Number and Numbers

John Kadvany reviews Alain Baidou’s book in Notre Dame Philosphical Reviews (via bookforum):

Like many philosophers, Alain Badiou relies on technical systems of mathematical logic as a foundation for philosophical exploration. Donald Davidson used Tarski’s theory of truth for formal languages to ground his approach to natural language semantics. Modal logic is frequently used to discuss problems of necessity, time, or belief. W. V. O. Quine made the reduction of mathematics to set theory a paradigm of “ontological commitment,” such that an idealized formalization of physical science identified the entities needed to ensure the theory as fundamentally “real.” Indeed, Badiou’s project is exactly in this Quinean mode, with set theory his preferred tool. While Badiou’s set-theoretic interpretations are not typical of those found in Berkeley or Princeton, the overall strategy is nonetheless “analytic.” Hence one’s response to Number and Numbers, and a similar earlier book, Being and Event, will depend on a) the philosophical narrative laid over the mathematics, and b) the treatment of the mathematics vis à vis the interpretation.

This review follows those two themes. But first a word on intellectual context. Badiou is one of several French philosophers pilloried by Alan Sokal and Jean Bricmont in Fashionable Nonsense: Postmodern Intellectuals’ Abuse of Science (1998). Sokal, of course, authored the sham 1996 Social Text article, concocted as a meaningless essay in high lit-crit-theoretic style, and buttressed by facile, often bogus or inconsistent, appeals to quantum physics or relativity, all undetected by the editors. The hoax was widely reported in the popular press and Sokal has since continued his manic crusade, though with much less panache and success. One of Sokal’s “tests” for intellectual value is whether one can detect a difference in meaning if key words are interchanged, say “being” and “other,” or “mediation” and “reification,” or whatever — sort of a Turing test for intellectual quality. For Badiou in Fashionable Nonsense, Sokal and Bricmont, as they so often do, just selectively quote him, rhetorically ask the reader if it makes sense (of course not, devoid of context), and then go on. The second trick could be applied to Number and Numbers, perhaps with some honesty, but I think ultimately unfairly, as explained below. Badiou’s ontological narrative is allusive, poetic, and deeply metaphorically inspired by his understanding of modern set theory. And Sokal’s “reversal” test also fails. Badiou’s vision is a wholly consistent one, built indeed from a stimulating historical account of the treatment of number by Gottlob Frege, Guiseppe Peano, Richard Dedekind, and Georg Cantor.