Concept and Form:
The Cahiers pour l'Analyse
and Contemporary French Thought

This project is funded by an Arts and Humanities Research Council (AHRC) research grant and is supported by the Centre for Research in Modern European Philosophy (CRMEP) and Kingston University's Faculty of Arts and Social Sciences.

The main question at issue in this, the first of Badiou’s two contributions to the Cahiers, is the abruptly ‘subversive’ i.e. revolutionary power of scientific formalisation, its capacity to interrupt the ideological categories of continuity, quality and temporality. Badiou affirms this power as essential to the operations of mathematical logic in general, and to Abraham Robinson’s ‘non-standard’ theorisation of ‘infinitely small’ quantities in particular. (See Synopsis of CpA 10.8 for more background on Badiou’s general conception of science).

‘Infinitesimal Subversion’ is notable not only for its seamless integration of a relatively austere topic in the foundations of differential calculus with a Lacano-Althusserian philosophical agenda, but also for the several ways in which it anticipates, perhaps more than any other of Badiou’s early texts, many of the distinctive concerns of his later philosophy: a privileging of axiomatic formalisation over any sort of intuitive or constructivist approach to mathematics; a valorisation of abrupt breaks or interventions over any appeal to continuity; a concern with the primacy of the ‘void’, the ‘constitutive silence’ or ‘blank empty space’ as the space in which a ‘decisive reworking’ might begin; a concern with the ‘real’ of a situation understood, after Lacan, in terms of its determinate ‘impossibility’, an impossibility which formalist invention can, on occasion, abruptly convert into possibility.

Badiou summarises what is at stake in ‘Infinitesimal Subversion’ in the stark juxtaposition that frames his conclusion, separating ‘quality, continuity, temporality and negation: the oppressive categories of ideological objectives’ from ‘number, discreteness, space and affirmation: or, better, Mark, Punctuation, Blank Space and Cause: the categories of scientific processes. These are the formal indices of the two “tendencies” that have been in struggle, according to Lenin, since the origins of philosophy’ (CpA 9.8:136). Berkeley and Hegel figure here as the main representatives of the ideological camp (though Badiou’s argument is framed in such a way that it might also include Bergson and Deleuze). In anticipation of Robinson, the mathematicians Galois and Cantor exemplify the subversive force with which science disrupts the categories of ideology; they also exemplify the sort of reactionary backlash and marginalisation that such disruption tends to provoke.

The first half of Badiou’s article makes a general point about the determination, 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 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. (Readers familiar with Badiou’s subsequent work, most notably the recent Logics of Worlds [2006], may recognise such a process as the prototype of his later theory of truth).

Badiou begins with a discussion of the distinction between the finite and the infinite. Whereas for Hegel, the (metaphysically) infinite can be thought in the figure of a circle, as a self-sufficient whole, the finite domain is a linear progression which ceaselessly transgresses its own limit. Finite natural numbers are generated through repeated operations of succession: 1 + 1 + 1 + 1... Take any finite number, however large: it is obviously possible, through the addition of a ‘supplementary inscription’ (n+1), to generate a still larger number.

Badiou’s opening question concerns the ‘place’ in which this inscription, so to speak, might come to be inscribed. The inscription of any finite number simultaneously ‘opens’ the empty place in which inscription of its successor might follow. ‘The numerical effect exhausts itself in the incessant shunting along of the empty place: number is the displacement of the place where it is lacking’ (CpA 9.8:118). (However, as Badiou will explain in ‘Mark and Lack’, CpA 10.8 – perhaps in order to forestall any possible misunderstanding of this point – such displacement of lack does not imply the marking or inscription of lack or non-identity as such).

Badiou’s opening and abiding question, here, concerns the nature of this empty place. He outlines his answer immediately: the endless generation of finite numbers through succession ‘presupposes a (unique) space of exercise, that is to say, an out-of-place blank where the [empty] place is displaced in the retroaction of the inscribed’, a place he describes, after Mallarmé, as an essentially ‘“gratuitous” blankness [...]: it is what is written that bestows upon it its status as place of the writing that takes place.’ Only insofar as we assume 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’ [CpA 9.8:118]). This presumed yet actual infinity, the empty space of all potential writing, is what Badiou calls the ‘infinity-support’ for any actual inscription.

