The priority of Alain Badiou’s first theoretical texts, written in the mid 1960s, is to assert the self-sustaining and self-enclosed sufficiency of alternatives to ideology. Following his mentor Louis Althusser, Badiou understands ideology in terms of a subject’s lived experience, a domain of illusions and of imaginary reflections of reality. The alternatives – scientific discourse (in particular mathematics) and aesthetic production (in particular literature) – offer true knowledge of their specific domain. The main task of theory, then, is to secure such knowledge against the persistent danger of ideological contamination, by defending its essential autonomy.
In this sense, a true or non-ideological discourse is one that encounters nothing other than itself. A true discourse would be one that can never be interrupted, neither by the intrusion of some external object nor by some inner gap or absence, some inner ‘fault’ or inconsistency that might elude articulation. A true discourse would be one whose outside – ideology, empirical reality, historical continuity, subjective experience, popular practice … – is conceived in such a way as to figure exclusively and unequivocally as outside, an outside that could never intrude into the discourse that excludes it. Insofar as the Cahiers pour l’Analyse set out to explore a ‘general theory of discourse’, Badiou’s insistence on such formal self-sufficiency (in his contributions to the journal’s final volumes) carries one aspect of the project to its logical conclusion, precisely as an attempt to purge this theory of its own ‘inner’ limit, that ‘extimate’ and elusive figure affirmed in its inaugural texts – the figure of the subject, whose place is marked by what discourse can only indicate as a gap or lack.1
What then happens, in May 68, is that something happens. The events of May interrupt Badiou’s scientific project once and for all, both from within and without. The ‘outside’ overcomes, in practice, its theoretical exclusion. His project interrupted, Badiou, like the other leading figures in the Cahiers, quickly abandons abstract theory for concrete practice and in the following years dedicates himself entirely to revolutionary politics, before attempting, in the later 1970s, to combine the two domains in his own ‘theory of the subject’.
I want to argue here that the best way to understand the overall movement of Badiou’s philosophy, in terms that account both for its inaugural assertion and its subsequent development, is as a form of fidelity to the logic of interruption as such. What occurs in the defining stages of this movement is a conversion, after 1968, of this logic from a merely transcendent one (whereby interruption can only be thought as if stemming from a mere absence or outside) to an immanent one (such that the exception that interrupts a discourse or situation can be thought as a sort of internal excess or gap, i.e. in terms that might be affirmed by a subject). This shift will allow Badiou to reorient his philosophy in terms that embrace rather than exclude the subject of interruption as such. Badiou’s philosophy acquires its distinctive shape when he is able to return to the inaugural thesis of the Cahiers, the thesis he himself tried to foreclose in its final issue, and find his own way of articulating the relation between subject and structure.
On the one hand, then, the moment of May 68 – for Badiou, the paradigmatic instance of what he will later conceive in more general terms as event, rupture, revolution, disruption, ‘grace’, and so on – indeed figures, at the level of both theory and experience, as a matter of interruption pure and simple. This aspect of Badiou’s philosophy, the aspect of subject and event, has the general status of an exception, of an excess or supplement. It is a matter of ‘supplement’ not in the deconstructive sense, as the indication of an irreducible lack or ‘deficiency’ in the situation or discourse that is supplemented,2 but on the contrary, as a pure addition, i.e. a pure intrusion or interruption from outside. And this is because, on the other hand, Badiou’s understanding of that which is interrupted is conceived in such a way as to ensure that whatever supplements it, whatever is added or happens to it, can indeed only be understood on the basis of an interruption or addition as such (rather than as an extension, intensification or transformation of existing features of the situation, e.g. as an alteration of its fundamental tendencies, capacities, and so on). This broad conception of ‘that which is interrupted’ will later apply as much to the ontology he develops in Being and Event (1988) and the theory of appearance he develops in Logics of Worlds (2006) as it does to the analysis of art and science proposed in his earliest theoretical articles (1966–68), which will be my chief concern here.
Badiou’s ‘mature’ theory of truth is essentially a theory of fidelity to a particular sort of interruption, one whose affirmation has universal and egalitarian consequences. But his philosophy is also faithful to a general theoretical perspective that obliges such interruption to be understood essentially as interruption, i.e. a perspective that understands reality or ‘what there is’ in such a way that for something new or true to happen it must happen to it. What takes place takes place in excess over its place. Badiou is faithful both to the interruption of any self-sufficient logic, and to the presentation of such logic as self-sufficient. It is in this sense that, contrary to what readers of his most purely political 1970s writings might suspect, Badiou’s later philosophy might be described as faithful both to the ultra-theoreticist work he wrote in the mid 60s and to the way he broke with such work in and after 68. Alone of all his contemporaries, Badiou has remained faithful to the theoretical project of the Cahiers pour l’Analyse, precisely in the way that he helped to interrupt it.
I FROM SARTRE TO ALTHUSSER
When Badiou left the École Normale Supérieure in the summer of 1961, already seasoned by years of campaigning against the Algerian War, he remained an enthusiastic admirer of Sartre and Sartre’s effort (in his 1960 Critique of Dialectical Reason) to think of the relation of subject and social structure, or of freedom and constraint, from the position of the subject. Even before he left the ENS, however, Badiou was already interested in many of the new formalist projects that helped to establish the nascent ‘structuralist’ paradigm in philosophy and the human sciences – new axiomatic approaches to the foundations of mathematics, new work in formal logic and linguistics, the legacy of Bachelard and Koyré, Lévi-Strauss’s work on kinship, and perhaps especially, Lacan – still very much a marginal figure outside psychoanalytic circles. By 1964, in the run-up to the Reading Capital seminar, Badiou was also reading Althusser with enthusiasm, and in the spring of 1965 he taught a broadly Althusserian course on the relation between literature and ideology back at the École Normale. By the end of the following year Badiou’s shift in orientation, from Sartrean freedom to Althusser’s structuring ‘process without a subject’, appeared definitive.
When in early 1967 Badiou joined the editorial team of the Cahiers pour l’Analyse, then, he joined as a convert to structuralism, with both the enthusiasm and the perspective that comes with conversion.3 This is one of several important differences between him and the slightly younger normaliens involved with the journal, who all came of age at the ENS directly through their engagement with Althusser and Lacan. Badiou’s range of reference and experience was wider than that of Jacques-Alain Miller, Jean-Claude Milner or Yves Duroux, since he had the ‘good fortune’, as he remembers, to have been fractionally older than the rest of his generation. Most of his younger contemporaries had ‘not had the time to be Sartreans’, or to engage in a political situation (like the struggle against colonialism) grasped through a Sartrean conception of commitment and principle.4
This first great interruption in Badiou’s thought – in brief, his conversion from Sartre to Althusser – already defines a fundamental problem that persists in all his subsequent work. How to think both the domain of rigorously structured objectivity and the domain of absolute freedom? The first effort aligns Badiou with Plato (and Althusser), and with the formal sufficiency of mathematics; the latter aligns him with Mao (and Sartre), and an equation of reason with revolt. Can these two efforts be thought together? Yes, Badiou will come to realize, precisely insofar as one interrupts the other. They can be thought together insofar as they can be considered aspects of an irreducible duality or ‘two’.5 Subject can be thought as exception, and objective structure as that which acknowledges and indeed ‘situates’ a kind of ‘impossible’ place for that exception – i.e., in Lacanian terms, a place for that ‘real’ whose affirmation appears impossible or inconceivable. Again like Althusser before him, Badiou draws here on the formalist, anti-empiricist approach to science developed by Bachelard, Koyré, and Canguilhem, with the proviso that the formalizing work of conceptual construction always leaves a space for the subject it excludes. In this sense, Badiou considers, ‘there is a general movement of thought’ common to this generation of French philosophers, ‘wherein we all agree on the fact that the world (or that which is) is arranged as a matter of formal objectivity, one that is foreign to consciousness and valid on its own terms.’ It ‘is precisely because this formal objectivity exists that one can search for and define the point [the gap or impasse] that exceeds it. And in this point the subject, or its possibility, takes place.’6 Badiou has dedicated much of his later philosophy to thinking this point that is both internal to and in excess of structure, the point that can be converted from objective impasse to subjective passe.7
But this project too will have come as an interruption, an interruption of Badiou’s ultra-formalist priorities in 1966–67. ‘In 1967’, he is the first to recognize,
I was indeed at the extreme point of a strict formalism […]. The fact that my thought is rooted in Platonism, which I’ve never denied, even when I was a convinced Sartrean, sometimes leads me to oscillate between a radical priority of the question of the Subject, on the one hand, and on the other a pre-eminence of the Idea, or of the truth, whose intelligible substructure, whose purest model, is to be found in the historical development or life of mathematics. Subjectively, for me this means that politics and mathematics constitute the two major ‘appeals’ on the side of what I call the ‘conditions of philosophy’, and that these two appeals are always in tension.8
Things will begin to fall into place when, after apparently banishing it without appeal, the neo-Sartrean domain of engagement and the subject returns, in May 68, to upset the theoretical edifice Badiou had been building in the previous years. We need then to review the construction of this edifice in more detail, before considering what suspended it.
