Chapter

Modality, Structure, Ontology

Stewart Shapiro

in Philosophy of Mathematics

Published in print October 2000 | ISBN: 9780195139303
Published online November 2003 | e-ISBN: 9780199833658 | DOI: http://dx.doi.org/10.1093/0195139305.003.0008
 Modality, Structure, Ontology

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There are a number of nominalist, or antiplatonist programmes in the philosophy of mathematics. Typically, defenders of these programmes maintain that mathematics is not about an independently existing realm of mathematical objects, but that, nevertheless, mathematical propositions have objective, nonvacuous truth conditions. This is accomplished with added ideology – typically a modal operator. I show that for the more prominent cases, there are straightforward translations between the set‐theoretic language of the realist and the nominalistic language with the added ideology. Since the translations preserve warranted belief (from each perspective), I contend that an advocate of any of the rival systems cannot claim an epistemological advantage over an advocate of any other. The treatment yields a structuralist illumination of the trade‐off between ontology and ideology.

Keywords: ideology; modality; modal operator; nominalism; objectivity; ontology; platonism; structure; translation

Chapter.  14982 words. 

Subjects: Philosophy of Mathematics and Logic

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