Clemens Freadman Uncorrected Proof.pdf

Genre, Text and Language Mélanges Anne Freadman Sous la direction de Véronique Duché, Tess Do et Andrea Rizzi PARIS CLASSIQUES GARNIER 2015 Véronique Duché est professeur de langue et littérature françaises à l’université de Melbourne. Spécialiste du roman à la Renaissance, elle s’intéresse tout particulièrement à l’histoire de la traduction. Elle a publié Si du mont Pyrenée / N’eussent passé le haut fait…, les romans sentimentaux traduits de l’espagnol en France au XVIe siècle (Paris, 2008). Tess Do est maître de conférence en études françaises à l’université de Melbourne. Spécialiste de littérature francophone et postcoloniale, elle s’intéresse aux rapports entre exil et identité, guerre et mémoire, culture et alimentation dans les œuvres des écrivains français d’origine vietnamienne. Elle a participé à l’ouvrage Exile Cultures, Misplaced Identities (Leyde, 2008), dirigé par P. Allatson et J. McCormack. Andrea Rizzi est maître de conférence en études italiennes à l’université de Melbourne. Spécialiste d’histoire de la traduction, il a obtenu de l’agence de la recherche australienne (ARC) la bourse de Future Fellowship. Il a publié « Living Well in Renaissance Italy : The Virtues of Humanism and the Irony of Leon Battista Alberti », dans le no 119 de l’American Historical Review. © 2015. Classiques Garnier, Paris. Reproduction et traduction, même partielles, interdites. Tous droits réservés pour tous les pays. ISBN 978-2-8124-3794-6 (livre broché) ISBN 978-2-8124-3795-3 (livre relié) ISSN 2103-5636 CAN MATHEMATICS THINK GENRE? Alain Badiou and Forcing The mathematician and the logician meet upon a common highway. But they face in contrary directions C.S. Peirce GENRES OF BEING? In her groundbreaking book on the American pragmatist C.S. Peirce, The Machinery of Talk, Anne Freadman notes that: ‘Genre is a rhetorical topic, and one in poetics; we take it for granted that it is a question legitimately asked of film or literature, of sculpture and of television, of painting and of journalism.’1 Is genre, however, a ‘question,’ let alone a ‘concept,’ that can be applied to mathematics? Is mathematics a genre? Are there genres in mathematics? Must the concept of genre be transformed in its application to mathematics; or does genre encounter its limits with respect to mathematics? To advert once again to Freadman’s thesis, part of the problem here is perhaps that ‘the major differend besetting scholarship at present is what came to be known as the “two cultures,”’2 and hence the ongoing inability of researchers to find any adequate way of negotiating the “science”/“humanities” divide without, it seems, privileging one side over the other, if not entirely misrecognising the true stakes of the dissensus. Is it then possible — as Alain Badiou has recently attempted—to rethink what “genre” is from the point of mathematics itself? This essay takes up the above questions by 1 2 Freadman 2004, xxxvi Freadman 2004, xxxi 204 JUSTIN CLÉMENS way of an exegesis of Badiou’s metaphysical account in Being and Event of Paul Cohen’s set-theoretical practice of ‘forcing.’ Surely an immediate objection will arise. How can “the genre” of metaphysics take “the genre” of mathematics to provide the only good account (read: “consistent concept”) of genre? Is this not the founding gesture of metaphysics par excellence? In Freadman’s terms: I take metaphysics to be a genre. This genre can be characterized in several ways: it is the genre that denies its own generic specificity; it is the genre that seeks its own foundations; it is the genre that stands outside experience in order to account for the conditions of possibility of all knowledge, knowledge of things, as well as knowledge of knowledge; it is the exercise of reason, directed to a transcendental account of reason; it is the knowledge of the essentially human, and so on1. Significantly, however, Badiou’s own metaphysics conforms to none of these restrictions, and does so by skewing the received acceptations concerning all the key terms. Firstly, if metaphysics is a genre, it is a genre that puts into question the very concept of genre itself. Second, metaphysics doesn’t therefore simply deny its generic specificity, but exposes, analyses and affirms its genericity as a problem. Third, Badiou maintains that the exception to genre of the genre of metaphysics is due to certain anti-foundational requirements: indeed, metaphysics is dependent upon its extra-philosophical conditions, that is, upon other genres. It is with respect to one of these conditions in particular—that of science—that Badiou seeks to rethink and reformulate what a genre is, in order to rethink philosophy itself as not generic but trans-generic. Even at a glance, the ‘doing ‘of philosophy cannot simply be said to be generic according to most received understandings of the term. As Badiou succinctly puts it in the ‘Author’s Preface’ to Theoretical Writings: ‘the philosophical corpus seems to encompass every conceivable style of presentation.’2 Between ‘Socrates, he who does not write’ and the endlessly-rewritten treatises of someone like Schopenhauer, philosophers are liable to produce dialogues, seminars, playlets, novels, poems—and so on and on. As we shall see, then, Badiou will hold mathematics generic, but philosophy trans-generic. Mathematics will be that genre that enables 1 2 Freadman 2004, 6. See also such works as Frow 2006 and Ferrell 2002 which examine this genre of claim in detail. Badiou 2004, xiii CAN MATHEMATICS THINK GENRE? 