## Infinity

L’infini

Some of the *Cahiers* authors found in the mathematisation of the infinite (undertaken by Cantor and other contributors to modern set theory) a
paradigmatic instance of the struggle between science and ideology.

The concept of infinity at issue in the *Cahiers pour l’Analyse* is not the metaphysical notion emphasised by Spinoza or Hegel so much as the mathematical notion at the heart of fundamental
debates in the development of modern mathematics. Before Cantor’s invention of transfinite numbers in the 1870s, the great majority of philosophers had agreed that application of the concept
of *actual* or self-embracing infinity should be reserved for an entity more or less explicitly identified with God: the most that mathematics
could do, it seemed, was describe something *potentially* infinite, the sort of thing illustrated by unending numerical succession: 1, 2, 3...*n*. Real or actual infinity was apparently destined to belong to a realm beyond number and thus beyond measurement - doomed
to remain, in short, an essentially indefinite if not explicitly religious concept.^{1}

Confronted with Zeno’s famous paradoxes concerning motion and division, Aristotle set a trend that would hold good for the
next two thousand years: if physical bodies *might* in principle (i.e. potentially) be infinitely divided, he argued, they never *are* so divided. Nothing existent is actually made up of infinitely small parts. As a result, if the infinite can be said to exist
at all, it must have an exclusively ‘potential existence’.^{2} The story of the actually infinite in modern mathematics, then, is the story of the slow subversion of this eminently ‘sensible’ Aristotelian approach. Most histories of mathematics distinguish three or four central episodes in this story: the discovery
of irrational numbers (by the Greeks); the algebraicisation of geometry (with Descartes); the discovery of the calculus and
the controversial status of ‘infinitesimals’ (Leibniz and Newton); the discovery of non-Euclidean geometries (Lobachevsky, Riemann), and the consequent search for a new
arithmetic foundation for mathematics (Cantor and subsequent contributors to the development of modern set theory). Cantor
and his followers provided precisely what previous mathematicians and philosophers had almost unanimously declared impossible:
a mathematically precise description of more-than-finite magnitudes *qua* numbers. Cantor 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.

### In the *Cahiers pour l’Analyse*

In his article ‘Suture’ (CpA 1.3), Jacques-Alain Miller links his Lacanian conception of the subject to the indefinite repetition of numerical succession, where the movement of 1+1+1...+*n* (which generates the infinite set of whole numbers) symbolises the infinite movement whereby one signifier (one ‘one’) represents a subject (a ‘zero’) for another signifier (or ‘one’), such that ‘the definition of the subject comes down to *the possibility of one signifier more* [un signifiant de plus]’ (CpA 1.3:48). On this basis, Miller presents 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). He links the indefinite or unending ‘excess’ of the succession of signifiers to Richard Dedekind’s contribution to the early development of set theory:^{3}

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.’

^{4}

In her ‘Communications linguistique et spéculaire’ (CpA 3.3), Luce Irigaray also evokes ‘that “flickering in eclipses” [“*battement en éclipses*”] of the subject who, at all times, wants to vanish in order to reappear as “one” [*un*], in a repetition irreducible to all temporal continuity, or to an infinity other than a denumerable, iterative succession’ (CpA 3.3:46).

Alain Grosrichard, in ‘Gravité de Rousseau’ (CpA 8.2), shows how the ‘idea of God’ emerges in Rousseau’s *Emile* at precisely the same moment as sexual desire. In Grosrichard’s formulation, ‘desire is finitude aware of itself, the opening to an other, to infinity’ (CpA 8.2:53): desire serves here as a sort of conduit to an infinity beyond itself. Bernard Pautrat’s article on Hume, ‘Du Sujet politique et de ses intérêts’ (CpA 6.5), considers the subject’s ‘infinity of desire’ from a very different point of view. On the assumption that ‘like all infinity’ this infinity of desire is ‘fictive’, Hume deprives the subject of its apparently ‘illusory autonomy’ (CpA 6.5:73), so as to allow for an analysis of the processes whereby subjects are led to obey the forms of authority that confront and
control them.

In his reading of Aragon’s *La Mise à mort* (CpA 7.2), Jean-Claude Milner considers the various ways a set of functions (‘love, depersonalisation, the novel’) are attributed to a series of figures or characters, in which ‘each term can be multiplied to infinity’ (CpA 7.2:46). Characters, insofar as they come to bear a given function, ‘challenge the work as the delimited space in which they move, and make it operative in an infinite world’ (47).