Our domain of natural numbers is thus composed of an endless series of places that successive members of this domain come to occupy. If then we are to posit a place that is unoccupiable by any member of the domain, it will have to be marked by an inscription that is ‘supplementary’ to the ordinary series of supplementary inscriptions which count out this domain. What Badiou calls an ‘infinity-point of the domain’ is a number which can be ‘forced’ to occupy this ‘unoccupiable empty space’ (CpA 9.8:119). Such an occupation will have to meet two conditions. On the one hand, if such an ‘impossible’ number is to be conceived as possible, it will have to figure as impossible for the original domain, and thus be constructed on the basis of the ‘initial procedures’ operative in the domain. On the other hand, as impossible it must of course be constructed as ‘exterior’ to the domain itself, precisely as a ‘supplement’ to the domain as a whole.

Badiou gives the example of the 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 ‘perfectly constructible, since the statement “for all x, x is less than y” is a well-formed statement of the system’. The ‘variable’ y here marks unoccupiable place; it thus indicates the limit of what can be written within the ‘infinity-support’. However, 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. The place appears as ‘trans-numeric’.

It might seem that there is nothing more to say: impossible is impossible, and we have reached the limit of number. But there is nothing to prevent a mathematician from ‘forcing’ the procedures operative in a domain to apply to ‘precisely that which they had excluded’ (CpA 9.8:120). In other words, we can 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 is less than i .’ By definition, this new number i will not ‘count’ as a natural number. But if the initial procedures which 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 natural number, or in Badiou’s jargon, ‘an infinity-point relative to the structure of order over the domain of whole natural numbers’ (CpA 9.8:120). (This is precisely what was at stake in the invention of ‘irrational’ and ‘imaginary’ numbers [CpA 9.8:123]).

By forcing the procedures of the domain to apply to something literally beyond their domain we of course modify and reconfigure the domain itself, dividing its new ‘non-standard’ (or ‘non-conformist’) zone in which the extended procedures apply from the older ‘standard’ zone in which they do not. The domain is now ‘stratified’: it has been ‘remoulded’ (in the sense of a term, refonte, that Badiou borrows from Regnault, who himself borrows it from Bachelard). If then the ‘infinity-support’ of the initial original domain is ‘required by the recurrent possibility of inscribing a mark in the empty place assigned by the primitive relation of the domain, conversely, it is the impossibility of a certain mark within that domain that gives rise to the infinity-point’ (CpA 9.8:120).

In the second section of his article, Badiou considers what is at stake in the ‘occupation’ of such a 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 power, within the original limits of the domain, to ‘occupy’ the empty place? Is the 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’ (CpA 9.8:121)? This is Hegel’s presumption (developed in a different context by Quine): the general logical concept of a variable itself seems to imply a ‘crossroads of infinities’, as if it subsumes all the possible places ‘contained’ within the infinity-support (CpA 9.8:121).

Badiou rejects this assumption. Like any other inscription, the variable presumes the infinity-support, and the mere inscription of a variable in an unoccupiable place (on the model, ‘for all x , x is less than 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’ (CpA 9.8:122).

Here Badiou follows Lacan’s lead: for any given ‘domain of fixed proofs, impossibility characterises the real’ (CpA 9.8:122). A specific and consistent axiomatic system proceeds on the basis of specific exclusions. Not all statements are possible within the system. The variable, then, insofar as it indicates a place that cannot be occupied but that nevertheless can be inscribed or ‘constructed’, figures as the ‘operator of the real for a domain.’ This means that a variable can be described as ‘inverse of the infinity-point, whose inscription it prepares’ (CpA 9.8:122). The infinity-point, by contrast, imposes itself as a new and hitherto impossible constant. It substitutes itself for the variable, via a writing which asserts the ‘possibility of the impossible’. Or again, if a variable ‘realises [réalise] the difference’ of a system by indicating the place of the term it excludes as impossible, the infinity-point ‘irrealises’ it by including (or ‘hallucinating’) this term. Such is Badiou’s compressed Lacanian refutation of Hegel: ‘the variable as mark cannot figure the Infinity of marks of a domain, since it is coextensive with their reality’ (CpA 9.8:123).

