Since last week I’ve been largely out of commission cognitively due to health issues, but I’m slowly regaining the ability to think. During this time, I happened to come across a brief statement by Badiou, explaining why he considers himself a materialist. Here my remarks will be less than elegant, though I hope to localize a problem that seems to emerge with regard to Badiou’s ontology, and a place where it might become possible to think questioningly with Badiou (in tandem with Deleuze and what might motivate a Deleuzian ontology).

In an interview accompanying Badiou’s Ethics: An Essay on the Understanding of Evil, Hallward asks Badiou to clarify the relation between his mathematical ontology and the nature of material reality. Despite my great love of mathematics and my tendency to advocate some form of Platonic realism where the ontological status of mathematical entities is concerned, I think this question naturally emerges insofar as mathematical entities are often thought as idealities and possibilities, independent of the actuality characterizing the material world. Consequently, there seems to be something of a gulf or chasm between the infinite possibilities of the mathematical world and the actualities characterizing this world. In response, Badiou remarks that,

If we accept that there exists a situation in which what is at stake for thought as being-as-being [viz., ontology]– and for me, this is simply one situation of thought, among others –then I would say that this situation is the situation defined by mathematics. Mathematics, because if we abstract all presentative predicates little by little, we are left with the multiple, pure and simple. The “that which is presented” can be absolutely anything. Pure presentation as such, abstracting all reference to “that which” –which is to say, then, being-as-being, being as pure multiplicity –can be thought only through pure mathematics.

To the extent that we abstract the “that which is presented” in the diversity of situations, to consider the presentation of presentation itself– that is to say, in the end, pure multiplicity– then the real and the possible are rendered necessarily indistinct. What I call ontology is the generic form of presentation as such, considered independently of the question as to whether what is presented is real or possible… Are they real, do they exist somewhere, are they merely possible, are they linguistic products…? I think we have to abandon these questions simply because it is of the essence of ontology, as I conceive it, to be beneath the distinction of the real and the possible. What we will necessarily be left with is a science of the multiple in general, such that the question of knowing what is effectively presented in a particular situation remains suspended. A contrario, every time we examine something that is presented, from the strict point of view of its objective presentation, we will have a horizon of mathematicity, which is, in my opinion, the only thing that can be clear. In the final analysis, physics– that is to say, the theory of matter –is mathematical. (Ethics: An Essay on the Understanding of Evil, pgs. 127-128, italics mine)

A moment later Badiou reminds us of what ontology is, remarking that, “all of this simply confirms a very old and somewhat inevitable ontological programme: that ontology always gathers up what remains to thought once we abandon the predicative, particular determination of ‘that which is presented'” (129). Ontology, then, is what remains once we subtract all other predicates characterizing an existent or an entity. According to Badiou, all that remains after such an operation of subtraction occurs is pure multiplicity or multiplicity qua multiplicity sans any other predicates or qualities:

Now, the existent qua existent is absolutely unbound [I read this as “un-related”]. This is a fundamental characteristic of the pure manifold as it is thought in Set theory. There are only multiplicities and nothing else. None of them on their own is connected to another. In Set Theory even functions have to be thought as pure multiplicities or manifolds. This is why we identify them by their graph [I’m not sure what he’s getting at here with his reference to the graph of a function]. The “beingness” of the existent does not presuppose anything else than its immanent composition, that is, that it might be a manifold of manifolds. Strictly speaking, this excludes the possibility that there might be a being of the relation. When thought as such, and therefore purely generically, Being is subtracted from any connection. (Briefings on Existence: A Short Treatise on Transitory Ontology, “Being and Appearing”, 162. This essay can also be found in Theoretical Writings)

The key sentence in this passage is “manifold of manifolds”, which should be translated as “multiplicities of multiplicities”. In thinking Being as manifold of manifolds Badiou is effectively claiming that there are no ultimate or irreducible terms of which sets would be composed, but only endless multiplicities without ultimate identities, and that these manifolds are unrelated or unconnected to one another. As such, Badiou’s ontology is an ontology of infinite dissemination without One or an overarching unity at the level of either the whole or the part. Now I take it that the great merit of Badiou’s mathematical thesis is two-fold:

