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Structuralism and Meta-Mathematics

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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.

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Notes

  1. See Resnik (1997, p. 201).

  2. See Hellman (2005, p. 537), the emphasis is Hellman’s.

  3. For the German original see (Frege 1976), an English translation can be found in (Frege 1980). It should be emphasized that Frege’s views on arithmetic and geometry are strongly different. Frege, while he aimed at a logicist reduction of arithmetic, insisted on a substantial role of intuition in geometry. Both his accounts on arithmetic and geometry, however, are in conflict with the Hilbertian approach according to which the axioms are used as implicitly defining the concepts they contain.

  4. This formulation is due to Parsons, see Parsons (1990, p. 303).

  5. See, for example, the claims brought forward in (Shapiro 2005) and the final section of (Parsons 1990).

  6. There already exists a study that focuses on criteria of identity for meta-mathematical objects, namely (Mühlhölzer forthcoming), and the ideas put forward there have helped to formulate the account of meta-mathematics presented here. Mühlhölzer explores the resources of the later Wittgenstein’s philosophy of mathematics to accommodate the existence of meta-mathematics where Wittgenstein seems to run into trouble. There are parallels between Mühlhölzer’s analysis and the one presented here for the case of structuralism. As an example, the structuralist with respect to meta-mathematics holds that the objects of meta-mathematics are individuated only according to their relations to other such objects (in a way to be clarified in Sect. 4), while similarly they are “not given in advance” with respect to the meta-mathematical “calculus”, or so Mühlhölzer’s later Wittgenstein holds. Mühlhölzer argues that the idea of studying criteria of identity can be found in the work of Wittgenstein (1958, p. 62).

  7. This terminology frequently appears in (Parsons 1990) as well as in his recent book (Parsons 2008) but his idea of the preeminent importance of quasi-concrete objects can already be found in his now classic Mathematical Intuition (Parsons 1980).

  8. See Gödel (1931) and Feferman (1960).

  9. The notion of concatenation as applied to the objects of a formal language can be spelled out formally in an axiomatic way. This has been done in (Grzegorczyk 2005) obtaining, as a result, the Gödel-Church theorem of the undecidability of logic the proof of which is usually based on arithmetization.

  10. See Shapiro (2005, p. 75).

  11. See Shapiro (2005, p. 75), further below.

  12. See Hellman (1989), e.g., p. 15.

  13. See Hellman (2003) and Hellman (forthcoming).

  14. See Hellman (2003), especially p. 9, and Hellman (2005).

  15. The most important contributions to structuralism in category theory include McLarty (1993), Awodey (1996), McLarty (2004) and Awodey (2004).

References

  • Awodey, S. (1996). Structure in mathematics and logic: A categorical perspective. Philosophia Mathematica , 4(3), 209–237.

    Google Scholar 

  • Awodey, S. (2004). Does category theory provide a framework for mathematical structuralism? Philosophia Mathematica , 12(3), 54–64.

    Article  Google Scholar 

  • Burgess, J. (1999). Book review: Stewart Shapiro. Philosophy of mathematics: Structure and ontology. Notre Dame Journal of Formal Logic, 40(2), 283–291.

    Article  Google Scholar 

  • Button, T. (2006). Realist structuralism’s identity crisis: A hybrid solution. Analysis, 66, 216–222.

    Article  Google Scholar 

  • Feferman, S. (1960). Arithmetization of metamathematics in a general setting. Fundamenta Mathematicae, 49, 35–92.

    Google Scholar 

  • Frege, G. (1976). In G. Gabriel, H. Hermes, F. Kambartel, & C. Thiel (Eds.), Wissenschaftlicher Briefwechsel. Hamburg: Felix Meiner.

  • Frege, G. (1980). In B. McGuiness (Ed.), Philosophical and mathematical correspondence (trans: Kaal, H.). Oxford: Basil Blackwell.

  • Gödel, K. (1931). Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme. I. Monatshefte Mathematics and Physics, 38, 173–198.

    Article  Google Scholar 

  • Grzegorczyk, A. (2005). Undecidability without arithmetization. Studia Logica, 79, 163–230.

    Article  Google Scholar 

  • Hellman, G. (1989). Mathematics without numbers. Oxford: Oxford University Press.

    Google Scholar 

  • Hellman, G. (2003). Does category theory provide a framework for mathematical structuralism? Philosophia Mathematica , 11(3), 129–157.

    Google Scholar 

  • 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 

  • Hellman, G. (forthcoming). What is categorical structuralism?, available online at http://www.tc.umn.edu/hellm001/.

  • Hilbert, D. (1899). Grundlagen der Geometrie. Leipzig: Teubner. Hilbert, D. (1959). Foundations of geometry (trans: Townsend, E.). La Salle, Illinois: Open Court.

  • Keränen, J. (2001). The identity problem for realist structuralism. Philosophia Mathematica , 9(3), 308–330.

    Article  Google Scholar 

  • Ketland, J. (2006). Structuralism and the identity of indiscernibles. Analysis, 66, 303–315.

    Google Scholar 

  • Ladyman, J. (2005). Mathematical structuralism and the identity of indiscernibles. Analysis, 65, 218–221.

    Article  Google Scholar 

  • Leitgeb, H., & Ladyman, J. (2008). Criteria of identity and structuralist ontology. Philosophia Mathematica , 16(3), 388–396.

    Article  Google Scholar 

  • 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 

  • MacBride, F. (2006). What constitutes the numerical diversity of mathematical objects? Analysis, 66, 63–69.

    Article  Google Scholar 

  • McLarty, C. (1993). Numbers can be just what they have to. Noûs, 27, 487–498.

    Article  Google Scholar 

  • McLarty, C. (2004). Exploring categorical structuralism. Philosophia Mathematica , 12(3), 37–53.

    Article  Google Scholar 

  • Mühlhölzer, F. (forthcoming). Wittgenstein and metamathematics. In P. Stekeler-Weithofer (Ed.), Wittgenstein, Philosophie und Wissenschaften. Hamburg: Felix Meiner.

  • Parsons, C. H. (1980). Mathematical intuition. Proceedings of the Aristotelian Society, 80, 145–168.

  • Parsons, C. H. (1990). The structuralist view of mathematical objects. Synthese, 84, 303–346.

    Article  Google Scholar 

  • Parsons, C. H. (2008). Mathematical thought and its objects. NewYork: Cambridge University Press.

    Google Scholar 

  • Resnik, M. D. (1997). Mathematics as a science of patterns. Oxford: Oxford University Press.

    Google Scholar 

  • Shapiro, S. (2001). Thinking about mathematics: The philosophy of mathematics. Oxford: Oxford University Press.

    Google Scholar 

  • Shapiro, S. (2005). Categories, structures, and the Frege-Hilbert controversy: The status of meta-mathematics. Philosophia Mathematica , 13(3), 61–77.

    Article  Google Scholar 

  • Shapiro, S. (2008). Identity, indiscernibility and ante rem structuralism: The tale of i and − i. Philosophia Mathematica , 16(3), 285–309.

    Article  Google Scholar 

  • Wittgenstein, L. (1958). The blue and brown books. Oxford: Blackwell.

    Google Scholar 

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Acknowledgments

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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  1. Institut für Theoretische Physik, Universität Heidelberg, Philosophenweg 16, 69120, Heidelberg, Germany

    Simon Friederich

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Friederich, S. Structuralism and Meta-Mathematics. Erkenn 73, 67–81 (2010). https://doi.org/10.1007/s10670-010-9210-x

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