1. Introduction

In Logics of Worlds, Badiou attempts to answer what had been left unattended in his previous works: how do multiples “appear” (become situated) if their existence is operated by an inconsistent multiple? In other words, how do multiples become consistent if their generic element, that grounds their existence, is nothing in itself? To do this, he acknowledges that the concept of Event must be reworked, because “the ontological break, whether mathematizing or vitalist, does not suffice” (Badiou, 2008, p. 8). However, things are not so simple, for the figure of Deleuze once more resurges as the giant that must be slain so a different concept of the Event in this bodily regard can be purported.

So, to articulate both his reversal of Deleuzian vitalism and stablish the grounds for his onto-logical ontogenesis, Badiou opts to axiomatize Deleuze, which means to subtract the formal core of Deleuze’s normative theory. In other words, “what gives value to the axiom -to the axiomatic disposition-, is precisely that it remains subtracted from the normative power of the one” (Badiou, 1998, p. 33). This means that Badiou extracts from Deleuze’s transcendental empiricism the pure thought that is arranged by the series of principles that sustain his theory of event and becoming. While Deleuze’s Event is based on the incorporeal effect the bodily encounters carry, from which derives the transcendental dimension of Time as pure virtuality that encompasses the whole of reality, Badiou claims the opposite, beneath the Event, beneath becoming there is an inconsistent void. Thus, Logics of Worlds aims to claim an ontology via method of axiomatization that refutes Deleuze’s vitalism, where the Event and the appearance of the bodies are dependent on a virtual, chaotic plane, that is Being.

Regarding this ontological discussion, the relationship between both philosophers has been extensively covered. However, few studies have focused on the metaphysical apparatus that come into clash in Badiou’s nuanced reading of Logic of Sense, given that most of them focus on Badiou’s The Clamour of Being. On what falls into the interest of this article, Jon Roffe’s work Badiou’s Deleuze is undeniably of high value. In it, the author engages with this problem by reconstructing Deleuze’s theory of the event and interspersing Badiou’s reading, highlighting the points of conflicts that arise as the latter workarounds Deleuze’s vitalist univocity. But this approach, although mostly loyal to Deleuzian metaphysics, intentionally proposes the concept of essence as key to understand Deleuze’s thought of the event. In doing so, Roffe is catering to Badiou’s core argumentation, that Logic of Sense and Difference and Repetition do not contend univocity, but rather reinvent it. Hence, while Roffe’s text serves well to comprehend up to what extent Deleuze’s theory of the event coincides with the Badiouian claim, it pales against the hypothesis that Badiou is utilising Deleuze to solve the problem of the multiple becomings of the One, which is not a Deleuzian problem, but rather a Badiouian inconsistency. Brief, Badiou is gathering all the counter points that would arise against his definition of the event, where appearances of bodies are determined by the eternal Truths and the inconsistent multiple that operates becoming.

Similarly, Todd May (2004) tackles the discussion around the multiple in both ontological systems, as he underlines how Badiou’s misreading of Deleuze’s reversal of Platonism shapes his conclusion on the univocity of Being in Deleuze’s work. According to May, Badiou correctly addresses an ontological problem that arises in Deleuze’s statement that both the univocity of Being and the multiplicity of beings are not in conflict:

In effect, the essential in univocity is not that Being is said in a single and same sense, but that it is said, in a single and same sense, of all its individuating differences or intrinsic modalities. Being is the same for all these modalities, but these modalities are not the same. It is ‘equal’ for all, but they themselves are not equal. It is said of all in a single sense, but they themselves do not have the same sense. The essence of univocal being is to include individuating differences, while these differences do not have the same essence and do not change the essence of being –just as white includes various intensities, while remaining essentially the same white. There are not two ‘paths’, as Parmenides’ poem suggests, but a single ‘voice’ of Being which includes all its modes, including the most diverse, the most varied, the most differenciated. Being is said in a single and same sense of everything of which it is said, but that of which it is said differs: it is said of difference itself. (Deleuze, 2001, p. 36)

As May points out, Deleuze’s biggest issue is to maintain the univocity of Being, as he “must be able to articulate Being both from the standpoint of Being and from the standpoint of beings, and do so in a single and same sense” (May, 2004, p. 70). Nonetheless, Badiou is purposefully reading Deleuze under a Spinozistic sieve to determine that Deleuzian ontogenesis, as it is dependant of the Idea, is a hypertrophied Platonism. From this stems Badiou’s conclusive statement, that Virtual is one of Being’s names, meaning that the Virtual/the actual, the plane of immanence/plane of composition, and so on, are one of the transcendental names of Being, which is one and only: “the real is the One, and beings are only real inasmuch as they are expressions of, and therefore participate in, that One” (May, 2004, p. 69). To solve this, Deleuze undertakes an analytic of the indistinguishable, which forces either the multiplicity or the univocity of Being to give way, something that Deleuze strains to avoid. By reclaiming this ontological conflict, Badiou is wiggling his own solution to the problem of virtuality and multiplicity of Being, the biggest hurdle in Deleuzian metaphysics: becoming is presupposed organically by a pure chaotic virtuality that is then structured or actualized by the Idea. Badiou artfully disregards all forms of a-subjectivity and a-significance that Deleuze (and Guattari) attend(s) to when considering Time as creative scission, or caesura. This incomplete picture of Deleuzo(-guattarian) ontology allows the Badiouian critique to be more resounding.

Jean-Jacques Lecercle’s Il y a événement et événement expands on this subject by providing a practical distinction between both philosophies of the event. On the one hand, the author states, there is the event-Badiou, an event that destroys, but also grounds: “the event is precisely dated, it interrupts the time in which it occurs, and to which puts an end, yet it does so in order to reestablish it -our calendar proves it” (Lecercle, 2005, p. 2). On the other hand, there is the event-Deleuze, diametrically opposed, as it does not resurge at all: “it shrouds and caresses the body through its affections. It is not the place of a violent act, but rather of an impassibility” (Lecercle, 2005, p. 2). While for Badiou the event is the “taking place” of the formal void common to all things, to Deleuze it is the vapour that circulates at the surface of the Real, or structuration of bodies.

