Brassier - Nihil Unbound - Remarks on Subtractive Ontology and Thinking Capitalism (Chap. 3 from Think Again - Alain Badiou and the Future of Philosophy)

3 NIHIL ON UNBOUND: SUBTRACTIVE AND THINKING REMARKS ONTOLOGY CAPITALISM Ray Brassier I C APITALISM AND T HE CONDITIONS OF PHIL OSOPHY As far as ‘nihilism’ is concerned, we recognize that our era pays witness to it insofar as nihilism is understood as the rupture of the traditional figure of the bond [lien], unbinding [dé-liaison] as the form of being of everything that assumes the aspect of the bond [. . .] [E]verything that is bound [relié] testifies that it is unbound in its being, that the reign of the multiple is the groundless ground of what is presented, without exception . . . (MP 37). Two basic yet apparently irreconcilable fidelities orientate Badiou’s thought. On the one hand, a fidelity to the Parmenidean axiom: ‘It is the same thing to think and to be.’ On the other, a fidelity to materialism.1 Against the phenomenological privileging of intentional Sinngebung (sensebestowal) and the concomitant denigration of the mathematical axiomatic as a sense-less combinatorial incapable of attaining to the dignity of ontological thought, Badiou invokes Parmenides to explain his decision to identify set theory with the science of being qua being: ‘In mathematics, being, thought, and consistency are one and the same thing.’2 Against the Deleuzian virtualization (and hence idealization) of multiplicity, Badiou reasserts the necessity of the materialist commitment to the unequivocal ‘univocity of the actual as pure multiple’.3 Yet there seems to be a conflict of fidelities here. For the Parmenidean identity of thinking and being, which underlies Badiou’s decision in favour of the set-theoretical identification of being as void, seems to assert the sovereignty of thought (a sovereignty underlined by Badiou’s endorsement of Mao’s ‘We shall know everything we did not know before’ [cf. TS 217]). How can this be reconciled with Badiou’s materialism, which would seem to require denying the ontological sovereignty of thought?

REMARKS ON SUBTRACTIVE ONTOLOGY 51 The answer is deceptively simple. It is the identification of being as void through axiomatic set theory that purges materialism of the methodological idealism whereby matter is reinscribed in a concept. By embracing a subtractive ontology, materialism requires only one name for being: that of the void or null-set, Ø. Being and thinking are ‘the same’ to the extent Badiou tends to define both of them subtractively – as void and truth respectively. Being is void as subtracted from presentation, while thought is truth as subtracted from knowledge.4 Thus, Badiou severs being from phenomenological plenitude in the same gesture whereby he dissociates thinking from the cognitive capacities of the human animal. But what are the conditions for this gesture? By what right does Badiou simultaneously subtract being from the element of phenomenological presencing and thought from the arena of intentional consciousness? Badiou’s materialism requires philosophical thought to be placed under extra-philosophical condition, that it be heteronomous rather than autonomous or causa sui, as it is for the idealist or phenomenologist. Thus, he maintains, a truly contemporary philosophy must operate under the condition of the disparate truths generated by post-Cantorian mathematics, poetry from Hölderlin to Celan, inventive politics from the Cultural Revolution to the contemporary struggles of illegal immigrant workers, and the Lacanian reconceptualization of the unconscious.5 But pre-eminent among these conditions is post-Cantorian set theory. Bearing in mind Badiou’s fundamental distinction between the order of knowledge (normative, verifiable) and that of truth (anomalous, unverifiable) – between legitimation and decision – it becomes apparent that the identification of axiomatic set theory with the long sought for ‘science of being qua being’ constitutes a decision, and hence an unverifiable subtraction from the order of knowledge. It affirms a fidelity to the Cantor-event. Thus, the evental condition for philosophy’s abdication from its ontological pretension is also that whereby it decides that being is nothing. Where ontology, in the shape of axiomatic set theory, sutures itself directly to the void of being,6 philosophy re-appropriates its rigorous systematicity by abjuring the claim to the kind of auto-positional self-sufficiency pursued in the systems of German Idealism (specifically those of Fichte, Schelling and Hegel). Philosophy becomes capable of re-assuming systematic consistency only insofar as it supervenes on the historial contingency of its evental conditions. It assumes its own groundlessness by deciding that it is another thought – Cantor’s – that has succeeded in suturing itself to the void of being, maintaining its independence from the norms of objective knowledge through the very gesture whereby it separates itself from the thinking of being qua being. In doing so, philosophy creates a space for the compossibilization of truths which have been subtracted from the realm of ontological consistency and the domain of objective verification. Badiou reconfigures philosophy’s claim to rational consistency by unbinding it from the myths of enlightenment and the superstition of teleological meta-narratives.7 For if, as Badiou insists, ‘History’ does not exist,8 then neither does ‘the History of Being’ as crypto-transcendental condition for thought. It is