205 a true knowledge of genre. Yet Badiou does not for all this think that genres are themselves mathematical, nor that they need to know what they are doing to be genres. On the contrary. But only mathematics is able to rigorously thematise and identify the points at which rationality is divided between consistency and inconsistency, and it does so as an integral part of its practice. It’s not that maths is the only acceptable form of rationality or of thought—on the contrary, Badiou is obsessed with demonstrating that love, politics and art are also rigorous forms of thinking—but that, without mathematics, one will not even know at which point one has fallen into an inconsistency which freights irrational forces, nor the real points where the real leaps of reason and unreason must be taken. To put this another way, mathematics as system is not truth, but it is the only acceptable paradigm of “true knowledge” for metaphysics (so to speak). It is to a discussion of Badiou’s account of mathematics that we therefore now turn. MATHEMATICS = ONTOLOGY or, Set-Theory as the Constrained Exposition of Being; or, Letter, Formalization, Affirmation In Being and Event, his magnum opus, Badiou makes a strong claim that is clear but which has proven extraordinarily controversial. It is that mathematics = ontology1. This claim is not, as it has very often been misread, a “Pythagorean” claim, which would treat existence itself as integrally mathematical, governed by proportion and harmony, reason and necessity2. On the contrary, as Badiou explains: 1 2 The secondary material to date has evidently been much concerned with this claim, albeit with ambiguous or unsatisfactory results. There is certainly understandable resistance to it, but it is at least necessary to get it right before responding. For helpful introductory English-language accounts, see Barker 2002; Clemens and Feltham 2003; Feltham 2008; Hallward 2003; Norris 2009; Pluth 2010. For various reasons—not least the narcissism of proximity—the French reception to Badiou has generally been less exegetical and more polemical. See, inter alia, Kacem 2011 and Laruelle 2011, but also Tarby 2005, and the collection Ramond 2002. On this point, see the polemic between Nirenberg and Nirenberg 2011; Badiou 2011; Bartlett and Clemens 2012; and Nirenberg and Nirenberg 2012. 206 JUSTIN CLÉMENS The thesis that I support does not in any way declare that being is mathematical, which is to say composed of mathematical objectivities. It is not a thesis about the world but about discourse. It affirms that mathematics, throughout the entirety of its historical becoming, pronounces what is expressible of being qua being1. It is rather to the extent that “being” can be properly thought at all, it must be thought mathematically. Why? What does this mean? First, mathematical thought must be extracted from linguistics and from all and any theory of natural signs, whether these are supposedly truly “natural” (à la ‘bio-signs’) or ‘natural languages.’ This is an unpopular position to take in a philosophical universe still dominated by rival versions of the linguistic turn. As Badiou himself notes of what he sees as the three major philosophical orientations active today—the hermeneutic, analytic and postmodern—they, despite the misunderstandings, polemics, and real differences, nonetheless all render “language” as ‘the great historical transcendental of our times.’2 Whether we think of Peirce, Nietzsche, Wittgenstein or Heidegger, the continentals or the analytics, the literary or scientifically-minded, the problematic of language qua logic of sense remains primary. The effect of this for Badiou is double: philosophy in the grand style is declared to be essentially over, and truth is subordinated to meaning. Badiou will maintain, quite to the contrary, that contemporary philosophy qua metaphysics must begin as a kind of rupture with the dominance of language. Yet it cannot do this in a naïve manner: there is no sense to simply reasserting the privilege of the real against the language that discloses it. Hence Badiou will agree with his enemies that ‘there are only bodies and languages’—but he adds ‘except that there are truths.’3 This assertion evidently necessitates a reconstruction of the notion of “truth” on Badiou’s part, and, as we shall see, he does this by returning to mathematics as giving a new, rigorous, consistent formal account of how to think genre itself. The power of exposing the exception of truths as generic is exactly the determining role that Badiou wishes to assign to mathematics and logic. But several distinctions then need to be made and justified. The relevant divisions here include: philosophy and its conditions; mathematics 1 2 3 Badiou 2005, 8 Badiou 2003, 46 Badiou 2009, 4. Italics Badiou’s. CAN MATHEMATICS THINK GENRE? 207 and language; being and formalization; consistency and inconsistency; knowledge and truth. These divisions are not simple oppositions; they are not directly aligned with each other; they do not totalise the field. What they do do, as we shall see, is enable a reconstruction of the ways in which metaphysics essentially relies on different genres to think at all, as it separates thinking from the dominance of any one genre in particular. Above all, the “exposure” of which I just spoke is not in the slightest reductionist, for reasons that I will try to elaborate. In order to do this, however, Badiou must essay to separate mathematics and logic from the dominance of language. These practices can hence no longer be considered subsets of language in general or to be understood on the model of language, even if they are de facto always and everywhere implicated with one or another natural language. As he responds to a question on precisely this point in an interview with Lauren Sedofsky: LS: Here’s the inevitable question: Isn’t logic a language? AB: My thesis is that it’s not reducible to a language. From the point of view of logic, you always have to make linguistic suppositions; there’s a linguistic manipulation. What I try to demonstrate philosophically, though, is that the essence of logic is not linguistic, no more so than that of mathematics or a scientific discipline. Like any discipline of thought, logic must finally settle in a language, but its essence concerns what the general form of a world is, which is the question I pose1. Difficult as it may be to accept the classical terms in which Badiou phrases his position here—“essence”, “form”, “discipline”—we cannot take this terms as if they were terms we already recognise and understand. On the contrary, part of Badiou’s import is that he rigorously re-establishes their significance so as to evade all existing critiques. This doesn’t of course mean that they cannot be criticised in their turn—only that existing criticisms will not be up to the task. For Badiou, there are a number of levels at which pure mathematics and pure logic function. Let us begin at the “lowest” level, that of the material sign itself. For if mathematics-logic are, like all other discourses, necessarily imbricated with materiality, one of their peculiarities is that they are, strictly speaking, indifferent to their materials. This feature, this power of indifference, derives from a triple knot, of 1 Badiou and Sedofsky 2006, 246-53 208 JUSTIN CLÉMENS material