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Philosophia Mathematica Advance Access published online on September 20, 2007
Philosophia Mathematica, doi:10.1093/philmat/nkm035
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Copyright © The Author 2007. Published by Oxford University Press.

Platonism and Aristotelianism in Mathematics{dagger}

Richard Pettigrew*

* Department of Mathematics, University of Bristol, University Walk, Bristol BS8 1TW, United Kingdom.

Correspondence: Richard.Pettigrew{at}bristol.ac.uk

Philosophers of mathematics agree that the only interpretation of arithmetic that takes that discourse at ‘face value’ is one on which the expressions ‘Formula ’, ‘0’, ‘1’, ‘+’, and ‘x’ are treated as proper names. I argue that the interpretation on which these expressions are treated as akin to free variables has an equal claim to be the default interpretation of arithmetic. I show that no purely syntactic test can distinguish proper names from free variables, and I observe that any semantic test that can must beg the question. I draw the same conclusion concerning areas of mathematics beyond arithmetic.

{dagger} This paper is a greatly extended version of my response to Stewart Shapiro's paper in the conference ‘Structuralism in physics and mathematics’ held in Bristol on 2–3 December, 2006. I would like to thank the conference organisers, Øystein Linnebo and James Ladyman, as well as Hannes Leitgeb, Stewart Shapiro, Stephen Williams, and an anonymous referee for this journal for helpful suggestions. However, my greatest debt is to John Mayberry, with whom I have had many extremely enlightening discussions on this topic.


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