Xavier Sabatier. Les formes du realisme mathematique. Paris: Vrin, 2009. ISBN 978-2-7116-2193-4. Pp. 304

Abstract

The book under review discusses the most standard forms of mathematical realism, a.k.a. ‘Platonism’, elaborated in the mainstream of twentieth-century philosophy of mathematics. Its purpose is therefore to introduce franco phone readers to an important set of ideas that have been, for the most part, cultivated by analytic philosophers. It originates from a doctoral dissertation at Université Paris 1. It comprises 287 pages of text divided into four main parts, completed by an eight-page bibliography. Beginning with Frege, Russell, and Gödel, Xavier Sabatier discusses the work of several contemporary Platonists, such as Quine, Resnik, Shapiro, and Maddy. Since the last no longer considers herself a Platonist in mathematics (p. 219), the author restricts his commentary to her Realism in Mathematics (published in 1990). The introductory first part ends with a discussion of the standard objections to realism. The fourth part of the book, which is also its conclusion, revisits those objections and shows how, in the light of what has been said, they can be overcome to some extent from within a naturalist stance. Following the first part, one finds two long sections entitled ‘A priori realism’ and ‘Naturalized realism’. The first focuses on the Frege-Russell brand of logical Platonism, prefigured in the work of Bolzano, and on Gödel’s version of realism. The second examines the renewal of realism emanating from Quine’s project of a naturalized epistemology. It is in this section that Sabatier considers some contemporary authors like Penelope Maddy, Michael Resnik, and Stewart Shapiro. Some might find it questionable to associate the structuralism of Resnik and Shapiro with Platonism, given the naturalistic stance of those philosophers (at least in Resnik’s case) but it is nevertheless an excellent idea to discuss their work in such a context. Indeed, one can give interpretations of Plato’s Sophist and of other late dialogues that are somewhat akin to mathematical structuralism.1 The author, though, has chosen not to discuss historical Platonism or even varieties of mathematical Platonism that do not fit well into the analytic tradition, such as one can find in the work of the philosopher Albert Lautman, or in the realist reconstruction of Husserl’s philosophy of mathematics given by G. Rosado Haddock. Moreover, he does not discuss in any detail the writings of Platonist mathematicians, such as René Thom, Roger Penrose, or Alain Connes. Therefore the book remains well within the confines of academic philosophy.