## Zero

Le zéro

A series of attempts are made in the *Cahiers pour l’Analyse* to relate the concept of zero to the concept of subjectivity, starting with Yves Duroux’s and Jacques-Alain Miller’s account of Frege’s genesis of the series of natural numbers.

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 “he_{0}” 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”, “you_{1}”, “you_{2}”, “he_{0}” - 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 ‘he_{0}’. 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.

## Notes

1. For a non-technical overview see Robert Kaplan, *The Nothing that Is: A Natural History of Zero* (London: Allen Lane, 1999). ↵

2. ‘[A] shift occurred from a stage when nothing had a meaning to another stage where everything had meaning […]. That radical
change has no counterpart in the field of knowledge, which develops slowly and progressively. In other words, at the moment
the universe became *significant*, it was none the better *known* for being so, even if it is true that the emergence of language must have hastened the rhythm of the development of knowledge.
So there is a fundamental opposition, in the history of the human mind, between symbolism, which is characteristically discontinuous,
and knowledge, characterised by continuity. Let us consider what follows from that. It follows that the two categories of
the signifier and the signified came to be constituted simultaneously and interdependently, as complementary units; whereas
knowledge, that is, the intellectual process which enables us to identify certain aspects of the signifier and certain aspects
of the signified, one by reference to the other – we could even say the process which enables us to choose, from the entirety
of the signifier and from the entirety of the signified, those parts which present the most satisfying relations of mutual
agreement – only got started very slowly. It is 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. The universe signified long before people began
to know what it signified; no doubt that goes without saying. But, from the foregoing analysis, it also emerges that from
the beginning, the universe signified the totality of what humankind can expect to know about it. What people call the progress
of the human mind and, in any case, the progress of scientific knowledge, could only have been and can only ever be constituted
out of processes of correcting and recutting of patterns, regrouping, defining relationships of belonging and discovering
new resources, inside a totality which is closed and complementary to itself’ (Claude Lévi-Strauss, *Introduction to the Work of Marcel Mauss*, 60). ↵

3. Ibid., 64. ↵

4. Ibid., 63-64. ↵

5. Alain Badiou, ‘Le (Re)commencement du matérialisme dialectique’ [review of Louis Althusser, *Pour Marx* and Althusser et al., *Lire le Capital*], *Critique* 240 (May 1967), 457n.23. ↵

6. For a comprehensive overview of the contents of this work, and its place in Frege’s more general thinking about the relation between mathematics and logic, see: http://plato.stanford.edu/entries/frege-logic/; note in particular section 5, on the zero. ↵

7. Miller’s distinction between an ‘absolute’ and ‘relative’ zero recalls the terms of Freud’s 1895 ‘Project for a Scientific Psychology’, where a ‘principle of neuronal inertia’ tending towards an absolute zero of excitation (what Freud will call ‘Nirvana’ in *Beyond the Pleasure Principle*; SE 18: 56) is checked by the ‘exigencies of life’: ‘In consequence, the nervous system is obliged to abandon its original trend towards inertia (that is, bringing the level of
Qή [quantitative stimulus] to zero). It must put up with [maintaining] a store of Qή sufficient to meet the demand for a specific
action’ (SE 1: 297). ↵

8. Leclaire gives a clearer explanation in his 1969 work *Psychoanalyzing*: ‘Excitation or excitability of the sexual type, which specifies the erotogenic zone and which we are attempting to characterize,
would thus be defined as the capacity of an area of the body to be the centre of an immediately accessible, felt difference
– pleasure or unpleasure – and to register in some way the mark of that difference […] To produce pleasure, something like
a perceptible rift must appear; an interval, a difference, a nothing has to open up that can, for the space of an instant,
offer an empty reflection of the absolute of *jouissance*, a moment in which tension is annulled or, better yet, in which the terms that maintain the interval of difference are effaced.
In that moment, which is the moment of pleasure, difference seems to annul itself in the illusion of a “pure difference”’ (46-47). ‘Let us not overlook that it is quite difficult to speak pertinently of this annulment since, by definition, the zero it evokes
is in turn really annulled as zero as soon as one speaks of it as a term. [A footnote cites Miller’s “Suture”]. This difficulty – which language, by its nature, must assume – would have little more than a speculative interest if *the zero in question were not in fact the reality of jouissance*. What is more, the difficulty highlights the major characteristic of what is called a subject: it is that alternative function
capable of engendering in turn its annulment and the effacement of this annulment itself. In other words, the subjective function
appears as the one that seems to tolerate or incite the vanishing of *jouissance*. There is no subject conceivable except in this relation of annulment with *jouissance* and no *jouissance* one can speak of outside of this relation of oscillation with the subject’ (97). ↵