IDEOLOGY ENTRY

Here it is:
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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

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[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.

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[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.

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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.

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[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.)

FORCING ENTRY

Hello, world.

I should begin with a quick word about what this blog is. The Form & Formalism Working Group began in November, 2009, in the wake the first annual "Form & Formalism" conference, held at the Jan Van Eyck Academie in Maastricht, and orchestrated by Tzuchien Tho of the Versus Laboratory research project. A second conference followed in 2010, and Versus is in the process of planning a third for the coming Fall. (Programmes for both FF conferences can be found here: http://versuslaboratory.janvaneyck.nl/events/view/5 and here: http://versuslaboratory.janvaneyck.nl/events/view/11.) From the conferences formed the group, and from the group now comes the blog. Nothing else needs to be said about this just yet.

To get the ball rolling, I've decided to make available here a few short texts that I've been working on, still in a somewhat rough state, for the Badiou Dictionary that Steve Corcoran is in the process of pulling together for Edinburgh University Press. Your comments, corrections, criticism, etc. are of course welcome.

I'll try to post an entry every day or so over the next week. Today, FORCING. Stay tuned for GENERIC, MODEL, SUTURE, IDEOLOGY, ONE, and VOID.
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DEFINITION OF FORCING

‘Forcing’ (forçage) is among Badiou’s signature expressions. From 1968 onwards, it has traced out an axial category in his work. Despite certain shifts in meaning, it has sustained a cluster of ideas and concerns that are, throughout, invariant. The core idea seems to be this: what Badiou calls ‘forcing’ is in each case a radical and systematic transformation of a situation by means of series of actions acting upon, or proceeding from, the real of the situation—that which, prior to the activity of forcing, subsists in the situation as an invisible, unoccupiable, or ‘impossible’ site, occluded by knowledge and cloaked by (the dominant) ideology. Invariant too is that it is in each case mathematics that conditions the category of forcing.

We can split Badiou’s use of the term into two periods, each of which presents a crucial variation on this central theme. The pivotal texts of each period, in which the concept of forcing is formulated or reformulated, are ‘Infinitesimal Subversion’ and Being and Event. (For completeness’ sake, Theory of the Subject should also be mentioned, but despite the importance that a whole array of related and analogous concepts of force, torsion, etc. play in that text, the idea of forcing, itself, appears only in a transitional capacity.)

Forcing in ‘Infinitesimal Subversion’

The context of ‘Infinitesimal Subversion’ is the project undertaken by several members of Le Cercle d’épistémologie—the working group behind Les Cahiers pour l’Analyse, and to which Badiou belonged in the last years of the 1960s—to develop a general theory of structural change, informed by both Lacanian psychoanalysis and Althusserian historical materialism—a project which gave Badiou’s enterprise its initial and lasting coordinates. With minimal violence, we can characterize it according to the following theses, which find their canonical expression in Jacques-Alain Miller’s ‘The Action of Structure’ (published alongside ‘Infinitesimal Subversion’ in Cahiers vol. 9):

1.     The structure of a situation always has at least one ‘empty place’, a place which cannot, according to the structure, be occupied. It is characterized by a certain structural impossibility, as the ‘Real’ of the situation. Jacques-Alain Miller calls this the ‘utopic point’ of the structure (97).

2.     The empty place is, in general, indiscernible. It is a ‘blind spot’, unstably masked or ‘sutured’ by ideological or imaginary illusion. 

3.     “Any activity which does not play itself out entirely in the imaginary but which is to transform the state of the structure,” Miller writes, “departs from the utopic point, the strategic post,” specific to the situation. (97)

If the reader of these remarks recognizes in the idea of a ‘utopic point’, not only an echo of Lévi-Strauss’ ‘floating signifier’ and a fellow traveler of Deleuze’s ‘empty square’ of structure, but a prefiguration of Badiou’s later category of the ‘evental site’, then she is on the right track. But first we must turn to the text where the idea of transforming a situation from the bias of its utopic point is first thought through under the condition of mathematics, for it by placing this notion under the mathematical condition that the category of forcing is won.