  1. Badiou’s ontology effectively allows us to escape any epistemological orientation in ontology, by sidestepping any questions of how it is possible for a subject to relate to being. That is, we need raise no questions of how a mind is able to know or represent being. This point might be obscure to those who have no background in philosophy of mathematics; however, ever since Frege and Husserl, questions of the psychology through which mathematics is known have been staunchly excluded from the thinking of the mathematical qua mathematical. Those unfamiliar with these arguments and the manner in which they trenchantly depose any psychologism would do well to refer to Frege’s Foundations of Arithmetic. This is why it’s so important to Badiou that ontology evade the distinction between actuality and possibility (as it must not be a matter of mind representing reality, but of a “common being” to possibility and actuality). It is also one reason that Badiou’s ontology diverges radically from Zizek’s Hegelian orientation. The consequences of this move are far reaching. In one fell swoop, Badiou is able to escape all questions revolving around the subject and anthropology. If mathematics truly is ontology, and if mathematics is independent of questions of knowledge, then all questions about differing subjective perspective on reality, different cultural perspectives on reality, etc., fall to the wayside as interesting psychological, anthropological, and sociological speculations, but speculations that are quite irrelevant to ontological researches. In short, ontology and philosophy no longer need concern themselves with cultural studies, linguistics, or the social sciences. This is what it means, for Badiou, to say that math inscribes the real.
  2. In a closely related vein, Badiou’s thesis allows us to depart, once and for all, from Heidegger’s endless preparations for properly posing the question of being. There is no need to engage in an elaborate hermeneutic of Dasein as that being that is “ontic-ontological” and who has a pre-ontological understanding of being, as the being of being is exhausted in its mathematicity.

To my thinking, these consequences cannot but come as a breath of fresh air to philosophy insofar as it has increasingly come to be dominated by cultural studies, rhetorical analysis, and pseudo-pious phenomenological discourses.

However, I wonder nonetheless whether Badiou hasn’t moved a bit too quickly. Badiou’s thesis regarding materialism seems to be that insofar as science always approaches matter mathematically, a mathematical ontology is necessarily a materialist ontology:

…it [the thesis that mathematics = ontology] is a fully materialist thesis, because everyone can see that the investigation of matter, the very concept of matter, is a concept whose history shows it to be at the edge of mathematicity… ‘Matter’ would simply be, immediately after being, the most general possible name of the presented (of ‘what is presented’). Being-as-being would be that point of indistinction between the possible and the real that only mathematics apprehends in the exploration of the general configuration of the purely multiple. Matter, in the sense in which it is at stake in physics, is matter as enveloping any particular presentation–and I am a materialist in the sense that I think that any presentation is material. (Ethics: An Essay on the Understanding of Evil, 130, italics mine)

What seems to be missing in Badiou’s account of materialism is precisely any discussion of this “concept at the edge of mathematicity”. I have highlighted these passages on how being-qua-being is the point of indistinction between the possible and the real to indicate that the moment we enter the realm of the material, we are no longer dealing with something merely possible, but rather with something actual. That is, something has been added to what we’re talking about, what we’re investigating, that isn’t strictly mathematical. Although mathematics is certainly an essential dimension of physics or any science (I would accept Kant’s thesis that mathematization is a necessary condition of scientificity), the object of any particular science is an object that cannot itself be mathematically deduced.

Doesn’t this edge of mathematicity, this element that is enveloped by mathematicity, itself have some claim to being? Hallward hits the nail on the head when he responds to Badiou’s elaboration of materialism, by remarking that, “It seems, however, that your most basic concept, the concept of a situation, oscillates somewhat between an essentially mathematical order and what appears to be a no less essentially eclectic order, combining heterogeneous elements of actuality” (129). The problem as I see it is that unlike being-qua-being, a situation (what Badiou now calls a “world”), does not straddle the distinction between the possible and the real. A situation is real, it is actual, it is this situation and no other.

