Here is my exegesis of Ray Brassier’s compelling conjectural reversal that moves from thinking Capital to the bold conclusion that Capital thinks. In his text ‘Nihil Unbound: Remarks on Subtractive Ontology and Thinking Capital’ he grapples with how it might be possible to think Capital by providing an objective determination-in-the-last-instance. Beyond the Marxist analysis of the dynamics of accumulation, Brassier draws on recent philosophical insights in continental thought, mathematics and informational theory that appear eminently useful supplementing our conceptualization of Capital as a dynamic that refuses totalization. In order to achieve this he examines the subtractive ontology of Alain Badiou — turning his attention to precisely that which Badiou’s axiomatic set-theory elides — reconciling it with the profound diagnosis of capitalism as auto-axiomatic by Deleuze and Guattari. He then supplements this with the innovative informational theory and epistemological sophistication of mathematician Gregory Chaitin — namely in his work on uncomputable reals — in turn making Capital’s unthinkability and uncomputability intelligible.
THE VOID: In Nihil Unbound –– principally, in the chapter ‘Unbinding the Void’ in which he offers an appraisal of the philosophy of Alain Badiou –– Ray Brassier outlines the nihilistic consequences ‘incurred by science’s disenchantment of the world and capital’s desecration of the earth’ [NU, p.97]. Under such a determination the twin vectors of science and capital reveal unbound or pure multiplicity as the ‘veritable figure of being’: a rupturing of the traditional figure of the bond [MfP, pp.35-9]. This unbinding is made possible under the auspices of post-Cantorian mathematics and the discrete formal object-language of Frege and Russell, which masters the multiple by rendering lexical terms axiomatic. The axioms of set theory are defined compositionally –– rather than conceptually –– since ‘the multiple does not allow its being to prescribed from the standpoint of language alone’ [B&E, p. 40]. Or more precisely, we cannot count-as-one, or count as ‘set’, everything that is subsumable by a property, denying the coherency of any linguistic institution of a universal all-encompassing ontological situation. In short, Badiou denigrates the power of intuition to totalize its objects. Consequently, we can claim that the axiomatic presentation of inconsistent multiplicity annihilates the logical consistency of language and inaugurates the anti-phenomenological reign of the pure multiple (i.e. the void/null-set, the multiple of multiples, or groundless ground of what is presented). This subtractive discipline broadens the discursive range of philosophy, abjuring any previous idealist claims of auto-positional self-sufficiency and deposes the precarious configuration of Oneness. In short, Badiou’s axiomatic decision requires that philosophy be ‘expropriated of its conditions, [and] deprived of the appeal to intuition’ [AR, p.2] which accords him the ability to claim that the One is not, denying the existence of the Whole.
Further to this, Brassier makes some critical remarks with regards subtractive ontology and thinking capitalism in his contribution to Think Again: Alain Badiou and the Future of Philosophy. Here Brassier examines Badiou’s unbinding of the ontological Being of History ‘as a crypto-transcendental condition for thought’ [TA, p.51]; an unbinding of the myth of the teleological train of history climbing to its inexorable progressive destination. Instead, Badiou acknowledges the aleatory contingency of philosophy’s historicity with regard to the future. Like Marx and Engels, Badiou applauds Capital –– which operates by number and value-form –– as a desacrilizing vector, saluting the genuine ontological virtue of non-being, whilst denouncing capitalism’s complete barbarianism. This allows Brassier to cautiously remodel the Badiouan description of Capital as an ‘over-event’ of universal unbinding; a ‘quasi-Truth’ of Capital as condition of conditions, situated outside of thought [TA, p. 52]. This truth can only be ‘quasi’ since it is only randomly and gratuitously true. Moreover, as Badiou himself points out –– the banal tyranny of numbers, outside of thought and under the law of Capital, obscures the ontological virtue of numbers and delivers us to numerical slavery:
“In our situation, that of Capital, the reign of number is thus the reign of the unthought slavery of numericality itself. Number, which, so it is claimed, underlies everything of value, is in actual fact a proscription against any thinking of number itself. Number operates as that obscure point where the situation concentrates its law; obscure through its being at once sovereign and subtracted from all thought, and even from every investigation that orientates it towards some truth.” [N&N, p,213]
THE AXIOMATIC: Accordingly, Capital is declared the historical medium for subtractive ontology, which Brassier claims is the imperative consequence of being as void. Brassier then risks a ‘hazardous analogy’ by comparing Badiou’s universal unbinding with Deleuze and Guattari’s understanding of the axiomatic automation of Capital. Here the parallel is drawn between Capital –– an impersonal pathological system experienced as the absolute unbinding of the schizophrenic condition –– and Badiou’s excessive void set. Under Deleuze and Guattari’s diagnosis the immanent axiomatic of Capital is purely functional and impersonal in its operations, operating directly on decoded flows whilst remaining thoroughly indifferent to their intrinsic content; just as in Badiou the axioms of set theory are defined compositionally rather than conceptually. Within universal history the errant automation of capitalism is activated when, as Karl Marx identifies, we are confronted the abstract configurations of objectified labour and independent capital, crossing the threshold of decoding and deterritorialization, rendering empty every social bond. The axiomatic demonstrates an extraordinary adapative ability to add and subtract axioms, continually modulating and reconfiguring its parameters ‘within relative limits that are sufficiently wide’ [A-Oe, p. 139]. This unprecedented suppleness allows it to take contingent positions with regards to the unpredictable future, since the axiomatic is never saturated and always capable of appending a new axiom to preceding ones. Consequently, it is able to ward off limits to its expansion and is never ‘stymied by its incompleteness’ [TA, p.53]. This dynamism is therefore a non-totality or open system that rapaciously thrives off the discontinuous breach to the outside; or limitless immanence. It is here that the comparison between the ‘role played by cosmic schizophrenia as locus of absolute unbinding […] and that played by the excess of the void for Badiou’ [TA. p.53] seems most apt.
