Structuralism and Meta-Mathematics
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The debate on structuralism in the philosophy of mathematics has brought into focus a question about the status of meta-mathematics. It has been raised by Shapiro (2005), where he compares the ongoing discussion on structuralism in category theory to the Frege-Hilbert controversy on axiomatic systems. Shapiro outlines an answer according to which meta-mathematics is understood in structural terms and one according to which it is not. He finds both options viable and does not seem to prefer one over the other. The present paper reconsiders the nature of the formulae and symbols meta-mathematics is about and finds that, contrary to Charles Parsons’ influential view, meta-mathematical objects are not “quasi-concrete”. It is argued that, consequently, structuralists should extend their account of mathematics to meta-mathematics.
KeywordsMathematical structuralism Meta-mathematics Quasi-concrete objects Criteria of identity
I would like to thank Wilfried Keller and Felix Mühlhölzer for stimulating discussions and suggestions of how to improve the paper. Furthermore, I am grateful to Stewart Shapiro and an anonymous referee of Erkenntnis for many helpful comments on an earlier version.
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