Meillassoux’s work History and Event in Alain Badiou allows to clarify this distinction, as he revises the main thesis that grounds Badiou’s metaphysics, that is, “being is multiplicity, and nothing but multiplicity”. This requires the exclusion of the One, or univocity as a subsistent immanence: “Being is not therefore a multiplicity composed of stable and ultimate unities, but a multiplicity that is in turn composed of multiplicities” (Meillasoux, 2011, p. 2). To reach this conclusion, Badiou identifies ontology to set theory, as well as mathematics to pure thought, so that any multiple is both multiple of multiples and set of sets. Just as mathematical sets have for their elements not unities but other sets, multiples are composed by other multiples (including itself). Thence, if not empty, any set (or multiple) is composed of multiple sets. As Bartlett and Ling summarize in their introduction to Mathematics of the transcendental,

ontology is ultimately a discourse which prescribes the rules by which something can be presented or ‘counted’ as one – its sole operation being that of the count – and the ‘one’ thing that necessarily fails to be counted is nothing other than inconsistent multiplicity, or being itself. […] Pure multiplicity is not something that can be known (or again, while we certainly know that being is, we cannot know what it is); rather, the nature of being is something that must be decided upon, in an axiomatic sense. (Badiou, 2014, p. 14)

Because Badiou rejects Deleuze’s theory of immanence and the coalescence of the virtual and the actual, he does not count with the Deleuzian “fix” to the problem of univocity: Time as creative scission or Difference. Indeed, May correctly addresses that while Badiou critiques Deleuzian ontology through the inconsistency of the object’s virtual/actuality, Deleuze tries to solve such discrepancy by purporting “the idea of a single time that has both virtual and actual aspects” (May, 2011, p. 76), hence there is no need to choose between a collapse of the virtual into the actual, or vice-versa, because there is a reintroduction of transcendence (Time as pure form). So, Badiou must find new grounds for a theory of Event, because Time ought not to be held as a persistent immanent form to which all beings are equally subjected. As Clayton Crockett points out in Deleuze beyond Badiou, Deleuze: the clamour of being remains one of the strongest readings on Deleuze that exists, where Badiou masterfully crafts an aristocratic, classical, and austere image of Deleuze, who remains a metaphysician in the traditional style, meaning someone who builds up a sufficient and all-encompassing apparatus that allows later divergent philosophical adventures. Badiou avows that this “Being is the One-All” (Crockett, 2013, p. 13) adagio is the key to distinguish their philosophies.

To understand Badiou’s axiomatization, it is important to summarize what Deleuze tries to do in Logic of Sense. As Hasegawa aptly points out, Deleuze uses structuralism to develop his own philosophy, a synthesising “philosophy of structure” (Hasegawa, 2020, p. 18). In fact, Hasegawa distinguishes what structure is in Deleuze to what it is in structuralism, which he accuses of being symbolic. The symbolic is the element of structure, of its genesis. Structuration is the incarnation of this opaque symbol, which manifests as a representation that does not derive from the structure itself. Rather, the symbolic representation suggests that the structure is deeper than its incarnations, that there is an elementary term, obscured by the impossibility to break free from the (linguistical) structures that serves as base to all real foundations as well as all imaginative heavens. In other words, becoming is the formative genesis of this unbeknown term. However, as symbolic elements incarnate in real beings and objects, differential relations are actualized in the real relationships between them, with singularities playing the role of places of the structure that distribute the roles that beings and objects come to occupy. Thus, singularity concerns the phase of appearance or manifestation of the structure, thereby overcoming the problem of symbolism in structuralism, and turning the structure into an expression and explanation of the sense that underlies the world as an event. Hence, Deleuze thinks of structure “as virtuality” (Hasegawa, 2020, p. 24), because it distinguishes itself from what it incarnates and what it constitutes as incarnated. This means that structure, although manifesting, can do so only as virtuality.

Unlike structuralism, Deleuzian “philosophy of structure” rejects any original sense or primordial depth; indeed, in line with Stoic philosophy, sense is an effect of the surface, or rather what promenades and encircles the structured world. This analysis of sense as a system is correctly explored by André Jacob in Sens, énoncé, communication, where Deleuze's proposal is regarded as closer to the notion of a decentralized system that philosopher Michel Serres observes in his interpretation of Leibniz. Jacob notes that sense and structure are to Deleuze a foundational or poetic organization, a play of surfaces “where only an acosmic, impersonal, pre-individual field unfolds, that exercise of nonsense and sense” (Jacob, 1969, p. 195). Badiou must restrict Deleuze’s capacity to establish the link between sense, structure, enunciation and event, for otherwise the Deleuzian univocity cannot be underpinned.

In short, the event-Deleuze is a set of singular points arrayed in series, which in turn compose the given. It is not the propositional designation or the individuality of a designated state of affairs, rather it is the pre-individual, non-personal, and a-conceptual dimension of Being that is “totally indifferent to the individual or collective, personal and impersonal, particular and general—and their oppositions” (Deleuze, 1990, p. 72). As the series are traversed by sense, they shift, redistribute, and transform, altering the whole: “if singularities are true events, they communicate in a single and same Event that continually redistributes them and their transformations to form a history” (Deleuze, 1990, p. 73). Singularities are distributed randomly over an intensive plane that Deleuze calls the Chaosmos, where they act as forces or cogitanda of pure thought. They are the distributed points that differentiate and fracture thought, dividing it into the two moments of an ideal, existential game by which sense or expression operates “a distribution of singularities” (Deleuze, 1990, p. 74). Each play is distributed as a single one that implies and expresses all the others, having the Same as the result of the Different. However, Badiou “fails to apply the same logic to this fundamental distinction between the actual and the virtual” (Crockett, 2013, p. 13). This subtlety entrusts Deleuze’s metaphysics with platonic transcendentalism, for Virtual becomes “the principal name of the Being” (Badiou, 1999, p. 43).