II THE AUTONOMY OF ART
Like his younger Cahiers colleagues, what is at stake in Badiou’s conversion to Althusserianism is the sufficiency of scientific theory as the means by which analysis can grasp a structuring dimension that remains inaccessible to those who simply experience or live its effects. Science can only proceed by overcoming the ideological obstacles that threaten its autonomy, i.e. by defending itself against interruption from the domain of non-science. Badiou’s first two theoretical essays aim precisely to secure the domain of analysis from such interruption, either from without (a ‘reality’ or event external to it) or from within (a lack or absence that might be inaccessible to it). In art as in science, the goal is to confirm foreclosure of all that is foreign to its self-demarcating domain. As we shall see in the penultimate section of this essay, Badiou’s critique of Miller in the final issue of the Cahiers pour l’Analyse is motivated by the same goal.
Badiou’s article on ‘The Autonomy of the Aesthetic Process’ (APE), dated June 1965 but published in the Cahiers Marxistes-Léninistes in July 1966, summarizes the course that Althusser had invited him to give, that spring, at the École Normale.9 His main concern is to show how a distinctively aesthetic process, unlike a merely ideological one, generates its own sphere or ‘reality’ and thus renders itself independent of anything external to it. He takes for granted Althusser’s elementary categories: a ‘real’ element is one grasped by science, say, a particular configuration of class forces in a particular place and time; an ‘ideological’ representation of such a reality is one that ‘reflects’ and distorts it according to positions determined by that configuration. Badiou’s opening question, then, is whether the aesthetic process that produces a genuine work of art is best understood merely as one of several ‘regional domains’ of ideology, i.e. as one among many ideological reflections of real elements external to it, or as an autonomous process operating only on elements internal to this process itself.
Classical Marxist literary criticism, for instance Lenin’s reading of Tolstoy, develops the first approach. Although it operates in ways that are no doubt convoluted and more or less inaccessible to its author himself, a literary work is supposed to reflect a real social situation, such that, in principle, the task of ‘true’ literature should be to reduce such convolutions to a minimum (as will be expected, for instance, of a truthful socialist realism). Badiou’s contemporary and fellow Althusserian Pierre Macherey offers a more sophisticated variant of this first approach, whereby to study a work ‘consists in demonstrating the relations it maintains with the historical structure’ that shapes the social reality of its day.10 Badiou applauds Macherey for recognizing that the indirect or convoluted quality of literary representation is in fact essential to (rather than destructive of) the peculiar effects it produces. Macherey shows how a text truly reflects a social reality precisely by what it fails to say about that reality, by the way it omits certain deplorable aspects, censors its descriptions, etc. A reader attentive to the gaps and silences in the text can thereby ‘see’ what subjects of the ideology itself cannot see. By rendering such absences present (as absences), by making these silences audible (as silences), this sort of Marxist literary criticism operates as a privileged domain of ideology critique.
What Badiou rejects is the assumption common to both Lenin and Macherey, that a text relates to and operates on a reality external to it (APE, 82). For Macherey, a work exists in ‘an internal relation to that which is other than it, as immanent contradiction’ or as ‘the phenomenon of an internal difference’, such that it renders ‘present’ i.e. visible or legible precisely ‘that which holds itself in absence’: insofar as all ideology reflects reality then the peculiar object of literary criticism will be the ‘absence of certain reflections’ in the distinctively ‘broken mirror’ that is a literary work.11 For Badiou, by contrast, an aesthetic process should not be understood with reference to such an immanent alterity, i.e. in terms of gaps or absences internal to its logic (and that by interrupting this logic, thereby point to a reality external to and other than it). In Badiou’s view, what art creates is not a mirror, be it broken or whole, of something else. Instead, art produces its own reality, according to its own logic, to the exclusion of any outside.
Badiou accepts, of course, that ‘ideology produces an imaginary reflection of reality’ (APE, 81), and agrees that ‘the aesthetic effect is certainly imaginary; but this imaginary is not the reflection of the real, since it is the real of this effect’ (77). Moving beyond Macherey’s notion that literature is ideology rendered visible (because its reflection is broken or incomplete), Badiou argues that we must conceive of the aesthetic process not as any sort of reflection or ‘redoubling [of the real] but as a reversal [retournement]’ of the reflecting relation between ideology and reality, i.e. as the conversion of a mere ideology into a kind of self-sufficient reality (albeit an imaginary one). If ideology produces an imaginary reflection of reality, a distinctively aesthetic process ‘produces in return [en retour] ideology as imaginary reality. We might say that art repeats in the real the ideological repetition of this real. Nevertheless this reversal does not reproduce the real; it realizes its reflection’ (81). In other words: ideology reflects reality but art, reversing ideology, does not merely reflect reality; instead it renders real or lends a sort of reality to the reflections produced by ideology.12
If we are to ‘take its autonomy seriously’ (85), Badiou continues, then by definition an autonomous art cannot merely reflect and thus depend on a reality external to it. Instead it must produce its own reality-effect, in the same way that a structuring cause generates the immanent field of its effects. By submitting them to the local ‘efficacity of structural causality’ (rather than the global indeterminacy of an ‘expressive causality’) and thus assigning them ‘a place in a structure’, the aesthetic process converts the elements of mere ideological reflection into an effectively self-sufficient (albeit imaginary) reality. Affirmation of the autonomy of the aesthetic process ‘blocks us from conceiving it as relation’, and consequently ‘the problem of the passage from ideology to art cannot be formulated as such’ (83). As soon as it begins, the aesthetic process already operates exclusively in its own self-enclosed domain, such that if ideological statements appear within it then they too will have been ‘produced as ideological’ by the aesthetic process itself (85). The regional autonomy (or ‘regional structural causality’) of the aesthetic process is such that ‘the “raw material” of aesthetic production is already itself aesthetically produced.’
