Thinking Nature v. 1 /2/ - Being and Counting: Speculative Materialism and the Threshold of the Given David Lindsay In After Finitude, Quentin Meillassoux uses the axioms of set theory to great effect to establish the excess of the universe over the necessary stability of phenomena, yet he seems to neglect them entirely when he engages the riddle of time prior to thought. I try in this essay to introduce a level of uniformity to his argument by applying the axiom of foundation to the problem of ancestrality. In doing so, I am able to submit this concept to a critical interrogation from within a mathematical discourse until it yields implications similar those Meillassoux himself has recently developed – again by other means – regarding a “sign without significance.” Finally, without contravening on those preliminary findings, I continue my analysis, still drawing on the same axiomatics, to advance a possible formal basis for demonstrating being – and therefore nature – beyond thought. The argument that Quentin Meillassoux develops in After Finitude1 is a bracing one. Against correlationism – his term for the general restriction on objectivity independent of the subject’s intervention – he advances a direct counterclaim: there is a thing in itself, and we can know something about it without adding the codicil of its correlation “for us.” Given its wide scope and rigorous argumentation, Meillassoux’s thesis bears close investigation for its implications for much of contemporary philosophy, especially where it engages questions regarding the reach of mathematics and the possibility of access to the natural world. In its dense chain of argumentation, AF presents the correlationist with two major challenges. The first of these is to explain scientific evidence pre-dating all human endeavor without sacrificing the correlation between subject and object. The second is to maintain this same correlation without relying on an implied absolute. Both of these challenges proceed from a distinction between a time devoid of humanity and what could be titled the time of the given. Meillassoux’s methods differ when developing these two arguments however, and by his own admission, their reconciliation remains to be achieved at the close of AF. To that end, my approach will be to apply the most decisive method of argumentation in favor of the absolute to the question of givenness. That is, I will consider the divide between given and pre-given as a problem for set theory. In AF, Meillassoux defines the pre-given, which he also calls the ancestral, and the attending concept of the arche-fossil appear very early in his argument: I will call “ancestral” any reality anterior to the emergence of the human species – or even anterior to every recognized form of life on earth. I will call “arche-fossil” or “fossil matter” not just materials indicating traces of past life, according to the familiar sense of the term “fossil,” but materials indicating the existence of an ancestral reality or event; one that is anterior to terrestrial life. An arche-fossil thus designates the material support on the basis of which the experiments that yield estimates of ancestral phenomena proceed – for example, an isotope whose rate of radioactive decay we know, or the Meillassoux, Q. (2008) After Finitude: An Essay on the Necessity of Contingency (London: Continuum). Hereafter referred to as AF. 1
luminous emission of a star that informs us as to the date of its formation.2 Everything here is designed to maximize the distance between the ancestral and emergence of the human species. Ancestrality is positioned as more remote than anteriority, which can apply to recent and distant past alike. Similarly, the arche-fossil pre-dates human life – or even all terrestrial life. (We will return to the ambiguity imposed by his use of the connective “or.”) Throughout AF, Meillassoux favors this distal formulation, often referring specifically to the example of radioactive decay, which measures magnitudes of time far greater than that of the duration of the human species. The purpose of drawing the contrast is clear enough: He wishes to underscore the existence of facticity prior to thought, the better to refute every correlationist claim of an undecidable priority between them. His approach does not rest entirely on a temporal remove, however. Also very close to the outset, he writes: [W]e shall therefore maintain the following: all those aspects of the object that can be formulated in mathematical terms can be meaningfully conceived as properties of the object in itself. All those aspects of the object that can give rise to a mathematical thought (to a formula or to digitalization) rather than to a perception or sensation can be meaningfully turned into properties of the thing not only as it is with me, but also as it is without me.3 This claim extends his argument over a wider domain, rather than limiting it to scientific evidence from eons gone by. As a result, Meillassoux leaves open the possibility of a determinable meeting point between the ancestral and the given. Our first task, then, will be to establish a clearer idea of what is meant by givenness in the context of speculative materialism, and by extension to approach with greater confidence the matter of its first appearance in the pre-given universe. As a point of entry, we might gloss the concept as that of the phenomenal: the