A MATERIALIST TRANSCENDENTAL: on the onto-logy of logics of worlds

Abstract

Alain Badiou's 2006 Logics of Worlds presents a fascinating attempt to construct an avowedly materialist analysis of the conditions of the intelligibility of appearance. This paper sketches the outlines of the materialist transcendental developed there in order to isolate just what Badiou takes its materialist credentials to consist in. This effort leads to an examination of the analysis of mathematical formalism provided in Badiou's earlier The Concept of Model and ultimately to the conclusion that Badiou's defense of the materialism of the process of mathematical formalization in that text does not supply a satisfactory explanation of the purported materialism of Logics of Worlds.

Keywords:

• Alain Badiou
• Logics of Worlds
• The Concept of Model
• materialist dialectic

Notes

I would like to thank an anonymous reviewer for providing a series of helpful and challenging remarks that helped me to develop the ideas presented here. I would, moreover, like to thank John Bova, with whom I first read and discussed Logics of Worlds in the summer of 2009. The shortcomings of this paper remain, of course, my own.

For a more detailed description of these structures see Being and Event 93–101.

Badiou, Second Manifesto 41.

Ibid. 55.

Idem, “Being by Numbers” 123.

That neither being nor beings can be defined ontologically is a consequence, according to Badiou, of the axiomatic nature of post-Cantorian set theory. In order to avoid Russell's paradox, set theory adopted an axiomatic method, which abandons the task of defining what a set is in favor of prescribing rules for the production of new sets given some previously existing set (with the notable exception of the axiomatic assertion of the existence of the empty set). Badiou writes:

What is counted as one is not the concept of the multiple; there is no inscribable thought of what one-multiple is. The one is assigned to the sign ∈ alone; that is to the denotation for the relation between the “something” in general and the multiple. (Being and Event 44)

As a result, according to Badiou, the ontological study of being and beings is distinct from a description of what being is.

Badiou has come under considerable criticism for his claim that the mathematical description of being qua being is or can be properly materialist. Although I will not pursue the question here – an examination of the tensions involved in attempting to grasp the ontological nature of beings apart from their worldly or ontic relations and entanglements outstrips the scope of this limited consideration of the materialist credentials of the mathematico-logical reworking of transcendental philosophy in Logics of Worlds – I refer the reader to Johnston's (“Alain Badiou” 140–81) and Hallward's (Badiou 314–15) criticisms of Badiou's dismissive attitudes toward the life and natural sciences and accompanying willingness to insist on the intelligibility of beings in isolation from their social, political, and environmental contexts.

Badiou, Logics of Worlds 101. The stated purpose of Logics of Worlds is to explain how a universal truth is subjectively produced, and so appears, within a concrete historical context, and so the category of subjectivity is central to the argument of the text as a whole (see Logics of Worlds 36–37). I am focusing here on the more general analysis of the conditions of appearing provided in what he calls “The Greater Logic” of appearing, and will accordingly not address how the subject is a “late and problematic construction” in Badiou's project. His point in this passage is not to claim that the late appearance of the subject is the positive source of his materialism, but that it is one of its negative conditions.

I will not address the failures of subjective idealism to account satisfactorily for the consistency of appearing. For this argument consult Quentin Meillassoux's indictment of correlationism, which Badiou enthusiastically endorses, in chapters 1 and 2 of After Finitude.

Kant Bxxvii.

Badiou, Logics of Worlds 118.

Ibid.

Ibid. 36.

Badiou reiterates this point elsewhere:

it's clear that you cannot pass directly from being-as-being to being-there. Were I to pass from one to the other by rational deduction I would simply be engaged in a reconstruction of Hegelianism. I would be drawing a figure of being-there from the being of multiplicity. Against this Hegelian inspiration, I assume the contingency of being-there. (“Beyond Formalisation” 129)

To consider a being ontologically, or according to its set-theoretical multiple-being, is different than the proper task of ontology, according to Badiou, which is to think being qua being. Being qua being, as he argues in Being and Event, is the inconsistent multiplicity, named by the void, that precedes the operational unity of the being of a being presented in a given situation. Being as such is neither, strictly speaking, one nor multiple:

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 is it multiple (because the multiple is solely the regime of presentation). (Being and Event 24)

The mathematical being of a particular being, what Badiou refers to as a thing in the Second Manifesto (46–53), is a unified, or consistent, multiple, whose identity or difference to other things is determined absolutely by the axiom of extension (cf. Being and Event 60–61). What remains unexplained in Badiou's account of the dialectic between being and appearing, as Johnston rightly argues, is the source of the unifying function of the count-as-one, which is responsible for rendering the inconsistent multiplicity of being qua being into static and manageable, if infinite, multiples (see Johnston, “Phantom Consistency” and “Alain Badiou” 196–210). For Kant, of course, the source of this necessary moment of unification was the synthetic spontaneity of transcendental subjectivity. Without identifying the origin of the synthetic function of the count-as-one, Badiou remains open to the criticism that he has maintained the transcendental subjectivity of Kantian idealism despite his materialist rhetoric. I accept and leave Johnston's criticisms to the side here in order to pursue a distinct limitation of Badiou's materialist claims.

