Only up to isomorphism? Category Theory and the Foundations of Mathematics

  • Ö Ù # J W T P G E B
  • Published 2010

Abstract

Does category theory provide a foundation for mathematics that is autonomous with respect to the orthodox foundation in a set theory such as ZFC? We distinguish three types of autonomy: logical, conceptual, and justificatory. Focusing on a categorical theory of sets, we argue that a strong case can be made for its logical and conceptual autonomy. Its justificatory autonomy turns on whether the objects of a foundation for mathematics should be specified only up to isomorphism, as is customary in other branches of contemporary mathematics. If such a specification suffices, then a category-theoretical approach will be highly appropriate. But if sets have a richer ‘nature’ than is preserved under isomorphism, then such an approach will be inadequate. A number of philosophers of mathematics have recently debated the claim that category theory provides a foundation for mathematics that is autonomous with respect to the orthodox foundation in set theory (Mac Lane (1986), Feferman (1977), Mayberry (1977), Bell (1981), Hellman (2003), McLarty (2004), Awodey (2004)). The debate has yielded progress: after some initial confusion, the particular theories from within category theory that are proposed as foundations have been identified precisely, and in some cases the autonomy of these theories with respect to the orthodox foundation has been defended—at least for one sort of autonomy. However, there are other sorts of autonomy that have not been considered in much detail. We wish to introduce a distinction between three types of autonomy, which we call logical autonomy, conceptual autonomy, and justificatory autonomy. The debate so far has been concerned almost exclusively with the first sort of autonomy. Yet all three are required before a foundation can claim genuine independence from the set-theoretic orthodoxy. We focus on one of the putative category-theoretic foundations, and argue that it can claim logical autonomy with respect to orthodox set theory. We then explore the possible

Cite this paper

@inproceedings{B2010OnlyUT, title={Only up to isomorphism? Category Theory and the Foundations of Mathematics}, author={{\"{O} {\`U} J W T P G E B}, year={2010} }