given is anything one is conscious of. Kant's goal, famously, was to set apart the thing-in-itself as not given. If we are concerned with the advent of consciousness in the history of the cosmos, however, we will need – against Kant – to include within givenness some terms that would normally be considered a priori, particularly the intuition of time. After all, if this intuition began and is characteristic of consciousness, then its occurrence will be “given” to an entity in the same sense that consciousness itself is given, even if we rule out any agent of donation. What are we able to glean from AF in this regard? To position the ancestral prior to all forms of life (as Meillassoux does, albeit tentatively) and the given somewhere in the midst of biological evolution seems heuristically correct, but leaves the question of the actual dividing line unexamined. When Meillassoux does address the limit of givenness directly, it is to introduce it as an aporia rather than to offer an explanation. How, he asks, are we "to think a world wherein spatio-temporal givenness itself came into being within a time and a space which preceded every variety of givenness?"4 His main strategy in this passage is to disqualify the transcendental subject, thereby setting the stage for the conclusion that consciousness arose for no reason, simply because it could. I will maintain the same conclusion regarding the emergence of consciousness. In developing his case, however, Meillassoux AF, p. 10 (italics in original). AF, p.3 (italics in original). 4 AF p. 22, and pp. 22-24 passim. 2 3
fails to explain what the phrase “every variety of givenness” may encompass, so I will enter the fray at this point with an assertion: The advent of mathematical thought in the universe and the threshold of givenness are one and the same. The statement has the ring of the arbitrary, no doubt. On the other hand, if we are trying to identify the status of any meaningful conceptions for mathematical discourse, which Meillassoux ultimately privileges above language, then the advent of mathematical thought will be a necessary condition for that discourse, so long as we maintain the fact of ancestrality. The proviso is crucial: If we did not affirm ancestrality, we would not need to identify a beginning for mathematical thought, because we would cease to insist on anything existing before it. With this assertion in hand, we might now ask how mathematics approaches the concept of beginnings, and, in turn, attempt a more critical question: Can mathematical thought describe it own advent? It would seem that an affirmative answer to the latter will also be necessary for any mathematical discourse regarding events prior to its advent, because if this starting point cannot be established, then it will not be possible to articulate that which is prior to that advent on its own terms. And of course, it is precisely the proclaimed coherence of any holding forth on anteriority in mathematical terms that supports the project of speculative materialism writ large. How then to begin? The branch of mathematics that has interrogated the problem of the starting point with perhaps the greatest vigor is that of set theory, which Meillassoux himself grants pride of place – at least in the form introduced into philosophy by Alain Badiou in Being and Event.5 When launching his attack on Kant’s objective deduction, Meillassoux turns in particular to Cantor’s theorem, which allows him to explain the apparent stability of objects through the excess of the universe over every probability. It is true that Meillassoux does not entirely agree with the conclusions that Badiou draws from set theory. It is also true that both lay marked emphasis on the implications of transfinite sets. This essay will concern itself with neither the overt disagreements nor the shared preoccupations of these two philosophers. Granting the remarkable contributions both have made in rendering the infinite thinkable, we may note simply that Meillassoux employs the same set of axioms in establishing his case for factuality in AF as does Badiou for the event in BE. We will therefore be within the bounds of the speculative materialist’s argument if we direct these same axioms, as laid out in great detail in Badiou’s magnum opus, toward the problem of givenness. 6 Especially useful in this regard will be the axiom of foundation, which addresses not the limitlessness of all things but rather he very issue of beginnings. 5
Badiou, A. (2006) Being and Event, tr. O. Feltham (London: Continuum), p. 43. Hereafter referred to as BE. Badiou favors the variant of the ZermeloFrankel axioms known as ZFC, because they include the sometimes controversial axiom of choice. Although he does not explicitly announce this decision in AF< Meillassoux also allows the axiom of choice insofar as he employs Cantor’s theorem, which cannot be validated unless this particular axiom is affirmed. 6 In AF, Meillassoux acknowledges Badiou as providing the “inaugural” gesture of introducing set theory into philosophy and cites BE, but not its sequel, Logics of Worlds. (One imagines that AF and Logics of Worlds were simultaneously in press.) For his own part, Badiou praises Meillassoux as an unpublished writer in Logics of Worlds without any reference to his views on set theory per se. Given these lines of influence, I will draw only upon BE for my explication of set theory.