This forcing interpretation derives from Badiou’s necessity to arrogate Deleuze’s transcendentalism and abandon his material and vitalistic immanentism. The Deleuzian Virtual becomes then an operational form that remains outside all becoming. However, this is not supported by Deleuze’s work, for he does not ground the actual in the Virtual, nor in its constant conflagration and differenc/tiation. On this vein, Simont suggests that on both cases it is about questioning the stability of that which the representation operates, the consistency of the predicative links that it mobilizes and “the legitimacy of its ‘objects’” (Simont, 2002, p. 457), a point where Badiou and Deleuze could not be any closer, as they both sustain a “marvellous convenance between being and thought” (Simont, 2002, p. 457). For example, Badiou observes that conceiving the present as a fluent passing of a representation does not account for the History that gathers all presents that have occurred. In fact, History is “chanted [scandée] by interventions, by evental [événementielles] recurrences” (Simon, 2002, p. 458) tensed by the fidelity of the body towards the Truth(s). Concordantly, Deleuze points out the deficiency of history considered chronologically, for there is the Aion, “this long line on which a creative repetition runs, the eternal return of the same, understood as the return of the different” (Simon, 2002, p. 458). It is correct to address, just as Simon does, that at this level exiled are the procedural differences that rift apart both philosophies, because what truly separates them is the way in which the “clamour of being” will be interpreted: to Deleuze, a virtual matter that persist beneath the actual; to Badiou, a pure voided operation outside the realm of evental becoming, that insists as the common, inexistent element to all existence.

2. The metaphysical stand-off: the univocity of Being

Essentially, as per Badiou, Becoming is operated by the Same, and not Difference. Otherwise, Being would be still considered a formal One-All (albeit organic and materialistic) to which the differential or disparity is always being added. Brief, whilst to Deleuze the One is an intensive, immanent, chaotic and a-subjective multiplicity or plane of existence, to Badiou the One is an operation, an inconsistent multiplicity that allows all multiples to be counted. Thus, as long as considered as a pure multiple, that is, an empty set, Being becomes an in-existent nothing from which all becoming stems, as it remains the element that all things (themselves also multiple) contain. Basically, Badiou asks Deleuze why is the Event persistent, why is it a dimension of Being. If the Event, and the recurrent conflagration of the world, seemingly distends from a pure Outside, it must be considered as the actual performance of a pure operational form that remains the purest multiple of multiples. This, Simon summarizes perfectly:

No, no presence, and against the vitalist thread which continuously ties the actual to the virtual intensity from which it comes, against the liturgical, or poetic thread which ensures that ‘where that which loses grows, that which saves also grows’, it is necessary to stick to the rigor of subtraction: no great vital whole, no saving abyss either, nothing but the harsh law of structure, which is also the implacable constraint of the present: there is only what there is, no sign, no genetic effervescence, nothing hidden, nothing lost, nothing found, no call or sheepfold of Being, there is only the One, which prohibits us from being except what escapes it, and what we do not have access to. (Simont, 2002, p. 462)

To understand this metaphysical bout, Vartabedian’s Multiplicity and Ontology in Deleuze and Badiou grants a superb reading on the central concept at stake: the multiple. Because Deleuze aims to proclaim becoming as a process of actualisation by assemblage, it is required that Being were at least twofold, an actual state of existence and a virtual one. Becoming is the process where Being is constantly added to itself, as if plied or composed immanently, so a new structured reality becomes factual. Contrary to this, Badiou’s multiple ascribes to Cantor’s attempt to resolve the Burali-Forti paradox, where only the orderly structure of numerical sets is considered. Cantor gives Badiou a way to define the multiple not in terms of topology or intensive spatiality, but rather in terms of consistency, a formal dimension that remains out of the bounds of Being and can be axiomatized.

As Vartabedian underlines, Riemann’s geometrical inquiries, particularly his re-definition of spatiality through the manifold and intensive continuous magnitudes, is employed by Deleuze to support the claim that Being is a n+1 occurrence, or an “n-dimensional, continuous, defined multiplicity” (Deleuze, 2002, p. 182). Whence, the Event is when the virtual plane of existence is assembled in new ways, forming new linguistic substances, welcoming new sensations into the world. In short, the Deleuzian chaosmos is modelled by Riemann’s “reiteration and expansion of continuous quantity as manifoldness, the procedure by which these manifolds become multiply-extended, and the consequences of these innovations for the comprehension of physical space” (Vartabedian, 2018, p. 64). Virtuality is an instrument of internal genesis that “depends on a structure afforded by a continuous multiplicity” (Vartabedian, 2018, p. 64), and so it is implied in Deleuze’s metaphysics that the intensive spatium, or complied multiplicity from which the pre-individual singularities that conform the actual emerge, is informed by hazard. The concept of multiplicity intends to answer the question of possibility of becoming and, furthermore, how new pathways may be added to the manifold of existence, by increasing the available paths to travel. According to Deleuze, the logic of event, although Badiou ascribes it to fatality, is that of the necessary affirmation of hazard.

To sum up, Deleuze considers multiplicity as definite. All singularities are defined by the elements that are reciprocally determined by the relations they sustain, a difference that cannot change “unless the multiplicity changes its order and its metric” (Deleuze, 2002, p. 183). A definite multiplicity captures a specific portion of the plane of intensity, or the virtual dimension of Being, “it occupies part of this ‘intense world’ and reads for the relationships in that part” (Vartabedian, 2018, p. 66). Since Being is a continuous multiplicity, to which “there is no ‘without’” (Vartabedian, 2018, p. 66), nor external conditioning, a portion of existence remains immanent as an operation of internal genesis. The reason why Deleuze voids the virtual multiple or considers actualisation as the information via the pure and empty form of Time, is because all its infinite depth is measured intensively. The intensive Deleuzian spatium lacks homogeneity. In terms of intensity, none of the elements arranged in the chaosmotic plane of existence lose their singular determinations, even if connected to an abysmally larger space, because they connect to each other by principle of assembly and agency:

The analysis of dimensionality in a Riemannian multiplicity identifies a complex and contained site from which an idea might arise, and its emphasis on locality or its status as definite focuses on a portion of space, rather than the space against a larger background or in a global configuration. (Vartabedian, 2018, p. 66)

