Fink, B. (1996) Alain Badiou. Umbr(a): One, 1: pp. 11-12.
Hegel marks, according to Badiou, a romantic, historicist tum in Western thought away from the mathematical concept of the infinite, Hegel seeing the latter as “interesting” but fundamentally unsubstantiated compared to the “properly philosophical” view that man’s existence is essentially finite.
Whereas Plato views mathematics as establishing a realm of discourse which has no need to resort to myth or any other discourse to found its conclusions — which thus breaks with mythopoetic discourse and can serve as the foundation or precondition of another discourse (philosophy or dialectics) with its own grammar and methods — Hegel opposes philosophy and mathematics, invalidates mathematical notions of infinity, and rather than effecting an Aufhebung (sublation), returns to a pre-Platonic view whereby a certain myth (of man’s finitude) rules philosophical “reason.”
This romanticism can be seen, according to Badiou, in both the Heideggerian project and co the “postmodem” project, both of which refuse the notion of the infinite nature of every human situation, subjecting infinities to the horizons of the “human.”
The lack of serious attention devoted to mathematics — whether by Carnap, Heidegger, or Derrida — is indicative of a philosophical tendency which is ultimately conservative and romantic in content, Badiou argues, and which leads directly to the view that philosophy has reached its end.
Badiou proposes that we rethink the very notion of endpoint and limit using modern developments in mathematics, a project which reestablishes mathematics as a condition of philosophy and allows us to think beyond the latter’s supposed end.
The importance of Badiou’s insight here goes far beyond the “simple” conditioning of philosophy upon mathematics. For what he refers to as romanticism is essentially defined by what he terms the “regime of the One,” that is, the regime or rule of a discourse of totalization.
As antitotalizing as certain modem discourses may claim to be, they too succumb, Badiou sustains, to a Parmenidean view of the whole, “counted as One,” complete with notions of limits which predate modern mathematics.
Badiou’s emphasis on the modem concept of multiple infinities, of infinities of infinities, subverts traditional views of limits and horizons, and moves in a radically non-logocentric direction. Badiou, in a sense, moves from the post-modem to the post-finite, from the end of philosophy to the beginning of philosophical multiplicity: towards the liberation of philosophy from the regime of the finite.