Mary Leng

Naturalism in Mathematics PENELOPE MADDY, 1997 Oxford, Oxford University Press viii + 254 pp., $CAN91, ISBN 0–19–823573–9 Bohmian Mechanics and Quantum Theory: an Appraisal JAMES T. CUSHING, ARTHUR FINE & SHELDON GOLDSTEIN, 1996 Dordrecht, Kluwer viii + 403, pp., US$159.00, ISBN 0–7923–4028–0 Pragmatism as a Principle and Method of Right Thinking: the 1903 Harvard Lectures on Pragmatism CHARLES SANDERS PEIRCE, 1997 Edited and introduced, with a commentary, by PATRICIA ANN TURRISI Albany, State University of New York Press xi + 305 pp., $26.53 ISBN 0–7914–3265–3 ISBN 0–7914–3266–1

This paper argues that it is scientific realists who should be most concerned about the issue of Platonism and anti-Platonism in mathematics. If one is merely interested in accounting for the practice of pure mathematics, it is unlikely that a story about the ontology of mathematical theories will be essential to such an account. The question of mathematical ontology comes to the fore, however, once one considers our scientific theories. Given that those theories include amongst their laws assertions that imply the existence of mathematical objects, scientific realism, when construed as a claim about the truth or approximate truth of our scientific theories, implies mathematical Platonism. However, a standard argument for scientific realism, the 'no miracles' argument, falls short of establishing mathematical Platonism. As a result, this argument cannot establish scientific realism as it is usually defined, but only some weaker position. Scientific 'realists' should therefore either redefine their position as a claim about the existence of unobservable physical objects, or alternatively look for an argument for their position that does establish mathematical Platonism.

This paper responds to John Burgess's ‘Mathematics and _Bleak House_’. While Burgess's rejection of hermeneutic fictionalism is accepted, it is argued that his two main attacks on revolutionary fictionalism fail to meet their target. Firstly, ‘philosophical modesty’ should not prevent philosophers from questioning the truth of claims made within successful practices, provided that the utility of those practices as they stand can be explained. Secondly, Carnapian scepticism concerning the meaningfulness of _metaphysical_ existence claims has no force against a _naturalized_ version of fictionalism, according to which our ordinary standards of scientific evidence may show that we have no reason to believe the mathematical existence claims made within the context of our mathematical and scientific theories.