Badiou will oppose this by basing his ontology on the mathematics of the multiple pioneered by Cantor. Against the topological geometry of the Deleuzian multiplicity, Badiou posits the various operations that the existent multiples (sets of elements) may or may not have:

In sum: the multiple is the regime of presentation; the one, in respect to presentation, is an operational result; being is what presents (itself). On this basis, being is neither one (because only presentation itself is pertinent to the count-as-one), nor multiple (because the multiple is solely the regime of presentation). (Badiou, 2007, p. 24)

This can only be fathomed if mathematics is the instance where thought and Being converge. For Badiou, there is a point where mathematical thought returns unto itself by cause of a forcing encounter with the real, or by the necessary resurgence of an impossible. This halt is paradoxical, transversal, for it causes an inconsistency within mathematical discourse; and as this oxymoronic term is presented, arises mathematics inability to move forward and structure its excess, because it brings forth the indefinite and unconstructible:

What then is the nature of mathematics as it twists unto itself beneath the injunction of an intrinsic sticking point? What surfaces concerns everything pertaining to an act or to decision making within the scope of mathematical thought. By the same token, a position has to be taken. For we stand actually as an act (au pied de l’acte), if I dare say, upon the very norm of the decision the act accomplishes. At any rate, what is referred to in this obligation to decide is Being. (Badiou, 1998, p. 52)

In short, mathematics run into the inconsistency and paradox of the excess, or rather the indefinite condition of thought. At this point, it is mandatory to take an ontological stance and decide on the attribution of Being to this inconsistent element. Badiou considers this as an act that durably engages the reality of Being, a particular instance where all ontology becomes axiomatized. This, Badiou will later name the “axiomatic prescription”.

The existence of a given set can be inferred first only from the void as originally affirmed or from operations allowed by the axioms. And the guarantee of that existence is only the principle of non-contradiction applied to the consequences of the axioms. Of course, whether these axioms, which have been historically selected by the mathematics community, are the best ones, or especially whether they are sufficient for thinking multiple-being qua being, is a question that has no a priori answer. It is the history of mathematical and philosophical ontology that will decide. All we can do is accept a principle of openness. (Badiou, 2022, p. 42)

In essence, an axiom of choice occurs where thought is summoned to abruptly decide upon the indeterminate infinite that drives all thought and situation of Being. Therefore, this conflicting point is an orientation in thought that regulates the assertions of existence, and it is either what formally authorizes the inscription of an existential quantifier at the head of a formula, or what ontologically sets up “the universe of the pure presentation of the thinkable” (Badiou, 1998, p. 53). Insofar as this operative instance, to which all becomings are subjected, the one is not. Or rather, “there is no one, only the count-as-one” (Badiou, 2007, p. 24). And, because it is the operational result of the presentation, Being is not One (considered Deleuzianly), but rather nothing. This indexation rests upon the concept of situation, which is “the place of taking-place” (Badiou, 2007, p. 24). All situations admit their own operator, or paradoxical encounter, as the count-as-one. This means that all multiples are presented as the set they are in a given situation: for example, my body is at this given situation the set that gathers the elements of a notebook, the desk, the chair, the books around me, the lamp, its light, the alight keyboard, the bird that chirps outside the window, etc.; all these are counted-as-one in the multiple that is Felipe at the current situation that writes this paper. This count that takes place, is what Badiou defines as structure, which is what prescribes, for a presented multiple, the regime of its count-as-one. Further from this, when anything is counted as one in a situation, it means that it belongs to the situation in the mode particular to the effects of the situtation’s structure. Meaning, for me to be situated as I am right now, my elements (for I am a multiple) are presented as accommodated to the actual situation that operates me. Thus, every situation is structured. Now, because all multiples are multiples of multiples, and the after-effect of the count, or presentation, is “uniquely thinkable as multiple” (Badiou, 2007, p. 24), the presentation as such is also a multiple:

The multiple is the inertia which can be retroactively discerned starting from the fact that the operation of the count-as-one must effectively operate in order for there to be Oneness. The multiple is the inevitable predicate of what is structured because the structuration-in other words, the count-as-one-is an effect. The one, which is not, cannot present itself; it can only operate. As such it founds, 'behind' its operation, the status of presentation-it is of the order of the multiple. (Badiou, 2007, p. 25)

At this point Badiou splits apart the multiple, that can either be a consistent multiplicity (counted), or an inconsistent one (that which counts, or operates the counting without being counted). Situation, or the structured presentation, is relative to these terms. Any situation is dual, it is both inconsistent and consistent. This duality is established in the distribution of the count-as-one, or the event, where what is situated is an inconsistent before, and a consistency afterwards. Hence, there is nothing apart from situations, reason why the presenting multiple, as it is non-situated, is nothing. So, if the One is the regime of presentation, then Being is not, or rather, it is not One. Essentially, as an operative multiple, Being is subtracted from all accounts: “it is in being foreclosed from presentation that being as such is constrained to be sayable, for humanity, within the imperative effect of a law, the most rigid of all conceivable laws, the law of demonstrative and formalizable inference” (Badiou, 2007, p. 27).

Consequently, Badiou remarks, if there cannot be any presentation of Being, because Being is what happens in every presentation, then the ontological situation is the presentation of presentation, just as any multiple is multiple of multiples. Then, a situation whose presentative multiple is that of presentation itself could constitute the place from which all possible access to being is apprehended. If there is a situation where the multiple presented is the inconsistent multiple, as paradoxical as it may seem, then what is presented is the situation to which all compossible presentations can be grasped. Such occurrence is the Event:

Because the law is the count-as-one, nothing is presented in a situation which is not counted: the situation envelops existence with the one. Nothing is presentable in a situation otherwise than under the effect of structure, that is, under the form of the one and its composition in consistent multiplicities. The one is thereby not only the regime of structured presentation but also the regime of the possible of presentation itself. In a non-ontological (thus non-mathematical) situation, the multiple is possible only insofar as it is explicitly ordered by the law according to the one of the count. Inside the situation there is no graspable inconsistency which would be subtracted from the count and thus a-structured. Any situation, seized in its immanence, thus reverses the inaugural axiom of our entire procedure. It states that the one is and that the pure multiple-inconsistency-is not. This is entirely natural because an indeterminate situation, not being the presentation of presentation, necessarily identifies being with what is presentable, thus with the possibility of the one. (Badiou, 2007, p. 52).