52 THINK AGAIN precisely by acknowledging the aleatory contingency of its historicity, its evental conditioning, that philosophy frees itself from the myth of its uncircumventable historial destination, whether the latter be construed in terms of an ineluctable progress according to ‘the History of Spirit’, or that of an irrecusable decline according to ‘the History of Metaphysics’. Rare, fragmentary and discontinuous, historicity is constituted through those evental contingencies in which philosophy finds its occasioning conditions. Yet it is precisely on account of its constitutive historicity that the decision to identify being as nothing remains philosophical rather than mathematical. It is utterly foreign to Cantor’s mathematical heirs – just as it is foreign to the poetic practices, political procedures and psychoanalytical discourses that delineate the space of possibility within which this decision functions as coordinating vertex. But if it is underdetermined by its conditions, from where does this decision derive its imperative character? Badiou hints at an answer of sorts: This is obviously the only thing that can and must be saluted in capital: it exposes the pure multiple as the ground of presentation, it denounces every effect of Oneness as a merely precarious configuration, it deposes those symbolic representations in which the bond found a semblance of being. That this deposition operates according to the most complete barbarism should not distract us from its genuinely ontological virtue. To what do we owe our deliverance from the myths of Presence, from the guarantee it provided for the substantiality of bonds and the perenniality of essential relations, if not to the errant automation of capital? (MP 37). My contention here is that the condition whereby philosophy embraces the necessity of a subtractive ontology (and simultaneously abjures its ontological pretension) is provided by a quasi-condition that is transversal to the four regimes of truth acknowledged by Badiou: capitalism as ‘over-event’ of universal unbinding. There is something like a ‘quasi-truth’ of Capital as condition for conditions, rendering the philosophical identification of being as void not merely possible but imperative. II CAPITAL ISM, UNIVERSAL UNBINDING AN D THE VOID For Marx, as for us, desacralization is in no way nihilist if by nihilism one means that which declares that access to being and truth is impossible. On the contrary, desacralization is a necessary condition in order for the latter to become accessible to thought [. . . The] residues of the empire of the One, because they obstruct truth procedures and designate the recurrent obstacle to the subtractive ontology for which capitalism is the historical medium, constitute an anti-‘nihilist’ nihilism . . .9 Capital, the ‘historical medium’ for subtractive ontology, unbinds nihil from the fetters of Presence, pulverizing the domain of phenomenological senseful-

REMARKS ON SUBTRACTIVE ONTOLOGY 53 ness and exposing the insignificant neutrality of the multiple as ground of presentation.10 The most powerful recent philosophical characterization of that unbinding is provided in Deleuze and Guattari’s Capitalism and Schizophrenia: As far as capitalism is concerned, we maintain at once that it has no external limit and that it has one: it has one in the shape of schizophrenia, which is to say, the absolute decoding of flows, but it operates only by pushing back and warding off that limit. It also has and doesn’t have internal limits: it has them in the specific conditions of capitalist production and circulation, which is to say, in capital as such, but it operates only by reproducing and widening those limits at an ever larger scale. The power of capitalism resides in the fact that its axiomatic is never saturated, that it is always capable of adding a new axiom to the preceding ones.11 Integrated global capitalism is a machine – and a machine is nothing other than an automated axiomatic system – but an astonishingly supple and adaptive one, singularized by its fluidity, its metamorphic plasticity. Whenever confronted by a limit or anomaly, capitalism has the wherewithal – the intelligence? – to invent a new axiom in order to incorporate the unexpected, constantly reconfiguring its parameters by adding a supplementary axiom through which it can continue expanding its own frontiers. Far from being stymied by its incompleteness, the capitalist axiomatic lives off it. Far from being threatened by its ‘contradictions’, capitalism thrives on them. It is an open system, an aleatory axiomatic, continually redefining its own structural boundaries, perpetually living off its own impossible limit. Let us, for the sake of argument, risk a hazardous analogy between the role played by cosmic schizophrenia as locus of absolute unbinding (or deterritorialization) for Deleuze and Guattari and that played by the excess of the void for Badiou.12 According to Deleuze and Guattari, what renders the capitalist socius unique and unprecedented among all other social formations is the fact that it reveals the common ontological source of social and desiring production even as it perpetuates itself by continuously warding off the threat of their convergence. Similarly, for Badiou, the void, the ‘material’ from which every consistent presentation is woven, is included in every multiple presentation, while the threat posed by its errant inconsistency is foreclosed to presentation through the re-presentation that neutralizes and configures its excess.13 If capitalism is the name for that curiously pathological social formation in which ‘everything that is bound [relié] testifies that it is unbound in its being, that the reign of the multiple is the groundless ground of what is presented, without exception’, it is because it liquidates everything substantial through the law of universal exchangeability, simultaneously exposing and staving off the inconsistent void underlying every consistent presentation through apparatuses of ‘statist’ [étatique] regularization. ‘Capital’ names what Deleuze and Guattari call the monstrous ‘Thing’, the cancerous, anti-social anomaly, the catastrophic overevent through which the inconsistent void underlying every consistent pre-

