New directions for nominalist philosophers of mathematics |
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Authors: | Charles Chihara |
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Institution: | (1) Philosophy, School of Social Sciences, University of Manchester, Arthur Lewis Building, Oxford Road, Manchester, M13 9PL, UK |
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Abstract: | The present paper will argue that, for too long, many nominalists have concentrated their researches on the question of whether
one could make sense of applications of mathematics (especially in science) without presupposing the existence of mathematical objects. This was, no doubt, due to the enormous
influence of Quine’s “Indispensability Argument”, which challenged the nominalist to come up with an explanation of how science
could be done without referring to, or quantifying over, mathematical objects. I shall admonish nominalists to enlarge the
target of their investigations to include the many uses mathematicians make of concepts such as structures and models to advance
pure mathematics. I shall illustrate my reasons for admonishing nominalists to strike out in these new directions by using Hartry Field’s
nominalistic view of mathematics as a model of a philosophy of mathematics that was developed in just the sort of way I argue
one should guard against. I shall support my reasons by providing grounds for rejecting both Field’s fictionalism and also
his deflationist account of mathematical knowledge—doctrines that were formed largely in response to the Indispensability
Argument. I shall then give a refutation of Mark Balaguer’s argument for his thesis that fictionalism is “the best version
of anti-realistic anti-platonism”. |
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