Deleuze and the History of Mathematics by Duffy Simon;

Author:Duffy, Simon;
Language: eng
Format: epub
Publisher: Bloomsbury Publishing

The Riemannian concept of multiplicity and the Dedekind cut

In addition to the explicit role played by the infinitesimal calculus in Bergson’s philosophy, there are two other examples in his work where he implicitly draws upon particular mathematical developments to characterize the continuity of duration that I’d like to draw attention to. Neither of these is directly acknowledged by Bergson; however, by virtue of the terminology used when describing these examples, it is readily discernable that they are drawn from recent developments in mathematics that Bergson would have been aware of to some degree. The specific developments that Bergson implicitly draws upon are the concept of multiplicity developed by Bernhard Riemann (b. 1826–1866) in 1854, published in 1868 (Riemann 1963), and the idea of the Dedekind cut advanced by Richard Dedekind (b. 1831–1916) in 1872 (Dedekind 1963).

Before giving an account of the importance of Riemann’s work on multiplicity to Bergson’s concept of duration, which I foreshadowed above, I’d first like to briefly characterize the importance of the idea of the Dedekind cut to Bergson’s ontology. The Dedekind cut demonstrates how the real numbers can be constructed from the rational numbers. It resolves the apparent contradiction between the continuous nature of the number line and the discrete nature of the numbers themselves by combining an arithmetic formulation of the idea of continuity with a rigorous distinction between rational numbers—such as m/n—and irrational numbers, which can’t be expressed in a ratio—such as π, e, and √2. The idea of the Dedekind cut is rooted in Euclidean geometry and characterizes the point at which two straight lines, one of which is the real number line, cross, or “cut,” one another. At that one point on the number line, if there is no rational number, then an irrational number is constructed. Wherever a cut occurs on the number line that is not a rational number, an irrational number is constructed. The result is that a real number, whether rational or irrational, is constructed at every point on the number line. The Dedekind cut therefore proves the completeness or continuity of the real number line.

Bergson employs the imagery of the Dedekind cut to characterize the way that extensive magnitudes or objects are extracted from the dynamic flux of experience which, he argues, is in constant change. In CE, Bergson speaks of “objects cut out by our perception” (CE 12) and claims that “Things are constituted by the instantaneous cut which the understanding practices, at a given moment, on a flux of this kind” (CE 262). In The Creative Mind, he maintains that “For intuition the essential is change: as for the thing, as intelligence understands it, it is a cutting which has been made out of the becoming and set up by our mind as a substitute for the whole” (CM 39). Just as the cut is constitutive of a real number on the number line in mathematics, so too is the cut constitutive of an extensive magnitude in perception in relation to the continuous dynamic flux of experience.

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