So, given a presented multiple, the multiple-one (or the inconsistent multiple) is part of it. This ontological theorem disgorges into an impasse: that the void, or inconsistent multiple is larger than the presented multiple. The ontological passage from the presented set onto the bigger sub-sets is an operation that exceeds the given situation. Thence, the multiple of the subsets of a situated multiple comprehends, forcefully, a multiple that does not belong to the initial set. This means that, because everything is multiple of multiples, and because counting-as-one depends on the accordance between the situation that evokes the multiple and the elements of the (un)presented multiplicity, all multiples have as elements what does not belong to the situation. If it were not the case, no occurrence would ever be possible. This, Badiou names theorem of the point of excess.

The one is the result of the count, or the effect of the structure. The void is a “voided” multiple (it is itself its own element), and so unpresentable. To the empty set, or void, Badiou ascribes the operative quality of presenting, it is the name he gives to the suture-of-being, or the axiomatic presentation of the pure multiple, that is, the presentation of presentation. This does not mean that the void is itself Being, for Being it is not, it is instead the act of subtraction that, as common element and inconsistent multiple, all multiples are subjected to. Then, the empty set, or count-per-one consists in indexing the negation of all presented multiple: it is the name of the unpresentable multiple. The void is the generic subset of the totality of worlds that is not constructible, or that cannot be counted: “The generic is, overall, the superabundance of being such as it evades the hold of language, when an excess of determinations brings about an effect of indetermination” (Badiou, 1992, p. 118). In other words, what escapes counting is counting in itself:

The count-as-one is, to my eyes, the evidence of presentation. It is the event which belongs to conceptual construction, in the double sense that it can only be thought by anticipating its abstract form, and it can only be revealed in the retroaction of an interventional practice which is itself entirely thought through. (Badiou, 2007, p. 178)

In essence, an event can always be localized, because it is always a point of a situation (it concerns a presented multiple). Moreover, events are uniquely in situations which present at least one site, they are attached “to the place, to the point, in which the historicity of the situation is concentrated” (Badiou, 2007, p. 179). This means that the site designates the local type of the multiplicity that is concerned, or remains loyal, to the concatenation (or conjugation) of elements the event reclaims. The site is what constitutes the possibility of an event to be, because there must be a local determination of site, a situation “in which at least one multiple on the edge of the void is presented” (Badiou, 2007, p. 179). Hence, the event is the multiple which both presents its entire site and the presentation in itself: “It 'mobilizes' the elements of its site, but it adds its own presentation to the mix” (Badiou, 2007, p. 182).

But, per which elements, appertaining to the situation, can this paradoxical multiple be connected to the signifier, or name, that locates it and makes it belong to a given situation? In other words, what does allow Badiou to posit the Event in the real, to suggest that the count-as-one discourses historically? Not a single presented term, or multiple, can offer such, for the site cannot name the event, even if it could limit or qualify it. As Badiou remarks, although it is called the “French” revolution, it is not France that engenders and names its eventality. So, if the unperceived of the site is what grounds the nomination of the event, then the elements of a given situation that permits an event to be named (or localized) are not what is there presented, but rather what is im-[un-]presented, or the void: “Its proper name is thus the common name 'belonging to the site'. It is an indistinguishable of the site, projected by the intervention into the two of the evental designation” (Badiou, 2007, p. 205). This way, the Event is an ultra-counted multiple, a subset of all multiples to which belongs the elements of its own site, as well as the event in itself. In short, the Event is the occurrence that enunciates that something must have had happened, or that what had had discoursed necessarily had to. The event only exists in so far it is subdued by an intervention that demands recurrence to the regulated structure of the situation, and that, in doing so, all novelty is relative.

Hence, as Meillassoux points out, a pure rupture in the situation must take place for an event to occur; nothing belonging to the situation should let one classify, under a list of facts, why has the event just happened: “the novelty of an event is expressed in the fact that it interrupts the normal regime of the description of knowledge, that always rests on the classification of the well-known, and imposes another kind of procedure on whomever admits that, right here in this place, something hitherto unnamed really and truly occurred” (Meillassoux, 2011, p. 2). The key here is Truth. Truth(s) happen in the world, leaving behind a trail, for they belong to history and are conditioned by the events. What is designated in the event is none other than the general truth of the situation, that is, the truth that fundaments all of what is to come. What is possible is what has not yet been situated and remains eternally actual as loyal to the truth that circumscribes the situation-whatsoever. This is the core of Badiouian logic of appearance: all bodies that appear in a situation, do so according to the truth to which they remain more or less loyal. In this sense, just as eternally actual, the truth, itself a multiple, remains an infinite part of the situation. But, how do bodies become part of a situation? This is the subject, almost entirely, of Logics of Worlds:

I give the name ‘truths’ to real processes which, as subtracted as they may be from the pragmatic opposition of bodies and languages, are nonetheless in the world. I insist, since this is the very problem that this book is concerned with: truths not only are, they appear. […] It is a question of presenting, in each of the orders in which it is set forth, the ‘except that’ of truths and of the subjects that these truths elicit. (Badiou, 2008, 9-10)

Badiou must articulate how the addition of truths to the Deleuzian equation “bodies+propositions/enunciations=incorporeals” refutes the logic of Sense and the fatality of the Event. Hallward summarizes it as follows:

An event is simply an opportunity for some members of a situation, if they so decide, to affirm that which they can never experience or observe, namely the inconsistency that they and all other members of the situation indifferently and indiscernibly are. If these members take up this implication in a consequential way (and thus become ‘subjects’ in its wake), it will entail fundamental transformation of the way a situation discerns its elements: such a transformation, of course, is what Badiou calls a truth. (Hallward, 2005, p. 14)

In sum, what belongs to the situation is mandated by the count-as-one operation of the pure multiple, or Truth of the situation. Moreover, “there is a necessary point of excess between belonging and inclusion” (Ronald Shaw, 2010, p. 434) that takes place once the event is effectuated. How can this point of excess regulate the appearance of bodies? To understand this, it is important to reprise Badiou’s axiomatization of Deleuze, for he concludes his logic of manifestation after refuting Deleuzian information of the virtual bodies.