The most important meditations on infinity in the *Cahiers* appear in Volumes 9 and 10, most notably Cantor’s own ‘Fondements d’une théorie générale des ensembles’ (CpA 10.3), and Alain Badiou’s article on the infinitesimal (CpA 9.8). Cantor’s article provides a sort of philosophical overview of the logic whereby Cantor came to accept the ‘point of view which considers the infinitely great not merely in the form of something growing without limit’ but as something that can be ‘fixed mathematically by [distinct] numbers in the determinate form of the completed infinite’ (CpA 10.3:42). Cantor distinguishes what he terms the ‘infinite improperly understood’ from the ‘infinite properly understood’. The ‘improper’ infinite is conceived as an infinite *indetermination*. Understood in this way, the infinite remains derivative of the finite; the infinite is only ever understood as an extension
of the finite, as an unending sequence of addition. This concept of the infinite is intrinsically indeterminate, and it inheres
in the concept itself that it remain forever open-ended. By contrast, for Cantor the infinite proper is defined by the fact
that it is always presented under a determinate (36). By inventing new ways of determining the ‘class’ or ‘power’ of numbers, Cantor demonstrates that ‘following the finite there is a transfinite (*transfinitum*), which might also be called supra-finite (*suprafinitum*); that is, there is an unlimited ascending ladder of modes, which in its nature is not finite but infinite, but which can
be determined as can the finite by determinate, well-defined and distinguishable numbers’ (43).

By contrast, Jacques Bouveresse’s article on Wittgenstein’s philosophy of mathematics, published in the same tenth issue of the *Cahiers*, includes a discussion of his anti-Cantorian finitism and constructivism. According to Wittgenstein, the statement that ‘there is no biggest cardinal number’ simply means (in keeping with his ‘behaviourist’ endorsement of the linguistic turn) ‘that the authorization to “play” with cardinal numbers does not have an end’ (CpA 10.9:191). The mistake that Cantor makes, Wittgenstein claims, is to suppose that an ability to apply such a ‘technique without end’ might be understood in ontological terms, and correlated with an actually infinite set: Cantor confuses the technique through
which real numbers are constructed for use in calculations with the actual being or extent of the set of real numbers itself.

In ‘La Subversion infinitésimale’ (CpA 9.8) Badiou defends (in keeping with what will be long-term anti-constructivist polemic) a post-Cantorian position, and like
Cantor, begins with a discussion of the conventional distinction between the finite and the infinite. Whereas for Hegel, the
(metaphysically) infinite can be thought in the figure of a circle, as a self-sufficient whole, the (mathematically) finite
domain is determined as a merely linear progression which ceaselessly transgresses its own limit. Take any finite number *n*, however large: it is obviously possible, through the addition of a ‘supplementary inscription’ (*n*+1), to generate a still larger number. Rather than consider a variable *n* simply as the mark of a potentially unending succession, Badiou argues that such a succession already ‘presupposes a (unique) space of exercise’, a space that we tacitly assume as actually and *already* endless. ‘This is why the “potential” infinite, the indefiniteness of progression, testifies retroactively to the “actual” infinity of its *support*’ (CpA 9.8:118).

Badiou turns, in sections three and four of his article, to consider a recent and especially significant instance of this
general approach to the infinite – not in the domain of the infinitely large but of the infinitely small. From Zeno and Aristotle
through to Berkeley and Hegel (and indeed right through to Skolem’s path-breaking work of the 1930s), both philosophers and
mathematicians tended to accept that the notion of an infinitely small number was a self-evident ‘absurdity’ (123). Abraham Robinson, however, in work first published in 1961, was able to validate the affirmation of infinitesimal *numbers*.^{5} Robinson’s ‘non-standard’ approach serves to ‘reconstruct all the fundamental concepts of analysis’ in terms that are, for the first time, fully ‘systematic’. Robinson’s approach exemplifies, for Badiou, the age-old ideological investment in the association of infinity with quality and continuity (and ultimate with a divine or meta-physical substance)
- an investment which dominated the early development of mathematical analysis and ‘structural’ thought (135).

### Select bibliography

- Badiou, Alain.
*L’Etre et l’événement*. Paris: Seuil, 1988.*Being and Event*, trans. Oliver Feltham. London, Continuum Press, 2005. - Badiou, Alain.
*Le Nombre et les nombres*. Paris: Seuil, 1990, ch. 16.*Number and Numbers*, trans. Robin Mackay. London: Polity, 2008, ch. 16. - Boyer, Carl Benjamin.
*A History of Mathematics*. New York: Wiley, 1968. - Dauben, Joseph Warren.
*Georg Cantor: His Mathematics and Philosophy of the Infinite*. Cambridge: Harvard University Press, 1979. - Hegel, G.W.F.
*The Science of Logic*, trans. A.V. Miller. NY: Humanity Books, 1999. - Koyré, Alexandre.
*From the Closed World to the Infinite Universe*. Baltimore: Johns Hopkins University Press, 1957. - Lavine, Shaughan.
*Understanding the Infinite*. Cambridge, Mass.: Harvard University Press, 1994. - Moore, A.W.
*The Infinite*. London: Routledge, 1990. - Robinson, Abraham.
*Non-Standard Analysis*. Amsterdam: North Holland Publishing Company, 1966.

## Notes

1. Cf. Boyer, *A History of Mathematics*, 611. ↵

2. Aristotle, *Physics*, III, 4 and 5. ↵

3. See Erich Reck, ‘Dedekind’s Contributions to the Foundations of Mathematics’, *Stanford Encyclopaedia of Philosophy*, http://plato.stanford.edu/entries/dedekind-foundations/. ↵

4. Miller here 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. ↵

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