## Number

le nombre, le chiffre

One of the main ambitions of several of the *Cahiers* authors is to follow through on Lacan’s ambition to ‘mathematise’ elementary dimensions of the unconscious, and they draw on various aspects of modern number theory in order to develop formal
accounts of science and of the subject.

Number is the elementary mathematical concept for measuring quantities or counting elements. The most familiar or ‘natural’ form of number serves to count out the discrete quantity of elements in a collection: none, one, two, three... A significant
portion of the agenda which informs the *Cahiers pour l’Analyse* is bound up with the effort to apply such discretion to concepts and beings conventionally considered to be resistant to
it: the subject, the unconscious, the non-self-identical, and so on. Alain Badiou, in a statement that summarises this aspect of the *Cahiers*’ project, contrasts ‘quality, continuity’ and other ‘enslaving categories of ideological objectives’ with ‘number, discretion’ and ‘categories of scientific processes’ (CpA 9.8:136). Scientific representation appears to offer an alternative to the illusions of ordinary lived experience - as Lacan put
it in his second seminar, ‘all scientific progress consists in making the object as such fade away’ and replacing it with symbolic-mathematical constructions.^{1}

Numbers serve to answer the elementary question, confronted with whatever sort of being, ‘how many?’ The question, ‘what is the being of numbers *themselves*?’ has received a wide range of answers, from transcendent realities to merely formal conventions. The answers most relevant
to the *Cahiers* authors are those given by Plato, Cantor, Frege and some post-Fregean philosophers of logic and mathematics, notably Russell and Gödel. Today perhaps the most widely accepted view remains the one, associated with Frege and Russell, whereby ‘a number is something that characterises certain collections’, or ‘classes’ - i.e. collections of twos, threes, fours.... Thus defined, ‘numbers do not have being - they are, in fact, what are called “logical fictions”’.^{2} Thirty years after the *Cahiers* ceased publication, Badiou, drawing in large part on Platonist and Cantorian questions addressed in the journal, came to
the opposite conclusion when in *Being and Event* (1988) he equated mathematics with ontology itself. From this perspective, the analysis of number is not merely a matter
of formal representation, ‘it is a matter of realities […]. A number is neither part of a concept, nor an operational fiction, nor an empirical given,
nor a constituent or transcendental category, nor a syntax, nor a language game, nor even an abstraction from our idea of
order. Number is a form of Being.’^{3}

### In the *Cahiers pour l’Analyse*

Yves Duroux’s contribution to the inaugural issue of the *Cahiers* (CpA 1.2) provides 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). Building on Duroux’s arguments in his ‘Suture’ (CpA 1.3), Jacques-Alain Miller goes on to claim that Frege’s genesis of the progression of whole natural numbers provides the logical basis for an analysis
of the subject and of the way a signifier ‘represents a subject for another signifier’.

In his ‘Note sur l’objet de la psychanalyse’ (CpA 2.5), Serge Leclaire observes that there is a grain of truth in the notion that the new dimension opened up by psychoanalysis is ‘irrational’, in the mathematical sense of the term, just as Pythagoras recognised that ‘there is no common measure, in the order of the rational numbers, between the diagonal of a square and its sides’, so psychoanalysis must also be alert in an analogous way to the intrinsic difficulties of grasping hold of its object (CpA 2.5:126).

Jean-Claude Milner’s analysis of Plato’s *Sophist* (CpA 3.5) begins with a reflection on the historical relation between ontology and number. Milner cites a text by Isocrates that distinguishes
between Greek ontologies on the basis of the number of entities they assume: the ancient sophists say that there are an *infinity* of beings, Empedocles says that there are *four*, Ion *three*, Alcmaeon *two*, Parmenides *one*, and Gorgias *none*. Milner suggests that Plato’s own position lies between the One of Parmenides (who is represented by the Eleatic Stranger
in the dialogue) and the absolute negation of Gorgias (CpA 3.5:73-75).^{4}

In ‘La Subversion infinitésimale’ (CpA 9.8) Alain Badiou considers the new ‘non-standard’ conception of infinitesimal (infinitely small) numbers developed by Abraham Robinson in work first published in 1961.^{5} Robinson’s ‘non-standard’ approach, Badiou argues, serves to ‘reconstruct all the fundamental concepts of analysis’ in terms that are, for the first time, fully ‘systematic’; it exemplifies the age-old ideological investment in the association of infinity with quality and continuity (and ultimately with a divine or meta-physical substance)
- an investment which dominated the early development of mathematical analysis and ‘structural’ thought (135).

