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

The historical emergence of zero as an actual number (rather than merely the absence of number) is one of the most complex topics in the history of mathematics. The number zero only came into use in Western Europe, via adaptation of the Hindu-Arabic number system, over the course of the Middle Ages.¹ Mathematical debates surrounding the function and significance of the concept of 0, its relation to 1 and to numerical series, were of fundamental importance for the Cahiers pour l’Analyse.

The concept of zero had assumed a novel significance within the immediately preceding context of 1950s French structuralism. In his Introduction to the Work of Marcel Mauss (1950), Claude Lévi-Strauss identified a ‘floating signifier’, or ‘zero symbol’, that he claimed functioned as a mediator between the network of signifiers and the domain of the signified. He gives an anthropological argument for the existence of this zero symbol. With the acquisition of a basic symbolic network, it was ‘as if humankind had suddenly acquired an immense domain and the detailed plan of that domain, along with a notion of the reciprocal relationship of domain and plan; but had spent millennia learning which specific symbols of the plan represented the different aspects of the domain’.² Hence the need for ‘floating signifiers’ such as mana, which could serve as symbols in their pure state, ‘liable to take on any symbolic content whatever’.³ The floating signifier is a sign of ‘zero symbolic value, that is, a sign marking the necessity of a supplementary symbolic content over and above that which the signified already contains, which can be any value at all, provided that it is still part of the available reserve, and is not already, as the phonologists say, a term in a set’. Because of the primacy of this ‘surplus’ signifier, however, ‘there is always a non-equivalence or “inadequation” between the [totality of signifiers and totality of signifieds], a non-fit and overspill which divine understanding alone can soak up’. If the floating signifier is ‘the disability of all finite thought, it is also the surety of all art, all poetry, every mythic and aesthetic invention’.⁴

This conception of the ‘zero symbol’ in turn influenced Jacques Lacan’s thinking about the role of ‘pure’ signifiers in the grounding of the symbolic order. In ‘Subversion du sujet et dialectique du désir dans l’inconscient freudien’ [‘The Subversion of the Subject and the Dialectic of Desire in the Freudian Unconscious’], referring to Lévi-Strauss’s theory, Lacan suggests that ‘Claude Lévi-Strauss, commenting on Mauss’s work, no doubt wished to see in mana the effect of a zero symbol. But it seems that what we are dealing with in our case is rather the signifier of the lack of this zero symbol’ (E, 821). The problem for psychoanalysis, as Lacan sees it, is accounting for the emergence of a pure signifier that transcends the dialectic of imaginary experience and founds the symbolic order for the child. Rather than the actual presence of a zero symbol (such as mana), it can only be a radical and irreducible lack or absence that can fulfil the role. He goes on to explain that ‘this is why, at the risk of incurring a certain amount of opprobrium, I have indicated how far I have gone in distorting mathematical algorithms in my own use of them: for example, my use of the symbol √ - 1, also written i in the theory of complex numbers, can obviously only be justified if I give up any claim to its being able to be used automatically in subsequent operations’.

Alain Badiou notes that ‘it is the immense merit of Lévi-Strauss to have recognised, in the still mixed form of the Signifier-zero […the] fundamental problem of all structuralism’: 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)’⁵ Jacques-Alain Miller’s ‘La Suture’ (CpA 1.3) is an extended meditation on the nature of zero, and several papers in the Cahiers pour l’Analyse take up, elaborate or criticise Miller’s conception. However, Badiou himself, while keeping hold of the problem of structural causality, dismisses Lacan’s and Miller’s hypotheses about the founding function of zero in the symbolic order, arguing instead for a concept of zero as a mark that is wholly identical to itself qua mark within the operations of logic and that, as such, the mark, or sign, of a lack should not be confused with the lack of a lack.

