VOID ENTRY

This entry completes the series of BADIOU DICTIONARY drafts that Form & Formalism has been bringing you for the last week (unless Tzuchien feels like tossing a few onto the blog), but keep an eye out for the book itself, which should (I think) be coming out at some point within the year. In it, you'll find entries on a slew of Badiousian vocabulary, written by some of the best scholars in the field. If nothing's changed since I last heard, we can expect entries from Nina Power, Tzuchien Tho, Anindya "Bat" Bhattaryya (who I've been waiting to see publish something on Badiou for ages; back in the days of the "Badiou-Dispatch" mailing list, everything he had to say about Badiou's work, and especially its use of mathematics, was profoundly clarifying), Alberto Toscano, both Brunos (Bosteels and Besana), Justin Clemens, Jelica Riha, A.J. Bartlett, Fabien Tarby, Michael Burns, Alenk Zupancic, Dominiek Hoens, Ozren Pupovac, and the editor of the volume, Steve Corcoran, along with a few I've probably forgotten or overlooked. (I welcome corrections on this point.)

That said, here's the entry: last but least, the VOID...

====================

BADIOU DICTIONARY

VOID ENTRY

Z.L. FRASER

            The word ‘void’ is surprisingly equivocal in Badiou’s writings. Leaving aside the non-ontological, ‘operational’ notion of the void that Badiou discusses in his first Manifesto (where ‘the void’ figures as the empty place in which a philosophy receives the truths of its time), we can discern, in Badiou’s work, at least four distinct senses of ‘void’:

1.     The void as the ultimate ground of ontological identity.

2.     The void as pure, non-self-identical, inconsistent multiplicity.

3.     The void as the emptiness of the count-as-one, itself.

4.     The void as the ‘gap’ between presentation and what-is-presented.

These senses are not always easy to untangle. Sense 1 is the one most readily divined from the empty set (Ø) of ZF. The identity between any two sets is determined by the axiom of extensionality, which simply states that ‘two’ sets are identical if they have all the same elements. Since a set is never anything but a set of sets, this criterion implies a regress: before the identity of a set can be established, we must first establish the identity of its elements by looking at its elements, and so on. The only stopping point to this regress is Ø, whose existence is asserted by the axiom of the empty set, and which, alone, is immediately self-identical. The void, in this sense, figures as the ‘prime matter’ from which presentation is composed; every presentational multiplicity is conceived as ‘a modality-according-to-the-one [selon-de-l’un] of the void itself’ (BE 57).

According to Sense 2, the void is thought as a sheer chaos of self-differentiation – the plethos of being-without-unity that appears in the ‘dream’ evoked at the end of the Parmenides. The void is that aspect of every situation that is still-uncounted, and which inheres in every presentation as the invisible but ineffaceable residuum of inconsistency, a ‘yet-to-be-counted, which causes the structured presentation to waver towards the phantom of inconsistency’ (BE 66). This sense, too, resonates with the set-theoretic Ø – as witnessed by the theorem that Ø is a subset of every set – but at the price of a slight ambiguity: Ø is, mathematically speaking, a perfectly consistent set, neither eluding identification nor threatening the stability of the sets in which it inheres. And so it is that, ‘with the inconsistency (of the void), we are at the point where it is equivocally consistent and inconsistent […] the question of knowing whether it consists or not is split by the pure mark (Ø).’[i]

While these first two senses of ‘void’ seem capable of enjoying a strained harmony as interpretations of the empty set, Sense 3 seems to speak of something else altogether. There is no straightforward set-theoretic formulation of the notion of ‘the emptiness of set-formation’. If there is dissonance here, it is unheard by Badiou, for whom

it comes down to exactly the same thing to say that the nothing [the void] is the operation of the count – which, as source of the one, is not itself counted [Sense 3] – and to say that the nothing is the pure multiple upon which the count operates – which ‘in itself’, as non-counted, is quite distinct from how it turns out according to the count [Sense 2].

            The nothing [the void] names that undecidable of presentation which is its unpresentable, distributed between the pure inertia of the domain of the multiple, and the pure transparency of the operation thanks to which there is oneness [d’où procède qu’il y ait de l’un]. The nothing is as much that of structure, thus of consistency, as that of the pure multiple, thus of inconsistency. (BE 55; emphasis added)

Here, yet another thought of the void emerges, playing on the equivocity of ‘between’ (which can indicate either a distribution or an interval): The void is, now, ‘the imperceptible gap, cancelled then renewed, between presentation as structure and presentation as structured-presentation, between the one as result and the one as operation, between presented consistency and inconsistency as what-will-have-been-presented’ (BE 54; trans. modified): this is Sense 4 in our list. It is not at all clear how this conception of the void is related to the empty set (unless we imagine it to mark a suture in Miller’s sense). Here, Badiou seems to be naming something like the sheer moment of differentiation that must be supposed to hold sway between a set and its elements—a moment which, structurally, recalls nothing so much as the nothingness that, to Sartre’s eyes, interposes itself between impersonal consciousness and its objects, and which arises as an ‘impalpable fissure’ that arises in the heart of consciousness in the event of its reflexive presentation.[ii]

I would argue that it is this notion of the void that allows us to make productive and non-trivial sense of Badiou’s thesis that

for the void to become localisable at the level of presentation, and thus for a certain type of intrasituational assumption of being qua being to occur, there must be a dysfunction of the count, resulting from an excess-of-one. The event will be this ultra-one of chance, on the basis of which the void of a situation may be retroactively discerned (BE 56; trans. modified).