54 THINK AGAIN sentation becomes unbound and the ontological fabric from which every social bond is woven is exposed as constitutively empty. Thus, although capitalism invests the operations of the state, it seems to me that contrary to what Badiou generally suggests, its effects cannot be summarily reduced to those of the state. The errant automation of Capital will not be explained by referring it to the excess of the state. Although the political truth procedure assigns a fixed measure to the excess of the state, the extent to which it thereby measures the unlocalizable excess of Capital is questionable.14 In fact, I would like to suggest that the void’s excessive or undecidable inconsistency finds objective determination in the errant automation of Capital as well as subjective measure in the political truth procedure. In order to do this, let us first examine what Badiou means by ‘truth procedure’. Badiou distinguishes between the infinite but indeterminate cardinality of the state and the infinite but determinate cardinality whereby the political truth procedure measures the excess of the state (cf. AM 162). That measure or determination of an indeterminate infinity is effected through forcing, a procedure that ‘constrains the correctness of statements according to a condition that anticipates the composition of an infinite generic subset’.15 Forcing describes the process whereby a truth procedure hazards assertions on the basis of the supposition that, although unverifiable within the situation as it stands, they will prove verifiable according to an extension of this situation that can and will exist even though it does not exist as yet. Through forcing, the knowledge that constrains the possibilities of thought within an actual situation is supplemented and those possibilities reconfigured by the situation’s generic extension, which is brought about by statements made according to a condition anticipating the existence of the elements that will legitimate them. Thus, while the elements of a generic sub-set cannot be named – since the latter is incomplete on account of its infinity and indiscernible because its components cannot be enumerated by means of predicative definition – the generic extension can be brought into being according to a process whereby statements are made about these indiscernible elements according to the hypothesis that if this or that element existed in the putatively complete generic sub-set, then this or that statement about this or that element would be correct. Every truth procedure supported by a subject has two crucial characteristics according to Badiou: 1. It is random or aleatory. Chance provides the aleatory substance of subjectivation because the subject of the truth procedure forces the generic extension through a series of entirely random choices; distinguishing x from y without recourse to a principle or concept by which to differentiate x from y.16 2. It is interminable because the generic sub-set is infinite. But since the subject proceeds via a series of finite discriminatory steps (a or b, c or d, etc.), it is an infinity woven from finite series of discrete sequences. This interminability delineates the generic infinite’s composition out of the finite, and hence its immanence to finite situations.17 Thus, Badiou writes

