Geoffrey Hellman

in The Oxford Handbook of Philosophy of Mathematics and Logic

Published in print June 2007 | ISBN: 9780195325928
Published online September 2009 | e-ISBN: 9780199892082 | DOI:

Series: Oxford Handbooks


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The main types of mathematical structuralism that have been proposed and developed to the point of permitting systematic and instructive comparison are four: structuralism based on model theory, carried out formally in set theory (e.g., first- or second-order Zermelo–Fraenkel set theory), referred to as STS (for set-theoretic structuralism); the approach of philosophers such as Shapiro and Resnik of taking structures to be sui generis universals, patterns, or structures in an ante rem sense (explained in this article), referred to as SGS (for sui generis structuralism); an approach based on category and topos theory, proposed as an alternative to set theory as an overarching mathematical framework, referred to as CTS (for category-theoretic structuralism); and a kind of eliminative, quasi-nominalist structuralism employing modal logic, referred to as MS (for modal-structuralism). This article takes these up in turn, guided by few questions, with the aim of understanding their relative merits and the choices they present.

Keywords: mathematical structuralism; model theory; set theory; sui generis structuralism; category-theoretic modal-structuralism structuralism

Article.  12500 words. 

Subjects: Philosophy of Mathematics and Logic ; Metaphysics

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