3. Onto-logy, onto-logy: the axiomatization of Deleuze and the evental appearance of a body

So, why does Badiou axiomatize Deleuze? As seen above, the interest at hand is to determine how are bodies counted-as-one in any given situation. In this regard, Deleuze conceded too much to the metaphysical category of life. This forced his ontology to be sucked up “by the doxa of the body, desire, affect, networks, the multitude, nomadism and enjoyment” (Badiou, 2008, p. 35). Hence, to bring “life” back to the centre of philosophical thinking one must methodically respond to “what is to live?”, or rather “what is to come to life?”. Thus, it must be explored the retroactive pressure that “the ‘except that’ of truths exerts on the very definition of the word ‘body’”. This must be consistent with the ontology of set theory, which is “is the effect of a decision regarding the intelligible” (Badiou, 2014, p. 15), a choice that takes the form of axioms that specify the Ideas that organize all ontology.

In other words, Badiou wishes to conclude that to live is to participate, point by point, “in the organization of a new body, in which a faithful subjective formalism comes to take root” (Badiou, 2008, 35). This is elucidated after bodies a redefined, understood as bodies-of-truth, or subjectivizable bodies, a definition that forbids the hegemony of democratic materialism, which main figure is Deleuze. So, bodies appear as borne by the eternal truths that index the possibilities of the events. This means that axioms are what describe the procedure by which bodies are more or less loyal to what the truth prerequisites for them to be eventaly situated. Essentially, set theory freely uses the ontological axiom of choice that posits the existence of a set that “chooses”, or counts, one element from each set of a given collection of infinite sets. One must keep in mind that, for Badiou, the process by which, in a situation, the multiples whose existence is linked to the name of the event is fidelity. As the truth of the event indexes a determinate set of parts of a given multiple (which is also a set), these parts are distinguished and gathered according to the becoming that is connected to the event:

An axiomatic presentation consists, on the basis of non-defined terms, in prescribing the rule for their manipulation. This rule counts as one in the sense that the non-defined terms are nevertheless defined by their composition; it so happens that there is a de facto prohibition of every composition in which the rule is broken and a de facto prescription of everything which conforms to the rule. An explicit definition of what an axiom system counts as one, or counts as its object-ones, is never encountered. (Badiou, 2007, p. 29-30)

Brief, an axiom system structures a situation in which what is presented is presentation, because it avoids making a one out of the multiple, which is left to the indexed consequences to derive and manifest. So, by means of axiomatization, an ontology inverts the consistency-inconsistency relationship as it appears in the living world: what is regulated by the axioms is the presentation of the formal act of presentation: “before its operation, what it prohibits -without naming or encountering it- in-consists” (Badiou, 2007, p. 30). What in-consists is the impure multiplicity, the composable according a defined structure, albeit formal and ontological:

Ontology, axiom system of the particular inconsistency of multiplicities, seizes the in-itself of the multiple by forming into t consistency all inconsistency and forming into inconsistency all consistency. It thereby deconstructs any one-effect; it is faithful to the non-being of the one, so as to unfold, without explicit nomination, the regulated game of the multiple such that it is none other than the absolute form of presentation, thus the mode in which being proposes itself to any access. (Badiou, 2007, p. 30)

Thus, to axiomatize Deleuze means to purport a structural system that rejects all organic immanentiality: Badiou installs an axiomatic decision in Deleuze’s discourse, that the one is not. Derivately, the first step in Badiou’s formalization of the upsurge of the bodies is to neglect any Deleuzianism that may still interleave his ontology. This means that, instead of a vital notion of the multiple as what grounds the event (the situating of any-body), Badiou posits a formal, structuralising incoherent multiplicity that is part of all bodies, as they are also part of it, which is the void or non-existent.

To complete his rebuttal of Deleuzian ontogenesis, Badiou aims at what he considers the mainframe of Deleuze’s metaphysics: all becoming obeys the law of structuration. Against this, Badiou interjects the concept of pure multiplicity: there is no “one-ness” beneath the surface of the actual, there is no “all-encompassing” codifying structure. The evental site that counts the multiples is the multiple as composed by the elements of the site and by itself (the event). Thus, the self-belonging property is what constitutes the Event, for it itself is element of the multiple it is. The inconsistent multiple cannot be counted because, as soon as the multiple is presented, the inconsistency ebbs into the unbeknownst, its presenting work already been done. The pure multiple is no place, it is inorganic: its virtuality is rather the pure possibility of placement, the compossibility of all situations, which arise once the multiples are counted. This enables Badiou to reclaim that the event is condition to all thinking and appearance of bodies, because it happens outside the realm of thought as the unpresentable, or the unthinkable:

I term void of a situation this suture to its being. Moreover, I state that every structured presentation unpresents ‘its’ void, in the mode of this non-one which is merely the subtractive face of the count. (Badiou, 2007, p. 55)

Consequently, Badiou attempts to rid the deleuzian ontology of the immanent “one-ness” that encompasses all existence as a virtual and vital dimension of Being. Instead, he defines the One as a formal and inconsistent multiple that can never be situated (since it does not exist) and operates the situation of all multiples. As pointed out by Brassier, Badiou is right in insisting that, to Deleuze, there can only be one real event, Eventum Tantum, provided that “the uniqueness of this ‘One’ is no longer an index of numerical unity” (Brassier, 2000: 201). In essence, Logics of Worlds aims to confront a problem left unaddressed by Being and Event: “how is it that being -pure inconsistent multiplicity- can come to appear as a consistent world?” (Meillassioux, 2011, p. 4). This unavoidably beckons the Deleuzian problem of the univocity of Event, which Badiou clearly rejects. So, to solve this, Badiou undertakes the constitution of a new plethora of concepts that allow the consistent multiples to appear per the formal operation of the inconsistent multiple, whilst retorting Deleuze’s claim that the structured world is composed by propositions and bodies.

