The Ontological Requisite

Nathan Brown

In the Introduction to Being and Event, Alain Badiou offers a note situating himself amid his contemporaries in France. Here we find a remark of the utmost importance for grasping the manner in which he understands his philosophical enterprise and its intellectual-historical position, yet which may surprise those who view Badiou as the arch enemy of deconstruction. The very first of his acknowledgements is as follows: “On that which concerns the ontological requisite and the interpretation of Heidegger, one must certainly name J. Derrida.” How is this laconic formulation to be understood? What is this “ontological requisite”? How is it related to the interpretation of Heidegger? And what are the consequences of this note not only for how we understand the relationship between Badiou and Derrida, but also the significance of that relationship for the development of philosophy after Being and Event?

There is a simple answer to the first of these questions, which can be stated forthrightly: ontological inconsistency is the requirement imposed by Derrida upon philosophy after deconstruction. Derrida’s intervention in the philosophical field requires the dissemination of being qua being. And we can say, provisionally, that this ontological requisite does indeed follow from an interpretation of Heidegger. In Being and Time, Heidegger shows that grasping and sustaining the ontological difference—the being of beings is not a being—requires a rejection of every metaphysics of substance and any thinking of being as presence. I will return to Heidegger’s thinking of being as ekstatical temporalization, as the constitutive exteriority of time, and to the complex way in which this thinking is both overlooked by and compatible with Badiou’s philosophy. But, for now, we can say that Derrida’s deconstruction of the metaphysics of presence carries forward Heidegger’s thinking of the ontological difference through the theory of différance, through the theory of the supplement as that which neither is nor is not, and through the grammatological theory of arche-writing as thetracing of the inconsistency of being: the tracing of what Derrida calls, in “Plato’s Pharmacy,” the “disruptive intrusion of otherness and nonbeing, of nonbeing as other in the unity of being” (163). Badiou, it seems, recognizes Derrida’s intervention as the imposition of an “ontological requisite” that must be acknowledged as a condition of his own thought. 

In Being and Event, Badiou meets and obeys this condition. He is the thinker, par excellence, of inconsistent multiplicity, of the ineliminable dissemination of the one, of “nonbeing as other in the unity of being,” of a supplementarity which renders irreducible the surplus or reserve of exteriority to the closure of number or identity. Indeed, Badiou is a thinker of writing, of inscription, as that techne which is not only indivisible from the eidos but which renders the mathematical inscription of being prior to its philosophical ideation, as a condition of possibility for the latter. He is also, and above all, the thinker who deconstructs the opposition of the finite and the infinite which had structured the history of ontology from Plato through Descartes and Spinoza, Kant and Hegel, to Heidegger and Derrida—and in this respect we may find that his reflections upon ontology go further than Derrida’s own upon the path the latter opens, fulfilling the vocation of deconstruction more adequately than could Derrida himself. Let me work through these characterizations of Badiou’s thought, briefly elaborating its adequacy to the “ontological requisite” concerning which J. Derrida must certainly be named.  

Being and Event opens with a discussion of the paradoxes of the one and the multiple endemic to philosophy since Parmenides, and it sets out from an ontological decision: the one is not. As Badiou writes, “my entire discourse originates in an axiomatic decision: that of the non-being of the one.” I take it that Badiou’s note on Derrida specifies the ground of this decision: the historical development of philosophy which constitutes the situation of Badiou’s discourse, after Heidegger and Derrida, imposes the nonbeing of the one as an ontological requisite. Heidegger’s critique of substance ontology and the metaphysics of presence is uncircumventable, as Derrida notes in “Différance.” The history of philosophy develops through such irreversible developments: the impossibility of substance dualism after Spinoza; the requisite binding of consciousness to time after Kant; the displacement of transcendental critique by Hegel’s immanant critique of the determinations of thinking; Heidegger’s deconstruction of the substantial unity of the subject through the existential analytic; the undoing of the subordination of writing to voice by Derrida’s critique of phonocentrism.

