Earlier Badiou: mathematical ontology
While one may characterise the turn from medieval to modern philosophy in terms of a shift from the Scholastic definition of the “transcendental” as the ontological structure in which entities exist to Kant’s definition as the epistemological conditions for our mode of cognition of things,Footnote 16 for Badiou, the turn to modernity is marked by a change in the ontological status of “the One”:
Modernity is defined by the fact that the One is not… that “God is dead”… for we moderns, the Multiple-without-One is the last word on being qua being. Now the thought of the pure multiple, of the multiple considered in itself, without consideration of what it is the multiple of, is called: “mathematical set theory.” Therefore every major concept of this theory can be understood as a concept of modern ontology.Footnote 17
In Badiou’s view, to ontologically posit that “the One is not” is to say that God is dead, and to truly move ontological inquiry away from its historic theological captivity—to move beyond so-called onto-theology.Footnote 18 As Badiou puts it:
There is no God. Which also means the One is not. The multiple “without-one”—every multiple being in its turn nothing other than a multiple of multiples—is the law of being. The only stopping point is the void.Footnote 19
All things—all beings—are multiples. For Badiou, multiples cannot be broken down or divided into ones, but only into further multiples: “The multiple is only ever composed of multiples. Every multiple is a multiple of multiples.”Footnote 20
Such an account of multiples consisting of further multiples—of the multiple that is “without-One”—is something that Badiou finds in modern “mathematical set theory”: “Any multiple is intrinsically multiple of multiples: this is what set theory deploys.”Footnote 21 It is for this reason that Badiou famously identifies mathematics—and specifically Cantorian set theory—as ontology itself in Being and Event (1988). This understanding of mathematics as ontology provides Badiou with a way to overcome the (onto-)theological focus on the One:
By initiating a thinking in which the infinite is irrevocably separated from every instance of the One, mathematics has, in its own domain, successfully consummated the death of God.Footnote 22
For Badiou, the most distinctive and profound ontological insight of set theory is that things (as sets) are not reducible to the one, but to zero or the “not”—the “stopping point” which Badiou calls “the void.”Footnote 23 Badiou notes: “Everything can potentially be reduced to a multiple without quality, made of the void alone.”Footnote 24
Drawing on von Neumann’s set-theoretical formulation of ordinal numbers, Badiou speaks of the empty set—denoted as “0” or “Ø” (what Badiou calls “the void”)—as “the inaugural point of being.”Footnote 25 All subsequent numbers or “successor ordinals” after 0—such as 1, 2, 3, 4, and so on—can be formulated set-theoretically in terms of the empty set, i.e. in terms of the number 0 or Ø:
$$\begin{array}{*{20}l} 0 \hfill & { = \emptyset } \hfill \\ {1 = \left\{ 0 \right\}} \hfill & { = \left\{ \emptyset \right\}} \hfill \\ {2 = \left\{ {0,1} \right\}} \hfill & { = \left\{ {\emptyset ,\left\{ \emptyset \right\}} \right\}} \hfill \\ {3 = \left\{ {0,1,2} \right\}} \hfill & { = \left\{ {\emptyset ,\left\{ \emptyset \right\},\left\{ {\emptyset ,\left\{ \emptyset \right\}} \right\}} \right\}} \hfill \\ {4 = \left\{ {0,1,2,3} \right\}} \hfill & { = \left\{ {\emptyset ,\left\{ \emptyset \right\},\left\{ {\emptyset ,\left\{ \emptyset \right\}} \right\},\left\{ {\emptyset ,\left\{ \emptyset \right\},\left\{ {\emptyset ,\left\{ \emptyset \right\}} \right\}} \right\}} \right\}} \hfill \\ \end{array}$$
While it is not our main purpose to explicate Badiou’s (admittedly contested) use of mathematics here, this brief presentation of his account of the composition of ordinal numbers can highlight two key features of Badiou’s set-theoretical ontology.Footnote 26
First, understood as a set, the number 1 is denoted by “{0}” or “{Ø}” (which is to be distinguished from the number zero which is an empty set represented by “Ø” [without the braces to denote the set]): number 1 is a set that is composed by the number 0 or Ø, a set that contains the number zero as an element.Footnote 27 It is in this sense that the one may be said to be reducible to zero or the void: All things are fundamentally reducible not to the one but the void—“the only stopping point.” To the extent that all numbers can be regarded as sets, they may be described as being “composed of the void.”Footnote 28 Second, given that every “one” always already includes the void—that every {Ø} includes Ø, and that all things (as sets) are ultimately composed of the void, the void is accordingly intrinsically included in all numbers and indeed in all sets, i.e. in all beings. As Badiou puts it:
The void is a subset of any set: it is universally included… [This] testifies to the omnipresence of the void. It reveals the errancy of the void in all presentation: the void, to which nothing belongs, is by this very fact included in everything.Footnote 29