One way of explaining this thesis would be to show how an ‘event’, a moment when presentation folds back on itself, breaking with the axiom of foundation and foiling the extensional regime of ontological identity, formally replicates what Sartre called ‘the immediate structures of the for-itself’.[iii] The void—the ‘nothingness’—which erupts in such circumstances is not described by a set theoretical ontology, but a Sartrean one—one capable of systematically articulating the relation between the ‘void’ exposed by an event and the form of subjectivity to which an event gives rise (including its dimensions of temporality, possibility, normativity, and liberty). What remains lacking is a univocal concept through which these various senses of ‘void’ can be synthesized.

See also: EVENT, ONTOLOGY, ONE, SUBJECT, STRUCTURE, SET THEORY, SUTURE, SARTRE

---

[i] Badiou & Tzuchien Tho, ‘The Concept of Model, Forty Years Later: An Interview with Alain Badiou’, in CM, p. 99.

[ii] Jean-Paul Sartre (1956), Being and Nothingness: An Essay in Phenomenological Ontology, trans. Hazel E. Barnes, New York, Philosophical Library, pp.77-8

[iii] Cf. Sartre, Being and Nothingness, esp. Part II, Chapter I, Section I.

ONE ENTRY

Today, Form & Formalism brings you an entry on the notion of 'The One' in Badiou's work (focussing more or less exclusively on Being and Event). This text may or may not turn up in the BADIOU DICTIONARY, alongside the others I've been posting. An entry on 'The One', it turns out, had already been commissioned from another author (I'm not sure who), and so this entry is really just an understudy. I wasn't planning on preparing an entry on this concept at all, in fact, but just happened to stumble across it in a pile of scraps I had left over from my contribution to BADIOU: KEY CONCEPTS (the word limit on that text had been lowered at the last minute, and so I ended up with plenty of scraps on the cutting room floor---just as well, I suppose, as the short essay was starting to bristle with tangents). After a bit of pruning and scrubbing, the piece seemed to stand up fairly well as a dictionary entry, and so here it is:

========================

BADIOU DICTIONARY

ENTRY ON ‘THE ONE’
Z.L. FRASER

Being and Event begins with an announcement of the book’s inaugural decision: that ‘the one is not’ (23). There is no One, no self-sustaining unity, in being, but only the count-as-one, the non-self-sufficient operation of unification. There is no unity-in-itself, because every unity is a unity of something, something that differs from the operation of unification. This decision is no less fundamental for Badiou’s philosophy than his well-known equation of mathematics with ontology, and their metaontological meanings are deeply entangled; any attempt to isolate one from the other would mutilate the sense that Badiou gives its twin.

Through the prism of set theory, the non-being of the One refracts into three distinct ontological bans: (1) the prohibition of an ‘All’, or a set of all sets (by the ZF axioms, the supposition of such a set’s existence leads directly into the embrace of the Russell paradox)—an important corollary of this ban is that is no ‘set theoretical universe’ against which the various models of set theory can be measured, and so every coherent interpretation of the axioms will be pathological or ‘non-standard’ to some extent—there is no such thing as the standard model of set theory—a fact we must bear in mind when grappling with Cohen’s results, among others; (2) the ban on atomic elements, or units that are not themselves sets (ZF makes no provisions for unities that are not unities-of-something—with the possible exception of the void, into which the axiom of extensionality would collapse any putative ‘atoms’); and, we could add, (3) a self-unifying unity, a set that counts-as-one itself alone; schematically, the set W such that W = {W} (this set, which would in any case evade identification by the axiom of extensionality, is expressly forbidden by the axiom of foundation). 