For this argument to work, Badiou needs a way of accounting for the incorporation of patently ideological statements within a literary text, statements like the expression of a maxim or a political opinion. In keeping with a logic that will recur in both the Cahiers Marxistes-Léninistes and the Cahiers pour l’Analyse, he accommodates the inclusion of such ‘separable ideological statements’ by distinguishing between the given or structured work on the one hand and its structuring or aesthetic process on the other. A work may well include separable statements, but these will ‘function in isolation’ and ‘owe nothing to the structure of the work’, i.e. to its aesthetic process per se, which is the sole concern of analysis (84). Unlike a separable ideological statement that only figures in a work as external and unrelated to the process that produces it, aesthetic statements or ‘generalities’ are formulated as internal to the imaginary reality produced. Badiou gives as an example a sequence in Dostoyevsky’s The Demons, in which an opinion (to the effect that suffering ennobles the person who suffers) that might be understood as a ‘separable’ maxim if read in isolation is in fact formulated in such a way as to require us to read it as an integral part of the narrated sequence (one that culminates in an actually noble gesture on the part of our sufferer).
The distinctive power of the aesthetic process is thus its capacity to generate an imaginary scenario (here, a compelling representation of an action) as self-sufficient, i.e. as real, such that ‘generalities of another order cannot enter or remain’ (87). The more general task of the analyst of such production will then be to determine precisely how a specific mode of the aesthetic process operates, how it forms and thus renders ‘present’ certain kinds of significations. (A specific ‘mode of aesthetic production’, Badiou suggests in his tentative conclusion, is an ‘invariant and invisible structure that distributes ways of linking real elements in such a way that these elements can function as ideological’; his examples of such modes include the ‘figurative space’ of representational art, the tonal system of classical music, the metric system of ancient Greek verse, and ‘the system of subjectivity in the novel’ [88]).
The domain of the effects produced by an autonomous aesthetic process, in sum, can be neither hollowed out from within nor interrupted from without. Understood along these lines, art is thus ‘structurally closer to science than to ideology’ because, by ‘realizing’ or rendering-real ideology, it ‘produces the imaginary reality of that which science appropriates, i.e. real-reality’ (77).
III THE AUTONOMY OF SCIENCE
Unhindered by art’s imaginary orientation, science itself proceeds more directly. Badiou’s next article, a substantial review of Althusser’s For Marx and Reading Capital, published in May 1967 under the title ‘The (Re) Commencement of Dialectical Materialism’ (RMD),13 pivots on the Bachelardian assertion that, unlike ideology and the reflections of lived experience, ‘science is precisely the practice that has no systematic sub-structure other than itself, no fundamental “ground”.’ Whereas ideologies are concerned with the way objects are lived, represented and experienced, the ‘proper effect of science – the effect of knowledge – is obtained by the rule-governed production of an object that is essentially distinct from the given object’ (RMD, 443n.10, 449). In the domain of science, as Bachelard recognized, ‘nothing is given, everything is constructed’,14 on the basis of an absolute and ongoing break with the given and the ideological. Genuine i.e. post-Galileen physics, for instance, has no relation with the common-sense understanding of nature associated with Aristotelian philosophy; rather than criticize or even deny the existence of nature in this sense, it has ‘nothing to say about it’ (443n.9). Once Marx establishes his science of historical materialism, likewise, he is no longer required to ‘invert’ or ‘subject to critique’ the ideological categories associated with Hegel’s philosophy ‘for the simple reason that they are no longer encountered, they cannot be found, so much so that we cannot even refer to their expulsion, since the space of science is constituted on the basis of their radical absence’ (443).
The radicality of the ‘epistemological break’ that establishes it allows science itself to escape reduction to a mere element of the ‘super-structure’ of a social order, and thus to escape the general constraint of historical determination in the usual sense. As Althusser puts it, a science ‘may well arise from an ideology, detach itself from its field in order to constitute itself as a science, but precisely this detachment, this “break”, inaugurates a new form of historical existence and temporality which together save science […] from the common fate of a single history: that of the “historical bloc” unifying structure and superstructure.’15 Badiou’s main reservation regarding Althusser’s effort to re-launch dialectical materialism as a general ‘science of the scientificity of the sciences’ (RMD, 446) is simply a concern that the project does not go far enough in establishing its central concepts (‘structure’, ‘determination’, ‘the dominant instance of a structure’, etc.) on the basis of a purely ‘formal’ and thus self-sufficient discipline. A general science of historical structure, Badiou suggests, actually requires as its foundation a ‘theory of historical sets’, to be worked out in a ‘strict dependence’ upon a wholly formal version of such a theory, namely mathematical set theory.16 Badiou speculates that refinement of such a theory should allow for ‘the concept of a conjuncture to be constructed’ in a suitably rigorous way (463), while regretting the hasty and half-hearted character of Althusser’s own recourse to formalization.17 Having made only an incomplete break with philosophy and ideology, Althusser is occasionally ‘blind’ to their persistence in his own work; as a result the status of his general theory of science remains uncertain, torn between a neo-Kantian reconstruction of the conditions of knowledge and a neo-Spinozist theory of structuring causality (466–467).
Purged of any ‘falsifiable’ reference to empirical reality or experience, the only practices that might qualify as unequivocally scientific are indeed mathematical, in the sense exemplified by David Hilbert’s formalist conception of geometry. For Hilbert, elementary geometric objects like points and lines are not ideal approximations to objects in physical space but undefined terms that conform to axiomatically asserted procedures governing their manipulation. ‘If the arbitrarily given axioms do not contradict each other with all their consequences, then they are true and the things defined by them exist’, with no need to verify this existence through some form of observation or experiment.18 ‘The axiomatic method’, Hilbert concludes, thus ‘guarantees complete freedom of investigation. To proceed axiomatically means nothing else than to think with knowledge of what one is about.’19 In line with such Hilbertian formalism, Badiou proceeds to insist, in his contributions to the Cahiers, that a suitably formal mathematical logic manipulates nothing other than the ‘marks’, letters or symbols that it prescribes for itself, in the absence of any external relation to objects or things. ‘Mathematical experimentation has no material place other than where difference between marks is manifested’, and ‘neither thing nor object have the slightest chance here of acceding to any existence beyond their remainderless exclusion.’20 No more than an aesthetic process, a scientific i.e. mathematical inscription in no sense represents, reflects let alone ‘imitates’ an object external to it.
Although there is not space to consider it here, the same principle governs the concept of mathematical model Badiou went on to defend in a pair of lectures that were quite literally interrupted by the events of May 68.21 As Zachary Luke Fraser explains, ‘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.’22 More precisely, what inclusion of the dimension of model offers here is a way to think the internal historicity of science’s relation to itself. In opposition to representational conceptions of model like that offered in Lévi-Strauss’s conception of kinship relations (whereby science is conceived ‘as the confrontation between a real object, about which one must inquire [ethnography], and an artificial object whose purpose is to reproduce the real object, imitating it in the law of its effects [ethnology]’) (CM, 10), Badiou proposes
to call model the status [statut] 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 category of model thus designates the retroactive causality of formalism on its own scientific history, the history conjoining object and use. And the history of formalism will be the anticipatory intelligibility of that which it retrospectively constitutes as its model.
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 (CM 54–55tm).
Perhaps even more than the (ideological) subject, the (empirical) object is the primary locus of imaginary illusion and the immediate occasion for scientific i.e. Platonist distrust.23
IV FORM AND SUBVERSION
The essential thrust of Badiou’s two contributions to the Cahiers, and of his early theoretical position in general, follows from his ultra-Althusserian assertion that ‘science is [a] pure space, without inverse or mark or place of that which it excludes’ (CpA 10.8:161). The main question at issue in the first of these contributions, ‘Infinitesimal Subversion’, is, as its title suggests, the abruptly ‘subversive’ power of scientific formalization, its capacity to interrupt and then exclude the ideological categories of continuity, quality and temporality (CpA 9.8:136). The first half of Badiou’s article is concerned with the status, in any numerical system, of an empty place for a ‘number’ that is excluded as impossible within the limits of that system; the second half of the article explores the process whereby Abraham Robinson’s non-standard analysis allows, in the domain of the standard number system, infinitesimal or ‘infinitely small’ numbers to come to occupy such an initially ‘impossible’ place, and thus render the impossible possible.24 As with Badiou’s critique of Macherey’s emphasis on quasi-visible absences in the literary work, everything depends here on the way we think of this internal gap or empty place.