A partially ordered set is a set (and so a presented or consistent multiple) that has minimal and maximal elements and a well-defined operation, ≤, to determine, given two elements within the set, whether the first element is greater than, less than, or equal to the second element.

John Bova presented, under the title “Participation and Chorism: On the Significance of the Topos Formalism for Phenomenological Ontology,” a compelling and nuanced analysis of the intuitive relationship between the transcendental of a world and the power set of the multiple being of that world. Bova's argument clarifies both the ontological status of a transcendental within Logics of Worlds and Badiou's occasional claims that it is of the essence of being to appear.

Badiou, Logics of Worlds 211.

Ibid. 101.

The transcendental, which determines not only the intensity of being's appearing in its world, but also its relations to other apparents in that world cannot be reduced to an index of phenomenal intensities (Badiou, “Beyond Formalisation” 130). An analysis of the logical relations that organize appearing is beyond the scope of this paper, and so I will limit my comments to the role of the transcendental function of ordering differences of phenomenal intensity (on which the other logical relations – the envelope, conjunction, compatibility, etc. – are based).

Badiou, Logics of Worlds 598.

Thus, given a multiple A, an element a ∈ A, a transcendental T and Id: A → T, Badiou defines the phenomenon of a relative to A to be ϕ(a/A) = {a, Id(a, x ₁), Id(a, x ₂), …, Id(a, x _n) | x _n ∈ A}.

The existence of x in a given world is measured, then, by the following function: E(x) = Id(x, x) = p.

The fact that we can logically think a multiple that belongs (ontologically) to but does not appear in a world supports his rejection of the teleological disposition of being toward appearing. If essence dirempts itself into existence, everything that is must necessarily exist. For Badiou's discussion of this issue, see his comments on Hegel (Logics of Worlds 141–52).

Ibid. 117.

Ibid. 247.

Ibid. 214. An atomic component is formally defined as follows: given a pure multiple A, a phenomenal component π and elements x and y ∈ a pure multiple A such that π(x) = M, where M is the maximum of T, π is an atom of appearing if and only if [π(y) = M] ⇒ [Id(x, y) = M].

More formally, a identifies a real atom π when π(x) = Id(a, x) for all x ∈ A.

Badiou, Logics of Worlds 218.

This is not to say that one can only know the ontological basis of a real atom after the appearance of that atom in a specific world. There are no ontological relations other than identity, difference, belonging and inclusion, and so the ontological basis of real atoms is only determined relative to the worlds in which those atoms appear. This retroactive revelation of the ontological basis of appearing is not only epistemological; it is onto-logical.

Badiou, Logics of Worlds 196.

The materialism of his position “depends therefore on an orientation in thought that is a philosophical choice and not the result of an argument” (Second Manifesto 60). The “speculative decision” that motivates this orientation of thought is not without argumentative support. The absence of a formal, mathematical necessity for the postulate of materialism does not, of course, indicate the absence of more properly philosophical support. This support is provided by a diagnosis of various idealisms' inability to respond to his guiding transcendental question: given that the natural sciences are capable of producing real knowledge from within specific historical and technological situations, what are the necessary conditions for this local production of universal truth?

Badiou, Logics of Worlds 220.

Ibid. 219.

That epistemological realism does not entail materialism is perhaps best illustrated by Berkeleyan idealism. It is not immediately clear that an ontological realism of the variety presented in Logics of Worlds in itself constitutes a genuine materialism. Though I am not inclined to believe that distinguishing between being and appearing itself vitiates Badiou's materialist pretensions, I argue below that even with the support of his early materialist account of mathematical formalism, the dialectic of being and appearing at issue here is not decisively materialist.

For Badiou's analysis of the connection between constructivism and language see Meditation 28, “Constructivist Thought and the Knowledge of Being” (Being and Event 286–94).

For Badiou's distinction between democratic materialism and his own materialist dialectic see Logics of Worlds 1–9.

Badiou and Hallward 126–27. For a more general statement of the relationship between traditional concerns in the philosophy of mathematics and the identification of ontology and mathematics see Being and Event 6–9.

Badiou, Being and Event 8–9.

For an excellent and more complete analysis than I can provide here of The Concept of Model see Brassier.

Badiou, The Concept of Model 42.

This correlation holds, more specifically, between the syntactic deducibility of theorems within the formal system and the semantic validity of those theorems in the model. For more detail see Badiou, The Concept of Model §7.

See Bachelard.

Badiou, The Concept of Model 43.

Ibid. 54.

Ibid. 48.

Quoted in Fraser lxi.

Badiou and Tho 103.

Badiou, Manifesto 37. Badiou repeats this point at Being and Event 4: “What philosophy must do is propose a conceptual framework in which the contemporary compossibility of these conditions [art, science, politics love] can be grasped.”

Badiou, Being and Event 27.

See ibid. 27–30 for the details of this argument.

Meillassoux adopts the related and ongoing project of establishing the ontological ground of fundamental logical laws (specifically the law of non-contradiction) in After Finitude. Although this aspect of Meillassoux's book has garnered considerably less attention than his diagnosis of a pervasive correlationism in post-Kantian Continental philosophy, the importance of this and similar projects for the possibility of articulating and defending a genuinely materialist metaphysics should not, in my opinion, be underestimated.