Note that there exist axiomatizations of set theory which violate each of the three impossibilities by which we have translated, ‘the One is not’. There are set theories with a universal set V, such that for all e, e Î V (the One as All, as set of all sets);[i] there are set theories with urelements, elements u such that no element e belongs to u, but which are nevertheless distinct from the empty set (the One as atom, as a unity that is not a unity-of-something);[ii] and there are set theories with hypersets, sets X such that X Î X, or X Î A₁ Î … Î A_n Î X (the One as counting-itself-as-one, as self-presentation).[iii]

It is the identification of ontology not simply with mathematics but with a particular version of set theory (ZF) that therefore helps to motivate the decision that the One is not. Observing the existence of other One-affirming set theories emphasizes the particularity of this decision. The converse motivation—of the decision to identify mathematics with ontology by the decision on the non-being of the one—is somewhat murkier, but Badiou insists upon it. It is because the One is not, Badiou argues, that we must resist any temptation to subject being qua being to the unity of a concept. Subtracted from unity, ontology can articulate the sayable of being only by means of a non-conceptual regime of axioms, which regulate the construction of pure multiplicities without having recourse to any definition of multiplicity (BE 29). But why insist that concepts cannot deploy themselves axiomatically, by way of definitions that are purely implicit?[iv] Even if this is granted, nothing prevents a reversal of the argument. What, for instance, keeps an opponent from objecting to the placement of being-without-oneness under the ‘formal unity’ of an axiomatic, rather than submitting it to ‘the mobile multiplicity of the concept’?

See also: MULTIPLICITY, MODEL, SET THEORY, ONTOLOGY, INFINITY, VOID

---

[i] See Willard Van Orman Quine, ‘New Foundations for Mathematical Logic,’ American Mathematical Monthly 44 (1937), pp. 70-80.

[ii] See Ernst Zermelo, ‘Investigations in the Foundations of Set Theory I’, trans. Stefan Mengelberg, in Jean van Heijenoort, ed. From Frege to Gödel: A Source Book in Mathematical Logic, 1879-1931, Cambridge, Harvard University Press, 1967, pp. 183–198. Zermelo’s axiomatization of set theory with urelements, originally published in 1908, actually pre-dates the Zermelo-Fraenkel axiomatic which does without urelements and recognizes only sets as existent.

[iii] See Aczel, Peter. Non-well-founded sets. Stanford: CSLI, 1988.

[iv] Replying to Frege’s accusation that an axiomatic operating with undefined terms is conceptually vacuous, the same Hilbert we met at the beginning of this chapter argues that, on the contrary,

If one is looking for other definitions of a ‘point’, e.g., through paraphrase in terms of extensionless, etc., then I must indeed oppose such attempts in the most decisive way; one is looking for something one can never find because there is nothing there; and everything gets lost and becomes vague and tangled and degenerates into a game of hide-and-seek. […T]o try to give a definition of a point in three lines [of text] is to my mind an impossibility, for only the whole structure of axioms yields a complete definition. Every axiom contributes something to the definition, and hence every new axiom changes the concept. (Hilbert, Letter to Frege, dated December 29^th, 1899, pp. 39-40; e.a.)

It’s strange that Badiou would, without argument, reshackle the concept to the Fregean notion of definition, after arguing (some twenty years prior to writing Being and Event),

that the concepts of a science are necessarily of undefined words; that a definition is never anything more than the introduction of an abbreviating symbol; that, consequently, the regularity of the concept’s efficacy depends on the transparency of the code in which it figures, which is to say, on its virtual mathematization. (Badiou, ‘(Re)commencement de la materialisme dialectique,’ p.464, n.28; e.a.)

Do ‘concepts’ deployed in this fashion refasten discourse to the One? On what grounds could we even give an answer to this question?

Our series of entries for the BADIOU DICTIONARY will conclude tomorrow with THE VOID, so stay tuned!

IDEOLOGY ENTRY

Here it is:
=========================================================

BADIOU DICTIONARY

IDEOLOGY ENTRY

Z.L. FRASER

            Badiou’s most explicit meditations on the topic of ideology appear in a series of texts written over the course of a decade or so, stretching from the late ’60s to the late ’70s. The series divides in two: the first sequence, all composed prior to the events of May ’68, aim to think ideology as that from which thought subtracts itself, impurely and interminably, whether through aesthetic process or epistemological break. The second sequence, in which a faithful articulation of the uprising’s consequences is at stake, and in which political rebellion comes to actively condition Badiou’s philosophy, aim to think ideology itself as a mode of struggle and process of scission.

Ideology: Before ’68

In 1967’s ‘The (Re)commencement of Dialectical Materialism’, Badiou distils a highly schematic concept of ideology from his teacher’s work, breaking ideology into the three imaginary functions of repetition, totalization and placement, which serve

(1)  to institute the repetition of immediate givens in a ‘system of representations […] thereby produc[ing] an effect of recognition [reconnaissance] rather than cognition [connaissance]’ (RMD 449);

(2)  to establish this repetitional system within the horizon of a totalized lifeworld, ‘a normative complex that legitimates the phenomenal given (what Marx calls appearance),’ engendering ‘the feeling of the theoretical. The imaginary thus announces itself in the relation to the ‘world’ as a unifying pressure’ (RMD 450-1).

(3)  to interpellate both individuals and scientific concepts (crossbred with ideological notions) into the horizons of that lifeworld (RMD 450, 450 n.19).

