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Philosophia Mathematica 2005 13(1):61-77; doi:10.1093/philmat/nki007
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Philosophia Mathematica (III), Vol. 13 No. 1 © Oxford University Press, 2005, all rights reserved

Categories, Structures, and the Frege-Hilbert Controversy: The Status of Meta-mathematics{dagger}

Stewart Shapiro*

* Department of Philosophy, The Ohio State University Columbus, Ohio, U. S. A. 43210 and Department of Logic and Metaphysics, University of St. Andrews Fife KY16 9AL, Scotland shapiro.4{at}osu.edu

There is a parallel between the debate between Gottlob Frege and David Hilbert at the turn of the twentieth century and at least some aspects of the current controversy over whether category theory provides the proper framework for structuralism in the philosophy of mathematics. The main issue, I think, concerns the place and interpretation of meta-mathematics in an algebraic or structuralist approach to mathematics. Can meta-mathematics itself be understood in algebraic or structural terms? Or is it an exception to the slogan that mathematics is the science of structure?


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