Deleuze goes further in providing an ontology for axiomatics with reference to competing mathematical tastes. Deleuze and Guattari make the explicit claim that their use of the term axiomatic is ‘far from a metaphor’ [TP, p. 455] and precisely stipulate that axiomatics is a ‘royal science’ synonymous with capitalism. This is something that French mathematician Nicolas Bourbaki has also noted, calling axiomatics ‘nothing but the “Taylor System” –– “the scientific management” of mathematics’ [TA, p.83]. Both the Taylor System and axiomatic set theory rise to prominence as foundational systems in the same historical moment. And here we discover a tension between the philosophical systems of Deleuze and Badiou, a political differend that arises from within mathematics itself. Yet, whereas Badiou favours the theorematic-axiomization of Cantor, Deleuze prefers the non-Cantorian problematics of differential calculus, with its emphasis on gradual quantitative transition, since philosophical novelty is engendered by the problem. He therefore designates problematics as a philosophical minor science which he then sets against the unifying royal mathematical tradition. Daniel W. Smith calls this the axiomatic-problematic distinction, a tension which is present in the very title of Capitalism and Schizophrenia. He further states that capitalism functions precisely on the basis of an axiomatic –– ‘not metaphorically, but literally’ –– since:
“…[Capital] as such is a problematic multiplicity: it can be converted into discrete differential qualities in our pay cheques and loose change, but in itself the monetary mass is a continuuous or intensive quality that increases and decreases without any agency controlling it. Like the continuum, the Capital is not masterable by an axiom; it constantly requires the creation of new axioms (‘it is like the power of the continuum, tied to the axiomatic but exceeding it’).” [TA, p. 91]
In short, Deleuze claims the ‘true’ axiomatic is social rather than scientific, since it generates new problematics (i.e. minorities or non-denumerable multiplicities) that are resistant to any axiomatic or discrete reduction.
GÖDEL, TURING & CHAITIN: Furthermore, Brassier is keen to cast suspicion on Badiou’s characterization of the State as the regulating force of re-presentation that configures and neutralizes the ‘inconsistent void underlying every presentation through apparatuses of ‘statist’ [etatique] regularization’ [TA, p. 54]. Brassier is wary that this constitutes a reductive description of the operations of Capital which cannot be explained simply through reference to the excess of the State’s regulating function. This requires him to explore the subjective political dimension of Badiou’s truth-procedures, the generic procedure that attempts to ascribe a fixed measure to the excess of the State, which in turn indexes the ‘unlocalizable excess of Capital’ [TA, p.54] in order to rationalize errancy. His suspicion is that the subjective truth-procedure –– which forces a determination in an unverifiable situation –– rests on a somewhat normative or common-sensical intuition of the random or aleatory. This affirmation of the decision on the undecidable within the parameters of aleatory rationalism is somewhat problematic, given Badiou’s founding disavowal of the power of intuition, and threatens a ‘relapse into superstitions of phenomenal voluntarism’ [TA, p.55]:
“[…] Badiou is curiously reliant on a suspiciously common-sense or intuitive notion of ‘chance’ or ‘randomness’. This suspicion is compounded by the eagerness with which Badiou wishes to dissociate the field of deductive fidelity concomitant with truth procedures from any ‘merely’ mechanical process of calculation. 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 inside mathematics itself.” [TA, p.55]
Badiou’s allegiance to chance as the substructure of the production of truth is therefore interrogated by Brassier by referencing the audacious post-Cantorian mathematics of real infinity. He turns to the renowned theoretical computer science of Alan Turing, in particular the halting problem, for a more rigourous account of randomness, proof and computability (also known as ‘effective calculability). Turing’s halting problem is principally a ‘decision’ problem with regards to deciding whether, as the result of a given input, a particular program will halt or continue interminably. Turing deepens Kurt Gödel’s incompleteness theorem –– under which no system for mathematics can be consistent, non-contradictory and complete (i.e. axiomatically provable) –– something that Badiou seems eminently familiar with. The halting problem as not Turing-computable was first proven by Turing in 1937 when he designed his machine to compute David Hilbert’s Entscheidungsproblem, or more precisely, show it was uncomputable. This problem is of historical importance since it was one of the first mathematical problems to be proved undecidable. Decidability requires there to be a method, definite procedure or test that can be applied to any given mathematical assertion, which will decide as to whether that assertion is provable. Consequently, Turing worsens Gödel’s