In the background of these three functions is what any Marxist analysis must take to be the ideology’s ultimate aim, which is ‘to serve the needs of a class’ (RMD 451, n.19) – by which is meant, however tacitly, the dominant class. Badiou’s earliest works have little to say about this most basic function of ideology, and even less to say about Althusser’s quiet conflation of ideology tout court with the category of dominant ideology – but this complacency (which, it should be noted, is not uninterrupted – The Concept of Model (1968) marks an important, but ultimately inadequate, exception) will not survive the rebellion mounted in Of Ideology, to which I will return in a moment.

In his first theoretical publication, ‘The Autonomy of the Aesthetic Process’ (1966 – written in ’65), Badiou describes how art, though it does not tear a hole in ideology as science does, nevertheless serves to subtract thought from ideological domination by capturing the latter in ‘the discordant unity of a form: exhibited as content, ideology speaks of what, in itself, it cannot speak: its contours, its limits,’ (APE 80) decentring the specular relation that ideology works to preserve, and exposing the audience to the ‘outside’ surface of ideology’s infinite enclosure:

If ideology produces the imaginary reflection of reality, the aesthetic effect responds by producing ideology as imaginary reality. One could say that art repeats, in the real, the ideological repetition of that real. Even if this reversal does not produce the real, it realizes its reflection. (APE 81)

If ideologies, as Badiou suggests in The Concept of Model, play themselves out as continuous variations on absent themes (CM 7), then the point of the aesthetic process is to expose those themes in their presence themes by capturing them in their form.

            The second mode by which thought subtracts itself from ideology is science, conceived as a sequence of epistemological breaks. Ideology confronts scientific practice in the form of what Bachelard termed epistemological obstacles. In ‘Mark and Lack: On Zero’ (1969 – written in ’67), Badiou contends that epistemological obstacles affect scientific discourse in the form of an unstable suture of the scientific signifier (see entry on suture). Epistemological breaks must therefore act on structure of the signifier itself: they demand a labour of formalization, desuturing and stratifying the scientific signifier, assembling it in an inhuman machine that tears through the fabric of ideological enclosure. The structure of the scientific signifier comes to foreclose every ideological recuperation, but this radical dissonance with ideology is not accidental. It is the constitutive engine of scientific practice:

it is not because it is ‘open’ that science has cause to deploy itself (although openness governs the possibility of this deployment); it is because ideology is incapable of being satisfied with this openness. Forging the impracticable image of a closed discourse and exhorting science to submit to it, ideology sees its own order returned to it in the unrecognizable form of the new concept; the reconfiguration through which science, treating its ideological interpellation as material, ceaselessly displaces the breach that it opens in the former. (MM 173)

Science thus proceeds in an endless dialectical alternation of scientific rupture and ideological recapture – a dialectic that structurally corresponds to that which Badiou will later describe as taking place between truth and knowledge.[i]

            Ideology is the ubiquitous medium of thought and practice, within and against which art and science operate. Philosophy’s task cannot, therefore, be one of purifying thought – whether scientific, artistic or philosophical – of ideology. Its task, as formulated in The Concept of Model, following the direction of Althusser’s ‘Philosophy and the Spontaneous Philosophy of Scientists’, is to draw abstract lines of demarcation between ideology and the subtractive practices it unstably envelops – but this demarcation is not an end in itself. It is carried out for the sake of new ideological-scientific syntheses. In fact, the Badiou of 1968 defines philosophy as ‘the ideological recovery of science,’ the manufacture of ‘categories, denot[ing] ‘inexistent’ objects in which the work of the [scientific] concept and the repetition of the [ideological] notion are combined’ (CM 9). It is clear that this vocation is futile so long as the category of ideology, itself, remains undivided – subsumed, root and branch, under the category of dominant ideology. The philosophical necessity of this division is already legible in The Concept of Model, whose attempt to trace ‘a line of demarcation’ between the scientific concept of model and its bourgeois-ideological recapture is explicitly oriented towards readying the concept’s ‘effective integration into proletarian ideology’ (CM 48). But the theory of this division is not yet clear, and so, for want of a clear articulation of the difference between dominant and resistant ideologies, The Concept of Model can only end with this promissory note.

Ideology: After ’68

            The reader of Badiou’s post-’88 works may recognize in the aesthetic process and the epistemological break an anticipation of the later conception of art and science as truth procedures. Only after ’68 does the third condition arrive in full force, and it is the entrance of political rebellion onto the scene that will force the division of the category of ideology that is needed if the philosophical fabrication of categories is to be justified. This fission comes to a head in a 1976 pamphlet, coauthored with François Balmès under the title, Of Ideology. Badiou and Balmès’ first (and powerfully Sartrean) move is to insist on the transparency of ideology: 

We must have done with the ‘theory’ of ideology ‘in general’ as imaginary representation and interpellation of individuals as subjects […] Ideology is essentially reflection, and in this sense, far from being an agent of dissimulation, it is exactly what it looks like: it is that in which the material order (which is to say, the relations of exploitation) is effectively enunciated, in a fashion that is approximate, but nonetheless real. (DI 19)