Badiou turns, in sections three and four of his article, to consider a recent and especially significant instance of this general theory of the infinite – not in the domain of the infinitely large, but of the infinitely small. From Zeno and Aristotle through to Berkeley and Hegel (and indeed right through to Skolem’s path-breaking work of the 1930s), both philosophers and mathematicians tended to accept that the notion of an infinitely small number was a self-evident ‘absurdity’ (CpA 9.8:123). The basic assumption, already formulated in Zeno’s paradoxes, was that we cannot distinguish a minimal differential element (or infinitely small number ) within the smooth fabric of continuity. A number is a discrete quantity. But since the general domain of number is infinitely and continuously divisible, surely it makes no sense to imagine its interruption at infinitely many yet discreet ‘infinitesimal’ points. The concept of the infinitely small, in other words, is by this logic – common to Aristotle, Berkeley and Hegel, but also Lagrange, Cantor, Fraenkel... – not a numerical or quantitative concept, but a qualitative one (CpA 9.8:124). Considered as a number or quantity, there seemed to be nothing that could clearly distinguish an infinitesimal from zero – hence the lasting force of Berkeley’s famous dismissal of infinitesimals as ‘the ghosts of departed quantities.’

Rejecting the notion of an infinitely small number, Hegel maintains instead the infinite divisibility of geometric continuity on the one hand, and measurable relations of size or quantity on the other. Thus conceived, the unit or ‘atom’ of mathematical measurement is an indivisible arithmetical element whose combinations do not generate the continuum but rather quantities or relations established on ‘the basis of the continuum’. If units or atoms can be combined ‘in the void, there can be no atoms of the void.’ Hegel rejects, in other words, the possibility that the divisibility of the continuum might be punctuated by its own indivisible quantities, or numbers (CpA 9.8:124).

Applied to the domain of the calculus or mathematical analysis, in which the ‘useful fiction’ of infinitely small numbers was first deployed, an infinitesimal ‘quantity’ (dx) will initially be marked not as an actual quantity, precisely, but as a differential relation between quantities. Berkeley’s early critique of the calculus pinpointed its apparently fatal inconsistency: calculus takes procedures that are legitimate for finite quantities, and then extends them to quantities it treats as ‘infinitely small’ (CpA 9.8:126). For a long time, no mathematician could devise a convincing answer to this objection, and by the middle of the nineteenth century the embarrassing and seemingly incoherent notion of an actually infinitesimal number had been replaced by the more easily acceptable idea of a ‘limit’ towards which an infinite series of numbers might tend.

Struck by the fact that this long proscription of infinitely small numbers endured long after Cantor’s revolutionary validation (in the 1870s) of the multiplicity of infinitely large numbers, Badiou interprets the resistance to infinitesimals as a paradigmatic instance of the sort of ideological ‘repression’ [refoulement] at work in the preservation of what Althusser called, after Bachelard, the ‘epistemological obstacles’ that ideology erects to hinder the development of a subversive science (CpA 9.8:126-128).

Returning now to the terminology presented in the first part of his article, Badiou formulates his main ‘epistemological thesis: in the history of mathematics, the marking of an infinity-point constitutes the transformation wherein are knotted together those (ideological) obstacles most difficult to reduce’ (128). In the case of the calculus of real numbers, such an obstructive infinity-point was marked by the concept of an infinitely small numerical element or point, the inscription of a number at the place marked as ‘smaller than all other numbers’. It is precisely this ‘punctualisation’ of the infinite that classical philosophy, here exemplified by Hegel, was determined to reject. More generally, as Hegel’s contemporary Galois anticipated, 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).

In brief, ‘in science as in politics, it is the unperceived which puts revolution on the agenda’ (128). (Badiou’s major books, Theory of the Subject [1982], Being and Event [1988] and Logics of Worlds [2006], might be most concisely understood as successive attempts to expand this formula into a theory).

Along with revolution comes, unsurprisingly, reaction or counter-revolution. The recasting [refonte] of a domain which follows the inscription of an infinity-point can have literally anarchic consequences.¹ By forcing the infinity-point of the real number system, for instance, we generate an extension or ‘macro-field [surcorps]’ of ‘complex numbers’ in which some basic arithmetic relations no longer apply. The structure of order itself is not valid for the remoulded domain, which thus presents a literally ‘disordered’ field of number. In the case of our natural numbers, one of its essential ‘standard’ properties is to remain ‘Archimedean’ – this means that for any ‘two positive numbers a and b, where a is less than b, there always exists another whole number n such that b is less than na.’ Once extended to include infinitesimals, however, the remoulded domain becomes non-Archimedean (CpA 9.8:129). It was to avoid such consequences that ‘epistemological prudence’ (on the part of mathematicians) came to conspire with ideological ‘repression’ (on the part of philosophers), to ensure that the infinitely small or ‘almost nothing’ remained without a numerical mark of its own (CpA 9.8:129).