Consider the exemplary case of the natural number sequence, generated through repeated operations of numerical succession (1 + 1 + 1 + 1 …). Take any finite number n, however large: it is obviously possible, through the addition of a ‘supplementary inscription’ (n+1), to generate a larger number, and then another, ad infinitum. Badiou’s opening question concerns the place in which this unending inscription, so to speak, might come to be inscribed. If through each operation of succession number shows itself to be ‘the displacement of the place where it is lacking’ (CpA 9.8:118), is inscription in the place of such lack a sort of ‘representation’ or reflection of something external to the inscription, a sort of indication whereby the lack itself might somehow be rendered ‘visible’? No, says Badiou, following Mallarmé. The endless generation of finite numbers through succession simply presupposes the blank or missing place required for its operation, such that ‘it is what is written that bestows upon it its status as place of the writing that takes place.’ In one and the same process, mathematical writing writes both its inscription and the blank space ‘on which’ (so to speak) it is inscribed. The space into which a further number in our sequence succeeds does not pre-exist this succession, like an inner gap in the fabric of scientific writing. There is nothing foreign, here, to this writing itself – neither object of reference nor surface of inscription. Only insofar as we assume, ‘internal’ to the domain of science, a space that is actually and already endless can we actually continue to count out an unending series of additional numbers. ‘This is why the “potential” infinite, the indefiniteness of progression, testifies retroactively to the “actual” infinity of its support’ (118).
Our domain of natural numbers is thus composed of an unending series of places that successive members of this domain come to occupy. If then we want to posit a place that is unoccupiable by any member of the domain, it will have to be marked by an inscription that is fully ‘supplementary’ to the ordinary series of inscriptions which count out this domain. Such a place will indeed be purely external to the series. What Badiou calls an ‘infinity-point of the domain’ will be a number which, by breaking the rules of the domain, can be ‘forced’ to occupy this ‘unoccupiable empty space’ (119), with the proviso that, even if it can be constructed using the mechanisms of the domain itself, such an ‘impossible’ number must nevertheless be constructed in such a way as to remain ‘exterior’ to the domain.
To illustrate this, Badiou gives the example of the simple relation of order, i.e. the relation which measures one quantity as ‘greater-than’ or ‘lesser-than’ another, applied to the domain of natural numbers. This relation allows for the construction of an unoccupiable place, i.e. the place of a number which would be larger than all others in the domain. This place itself is easily constructed, since the statement ‘for all x, x < y’ is a perfectly intelligible statement of the domain; it cannot be occupied, however, by any actual finite number, however large. The variable y here marks an unoccupiable place; it thus indicates the outside limit of what can be written on the basis of our ‘infinity-support’. No actual number or ‘constant’ belonging to the system can be attributed to this place, i.e. substituted for the variable y, without contradicting the rules operative in the domain. It might seem then that there is nothing more to say: impossible is impossible, and that is all there is to it. On formalist grounds, however, there is nothing to prevent the mathematician from ‘forcing’ the point of impasse, and obliging the procedures operative in a domain to apply to ‘precisely that which they had excluded’ (120). In other words, we can simply impose a new constant (call it i) and define its meaning in terms of its ‘occupation of this transnumeric place, positing that, for every number n, n < i.’ By definition, this new number i will not then ‘count’ as a natural number. It will forever remain a pure addition over and beyond the domain it supplements. But if the initial procedures that generate the domain of natural numbers can be made to apply to it – if we can define a successor i+1 for i, and add and subtract i, etc. – then in principle there is nothing to stop us from imposing i as a previously impossible term, i.e. as an infinite number. (This is precisely what was at stake in the invention of ‘irrational’ and ‘imaginary’ numbers [123]). A similar argument, as Badiou goes on to explain in more detail, applies to the domain of the infinitely small rather than the infinitely large.
How then should we understand this version of an epistemological break, this brutal ‘occupation’ of a previously unoccupiable place? Since the place occupied by an infinity-point has already been marked with a variable (y), might we simply say that the variable itself has the immanent power, within the original limits of the domain, to ‘occupy’ the empty place? Is the potential inscription of an infinity-point thus already implicit within the actuality of the infinity-support, ‘such that the true concept of the infinite would already be enveloped in the mobile inscription of x’s and y’s’ (121)? In order to generate our new number is it enough merely to develop or stretch the existing resources of the situation? No, says Badiou, against Hegel. Mere inscription of a variable in an unoccupiable place (on the model, ‘for all x, x < y’) in no way inscribes an actual number or constant in that place; it attests simply to the fact that the resources of the system enable us to write or formulate ‘the impossibility of the impossible’ (122). Analysis can identify the real of a situation, but it is not possible, by remaining within the limits of the situation, to render the impossible possible.
In order to exceed the rules that limit mathematical writing to the domain it defines as possible, in other words, these rules must be interrupted by something wholly external to their domain. The ‘impossible’ must itself intervene and interrupt the possible. The impossible must occur as something that happens to the domain. Here Badiou follows Lacan’s lead: for any given ‘domain of fixed proofs, impossibility characterizes the real’ (CpA 9.8:122). All that can be formulated of this impossibility, within the normal confines of the domain, is simply that it is impossible; the unoccupiable place is just that, unoccupiable, and the variable that indicates it as unoccupiable exhausts what can be said of it. Or to return to the terms of Badiou’s argument with Macherey in 1965: the unoccupied or absent space testifies to nothing other than impossibility, and is not the indication of some hidden reality or resource that itself contributes to changing the domain. ‘Non-standard’ extension of the domain requires the abrupt imposition of an ‘infinity-point’ (i.e. a new infinite constant or number) into the impossible place. The infinity-point is imposed as a substitute for the variable, by asserting without mediation the ‘possibility of the impossible’. Or again, if the mere variable ‘realizes [réalise] the difference’ of a system by indicating the place of the term it excludes as impossible, the infinity-point ‘irrealizes’ it by including (or ‘hallucinating’) this term. Once included or inscribed, the new term persists in a relation of pure excess in relation to the ‘standard’ terms of the domain, i.e. the infinity-point for our relation of order will remain a term that is incommensurably larger than every other term in the domain. In a further confirmation of Bachelardian-Althusserian logic, the reconfiguration or ‘recasting’ [refonte] of the domain that inscription of such an exceptional point entails expands and opens it up in such a way that the ‘causality’ of the recasting itself is ‘dissipated in its operation’. Such dissipation shows how a cause can be effaced in the ‘apparatus of a structure’ (CpA 9.8:132–133).
From the perspective of the initial domain itself, there is thus no more possibility of ‘explaining’ and justifying the impossible inscription than there is of ‘deducing’, to the satisfaction of those who oppose it, the fact and necessity of a political revolution. ‘In science as in politics, it is the unperceived which puts revolution on the agenda’ (128). It does so, however, not by helping us to understand those under-appreciated or misrecognized aspects of the situation that might lead to its ‘inner’ transformation, so to speak, but by identifying the point at which it might be interrupted, from without, by that which it excludes.