This all gets underway in ‘Infinitesimal Subversion’, where Badiou transports Miller’s schema into the laboratory of formal mathematics and model theory, in an analysis of Abraham Robinson’s invention of non-standard analysis. The situation’s structured space of possibilities here becomes the space of inscriptions allowed by a formal axiomatic—the formulae that it can demonstrate. The Real, the ‘utopic point’, becomes the place of the underivable, the space unoccupiable by formulae licensed by the formalism. That every consistent formalism is punctuated by such impossibilities is an iron necessity; if every place could be occupied, and every expressible formula written down as a theorem, formalism would become “an opaque body, a deregulated grammar, a language thick with nothing” (SI 122), which is to say, inconsistent. A formalism is only consistent—and so, in virtue of Gödel’s completeness theorem, interpretable in a model—if there is at least one formula which it can express but not demonstrate; it is “owing to the exclusion of certain statements, the impossibility of having the constants occupy certain constructible places, that an axiomatic system can operate as the system it is, and allow itself to be thought differentially as the discourse of a real” (122).

            The demarcation of an unoccupiable place is made precise in mathematics with its syntactical distinction between constants and variables. The system of finite arithmetic, for instance, allows no constant—no integer—to be substituted for the y in the expression ‘For all integers n, n ≤ y’—but by recourse to the variable ‘y’ it is able to mark this inoccupiable place, without, for all that, occupying it.[1] “A variable,” Badiou writes, “ensures that impossible equations are sufficiently legible to read their impossibility;” it is the "operator of the real for a domain, it in fact authorizes within that domain the writing of the impossible proper to it. The existent has as its category a being-able-not-to-be the value of a variable at the place it marks." (SI, 122)

            ‘Forcing’ is a procedure of radically transforming the structure by occupying one or more of its real, unoccupiable places, without for all that collapsing the structure into sheer inconsistency. It begins with an act of nomination, the definition of a constant that occupies an inoccupiable place, closing one of the open and formerly unsatisfiable sentences in which only variables could once be written. Robinson’s intervention consists in defining a new constant a and axiomatically stipulating it to be such that for all real numbers n, n ≤ a, a gesture which, by occupying the inoccupiable, marks an intrusion of formalization into the real that was its impasse.[2] (Badiou calls occupation the inscription of an ‘infinity-point’, though the general concept is meant to apply to constants like the ‘imaginary number’ with which Bombelli breached the x in ‘x² + 1 = 0’, which the existing algebra had declared inoccupiable.) The forcing procedure continues with a submission of the new constant to all the remaining operations of the initial system—a, for instance, can be added to, divided by, and so on, and so Robinson is able to define infinitesimals simply as multiples of 1/a.  In sum,

the infinity-point is the marking of something inaccessible for the domain; a marking completed by a forcing of procedures, constraining them to be applied to precisely that which they had excluded. Of course, this forcing entails a modification of the way in which the domain is set out, since the constructible objects in the higher domain are able to occupy places which those of the domain itself ‘inoccupy’. The new space in which the procedures can be exercised is disconnected from that which preceded it. The models of the system are stratified. (SI 120)

This brings about not merely an extension, but a transformation of the domain in question: new patterns are unleashed, old ones often destroyed. And so Badiou goes on to identify this forcing procedure as a “reforging” (refonte) of the structure, connecting it explicitly with the theory of epistemological breaks, a theory which he inherits, with modifications, from Althusser and Bachelard—a recognizable prototype of the theory of truths unleashed in Being and Event.[3]

Forcing in Being and Event

            Between the theory of forcing presented in ‘Infinitesimal Subversion’ and the one we find in Being and Event, intervenes the a new and decisive condition: a technique developed by Paul J. Cohen in his proof of the independence of the Generalized Continuum Hypothesis and the Axiom of Choice from the axioms of Zermelo-Fraenkel set theory (ZF), which likewise appears under the name of ‘forcing’. Before we address its incorporation into Badiou’s philosophical apparatus, we will take a quick look at forcing in its native, mathematical terrain.

It is, once again, set-theoretical model theory that provides Badiou with the requisite conceptual (scientific) material. Like Robinson’s procedure for the making of ‘non-standard’ models, Cohen’s forcing technique is, at bottom, a systematic way of generating a new model from a model already given. The main thrust of Cohen’s proof is to take a countable, transitive model[4] of ZF and ‘force’ the existence of a new model by supplementing it with a generic element included in, but not belonging to, to initial model—together with all the sets which can be constructed on the supplement’s basis by licensed by the ZF axiomatic. Considered in its logical structure, forcing is a relation of the form ‘a forces P’, where a is a set and P a proposition that will hold in the generic extension of the initial model—provided that a turns out to belong to the generic supplement on which that extension is based. In this respect, forcing resembles a logical inference relation, but one that differs markedly from the inference relation of classical logic—the law of the excluded middle, in particular, does not hold for the forcing relation, and the logic it generates is essentially intuitionistic.[5]

