Structuralism

Geoffrey Hellman

Research output: Chapter in Book/Report/Conference proceedingChapter

21 Scopus citations

Abstract

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 modalstructuralism. 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."

Original languageEnglish (US)
Title of host publicationThe Oxford Handbook of Philosophy of Mathematics and Logic
PublisherOxford University Press
ISBN (Print)9780195148770
DOIs
StatePublished - Jul 1 2005

Keywords

  • Axiom
  • Category theory
  • Frege
  • Hilbert
  • Indefinite extendability
  • Mathematics
  • Modality
  • Set theory
  • Structuralism
  • Structure
  • Universals

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