V MARKING THE LACK OF THE SUBJECT
Towards the end of ‘Infinitesimal Subversion’, referring to Hegel’s contemporary Évariste Galois, Badiou expands on the general significance of his argument. It is ‘by establishing oneself in the constitutive silence, in the unsaid of a domainial conjuncture, [that] one maintains the chance of producing a decisive reconfiguration’ of this domain (CpA 9.8:128). Might we then say (with Macherey and Miller) that the task of analysis is somehow to lend voice to this silence or unsaid itself? Does the gap left by the impossible place only mark the possible place of a supplement or interruption from without – or might its status as gap, as lack of possibility or lack of self-identity, somehow contribute to the internal (i.e. unsupplemented) transformation of the domain? Might a gap in the discourse of science itself testify to an immanent exception to this discourse, i.e. a place for precisely that subject excluded from the Althusserian analysis of structure? No, says Badiou, following Gödel. One of the main goals of Badiou’s second and last contribution to the Cahiers, ‘Mark and Lack’ (CpA 10.8), is to refute this possible line of interpretation, by reiterating once again the unconditional expulsion, from autonomous science (i.e. mathematics), of all that is external to science. This means that scientific discourse must exclude any notion of lack that is invested with the qualities of an immanent exception ‘within’ the domain of science, i.e. a lack invested with the dynamism of self-contradiction and self-dissociation, a lack associated with non-self-identity: it must exclude, in other words, the sort of lack or néant characteristic of, among other things, Sartre’s theory of the subject.
Looking back at the ‘basic operation’ at work in the Cahiers, Badiou describes it as inspired by the ‘idea – which also seduced me, as I’ve often admitted – that it’s not because one engages in the most extreme formal rigour and takes up the intellectual power of mathematics, of logic, etc., that one must necessarily erase or abolish the category of the subject.’25 What this perfectly accurate description leaves out is the fact that Badiou’s seduction took place after the closure of the Cahiers, after May 68 interrupted its most emphatically formalist investigations of science and logic. His final contribution to the journal actually constitutes its most forceful argument against precisely this combination of rigour and subject, in another dazzling deployment of the former in order to secure the exclusion of the latter. As Duroux remembers, at the time of his participation with the journal Badiou rejected its founding premise, that ‘subjectivity was included in the structure’, because he rejected the whole ‘theory of the structuring lack’.26 And as Badiou himself remembers, during his Cahiers years, ‘I was [still] freeing myself from Sartre, existentialism and phenomenology. I would say that what first seduced me in my mathematical education was the non-subjective, the making possible of a capacity to think outside of all intentionality and subjectivity.’ At this point, Badiou believed ‘that if mathematics were to achieve the secrets of thought, it was because of its a-subjectivity. It seemed like a psychosis; that is to say, it was the automatism, a characteristic of the automatism of thought, a mechanical conception of mathematics, that I was concerned with in those days.’27
Although in a sense it was ‘left behind’ even before its belated publication in 1969, the argument that separates Badiou’s ‘Mark and Lack’ (CpA 10.8) from Miller’s ‘Suture’ (CpA 1.3) deserves to be recognized as one of the pivotal moments in the history of French structuralism. The Lacanian ‘logic of the signifier’ that Miller had proposed turns essentially on the representation of an irreducible lack: a signifier represents (i.e. ‘sutures’, treats-as-identical or counts-as-one), for another signifier, that essential lack of self-identity which is all that can be expressed of the subject.28 In ‘Mark and Lack’ Badiou rejects Miller’s attempt to formulate a logic that both expresses and is engendered by the subject’s lack of identity with itself, in favour of a strictly formal conception of logic purged of any reference to lack, non-self-identity or subjectivity.29 According to Badiou, in other words, Miller’s logic of the signifier is not really ‘logic’ at all. A properly scientific logic is a wholly ‘positive’ process, one that – in this respect comparable to an autonomous aesthetic process – ‘lacks nothing it does not produce elsewhere’ (CpA 10.8:151). The pseudo-logic proposed by Miller and Lacan serves merely to blur if not to ‘erase the epistemological break’ by which science distances itself from the domain of ideology, i.e. the domain of the subject and lived experience. Their logic of the signifier remains a form of ‘metaphysics: a representation of representation, an intra-ideological process’ (151; 151n.2). The implications of Gödel’s famous incompleteness theorem, Badiou adds, actually undercut rather than reinforce Miller’s efforts and Lacan’s gestures on this score, in effect reasserting the integrity of formalized science and restricting the ‘logic of the signifier’ to ideology alone.
The basic argument of ‘Mark and Lack’ stems from its (neo-Hilbertian) ‘inaugural confidence in the permanence’ and self-identity or self-substitutability of logico-mathematical marks or graphemes (CpA 10.8:156). Given any mark x, i.e. any x that refers to a number or variable, logic must always treat x as strictly and unequivocally identical with itself. Badiou thus takes for granted a position that Miller associates, in ‘Suture’, with Leibniz and Frege (CpA 1.3:43): the articulation of scientific truth depends on the exclusion of the non-identical, with the proviso that ‘the concept of identity holds only for marks’, i.e. mathematical inscriptions, rather than for the ‘objects’ to which these marks might refer. Science as a whole ‘takes self-identity to be a predicate of marks rather than of the object’, a rule which applies to the ‘facts of writing proper to Mathematics’ as it does for the ‘inscriptions of energy proper to Physics’, along with the instruments used to measure them:
It is the technical invariance of traces and instruments that subtracts itself from all ambiguity in the substitution of terms. Thus determined, the rule of self-identity allows of no exceptions and does not tolerate any evocation of that which evades it, not even in the form of rejection. What is not substitutable-for-itself is something radically unthought, of which the logical mechanism bears no trace […]. What is not substitutable-for-itself is foreclosed without appeal or mark (CpA 10.8:157).
On the basis of this scriptural self-identity, Badiou goes on to show how the essential operation of (genuine or scientific) logic – for instance the sort of logic at work in formal propositional calculus – involves the ‘separation’ or distinction of statements as valid or invalid on several distinct levels, or according to several stratified ‘mechanisms’. The primary level or mechanism (M1), the mechanism of ‘concatenation’, produces arrangements of a discrete set of elementary marks or letters, as a sort combinatorial ‘alphabet’ – by way of illustration, consider for instance all the possible combinations of the letters used in the English language. At a second level, the level of ‘formation’ (M2) or ‘syntax’, certain expressions are deemed acceptable or ‘well-formed’, and separated from those that are rejected as ill-formed (152); to pursue our linguistic analogy, at this second level we might distinguish between syntactically valid (though not necessarily coherent or meaningful) sequences of words, as opposed to random series of letters. The third level (M3), the mechanism of ‘derivation’, will further separate sequences that can be derived or ‘proved’ as valid theses from sequences or statements that cannot be thus proved, i.e. from ‘non-theses’. We might call this third level the domain of intelligibility or coherence, the level at which coherent expressions are distinguished from unintelligible albeit ‘well-formed’ strings of words.
Now whereas the distinction of well-formed from ill-formed expressions at level two is absolute and straightforward, the relation between derivable and non-derivable expressions at level three may, in any system complex enough to formulate basic arithmetical expressions, be either decidable or non-decidable. An undecidable statement is formulated in such a way that ‘neither it nor its negation is derivable’ or provable (153) – and although the details need not detain us here, Badiou proceeds to show how Gödel’s demonstration of the incompleteness of any relatively complex logical system applies to this level three (M3) alone. The important thing is that admission of such incompleteness or undecidability, pace Miller and Lacan, in no way threatens the self-sufficient or subject-less status of science, i.e. its exclusion of all lack or non-self-identity. No matter what is expressed of it, any mark x must itself always be identical to itself: as a sign or mark, every x must be and remain this same x. In other words, expression of non-self-equality at level two does not carry any ontological implications of non-self-identity at level one. It is certainly possible to express a lack of equality between x and itself, with the well-formed (though unintelligible) statement ‘x ≠ x’ – both x = x and x ≠ x are legitimate expressions at level two, i.e. the level of formation, but the expression x ≠ x is excluded as non-derivable or meaningless from level three (CpA 10.8:158–159).