To summarize as mercifully as possible: The axiom of foundation was introduced into set theory in order to prevent any set from belonging to itself, an outcome that would jettison its efforts into absurdity. In its classical form, this axiom states that every non-empty set A contains an element B that is disjoint from A. An equivalent statement reads: For any non-empty set A there is some B belonging to A, where the intersection of A and B is nothing, or, as formally written: A∩B = ∅. If one is dealing with ordinals, their set will therefore be founded on that which is entirely other to ordinals – in other words, the empty set. (On scrutiny, the logic of this will be clear enough: the intersection between a number and “no number” will always be “no number.”) The attending result is that the axiom of foundation allows one to express a minimal point of origin or “downward halting point” for any set, past which one reaches disjunction.7 Many mathematical enterprises do not need to consider the problem of foundation, because they do not need to commit to a first instance of number. Badiou is in a different position, because his scope extends to the existence of numbers as such, and so he must locate what he calls “an absolutely initial point of being.”8 Speculative materialism requires foundation as well, because it has to assert the beginning of mathematical thought in a factical time, on an axis along which other events can be established as anterior to thought. In this much, then, Badiou and Meillassoux agree: Their mathematics must be founded. Because Badiou – and by implication Meillassoux – cannot rely on existing sets for his ontology, he articulates foundation (or “the suture to being”) around the count-as-one, which has the effect of having counted nothing. Here it is necessary again to dispatch some fairly arcane mathematics. The count-as-one, which can be written ∅ –> (∅), is always the retroactive result of having counted what Badiou calls the inconsistent or pure multiple, which is neither one (or it would already be counted) nor many (or we would be able to answer the question “how many?”) but rather the unstructured condition for presentation as such. The act of counting this inconsistent multiple generates a set with the single element of nothing, or (∅). At this point, another important axiom comes into play: the axiom of subsets, also called the power-set axiom, which establishes that the parts of a set can be rendered as sets of their own.9 Because (∅) is made up of two parts – (∅) and its only element ∅ – the count-as-one delivers ∅(∅), which can in turn be counted as well, through a persistence of the count that Badiou calls forming-asone, thus generating ∅(∅(∅)); ∅(∅(∅(∅)))... and onward through the series of natural whole numbers. Badiou resolves the remainder of the inconsistent multiple – the “uncounted nothing” that initiates the series – by declaring that it must be absolutely excluded from countable sets (which he terms consistent multiples), and thereby offers the inconsistent multiple that is never counted as the disjunct foundation of consistent multiples.10 7
BE, p. 500. BE, p. 48. 9 The axiom of subsets, also called the axiom of the power set, states that “[T]here exists a set whose elements are subsets or parts of a given set. This set, if a is given, is written p(a).” BE, p. 501. 10 BE, pp. 5259, 8692. I would be remiss if I did not mention that Badiou proposes a “count of the count” that enacts a movement from presentation to representation, and so to the state of a situation. However, it is difficult to see how the count of the count is not simply another instance of forming as one, which already allows him to “layer” distances from the void. Why double the count? Why not quintuple? See BE p. 94. 8
Badiou’s ontology would seem to be a rather abstract procedure for founding mathematical thought. How, after all, are we to locate the onset of numeracy through the investigation of numbers themselves? Fortunately, Meillassoux's premise comes to the rescue at this point: It is his stipulation that consciousness has a downward halting point – in time. As we are exploring the consequences of this premise, it is not a question of determining if mathematical thought is founded but rather how. And it is at this radically reductive point that we find our opening. If we suppose that mathematical thought began in the same manner as the series of natural whole numbers is derived, we would say that mathematical thought is founded, as in Badiou's ontology, on the inconsistent multiple. In so saying, however, we will have omitted any distinct value for thought. This will not do, because it destroys any opportunity for givenness. It allows for mathematics to enter into being but no way for mathematics to enter into thought. If, on the other hand, we founded mathematical thought on an original ability to think nothing, we would no longer be singling out nothing, which is the basis for mathematical fact, but rather that which forms sets. What would such an isolation entail? If we pose mathematical