REMARKS ON SUBTRACTIVE ONTOLOGY 55 of truth that ‘between the finitude of its [subjective] act and the infinity of its [generic] being, there is no measure’ (C 192). Now, as far as the first point is concerned, it seems to me that Badiou is curiously reliant on a suspiciously commonsense or intuitive notion of ‘chance’ or ‘randomness’. This suspicion is compounded by the eagerness with which Badiou wishes to dissociate the deductive fidelity concomitant with truth procedures from any ‘merely’ mechanical process of calculation.18 Yet it is precisely this venerable distinction between thinking and calculating – often a cipher for the familiar philosophical opposition between subjective freedom and objective necessity – which Alan Turing subverted from within mathematics itself. Turing showed how any deductive procedure could be defined in terms of recursive functions, algorithmically generated, and therefore automated as a computable function.19 And this automation of computable functions is entirely compatible with the straightforwardly intuitive characterization of ‘chance’ Badiou seems to invoke in his account of the deductive process that constitutes truth. For algorithms excel at unprincipled distinctions wherein nothing intrinsic to the terms themselves plays a role in effecting the discrimination. One merely has to specify a condition, any condition: if the sky is green, then rabbits are bigger than elephants. To believe that a discriminatory procedure, no matter how ‘arbitrary’, is truly ‘spontaneous’ and non-mechanical in anything but the most superficial of senses, is to relapse into the superstitions of phenomenological voluntarism. Turing demonstrated the identity of proof and computation,20 circumscribing the realm of proof through that of computable functions. Now, it is obvious that there is supposed to be a fundamental distinction between the deductive process involved in proof and that involved in what Badiou calls truth. But Turing did more than merely demonstrate the possibility of automating proof procedures. He used a technique similar to Gödel’s to delimit a realm of non-computable functions, and thereby a realm of non-provable mathematical statements. He constructed a function that could be given a finite description but that could not be computed by finite means in order to show how even a ‘universal computing machine’ capable of duplicating the operations of any possible computer could not compute in advance whether or not a given program would carry out its task within a finite length of time or carry on indefinitely. This non-computable function is known as the ‘halting function’: given the number of a computing machine and the number of an input tape, this function returns either the value 0 or the value 1 depending on whether the computation will ever come to a halt. Through the halting function, Turing showed that there exists no finite proof procedure whereby one can prove whether or not a given mathematical statement is provable. Thus, in order to show that the distinctions fuelling the truth procedure are aleatory and unverifiable in a sense that does not depend on some pre-theoretical notion of spontaneity, and in order to sustain his distinction between thought and calculation – as well as his even more fundamental distinction between

56 THINK AGAIN truth and knowledge – Badiou needs to show that truth procedures effectuate non-computable functions. Moreover, as far as the second point is concerned, if forcing delineates the cusp between the finitude of truth’s subjective act and the infinity of its generic being, it does not necessarily follow from this either that it must index some putatively unquantifiable upsurge of subjective freedom or that this cusp must be ‘without measure’. And as it happens, following in the footsteps of Gödel and Turing, Gregory Chaitin21 has recently elaborated an extremely interesting metamathematical characterization of randomness in a way which – at least in my admittedly non-expert opinion – seems both to undermine the distinction between the subjective measure of excess and its objective calculation, and to determine the abyss between the finitude of truth’s subjective act and the infinity of its generic being. III CHAIT IN: ALGORITHMIC RANDOMNESS What follows is no more than a clumsy philosophical sketch of Chaitin’s work. The latter finds its initial impetus at the intersection of Gödel’s incompleteness theorem and Turing’s ‘halting function’. In a move that clearly exhibits the latent affinity between these metamathematical concerns and those of transcendental epistemology, Chaitin provides a metacomputational specification of the uncomputable by determining the halting probability for a universal Turing machine as a real number lying somewhere between 0 and 1. And just as there can be no computable function determining whether or not a program will halt, there can be no computable function determining the digits of this halting probability, which Chaitin, with admirable dramatic flair, names . Unlike
, which can be compressed as a ratio and whose digits can be generated through a program shorter than the bit string it generates, is strictly uncomputable. This means that its shortest program-length description22 is as long as itself, which is infinitely long and consists of a random, i.e. incompressible string of 0s and 1s exhibiting no pattern or structure whatsoever: each digit is as unrelated to its predecessor as one toss of a coin is from the next. Not satisfied with having demonstrated ’s theoretical possibility in the abstract realm of universal Turing machines, Chaitin set about proving ’s actuality in the very concrete domain of elementary number theory, the cornerstone of pure mathematics. Other mathematicians23 had already shown how to translate the operations of Turing’s universal computer into a Diophantine equation – an equation involving only the addition, multiplication and exponentiation of whole numbers – thereby establishing a correlation between the interminability of a given program and the insolubility of a given algebraic equation. Following their lead, Chaitin saw how he could construct a link between the halting probability and number theory by encoding ’s bits in such an equation. But rather than trying to determine specific whole number solutions for his equation, Chaitin set about determining whether or not there was a finite or infinite number of them as a function of ’s first N digits.24 Chaitin’s equation is 200 pages long, with 20,000 variables, X1 to X20000, and