incompleteness theorem, recasting it in terms of computers (axiomatic logic machines). Incompleteness means that there will always be problems that even a universal computing machine can never solve, so a Turing machine fed one of these problems would perhaps never stop (halt). And so Turing proved that there was no way of telling which these problems were satisfiable beforehand. This is the definition of uncomputability that proves there are ‘non-provable’ statements and determines the range and limits for axiom systems for mathematics. Therefore there is ‘no finite proof procedure whereby one can prove whether or not a given mathematical statement is provable’ [TA, p.55]. Brassier states that under such determinations Badiou must demonstrate that his truth-procedures are not based in a ‘pre-theorectical’ intuition of the spontaneous. Otherwise, he threatens to uphold the unsullied division between thought and calculation (i.e. non-algorithmic and algorithmic computation); which could similarly be designated as truth and knowledge. In short, Badiou is required ‘to show that truth procedures effectuate non-computable functions’ that are immeasurable [TA, p.56]:
“[…] if forcing delineates the cusp between the finitude of truth’s subjective act and the infinity of its generic procedure, 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.” [TA, p.56]
Brassier continues by analysing Gregory Chaitin’s recent contributions to the tradition of Gödel and Turing. Chaitin’s digital philosophy derives inspiration from Turing’s 1937 paper on computable numbers. His work is of epistemological and ontological importance in physics, biology and philosophy, determining that particular areas of mathematics are only quasi-empirical. Chaitin’s novel contribution is to demonstrate that not only do we have no way of predicting whether a given programme will loop infinitely, he shows that the probability of a program’s halting is absolutely random. In other words, there is no axiom capable of predicting it –– since every bit of information in a binary system is disconnected from the last, be it 1 or 0 (like the fair toss of a coin) –– and can only be iterated as ‘an incompressible string […] exhibiting no pattern or structure whatsoever’ [TA, p. 56]. Subsequently, Chaitin calls this strictly random probability Ω, a metacomputational constant that is irreducible; or in his terminology incompressible. Unlike the irrational number π, which can be expressed as a fraction and is therefore a computable real, Ω cannot be compressed in explanatory terms. Consequently, it would require an infinite amount of information to precisely measure Ω’s length and consequently it constitutes a strictly non-denumerable infinity. According to Chaitin, every explanation is a compression of the original problem that must by definition be simpler and more precise than that which it explains. In short, to explain is to compress and the best informational theory, program, explanation or logique du monde is the shortest (i.e. the smallest in bit length). In order to produce an elegant explanation a theory must be more concise than the unstructured data-set you began with. Accordingly, most infinitely recursive strings of bits (i.e. uncomputable reals) cannot be compressed or produced by a program smaller than they are, making them an un-totalizable and incompressible sequence that is completely unstructured. Moreover, Ω’s length jumps from finite to infinite from iteration to iteration. It produces what Ray Brassier nominates as an ‘unpredictable burst of objective randomness [or] indecipherable noise’ [TA, p.57] that can only constitute a quasi-empirical fact that is randomly and accidently true. Since Ω’s finitude or infinitude is non-deducible, every iteration of Ω (each bit, 1 or 0) can only be incorporated into the computable axiomatic order with the supplementation of further axioms. This allows us to detect an uncanny similarity to Deleuze and Guattari’s description of the pathological functioning of the capitalist axiomatic that thrives off incompleteness through the addition of new axioms. Here Brassier seeks to index the real through reference to Ω as an empirical and scientific determination-in-the-last-instance that provides an objective measure of the gap between Badiou’s ontology of the void and requirement of a subjective truth procedure in the face of the inconsistency. Furthermore, Brassier concludes that capitalism itself is ‘fuelled by random undecidabilities’, just as we find in the flip-flopping digital iterations of Ω –– the stochastic unpredictability and incremental price differences from which value is machinically extracted. It harnesses dysfunction and crisis by ‘axiomatizing empirical contingency’ [TA, p.57]. Here he alludes to the unthinkable and ‘unameable real’ which Badiou cannot acknowledge, surmising that Chaitin’s constant provides us with ‘an objective determination of the excess of the void as embodied by the errant automation of Capital.’ [TA, p.58] And perhaps more acutely, Brassier speculates that Capital in fact thinks through the very micro-processes that comprise it.
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