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

Philosophia Mathematica 19 (1):95-103 (2011)
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 , the author restricts his commentary to her Realism in Mathematics .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 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 …
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