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A concept common to mathematics and philosophy, the ‘one’ was a site of theoretical engagement throughout the Cahiers pour l’Analyse, tying the inquiry into modern mathematics and logic to the investigation of Platonist ontology that was a recurring theme in the journal.

It is a conceptual irony that, throughout the history of philosophy, the ‘one’ has had multiple meanings and connotations. The term evinces singularity and uniqueness, as well as unity and completeness. Ancient Greek philosophy turned on the question of the one and the multiple, their relation, and the argument over which was ontologically primary. Where Parmenides held that being was in its essence singular, i.e. that the One was primary, Heraclitus emphasized the multiplicity of becoming or flux prior to the One. Plato’s dialogue, the Parmenides, is equally concerned with the relation of the one to the multiple and of being to non-being. This dialogue is a central point of reference in the Cahiers.

The French term ‘un’ translates both the number ‘one’ and the indefinite article ‘a’, and the concept of the one is bound up with the problematic of what constitutes a singular or discrete object, theoretical or otherwise. As such, the phenomenon of ‘oneness’ is also related to that of ‘otherness’, i.e. what is not part of the unity or one, but which seems to be the one’s necessary relational condition. The metaphysical argument concerning the primacy of being or non-being is manifested in mathematics as a dispute over the relation between the one and the zero, i.e. which is the ‘first’ number. Frege’s production of the one out of the concept of zero is itself one of the fundamental inspirations and sites of theoretical disputation within the Cahiers.

Debate on the status of the one is a significant part of the legacy of the Cahiers. In his Seminar XX, Lacan introduces the formula ‘there is Oneness’ [‘Y a-t-il de l’un’] in an effort to account for the operations of the register of the symbolic. Alain Badiou will affirm a similar formula in the ontology of pure multiplicity (a multiplicity ‘without one’) that he develops in Being and Event (1988), where the one figures as the result of the operation of a ‘count-as-one’ or ‘count-for-one’ [‘compte-pour-un’].

In the Cahiers pour l’Analyse

The function of the one in Frege’s Foundations of Arithmetic (1884) is first addressed in Yves Duroux’s ‘Psychologie et logique’ (CpA 1.2). Here Duroux introduces Frege’s critique of psychologistic accounts of mathematics. For Frege, the collapse of logic into psychology arises in part from the ambiguity of the term ‘unity’ (Einheit in German). 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. 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 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.

The gestation of the number One is given its fullest explication in the article that follows Duroux’s in this volume, Jacques-Alain Miller’s ‘La Suture’ (CpA 1.3:40-47, trans. 26-32). In his second section, ‘The Zero and the One’, Miller stages a ‘movement back’ into the logic of the signifier through a reading of Frege’s text. Miller’s claim is that Frege’s genesis of the progression of whole natural numbers involves a misrecognition of the function of the subject. 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.

Frege’s logicist system 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 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’. 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’ (CpA 44, trans. 30). 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’ (44/30).

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. 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). This structure of repetition sutures the subject to the chain of its discourse.

In her contribution to Volume 3 (CpA 3.3), Luce Irigaray explores the relation between the account of the one and the zero found in Miller’s reading of Frege and the semantics of the French impersonal pronoun ‘on’ [‘one’]. In the movement from the impersonality of the on to the person denominated by the proper name, we witness a movement analogous to the generation of the 1 out of the 0 via the designation of the latter via the proper name of the former. This analysis also applies to the web of nouns and pronouns that surreptitiously replace the impersonality of the on in the child’s experience, e.g., ‘he’, ‘him’, ‘son’. Irigaray claims: ‘The proper name best represents the paradox of the engendering of the “1” out of “zero”’ (CpA 3.3:42).

Jean-Claude Milner’s contribution to Volume 3, ‘Le Point du Signifiant’ (CpA 3.5) explicitly deals with the relation between ontology and number. Undertaking a reading of Plato’s Sophist inspired by Miller’s arguments in ‘La Suture’, Milner suggests that Plato’s position lies between the One of Parmenides (who is represented by the Eleatic Stranger in this dialogue) and the absolute negation proposed by Gorgias. In the Sophist, Plato’s ‘guiding desire is to establish the status of non-being’ (CpA 3.5:73). At the heart of Milner’s reading is an assessment of non-being’s dual status as both function and term; this vacillation integral to its status is that of the subject in the chain of signification. On the one hand, there is being as the order of the signifier, the ‘radical register of all computations’, totality of all chains, and on the other hand, the ‘one’ of the signifier, the unity of computation, the element of the chain, non-being, as the signifier of the subject (CpA 3.5/77). This latter reappears as such every time that discourse deploys its power to ‘annul’ signifying chains.

In their new questions for Michel Foucault, following upon his response to their first set of queries, the Cercle d’Épistémologie presses Foucault on what constitutes the ‘unity’ of a discourse and, moreover, what accounts for the ‘singularity’ of an event within (and without) discourse (CpA 9.3/43). This set of questions turning on themes of unity, unicity, and singularity, will permeate The Archaeology of Knowledge (1969), which can be read as Foucault’s book-length response to his exchanges with the editors of the Cahiers.

The most ambitious interrogation of the ontological and epistemological status of the One takes place in François Regnault’s ‘Dialectique d’épistémologies’ (CpA 9.4), also in Volume 9. Here Regnault reads the ontological discourse of Plato’s Parmenides, concerned with the relation between the one and its other via the category of being, in order to dialectically generate a series of possible relations between science and epistemology, the latter being the discourse of science. The basic matrix of the series of Hypotheses considered in the dialogue is determined here by the presumed existence or non-existence of the One: What follows if the One exists, for the One itself, and for that which is defined as not-One, i.e. ‘the Others’ (or ‘difference’, to heteron)? And then, what follows if the One does not exist, both for the One itself, and again for the Others or not-One? In particular, what follows if the existence of science is considered as one or as not-one (multiple)? ‘There will be as many epistemologies as there will be different conceptions of this existence and of this unity’ (CpA 9.4:51).

In his text ‘The Mathematical Analysis of Logic’, translated and reproduced in Volume 10 of the Cahiers (CpA 10.2), George Boole affirms his use of ‘the symbol 1 or unity in order to represent the universal class. We understand it as including any conceivable class of objects, whether it has an actual existence or not, positing it as the principle that the same individual can find itself in more than one class, just as it can have more than one quality in common with others’ (CpA 10.2:31).

Select bibliography

  • Badiou, Alain. L’Etre et l’événement. Paris: Seuil, 1988. Being and Event, trans. Oliver Feltham. London: Continuum, 2005.
  • ----. ‘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
  • Cornford, Francis MacDonald. Plato’s Theory of Knowledge. The Theaetetus and the Sophist of Plato, translated with a running commentary. London: Routledge, 1935.
  • Cornford, Francis MacDonald. Plato and Parmenides: Parmenides’ Way of Truth and Plato’s Parmenides, translated with a running commentary. New York: Harcourt, Brace, and Company, 1939.
  • Frege, Gottlob.The Foundations of Arithmetic (1884), trans. J.L. Austin. Evanston, Illinois: Northwestern University Press, 1980.
  • Hallward, Peter, ed. ‘The One and the Other: French Philosophy Today’. Angelaki 8:2 (2003).
  • Lacan, Jacques. Seminar XX, Encore: On Feminine Sexuality, the Limits of Love and Knowledge (1972-73), ed. Jacques-Alain Miller, trans. Bruce Fink. New York: W.W. Norton, 1999.
  • Plato. Sophist, trans. Harold North Fowler. Cambridge, MA: Harvard, Loeb Classical Library, 1921.