Badiou decides upon the nonbeing of the one, but it is not really “his”decision: he registers the philosophical illegitimacy of positing an ontological totality in the wake of deconstruction, and his project is to show that there is a discourse capable of formalizing inconsistent multiplicity, of rendering it coherent without rendering it consistent, and thus of integrally transmitting the thinking of inconsistency without falling back into the closure of the one or the evocation of the unsayable. If Derrida, after Heidegger, shows that the thought of being can only be inconsistent, Badiou shows how to think ontological inconsistency without thereby surrendering ontology to incoherence. Post-Cantorian set theory makes rigorously thinkable the infinite proliferation of differential infinities, through the excess of inclusion over belonging. Moreover, this dissemination of both finite quantities and the infinite Whole binds being to nonbeing, since the inconsistent proliferation of multiplicities formalized by transfinite set theory stems from the void. As a discourse, mathematical ontology structures the void, counts it as a set, names it as multiple of nothing, yet also registers an errancy of the void in excess of the count through the inconsistency of the multiples whose proliferation it structures. Thus, Badiou can say that ontology, as an axiomatic discourse, “seizes the in-itself of the multiple by forming into consistency all inconsistency and forming into inconsistency all consistency.” This discourse “thereby deconstructs any one-effect” entailed by formal presentation. There is a gap, a “nothing,” between “presented consistency and inconsistency as what-will-have-been-presented” (54). The logic of the supplement, in Being and Event, resides not only in the excess of parts over elements producing a non-foreclosable proliferation of differential infinities, transfinite sets of different cardinalities, but also in the fact that this excess stems from the disjunction between the unpresentable errancy of the void and the ontological presentation that structures it, the surplus of the former constituting a reserve inaccessible to the latter. Nonbeing is indeed woven as other into the unity of being, and disrupts that unity within the structure of ontological presentation. This ontological insistence of the void thus meets the condition articulated by Derrida in “Plato’s Pharmacy,” whereby “what is not has a being” while “what is in a way is not” (164).

By according transfinite set theory this role as the discourse of inconsistent multiplicity, Badiou displaces ontology from the field of philosophy per se. And while the thesis “mathematics is ontology” might seem at the furthest remove from Derrida's thinking, I would argue that it does in fact have deconstructive effects. The implications of Badiou's claim are that mathematical writing, chains of mathematical signifiers traversing and restricting the development of mathematical proofs, must condition any ontological thinking within philosophy. Mathematical formalization binds mathematical thinking to the rigorous articulation of written signs and to the history of their inscription—a theme that Derrida takes up in his introduction to Husserl's Origin of Geometry and which Badiou also emphasizes in his great text of the 1960s, “Mark and Lack.” There Badiou argues that mathematical inscription forecloses the unity of consciousness through its ramified stratification of discrepant orders of formal determination, such that “science is the veritable achi-theatre of writing: traces, erased traces, traces of traces: the movement where we never risk encountering the detestable figure of Man: the sign of nothing.” Philosophical reflection upon ontology encounters and is constrained by this movement, by the unfolding of a history wherein thinking subjects enter into a field of inscriptions radically exceeding the unity of consciousness. Philosophical thinking is exposed to an outside—mathematical writing—which not only impinges upon but subsumes the opposition of writing and thinking, displacing the priority of philosophy vis-à-vis the thinking of being.

Derrida argues that “the supplement is neither a presence nor an absence. No ontology can think its operation.” In a sense this is true. Transfinite set theory weaves the inconsistency of multiplicities from the void, such that the supplement of the void with respect to the presentation of multiples is “neither a presence nor an absence.” Mathematics, which is ontology, need not think this operation; it need only perform it, and insofar as it thinks the philosophical consequences of mathematical ontology, philosophy is not ontology. Ontology precisely cannot think the excess of what is unpresented (errancy of the void) to its presentation; philosophy can think this—above all by thinking the event, which is not being—but, on its own, it cannot think being-qua-being per se. Thus, when Derrida argues that différance is “irreducible to any ontological appropriation,” we may note that, for Badiou, the most radical consequences of the relation between philosophy and mathematics lie precisely in a différance which is indeed irreducible to ontology: in the movement between mathematics and philosophy themselves, each irreducible to the other insofar as one is a discourse upon being and the other a discourse upon the event and the subject, each producing or relying upon a supplement outside its purview. 

