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.

‘Suture: Elements for a Logic of the Signifier’ was first presented as a paper at the 9th session (24 February 1965) of Lacan’s Seminar XII: Crucial Problems for Psychoanalysis. It was intended as a companion piece to Yves Duroux’s paper ‘Psychology and Logic’, under the rubric ‘Number and Lack’. In the Cahiers pour l’Analyse, it follows Yves Duroux’s paper (CpA 1.2), under the new heading ‘Sur la logique du signifiant’ (‘On the Logic of the Signifier’). Following on from Duroux’s presentation of Frege’s account of the construction of number, Miller demonstrates its relevance for the Lacanian theory of the logic of the signifier. This piece, along with ‘Action de la Structure’ (CpA 9.6), is among several important articles written by Miller between 1964 and 1968, before he eventually became a psychoanalyst. Its publication in the first volume of the Cahiers indicates its foundational status for Miller.

‘Suture’ begins with an elaborate play on the anomalous character of Miller’s own position, as a non-analyst, facing Lacan’s audience in his seminar. There is apparently no reason why the audience should listen to him. But he says that a justification can be found in the topological conceptions Lacan has been developing in previous sessions of the current seminar (Seminar XII), which, Miller says, reveal that ‘the Freudian field is not representable as a closed surface’ (CpA 1.3:38; trans. 24). In certain topological spaces (such as the Moebius strip, the cross-cap and Klein bottle), there are ‘points’ on surfaces which are not assignable to the interior or the exterior. Thus Miller says that the mere fact of not being situated on the interior of a space (here, the space of analysts) does not imply that one is thereby completely on the outside of it; one may be occupying a ‘point’ ‘excluded from a two-dimensional topology’, where the ‘two surfaces join up and the periphery or outer edge crosses over the circumscription’ (38/24-5). The fact, he says, that he is able to ‘recognise and occupy that point’ is what will justify the audience lending him his ear.

The first of Miller's three sections, ‘On the Concept of the Logic of the Signifier’ claims that the ‘logic of the logicians’ presupposes a prior ‘logic of the signifier’, which ‘should be conceived of as the logic of the origin of logic’ (39/25). The status of the logic of signifier is thus ‘archaeological’, insofar as it attempts a ‘movement back’ that overcomes the ‘miscognition’ [méconnaissance] that is characteristic of the field of logic itself. Miller compares his strategy with that of Derrida in his commentary on Husserl’s ‘Origin of Geometry’ (1962). The difference is that for Miller, ‘misrecognition finds its point of departure in the production of meaning’, and that meaning ‘is constituted not as a forgetting, but as a repression’. It is this repression that Miller will soon designate as suture.

In his second section, ‘The Zero and the One’, Miller stages a ‘movement back’ into the logic of the signifier through a reading of Gottlob Frege’s Foundations of Arithmetic (1884). Miller’s 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 misrecognition of the function of the subject. Miller acknowledges that to claim that Frege is grounding his logical account of number on a theory of the subject will appear surprising to those familiar with Frege’s basic arguments against empiricist theories of number. For the empiricists, the ‘function of the subject’ is to bring about a synthesis, as a ‘support of the operations of abstraction and unification’. Frege does indeed exclude this notion of merely ‘psychological’ subjectivity, but Miller argues that this exclusion should not be extended to an exclusion of the subject tout court.

The primary instrument for Frege’s genesis of the sequence of whole natural numbers is his distinction between the concepts of Concept, Object, and Number. Apart from these three basic concepts, two relations are assumed: the subsumption of the object under the concept, and the assignation of a number to the concept. Miller contends that Frege’s logicist system of conceptual representation already involves a fundamental ‘disappearance’: ‘the disappearance of the thing [...] must be effected in order for it to appear as object - which is the thing insofar as it is one’ (41/27). Frege’s logicist system therefore deals with ‘redoubled concepts’: it presupposes ‘the concept of identity to a concept’. Before a concept can subsume an object, the latter must be ‘transformed into a unit’, it must be made ‘numerable’. ‘It is this one of the singular unit, this one of identity of the subsumed, which is common to all numbers insofar as they are first constituted as units’. Frege's conception of the unit is thereby directly connected with the concept of identity, to the degree that Frege adopts a classical definition of identity, as formulated by Leibniz: eadem sunt quorum unum potest substitui alteri salva veritate: ‘those things are identical of which one can be substituted for the other without loss of truth’.¹ Leibniz’s salva veritate or ‘saving of the truth’, argues Miller, thus rests on the concept of self-identity.

