Mathematics and Reality, by Mary Leng.

Abstract

The popularity of anti-realism in the philosophy of mathematics has been steadily increasing over the past several decades. This book is a valuable addition to the anti-realist cause. Beginning from a naturalistic standpoint familiar to any Quinean, Leng ends with the very non-Quinean view that the standards of justification operative in our best science do not ultimately give us reason to believe that there are mathematical objects, not even those low-level objects — in terms of set-theoretic rank — that she accepts as (very likely) indispensable to our best science (Sect. 3.3). Leng’s overarching strategy — namely, undermining the Quine-Putnam indispensability argument — is not new. Hartry Field is well known for taking this approach in arguing for nominalism (Science Without Numbers, Princeton: Princeton University Press, 1980). Neither is her primary strategy for accomplishing this — namely, undermining confirmation holism — entirely novel. For example, Penelope Maddy (‘Indispensability and Practice’, Journal of Philosophy, 89 (1992), pp. 275–89; Naturalism in Mathematics, Oxford: Oxford University Press, 1996, Sects II.6 and II.7), Elliot Sober (‘Mathematics and Indispensability’, Philosophical Review, 102 (1993), pp. 35–7), and Susan Vineberg (‘Confirmation and the Indispensability of Mathematics to Science’, Philosophy of Science, 63 (1996), pp. S256–S263) have all called confirmation holism into question. Even Leng’s insightful application of Kendall Walton’s theory of fiction (Mimesis as Make-Believe, Cambridge, MA: Harvard University Press, 1990) was prefigured by Stephen Yablo (‘Does Ontology Rest on a Mistake?’ Proceedings of the Aristotelian Society, Supplementary Volume 72 (1998), pp. 229–61; ‘Go Figure: A Path Through Fictionalism’, Midwest Studies in Philosophy 25 (2001), pp. 72–102; ‘The Myth of the Seven’, in Fictionalism in Metaphysics, ed. Mark Eli Kalderon, Oxford: Oxford University Press, 2005, pp. 88–115). That said, the care and detail with which she develops her fiction-based arguments against confirmation holism and explores the dialectic between responses and counter responses to these arguments make this book worthwhile for anyone seriously interested in the philosophy of mathematics.