thought as that which forms sets, we will have to articulate its foundation as disjunct from everything we understand as arithmetic, algebra, geometry and so on. This it turns out is possible, at least in hypothetical form. The articulation that meets this demand arrives under the title of the operation of the count.11 The operation of the count is described in Badiou's ontology as the act of establishing nothing as an empty set. Whenever the count-as-one obtains, the operation of the count upon a presumed nothing (the inconsistent multiple) produces “one.” This operation cannot itself be grouped as a set any more than the presumed nothing can. Indeed, the case is clearer here, because grouping is what the operation does. By asserting “one,” the action of assertion has escaped the formulation. As such, it takes on neither a single point of view, nor one with an inexhaustible purview of the world. In other words, the operation of the count explicitly never becomes a transcendental subject. Yet neither is it the same as the inconsistent multiple that precedes the count. Insofar as we grant that number proceeds from the operation of the count upon nothing, we would say that the operation is the unpresented, inconsistent verb of enumeration. My hypothesis, then, is that the operation of the count founds the temporal advent of mathematical thought as such. There are appealing reasons to make this assumption. By definition, the operation has no intersection with ordinals, yet the most elementary count cannot commence without it. Moreover, it is not identical to the inconsistent multiple (according to our assertion) and so permits speculation on its advent after the dawn of the universe. We are, in other words, able through this hypothesis to maintain the possibility of ancestrality within the axioms of set theory. Whatever the unorthodoxy of this move for Badiou or the exemplary set theorist, we have to risk it, because it retains the indispensable outlines of Meillassoux’s case for the ancestral. We can see fairly quickly, however, that something is wrong here. Although we have allowed that mathematical thought emerged in time, we have in no way secured temporal finitude for its foundation, because, as we have defined it, the foundation of thought eludes every presentation. The enumerating of a minimal time – the first “before” – is a mathematical operation that cannot be performed without having counted that operation. In other words, the first iteration of the count to be 11
BE, p. 55.
presented will follow no rule of passage “back” to the unpresentable operation that founds it. Proceeding by the axioms renders the same result. One can employ the axiom of subsets (as was done in taking ∅ and (∅) as separate parts after the count-as-one) and, from the count-as-one forward, generate a formula to describe the time of the accretion of the earth. Such a sequence could be demonstrated without too much difficulty, given patience and a sinecure. Unfortunately, the operation of the count cannot be enlisted to separate the instant of accretion from the instant that the formula was constructed, because the operation remains alien to every measurable interval. This ontic structure of set formation will pose a serious difficulty for any formula or digitalization derived from the study of an arche-fossil, simply because any formulation of temporal succession, which the concept of ancestrality manifestly requires, will fail to guarantee that the archefossil as described lies on the far side of givenness. It remains to be explained, in short, how access to the pre-given through mathematics is not itself relational, in the sense that it “spreads its fame” wherever it goes Nor is this the only hitch. If the succession of time is to be consistent, it will have to exhibit denumerability for every element of time that is counted. Yet as we have shown, in order to form a consistent thought regarding ancestrality (i.e., a mathematical formula or digitalization of an archefossil), it will be necessary to exclude at least one element – the unpresented operation of the count. However, it will also be necessary to include some element that stands for the occurrence of mathematical thought, insofar as the posteriority of some thought to something else will be necessary if one is to speak of ancestrality at all. Therefore, the operation of the count must, by default, be both included and excluded from any consistent set that describes the occurrence of mathematical thought distinct from being. Maintaining consistency in the face of this contradiction will initiate an unrequited effort that we may rightly call work. The endless tasks of exclusion and inclusion, which we might deem modes of work, bring us to the verge of historical traditions. On one hand, one can present a substitute foundation for mathematical thought as temporal proxy for the operation (in a kind of obtuse knowledge of its artifice). On the other, one can try to gain distance from every presumed foundation by putting a countable world in the way (in a type of