However, insofar as Frege is concerned with ‘the passage from the thing to the object’ (43/29), and claims to make an ‘autonomous construction of logic through itself’ (44/30), he must himself risk ‘for an instant’ the suspension or ‘failing’ of truth in the logical substitution of identical elements. In this sense, Frege’s construction bears within it a performative contradiction. On the one hand, if a thing is not identical with itself, then the field of truth is ‘ruined’ and ‘abolished’; but on the other hand, the non-identical must be invoked, if only for an instant, in order to found self-identity as an actual relation between object and truth. Truth is thus founded on the exclusion of the non-identical (43/29).

Frege’s construction of the series of whole natural numbers is built on his assignment of the concept ‘zero’ to the non-identical. In order for the single unit to be preserved as self-identical, we must assign the number zero (the number of no-unit, or no-thing) to the concept ‘not identical with itself’. Miller explains: ‘In effect, let there be the concept “not identical with itself”. This concept, by virtue of being a concept, has an extension, subsumes an object. Which object? None. Since truth is, no object falls into the place of the subsumed of this concept, and the number which qualifies its extension is zero’ (44/29). We thereby arrive at the crux of Miller's argument. ‘It is this decisive proposition that the concept of not-identical-with-itself is assigned by the number zero which sutures logical discourse’ (44/29). Miller acknowledges that he is bringing to light something in Frege’s text that is not fully articulated by Frege himself: ‘in the autonomous construction of the logical through itself, it has been necessary, in order to exclude any reference to the real, to evoke on the level of the concept an object not-identical-with-itself, to be subsequently rejected from the dimension of truth’. This movement of evocation and rejection, summoning and annulment is the movement that is essential to the logic of the signifier, and that grounds logic and language itself according to Miller. If one is not cognizant of this fundamental procedure of ‘evocation and revocation’, one will miss that there has been a conversion from an ‘absolute zero’, or a zero as ‘lack’, to the relative zero, or zero as number. 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’ (44/30; italic added). The 1 thus becomes the ‘ proper name’ (46/32) of zero: there is only one zero. The final operation necessary to generate the sequence of the whole natural numbers is then the ‘successor operation’. As soon as we grant the possibility of counting zero as one, this counting in turn implies the operation ‘n + 1’. ‘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 this 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’ (46-47/31). Whereas ‘logical representation’ tends to collapse this construction, the Lacanian concepts of metaphor and metonymy are capable of articulating this construction within a logic of the signifier.² The primary ‘metaphor’ of the substitution of 1 for 0 is the motor for ‘the metonymic chain of successional progression’. Miller contends that that we thus arrive at ‘the structure of repetition, as the process of the differentiation of the identical’ (46/32). He concludes:

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? (47/32)

It follows that the subject of the unconscious must be identified with this ‘impossible object’, which the discourse of logic ‘summons and rejects wanting to know nothing of it.’ This exclusion is an instance of suture, which Miller defines as follows:

Suture names the relation of the subject to the chain of its discourse; we shall see that it figures there as the element which is lacking, in the form of a placeholder [tenant-lieu]. For, while there lacking, it is not purely and simply absent. Suture, by extension - the general relation of lack to the structure of which it is an element, inasmuch as it implies the position of a taking-the-place-of (39/25).

