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.

Yves Duroux presented this paper at the seventh session of Lacan’s Seminar XII, Crucial Problems for Psychoanalysis (27 January 1965). It was intended to be the first part of a joint presentation with Jacques-Alain Miller on ‘Number and Lack’. Due to time constraints, Miller’s paper (‘Suture: Éléments de la logique du signifiant’, CpA 1.3) was presented in the 9th session of the seminar (24 February 1965).¹ In the first volume of the Cahiers, Duroux’s paper is presented immediately before Miller’s, under the new heading ‘Sur la logique du signifiant’ [‘On the Logic of the Signifier’].

Duroux’s paper is an exposition of Gottlob Frege’s Foundations of Arithmetic [Grundlagen der Arithmetik] (1884), Chapter IV (‘The Concept of Number’). In sections 72-80 of this chapter, Frege presents a logical construction of the series of whole natural numbers, derived from the definition of zero, the number one, and the ‘successor function’ (denoted by the + sign). Duroux’s paper expounds Frege’s basic moves, foregrounding the question of Frege’s break from the empiricist and psychological approaches to the nature of number, while also making way for Miller’s critique of Frege’s construction of number from the perspective of the logic of the signifier.

Duroux begins by making a distinction between the properties and the nature of number. He takes the properties of number to be given in the axioms (or postulates) stated by Giuseppe Peano. These state that 0 is a number that is not the successor of any number, that every number has just one successor and that no two numbers share the same successor. Duroux notes that these properties implicitly depend on three major assumptions about the nature of number.

1. What is a number? (Peano’s axiom takes for granted that one knows what a number is).

2. What is zero?

3. What is the successor?

Duroux states that he will focus on Frege’s response to these particular questions, and on his critique of the previous tradition of thought about the nature of number. The exposition of this response and critique will then provide the basis for Miller’s exposition.

Frege situates his inquiry into the nature of number in relation to previous thought about number by way of two further questions:

1. How is one to pass from a mere collection of things to a number that serves as the number of these things?

2. How is one to pass from one number to another?

His answers to these questions distinguish his approach from ‘the empiricist tradition’, which refers these operations back to the activity of a ‘psychological subject’ (CpA 1.2:32). According to Frege, the collapse of logic into psychology arises in part from the ambiguity of the term ‘unity’ (Einheit in German).² An Einheit may be understood as a unit or component in a set, but it may also denote the name One, that is, the name of the number 1. But a collection or set of things does not yet yield up the concept of number. In a collection (that we can represent, for instance, as ‘one horse + one horse + one horse’), the one legitimately refers to the individual units, but it is not possible to jump from there to the attribution of the number three to the collection. In order to do this, two modifications are necessary one must be conceived as a number, and the ‘and’ must be transformed into the plus sign. These modifications are not straightforward. According to Duroux, Frege recognises the need for a detailed account of both how the unit ‘1’ acquires its additional significance as a number, and how the ‘return’ of a number involves more than ‘a simple repetition of a unit’. The empiricist tradition was content to relate the emergence of the additional significance of number (beyond the units of a collection) to the ‘function of inertia’ of the psychological subject, so that number would emerge from adding units along a temporal line of succession, and naming them as consecutive numbers. But the real problem is masked by referring the concept of number back to the psychological operations of collecting, adding and naming. Frege saw that the problem was rather to discover ‘what is specific in the plus sign’ and in the one as a number, outside of the activities (or inertias) of the psychological subject. Frege’s original enterprise was thus to eliminate the role of psychology in the determination of the nature of number.

His reduction of the psychological proceeds in two steps. First, he takes up and modifies the concept of Vorstellung (representation), inherited from the German philosophical tradition. Frege makes a division between merely subjective representations, and ‘objective’ representations, which can be treated purely according to logical laws. Objective representations must themselves be divided into a concept on the one hand, and an object on the other. ‘The function Frege assigns to these’, says Duroux, ‘is no different to the function of a predicate in relation to a [logical] subject’. Within the context of modern logic, the relationship is what Frege denotes elsewhere as the relation between a ‘function’ and its ‘argument’.³ The ‘objects’ that are correlative to concepts are not necessarily empirically real objects, but may be entirely ideal; what is important is their logical structure.

Frege goes on to generate the notion of number from this minimal framework of concepts and objects. ‘The diversity of possible numerations cannot be supported by the diversity of objects’, but it can be supported if it is taken as the ‘index of a substitution of the concepts on which number bears’ (33; italic added). To illustrate Frege’s initial argument that number is first of all something that is ‘assigned’ to concepts, he takes Frege’s example from § 46 of the Foundations: ‘Take a phrase, ‘Venus does not possess any moon’. What does the enumerating term ‘any’ [aucune] refer to?’ (33). It does not make sense to attribute it to the concept ‘moon’, because there is no moon of Venus. Rather, that to which one attributes the term ‘any’ can only be the concept ‘moon of Venus’. ‘The concept ‘moon of Venus’ is related to an object (the object ‘moon’), and this relationship is such that there is no moon’ (33). Hence one attributes the number zero to the concept ‘moon of Venus’. This, comments Duroux, is Frege’s first definition of number: that ‘number belongs to a concept’.

Nevertheless, this definition still does not provide us with a way to obtain individual numbers, that is, ‘numbers that possess a definite article: the one, the two, the three, which are unique as individual numbers (there are not several ones, there is one one, one two)’ (34). For this, one needs further steps: first, numerical relationships must be defined according to a ‘pure logical relationship … of equivalence’; second, the equivalents to a concept can be defined as that concept’s extension (34). Frege will generate an account of the individual numbers from this ‘relational machine’ (35), composed of a ‘horizontal axis along which there operates the relation of equivalence, and a vertical axis that is the specific axis of the relation between the concept and the object’. Again, the object here is not a ‘real’ object: ‘as soon as one has a concept, one can always transform it into the object of a new concept, since the relation of concept to object is a purely logical relation’.