hyperbolic banishment of the gap). In either case, a disjunct foundation is clearly implied, yet the onset of givenness cannot be stabilized around it. Or rather, where it is stabilized, it will have to be done arbitrarily, which is to politicize in advance the category of the human.12 We see now why Meillassoux places anteriority so far back in time, and also why he vacillates between strict ancestrality and the larger anteriority that subsumes it. Whenever ancestrality is sought close to givenness, thought will encounter the narcissism if its endeavor. Yet to maintain a vast span of time between a fact and its meaningful conception only redoubles the problem by assuming a consistency for the interim between the former and the latter. And if that is the case, on what basis can the arche-fossil be considered ancestral? It is not only that mathematical thought brings its own correlation to bear on phenomena. It is also that the access it provides does violence to its own methodology. One might, for example, suppose a species killing off its immediate quasirelatives in order to achieve the power of consistent formulaton. This is not far from Freud’s theory of the totemic taboo, purged of resentment as a primary motivation. 12
What Meillassoux calls the diachronic – the split in time between the given and the pre-given 13 – will sustain this problem so long as one admits both the evidence of the arche-fossil and the tenets of set theory. The very mathematical axioms that allow him to argue the necessity of contingency enact a brand of correlation that is at least as strong as the Kantian variety, if not stronger. Nevertheless, our analysis does not augur a return to gratuitous philosophies of foreclosure. If the reach of mathematics prevents the naïve realist from finalizing any objective distance from the ancestral (thereby problematizing any distinction for nature), it also prohibits the correlationist from defining the “us” to whom givenness is given (thus dissolving the borders of humanity). This correlation hardly invites the sort of complacency for which the term is coming to be associated. In a discursive mood, we might on the contrary hazard a foray into a “dark” negative theology. Mathematical thought entails a reflexive structure of dominion – the extension of number without impediment (whether accurate or not) – and an agency of that dominion without any structure at all. No doubt some of these difficulties can be readily resolved, as Meillassoux is correct to insist,14 by orienting the operation of the count toward the future, which is to admit that mathematical thought, in its hypothesizing and re-calculating, is actually happening, is actually adding its own facticity to the passage of time, and so helping to predict outcomes still in store. To render the count-as-one as imminent is, of course, a hallmark of messianism and modernism alike. Yet it would be the height of naivete to think that a postponed count-as-one is not drawn in some way from an accomplished countas-one – indeed, this in a nutshell is Meillassoux’s argument for the persistence of the absolute and constitutes the kernel for his position that the correlation must be radicalized. If the future is not to be decided in advance, the same must be the case for the past. In order to maintain the existence of the future, then, the speculative materialist will be compelled to “chase the limit” of ancestrality back to a minimum unpresented ancestrality at the foundation of being, which, as we have said, can be declared as the inconsistent multiple of nothing. There is, I believe, a case to be made along these lines. We have argued that the count-as-one establishes the structure of anteriority from which the inconsistent multiple is severed. We have also argued that both the operation of the count and the inconsistent multiple are never presented (except insofar as there is presentation). Yet we have also argued that the operation and the inconsistent multiple are not identical. Thus, we are still entitled to speculate on the possibility of a manifestation, however fleeting, of the independence of the inconsistent multiple from the operation of the count. It should come as no surprise if we seem to be drifting into the realm of contingency. Indeed, the inconsistent multiple that “will have been” prior to the count would seem to be precisely that which Meillassoux identifies as the factial – that which alone is not contingent in a universe otherwise populated by necessarily contingent entities.15 Bearing in mind that Meillassoux frames his argument for the factial in terms of posterity rather than origin,16 the following parallels present themselves. We find in both the terms of necessity, absoluteness and other-than-us. In each case, what is posed is an AF, p. 112. AF, p. 12. 15 AF, pp. 5460. 16 There is no structural constraint on rendering the series of whole natural numbers as [(∅) ∅, ((∅) ∅) ∅, (((∅) ∅) ∅) 13 14
∅…] rather than as [∅(∅), ∅(∅(∅)), ∅(∅(∅(∅)))…], thereby implying that the pure multiple haunts the future rather than the past.