Following a merciless critique of the Althusserian theory of ideology (within which Badiou’s initial reflections on the topic took shape), Balmès and Badiou lay down the rudiments of a properly Marxist and militant theory of ideology. They begin by drawing a line between the ideology of the exploiters (the ‘dominant ideology’) and the ideology of the exploited. There can be a ‘dominant ideology’ only where there are people who are dominated, and those who are dominated will resist, whether powerfully or weakly: It is from the standpoint of this resistance that the concept of ideology must be formulated. In resisting domination, the exploited form a more or less systematic representation of the real and antagonistic class relations that exploit them. This representation contains the germ of the ideology of the exploited class – the germ of an ideology of resistance. It is in a resistance to the ideological resistance of domination that the dominant ideology takes shape, struggling, not to deny the existence of contradictory class relations – which could only be a product of blindness or stupidity – but to downplay their antagonistic character. Its platform is threefold:

(i)    Its first move is to contend that ‘[e]very apparent antagonism is at best a difference, and at worst a non-antagonistic (and reconcilable) contradiction.’ (DI 40)

(ii)  Its second is to maintain that ‘[e]very difference is in itself inessential: identity is the law of being, not, of course, in real social relations, but in the ceremonial register of regulated comparisons before destiny, before God, before the municipal ballot-box.’ (DI 40)

(iii) Its ‘third procedure is the externalization of the antagonism: to the supposedly unified body politic [corps social] a term ‘outside of class’ [hors-classe] is opposed, and posited as heterogeneous: the foreigner (chauvinism), the Jew (anti-Semitism), the Arab (racism), etc. The procedures of transference are themselves riveted [chevillées] over an exasperation of the principal contradiction.’ (DI 40; n.27)

Resisting this resistance of resistance to domination, the ideology of the exploited may become an active ideology of rebellion. To do so, ‘revolt must produce an inversion and reversal of values: for it, it’s the differential identity of the dominant ideology that’s the exception, and it is antagonism that is the rule. It is equality that’s concrete, and hierarchy exists abstractly’ (DI 41). In this exponentiation of resistance the communist invariants take shape: egalitarian, anti-proprietary and anti-statist convictions, which, Badiou and Balmès argue, are not specific to proletarian revolt, but genuinely universal, legible in every real mass revolt against class exploitation (DI 66-67). These invariants comprise the contents of resistant ideology, and not necessarily its form, which it as a rule is inherits from the ideology of the dominant class (the communist invariants inscribed in Müntzer’s peasant rebellion, for instance, were couched in a religious form inherited from the ideology of the landowning class).

This division between content and form – with the form of an ideology deriving from the ideology it resists, and its contents reflecting the real class forces that drive it – supplies Badiou and Balmès with a straightforward way of accounting for false consciousness. ‘Illusion and false consciousness,’ they write,

concern the form of representations, and not their content. That a small-time union boss might hold the sincere conviction that he speaks in the name of the working class, and even has the backing of a tawdry Marxism, when he bends over backwards to liquidate a mass revolt, that’s false consciousness – but only so far as the formal side of the question goes. The truth is, our little revisionist is invested by the force of the bourgeois class, which his thought quite adequately reflects. (DI 32)

It is here that the Marxist formation of a proletarian party becomes crucial to the organization of revolt, in its function of welding the correct ideas of the masses – the invariant, communist contents of mass revolt – to the scientific form of Marxism. It is this that sets the proletariat – the organized proletariat – apart from the exploited classes of the past, for while it ‘is not the inventor of ideological resistance, it is its first logician’ (DI 128).

SEE ALSO: SUTURE, FORCING, MODEL, SPLACE, ENCYCLOPAEDIA, REPRESENTATION, STATE, HISTORY, CAPITALISM

---

[i] For details on this correspondence, see Z.L. Fraser, Translator’s Introduction to Alain Badiou, The Concept of Model, (Melbourne: re.press, 2007), § VII in particular. 

AN OVERVIEW OF MEDIAEVAL LOGIC

Looking for a helpful overview of the Mediaeval study of logic? Well, look no further. A Mediaevalist friend of mine, Adam Langridge, just sent me a link to what looks like a fascinating overview of the topic---P.V. Spade's very helpful text, "Thoughts, Words and Things: An Introduction to Late Mediaeval Logic and Semantic Theory"---which you can find here: http://pvspade.com/Logic/docs/Thoughts,%20Words%20and%20Things1_2.pdf

I'll be back tomorrow with another entry from the forthcoming BADIOU DICTIONARY, on the topic of IDEOLOGY.

MODEL ENTRY

Continuing our series of roughly-drafted entries for the upcoming Badiou Dictionary (a massively collaborative effort being edited, as I write, by the indefatigable Steve Corcoran), here's a freshly cut-and-pasted entry on Badiou's use of the word "MODEL":

BADIOU DICTIONARY

MODEL ENTRY

Z.L. FRASER

The Concept of Model is the first book Badiou published in philosophy, and in it he initiates a lifelong concern not only with mathematics and mathematical logic, but also with the ways in which philosophy can receive these disciplines as a condition for the philosophical thinking of truth and change. The concept of model, itself, will go on to occupy a pivotal position in Badiou’s work, orienting in productive and problematic ways his later use of mathematical set theory, and, as Oliver Feltham has argued, giving him an apparatus by which to think the compossibilization and interaction between various truth procedures in addition to mathematics. But what is a model?

