal language itself. From an internal perspective, the axioms are sentences which have been *decided* to be true.

[8] That said, the investigation of the consequences of axioms *extending* ZFC is a significant part of the development of set theory and of considerable importance to Badiou’s late work. Also, a lot of mathematics is done using concepts which are not *prima facie *set-theoretical. But nonetheless, these objects are still regarded as ‘living’ in some universe of sets. Set theory remains the default ontology for mathematics.

[9] Compare parts VI and VII of Badiou, *Being and Event*. I save the development of these themes for later essays.

[10] This aligns Badiou quite strongly with the philosophy of mathematics that has been developed in analytic philosophy in the tradition Frege – Russell – Carnap – Gödel – Tarski. It is clear from his early writings that he had been studying these figures quite intensely. But there are also important reasons why Badiou stands outside this tradition to which I will return later in this article. These in turn explain to some extent the hostility Badiou has received from contemporary analytic philosophers. For Badiou, the real ontologists have become the set theorists and not the philosophers engaging in linguistic analysis.

[11] For example, on the latter level, one is familiar with talk of the set of all natural numbers, whereas on the former, one knows how the numbers themselves are constructed as sets and are defined and shown to exist as simple consequences of the ZFC axioms. Most introductory textbooks on mathematical logic make all this precise.

[12] Ibid., 23.

[13] Before Cantor, there was no mathematics of set formation. If related concepts were studied, it happened in ontology and in a philosophical language not concerned with axioms, proofs, and theorems. It is never quite clear what author/which authors Badiou means when he speaks of ontology’s impasses in the first chapter of *Being and Event*. Presumably, he is positing a tendency common to the entire tradition, without wanting to single out any one author as a representative. Nevertheless, it would be highly interesting to evaluate his claim more closely, also because his whole endeavor gains traction from this highly general characterization of his predecessors.

[14] Ibid., 27. “Presentation” is Badiou’s technical (ontological) term reserved for what I have loosely described as the setting of grouping and/or categorization. Since I am focusing on the role of mathematics here, I will leave the discussion of the ontological vocabulary to a later essay. For Badiou, the original ontological place is what he terms a situation, which is the presentation of a multiple. Ontology, as inquiry, is then itself a presentation/situation in which a multiple is presented. This multiple is the theory of presentation. Hence the slogan that ontology is “the presentation of presentation.”

[15] A universally quantified sentence starts with the universal quantifier. E.g. for all *x* and for all *y*, there exists a *z* such that *z* is the union of *x* and *y*.

[16] Ibid., 58.

[17] Ibid., 89–92.

[18] However, this remains a choice. That such a choice becomes imperative is connected to Badiou’s definition of philosophy itself. He strongly stresses that philosophy must situate itself at the crossroads of contemporary innovations in science, art, and politics. Other philosophers who consider philosophy’s task more autonomous may reject this imperative to rework ontology through the innovations of 20^{th} century mathematics. A good example is chapter 4 of Markus Gabriel, *Fields of Sense: A New Realist Ontology* (Edinburgh: Edinburgh University Press, 2015). It will be a very important task to compare these ontologies through the lens of the underlying difference regarding the very definition of philosophy.

[19] It thus comes as no surprise that the last volume of *Being and Event, *due for publication in France in 2017 will contain exactly that: an investigation into recent set-theoretical work regarding the higher infinite.

[20] It would be interesting to see whether *early *analytic philosophers were not more in agreement with Badiou than their contemporary heirs.

[21] Alain Badiou, *In Praise of Mathematics*, trans. Susan Spitzer (Cambridge: Polity Press, 2016).