How to be a structuralist all the way down

Synthese 179 (3):435 - 454 (2011)
This paper considers the nature and role of axioms from the point of view of the current debates about the status of category theory and, in particular, in relation to the "algebraic" approach to mathematical structuralism. My aim is to show that category theory has as much to say about an algebraic consideration of meta-mathematical analyses of logical structure as it does about mathematical analyses of mathematical structure, without either requiring an assertory mathematical or meta-mathematical background theory as a "foundation", or turning meta-mathematical analyses of logical concepts into "philosophical" ones. Thus, we can use category theory to frame an interpretation of mathematics according to which we can be structuralists all the way down.
Keywords Mathematical structuralism  Category theory  Algebraic structuralism  Philosophy of mathematics  Hilbert  Frege  Shapiro  McLarty  Marquis  Hellman  Mac Lane
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DOI 10.2307/41477430
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