literalisation, epistemic formalization, and existential assertion. In what follows, I will restrict myself to outlining what I believe to be Badiou’s arguments, without always being able to argue fully for their veracity; nonetheless, I will also attempt to give some indications of how and why he supports these arguments, as well as how they depart from several currently still-dominant interpretations of mathematical practice. The triple of which I speak is not all that is important in Badiou’s work on mathematics—far from it—but for the present context it provides the necessary background features to the rethinking of the problem of genre that is at stake here. Regarding the interpretation of the status of the materiality of letters, Badiou has been strongly influenced by the work of the French psychoanalyst Jacques Lacan and his school, whose propositions have been perhaps most thoroughly developed and justified by the linguist Jean-Claude Milner1. However, Badiou also contests the general Lacanian approach at certain points2. For Badiou, a letter is not a signifier or a sign; it has in itself neither sense nor reference; it marks nothing in particular3. It is simply and primarily a mark of any kind, and any sort of mark will do. A letter is at once eminently material, yet, in being so, requires the necessity of a difference between itself and its ground. If this difference is undeniably a form of suture, it is also the case that the logical or mathematical mark precisely deploys itself as independent of all and any empirical grounds4. Why? Because a logical or mathematical mark can in principle take any form whatsoever without loss of function; it can 1 2 3 4 For the key texts of J.-C. Milner, see Milner 1995a and 1995b. Today, the influence of the Lacanians upon Badiou has literally blossomed into controversy: see Badiou, Milner and Petit 2012. Although the scholarship on this point remains minimal to date, see Bartlett et al. 2013. See also Bartlett and Clemens 2012, as well as the entry on “Lacan” in Bartlett and Clemens 2010; Ronen 2010; Hoens and Pluth 2004; Grigg 2005; Chiesa 2006. For the most part, card-carrying Lacanians have responded critically to Badiou’s work, e.g., MacCannell 2009. The same goes, a fortiori, for Deleuzians: in addition to the essays in the aforementioned Badiou and Philosophy, the summa of such attempts would have to be Roffe 2011. On this point, see Clemens 2003; for a radical post-Badiouan take on the problematic of the letter, see Meillassoux 2012. For a different account of the role of formalization, see Livingston 2011. If there are of course many different types of mathematical signs (e.g., variables, connectors, quantifiers, relations, punctuations, etc.), it is not necessary to go into the arcane complexities of their differentiation here: all that is necessary is to show that such “signs” share a CAN MATHEMATICS THINK GENRE? 209 take on any function without predetermining any form. This is one integral aspect of the Lacanian insistence on the priority of the matheme: letters that are integrally transmissible beyond any particular meaning, and indeed beyond any particular language, precisely to the extent that they are expressly contingent in the form of their materialization. This precisely distinguishes them from all other kinds of expression, which otherwise continue to partake of ‘the machinery of talk,’ in Peirce’s expression. The inscriptions of mathematics have nothing whatsoever to do with such machinery, insofar as their patent formal contingency is explicitly and integrally directed against any substance, meaning or reference inhering in their empirical apparition. Letters in this sense, and this sense alone, are “transmaterial” entities. One can denominate a variable a or a variable x: the forms of inscription remain indifferent to this formalization. Whether one uses a, b and c or x, y, and z or, indeed, É, ®, or Þ (for material implication)—again, the indifference of the mark to what it “is” or “designates”—the manipulations of these letters must sustain and exemplify their own consistent deployment throughout a demonstration. Formalization, as opposed to form, requires the indifference of all or any particular set of letters to what they can be deployed to demonstrate. Yet it is precisely for this reason that the principles of their deployment must be explicitly and essentially manifested in and by their deployment itself. Moreover, this must be done immanently to the demonstration: at every point of a mathematical demonstration the linkages between steps of a proof and the principles of such linkages must be practically shown as inseparable. Indeed, formalization can be considered the immanent rendering-consistent of the ramshackle indifference of its own contingent materials. In mathematical and logical systems, this rendering-consistent — at least in classical and intuitionist logics — is equivalent to the impossibility of producing any contradiction within that system. So consistency is what mathematics and logic inseparably do and what they think in and by this doing. To put this yet another way, the difference between appearing and doing or between showing and proving is itself indifferent in mathematical practice — and only in this practice. This is equivalent to what Brian Rotman, from an essentially Peircean perspective, derisively refers to peculiar feature and are the only signs that do so: their material contingency qua patent indifference to their own materiality. 