inconsistent chaos void of time and reason from which differentiation arises. Most important perhaps, both the factial and the inconsistent multiple (and here is where the latter truly deserves the term of “pure multiple”) are without parts. It is beyond the present purview to declare Meillassoux’s twin arguments successfully joined in this observation. The wager of their identity does allow us, however, to consider the factial in terms of set theory, and to ask what it may mean to talk about an entity that, being without parts, does not lend itself to forming as one. Is it even possible to talk about such an entity? To recapitulate, gathering our disparate terms as we go: The inconsistent multiple is pre-given. The count-as-one is given. The reader will notice that the operation of the count is missing from these statements. For reasons that we have outlined above, it is clear that Meillassoux must retain a theoretical independence for the operation of the count in relation to the inconsistent multiple. This is where he parts ways with Badiou, who identifies the operation of the count as the inconsistent multiple that precedes the countas-one, and then treats the resulting “phantom of inconsistency” as an indiscernible to be forced locally from the greater architectures of history.17 Meillassoux certainly has the weight of science on his side. Yet by insisting on a time when facts existed during which the operation of the count was absent, he is still constrained to demonstrate a limit to givenness. What is at stake, then, is the surmounting of a familiar correlationist position, now visible through the prism of axiomatic set theory. One can, on the basis of everything we have laid out, make the decision “there is pre-givenness.” But the statement “there is pre-givenness” will always be given, even when rendered in the most rigorous mathematical formula, because it has parts that are subject to separation. That is, the statement already offers a rule of passage for formulating related statements by virtue of its including the term “pre-givenness” within givenness. Because there is no rule by which the operation of the count can pass into presentation, it can only be that statements that are partly but not exclusively about the pre-given are false. As the preceding sentence is necessarily false (because it contains parts, and makes claims about both the given and the pre-given), we are abruptly brought up short. Of course, if there really is a not-for-us, if there is an exterior to thought, this must still be the case without our claiming it to be so. The question accordingly becomes: What are we to make of the claims we do make? One could declare that all claims are for us – since that is the essence of a claim – and then bar any nomination of the not-for-us in order to avoid inconsistency. This, one is inclined to believe, is the point at which Wittgenstein counseled his division of silence and speech. Yet we do not get away so cleanly. As we have shown, barring the not-for-us can only mean assigning it again by the default assignment of an un-finishable minimal instance of givenness. Speculative materialism is accordingly compelled to demonstrate, which is not the same as to prove, a fact over which mathematical thought has no claim. 17
“Thus it comes down to exactly the same thing to say that the nothing is the operation of the count – which, as the source of the one, is not itself counted – and to say that the nothing is the pure multiple upon which the count operates – which ‘in’itself,’ as noncounted, is quite distinct from how it turns out according to the count.” BE, p. 55.
Clearly a metaphysical solution cannot serve us here, because it can only reinstate the One, which is a mathematical thought par excellence. Nor will language help, insofar as it too relies on the forming of sets, albeit according to a different procedure.18 What then? If every comprehension of exteriority or anteriority is itself a mathematical thought, what could possibly offer the means of breaking this impasse? The answer would seem to be: nothing. Then again, what if this very nothingness turned out not to be a liability but rather our greatest resource? Meillassoux himself has recently developed the case for “a sign devoid of significance”19 as an instance of semiotic contingency. There can be no question of my judging the merits of this concept, given its fragmentary translation as of this writing. It will be interesting, however, to see what the implications of a “pointless sign” may become in his hands, because it is precisely the prospect of pointlessness that our own investigation of parts leads us to consider. The key, as I see it, lies in our analysis of statements that refer only partly to the pre-given, which suggested that the pre-given can only be excluded from expression. While it is true that the pregiven will, by definition, resist the axiom of separation, we can approach the problem as residing not in the sign as such, but rather in the capacity of the sign to be related to other signs. If the pre-given were actually the reference of this sign, it would not have any such relation. The challenge would therefore become to construct or identify a sign, or more correctly a multiple, that provides the operation of the count with no rule of passage to any other multiple that may be formed as one. On first consideration, we might assume that such a multiple would simply fail to have any parts available for separation. But this is not quite right, because a multiple without parts would simply evade recognition. In fact, a multiple that truly provided no rule of passage to other multiples in the same situation would consist only of parts that were made to fail upon separation. The axiomatic terminology is even more exact. We are contemplating a finite multiple, each part of which is constructed to be intransitive20 to the operation of the count in the situation to which that multiple belongs. As for how this intransitivity could be accomplished, the situation would have to contain the parts of the multiple only insofar as those parts are negated. Where the multiple contained a, the larger situation would contain ~a. Thus the claim of a would fail to carry over into the situation whenever the operation of the count tried to form as one the multiple and anything other multiple in its situation.