REMARKS ON SUBTRACTIVE ONTOLOGY 57 a single parameter, N.25 It has finitely or infinitely many natural number solutions depending on whether the Nth bit of is respectively a 0 or a 1. Since the number of solutions to the equation jumps from finite to infinite in a completely arbitrary fashion as a function of N, Chaitin argues that ‘determining whether this equation has finitely or infinitely many solutions is just as difficult as determining the bits of Omega’.26 Moreover, each of ’s bits, each number of solutions, constitutes an irreducible, separate mathematical fact; one that cannot be deduced unless it is added as an axiom – it takes N bits of axiom to prove N bits of . Since a formal axiomatic system amounts to a computation in the limits of infinite time, ’s algorithmic incompressibility shows that although the set of theorems implied by an axiomatic system can be algorithmically generated (in some arbitrary order), no algorithm can determine whether or not a given theorem belongs to that set. Thus, Chaitin concludes, incompleteness is far more than a marginal, metamathematical anomaly. It is a central, possibly even ubiquitous mathematical predicament. There are non-deducible, un-provable mathematical truths everywhere, quasi-empirical ‘facts’ that are gratuitously or randomly true and that can only be integrated by being converted into supplementary axioms. Isn’t there a case then for maintaining that indexes the ‘not-all-ness’ (pastout), the constitutive incompleteness whereby the Real punctures the consistency of the symbolic order, at least as much as the excess of the void does for Badiou? But that it does so as a mercilessly unpredictable burst of objective randomness – undecipherable noise – rather than as a ‘grace accorded to us’, a liberating upsurge of subjective freedom?27 Doesn’t the evental excess which, for Badiou, indexes the inconsistency of the Real and necessitates the uncomputable freedom of axiomatic decision,28 find embodiment in ’s objective randomness at least as legitimately as in subjective intervention? Moreover, doesn’t the pathological peculiarity of the capitalist machine consist in its ability to do just this: convert random empirical facts into new axioms? Integrated global capitalism is constitutively dysfunctional: it works by breaking down. It is fuelled by the random undecidabilities, excessive inconsistencies, aleatory interruptions, which it continuously reappropriates, axiomatizing empirical contingency. It turns catastrophe into a resource, ruin into opportunity, harnessing the uncomputable. IV THE REAL AS AUTOMAT ED RANDOMNESS OR THINKING CAPIT AL There cannot be two or more unnameables for a singular truth. The Lacanian maxim ‘there is Oneness’ is tied here to the irreducibility of the real, to what could be called the ‘grain of real’ jamming the machinery of truth, which in its power is the machinery of forcings, and hence the machinery for producing finite veridicalities at the point of a truth that cannot be accomplished. Here, the jamming effected by the One-real is opposed to the path opened up by veridicality (C 209).

58 THINK AGAIN Here is my conjecture: ’s incompressible algorithmic randomness indexes the uncomputability of the real, which Capital’s errant automation at once exposes and wards off, unleashes and regularizes. But if Capital functions as the real condition through which philosophy simultaneously identifies the void of being, abjures its ontological pretensions, and becomes the harbinger of truths, might its automated randomness not also function as that unnameable Thing which Badiou’s philosophy cannot acknowledge: the unthinkable determinant for its own identification of being as void? Might – the real as inconsistent randomness – not also provide an entirely objective determination of the excess of the void as embodied in the errant automation of Capital? What if the void’s undecidable excess found embodiment in the objective randomness of Capital’s non-computable dysfunctions, as opposed to the subjective ‘freedom’ of evental decision? What if furnished a real, objective measure of what Badiou describes as the ‘abyss’ between the finitude of truth’s subjective forcing and the infinitude of its generic being? Perhaps the condition for Badiou’s subtractive ontology is a thought of Capital, or more precisely, an acknowledgment that capitalism – blind, monstrous, acephalic polymorph – thinks. What if it were precisely the thought that this Thing thinks that was still unthinkable for this philosophy?