210 JUSTIN CLÉMENS in mathematical formalization as being ‘completely without indexical expressions.’1 Mathematical indifference to empirical criteria is ensured precisely through the effective suppression of any markers of the taking-place of the ‘utterance,’ which is therefore not an utterance in any existing semiotic sense. If, at the turn of the twentieth century, Gottlob Frege could still maintain that the three required properties for any mathematical theory were consistency, completeness and decidability, it is now the case that at least the latter two have been demonstrated to be, strictly speaking, external to mathematics or, rather, that mathematics must mathematically limit the extension of its own claims with respect to them: Gödel’s incompleteness theorems show that there is any sufficiently powerful axiomatized system can only be consistent if incomplete, and that any such system will produce at least one undecidable proposition. Twentieth-century developments have therefore pared-away the various inessential mathematical elements that inhered in prior mathematics, to expose the problem of consistency itself as the singular domain of mathematics and mathematical logics. I say ‘the problem of consistency,’ since consistency is therefore identified and isolated as such, as the fundamental concern of all mathematical/ logical thinking, yet without its own status being thereby able to be resolved in general by such thinking. Otherwise put, this is one reason why mathematics is the paradigm of knowledge for Badiou: not because it knows more or better than anything else, but because it alone apodictically knows the points that it doesn’t know in order to give a new knowledge of what it does. Hence, as Badiou puts it in Briefings on Existence: ‘Mathematics…. forges a fiction of intelligible consistency.’2 I underline Badiou’s careful use of the term “fiction” here, not because it is thereby intended to link or equate the fictions of mathematics with the fictions, say, of literary 1 2 Rotman 1993, 7. Rotman’s very interesting book nonetheless suffers from a number of serious problems, evident in the current context. Above all, his understanding of what constitutes “Platonism” is quite naïve, and certainly unable to account for Badiou’s peculiar interpretation. Another problem is that, despite Rotman’s emphasis on the materiality of mathematical signification, he doesn’t quite separate—as the French thinkers in Lacan’s wake have—the “signifier” and the ‘letter,’ such a distinction enabling a more detailed, materialist and persuasive account of the issues. Badiou 2006, 48 CAN MATHEMATICS THINK GENRE? 211 writing, but quite the reverse. It is rather to qualify the particular discursive powers and problems of mathematics: its operations establish and model the pure limits of reason, without thereby claiming they are the final take on the question. After all, subsequent developments in mathematics might pinpoint hitherto-inseparated differences within the field (hence the “fictionality” of maths insofar as it is always in principle susceptible to radical new formalizations). In fact, the exposure of pure consistency as the essential problem of mathematical system is confirmed by the recent development of paraconsistent logics which contravene the principle of non-contradiction, and for which consistency no longer simply means the impossibility of deriving a contradiction within the system: for such logics, “true contradictions” are sometimes possible. What this shows is that contemporary mathematics and logic negatively limn inconsistency as such, exposing with the most rigorous clarity and distinctness the contested points on the borders between pure consistency and the inconsistent. This “consistency” is outlined without recourse to any unifying figure of unity or totality as such, since it is a trans-arithmetical program. It rigorously theorizes its own limits, and is thus also a radical program of self-restraint. Yet the consequences of the exposure of the problem of in-consistency itself as restrictively yet essentially mathematical does not render mathematics simply a formalism, that is, a rule-governed game with letters for which there is no particular meaning or reference. Mathematics as pure formalization is therefore not a metaphor. It is not a metaphor, because it is a literalisation of the consistent without possible figure: the little letters and lines of mathematical formalisation have no inherent apparition-value. Rather, we find there a formal inscription of the limits of reason, of pure reason at and as its limits. Given then that form ≠ formalism ≠ formalization for Badiou, the problematic of mathematical formalization necessarily requires a rethinking of the terms and conditions of reference1. If in-consistency becomes the problem for mathematical practice, then a new double question emerges as a result: i) what is the relation between logic and mathematics? ii) what 1 Note that, “while mathematical and logical formalization is a paradigm for formalization, formalization is not identical with this,” Badiou, Fraser, and Tho, 2007, 89. As we shall see, different kinds of formalization— generic!—can be accomplished in the domains of love, politics and art as well. 212 JUSTIN CLÉMENS is the relation between mathematics and “being” and mathematics and “the world”? The classical problems of metaphysics, of philosophical ontology, can therefore be rephrased as: how can letters discern and inscribe the status of the relation between letters themselves and their ground? This is where mathematical axiomatics become crucial: it is at once the point at which mathematics can be separated from logic, and the point at which mathematics can return to the world. If both mathematics and logic share an indifference to the contingent form of their materiality, and both deploy this indifference in the service of a pure becoming-consistent that is also their raison d’être (the stringent elaboration or practice of laws of thought as the rendering of materiality to the absolute limits of consistency), unlike logic—which ultimately deals with regulating the consistency of thinking, delimiting, defining and describing what can and cannot count as such—mathematical axioms also make fundamental declarations about what is and is not1. There is an ‘originary belonging of mathematical deductive fidelity to ontological concerns.’2 Mathematics enacts consistency in inscribing “some” extra-mathematical reference at its heart, albeit inscribing it as foreclosed; logic formalizes the laws of possible consistencies, without necessarily making any actual referential claims. Mathematics inscribes rational actuality, logic rational virtuality. Paradoxically, this means that mathematics has something 1 2 This is perhaps the central and most radical position elaborated in the opening sections of Being and Event. For a different, but in my opinion, quite wrong-headed account of the role of mathematics and logic for Badiou, see Plotnitsky 2012, 351-68. Although there is obviously not the space to respond to Plotnitsky adequately here, the points of his confusion hinge on his complete failure to understand the particular role Badiou assigns to axiomatization in mathematics (as opposed to the constructive establishment of logical spaces), to the difference he sees regarding classical compared to post-classical