文档介绍：Does Category Theory Provide a
Framework for Mathematical Structuralism?t
GEOFFREY HELLMAN*
1. Introduction
In previous work (Hellman [2001]), we pared three varieties of
mathematical structuralism, which we called 'set-theoretic', 'sui generis',
and 'modal'. It was noted that a fourth variety based on category the-
ory parable systematic consideration. This paper is aimed at
providing that.
The suggestion that category theory provides a way of realizing struc-
turalism as a foundational framework and philosophical interpretation of
mathematics can be found in writings of Mac Lane, Moerdijk, Bell and oth-
ers, and recently Awodey. Having described how category theory provides
a systematic notion of mathematical structure based essentially on families
of structure-preserving mappings, Awodey summed up:
The structural perspective on mathematics codified by categorical methods
might be summarized in the slogan: The subject matter of pure mathematics
is invariant form, not a universe of mathematical objects consisting of logical
atoms. ... My aim here was not to make the case for philosophical struc-
turalism, but to suggest that it be pursued using a technical apparatus other
than that developed by logical atomists since Frege, one with a mathemati-
cal heritage sufficiently substantial, and mathematical applications sufficiently
uniform, to render significant a view of mathematics based on the notion of
'structure'. (Awodey [1996], pp. 235-236)
This is an intriguing suggestion. It is naturally viewed in the context of
Mac Lane's repeated claim that category theory provides an autonomous
foundation of mathematics as an alternative to set theory. The reason for
this should be clear: if category theory is not autonomous but rather must
I I am indebted to John Bell and to Solomon Feferman for very helpful correspondence
ments on an earlier draft of this paper.
" Department of Philosophy, University of Minnesota, Minneapolis,