Why inference to the best explanation doesn’t secure empirical grounds for mathematical platonism

Abstract

Proponents of the explanatory indispensability argument for mathematical platonism maintain that claims about mathematical entities play an essential explanatory role in some of our best scientific explanations. They infer that the existence of mathematical entities is supported by way of inference to the best explanation from empirical phenomena and therefore that there are the same sort of empirical grounds for believing in mathematical entities as there are for believing in concrete unobservables such as quarks. I object that this inference depends on a false view of how abductive considerations mediate the transfer of empirical support. More specifically, I argue that even if inference to the best explanation is cogent, and claims about mathematical entities play an essential explanatory role in some of our best scientific explanations, it doesn’t follow that the empirical phenomena that license those explanations also provide empirical support for the claim that mathematical entities exist.

This is a preview of subscription content, access via your institution.

Fig. 1

Notes

  1. 1.

    For a general overview of such arguments see (Colyvan 2015).

  2. 2.

    See (Quine 1948, 1951, 1981a, b) and (Putnam 1971, 1975) for classical sources of inspiration for this argument. See (Liggins 2008), however, for a challenge to the claim that either Quine or Putnam actually advocated the argument that has come to bear their name.

  3. 3.

    For some discussions of this liability, see (Busch 2011, 2012), (Leng 2010), (Maddy 1992, 1995, 1997), (Morrison 2010, 2012), (Peressini 2008), (Sober 1993), and (Vineberg 1996).

  4. 4.

    Baker (2005, 2009, 2016) and Colyvan (1999, 2001, 2002, 2006, 2010, 2012) provide the most influential articulations of this argument.

  5. 5.

    See (Field 1980) for an influential challenge to this claim.

  6. 6.

    See for example (Azzouni 2004, 2012), (Melia 1995, 2000), and (Yablo 2005).

  7. 7.

    See (Batterman 2002), (Leng 2010), and (Maddy 1992, 1995, 1997, pp. 143–154).

  8. 8.

    See (Melia 2000, 2002, 2008), (Saatsi 2011), and (Yablo 2005).

  9. 9.

    For a similar point see (Morrison 2012, pp. 265–266).

  10. 10.

    See (Chakravartty 2017) and (Psillos and Ruttkamp-Bloem 2017) for some recent overviews of selective scientific realism.

  11. 11.

    See for instance (Balaguer 1998, pp. 132–136, 2009), (Leng 2005b, 2010), and (Vineberg 1996).

  12. 12.

    See (Busch 2012), (Maddy 1992, 1995, 1997, pp. 138–143), (Parsons 1983, pp. 195–197), and (Sober 1993, 2011, pp. 200–211).

  13. 13.

    For a historically and philosophically informative discussion of the role that Brownian motion played in convincing the scientific community of the reality of atoms see (Maddy 1997, pp. 133–157).

  14. 14.

    See (Colyvan 2001, pp. 76–86) for an argument that the sense in which mathematics may be regarded as “indispensable” to our best scientific theories is best understood in terms of its contributions to those theories’ explanatory virtues.

  15. 15.

    Or in Baker’s (2009, p. 613) own wording, it is an example of a case in which “mathematical objects play an indispensable explanatory role in science.”

  16. 16.

    Frege (1884), at least by some interpretations, serves as a historically prominent example of someone who held this view. Bengson (2015), Hale and Wright (2001, 2002), and Marcus (2015) serve as contemporary examples.

  17. 17.

    See for example (Baker 2016, p. 334) and (Colyvan 2006, pp. 227–228, pp. 234–235) for suggestions along these lines.

  18. 18.

    An argument along these lines from the cogency of IBE to a principle like Explanatory Consequence is discussed by Morrison (2012, pp. 274–275).

  19. 19.

    See (Lipton 2004, p. 63) and (McCain 2016, pp. 160–161) for discussions of this requirement.

  20. 20.

    For helpful comments on previous drafts, I would like to thank Mark Balaguer, Sarah Boyce, Lindsay Brainard, Matt Duncan, Luke Kallberg, various anonymous referees, and the audience members of numerous venues in which I presented previous versions of this paper. I would also like to thank the University of Missouri Research Board and the University of Missouri Philosophy Department for the generous provision of a research leave that allowed me to focus on this project.

References

  1. Azzouni, J. (2004). Deflating existential consequence. NY: Oxford University Press.

    Google Scholar 

  2. Azzouni, J. (2012). Taking the easy road out of dodge. Mind, 121, 951–965.

    Google Scholar 

  3. Baker, A. (2005). Are there genuine mathematical explanations of physical phenomena? Mind, 114, 223–238.

    Google Scholar 

  4. Baker, A. (2009). mathematical explanation in science. British Journal for the Philosophy of Science, 60, 611–633.

    Google Scholar 

  5. Baker, A. (2016). Parsimony and inference to the best mathematical explanation. Synthese, 193, 333–350.

    Google Scholar 

  6. Balaguer, M. (1998). Platonism and anti-platonism in mathematics. NY: Oxford University Press.

    Google Scholar 

  7. Balaguer, M. (2009). Fictionalism, theft, and the story of mathematics. Philosophia Mathematica, 17, 131–162.

    Google Scholar 

  8. Bangu, S. I. (2008). Inference to the best explanation and mathematical realism. Synthese, 160, 13–20.

    Google Scholar 

  9. Batterman, R. W. (2002). The devil in the details: Asymptotic reasoning in explanation, reduction, and emergence. NY: Oxford University Press.