In the third and final section, ‘The Relation of the Subject and Signifier’, Miller returns to Lacanian psychoanalysis to situate the full significance of the preceding deductions. ‘In effect, what in Lacanian algebra is called the relation of the subject to the field of the Other (as the locus of truth) can be identified with the relation which the zero entertains with the identity of the unique as the support of truth’ (47/32). Miller also relates his conception to Lacan’s ideas about a ‘unary trait’ in the unconscious (developed in Seminar IX [1961-62] on Identification). He concludes that his re-framing of the concept of repetition implies that ‘the definition of the subject comes down to the possibility of one signifier more [un signifiant de plus]’ (48/33). This ‘excess’ of the signifier, he claims, can nevertheless be connected with the most fundamental principles of mathematics:

Is it not ultimately to this function of excess that can be referred the power of thematisation, which Dedekind assigns to the subject in order to give to set theory its theorem of existence? The possibility of existence of an enumerable infinity can be explained by this, that ‘from the moment that one proposition is true, I can always produce a second, that is, that the first is true and so on to infinity’ (48/33>).³

In subsequent articles such as ‘U ou “Il n’y a pas de métalangage”’ (1967) and ‘Matrice’ (1968), Miller would go on to elaborate the connections between the logic of the signifier, set theory, and axiomatisation, engaging with some of the issues raised by Alain Badiou’s criticisms of the concept of suture in ‘Marque et manque: A propos du zéro’ (CpA 10.8).⁴

In some final remarks that link up with his account of structuration in ‘Action de la structure’ (CpA 9.6), Miller suggests that a distinction between circular and linear time arises from his theory of the signifying chain. Although the enunciation of the signifier is limited by the linearity of the signifying chain, the retroactive nature of signification is circular and in fact conditions ‘the birth of linear time’ (49/34). In conclusion, Miller says that ‘by crossing logical discourse at its point of least resistance, that of its suture, you can see articulated the structure of the subject as a “flickering in eclipses”, like the movement which opens and closes the number, and delivers up the lack in the form of the 1 in order to abolish it in the successor’ (49/34). The ‘division of the subject’ - which is ‘the other name for its “alienation”’ - is hereby made manifest. Making way for his development of this model in ‘Action of the Structure’, Miller states that ‘it will be deduced from this that the signifying chain is the structure of structure’, from which may be developed a new theory of structural causality.

Notes

1. Leibniz, Gottfried Wilhelm. Die philosophischen Schriften von Gottfried Wilhelm Leibniz, ed. Carl J. Gerhardt. Berlin: Weidmann, 1875-90, vol. VII, 228, 236. ↵

2. Miller notes that the logicist ‘hesitation’ concerning the localization of zero (as interior or exterior to the series of numbers) continues in the work of Bertrand Russell (CpA 1.3:46). On the concept of zero in Russell, cf. The Principles of Mathematics (1903), 184-187. ↵

3. Miller cites Jean Cavaillès, Remarques sur la formation de la théorie abstraite des ensembles, in Philosophie Mathématique, chapter III: ‘Dedekind et la chaîne. Les Axiomatisations’, 124. ↵

4. In ‘U ou “Il n’y a pas de méta-langage”’, Miller discusses Russell’s theory of types in the light of Lacan’s remarks about metalanguage in Seminar XII, arriving at a conception of the ‘elementary stratification of languages’ (Un debut dans la vie, 128). ‘The elementary stratification of languages is nothing other than an ordered sequence which, if it has a lower limit, does not have a superior’. In the 1967 piece, Miller analyses the relation between the axioms of set theory and the distinction between ‘object language’ and ‘metalanguage’ proposed by Alfred Tarski and taken up by Russell in his remarks on the theory of types in An Inquiry into Meaning and Truth, 60). In ‘Matrice’, Miller uses the logic of stratification to reconceive the relation between the All [le Tout] and the Nothing [le Rien] (Un debut dans la vie, 135-36; trans. 45-46), transforming Badiou’s claims about the nature of stratification (in CpA 10.8:159-62) for his own ends. ↵