How Not to Enhance the Indispensability Argument

Abstract

The new explanatory or enhanced indispensability argument alleges that our mathematical beliefs are justified by their indispensable appearances in scientific explanations. This argument differs from the standard indispensability argument which focuses on the uses of mathematics in scientific theories. I argue that the new argument depends for its plausibility on an equivocation between two senses of explanation. On one sense the new argument is an oblique restatement of the standard argument. On the other sense, it is vulnerable to an instrumentalist response. Either way, the explanatory indispensability argument is no improvement on the standard one. 1. EXPLANATIONS AND THEORIES As a preliminary, I make two claims. The first claim is relatively uncontroversial: not all uses of mathematics should compel our belief in mathematical objects. Consider the claim: ‘There are three mangoes on the table.’ This claim uses a mathematical term, ‘three’. A naive argument to the existence of mathematical objects, let us call it the applicability argument, concludes that such mathematical terms should be taken to refer to numbers. It follows fairly directly from the applicability argument that there are abstract objects located outside of space and time and inaccessible to sense perception. The applicability argument, were it sound, would quickly settle core debates in epistemology over the existence of a priori justification as well as metaphysical debates about abstracta. But the applicability argument is tooweak to achieve such lofty goals. Our adjectival uses of mathematical terms are easily understood as non-referring terms. Sentences which employ them can be seen as convenient shorthand for less extravagant claims like: ‘Here is a mango, and here is another mango unidentical to the prior one, and †Thanks toAlanBaker, Alex Paseau, Sarah Scott, and an anonymous referee for this journal, as well as to audiences at the First Colombian Conference in Logic, Epistemology, and Philosophy of Science and Third Cambridge Graduate Conference on the Philosophy of Logic andMathematics, for helpful comments. Philosophia Mathematica (III) Vol. 00 No. 0 C ©The Authors [2014]. Published by Oxford University Press. All rights reserved. For permissions, please e-mail: journals.permissions@oup.com • 1 Philosophia Mathematica Advance Access published March 24, 2014 by gest on M arch 6, 2014 htt//philm at.oxfournals.org/ D ow nladed from