            The simplest, and least adequate, answer is that a model is a pair, consisting of (1) a structure that a given formal theory can be taken to be theory ‘about’, and (2) an interpretation that systematically, and functionally, links the terms of the theory to the structure in question, in such a way that we can say that the axioms of the theory are ‘true’ or ‘valid’ for the model, and in such a way that the rules by which the theory transforms its axioms into theorems ‘preserve truth’. This simple idea can give rise to numerous misapprehensions, so it is best to go over things more carefully.

First off, we should resist any temptation to view the model/theory distinction as the distinction between an object and its discursive representation. This, by Badiou’s lights, is the error of the empiricist epistemology of models (CM 18-22). It is inadequate on two counts: To begin with, the model/theory distinction is, strictly speaking, internal to mathematical practice: both a formal theory and its models are mathematical constructions, and no structure can ‘deploy a domain of interpretation’ for a mathematical theory if it is not already embedded ‘within a mathematical envelopment, which preordains the former to the latter’ (CM 42). The point of interpreting a structure as a model for a theory (or interpreting a theory as the theory of a structure) is not to mathematically represent something already given outside of mathematics, but to generate a productive interaction between already-mathematical constructions, opening each to new, essentially experimental techniques of verification and variation: determining the relative intrications and independences among concepts, establishing the extent of a concept’s mobility and applicability, sounding out unseen harmonies between apparently heterogeneous domains, and exposing what the logician Girard has called ‘disturbances’ and points of ‘leakage’, the ‘cracks in the building’ which ‘indicate what to search and what to modify.’[i] Freeing it from the doublet that binds representations to their objects, Badiou proposes

to call model the ordinance that, in the historical process of a science, retrospectively assigns to the science’s previous practical instances their experimental transformation by a definite formal apparatus. […] The problem is not, and cannot be, that of the representational relations between the model and the concrete, or between the formal and the models. The problem is that of the history of formalization. ‘Model’ designates the network traversed by the retroactions and anticipations that weave this history: whether it be designated, in anticipation, as break, or in retrospect, as reforging. (CM 54-5; trans. modified)

That it is indeed a network of relations that are at stake in the concept of model, and not the bilateral mirror-play of object and representation, is pressed on us by the fact that, in general, no privileged relation obtains between a syntactically formulated theory and a structure interpreted as its model: more often than not, a theory admits of a vast multiplicity of models, which only in the rarest of cases map on to one another in any strict sense (where a strict mapping—or, precisely, an isomorphism—exists between all the models of a theory, that theory is said to be categorical, but this is quite uncommon); similarly, a given structure can in most cases be equipped with distinct interpretations, each of which making of it a model for quite different formal theories. It is even possible, with a bit of tinkering, to interpret the literal structure of a formal theory as a model for the theory itself—a technique which often proves useful in logic (an example of this technique is given in the Appendix to The Concept of Model).[ii]

The (‘ideologically’ motivated) intuitions that push us to see the mirror-play of object and representation in the model/theory distinction are strong ones. It is instructive to learn that even Paul Cohen—to whom we owe some of the most significant proofs that have ever been written regarding the relation between Zermelo-Fraenkel Set Theory and its models, including his proof of the independence of the continuum hypothesis, in which the concepts of forcing and the generic, so decisive for Badiou’s philosophy, first see the light of day—would confess that

The existence of many possible models of mathematics is difficult to accept upon first encounter […]. I can assure you that, in my own work, one of the most difficult parts of proving independence results was to overcome the psychological fear of thinking about the existence of various models of set theory as being natural objects in mathematics about which one could use natural mathematical intuition.[iii]

An avatar of this prejudice—which Cohen magnificently overcame—is the distinction between standard and non-standard models, which is even today commonplace in mathematical literature. The ‘standard’ model of a theory, in a nutshell, is simply the structure that the theory is intended to describe, together with an interpretation that puts things together in the expected manner. A ‘non-standard’ model is a structure and interpretation that deviates, often wildly, from these educated expectations. (To put it another way, a ‘standard interpretation’ obeys the spirit of the law; a ‘non-standard’ one adheres only to its letter.) Though he rarely addresses this distinction head-on, ever since his remarkable study of Abraham Robinson’s non-standard analysis (the non-standard model that Robinson constructed for the infinitesimal calculus), Badiou has engaged with mathematics in such a way that the distinction between the standard and the non-standard can confront his readers only as an obstacle to understanding. Nowhere is this distinction less pertinent than in set theory, and no single insight does a better job of linking Badiou’s ontological use of set theory with his pronouncement that ‘the One is not’ than the realization that, in all rigour, a standard model for set theory does not exist. If there is anything that set theory is expected to be a theory about, it is the ‘universe of all sets’, but it was a theorem already known to (and considered to be of tremendous importance by) Georg Cantor that the set of all sets cannot exist, on pain of inconsistency.[iv]

