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The concept of ‘identity’ is in play both in mathematics and logic, where it names either a qualitative or quantitative sameness between concepts or variables, and in discourses of the human sciences, including psychoanalysis, where it names a quality of a unique individual or subject. The relation and tension between these two meanings were themes explored throughout the Cahiers pour l’Analyse.

Whether the discourse is one of logic or of personhood, the concept of ‘identity’ is inextricable from its opposite ‘difference’. In logic, two variables are said to be identical in either a qualitative or numerical sense, i.e., insofar as poodles and Labradors are both dogs they are qualitatively identical qua dogs, whereas if we have five poodles and five Labradors we have an identical number of dogs. In both cases, however, identity masks an inherent difference - between concept (e.g., ‘dog’ or ‘five’) and object - that makes establishing an identity possible in the first place. Likewise, in the case of personal identity, the identity of a unique subject or person requires the coordination of two terms, viz. the unique subject or person and the concept, pronoun or proper name that designates. In the familiar discourse of personal identity, there is an emphasis on uniqueness and discreteness that seems to be a rejection of difference, i.e., my identity is unique to me; it is singularly mine. But here again the element of difference is unavoidable, grammatically and metaphysically, in that my identity requires, in effect, two terms, a subject and predicate, a phenomenon captured in the apparent pleonasms of ‘I am myself’ or ‘I am [my proper name]’.

Essential to both concepts of identity - the logical and the personal - is a process of identification, a sequence that establishes identity as a quality. In contemporary identity politics, subjects ‘identify’ with a generic concept (e.g., French, female, Christian, etc.) that differentiates the subject from other groups through identification with a discrete grouping. If difference is at the heart of identity, insofar as identity results from a disparity brought into parity, then identity itself also plays a role in the establishment of new differences; through different processes of identification, differences are accentuated or produced.

‘Identity’ has been a central and contested concept in the history of psychoanalysis. In the ego-psychology of Freud’s American inheritors, re-establishing and securing the self-identity of the ego, conceived as the seat of personal identity, has long been the goal of analytic practice. The scandal of Jacques Lacan’s intervention in the psychoanalytic field was announced as early as 1936, in his lesson on the ’mirror stage’ in the formation of the ego. For Lacan, the establishment of the ego in childhood results from a process of identification that is essentially imaginary. When the child sees its body in a mirror, it recognizes this specular image in its wholeness or completeness as different from its subjective experience. And yet, the child comes to recognize the image as its own, as ‘mine’. Lacan claims that this moment of recognition that establishes the ego-ideal (or self-image) is essentially a misrecognition (méconnaissance) in that the identity achieved is spurious or incomplete, especially given the child’s inadequate mastery over the image. Identifying with the image, the child covers over the essential difference or lack separating its subjective regard from the image it has of itself.

Lack is at the heart of the Lacanian interrogation of identity. In the Cahiers pour l’Analyse, we witness an effort to explore the relation between the discourse of identity in logic and mathematics, on the one hand, and that of psychoanalysis on the other. The role of identity in scientific writing (e.g., in numbers and zero) will be a bone of contention in the Cahiers, chiefly between the positions developed by Jacques-Alain Miller and Alain Badiou.

In the Cahiers pour l’Analyse

The concept of identity occupies a crucial position in the argument of one of the foundational articles for the Cahiers, Jacques-Alain Miller’s ‘La Suture’ in Volume 1 (CpA 1.3). Miller’s goal in this essay is to examine Gottlob Frege’s account of the foundations of arithmetic with an eye toward establishing its consistency with a Lacanian logic of the signifier. What is more, Miller accords Frege’s discourse an exemplary status in that the subject in play in this discourse is not a psychological subject, synthesizing a field of operations, but rather a subject in the Lacanian sense.

Miller’s argument is that when Frege establishes the sequence of whole numbers out of the zero, an inaugural identity is established that has a non-identity as its fundamental ground. Miller’s reading of Frege focuses on two operative relations in his argument, the subsumption of an object under a concept and the assignation of number to a concept. The concept of object, however, presupposes a unity, something making this object a discrete (theoretical) object. In this operation, however, the ‘thing’ that serves as the object’s condition is forgotten or covered over. The concept of zero presupposes a non-identity, more specifically a set of things that are not identical to themselves. Insofar as the assignation of zero in turn. With zero and one established, we now have two; the sequence of numbers is now generated as a matter of course.

The crux of Miller’s argument is that the identity established at the base of Frege’s analysis - between the number zero and the concept of ‘things which are not-identical to themselves’ - is achieved only through the suture over an essential lack (i.e., non-identity, or non-being). In this operation of suture, which is forgotten or repressed in the surety of Frege’s discourse, we witness the emergence and effect of the subject. Suture is at once the operation of a signifier, i.e., that which represents a subject for another signifier, in this case zero, and the gestation of the subject, in that the subject is ‘carried forth’ in the infinite procedural sequence of whole numbers, notwithstanding the lack that serves as its primordial ground.