In the Cahiers pour l’Analyse

The immediate source for several of the Cercle d’Épistémologie’s interrogations of the mathematical concept of zero was Gottlob Frege’s seminal text, The Foundations of Arithmetic (1884).⁶ Yves Duroux’s ‘Psychologie et logique’ (CpA 1.2) presents Frege’s basic argument for a logical construction of the concept of zero in The Foundations of Arithmetic: ‘In order to give himself the number zero, Frege forged the concept of “non-identical to itself”, which is defined by him as a contradictory concept, and he declared that for any contradictory concept (and here he referred to the contradictory concepts received from traditional logic, such as the square circle), for any concept under which no object falls, the name “zer” could be attributed to it. The zero is defined by logical contradiction, which serves as the guarantee of the non-existence of the object’ (CpA 1.2:35). The concept of contradictory things is a concept under which no object can fall, and can therefore be assigned the number zero. However, Duroux’s insistence on the importance of contradiction for the definition of zero is not taken up in Jacques-Alain Miller’s presentation of Frege’s ‘engendering of the zero’ (CpA 1.3:44; trans. 29); instead, Miller will stress that ‘if no object falls under the concept of non-identical-with-itself, it is because truth must be saved. If there are no things which are not identical with themselves, it is because non-identity with itself is contradictory to the very dimension of truth’ (CpA 1.3:44; trans. 29).

Jacques-Alain Miller’s extended analysis of Frege’s concept of zero takes place in the second section of ‘La Suture’ (CpA 1.3), ‘The Zero and the One’. Miller’s main claim is that Frege’s genesis of the progression of whole natural numbers, as carried out in section IV of the Foundations, ‘The Concept of Number’ (specifically §§74-79), involves a miscognition of the function of the subject. ‘It is this decisive proposition that the concept of not-identical-with-itself is assigned by the number zero which sutures logical discourse’ (CpA 1.3:44; trans. 29). Frege occludes the fundamental procedure of ‘evocation and revocation’, thus missing the fact that there has been a conversion from an ‘absolute zero’, or a zero as ‘lack’ to the ‘relative’ zero of zero as number.⁷ The zero, understood as a number, can thus claim to be ‘the first non-real thing in thought’ (44/30).

In §77 of the Foundations, Frege goes on to generate the number 1 from this preliminary operation. As Miller puts it, ‘if of the number zero we construct the concept, it subsumes as its sole object the number zero. The number which assigns it is therefore 1’ (44/29, italic added). The 1 thus becomes the ‘proper name’ (46/32) of zero. ‘Frege’s system works by the circulation of an element, at each of the places it fixes: from the number zero to its concept, from this concept to its object and to its number - a circulation that produces the 1’ (44-45/30). Miller argues that the ‘verticality’ of the movement from zero to one, by which ‘the 0 lack comes to be represented as 1 […] indicates a crossing, a transgression’; the successor operation installs a ‘horizontal’ sequence of numbers on the basis of this primary ‘verticality’ (46-47/32). Whereas ‘logical representation’ collapses this construction, the Lacanian concepts of metaphor and metonymy can be used to rearticulate this construction within a ‘logic of the signifier’. The primary ‘metaphor’ of the substitution of 1 for 0 is the motor for ‘the metonymic chain of successional progression’. Miller contends that we thus arrive at ‘the structure of repetition, as the process of the differentiation of the identical’ (46/32). He concludes that ‘if the series of numbers, metonymy of the zero, begins with its metaphor, if the 0 member of the series as number is only the standing-in-place suturing the absence (of the absolute zero) which moves beneath the chain according to the alternation of a representation and an exclusion - then what is there to stop us from seeing in the restored relation of the zero to the series of numbers the most elementary articulation of the subject’s relation to the signifying chain?’ (47/33).

After his earlier rejection of Miller’s logic of the signifier in ‘L’Analyste à sa place’ (CpA 1.4), Serge Leclaire begins to engage with Miller’s ideas in his ‘Compter avec la psychanalyse’ seminars (CpA 1.5, 3.6, 8.6). He claims that the ‘marking’ of the subject’s erogenous zones in its early encounters with the Other is responsible for the ‘institution of the zero of lack’ (CpA 1.5:68). The child’s ‘detachment’ of a piece of its body for erogenous satisfaction ‘incarnates’ the signifier, ‘inasmuch as the cut makes the zero of lack emerge and the polarising one of the trait’ (cf. unary trait). ‘It is there alone that the zero of lack as zero and not just as lack appears’.⁸