logics in the assignation of existence, to the non-mathematical status of the event (which Plotnitsky doesn’t seem to get at all), and to Plotnitsky’s failure to separate the concept of the generic provided by mathematics from Badiou’s generically political claims. Even Plotnitsky’s claim that topos theory ‘inscribes the plurality of possible ontologies’ as if it were a rebuke to Badiou at once repeats Badiou’s own position without understanding its import. Other, more rigorous critical accounts of Badiou’s mathematics can be found in Brassier 2007; Mount 2005; Fraser 2006. It is, however, symptomatic that this latter crew, Livingstone op. cit. included, are essentially launching their critiques on the basis of their own speculative improvisations upon an unconfessed intuitionist decision of their own. Hence their positions ultimately remain finitist, realist, and analytic, in the pejorative senses of these terms—and hence, in Badiou’s account, pre-Cantorian. Badiou 2005, 250 CAN MATHEMATICS THINK GENRE? 213 extra-logical about it: the practice of inventive mathematics can embody forms of reason that go beyond the contemporaneous closure of reason acceptable to logic. Mathematical axioms do this for Badiou—quite to the contrary of the usual claims and presuppositions about their status—not by defining basic terms, but by refusing to define what they affirm. If received interpretations about the status of axioms usually hold the axioms themselves to constitute “indubitable” ‘clear,’ “distinct” and/ or ‘self-evident’ “truths” — a position that could be (relatively) plausibly assigned to mathematicians and logicians from Euclid through Descartes to Frege — there is also a lineage that sees the axioms as fundamentally game-like presuppositions, without necessitating any absolute grounding. For example, the independence of Euclid’s parallel postulate became clear in the 19th century, when radically new-yet-consistent forms of geometry were developed by tampering with this axiom: a development which naturally founded another intense spurt of interrogations of the foundations of mathematics. Hence there developed plausible pragmatic interpretations of the status of axioms, whether as what’s good in the way of belief, as evidence of unplumbable ‘ontological relativity,’ or under other descriptions. None of this conforms to Badiou’s position. Axioms rather constitute decisions about what exists, but without having any grounding in the incontrovertible. On the contrary, they declare the basic constituents of existence, but without providing definitions of what they declare to be the case; they prescribe connections, but without defining what it is they are working with. As such, they comprise what Badiou calls in Platonizing vein ‘the grand Ideas of the multiple.’1 In doing so, axioms pinpoint the ways in which they are themselves decisions which cannot ever be fully rationalised. A new account of the dialectic between freedom and necessity is broached. If mathematics has undergone many fundamental transformations, the development of set-theory in the late nineteenth century is of crucial significance in Badiou’s account, precisely because in it the practice of axiomatization is from the first an explicit problem for that work of axiomization itself. On the basis of his interpretation 1 Ibid., 60 214 JUSTIN CLÉMENS of set-theory, Badiou at once wants to account for the fact that there are rationally many different, even incommensurable systems able to be constructed by mathematic reason — in fact, there are several different rival axiomatizations of set-theory itself — but what all the different systems share is an agreement as to which points are the ones one can rationally differ in regards to. Axioms designate the points at which extra-rational decisions about rationality have had to be made, and every axiom is thus necessarily polemical, in an extra-rhetorical (i.e., extra-linguistic) sense. Rather than an axiom simply being or stating an incontrovertible principle of reason, then, it is a statement which, in its very declarativeness and prescriptiveness, shows that it is rationally incontrovertible that a decision about what will have to count as rational has had to be made at and about a particular point. This is one reason why Badiou’s thought is at once righteously unbending and authentically pluralist: it demonstrates that there is a rational necessity for incommensurable decisions about how reason must proceed. There must be axioms for systematic rational knowledge to be able to practice self-knowing consistency as such; but those axioms are themselves the outcomes of (necessarily) extra-rational decisions; as such, their very elaboration affects something new at the limits of reason itself. But this is only one, very general consequence of Badiou’s stunning interpretation of set theory1. There are very many aspects of interest, which I can only list here, but which provide the necessary terms for the account of genre that follows. Perhaps the most important of these is the thinking of the infinite, and the concomitant development of mathematics’ self-grasping of its own systematic destiny. Indeed, Georg Cantor, the inventor of set-theory, indexes the moment for Badiou at which mathematical thought shifts from a restricted to a general economy. Why? Because for the first time, a rigorous mathematical concept was given of the infinite, as opposed to it simply functioning as the unthinkable exception to the finite, as totality, or as unique. Cantor demonstrated that “infinity” was not only not an ultimate limit of mathematical thinking, but that there were infinite infinities, that is, sets that could not be put into one-to-one correspondence with each 1 In addition to the works already cited, see the brilliant little volume Badiou 2008. CAN MATHEMATICS THINK GENRE? 