As Cohen has shown, the consequences this supplementation can be quite extraordinary, and go far beyond simply adding a new set’s name to the census. The generic supplement, for instance, may be structured so as to induce a one-to-one correspondence between transfinite ordinals that, in the initial model/situation, counted as distinct orders of infinity, thereby collapsing them onto one another and making them effectively equal. Cohen exploited this possibility to great effect by taking the model that Gödel had built in order to show that the Generalized Continuum Hypothesis (GCH)—the thesis that the size of the set of subsets set of any transfinite cardinal number À_n is equal in size to the next greatest cardinal À_n+1—is consistent with ZF (a model in which the continuum hypothesis holds), and on its basis forcing a generic extension in which the continuum hypothesis fails (the extension being a model in which the set of subsets of À_n is demonstrably equal to almost any cardinal whatsoever, so long as it’s larger than À_n), thereby demonstrating the consistency of GCH’s negation with the theory, and hence the independence, or undecidability, of GCH with respect to ZF.

Being and Event recovers Cohen’s concept and enlists it in a re-articulation of the existing category of forcing: the set underlying the model is now seized upon as the situation that forcing will transform, and faithful Miller’s cartography of change, Badiou adds that the whole procedure—both the articulation of the generic truth and the forcing of its consequences for the situation to come—must in every case proceed from an anomalous occurrence in the ‘utopic point’ of the situation in question, now rechristened ‘evental site’. Though it is now Cohen rather than Robinson whose mathematics condition Badiou’s theory of change, the new category of forcing preserves most of the features familiar to us from “Infinitesimal Subversion.” One crucial difference, however, is that the whole process is now seized as a logic of subjective action: Forcing is now names “the law of the subject” [CITE], the form by which a subject faithful to an event transforms her situation into one to which a still-unknown[6] truth (understood as a generic subset of the initial situation) well and truly belongs, by deriving consequences that the inscription of this new constant will have brought about.

In light of Being and Event’s decision to interpret ZF as the theory of being qua being, and forcing as the form of a subject’s truth-bearing practice, Badiou extracts two lessons from Cohen’s proof of the undecidability of GCH: first, that it demonstrates the existence of a radical ontological gap or ‘impasse’ between infinite multiplicities and the sets of their subsets (to which Badiou associated the notions of ‘representation’ or ‘state of a situation’), the exact measure of which is indeterminate at the level of being-in-itself; second, that this ontological undecidability is nevertheless decidable in practice, but only through the faithful effectuation of a truth, suspended from the anomalous occurrence of an event.

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[1] It goes without saying that variables can also be used to mark occupiable places.

[2] This is a bit of a simplification. Robinson’s procedure is carried out not only with respect to ≤ but for every relation R such that for every finite set of constants {a₁, …, a_n}, there exists a y such that a₁Ry & … & a_nRy. (Robinson calls these relations ‘concurrent’; Badiou, with typical pizzazz, calls them ‘transgressive’.) For each of these relations R_i, the idea is to introduce a new constant a_i which is axiomatically stipulated to be such that yR_ia_i for all y. Amongst these relations, and certainly the most interesting for the present case, is ≤.

[3] In describing this procedure, Robinson himself does not use the word ‘forcing’—which, at the time of his invention of non-standard analysis, had recently been given a technical coinage by Paul Cohen, whose ‘forcing’ concept would (by 1988) become a decisive condition for Badiou’s project of developing the philosophical category of forcing that we already see in motion here. Interestingly enough, a year or so after the publication of “Infinitesimal Subversion” in the Cahiers pour l’analyse, Robinson turned his attention to Cohen’s work and, on its basis, developed a new form of forcing, in Cohen’s sense, which is now known as ‘infinite model theoretic forcing’ or, simply, as ‘Robinson forcing’. Badiou, to my knowledge, has not commented on this device.

[4] A model structure M is transitive if x belongs to M whenever x belongs to y and y belongs to M (for all x, y). For a quick explanation of how ZF can have countable models, see the entry for model.

[5] For details, see Z.L. Fraser, “The Law of the Subject: Alain Badiou, Luitzen Brouwer and the Kripkean Analyses of Forcing and the Heyting Calculus”, in The Praxis of Alain Badiou, Melbourne, re.press, 2007: pp. 23-70.

[6] According to Hao Wang, Kurt Gödel once described forcing as “a method to make true statements about something of which we know nothing.” (Hao Wang, Kurt Gödel: A Logical Journey, (Cambridge, Mass.: MIT Press, 1996), 252.)