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University Press Scholarship Online

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Constructibility and Mathematical ExistenceCharles S. Chihara

Published in print: 1991 Published Online:November 2003ISBN: 9780198239758 eISBN: 9780191597190Item type: book

Publisher: Oxford University PressDOI: 10.1093/0198239750.001.0001

A continuation of the study of mathematical existence begun in Ontologyand the Vicious‐Circle Principle (published in 1973); in the present work,Quine's indispensability argument is rebutted by the development of anew nominalistic version of mathematics (the Constructibility Theory)that is specified as an axiomatized theory formalized in a many‐sortedfirst‐order language. What is new in the present work is its abandonmentof the predicative restrictions of the earlier work and its much greaterattention to the applications of mathematics in science and everyday life.The book also contains detailed discussions of rival views (MathematicalStructuralism, Field's Instrumentalism, Burgess's Moderate Realism,Maddy's Set Theoretical Realism, and Kitcher's Ideal Agent account ofmathematics), in which many comparisons with the ConstructibilityTheory are made.

Strategies in the LiteratureJohn P. Burgess and Gideon Rosen

in A Subject With No Object: Strategies for Nominalistic Interpretation ofMathematics

Published in print: 1999 Published Online:November 2003ISBN: 9780198250128 eISBN: 9780191597138Item type: chapter

Publisher: Oxford University PressDOI: 10.1093/0198250126.003.0007

Relates the ideal types of strategies presented in earlier chapters toactual proposals in the literature. We begin with the geometric strategyof Harty Field, and claim that our version better reveals what is reallygoing on than do his own claims about mathematically formulatedtheories always being conservative extensions of correspondingnominalistic theories. We also consider strategies using modal logic,as found in the works of Charles Chihara and Geoffrey Hellman,

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emphasizing that one must distinguish metaphysical necessity andpossibility from logical necessity and possibility, and use all the resourcesof cross‐comparison provided by ordinary language, in order to makethese strategies work. A potpourri of further strategies is brieflyexamined.