By attributing “omnipresence” to the void, one might wonder whether there is still a certain “theological” character to Badiou’s set-theoretical ontology. But for Badiou, this is not an “omnipresence” of some transcendent divine figure, but rather an omnipresent declaration of an absence or “lack”: “the void is presented everywhere in its lack.”Footnote 30 According to Badiou, set theory’s “omnipresent” presentation of the void is nothing other than a declaration and affirmation of the death of God: “Ultimately, this halting point is the void, not the One. God is dead at the heart of presentation.”Footnote 31 The empty set Ø (i.e., “zero” or the “not,” instead of the “one”) is not just the “halting” or “stopping point” but also “the absolute initial point of being” from which all the other sets are unfolded.Footnote 32 Thus, for Badiou, the void Ø—the “not”—is both the initial point and the halting point of all being: to put it in Christian theological language, the void is the archè and telos of all things.Footnote 33
If all things or all beings are sets, then there is, so to speak, set-ness in all things.Footnote 34 The empty set—what Badiou calls the void—is something that formally predicates or indeed “sets” all things.Footnote 35 As such, what we have here is in this regard something rather similar to the classical transcendentals: Common notions which are essential properties of all things. Moreover, as if echoing Thomas Aquinas’s designation of “Being” as the most proper name of God,Footnote 36 Badiou speaks of the void as “the proper name of Being.”Footnote 37 Just as all creatures are said to bear a likeness to God’s perfect oneness, truth and goodness in Aquinas’s theological metaphysics,Footnote 38 all beings are said to be marked by the void that is “included in everything” in Badiou’s atheistic ontology.Footnote 39 As opposed to all things reflecting the perfect oneness, truth, and goodness of God, Badiou’s universal and omnipresent—and indeed, “transcendental”—notion of the void is one which signifies the death and absence of God—that the divine is not. It is this void—this not and absence of God—that underlies all being.Footnote 40 To paraphrase Psalm 19: All beings declare the death of God, all creatures proclaim his absence.Footnote 41
Later Badiou: logical phenomenology
For Badiou, the “omnipresent” declaration of the death of God by the void is formally an ontological thesis as opposed to a phenomenological one.Footnote 42 Positioning himself against the phenomenological tradition, Badiou draws a peculiar distinction between ontology and phenomenology: Whereas ontology is mathematics, phenomenology is logic.Footnote 43 As Badiou remarks in Logics of Worlds (his 2006 sequel to Being and Event): “just as being qua being is thought by mathematics (a position that is argued for throughout Being and Event), so appearing, or being-there-in-a-world, is thought by logic [in Logics of Worlds].”Footnote 44 As we can observe from Badiou’s use of the plural for both “logic” and “world” in title of Logics of Worlds (and his use of the indefinite article in “being-there-in-a-world”—as opposed Heidegger’s “being-in-the-world”), there is a deep commitment to the plurality or multiplicity of “worlds” (or “situations”) in Badiou’s project—something that is already developed in Being and Event.Footnote 45
Following the central ontological thesis that “the One is not”—that there can be no oneness, unity or indeed “wholeness” in being—in Being and Event, Badiou fervently asserts that “there is no Whole” in Logics of Worlds:
we will call universe the (empty) concept of a being of the Whole. We will call world a “complete” situation of being. Obviously, since we show that there is no universe, it belongs to the essence of the world that there are several worlds, since if there were only one it would be the universe.Footnote 46
According to Badiou, the thesis that that “there is no universe” or “no Whole” is simply an extension of “the conceptual consequences of Russell’s paradox.”Footnote 47 In Badiou’s reading, Russell’s paradox (which famously shows that there can be no universal set of all sets that are not members of themselves) provides a “logical demonstration” that the existence of an absolute totality is inconsistent and self-contradictory—which, for Badiou, means that “a set of all sets cannot exist.”Footnote 48
Before explicating the full implications of this assertion that there can be no “set of all sets,” it is important to highlight why the “inexistence of the Whole” is integral to the formulation of Badiou’s “logical phenomenology” in Logics of Worlds. According to Badiou, since “there is no Whole,” there can be “no uniform procedure of identification and differentiation of what is.”Footnote 49 For Badiou, given that there is no absolute totality—no “set of all sets” or “world of all worlds” (what he calls a “universe”), each world or situation has its own “logic”—what Badiou calls “transcendental indexing”—which objectively describes, regulates and organises the appearances of things within those respective worlds.Footnote 50 There are as many transcendental logics as there are worlds—hence the book title Logics of Worlds, which is why Badiou argues that there cannot be a “unified ‘centre’ of transcendental organisation, such as the Subject is for Kant.”Footnote 51 As such, at the heart of Badiou’s so-called “objective phenomenology” is a replacement of Kant’s subjective account of the transcendental with a logical conception of the “transcendental”—as Badiou admits: “The [non-Kantian] concept of ‘transcendental’ is without doubt the most important operational concept in the whole of [his] theory of appearing.”Footnote 52
