Geoffrey Hellman

in The Oxford Handbook of Philosophy of Mathematics and Logic

Published in print March 2005 | ISBN: 9780195148770
Published online July 2005 | e-ISBN: 9780199835560 | DOI:

Series: Oxford Handbooks in Philosophy


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With developments in the 19th and early 20th centuries, structuralist ideas concerning the subject matter of mathematics have become commonplace. Yet fundamental questions concerning structures and relations themselves as well as the scope of structuralist analyses remain to be answered. The distinction between axioms as defining conditions (Hilbertian conception) and axioms as assertions (traditional Fregean conception) is highlighted as is the problem of the indefinite extendability of any putatively all-embracing realm of structures. This chapter systematically compares four main versions: set-theoretic structuralism, a version taking structures as sui generis universals, structuralism based on category theory, and a quasi-nominalist modal-structuralism. While none of the approaches is problem-free, it appears that some synthesis of the category-theoretic approach with modal-structuralism can meet the challenges set out, given the notion of “logical possibility.”

Keywords: mathematics; structure; axiom; Hilbert; Frege; indefinite extendability; set theory; structuralism; universals; category theory; modality

Chapter.  12570 words. 

Subjects: Philosophy of Mathematics and Logic

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