The semantic plights of the ante-rem structuralist

Article
  • 76 Downloads

Abstract

A version of the permutation argument in the philosophy of mathematics leads to the thesis that mathematical terms, contrary to appearances, are not genuine singular terms referring to individual objects; they are purely schematic or variables. By postulating ‘ante-rem structures’, the ante-rem structuralist aims to defuse the permutation argument and retain the referentiality of mathematical terms. This paper presents two semantic problems for the ante-rem view: (1) ante-rem structures are themselves subject to the permutation argument; (2) the ante-rem structuralist fails to explain reference in a way that makes her account different to, and privileged over, that of her eliminativist rivals. Both problems undercut the motivation behind ante-rem structuralism.

Keywords

Permutation argument Mathematical structuralism Indeterminacy of reference Arbitrary reference Model-theoretic arguments 

Notes

Acknowledgements

This paper draws on Chapter 2 of my Ph.D. dissertation, Assadian (2016). For extremely helpful discussion and written comments, I am very grateful to Tim Button, Simon Hewitt, Keith Hossack, Daniel Isaacson, Jeff Ketland, Hannes Leitgeb, Øystein Linnebo, Jonathan Nassim, Richard Pettigrew, J. Robbie G. Williams, Jack Woods, Mohammad Saleh Zarepour, an anonymous referee of this journal, and the audiences at the universities of London, Manchester, and Munich.