Bodies and subjects emerge once convoked by a Truth that beckons their resurgence. Instead of conceiving the event as an effect of the encounter of bodies, that is, an incorporeal, Badiou sustains the trans-existence of the eternal truths, for truths exceed the perishable existence of material by which it comes to light. In short, as an event, becoming is the intense manifestation of something that is already there, the eternal truth. The event, then, is the disruption of the current situation of which the new was subtracted and denied (it was not counted for). For Badiou, throughout history the Event has been theorised as having both a structural dimension, and, because it cuts through the structured world, a historicity, or chronology. In the first case, the Event is seemingly unbound from the One, or unity that synthesizes and holds together the world, as it represents the abhorrence of such unification: the Event is the non-sense, the void or caesura that differentiates the composing elements of the structure, allowing the new to appear. The second dimension of the Event conceives this Difference as internal to the One, as pure multiplicity or virtuality that subsists within the structure as an empty form that is crystalized in becoming, in other words, an emptiness that garners all differentiations immanently.

Logic of Sense is Deleuze’s effort to clarify such ambiguity by claiming that the event complies to “the inflexible discipline of the All” (Badiou, 2008, p. 382). This suggests that between the event and fate, that is, between the necessity of becoming and its hazardous fundament, there must be a subjective reciprocation. Thence, by axiomatizing Deleuze, Badiou can reclaim Deleuze’s philosophical advances, while at the same time criticize and drop the most radical aspect of Deleuze’s philosophy: his so-called immanent vitalism, to which acceptance Badiou remains reluctant. Badiou states that there are four main axioms that index Deleuze’s theory of the Event, to which he responds respectively by formalizing his own theory.

Axiom I – “unlimited-becoming becomes the event itself”

As per Badiou, the Deleuzian Event is first and foremost the ontological realization of the eternal truth of the One, for it exposes the One of which all becoming are the expression. In a sense, the Event is “the becoming of becoming” (Badiou, 2008, p. 382), or the becoming-one of the unlimited becoming (of the One), it is an intrinsic difference or dimension of the One that entails its own perpetual coming-to-be.

Axiom II – “the event is always what has just happened, what will happen, but never what is happening”

As pure formality, the Event is the synthesis of past and future, it is the formal identity of future and past, the encroachment of all iterations of the Same. In other words, the Event safeguards all possible connections that bodies may require to appear. Hence, the event “exhibits the unity of the passage that binds the a-little-after to the a-little-before” (Badiou, 2008, p. 382), because it is not what is happening, nor what is coming to pass, it is voided of any appearing body or instantiation of the multiple in a given situation of the world. Badiou is separating the concept of Aion as an expressive formality of Time from the organic becoming of the One, he is not conceiving the Event as the multiplicity that garners all singular-events and grounds all possible bodily encounters.

Axiom III – “the nature of the event is other than that of the actions and passion of the body. But it results from them”

This is a key aspect in Deleuze’s logic of the event, for it states that the Event (or sense) is both extra-linguistic and extra-Being, meaning that it resides at the absolute Outside of all structurisation, while remaining susceptible to be structured. As an empty form of becoming, the event (or disjunctive eternity) intensifies bodies by concentrating their constitutive multiplicity. In other words, the operation of the everlasting event on the bodies consists of singling out a portion of their multiplicity constantly, thus forming their present constitution without erasing or imposibilitating their future or past. Therefore, the event is neither of the same kind of the action and passion of the bodies, nor it is above them, its “outsideness” is the transversality by which it operates on the All, or One. By being itself a bodily being, although empty and formal as “it also cannot be said to differ (ontologically) from bodies” (Badiou, 2008, p. 383), the Events is the differentiator of the actions and passions of the bodies.

Axiom IV - a life is composed of the same single Event, despite all the variety of what happens to it

At this fourth and final axiom, Badiou makes clear what he considers key to understand the deleuzian event. Instead of being the One expressed via the vital unfolding or becoming of the world, the event is that which composes life, “a little like a musical composition is organized by its theme” (Badiou, 2008, p. 383). In a sense, the Same outweighs Difference as protagonist of Becoming: “when it comes to any multiplicities whatever, it is of the essence of the Event to compose them as the One that they are, and to exhibit this unique composition in potentially infinite variations” (Badiou, 2008, p. 384).

According to Badiou, Deleuze resolves the ambiguity of the Event by opting for fate, as the event is not “the change-laden passage from one state of affairs onto another” (Badiou, 2008, p. 384), but rather the immanent mark of the One-result of all becomings. This means that, even if Deleuze tries to vouchsafe a certain empiric dimension in his transcendental philosophy by asserting the hazardous actualisation of the disperse singularities in a given state of affairs, becoming is the fate of the One, or Same that perpetually repeats itself in the information of the world. From this, Badiou’s own definition of the event follows as a form of reversal: event is “a site which, having appeared according to the maximal intensity, is equally capable of absolutizing in appearing what hitherto was its own proper inexistent” (Badiou, 2008, p. 384). Indeed, for Badiou the event is the point of no return where the appearance of certain bodies coincides with the maximum, or maximal intensity, of a given situation, as the system of axioms prescribe the existence of a given set that is proven possible by the event that bares its name (the principle of maximality). That is, considering the current state of affairs, the interacting and coalescent bodies coincide in a specific site which is then granted the capability of appearing (forcing into appearance or manifestation) its own inexistent. This Badiou summarizes by axiomatizing his own system.

Axiom I – The event is the pure cut in becoming as made by a body through its own appearance

The badiouian event is never the concentration of vital continuity, or immanent intensification of becoming: “it is never coextensive with becoming” (Badiou, 2008, p. 384). This means that the event is not what guarantees the emergence of a new body, or actualisation of the virtual matter. Instead, the body that has surged and appeared expresses the evental capacity of the inconsistent multiple to auto-appear, to instantiate in a determinate situation through its coalescent points; this gives away that the event has happened. In addition, the event is what supplements the appearing body “through the upsurge of a trace” (Badiou, 2008, p. 384). It is not the whole of the world that becomes, but that element common to everything that interconnects logically all worlds per the eternity of Truth. There is an impossibility of auto-appearance without interrupting the authority of the mathematical laws of Being and the logical laws of appearing.