In ‘Communications linguistique et spéculaire’ (CpA 3.3), Luce Irigaray grounds Miller’s conception of the zero as signifier of the subject in the Oedipal situation. She suggests that the subject, as a ‘zero’ or ‘placeholder’, comes into being through its exclusion from the communication of the parents. ‘What is “he” at this point, if not “zero”, condition of the permutation of “I” and “you”, and the empty form that guarantees the structure. Evocative of, without being similar to, that empty space in draughts or chess that allows one pawn to move into another’s space. The status of “he₀” is nothing like that of “I” or “you”, despite the ambiguity that reifies it and classifies it with the latter, the personal pronouns. It is nothing and nobody, but rather the possibility of identification and of permutation of “I” and “you”, of “the sender” and “the receiver”, the only terms that effect communication. Implicated in this communication as its condition of possibility, this third, or, even more accurately, fourth number - “I”, “you₁”, “you₂”, “he₀” - is a blank, a void, the space left by an exclusion, the negation that allows a structure to exist as such’ (CpA 3.3:41/10). The zero here is the ‘empty form that guarantees the structure’, analogous to a place on a chessboard, the ‘mere possibility of identification’. The child therefore enters the symbolic order as a ‘he₀’. Irigaray contends that the task of the analyst is to ‘restore [the patient’s] non-identity to himself, the identity to “zero” that permits the inversion of the sign’ (53/23).

In his reading of Plato’s Sophist, ‘Le Point du signifiant’ (CpA 3.5), Jean-Claude Milner contends, against Xavier Audouard’s reading of the same text in his ‘Le Simulacre’ (CpA 3.4), which presents Plato as successfully transforming the notion of non-being into a species of falsity or appearance, that such a transformation, if it occurs in Plato’s text, is rather symptomatic of a repression of non-being of which Plato himself is not explicitly aware. Milner claims that the Eleatic Stranger’s minimal ontology of five elementary kinds (being, change, rest, sameness, difference or ‘the other’ 254b-257a) contains the rudiments of the logic of the signifier, but that Plato himself is guilty of ‘ignoring the structure of zero’ (CpA 3.5:82) that underlies his formulations.

Alain Badiou’s ‘Marque et manque: À propos du zéro’ (CpA 10.8) is, amongst other things, a thoroughgoing critique of Jacques-Alain Miller’s reading of Frege and his conception of zero. On the basis of an epistemology of logic, Badiou arrives at ‘a more precise determination of the meta-theoretic function of the zero’ (CpA 10.8:155). According to Badiou, there is no ‘absolute’ zero outside the chain of marks or signifiers. ‘The zero is not the mark of lack in a system, but the sign that condenses the lack of a mark’ (156). Zero is a secondary concept within the scheme of stratification, and merely serves to designate the field of non-derived propositions in a logical system. ‘The zero does not name a blank space but the erasure of a trace’. According to Badiou’s theory of the three ‘mechanisms’ of logical order (concatenation, syntax and derivation), the zero is a mark of a mark at the syntactical level that is lacking only insofar as it refers to something non-derivable by the third mechanism (161).

Select bibliography

• Freud, Sigmund. ‘Project for a Scientific Psychology’ [1895]. In Standard Edition the Complete Psychological Works of Sigmund Freud, ed. James Strachey et al. London: Hogarth Press, 1953-1974, vol. 1.
• ---. Beyond the Pleasure Principle [1920], SE 18.
• Lacan, Jacques. ‘The Subversion of the Subject and the Dialectic of Desire in the Freudian Unconscious’. In Écrits, trans. Bruce Fink. New York: W.W. Norton, 2006.
• Leclaire, Serge. Psychanalyser. Paris: Seuil, 1969. Psychoanalyzing, trans. Peggy Kamuf. Stanford: Stanford University Press, 1998.
• Lévi-Strauss, Claude. Introduction to the Work of Marcel Mauss [1950], trans. Felicity Baker. London: Routledge, 1987.