It is only necessary to replace every letter in an alphabet with a number to show that language forms sets that are fundamentally positional and so organized around ordinality rather than the cardinality usually emphasized in the Zermelo Fraenkel axioms. 19 http://hypertiling.wordpress.com./2010/02/07/translationofpartofmeillassouxscontingenceetabsolutisationdel”un/ . Meillassoux, Quentin. Contingence et Absolutisation de l’Un,“5. Two Ones” transl. Fabio Cunctator, February 10, 2010. It is a matter of some satisfaction that my interest in signs and marks without reference drew me to Meillassoux prior my knowledge of this paper, rather than the other way around. Moreover, that I developed my argument in blithe ignorance of its faroff confluence. In any event, I would happily cede originality to Meillassoux, if only it didn’t destroy the absence of significance that he has reserved for such a sign! 20 I use the terms “transitivity” and “intransivitiy” in their mathematical sense – as a measure of the ability of an element to extend from one multiple to another. 18
Such a multiple would be countable (and therefore not instantly lost) and yet still be a claim of nothing, free not only of claims but of claimants as well.21 This latter independence is paramount: Insofar as it exhausts the attempt at separating parts, an intransitive claim could, in theory, expose the operation of the count to its inability to form itself as one, and so dash any claims of a transcendental subject. I say “could” advisedly. A multiple so defined will not be at all easy to construct, if by construction we mean to indicate durability. The axiom of replacement, which permits the substitution of one sign for another, quickly reveals the jeopardy in which this claim will be placed. So long as all of the other axioms are observed, a physical mark for, say, the void can be replaced with a hexagon, the image of a bald eagle, or anything else, without violating set theory in the least. The ability to substitute one sign with another the leaves us with the possibility that the shape ∅, although apparently the perfect intransitive claim, could be expressed in a form with separable parts, which would in turn require these parts to be successively removed from circulation (and guarded by limit terms that would be identical to juridical laws). The symbol reproduced on the cover of the paperback edition of BE, for example, could easily be rendered in forms other than a cairn of stones – as a drawing in the sand, or on a billboard the size of a building. Then again, the very vulnerability of an intransitive claim only shows that, in its potential repeatability and its subsequent repetition, it will necessarily be contingent. Insofar as there is no reason, sufficient or not, to repeat such a sign, it could not be otherwise. For the speculative materialist, the real problem with assigning the void symbol as an intransitive claim is not that it can be arrogated so easily by mathematical thought, but that the situation in which it appears is too poorly defined. By way of contrast, we might consider an example in which the situation is defined too well. Imagine that someone takes a blank sheet of paper, tears it in half and hands one half to a second person. Each person now holds an object that matches the other, in the same sense that documents of indenture used to be drawn up. The ragged edges are “indented” so as to permit each party to verify the authenticity of the agreement by joining them together again. In this case, however, the paper is blank. One could say that a promise has been made that cannot be betrayed, because the terms available – paper, indenture, match – fail to carry through on their own implications. There is no clear requirement to protect either sheet from theft or decay, nor is there any indication that the indenture entitles either party to its authentication. Again, the contingency of the arrangement is exposed, this time in the local modality of a pact. But is it only the multiple that is contingent? It seems that we have introduced a circularity between the parts of the multiple and the situation to which the multiple belongs. If, for example, one party insisted on verifying a physical match between the two indented sheets, it would be tantamount to changing the situation. The same would go for a void symbol forty feet high – the affront was neither part of the deal nor explicitly disallowed at the outset. Thus we must entertain the prospect that 21
Quite apart from Meillassoux’s “sign devoid of significance,” there is a growing philosophical literature concerning the viability of an empty signifier. Without pretending to have exhausted this literature, we could reasonably venture that a claim of nothing would be subject to greater stringency than many of the proposals offered thus far, which tend to conceive such a signifier as an emptiness with the capacity to be filled. (See especially Laclau, E. (2007) Emancipation(s) (London: Verso.) What is on the table is rather a form of givenness that can in no way be filled, but on the contrary is explicitly “all the way” closed.