It thus seems to me that for Badiou there is a tremendous gap between ontology as the “presentation of presentation” or pure multiplicity “without connection”, and the ordered situations of the world. One might respond by arguing that ontology is not in the business of explaining situations as it only studies pure multiplicities, not “consistent multiplicities”. However, Badiou himself says otherwise:

What links a being to the constraint of a local or situated exposure of its manifold-being is something we call this existent’s “appearing.” It is the existent’s being to appear insofar as Being as a whole does not exist. Every being is being-there. This is the essence of appearing. Appearing is the site, the “there” of the multiple-existent insofar as it is thought in its being. Appearing in no way depends on space or time, or more generally on a transcendental field. It does not depend on a Subject whose constitution would be presupposed. The manifold-being does not appear for a subject. Instead, it is more in line with the essence of the existent to appear. As soon as it falls short of being localized according to the whole, it has to assert its manifold-being from the point of view of a non-whole, that is, of another particular existent determining the being of the there of being-there [incidentally, Hegel already conceives appearing as appearing to another existent or Relation in the “Doctrine of Essence”, Science of Logic].

Appearing is an intrinsic determination of Being. The localization of the existent, which is its appearing, involves another particular being: its site or situation. This is why it can be seen immediately that appearing is as such what connects or reconnects an existent or its site. The essence of appearing is the relation. (Briefings on Existence, pg. 162)

I confess that I find these remarks exhilerating, though I am unable to understand Badiou’s thesis or the logical entailment necessitating that “because the whole is not, the existent must appear. ” It seems to me that there is a fundamental ontological axiom here that is currently the is a central theme of a good deal of contemporary theory (Deleuze’s thesis that the Whole is not giveable in Cinema 1 and that this is a condition for the given, Zizek’s thesis that the One is not, Lacan’s thesis that the world does not exist, etc. One of my central questions is that of how to understand the relation between the in-existence of the Whole (not simply that we cannot know the whole, but that the whole does not exist –and the givenness of the given. I am not sure why this question strikes me as so important, but there’s something there. Now, when Badiou glosses category theory in an earlier essay “Group, Category, Subject”, he largely describes my own ontological project:

In category theory, the initial data are particularly meager. We merely dispose of undifferentiated objects (in fact, simple letters deprived of any interiority) and of ‘arrows’ (or morphisms) ‘going’ from one object to another. Basically, the only material we have is oriented relations. A linkage (the arrow) has its source in one object and target in another. Granted, the aim is ultimately for the ‘objects’ to become mathematical structures and the ‘arrows’ the connection between these structures. But the purely logical initial grasping renders the determination of an object’s sense entirely extrinsic or positional. It all depends on what we can learn from the arrows going toward that object (whose object is the target), or of those coming from it (whose objects is the source). An object is but the marking of a network of actions, a cluster of connections. Relation precedes Being. This is why at this point of our inquiry we have established ourselves in logic, and not ontology. It is not a determined universe of thought we are formalizing, but the formal possibility of a universe” (Briefings on Existence, “Group, Category, Subject”, 144).

From my perspective, this is the place to begin ontologically, as I see no way that can account for the emergence of Relation on the basis of pure, unconnected multiplicity as described by Badiou. I am not certain why Badiou refers to this as a logic (generally I’m fuzzy on his conception of logic overall and why he is so hostile to placing math under logic), nor am I sure why he refers to this as the articulation of a possible universe, rather than a determined universe. However, when Badiou suggests that Relation precedes being, and proposes that we conceive entities as “networks of action”, “clusters of connections”, “bundles of relations”, I think this is the right direction to move in ontologically. It is this move towards conceiving beings as activities, as networks, as doings, that allows us to do away with substance ontology. This, then, is the central difficulty. One the one hand, Badiou wishes to claim that ontology is indifferent to the distinction between the possible and the real. Yet on the other hand, Badiou wishes to argue that it belongs to the essence of Being, that it is intrinsic to being, to appear. How can this be? How can we simultaneously affirm both of these theses without undermining the thesis that ontology is indifferent to the distinction between the possible and the real?

Could it be that this is the real source of Badiou’s hostility to Deleuze? It will be recalled that Deleuze defines his transcendental empiricism as that ontology that articulates the conditions of real being, and not all possible being. In developing this ontology Deleuze, following Bergson, advances a substantial critique of the category of possibility, arguing that the dialectic between the possible and the real is unable to give any account of how the real is realized insofar as the real in no way differs from the possible (Kant’s famous critique of the ontological proof for the existence of God, wherein he argues that “existence is not a real predicate”). That is, accounts of realization are unable to explain what the real contributes to being. It seems to me that Badiou finds himself in a very similar position and that for this reason it is difficult to identify his ontology as being genuinely materialist.