Mary Leng

For many philosophers not automatically inclined to Platonism, the indispensability argument for the existence of mathematical objectshas provided the best (and perhaps only) evidence for mathematicalrealism. Recently, however, this argument has been subject to attack, most notably by Penelope Maddy (1992, 1997),on the grounds that its conclusions do not sit well with mathematical practice. I offer a diagnosis of what has gone wrong with the indispensability argument (I claim that mathematics is indispensable in the wrong way), and, taking my cue from Mark Colyvan''s (1998) attempt to provide a Quinean account of unapplied mathematics as `recreational'', suggest that, if one approaches the problem from a Quinean naturalist starting point, one must conclude that all mathematics is recreational in this way.

A phenomenological approach to mathematical practice is sketched out, and some problems with this sort of approach are considered. The approach outlined takes mathematical practices as its data, and seeks to provide an empirically adequate philosophy of mathematics based on observation of these practices. Some observations are presented, based on two case studies of some research into the classification of C*-algebras. It is suggested that an anti-realist account of mathematics could be developed on the basis of these and other studies, locating the substance of mathematics in the various informal argument methods used by mathematicians.

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.