Badiou’s point, against Miller, is that the expression of such non-self-equality does not threaten the identity of any variable x itself. Indeed, although it makes no sense to say it, it is only possible to say that ‘x is unequal to x’ if x necessarily remains the ‘same’ x in both instances. ‘The mere convocation-revocation of x’s non-self-identity, the shimmering of its self-differing, would suffice to annihilate the scriptural existence of the entire calculus.’ In other words, we can formulate logically coherent statements of non-self-equality (on the model x ≠ x) only if we first exclude all that is ‘scripturally non-self-identical’, such that ‘the lack of the equal is built upon the absolute absence of the non-identical’ (158). Miller’s evocation of a non-identical and thus non-substitutable thing (i.e. a subject) is thus ‘foreclosed’ here in advance, ‘without appeal or mark’ (157).
Miller’s logic of suture, like Macherey’s reading of absence, is therefore condemned to apply only to the domain of ideology, i.e. to the domain of the subject considered, as much by Sartre as by Lacan, as not-identical-to-itself. For the same reason
there is no subject of science. Infinitely stratified, regulating its passages, science is pure space, without inverse or mark or place of that which it excludes. Foreclosure, but of nothing, science may be called the psychosis of no subject, and hence of all: universal by right, shared delirium, one has only to maintain oneself within it in order to be no-one, anonymously dispersed in the hierarchy of orders.30
Purged of any reference to a subject, science qua mathematics remains an austere ‘archi-theatre of writing: traces, erased traces, traces of traces […], marks indefinitely substituted for one another in the complication of their entangled errancy’ (CpA 10.8:164). In terms reminiscent not only of Mallarmé, Blanchot, and Foucault, but also of the Derrida whose reflections on archewriting had shaped most of the fourth volume of the Cahiers (CpA 4.1), science here prescribes a signifying movement in which ‘we never risk encountering the detestable figure of Man’, no more than God, Spirit or any other figure of the subject (CpA 10.8:164).
Once again, however, the exclusion of this risk indicates the point of its possible violation – the point where philosophy tries to intervene in science. The lack of a subject in science, i.e. its radical lack of any lack, marks the eternal ‘torment [supplice]’ of philosophy. An ideological practice, ‘philosophy’ figures here as the desperate effort to locate a subject (be it logos, God, revelation, man, speech …) at the very point, indicated by science, where every figure of the subject is proscribed in advance. ‘Through science we learn that there is something un-sutured; something foreclosed, in which even lack is not lacking, and that by trying to show us the contrary, in the figure of Being gnawing at itself, haunted by the mark of non-being, philosophy exhausts itself trying to keep alive its supreme and specific product: God or Man, depending on the case’ (163).
It is hard to imagine a more emphatic assertion of the inviolable autonomy of science as a weapon against ideology and its subject. It is all the more remarkable, then, that even before it was published this assertion had already been interrupted, not least for its author, by nothing other than an intrusion of the subject, sparked by a revolutionary though evanescent upsurge of the masses. Badiou’s subsequent reorientation from scientific closure to a philosophy of perseverance, moreover, will involve an affirmation of the subject in precisely the place where that science had excluded it.
Badiou has never tried to downplay the force of the second great conversion experience of his life. As he was to write several years after the fact, ‘I admit without any reticence that May 68 was for me, in the order of philosophy as in everything else, a genuine road to Damascus experience’ (TC, 9; cf. DI, 24). ‘May 1968 was for us first and foremost a formidable lesson. We felt ourselves woken up, contested, by the immense collective anger of the people […]. After May, nothing is nor should be “as before”.’31 May revealed once and for all that ‘it is the masses who make history, including the history of knowledge’ (TC, 9). The first thing to go, then, is precisely the old Althusserian relation of science and ideology. Badiou now rejects out of hand any general ‘theory of ideology’ understood as an ‘imaginary representation’ sustaining the lived illusions of a subject. Instead, precisely because it is ‘essentially reflection’ or mirroring of reality, so then ‘far from being an agent of dissimulation, ideology is exactly the way things look [ce qui se voit]’, i.e. ‘the material order of things’, expressed from the perspective of either the exploiters or the exploited (DI, 19). In 1969 the great task is now nothing other than ‘the ideological preparation of the masses’, ‘a vast campaign of ideological rectification, so as to guarantee the progressive preponderance of Marxism-Leninism’. Once banished to the outside of science, ideological questions have suddenly become ‘questions of life and death.’32
Just as May 68 occurred as an unpredictable ‘supplement’ to the relatively stable state configuration of late 60s France, Badiou’s successive conceptualizations of the subject have always retained an ‘excessive’, exceptional or supplemental status. The subject is never ‘given but must be found’: it isn’t something one is or has so much as something one ‘arrives at’ (TS, 294–295/278–279). This remains the essential point, in all of Badiou’s post-68 work: everything turns on the affirmation of the abrupt appearance of something that ‘deregulates’ the existing state of things, something ‘was not there before, so that there is a supplementation, or a creation, a positive dimension, and that remains the point around which everything hangs together […]. The apparition of what there isn’t, that’s the origin of all true subjective power!’33 The logic of a radical ‘epistemological break’ is thereby preserved, precisely by applying it to the very domain it was initially formulated to foreclose: the domain of the subject. The alterity that was first excluded as pure transcendence or externality can now be incorporated as immanent excess. (And as for ‘suture’, it too can be recuperated, through an inversion – by shifting it away from its association with the subject and lack, and thinking of it in terms of being and void).34
As a supplement that happens to and exceeds a pre-existing order of things, moreover, the subject must itself, over time, be supplemented. Sooner or later there must be a supplement to the supplement. That’s because the subject’s tendency over time is to fade away, towards reconciliation with the situation it supplements. Although the consequences of the (previously ‘impossible’) event unfold from within the situation it supplements, the main priority is thus ‘to make sure that the initial distance established between the event and the order of the world, its law, its state, be preserved. The main danger is always that of self-dissolution under the dominant law [of the world as it is], and that the truth process fall under resignation to what there is.’35 This is the main danger, since over time the consequences of an interruption (a ‘truth-process’, in the jargon of Being and Event) tend towards ‘saturation’ or exhaustion. ‘Everything that is born deserves to perish, in time.’36 In due course, then, it won’t be enough to be faithful to the interruption; this fidelity must in turn be supplemented, and so in a sense must itself be interrupted, through invention of a new ‘fidelity to the [old] fidelity’. The Maoist consequences of May 68 have themselves been saturated, Badiou concedes, for at least twenty years. ‘Since the mid 80s, more and more, there has been something like a saturation of revolutionary politics in its conventional framework: class struggle, party, dictatorship of the proletariat, and so on.’ The alternative is then either to conclude that the revolutionary sequence is simply finished (so as to ‘move on with the times’ and the prevailing counter-revolution), as most of Badiou’s contemporaries have done, or else to affirm a new supplement. ‘Fidelity to the fidelity is not a continuation, strictly speaking, and not a pure rupture, either. We have to find something new’, and to affirm those new interruptions or events that allow us to continue in the affirmation they supplement.37
The subject conceived in Badiou’s post-Maoist work is not simply rare and destined to saturation; it also comes to be precisely through the imposition of a new formalization – as a new way of formulating and thus occupying (to use the terminology of ‘Infinitesimal Subversion’) the position of the impossible, unoccupiable or undecidable place. In the works he published in the 1980s, unlike the works he published in the 60s and 70s, Badiou found a way of conceiving the movement of such occupation precisely by rethinking that category and position excluded as ‘impossible’ or unoccupiable by his own initial conception of science, i.e. the position of the subject, a subject understood now as the subject of inventive or revolutionary formalization itself. On this condition, Badiou can refute his initial insistence that ‘there is no subject of science’ (CpA 10.8:161). With Being and Event, he develops an account of science in which it is not merely the domain of writing or inscription but the domain of being itself that must be decided, and in the most radically ‘subjective’ sense of the word – on the basis of a free choice, independent of all ‘objective’ determination or influence (i.e. as an axiom). The scientific revolutionaries of the seventeenth century, for instance, understood that to think being as being-infinite is ‘necessarily an ontological decision […]. What it took was a pure courage of thought, a voluntary incision into the – eternally defendable – mechanism of ontological finitism’ (BE, 167/148). Pursuing the consequences of this decision, in work undertaken by the mathematicians Gödel, Cohen and Easton, Badiou can now show how ‘quantity, this paradigm of objectivity, leads to pure subjectivity’ and the ‘almost completely arbitrary situation of a choice’ (BE, 309/280).
