结构主义和元数学.pdf

Erkenn (2010) 73:67–81 DOI 10.1007/s10670-010-9210-x ORI GIN AL ARTICLE Structuralism and Meta-Mathematics Simon Friederich Received: 22 January 2009 / Accepted: 1 February 2010 / Published online: 2 March 2010 Springer Science+Business Media B.V. 2010 Abstract 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 structur- alism 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. Keywords Mathematical structuralism Meta-mathematics Quasi-concrete objects Criteria of identity 1 Structuralism and the Individuation of Objects The main idea of mathematical structuralism is, in the words of Michael Resnik, ‘‘that in mathematics the primary subject-matter is not the individual mathematical objects but rather the structures in which they are arranged.’’1 The structuralist denies mathematical objects the ontological independence which is ascribed to them by traditional platonist accounts of mathematics and focuses on the relations that obtain between them. According to the structuralist account of mathematical axiom 1 See Resnik (1997, p. 201). S. Friederich (&) ¨ ¨ Instit