Indeed, in his acknowledgement that he is “reprising the old word ‘transcendental’” in Logics of Worlds, Badiou explicitly states that this “old word” is “detached from its constitutive and subjective value” in his objective phenomenology.Footnote 53 What Badiou seeks to present with his “logical” formulation of the transcendental in Logics of Worlds is an “analytic of being-there [that] does not presuppose subject”—in other words, “a transcendental constitution (without subject).”Footnote 54 Underlying Badiou’s non-subjective “logical” account of the transcendental is ultimately a deeply anti-Kantian sentiment, as Badiou openly admits in Logics of Worlds:
Kant is the one author for whom I cannot feel any kinship… The critical machinery he set up has enduringly poisoned philosophy.Footnote 55
However, despite his strong anti-Kantian rhetoric and his aspiration to develop a subject-less phenomenological account of the transcendental, Badiou’s conception of “the transcendental” in phenomenological terms of appearance—as opposed to ontological terms of “Being” itself—nonetheless still resembles Kant’s critical idealist definition of transcendentality.Footnote 56
At this point, it is worth revisiting an earlier interview Badiou held with Peter Hallward in 1997. Hallward asks: “Isn’t it a kind of transcendental condition, an enabling condition of our existence, that we must always be specific to a situation?” To which Badiou responds:
I take it to be an ontological principle… I’ve no need to call it transcendental… I try to limit the use of the word “transcendental” to its Kantian meaning. “Transcendental” refers back to the subjective conditions of experience, and Kant never stops telling us that it is precisely not a law of being. It is a law of the unity of the phenomenon, not a law of being. If you want to extend the meaning of the word “transcendental” to the point that you call, in the end, transcendental the first or ultimate condition of thought in general, of existence in general, then at that point I’d agree: yes, it’s transcendental.Footnote 57
As we see from Logics of Worlds, Badiou inevitably moves away from understanding transcendentality in the Kantian meaning of the subjective conditions of experience, but nonetheless, like Kant, in Logics of Worlds Badiou still sees the “transcendental” as “a law of the unity of the phenomenon, [and] not a law of being.” However, if we understand “transcendentality” not in a Kantian way, but in a pre-Kantian if “classical” sense—in Badiou’s words, not as “the first or ultimate condition of thought in general” but “of existence in general”—then it seems that Badiou would be happy to say that his mathematical account of being (and not appearance) is indeed “transcendental.”
However, the “transcendental” attribution of the void to all things in Badiou’s ontology is not a metaphysical ascription of essences or properties (such as oneness, truth, or goodness) to things that one finds in Scholastic ontology and its corresponding theological conception of the transcendentals. Badiou’s ontology is deeply atheistic not just with its replacement of “the one” with the void or the not as a transcendental notion that is convertible with “Being.” Just as there can be no “set of all sets” according to his set-theoretical ontology, there is no God-like metaphysical structure such as an ideal “Form of all forms” or “value of values” by which we measure and evaluate things, nor is there a universal Platonic Good from which all things derived their being and goodness.Footnote 58 For Badiou, while all things (sets) have “set-ness,” there is no transcendent “set of all sets” in which these sets participate: All sets are all equally sets—no set is more perfect or less perfect as a set than any other set.Footnote 59
For Badiou, the inexistence of “a set of all sets” is one of the reasons why his set-theoretical ontology is “inessential”: “A set, in Cantor’s sense of the word, has no essence besides that of being a multiplicity; it is without external determination,” precisely because “there cannot be a set of all sets” to act as an external or indeed transcendent metaphysical principle to determine or prescribe essences or properties to beings.Footnote 60 It is for this reason that Badiou says that his “ontology attributes no other property to multiples than existence.”Footnote 61 Badiou’s set-theoretical ontology “makes no claims concerning the nature of being… it has nothing to say about the qualities of identity of any concrete situation.”Footnote 62 While Badiou is not transcendental thinker in the Kantian sense, his “transcendental” ontology is a distinctively modern one insofar as it consciously avoids prescribing essences or natures to things.Footnote 63 As we see in the following section, this “modern” attempt to develop an inessential transcendentality is something that is also—and perhaps more explicitly—found in the anti-essentialist ontology of one of Badiou’s contemporaries: Giorgio Agamben.