What Badiou will do in Being and Event with the suspended fragments of his early formalism is thus reconstitute them in an ontological form that remains faithful to both to their original autonomy and to their revolutionary disruption. The only difference – but it is precisely this difference that separates Badiou’s philosophy from his earlier scientism – is that he can now think interruption and formalization together, precisely as a logic of active, immanent, internally-excessive interruption. This allows him to uphold a fidelity to that fidelity which sustained the great formalizing projects of the twentieth century, in defiance of their apparent saturation. Against a traditional notion of form as the shaping of material or appearing of an object, what was at stake in the logic of the century was a conjunction of idea and act: formalization conceived through ‘the real grip of an act’ is ‘the great unifying power of all the century’s undertakings’ (LS, 225/159–160). In the domain of science, from Hilbert to Bourbaki, Badiou can now redescribe the formalizing project as an attempt ‘to reduce mathematics to its act’, an attempt to grasp this act as ‘the enunciation of the mathematical real and not an a posteriori form stuck onto an unfathomable material’ (LS, 229/162–163). In the domain of politics, at the furthest limit of its effort, Maoism overcame the gap between objective science and subjective commitment (the gap between Althusser and Sartre) by asserting that revolt itself is reason.38 There was then no longer a need to choose between scientific reason and the subject that suspends it, since revolt figured there (following Hegel) as both subject and as historical reason. Although this figure may be saturated today, Badiou’s post-Maoism remains faithful to the communist Idea it inspires.39
This is why, when Badiou stops to remember the legacy of the Cahiers today, he of all its contributors is the one who remains closest to its original effort. He is ‘the only one from the old team who remained faithful to the initial project’40 – i.e. precisely that project that he, at the time, sought to refute. On the one hand, only a formalizing science indifferent to lived experience can grasp the world in its objectivity, and only the impasse or deadlock of such formalization can touch its real; on the other hand, only the interruption of an event and the subject who affirms it, in immanent exception to objectivity, can transform impasse into passe, and propose through a new formalism a genuine ‘figure of universality’.41 From Sartre to Althusser to Mao to post-Maoism: Badiou’s philosophy has avoided the philosophical exhaustion evoked in ‘Mark and Lack’ (163) by finding a way to affirm each successive interruption as aspects of a continuing project.
1 For an overview see Peter Hallward, ‘Theoretical Training’, CF1.
2 See in particular Jacques Derrida, De la grammatologie (Paris: Minuit, 1967), 207–211; Of Grammatology, trans. Gayatri Chakravorty Spivak (Baltimore: Johns Hopkins University Press, 1997), 144–147.
3 See Yves Duroux, ‘Strong Structuralism, Weak Subject’, CF2.
4 Alain Badiou and Bruno Bosteels, ‘Can Change be Thought?’, in Alain Badiou: Philosophy and its Conditions, ed. Gabriel Riera (New York: SUNY, 2005), 242.
5 On the centrality of ‘two’ as a figure in Badiou’s philosophy, see Hallward, Badiou: A Subject to Truth (Minneapolis: University of Minnesota Press, 2003), 45–47.
6 Badiou, ‘Theory from Structure to Subject’, CF2, 283.
7 In Being and Event (1988), with a nod to Lacan, Badiou will show how the ‘impasse of being-as-being’ locates the ‘passe of the Subject’ (BE, 469/429tm).
8 ‘I only came to find the conceptual form of that tension’, Badiou continues, ‘once I understood that the most significant mathematical events might also provide the key to the subjective process of truths. That was the entire aim of Being and Event (1988), in the crossing, through the concept of genericity, of the mathematics of the pure multiple and the post-evental subjective trajectory that constructs a truth. In 1967, just before the political storm, my meditations were on the side of formal structures. For the ten years following it, I was rather on the side of political subjectivity. Philosophy really began for me after these oscillations, at the start of the 1980s’ (‘Theory from Structure to Subject’, 280). Already, with Maoism, Badiou had ‘found something that made it possible for there to be no antinomy between whatever mathematics is capable of transmitting in terms of formal and structural transparency, on the one hand, and on the other, the protocols by which a subject is constituted’ (Badiou and Bosteels, ‘Can Change be Thought?’, 243).
9 Badiou, ‘L’Autonomie du processus esthétique’, Cahiers Marxistes-Léninistes 12–13, special issue entitled ‘Art, langue: Lutte des classes’ (July 1966), 77–89. Althusser originally planned to include Badiou’s work on literature in his Théorie collection as a book on the ‘theory of the novelistic effect (l’effet romanesque)’ (Althusser, letter to Franca Madonia, 23 July 1966, in Lettres à Franca (1961–1973) (Paris: Stock/IMEC, 1998), 691; cf. 576).
10 Pierre Macherey, ‘Lénine critique de Tolstoï,’ La Pensée 121 (June 1965), 84; republished in Macherey, Pour une théorie de la production littéraire (Paris: François Maspero, 1966), part two, online at http://stl.recherche.univ-lille3.fr/sitespersonnels/macherey/machereyproductionlitteraire2.html#II1; Macherey, A Theory of Literary Production, trans. Geoffrey Wall (London: Routledge, 1978), 112.
11 Macherey, ‘Lénine critique de Tolstoï,’ 84/112.
12 Cf. Althusser’s interpretation of the ‘absence of relations’ at work in Bertolazzi’s El Nost Milan (Althusser, ‘The “Piccolo Teatro”: Bertolazzi and Brecht. Notes on a Materialist Theatre’ [1962], FM, 135ff.).
13 Badiou, ‘Le (Re)commencement du matérialisme dialectique’, Critique 240 (May 1967), 438–467.
14 RMD, 440n.1, citing Étienne Bachelard, La Formation de l’esprit scientifique, 14; cf. APE, 87n.19.
15 Althusser and Balibar, RC, 133; cf. Hallward, ‘Theoretical Training’, CF1, 12; Peter Dews, ‘Althusser, Structuralism, and the French Epistemological Tradition’, Althusser: A Critical Reader, ed. Gregory Elliott (Oxford: Blackwell, 1994), 124.
16 RMD, 461. Such a theory, Badiou tentatively suggests, would include a set P of places, and a set F of functions or practices, applied to these places; a subset H of F would then be ‘historically representable’ if its application is ‘determinant’ and ‘structured in dominance.’
17 ‘Althusser’s whole effort is to try to achieve right away, for a discipline without tradition, something that mathematicians have been laboriously striving to obtain via the newly developed theory of Categories – a direct determination of the concept of structure, without referring to a set that underlies it’ (464).