Badiou posits that, even if the Event is the paradoxical element that deconstructs and infinitely subdivides the structure of the given, it is part of it as its empty form, its purest and most static state. For Badiou, this is crucial, because “there is no contradiction between the unlimited character of becoming and the singularity of the event” (Badiou, 2008, p. 382), so he is guaranteeing that the emergence of a singular body, capable to act and transgress the world as an individualized subjectivity, does not conflict with the infinite, all-encompassing situation that allows it to become.

Axiom II – The event is a separating evanescence, or atemporal instant, which disjoins the previous state of an object from its subsequent state

Basically, Badiou is insisting on the formal a-temporality of the event, it does not pertain to time as chronological continuation, less so as pure form of time. The event can be considered as the effect time has on things, expressed by the sudden appearance of the bodies, for instance. Essentially, the event “extracts from one time the possibility of another time” (Badiou, 2008, p. 384).

Axiom III – The active (appearing) body is adequate to the new present, as an effect of the act of presentation the event is

At this point, Deleuze is completely reversed: “it is not the actions and passions of multiples that are synthesized in the event as an immanent result” (Badiou, 2008, p. 385), for the bodies express the Same (the eventual One) that perplicates through all possible worlds and operates the appearance of a given situation. It is the operatory formality sustained by the One what constitutes the multiples into subjectivizable bodies, leaving behind its trace, “itself incorporated into the new present” (Badiou, 2008, p. 385).

Axiom IV – The Event is a separation (subtraction) from other events

Indeed, Badiou underlines that, just as truths are multiple and multiform, events are the exception of worlds. However, even if the Event is the operation by which the One “makes worlds chime with one another” (Badiou, 2008, p. 385), it does not carry an organic resonance through the entire plane of immanence to which all possibilities are connected. Against the charming rhythmic ritournelle to which all machines are subjected when assembled, Badiou purports

the dull and utterly unresonant sound of what has locally cut through the appearing of a site, and which nothing brings into harmony -or disharmony- either with itself (be it as subsistent solitude) or with other becomings (be it as the absorption of contraries). (Badiou, 2008, p. 385)

3. Conclusion

What follows from this axiomatization? Badiou aimed to prove that, “like all the philosophers of vital continuity, Deleuze cannot maintain the gap between sense, the transcendental law of appearing, and truths as exceptions” (Badiou, 2008, p. 386). Because Sense serves as a category just as Truth does, the event falls into a vitalist logic where the actualizations of multiplicities follow the law (axiom) of the virtual One-All. This axiomatization strains that the place of the event is, to Deleuze, the summation of all ideal singularities that communicate in one and the same Event (Eventum Tantum). In other words, the event is the intensified and continuous result of the coming-to-be of a virtual Being: “Deleuze is an empiricist (which after all he always claimed to be). […] When he reabsorbs the event into the One of the ‘ulimited Aiôn, of the Infinitive in which it subsists and insists’, in the always-there of the Virtual, he has a tendency to dogmatism” (Badiou, 2008, p. 387). It is the task of the whole Badiouan system to break with this empiricism, to think the event as the advent of what subtracts itself from all experience: “the ontologically un-founded and the transcendentally discontinuous” (Badiou, 2008, p. 387). To do so, not only the axioms that govern Deleuze’s metaphysics must be subtracted from Logic of Sense, but also new axioms must be purported. Hence, Badiou breaks with dogmatism by removing the event from the empirically attainable, he subtracts it from Life “in order to deliver it to the stars” (Badiou, 2008, p. 387).

Essentially, the organic existence of the Event, as virtual and subsistent to the chaotic world, is unacceptable to Badiou: “the one of the truth that the event initiates presupposes its being without-One, its contingent dissemination” (Badiou, 2008, p. 386). Thus, while Deleuze, inasmuch according to this reading, asserts the fatidic dimension of the Event, as effect of the mixture of bodies, Badiou asseverates the complete opposite, it is formal and pure, contingent and sporadic. On the one side the Event is the paradoxical element of Sense, on the other side, the Event is the generic void of Truth, multiple and compossible to all truths and world. Deleuze mitigates the importance of the current state of affairs being truth (for it is real and nothing else, containing in its assemblage the germ of its own deterritorialization), Badiou requires it to be held as primordial, for all bodies are assembled truthfully in the situation:

I cannot accept the idea that events ‘are never anything but effects’, to the point that Deleuze ends up calling them ‘events-effects’. This is not because they are causes, or worse, ‘essences’. They are acts, or actants, material principles (the site) of a truth. For Deleuze, the event is the immanent consequence of becomings or of Life. For me, the event is the immanent principle of exceptions to becoming, or Truths. (Badiou, 2008, p. 385)

So, the concept of “sense-event”, by which Deleuze would argue that the event belongs to the register of sense, that is the linguistic dimension by which the chaotic world is structured, is to Badiou inconsistent at most. Because the event is a localized dysfunction of the transcendental of a world (and, by transitive properties shared by all other compossible worlds), it does not possess sense, nor it is sense. The Event is, as transcendental operation of the worlds and becoming, what pries open a space of consequences “in which the body of a truth is composed” (Badiou, 2008, p. 386): the transcendental law of appearing cannot marry the organic and immanent Difference that perplicates through the Same by actualisation. The Event cannot be both a material virtuality and a conceptual/formal operation, for truths are exceptions. Badiou needs his logic to follow exclusively an idealist ontology, where becoming is operated by the empty form of existence, Being or the One; Deleuze has no need for such a category of truth, as Sense is a sufficient name for it. Essentially, Deleuze’s interest is in demonstrating that all becoming follows a logic where the un-grounded and chaotic plane of immanence, where subjectivity dwells effaced and all truths are equally possible, is structured and assembled linguistically, then to be again conflagrated.

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Author notes