Much of the final issue of the *Cahiers* is concerned with modern conceptions of number and their logical implications (see individual article synopses for more details).
Georg Cantor’s revolutionary contribution to modern mathematics is especially significant here (CpA 10.3). Cantor was the first to propose a precise description of more-than-finite magnitudes *qua numbers*. He established that the concept of numerical *order* or succession is every bit as coherent in the realm of the actually infinite as it is in the realm of the finite. He showed
that it made perfect sense to speak of the *size* (or ‘cardinality’) of different infinite quantities, conceived as completed wholes or sets.^{6} The subsequent development of set theory has been concerned with the limits required to specify such ‘completion’, and with it, a precise conception of numerical size.

### Select bibliography

- Badiou, Alain.
*L’Etre et l’événement*. Paris: Seuil, 1988.*Being and Event*, trans. Oliver Feltham. London: Continuum, 2005. - Badiou, Alain.
*Le Nombre et les nombres*. Paris: Seuil, 1990.*Number and Numbers*, trans. Robin Mackay. London: Polity, 2008. - Benacerraf, Paul, and Hilary Putnam, eds.
*Philosophy of Mathematics: Selected Readings*. Cambridge: Cambridge University Press, 2nd ed., 1983. - Beth, Evert W.
*Mathematical Thought*:*An Introduction to the Philosophy of Mathematics*. Dordrecht: Reidel, 1965. - Dauben, Joseph Warren.
*Georg Cantor: His Mathematics and Philosophy of the Infinite*. Cambridge: Harvard University Press, 1979. - Davis, Philip J., and Reuben Hersh.
*The Mathematical Experience*. Harmondsworth: Penguin, 1980. - Devlin, Keith.
*Fundamentals of Contemporary Set Theory*. NY: Springer-Verlag, 1979. - ---.
*Mathematics: The New Golden Age*. Harmondsworth: Penguin, 1988. - Eves, Howard.
*An Introduction to the History of Mathematics*. NY: Holt, Rinehart and Winston, 1975. - Frege, Gottlob.
*The Foundations of Arithmetic*. Trans. J.L. Austin. Oxford: Blackwell, 1950. - Hallett, Michael.
*Cantorian Set Theory and Limitation of Size*. Oxford: Oxford University Press, 1984. - Robinson, Abraham.
*Non-Standard Analysis*. Amsterdam: North Holland, 1966. - Russell, Bertrand.
*Introduction to Mathematical Philosophy*. London: Allen and Unwin, 1920. - Shapiro, Stewart.
*Philosophy of Mathematics: Structure and Ontology*. Oxford: Oxford University Press, 1997. - Tiles, Mary.
*The Philosophy of Set Theory: An Introduction to Cantor’s Paradise*. Oxford: Blackwell, 1989. - Van Heijenoort, Jean, ed.
*From Frege to Gödel*. Cambridge: Harvard University Press, 1967.

## Notes

1. Lacan, Seminar II, 130/104. In the decade following his famous eleventh seminar, at the Ecole Normale Supérieure (1964-65), Lacan’s own thinking moved ever closer toward the ideal of pure mathematisation. Unlike Badiou’s later work, however, Lacan never ceased to believe that ‘the notion of being, as soon as we try to grasp it, proves itself to be as ungraspable as that of speech’ (Seminar I, 352/229) - if only because ‘the unconscious does not lend itself to ontology’ (Seminar XI, 38/29). ↵

2. Russell, *Introduction to Mathematical Philosophy*, 12; 132. ↵

3. Badiou, *Le Nombre et les nombres*, 11, 261. ↵

4. In his *L’Etre et l’événement* (1988) Badiou will come back to this cluster of questions, to ask, in the most general terms: which sort of number comes
first, the one or the many? unity or multiplicity? Ever since Plato’s *Parmenides*, Badiou maintains, classical ontology has been unable to reconcile these two categories. For if being is one, then that which
is not-one, i.e. the multiple, must not be. But the beings we can present to our minds are presentable precisely as multiple
and variable being*s*: presentation itself is clearly multiple. If be-ing is one, then it appears that presentation itself must somehow not be.
On the other hand, if be-ing is multiple then it would seem impossible to conceive of a presentation as *a* presentation, i.e. as *one* being. Badiou’s way out of this conceptual quagmire is to accept that being is not one, while recognising nevertheless that
ones are *made* to be. ‘*L’un n’est pas*’ - there is no being of the one - but ‘*il y a de l’Un*’, a phrase whose meaning might be best rendered as ‘there is a One-*ing*’. The one is not, but there is an operation that one-ifies or makes-one. There is no one, there is only an operation that
counts-as-one (Badiou, *L’Etre et l’événement*, 31-32). And if the one is not, then only the multiple is. ↵

5. See ‘Abraham Robinson’, *Wikipedia*, http://en.wikipedia.org/wiki/Abraham_Robinson. ↵

6. See Dauben, *Cantor*, 124-125. Important preliminary steps were taken, as Cantor acknowledged, in Bolzano’s *Paradoxes of the Infinite* (1851). ↵