With these steps in place, the series of individual whole numbers can be generated by turning to the minimal relation between zero, one, and the successor. As Duroux puts it in a passage omitted from the Cahiers version of the text, ‘the problem is to know whether one is going to be able to define the zero otherwise than by tautological reference to non-existent objects falling under concepts’.⁴ Frege does not want to be left with a merely empirical definition of zero (where the attribution of zero to the concept ‘moon of Venus’ depends on my empirical knowledge of the Venus’s moonless status). He must rather account for the logical necessity for the concept of zero.

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 or the golden mountain), for any concept under which no object falls, the name ‘zero’ could be attributed to it. The zero is defined by logical contradiction, which serves as the guarantee of the non-existence of the object (35).

So we have a concept of zero: what is contradictory to itself. The class of self-contradictory things contains no members. The concept of contradictory things is a concept under which no object can fall. Duroux suggests that the non-existence of centaurs and unicorns also derives from the contradiction in their concepts.⁵ His insistence on the importance of contradiction for the definition of zero is not taken up in Miller’s presentation of Frege’s ‘engendering of the zero’; instead, Miller will st ess 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).

The second operation necessary for the engendering of the series of individual numbers according to Frege is the successor operation. Duroux says that ‘Frege simultaneously gives the definition of the one and the definition of the successor function’ (CpA 1.2:35). The successor operation is first given a formal definition:

One says that a number naturally follows after another number if this number is attributed to a concept under which there falls an object (x), such that there is another number (i.e. the number that the first number follows) to which the concept ‘falling under the preceding concept, but not identical to (x)’ can be attributed (35).⁶

But, according to Duroux, Frege provides a more substantive definition in his account of the construction of the one. He generates this in a purely conceptual way, on the basis of his concept of zero: ‘Assume the concept “equal to zero”’. What object falls under this concept? The object zero. Frege thus says that ‘1 follows 0 to the extent that 1 is attributed to the concept “equal to zero”’ (36). If no objects fall under the concept of non-identical or self-contradictory things, then nevertheless the concept of non-identical things, i.e. the concept of zero, is itself one. Zero might be nothing, but there is only one concept of zero. Frege therefore generates the concept of ‘one’ from the concept of zero.

Duroux wants to account for this reciprocal dependency between the definitions of the one and the successor function by appealing to an implicit ‘double contradiction’ or ‘negation of the negation’ at work in Frege’s text:

The successor operation is engendered by a double play of contradiction in the passage from zero to one. One can say without going too far beyond the field of Frege that the reduction of the successor operation is carried out by an operation of double contradiction. Zero being given as contradictory, the passage from zero to one is given by the contradictory contradiction. The motor which animates succession in Frege is purely a negation of negation (36).

Thus if the concept of zero is first of all the concept of self-contradictory or non-identical things, then the generation of unity out of zero amounts to a double contradiction. For Duroux (if not for Frege or Miller), it is the movement of contradiction that will ground Frege’s construction of number.⁷

Duroux then adds a question that leads into Miller’s presentation: ‘The apparatus that has permitted the definition of number functions very well. But is it capable of responding to the question ‘How does 1 come after 0?’ I will not interrogate further the legitimacy of this operation. I will leave this to the care of Jacques-Alain Miller’ (36).

He concludes with two remarks that problematise his own eponymous distinction between psychology and logic. 1. That in the empiricists and Frege alike, ‘the name of number (which Frege treats as an individual name) is only obtained through an act of force’; and 2. In Frege as with the empiricists, ‘number is always captured by a function of making complete [fonction de faire le plein], whether through a collection [rassemblement], or through the operation of bi-univocal correspondence that has the function of exhaustively collecting a whole field of objects’. Thus, despite the differences between the empiricist and Fregean approaches to number, the activity of the subject in the former case and the logic of equivalence in the latter ‘have the same function’. He finishes on this unresolved note, remarking that the consequences of this similarity must be worked out in the future.

Notes

1. In response to Lacan’s curtailment of the session, Duroux insists that his and Miller’s papers are to be heard together, and protests that ‘the benefit of the paper will be zero if we do not follow upon one another in a single continuity’ (Lacan, Seminar XII, Crucial Problems for Psychoanalysis, 7th session, 27 January 1965, trans. Cormac Gallagher, 7). ↵

2. Frege, Foundations of Arithmetic, § 29, § 54. ↵

3. See Frege’s ‘Concept Writing’, in Conceptual Notation and Related Articles, 107. ↵

4. See Jacques Lacan, Seminar XII, Crucial Problems for Psychoanalysis, 27 January 1965, 12. ↵

5. In Duroux’s presentation in Lacan’s seminar, Lacan himself adds ‘unicorns’ to Duroux’s ‘centaurs’ (Seminar XII, 7th session, 12), alluding to Serge Leclaire’s analysis of Philippe’s Dream of the Unicorn. Duroux makes no distinction between the non-existence that follows from the contradiction involved in the concept of a square circle, and the non-existence of golden mountains, centaurs and unicorns (which are logically possible). ↵

6. . Frege’s own definition of the successor function or the function of ‘following in a series’ [der Folgen in einer Reihe] in Foundations of Arithmetic, § 79. ↵

7. The claim that ‘the motor that animates succession in Frege is purely a negation of the negation’ appears to make appeal to a quasi-Hegelian explanation of Frege’s procedure that goes beyond the text of the Foundations of Arithmetic. ↵