situations are contingent as well. And if that is the case, how will it be possible to choose one situation over another without introducing a transitive element into the claim? No doubt the problem cannot be addressed adequately here. We might lay some preliminary groundwork, however, by approaching the circular argument through a general elaboration of mathematical thought as a fact in its own right. We are now better equipped, for example, to embark on Meillassoux's critique of Kant’s objective deduction from within, and to ask why, if the production of consistent thought takes the effort of inclusion and exclusion, do we not let thought collapse, such that presentation descends into chaos on our own account, quite apart from any contingency of the external facts? Why do we bother to maintain the consistency of multiples at all? Or rather: Why are we compelled to maintain them?22 This is another way of asking: For what do we work? In this regard, we can see that the operation of the count acts as a kind of a hinge. From without, it is simply contingent, another fact without cause in the cosmos. From within, as the foundation of mathematical thought, it is a necessity that saturates every conceivable enterprise. If we did not count, we would become other-than-ourselves – insane, inert, some state for which we have no report. Because we are unable to measure this ability, we remain in the dark as to where any otherness may truly commence. Nevertheless, the act of forming sets is necessary for us. If what I am saying is correct, and to interrupt the operation of the count is to venture contact between a counting being and being (the in-itself, nature), then we have established not only a formal basis for this interruption, but a point of departure as well. Suppose a situation is necessary-for-us. Better, let it be “absolutely necessary-for-us”. It will be precisely in those situations where this position is held, where the situation seems immutable, that the intransitive claim will be most effective at breaking the circle of thought. *
Coda: In supposing we can introduce an intransitive claim into an existing situation, we have neglected what is perhaps the most important point – the problem of the introduction itself. After all, if the multiple is to preserve intransitivity, the situation from which it is drawn will have to lack any multiple for the operation of the count. It will have to be a “no man’s land” that nonetheless has contents to yield. To make being evident, one would then bear home a thing from a dead world. Is this what is meant by adventure, or, more saliently, by access to the great outdoors? If so, we come to an epistemological test – to be able to draw an intransitive claim from a situation that we know to be unoccupied, rather than to be haunted by the possibility of a once and future inhabitant who may come to seize custody. We also begin to see the structure by which the familiar hero emerges. (And the category of heroic narrative includes legends from domains as arcane as that mathematics, Galileo being the current exemplar.) Insofar as we seek a universally intransitive claim, the situation would then be subject to a second condition of being free from any ordeal – of the returning hero as much as the violated stranger. What is required, then, amounts to structuring the claim as an ordinary math problem, in which the work done to exclude any operation from the dead world and the work done to include it in the This was Nietzsche’s problem, of course, in more ways than one.
necessary situation are paired with the smallest measure of effort in the necessary situation. In this way every inhabitant of the necessary situation would enjoy consistent access to a claim that is consistently determinable to be lacking any proxy to an enumerating verb except the proxy that everyone has. (No one knows who invented indenture, if indeed anyone did; nor is it difficult to rip a sheet of paper in two.) It seems odd that the experience of nature (being, otherness, the in-itself, whichever term precipitates first) should depend on a retrieval from lifelessness. Perhaps the argument remains flawed on some critical level. Nevertheless, when stated this way, it reveals a curious integrity to Meillassouxâ€™s extant works: Given that we are simultaneously engaging the effectivity of the factial and the defeat of hauntology, it may turn out that the question of the great outdoors and the spectral dilemma are not so very far removed from each other at all. *
Reworks Meillassoux's utilization of mathematics to justify a nature beyond the given in relation to Badiou's set theoretical ontology.