215 other (e.g., the infinity of natural numbers is demonstrably smaller than the continuum of all the numbers between 0 and 1). In doing this, Cantor inspired enormous resistance among even the greatest mathematicians, and very quickly certain paradoxes started to show up in his and related programs. Perhaps the most famous of these is Russell’s paradox, which showed that the concept of a set of all sets—an ultimate or total set which contains all others—is inconsistent. (Notoriously, this set belongs to itself if it doesn’t and doesn’t if it does)1. As Badiou notes, the two major effects of these paradoxes was: i) ‘to abandon all hope of explicitly defining the notion of set’; ii) “to prohibit paradoxical multiples.”2 This led to the work of axiomatization over the next 40 or so years, which produced various rival systems. Badiou’s own decision is to run with the standard Zermelo-Fraenkel (ZF) system of nine axioms, which, he declares “concentrate the greatest effort of thought ever accomplished to this day by humanity.”3 To give one reviewer’s useful general summary: ZF is a mainstream focus of modern set theory and principally used to study varieties of the mathematical infinite. Additionally, most “ordinary” mathematics (calculus, algebra, probability, number theory, geometry, etc.) can be thought of as codified as sets, sets of sets, and so on. In this way, ZF (defined formally in standard predicate logic) may be thought of as a foundation for all types of mathematical representation. Intuitively, ZF starts with nothing, the empty set, Ø. From that, you can form a new set consisting of all of the subsets of Ø; since the only subset of Ø is Ø itself, this new set of subsets is just {Ø}. Then you can take that set’s subsets (i.e. {Ø, {Ø}}), and so inductively generate partial universes of sets Vn, each mini-universe Vn defined as all subsets of Vn-1. That only leads to finite sets, so a further axiom is needed to guarantee the existence of infinite sets. With that, additional ZF axioms for collecting and redefining sets allow construction of transfinite ordinal numbers α which continue the natural, or counting, numbers into the infinite, and hence the process of set formation in further collections Vα4. Badiou gives an extraordinary interpretation of each of ZF’s axioms and their philosophical import: picking up on what I have said above, 1 2 3 4 See Badiou’s various discussions of this, e.g., in Badiou 2009, 109-11, 153-5. Badiou 2005, 43 Ibid., 499 Kadvany 2013. The review, unfortunately, concludes in error by imputing several theses to Badiou that he does not hold, including the incomprehensible claim that Badiou thinks that there is a set-theoretical foundation for natural science. 216 JUSTIN CLÉMENS and as I have shown more extensively elsewhere, he literally reads these axioms literally1. To give a single example, crucial to Badiou: among the nine axioms of standard ZF set theory, ‘one alone, strictly speaking, is existential; that is, its task is to directly inscribe an existence, and not to regulate a construction which presupposes there already being a presented multiple.’2 This axiom is usually called ‘the axiom of the empty-set,’ and its canonical formulation is ‘there exists a set which has no members’: to advert to the terms provided just above, this axiom decides that something exists; it declares that this something is a set that is empty; it prescribes that all (mathematical) existence be thought on the basis of this vacuity. Since we are dealing with pure mathematics here (i.e., “pure” in the sense of the above-noted indifference to all empirical content), what sort of referent can be given to such an axiom? This is one great metaphysical moment within Badiou’s work: set theoretical mathematics thinks being as void. Not only does this evade the problems of substance that have always haunted ontology, but it gives a basis to mathematics that is quite extraordinary. The empty set is the most rigorous possible formulation of the place at which thought (in this case, mathematical thinking) is sutured to being (in this case, the void). Or, to be even more precise, the empty set for Badiou becomes “the proper name of being.” Axioms are thus ultimately formalisations of pure reference. Not only are they pure because non-empirical in a general sense, but because what they refer to is without any possible definition, substance or meaning. Yet this is also precisely what gives them their stringently rational character, their “extimate” bond to a foreclosed interior. There are three further points to make first about this triplet of literalisation, formalization, and existentialisation. First, it’s crucial that, for Badiou, these are not strictly mathematical, but meta-mathematical theses. As such, they are metaphysical, external to mathematical practice proper. Yet they are entirely dependent upon such a practice. As I’ve already flagged, Badiou’s metaphysics therefore does not conform to the charges levelled by Freadman above. Precisely because metaphysics relies on mathematics for its ontology, metaphysics cannot be a 1 2 See Clemens 2005, 21-35. Badiou 2005, 60 CAN MATHEMATICS THINK GENRE? 217 self-grounding genre. But there is more. Mathematics is not the only practice upon which metaphysics is dependent for its own ‘grounds.’ For, second, Badiou wants to claim that there are four (if only four!) kinds of genre: science, art, politics and love. He explicitly nominates these genres ‘truths,’ and therefore considers truths to be essentially generic. If this is the case, then, philosophy cannot itself simply be generic: its “genre” is to be trans-generic insofar as it relies upon all four fundamental genres simultaneously. Why is “generic” the appropriate name for truth? Because something generic is precisely something that belongs as such, but without any particularities to compromise the belonging; the generic is something that exemplifies a kind of universal pure belonging. But, then, third, what enables us to think genre as such? Nothing but mathematics has the necessary rigour to provide a consistent formalization of the concept of genre. The paradoxical essence of the genre of mathematics is truly knowing what it doesn’t know about the thing it declares to be nothing—even or especially since that “thing” is literally ‘nothing.’ Yet things are more complicated still, for the mathematical genre formalises how it necessarily exceeds its own knowledge in its self-elaboration. COHEN’S FORCING AS THE FORMALIZATION OF THE THOUGHT OF THE BECOMING-GENERIC OF BEING Having established mathematics as the “knowing genre” qua paradigm of knowledge-of-being—what Nick Heron has called, in a private communication to me, Badiou’s ‘onto-mathesis’—we can now turn to the problem of how Badiou holds set-theoretical mathematics thinks genre. I mentioned above that the distinction between “knowledge” and “truth” is crucial to Badiou, and it is precisely here that it has its most fiendish impact. For if mathematics is the paradigm of knowing, it also, as I have explained, knows by not knowing in a very specific sense. Moreover, it can formalize the ways in which thought can proceed rigorously in and by its own not-knowing. But it must do so by foreclosing inconsistency in its theory and practice. This opens onto the problem 218 JUSTIN CLÉMENS for Badiou of the “generic” or, as he will also nominate it for reasons to be discussed, the ‘indiscernible.’ To give a coherent formalization of the generic, Badiou picks up the work of Paul Cohen on ‘forcing,’ which deals with, to begin as simply as possible, a process that, while conforming absolutely to the strictures regarding consistency required by mathematics, produces something new, something “impossible” in the terms of the initial situation into which it intervenes. And while any actual such process cannot be mathematically captured (by definition), it turns out that there is indeed a way to think the in-principle compatibility of the existence of such a process with mathematical knowledge. If one imagines an infinite situation established by the most stringent mathematical procedures, it is possible to say that everything in this situation is known, or knowable: nothing goes unaccounted for; everything is necessary. Every element in a situation can be mixedand-matched with others, “represented” by classifying different kinds of subsets of the basic situation. Badiou will even term this aspect of knowledge “encyclopedic”; the knowledge of a situation “discerns” and “classifies” everything that can appear within it. The problem of nomination and reference arises again with a vengeance. The generic, to be truly generic, must therefore evade the language of the situation—otherwise it will be named and localised in an incontrovertible frame. But even if naming is evaded altogether (and it is very uncertain whether such an eventuality is possible), then the subsequent “unnameable” could not, strictly speaking, be said to exist, since existence is, as we have seen, established axiomatically, at the most fundamental level. But what if a name might be able to refer to something that has no name-being in the situation as such, or nomination might be able to be separated from predication and linked to a pure existence without identity? We are clearly in the realm of impossibility and paradox. Can this paradox be resolved? Abstractly speaking, one can immediately see that this can be done if one can find a way to separate and re-stratify the intra-mathematical relationships between nomination, predication and existence. For every particularity in mathematics must be able to be named as such, i.e., can be given a predicate. To be generic, then, predication must be evaded insofar as predication is equivalent to particularization. But surely evading predication is precisely impossible in the mathematical environment? In set theory, such an impossibility CAN MATHEMATICS THINK GENRE? 219 would require that something be able to be designated without being discernible; it would mean finding a way to identify an existence without any certification of that existence; it would certainly mean contravening the Leibnizian principle of the identity of indiscernibles. As Badiou puts it, ‘it all comes down to this: can ontology produce the concept of a generic multiple, which is to say an unnameable, un-constructible, indiscernible multiple? The revolution introduced by Cohen in 1963 responds in the affirmative.’1 As one might expect, the details of Cohen’s forcing are of a stunning complexity2. I will try to avoid as far as possible any recourse to technical language here; the aim is merely to give an ordinary language translation that points to the key intellectual operations and consequences at stake. Cohen forged his new techniques in his attempt to resolve some fundamental questions that emerged very early in the set-theoretical enterprise. If, as Cantor showed, there can be infinite infinities of different “sizes”—which he denominated as ‘cardinality,’ to distinguish the pure multiplicity of infinity from ‘ordinality,’ an ordered sequence—how are these infinities to be ordered? Designating the sequence of infinities by the Hebrew aleph, À, and their increasing order by subscripts, that is, À0, À1, À2…, Cantor, having demonstrated that the cardinality of the real numbers (the numbers on a real line) was “larger” than the cardinality of the natural numbers (the infinity 1, 2, 3…), he wanted to propose an ordering of these massive sets. As he had also shown that the “power set” of a set is always “larger” than the initial set, Cantor’s so-called “continuum hypothesis” posited that the cardinality of the power set of the natural numbers was equal to the cardinality of the real numbers. This would, in essence, give a nice, orderly reordering to the hallucinatory consequences opened up by the demonstration that there were indeed infinite infinities that were mathematically thinkable. Cantor, however, could not prove his continuum hypothesis; Kurt Gödel showed that, at least, it cannot be disproved given the presumption of the consistency 1 2 Ibid., 355 In his excellent presentation “A beginner’s guide to forcing,” T.Y. Chow proposes the concept of “an open exposition problem”: given the difficulty of forcing even for mathematicians, the problem of teaching what’s going on becomes paramount. Paper downloaded from <http://arxiv.org/abs/0712.1320> 8 April 2013. Paul Cohen’s own report of his findings is Cohen 1966; English readers of Badiou should, first and foremost, refer to the short account in Hallward 2003. 220 JUSTIN CLÉMENS of the axioms of set theory; Cohen’s demonstration, however, showed that neither can the continuum hypothesis be proved from those axioms. Cohen had in fact shown the independence of the continuum hypothesis. But to do so, he had had to invent a new mathematical technique, the aforementioned ‘forcing.’ The key operations of forcing require, first, the establishment of what is known as a ‘quasi-complete situation’ or ‘ground-model’ that is “countably infinite” (that is, it has the cardinality, the size, of the infinity of natural numbers, À0). What Cohen calls a “generic extension” will be a subset of this situation, i.e., it will be included in a situation. Note that in set-theory, there is a distinction between elements, which “belong” to sets (belonging being the fundamental relation of set theory), and subsets, which are “included” in sets, insofar as every element that belongs to the subset also belongs to the set in question (inclusion being therefore a derivative relation). This generic extension will necessarily be an unnameable subset in a very particular sense. The elements of the generic extension (this special subset of our initial set) are given a double aspect: first, they constitute the material of this subset; second, they also provide information about this subset. These elements are called the “conditions” of the generic set. As Badiou so brilliantly summarizes: a ‘generic subset is identical to the whole situation in the following sense: the elements of this subset—the components of a truth—have their being, or their belonging to the situation, as their only assignable property.’1 The elements have to be able to be grouped together in such a way that, even though every particular condition will have to be able to be named in the language of the existing situation, taken together, there will always be at least one element in any grouping of conditions that prevents that grouping from falling under a name already available in and for that situation. In other words, any grouping of conditions (an ordered set of elements of the generic subset) cannot permit the “next” element in the string to be able to be predicted no matter how long or large that grouping has become. Badiou is blunt: ‘The striking paradox of our undertaking is that we are going to try to name the very thing which is impossible to 1 Badiou 2009, 36, translation slightly modified. CAN MATHEMATICS THINK GENRE? 221 discern. We are searching for a language for the unnameable. It will have to name the latter without naming it, it will instruct its vague existence without specifying anything whatsoever within it.’1 To do so, requires enriching the language of the situation by reconstructing a hierarchy of names already available within that situation, but whose referent will no longer be directly tied to this or that element of the situation. A linguistic paradox is thereby to be avoided—how does one define the name “name” without circularity?—and set theory does so by reference to the above-discussed ‘empty set.’ The technique involves defining names as pairs, built upon a coupling of the empty set (the minimal condition) and another mark (designating an indeterminate condition). The definitional circle can thus be cut by reinscribing names as indices of a hierarchy all built up on the basis of the void, the most minimal and universal of conditions (it is “absolute”); this means that every name can be constructed on the basis of an already-established definition. As Bartlett puts it, this project depends ‘on establishing a form of nomination which allowed Cohen to talk about the elements of this non-constructible thus unknowable set, before knowing them. In short, the referent for this name (a name including within it conditions (terms) and statements that are part of the original set), which counts out the elements that belong to the “generic set” without knowing them, is contained in the supposed or quasi-complete “generic extension.”’2 Technical as this necessarily is, the import is hopefully perspicuous. The hierarchy of names becomes literally hypothetical or conditional, whereby we can repurpose names drawn from our existing ground-model that now name elements that belong, not to the original ground-model, but to the generic extension. In doing so, these names have a referent, but an indeterminate one, a referent which may exist but cannot be picked-out as such by their nomination. Moreover, in Badiou’s own words: the force of a generic extension is to obtain models that validate propositions other than those that validate the ground model and not to conserve the same semantics!! For this, it is necessary that what supplements the ground model is generic and therefore, in a precise sense, not constructible on the basis of 1 2 Badiou 2005, 376 Bartlett 2011, 211-2 222 JUSTIN CLÉMENS that model. Otherwise one remains in the same semantics. It is precisely this characteristic, at once supplementary and nonconstructible, that makes the generic the formal schema of truths; they are added to the situation without being able to be directly deduced from it, and their deployment thus unveils unknown possibilities of the primitive situation. This attests that they aim at, in the situation, what the language of this situation cannot separate in it before the supplementation1. Mathematics thus formalizes the rational compatibility of knowledge and truth, though the latter by definition transforms established knowledges (all we could know from and about the ground model differs after forcing) and therefore cannot be captured by these knowledges. Truth is generic (universal) because it escapes being known (particularism), and it escapes being known by nominating something that could not previously be recognised in and by any knowledge within its situation. Yet it can be named, even if this name cannot identify any particular element as such: hence, mathematics enables the rigorous demonstration of the possibility of reference without identification. The nominated indiscernible novelty: the hallmark of genre. CONCLUSION What I have sought to elucidate in this presentation is how Badiou uses mathematics to give a rigorous formalization of the generic. If such an elucidation has been, by necessity, truncated and abstract, I nonetheless hope that the basic waystations of Badiou’s itinerary have been at least somewhat clarified. In short, mathematics is a genre whose own generic singularity is bound up with its rigorous elaboration of the disjunctive syntheses between consistency and inconsistency, truth and knowledge. In doing this, mathematics is the only genre that can consistently formalise ‘the concept of genre’ itself, as a particular relation between truth, knowledge and praxis. As a result of its deployment of mathematics in this fashion, metaphysics is not so much itself a genre as at once meta—and trans-generic—not in the sense of being a superior 1 Badiou 2012, 363 CAN MATHEMATICS THINK GENRE? 223 or higher form of genre, or something external to or beyond genre, but as entirely dependent upon genre, and upon more than one genre. What, in the end, is exceptional about the generic is that the generic is exceptional in its rigorous evasion of particularity—and that metaphysics must deploy mathematics to truly know this. Justin Clémens University of Melbourne