The last and longest section of Badiou’s article is mostly a semi-technical summary of the reasoning whereby Abraham Robinson, in work first published in 1961, was able to demonstrate that infinitesimals are indeed numbers after all. In brief, Robinson shows how ‘we can entirely reconstruct classical analysis by “immersing” the field of real numbers in a non-Archimedean field, by the inaugural marking of an infinity-point – an infinitely large number – and the correlative free recourse to infinitesimal elements’ (CpA 9.8:130). Badiou sees in Robinson’s discovery impressive and ‘convincing proof of the productive capacity of formal thought’.

The demonstration might be outlined as follows. Take a formal numerical system S, and extend it by inscribing the supplementary mark (or infinity-point) i (CpA 9.8:131), along the lines indicated above. The extended system is just that – an extension of the original system, whose initial rules and axioms (e.g. the rules that allow for the generation of successive finite numbers) remain unchanged. Is the extended or ‘transgressive’ system still ‘coherent’? Pure logic, drawing on ‘the theorem of compactness’ (n.27), establishes as a general principle that, ‘if a system is coherent, its transgressive extension is also coherent’ (CpA 9.8:132). This authorises the inscription of an infinity-point for our finite domain, which then defines a ‘non-standard’ extension of the ‘standard’ or normal domain. (Badiou translates Robinson’s phrase ‘non-standard’ with the more politically charged ‘non-conforme’).

The infinity-point α for the relation of order, for instance, will be an ‘infinitely large’ term, i.e. a number that is larger than any and every other (within the system). Inscription of this point α does not determine its position in the system, relative to other infinite numbers; the ‘remoulding’ of the system which this inscription entails expands and opens the system up in such a way that the ‘causality’ of the remoulding itself is ‘dissipated in its operation’. Badiou finds in this dissipation an example of the way a cause (in keeping with the logic of ‘structural causality’ put forward by Althusser and his students) can be ‘effaced in the apparatus of a structure’ (CpA 9.8:132-133). Badiou suggests, rather quickly, that the ‘marking of an infinity-point is an operation of the signifier as such’, insofar as ‘the causality of the mark α is here, in the domainial effacement of that which it designates, the omnipresence marked for every occupation of a place where only “new” infinite numbers can come’ (CpA 9.8:133).

The numbering of the infinitely small can now be written, very simply, as 1 divided by α: the result is a number that is infinitely small relative to all the constants included in the initial system or domain. The extended system now contains infinitely many elements that are both infinitely large and infinitely small, and it is no longer Archimedean.

On this basis, Badiou concludes, Robinson’s work serves to ‘reconstruct all the fundamental concepts of analysis’ in terms that are, for the first time, fully ‘systematic’. The previously vague (i.e. ideological) notion of ‘infinite proximity’ can now be numbered precisely: ‘a number a is infinitely close to a number b if the difference a – b is an infinitesimal number’ (CpA 9.8:134). The previously vague notion of an approximation or ‘tending towards a limit’ can be replaced with a precise inscription of ‘being’, since ‘to be infinitely large (or small) means: to be an infinite (or infinitesimal) number’. By the same token, this ‘punctualisation’ or numbering of the domain of the infinite undercuts Hegel’s resistance to its ‘quantification’, and with it the resistance of classical philosophy more generally.

The age-old ideological investment in the association of infinity with quality and continuity – an investment which dominated the early development of mathematical analysis itself – is thus broken or interrupted by formal i.e. ‘inscribed’ or ‘structural’ thought (CpA 9.8:135).

Notes

1. The primary meaning of refonte refers to the remoulding or recasting of a metal. Gaston Bachelard used the term in order to describe revolutionary scientific change. ‘Crises in the development of thought imply a total remoulding [refonte] of the system of knowledge. The mindset [la tête] must at such a time be remade. It changes species [...]. By the spiritual revolutions required by a scientific invention, man becomes a mutable species, or better, a species that needs to mutate, one that suffers from failing to change’ (Bachelard, La Formation de l’esprit scientifique [1938] [Paris: Vrin, 2004], 18). ↵