18 David Hilbert, letter to Gottlob Frege, 29 December 1899, in Frege, Philosophical and Mathematical Correspondence, ed. Brian McGuinness, trans. Hans Kaal (Oxford: Blackwell, 1980), 39–40; cf. Hilbert, The Foundations of Geometry [1899] (Chicago: Open Court, 1977).
19 Hilbert, 1922, cited in Morris Kline, Mathematical Thought From Ancient to Modern Times (Oxford: Oxford University Press, 1972), 1027.
20 CM, 30; CpA 10.8:156. The basic idea, as Zachary Luke Fraser notes, is that ‘the material inscription of mathematical thought – its reduction to bare letters and axiomatic rules of manipulation – is a priori adequate to its essence’ (Fraser, ‘Introduction’, CM, xxix).
21 ‘The story of that lecture course on the concept of the model is itself a veritable allegory of the moment. There were supposed to be two sessions: the first took place and the second didn’t, because it was supposed to take place right at the beginning of May 68!’ (Badiou, ‘Theory from Structure to Subject’, CF2, 286).
22 Zachary Luke Fraser, ‘Model’, in Badiou Dictionary, ed. Steven Corcoran, forthcoming.
23 As Badiou puts in 1990, defending his affirmation of a ‘subject without object’, ‘the true does not speak of the object, it speaks of nothing but itself’ (Badiou, ‘Saisissement, dessaisie, fidélité’, Les Temps modernes 531–533, vol. 1 [1990], 20). Again, ‘thought is not a relation to an object, it is the internal relation of its real’ (Badiou, Abrégé de métapolitique [Paris: Seuil, 1998], 37); cf. Badiou, ‘On a Finally Objectless Subject’ [1989], trans. Bruce Fink, in Who Comes After the Subject?, ed. Eduardo Cadava et al. (London: Routledge, 1991), 24–32; CM, xxxv.
24 For a more detailed account of this article see the synopsis posted on the Concept and Form website, http://cahiers.kingston.ac.uk/synopses/syn9.8.html.
25 Badiou, ‘Theory from Structure to Subject’, 278; cf. ‘Can Change Be Thought?’, 243–244.
26 Duroux, ‘Strong Structuralism’.
27 Badiou and Tzuchien Tho, ‘The Concept of Model, Forty Years Later’, in CM, 103.
28 For more on Miller’s ‘Suture’ see the synopsis posted on the Concept and Form website, http://cahiers.kingston.ac.uk/synopses/syn1.3.html, and in the present volumes, Hallward, ‘Theoretical Training’, CF1, and Žižek, ‘ “Suture”, Forty Years Later’, CF2.
29 In RMD Badiou had already acknowledged the significance of Miller’s contribution to structuralist epistemology. ‘The fundamental problem of every structuralism is that of a term with a double function, which determines the belonging of the other terms to the structure insofar as it itself is excluded from it by the specific operation that makes it figure there only in the form of its representative (its place-holder [lieu-tenant], to use Lacan’s concept).’ The ‘determination, or “structurality”, of the structure’ is thus governed by the ‘location of the place occupied by the term indicating the specific exclusion, the pertinent lack.’ Badiou applauds Miller’s article ‘Suture’ as an ‘essential reference’ in the conceptualization of such determination. Although ‘extraordinarily ingenious’, however, Badiou warns that Miller’s approach is ‘epistemologically inadequate’ and involves a ‘duplication of the structure of metaphysics’ (RMD, 457n.23).
30 CpA 10.8:161–162. The ‘science of psychoanalysis’, consequently, can have ‘nothing to say about science’ per se precisely because it serves to analyze the ‘functioning’ and ‘efficacy’ of ideologies. Psychoanalysis helps to establish ‘the laws of input and connection through which the places allocated by ideology are ultimately occupied.’ It is on this basis that psychoanalysis and historical materialism might be articulated together, as a double ‘determination of the signifiers’ at work in lived or ideological discourse, thereby ‘producing the structural configuration wherein ideological agency takes place’ (162). As if to refute Badiou, Lacan would later argue, in a 1975 lecture, that ‘psychosis is a trial in rigor. In this sense, I would say that I am psychotic. I am psychotic for the sole reason that I have always tried to be rigorous. This plainly takes us quite far, since it implies that logicians, who tend toward this goal, as well as geometricians, would in the final analysis share in a certain form of psychosis’ (Lacan, ‘Conférences et entretiens dans les universités nord-américaines’, Scilicet 6–7 [1975], 9, cited in Peter Starr, Logics of Failed Revolt: French Theory After May ’68 [Stanford: Stanford University Press, 1995], 52).
31 Badiou et al., Contribution au problème de la construction d’un parti marxiste-léniniste de type nouveau (Paris: Maspero, 1969), 4; cf. TC, 9; LS, 178/125.
32 Badiou, Contribution au problème, 15, 26, 39.
33 Badiou, ‘Politique et vérité’ [interview with Daniel Bensaïd], Contretemps 15 (February 2006), 49.
34 For Badiou against Lacan and Miller, ‘the void is not at all the subject but rather the “object” of the procedure’, void is on the side of being rather than subject (Badiou and Tzuchien Tho, ‘The Concept of Model, Forty Years Later’, CM, 100). In Badiou’s ontology, the void of a situation, i.e. that which counts for nothing according to the criteria of a situation, is what presents its ‘suture to being’ per se. ‘Suture’ in this anti-Millerian sense marks the point of intersection between the consistent structure of a situation and the inconsistent, unpresentable multiplicity of pure being as being – leaving Badiou free to think the category of the subject in terms that interrupt the logic of its situation, an interruption that is not itself grounded in lack or the void so much as occasioned by an ephemeral supplement (a ‘supernumerary’ event) which indicates this void (cf. Hallward, Badiou, 90–93). We might say that Miller’s approach allows him to think of a universal or ubiquitous subject, a subject of the signifier as such – but literally as a subject that itself counts for nothing, a subject whose ‘zero’ is represented as one in the movement from one signifier to another; Badiou, on the basis of an event which reveals that which counts for nothing, develops instead the theory of a subject that can come to count for everything. In a 1990 note on ‘Suture’, Badiou contrasts Miller’s argument that number is engendered through the ‘function of the subject’ with his own position in which number, on the contrary, is a ‘form of being’, and in which non-self-identity can only be attributed to an ‘event’ (Badiou, Le Nombre et les nombres [Paris: Seuil, 1990], 36–37, 40).
35 Badiou, ‘Politique et vérité’, 50.
36 Badiou, ‘Politique et vérité’, 52.
37 Diana George and Nic Veroli, ‘Interview with Badiou’, Carceraglio, October 2006, http://scentedgardensfortheblind.blogspot.com/2006_10_15_scentedgardensfortheblind_archive.html#116103479719156657.
38 TC, 21, cf. 15, 58; DI, 16; Contribution au problème, 44, 47; TS, 123–124/106.
39 See in particular Badiou, The Communist Hypothesis, trans. David Macey and Steve Corcoran (London: Verso, 2010); Badiou, The Rebirth of History, trans. Gregory Elliott (London: Verso, 2012); Bosteels, Badiou and Politics (Durham, NC: Duke University Press, 2011).
40 Badiou, ‘Theory from Structure to Subject’, 288; cf. Duroux, ‘Strong Structuralism’.
41 ‘My fidelity to what happened in that period [of May 68 and its immediate consequences] is unshakable but it is also profound, because a large part of my philosophy in reality is an attempt fully to come to terms, including from my own experience, with what happened there and then, while at the same time explaining the reasons for remaining loyal to those events’ (‘Can Change be Thought?’, 237).