If we return, however, to the other aspect of Badiou’s acknowledgement of Derrida, concerning “the interpretation of Heidegger,” we can specify the exact point on which we must delimit our reconstruction of Badiou’s obedience to “the ontological requisite.” Heidegger’s critique of substance ontology and the metaphysics of presence is grounded in his analysis of ekstatical temporalization, of the exteriority of time. The figures of the trace and of différance in Derrida stem from Heidegger’s thinking of temporality as ekstasis, and the criterion of dissemination—the ontological requisite of inconsistency—is inseparable from the ruptural movement of time. On the other hand, Badiou’s meta-ontological thinking of being as inconsistent multiplicity is strictly atemporal. Time, in Badiou, pertains to existence, not to being, to the relationality of worlds and to the post-evental fidelity of subjects, not to the pure multiple which subtends any context of historical or phenomenal presentation. Indeed, for Badiou, the power of post-Cantorian set theory is its capacity to subtract the theory of being from existential, historical, and phenomenal determinations, and thus to think being-qua-being in its distinction from existence. In this sense he registers the import of the ontological difference (the being of beings is not a being), but he does so in a manner quite refractory to Heidegger and Derrida.

But to understand exactly how time is related to the theory of the ontological difference in Heidegger, we must understand that his thinking of time, grasped as ekstatical temporalization, cannot be delimited by the parameters of the existential analytic. We have to confront the radicality of the formulation Heidegger arrives at in Chapter III of Division Two of Being and Time: “Temporality is the primordial ‘outside-of-itself’ in and for itself” [Zeitlichkeit is das ursprüngliche »Außer-sich« an und für sich selbst] (§65, 329). Note that this statement pertains not to what is the case for us, but to what is the case in and for itself. Heidegger is not a correlationist. Time is the being of beings—and not a being—because it must be thought as pure exteriority: “temporality ‘is’ not a being at all,” Heidegger insists. The constitutive exteriority of time never has the self-identical unity of a substance, but only the unity of a synthesis transpiring through the intersection of the not-yet and the already with the horizonal constitution of the present as the in-order-to. Heidegger shows that this ekstatical unity of temporality, “the unity of the outside of itself” is “the condition of the possibility that there can be a being that exists as its there” (§68, 350). That is: the modality of being called Dasein (existence) is possible on the basis of the exteriority of time; it is not the exteriority of time that is possible on the basis of Dasein. The priority of the existential analytic is methodological, not ontological. By asking how it is that we can be the kind of being that exists as its there, Heidegger proceeds toward the thinking of ekstatical temporalization. But in Division Two of Being and Time, this methodological directionality undergoes a transcendental displacement: what the existential analytic actually shows is that the pure exteriority of time is the condition of possibility for there to be an existential analytic at all (not the other way around). And this was of course the point: Heidegger does indeed achieve the thinking of being-qua-being, in and for itself, when he thinks temporality as the primordial outside of itself. He thus meets the condition of the ontological difference and establishes what Badiou will call “the ontological requisite”: being cannot be thought as presence or substance, cannot be thought as one, cannot be thought as a being. Rather, it must be thought as time, grasped as ekstasis itself. Badiou sustains the thinking of being as inconsistency, as dissemination, but he rejects on the temporal determination of being-qua-being established by Heidegger—perhaps because he misrecognizes it as delimited by the phenomenality of the for us (as do most readers).

It is Quentin Meillassoux was has carried forward the consequences of both Badiou’s and Heidegger’s approaches to ontology. Meillassoux is Badiou’s apprentice insofar as he thinks being as transfinite inconsistency: “One of the most forceful aspects of Being and Event,” he writes, “is the way in which it uses mathematics itself to produce a liberation from the effects of calculatory reason, a gesture altogether more powerful than any external critique of calculation” (103). But he is Heidegger’s apprentice insofar as he thinks being as time.

In After Finitude, a theory of time is only briefly alluded to: the absolute contingency of being—the necessary contingency of all determinate beings—is described as a time “that would be capable of bringing forth or abolishing everything.” In his doctoral dissertation, Meillassoux conceptualizes such a time as “absolute becoming,” unsubordinated to the laws which structure the regularities of becoming in any given world. Here the philosophical question is quite precise: we understand that there are laws of becoming, but is there a becoming of laws? If so, this would necessarily belong to a time unregulated by the laws of becoming, since it would be a time in which such laws themselves come to be or pass away.