            If set theory is ontology, but an ontology which, ungrounded by the annulment of the One, has no standard model, then there is every reason to expect that the rules for its interpretation cannot be given in one stroke—a fact which has caused no end of frustration for Badiou’s exegetes—and that they must (within strict but underdetermining constraints) be reinvented situation by situation. The difficulty that remains, of course, is that of escaping the iron strictures of The Concept of Model, which forcefully argues that only an already-mathematical structure can model a mathematical theory. This may be true for mathematics qua mathematics, but it cannot (on pain of a philosophical suture) be maintained for mathematics qua condition for philosophy. The philosophical category of model, conditioned by the mathematical concept, can not remain (as it does in ’68), a purely epistemological category. What is needed, as Oliver Feltham has forcefully argued, is a category of ‘modelling’ that

is the inverse of the procedure of conditioning. In modeling the syntax is constructed in philosophy and then tested in diverse semantic fields such as revolutionary politics or Mallarmé’s poetry. In contrast, with conditioning it is a particular generic procedure such as set theory that provides the syntax and philosophy provides the semantic domain: hence ‘metaontology’ is a model of set theory. (Feltham, 132)

This inverse operation is not contrary to, but demanded by philosophy’s mandate to compossibilize radically heterogeneous conditions, for

if it must circulate between a multiplicity of artistic, scientific, political and amorous conditions, [philosophy] can never be perfectly faithful to one truth procedure alone. Thus, with regard to the comparison between modeling and conditioning, one cannot simpy assert that it is always a truth procedure alone that furnishes the syntax for the model; sometimes it is also philosophy that provides part of the syntax, based on its encounters with other conditions. (Feltham, 132)

It is in this light that we should see in the concept of model the first condition, issuing from the truth procedure of mathematics, of Badiou’s philosophy, the philosophical effects of which make it possible for Badiou, many years later, to put into practice a full and unsutured philosophy under conditions.

See also: Conditions, Suture, Forcing, Generic, Ontology, Mathematics, Set Theory, Ideology.

---

[i] See Jean-Yves Girard, ‘Linear Logic,’ in Theoretical Computer Science, vol. 50 (1987), p.14, and ‘Locus Solum,’ in Mathematical Structures in Computer Science, vol. 11 (2001), pp. 441, 485.

[ii] This is not to suggest that the relations between theories and models are so loose and wooly that they tell us nothing of interest about either structure. Quite the contrary: since Gödel’s famous completeness theorem (which is a bit less famous, perhaps, than his celebrated incompleteness theorems, at least outside of mathematics), we have known that there is a strict equivalence between saying that a theory has a model (that it is ‘true’ of a certain structure) and that the theory is consistent (that it doesn’t prove everything).

[iii] Paul J. Cohen, ‘The Discovery of Forcing,’ Rocky Mountain Journal of Mathematics, Vol. 32, Nº 4 (Winter 2002), p.1072.

[iv] Negligence of this fact, I believe, is responsible for the widespread impression that Cohen’s proof is something altogether ‘artificial’—impressions which draw their apparent strength from the fact that Cohen’s key procedure—the forcing of a generic extension to a model—depends on his decision to take as his starting point a countable model of set theory. ‘Countable’ here means that the elements of the model-structure can, if immersed in a sufficiently large super-structure, be shown to be in a one-to-one correspondence with the set of natural numbers—but which, in itself, lacks all the ties that would bind its terms to such tiny infinities, and so is not ‘countable for-itself’ and capable of harbouring interpretations of any theorem about transfinite sets that the theory can throw at it. The astonishing fact that such a creature could be a model for a theory of sets of infinitely many different, and ascending, orders of infinity is what was shown to be true by the Löwenheim-Skolem theorem—whose authors, it should be noted, saw their result as something approaching a reduction to the absurd of the formalistic and axiomatic methods of set theory’s pioneers, and as the eternal inadequacy of the letter to the spirit. History would retain the theorem, but invert its moral, winning for formalistic and model-theoretic methods an unprecedented array of freedoms.

========================================================

Stay tuned for "SUTURE"!