In his response to Miller, published in this same volume, Serge Leclaire questions Miller’s claim that Frege ‘sutures’ logical discourse by assigning the number zero to the concept of the non-identical:

The introduction of this concept of non-identity to itself follows on from the Leibnizian concept of identity-to-self advanced by Frege, namely: ‘those things are identical of which one can be substituted for the other without the truth being lost’. It is starting from there that one arrives at this other proposition: ‘The truth is: each thing is identical to itself’. What is this thing that is identical to itself? It is the thing insofar as it is one, namely, the object. That everything is identical to itself is what permits the object (the thing insofar as it is one) to fall under a concept. It is necessary that the thing should be identical to itself in order that truth can be saved: here, we might discover the major accent not only of Frege’s book, but also of Miller’s exposition, namely, the saving of the truth (CpA 1.4:51).

For Leclaire, the ‘saving of truth’ is not the essential task of the psychoanalyst, as it may be for the logician. As such, Leclaire questions the utility of Miller’s analysis for analytical practice.

The psychoanalyst André Green also puts pressure on Miller’s account of identity in his own contribution to Volume 3 of the Cahiers (CpA 3.2). For Green, truth is more than a matter of the possibility of establishing or preserving self-identity, even as he grants the role non-identity plays in the emergence of the subject. ‘Not only does the subject cut itself off from the scene and chain of signification, but the first of these objects functions simultaneously as a concept and as an object, not represented but named as single object and as concept of non-identity to itself, a concept that threatens truth, insofar as it is out of play [hors-jeu], or beyond the I [hors-je]’ (CpA 3.2:23, trans. 172). This ‘truth-threatening situation’ Miller presents requires further analysis, however. What must be involved is a particular kind of ‘encounter with truth’, where truth is dissociated from the possibility of demonstration [manifestation] and self-identity, and persists only through ‘the blank or the trace that negates it’ (CpA 3.2:23, trans. 173).

The fourth and final section of Green’s article is titled: ‘Identity and Non-Identity to Self: The Death Drive’ (CpA 3.2:34-37, trans. 186-190). Here Green lauds Freud’s account of dissimulation against Lévi-Strauss’s account of the dual nature of masks. He also locates the death drive in the non-identity of the signifier to itself, locating this phenomenon in turn in the primordial non-identity of the Name-of-the-father at the basis of monotheisms.

In her assessment in Volume 9 of the epistemological break in the sciences accomplished by Galileo, Judith Miller notes how the new concept of speed operative in Galilean science put paid to an Aristotelian concept of identity. In the Aristotelian view, the speeds of two objects could be said to be identical only if the objects in question traversed the same space in the same time. Galileo produced a new concept, grounded in a mathematics of proportionality, which stated that the speeds of two moving bodies are ‘said to be equal when the spaces they traverse stand in the same proportion to each other as the times required to traverse them’ (CpA 9.9:140).

In his article in Volume 10, ‘Marque et manque: à propos du zéro’ (CpA 10.8), Alain Badiou develops a case for the self-identity of marks in scientific, i.e., logico-mathematical, writing, arguing that the integrity or truth of science itself is dependent upon these marks, e.g., ‘x’, retaining their self-identity across multiple instantiations or appearances. He writes:

It is the technical invariance of traces and instruments that subtracts itself from all ambiguity in the substitution of terms. Thus determined, the rule of self-identity allows of no exceptions and does not tolerate any evocation of that which is withheld from it, not even in the form of rejection. What is not substitutable-for-itself is something radically unthought, of which the logical mechanism bears no trace […]. What is not substitutable-for-itself is foreclosed without appeal or mark (CpA 10.8:157).

Mobilizing an appeal to Gödel’s undecidability theorems, Badiou holds that it is the non-identical to itself, e.g., x ≠ x, that is precisely excluded from a logical discourse. This is so since such a formulation, while syntactically sound, is inconceivable and ultimately non-derivable. The non-identity that forms the ground of the subject’s discourse in Miller’s venture is relegated solely to the domain of ideology in Badiou’s reading, which, following Althusser upholds not only the possibility, but the necessity of an epistemological break which establishes and secures scientific discourse.

Select bibliography

  • Badiou, Alain. ‘Note complémentaire sur un usage contemporain de Frege’, Le Nombre et les nombres. Paris: Seuil, 1990. 36-44. ‘Additional Note on a Contemporary Usage of Frege’, in Number and Numbers, trans. Robin Mackay. London: Polity, 2008. 24-30.
  • Deleuze, Gilles. Différence et répétition. Paris: PUF, 1968. Difference and Repetition, trans. Paul Patton. New York: Columbia University Press, 1994.
  • Frege, Gottlob. Foundations of Arithmetic [1884], trans. J.L. Austin. Oxford: Basil Blackwell, 1980.
  • Lacan, Jacques. Écrits. Paris: Seuil, 1966. Écrits, trans. Bruce Fink, in collaboration with Héloïse Fink and Russell Grigg. New York: W.W. Norton, 2006.