Erkenntnis

, Volume 73, Issue 1, pp 67–81 | Cite as

Structuralism and Meta-Mathematics

Original Article

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 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 

References

  1. Awodey, S. (1996). Structure in mathematics and logic: A categorical perspective. Philosophia Mathematica , 4(3), 209–237.Google Scholar
  2. Awodey, S. (2004). Does category theory provide a framework for mathematical structuralism? Philosophia Mathematica , 12(3), 54–64.CrossRefGoogle Scholar
  3. Burgess, J. (1999). Book review: Stewart Shapiro. Philosophy of mathematics: Structure and ontology. Notre Dame Journal of Formal Logic, 40(2), 283–291.CrossRefGoogle Scholar
  4. Button, T. (2006). Realist structuralism’s identity crisis: A hybrid solution. Analysis, 66, 216–222.CrossRefGoogle Scholar
  5. Feferman, S. (1960). Arithmetization of metamathematics in a general setting. Fundamenta Mathematicae, 49, 35–92.Google Scholar
  6. Frege, G. (1976). In G. Gabriel, H. Hermes, F. Kambartel, & C. Thiel (Eds.), Wissenschaftlicher Briefwechsel. Hamburg: Felix Meiner.Google Scholar
  7. Frege, G. (1980). In B. McGuiness (Ed.), Philosophical and mathematical correspondence (trans: Kaal, H.). Oxford: Basil Blackwell.Google Scholar
  8. Gödel, K. (1931). Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme. I. Monatshefte Mathematics and Physics, 38, 173–198.CrossRefGoogle Scholar
  9. Grzegorczyk, A. (2005). Undecidability without arithmetization. Studia Logica, 79, 163–230.CrossRefGoogle Scholar
  10. Hellman, G. (1989). Mathematics without numbers. Oxford: Oxford University Press.Google Scholar
  11. Hellman, G. (2003). Does category theory provide a framework for mathematical structuralism? Philosophia Mathematica , 11(3), 129–157.Google Scholar
  12. Hellman, G. (2005). Structuralism. In S. Shapiro (Ed.), The oxford handbook of the philosophy of mathematics and logic (pp. 536–562). Oxford: Oxford University Press.Google Scholar
  13. Hellman, G. (forthcoming). What is categorical structuralism?, available online at http://www.tc.umn.edu/hellm001/.
  14. Hilbert, D. (1899). Grundlagen der Geometrie. Leipzig: Teubner. Hilbert, D. (1959). Foundations of geometry (trans: Townsend, E.). La Salle, Illinois: Open Court.Google Scholar
  15. Keränen, J. (2001). The identity problem for realist structuralism. Philosophia Mathematica , 9(3), 308–330.CrossRefGoogle Scholar
  16. Ketland, J. (2006). Structuralism and the identity of indiscernibles. Analysis, 66, 303–315.Google Scholar
  17. Ladyman, J. (2005). Mathematical structuralism and the identity of indiscernibles. Analysis, 65, 218–221.CrossRefGoogle Scholar
  18. Leitgeb, H., & Ladyman, J. (2008). Criteria of identity and structuralist ontology. Philosophia Mathematica , 16(3), 388–396.CrossRefGoogle Scholar
  19. MacBride, F. (2005). Structuralism reconsidered. In S. Shapiro (Ed.), The oxford handbook of the philosophy of mathematics and logic (pp. 563–589). Oxford: Oxford University Press.Google Scholar
  20. MacBride, F. (2006). What constitutes the numerical diversity of mathematical objects? Analysis, 66, 63–69.CrossRefGoogle Scholar
  21. McLarty, C. (1993). Numbers can be just what they have to. Noûs, 27, 487–498.CrossRefGoogle Scholar
  22. McLarty, C. (2004). Exploring categorical structuralism. Philosophia Mathematica , 12(3), 37–53.CrossRefGoogle Scholar
  23. Mühlhölzer, F. (forthcoming). Wittgenstein and metamathematics. In P. Stekeler-Weithofer (Ed.), Wittgenstein, Philosophie und Wissenschaften. Hamburg: Felix Meiner.Google Scholar
  24. Parsons, C. H. (1980). Mathematical intuition. Proceedings of the Aristotelian Society, 80, 145–168.Google Scholar
  25. Parsons, C. H. (1990). The structuralist view of mathematical objects. Synthese, 84, 303–346.CrossRefGoogle Scholar
  26. Parsons, C. H. (2008). Mathematical thought and its objects. NewYork: Cambridge University Press.Google Scholar
  27. Resnik, M. D. (1997). Mathematics as a science of patterns. Oxford: Oxford University Press.Google Scholar
  28. Shapiro, S. (2001). Thinking about mathematics: The philosophy of mathematics. Oxford: Oxford University Press.Google Scholar
  29. Shapiro, S. (2005). Categories, structures, and the Frege-Hilbert controversy: The status of meta-mathematics. Philosophia Mathematica , 13(3), 61–77.CrossRefGoogle Scholar
  30. Shapiro, S. (2008). Identity, indiscernibility and ante rem structuralism: The tale of i and − i. Philosophia Mathematica , 16(3), 285–309.CrossRefGoogle Scholar
  31. Wittgenstein, L. (1958). The blue and brown books. Oxford: Blackwell.Google Scholar

Copyright information

© Springer Science+Business Media B.V. 2010

Authors and Affiliations

  1. 1.Institut für Theoretische PhysikUniversität HeidelbergHeidelbergGermany

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