GENERIC ENTRY

BADIOU DICTIONARY:

GENERIC ENTRY

Z.L. FRASER

The concept of ‘the generic’, which Badiou first deploys in Theory of the Subject in an essentially metaphorical reflection on the subjectivizing production of excess (271-4), comes into full philosophical force in Being and Event, where it is taken up to describe the ontological – set-theoretical – structure of a truth-procedure: the total multiplicity that will have been composed of all the elements in the situation that a faithful subject positively links to the name of an event (by way of a ‘fidelity operator’), from the perspective of this multiplicity’s always-futural and infinite completion, takes the form of a generic subset of the situation in which the subject of truth operates. As a consequence of their genericity, truth-procedures exhibit at least five critical traits: (1) their indiscernible, unpredictable and aleatory character; (2) their infinitude; (3) their excrescence relative to the situation; (4) their situatedness, and (5) their universality. The concept of a generic subset, itself, was first formulated by the mathematician, Paul J. Cohen, in his 1963 proofs of the independence of the Generalized Continuum Hypothesis (GCH) and the Axiom of Choice (AC) relative to the axioms of Zermelo-Fraenkel set theory (ZF). The problem Cohen faced was this: Kurt Gödel had already shown (in 1940) that both GCH and AC are consistent with ZF by showing that if ZF has a model, then a model can also be produced which satisfies ZF supplemented by GCH and AC. This means that one can never prove the negation of GCH or AC on the basis of ZF, but it does not imply that the statements themselves can be proven. To show that ZF is no more able to entail these theses than their negations, Cohen sought to construct a model in which AC and GCH fail to hold. This would show that they are independent – or undecideable – relative to ZF. Cohen’s strategy was to alter Gödel’s model S (in which GCH and AC do hold) by supplementing it with (i) a single element ♀ and (ii) everything that can be axiomatically constructed on its basis. The supplemented construction S(♀) must both be capable of satisfying the ZF axioms (and so remain a model of ZF), while encoding the information needed to falsify GCH or AC (information which can be extracted by the forcing procedure). The difficulty is this: though it suffices to encode the many ZF theorems concerning transfinite sets, ‘from the outside’ (when embedded in a sufficiently rich super-model, that is) Gödel’s model-structure appears to be countable (it can be placed in a one-to-one correspondence with the set of natural numbers). (The surprising fact that set theory has such models, if it has any at all, is guaranteed by the Löwenheim-Skolem Theorem.) Any supplement carrying that kind of information would spoil the structure’s claim to be a model of ZF, and so

♀ must have certain special properties if S(♀) is to be a model. Rather than describe it directly, it is better to examine the various properties of ♀ and determine which are desirable and which are not. The chief point is that we do not wish ♀ to contain ‘special’ information about S, which can only be seen from the outside […]. The ♀ which we construct will be referred to as a ‘generic’ set relative to S. The idea is that all the properties of ♀ must be ‘forced’ to hold merely on the basis that ♀ behaves like a ‘generic’ set in S. This concept of deciding when a statement about ♀ is ‘forced’ to hold is the key point of the construction.[i]

Leaving technical subtleties aside, the idea is to construct ♀ in such a way that for every predicate or ‘encyclopaedic determinant’ restricted to S (where ‘restricted’ means that its constants and quantified variables range only over elements of S), ♀ contains at least one element which fails to satisfy this predicate. This suffices to determine the generic: (1) as indiscernible, insofar as no predicate can separate it from the swarming multitudes of S, and for this reason the generic must present itself in time as unpredictable and aleatory, its lawless composition impossible to forecast; (2) as infinite, since it remains essentially possible to determine any finite multiplicity by means of a complex predicate, even if this is only a list of its constituents (the syntactic constraints of set theory, if nothing else, prevent us from ever writing an infinitely long formula); (3) as excrescent, meaning that it is a subset but not an element of the ‘situation’ (the model in which the generic is articulated), the reason for this being that if ♀ was an element of S, then the predicate ‘x Î ♀’ alone would be enough to capture it; (4) as situated or immanent, since genericity is by no means an absolute property, but one which is relative to the model in which it is articulated; (5) as universal, since the generic outstrips every mark of particularity to the extent that no element of the model is excluded from entering into a generic subset by reason of the predicates it bears. Finally, though it must be connected to an essentially non-mathematical (non-ontological) theory of the event in order to do so, genericity helps to capture the idea that truths are effected through the work of a subject whose existence precedes and outstrips its essence. The ‘existentialist’ resonance that the concept of genericity brings to the Badiousian theory of the subject must be taken seriously, for it bears directly on obstacles accompanying trait (2): insofar as every actual truth-procedure unfolds undeterministically in time, each procedure is at any actual moment, finite, and can lay claim to genericity only by projecting itself ahead of itself, by being the future it factically is not: the infinite truth-multiple that it seeks to complete but which it cannot fully determine in advance.

See also: MODEL, FORCING, ONTOLOGY, SET THEORY, MATHEMATICS, SUBJECT, TRUTH, EVENT.

Further reading:

BE: Meditations 26–36.

TS: Seminar of May 15, 1978, ‘Logic of the Excess’.

Paul Cohen (1966), Set Theory and the Continuum Hypothesis, New York, W.A. Benjamin.

---

[i] P.J. Cohen, Set Theory and the Continuum Hypothesis, (New York: W.A. Benjamin, 1966), p.111. (Notation modified to parallel Badiou’s in Being and Event.)