References

  1. Assadian, B. (2016). Displacing numbers: On the metaphysics of mathematical structuralism (Ph.D. Dissertation). Birkbeck College, University of London.Google Scholar
  2. Awodey, S. (2014). Structuralism, invariance, and univalence. Philosophia Mathematica, 22(1), 1–11.CrossRefGoogle Scholar
  3. Balaguer, M. (1998). Platonism and anti-platonism in mathematics. Oxford: Oxford University Press.Google Scholar
  4. Benacerraf, P. (1965). What numbers could not be. Philosophical Review, 74(1), 47–73.CrossRefGoogle Scholar
  5. Boccuni, F., & Woods, J. (ms.). Structuralist neo-logicism.Google Scholar
  6. Breckenridge, W., & Magidor, O. (2012). Arbitrary reference. Philosophical Studies, 158(3), 377–400.CrossRefGoogle Scholar
  7. Burgess, J. P. (1999). Review of Stewart Shapiro’s philosophy of mathematics: structure and ontology. Notre Dame Journal of Formal Logic, 40(2), 283–291.CrossRefGoogle Scholar
  8. Burgess, J. P. (2015). Rigor and structure. Oxford: Oxford University Press.CrossRefGoogle Scholar
  9. Button, T., & Walsh, S. (2016). Structure and categoricity: Determinacy of reference and truth-value in the philosophy of mathematics. Philosophia Mathematica, 24(3), 283–307.CrossRefGoogle Scholar
  10. Cameron, P. J. (1999). Permutation Groups. London Mathematical Society Student Texts 45. Cambridge: Cambridge University press.Google Scholar
  11. Carnap, R. (1961). On the use of Hilbert’s \(\varepsilon\)-operator in scientific theories. In Y. Bar-Hillel, et al. (Eds.), Essays on the foundations of mathematics (pp. 156–164). Jerusalem: The Magnus Press.Google Scholar
  12. Dedekind, R. (1888). Was Sind und Was Sollen die Zahlen? Vieweg, Braunschweig. Translated by W. W. Behma as ‘the nature and meaning of numbers’. In R. Dedekind (Ed.), Essays on the theory of numbers (pp. 31–115). New York: Dover.Google Scholar
  13. Dedekind, R. (1890). Letter to Keferstein. In J. van Heijenoort (Ed.), From Frege to Gödel (pp. 98–103). Cambridge MA: Harvard University Press.Google Scholar
  14. Field, H. (1975). Conventionalism and instrumentalism in semantics. Noûs, 9, 375–405.CrossRefGoogle Scholar
  15. Fine, K. (1983). A defence of arbitrary objects. Proceedings of the Aristotelian Society, Supplementary Volumes, 57, 55–77+79–89.CrossRefGoogle Scholar
  16. Fine, K. (1998). Cantorian abstraction: A reconstruction and defense. Journal of Philosophy, 95(12), 599–634.Google Scholar
  17. Hale, B. (1987). Abstract objects. Oxford: Blackwell.Google Scholar
  18. Hellman, G. (1989). Mathematics without numbers: Towards a modal-structural interpretation. Oxford: Oxford University Press.Google Scholar
  19. Hellman, G. (2001). Three varieties of mathematical structuralism. Philosophia Mathematica, 9(3), 184–211.CrossRefGoogle Scholar
  20. Hodes, H. T. (1984). Logicism and the ontological commitments of arithmetic. Journal of Philosophy, 81(3), 123–149.CrossRefGoogle Scholar
  21. Horsten, L. ms. Generic structuralism.Google Scholar
  22. Isaacson, D. (2011). The reality of mathematics and the case of set theory. In Z. Novák & A. Simonyi (Eds.), Truth, reference, and realism (pp. 1–75). Budapest: Central European University Press.Google Scholar
  23. Keränen, J. (2006). The identity problem for realist structuralism II: A reply to Shapiro. In F. MacBride (Ed.), Identity and modality (pp. 146–163). Oxford: Oxford University Press.Google Scholar
  24. Ketland, J. unpublished ms. Abstract structure. https://www.academia.edu/7673229/Abstract_Structure.
  25. Leitgeb, H. (forthcoming). On mathematical structuralism: A theory of unlabeled graphs as ante rem structures. Philosophia Mathematica.Google Scholar
  26. Leitgeb, H., & Ladyman, J. (2008). Criteria of identity and structuralist ontology. Philosophia Mathematica, 16(3), 388–396.CrossRefGoogle Scholar
  27. Lewis, D. (1984). Putnam’s paradox. Australasian Journal of Philosophy, 62(3), 221–236.CrossRefGoogle Scholar
  28. Linnebo, Ø., & Pettigrew, R. (2014). Two types of abstraction for structuralism. Philosophical Quarterly, 64, 267–283.CrossRefGoogle Scholar
  29. MacBride, F. (2005). Structuralism reconsidered. In S. Shapiro (Ed.), The oxford handbook of philosophy of mathematics and logic (pp. 563–589). Oxford: Oxford University Press.CrossRefGoogle Scholar
  30. Nodelman, U., & Zalta, E. N. (2014). Foundations for mathematical structuralism. Mind, 123(489), 39–78.CrossRefGoogle Scholar
  31. Pettigrew, R. (2008). Platonism and Aristotelianism in mathematics. Philosophia Mathematica, 16(3), 310–332.CrossRefGoogle Scholar
  32. Putnam, H. (1967). Mathematics without foundations. Journal of Philosophy, 64(1), 5–22.CrossRefGoogle Scholar
  33. Reck, E. H. (2003). Dedekind’s structuralism: An interpretation and partial defence. Synthese, 137(3), 369–419.CrossRefGoogle Scholar
  34. Resnik, M. D. (1981). Mathematics as a science of patterns: Ontology and reference. Noûs, 15(4), 529–550.CrossRefGoogle Scholar
  35. Resnik, M. D. (1997). Mathematics as a science of patterns. New York: Oxford University Press.Google Scholar
  36. Russell, B. (1919). An introduction to mathematical philosophy. London: George Allen & Unwin.Google Scholar
  37. Schiemer, G., & Wigglesworth, J. (forthcoming). in British Journal for the Philosophy of Science. The structuralist thesis reconsidered.Google Scholar
  38. Schiemer, G., & Gratzl, N. (2016). The epsilon-reconstruction of theories and scientific structuralism. Erkenntnis, 81, 407–432.CrossRefGoogle Scholar
  39. Shapiro, S. (1997). Philosophy of mathematics: structure and ontology. Oxford: Oxford University Press.Google Scholar
  40. Shapiro, S. (2006). Structure and identity. In F. MacBride (Ed.), Identity and modality (pp. 34–69). Oxford: Oxford University Press.Google Scholar
  41. Shapiro, S. (2008). Identity, indiscernibility, and ante remstructuralism: The tale of \(i\) and \(-i\). Philosophia Mathematica, 16(3), 285–309.CrossRefGoogle Scholar
  42. Shapiro, S. (2012). An “\(i\)” for an \(i\): Singular terms, uniqueness, and reference. Review of Symbolic Logic, 5(3), 380–415.CrossRefGoogle Scholar
  43. West, D. (2001). Introduction to graph theory (2nd ed.). Upper Saddle River, NJ: Pearson.Google Scholar
  44. Woods, J. (2014). Logical indefinites. Logique et Analyse, 57(227), 277–307.Google Scholar
  45. Williamson, T. (1994). Vagueness. London: Routledge.Google Scholar

Copyright information

© Springer Science+Business Media B.V. 2017

Authors and Affiliations

  1. 1.Department of PhilosophyBirkbeck CollegeLondonUK

Personalised recommendations