Indeed, this distinction between the laws of becoming and the becoming of laws implicitly constitutes a thinking of the ontological difference. A time exterior to the laws of becoming would be an ontological time rather than the ontic time structured by phenomenological or physical regularities. Beings conform to the laws of becoming, while the being of beings is conceived as the absolute contingency of a time in which the laws of becoming may come to be or pass away, and in which there cannot be a necessary being. Beings are contingent; their contingency is ontic. But the necessity of contingency itself is ontological: it is not a being, but the being of beings. (Thus, there is a sense in which it is nothing—a question Meillassoux takes up at length in his dissertation). Just as Heideggerian time, conceived as the “absolute ‘outside of itself’ in and for itself” cannot be conceived as a being, as a substance or as an identity, so absolute contingency, conceived as a time unregulated by the laws of becoming, cannot be conceived as a determinate being which comes into existence and passes away. I.e., such contingency is not itself contingent but rather necessary, and only such contingency is necessary.

We can thus conceive four orders of time: transcendental; phenomenal; cosmological; and absolute. Transcendental time determinations structure the field of experience given in phenomenal experience. Phenomenal time is the field of that experience itself. Cosmological time exceeds the existence of subjects of experience. And absolute time is exterior to, and punctuates, the regularities which govern the becoming of cosmological time. Thus we can see why absolute time—understood as being qua being—makes no sense from the perspective of transcendental philosophy, phenomenology, or physics: it cannot be experienced, it does not structure experience, and it does not obey the regularities of physical theory. It can only be deduced by a subtractive movement of thought. It is indeed a concept without an intuition, and therefore “empty.” But this apparent problem precisely specifies the encounter of thought with the ontological difference: that which is not a being, but the being of beings, can only be thought, yet this thought can have no determinate content according to the regularities of our experience. It is in this sense that ontology is ungrounded, and its determinations can only be forced by the history thought: by the unfolding of “the ontological requisite.” From the perspective of the existential analytic, Heidegger’s thinking of time as the primordial outside of itself in and for itself cannot be directly grounded; yet the traversal of the existential analytic can nevertheless show why such an ekstatical temporality must be the condition of possibility for the existential analytic itself, i.e. for a being to exist as its there.

The ontology of inconsistent multiplicity, elaborated by Being and Event following the requirement imposed by Derridean deconstruction and rejoined with the thinking of time by Meillassoux, returns us to the project of Being and Time from a perspective enriched by the subsequent history of philosophy. While it is post-Cantorian set theory that enables a rigorous exploration of ontological inconsistency, in my view it is Heidegger’s philosophy, above all, which informs us as to the why of inconsistency: the pure exteriority of time, riven between the already and the not-yet, destroys the being present not only of any finite entity, but also of any figure of the Whole. If there is an “errancy of the void” which agitates any consistency of numerical unity, which imposes inconsistency upon the operation of the count and which forces any infinite set to exceed its own closure, it is because time is the being of beings which is not a being, and which thus renders every determinate being exterior to self-identity. Even if Badiou’s philosophical theory of mathematical ontology expels temporality from its purview, in my view it implies its ekstatical force through the figure of errancy, which requires an explanation, in itself, beyond the disjunction between presentation and the unpresentable. The question is why the unpresentable is errant, why the void cannot consistently be structured into a one, and at the limit of the existential analytic Heidegger arrives at a theory of this errancy: the impossibility of thinking or presenting ekstatical temporalization through any presentation of the one. In my view, this is the ontological reserve, in excess of any determinate situation and in excess of any law of becoming, which may rupture the regularity of the given: that is, which holds in reserve the contingency of laws, and thus the eternal possibility that new laws may come into being. An adequate reading of Being and Time—informed by a reading of Being and Event and After Finitude—will show that Heidegger is far from a correlationist thinker. On the contrary, he is the most important thinker of absolute exteriority in the entire philosophical  tradition, and it is this aspect of his significance that I find illuminated by the development of speculative philosophy in the late twentieth and early twenty-first century.