David Bourget (Western Ontario)
David Chalmers (ANU, NYU)
Rafael De Clercq
Jack Alan Reynolds
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Erkenntnis 73 (1):67 - 81 (2010)
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
|Keywords||Mathematical structuralism Meta-mathematics Quasi-concrete objects Criteria of identity|
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References found in this work BETA
Michael D. Resnik (1997). Mathematics as a Science of Patterns. New York ;Oxford University Press.
Charles Parsons (2008). Mathematical Thought and its Objects. Cambridge University Press.
Ludwig Wittgenstein (1958). The Blue and Brown Books. Harper and Row.
Hannes Leitgeb & James Ladyman (2008). Criteria of Identity and Structuralist Ontology. Philosophia Mathematica 16 (3):388-396.
Geoffrey Hellman (1989). Mathematics Without Numbers: Towards a Modal-Structural Interpretation. Oxford University Press.
Citations of this work BETA
Felix Mühlhölzer (2010). Mathematical Intuition and Natural Numbers: A Critical Discussion. [REVIEW] Erkenntnis 73 (2):265–292.
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