    Google Scholar 

  10. Bengson, J. (2015). Grasping the third realm. In T. S. Gendler & J. Hawthorne (Eds.), Oxford studies in epistemology (Vol. 5, pp. 1–38). NY: Oxford University Press.

    Google Scholar 

  11. Busch, J. (2011). Is the indispensability argument dispensable. Theoria, 77, 139–158.

    Google Scholar 

  12. Busch, J. (2012). Can the new indispensability argument be saved from euclidean rescues. Synthese, 187, 489–508.

    Google Scholar 

  13. Busch, J., & Morrison, J. (2016). Should scientific realists be platonists? Synthese, 193, 435–449.

    Google Scholar 

  14. Chakravartty, A. (2017). Scientific realism. In E. N. Zalta (Ed.) The Stanford encyclopedia of philosophy. https://plato.stanford.edu/entries/scientific-realism/.

  15. Colyvan, M. (1999). Confirmation and indispensability. Philosophical Studies, 96, 1–19.

    Google Scholar 

  16. Colyvan, M. (2001). The indispensability of mathematics. NY: Oxford University Press.

    Google Scholar 

  17. Colyvan, M. (2002). Mathematics and aesthetic considerations in science. Mind, 111, 69–74.

    Google Scholar 

  18. Colyvan, M. (2006). Scientific realism and mathematical nominalism: A marriage made in Hell. In C. Cheyne & J. Worrall (Eds.), Rationality and reality: Conversations with Alan Musgrave (pp. 225–237). Dordrecht: Springer.

    Google Scholar 

  19. Colyvan, M. (2007). Mathematical recreation versus mathematical knowledge. In M. Leng, A. Paseau, & M. Potter (Eds.), Mathematical knowledge (pp. 109–122). NY: Oxford University Press.

    Google Scholar 

  20. Colyvan, M. (2010). There is no easy road to nominalism. Mind, 119, 285–306.

    Google Scholar 

  21. Colyvan, M. (2012). Road work ahead: Heavy machinery on the easy road. Mind, 121, 1031–1046.

    Google Scholar 

  22. Colyvan, M. (2015). Indispensability arguments in the philosophy of mathematics. In E. N. Zalta (Ed) The Stanford encyclopedia of philosophy. https://plato.stanford.edu/entries/mathphil-indis/.

  23. Daly, C., & Langford, S. (2009). Mathematical explanation and indispensability arguments. The Philosophical Quarterly, 39, 641–658.

    Google Scholar 

  24. Field, H. (1980). Science without numbers: A defense of nominalism. Princeton, NJ: Princeton University Press.

    Google Scholar 

  25. Frege, G. (1884). The foundations of arithmetic: A logico-mathematical enquiry into the concept of number (J. L. Austin, Trans.), (second revised ed.). NY: Harper & Brothers 1950.

  26. Hale, B., & Wright, C. (2001). The reason’s proper study: Essays towards a Neo-Fregean philosophy of mathematics. NY: Oxford University Press.

    Google Scholar 

  27. Hale, B., & Wright, C. (2002). Benacerraf’s dilemma revisited. European Journal of Philosophy, 10, 101–129.

    Google Scholar 

  28. Hempel, C. G. (1945). Studies in the logic of confirmation (II.). Mind, 54, 97–121.

    Google Scholar 

  29. Leng, M. (2005a). Mathematical explanation. In C. Cellucci & D. Gillies (Eds.), Mathematical reasoning and heuristics (pp. 167–189). London: King’s College Publications.

    Google Scholar 

  30. Leng, M. (2005b). Platonism and anti-platonism: Why worry? International Studies in the Philosophy of Science, 19, 65–84.

    Google Scholar 

  31. Leng, M. (2010). Mathematics and reality. NY: Oxford University Press.

    Google Scholar 

  32. Liggins, D. (2008). Quine, Putnam, and the ‘Quine-Putnam’ indispensability argument. Erkenntnis, 68, 113–127.

    Google Scholar 

  33. Lipton, P. (2004). Inference to the best explanation (2nd ed.). NY: Routledge.

    Google Scholar 

  34. Maddy, P. (1992). Indispensability and practice. The Journal of Philosophy, 89, 275–289.

    Google Scholar 

  35. Maddy, P. (1995). Naturalism and ontology. Philosophia Mathematica, 3, 248–270.

    Google Scholar 

  36. Maddy, P. (1997). Naturalism in mathematics. NY: Oxford University Press.

    Google Scholar 

  37. Marcus, R. (2014). The holistic presumptions of the indispensability argument. Synthese, 191, 3575–3594.

    Google Scholar 

  38. Marcus, R. (2015). Autonomy platonism and the indispensability argument. NY: Lexington Books.

    Google Scholar 

  39. McCain, K. (2016). The nature of scientific knowledge: An explanatory approach. Berlin: Springer.

    Google Scholar 

  40. Melia, J. (1995). On what there’s not. Analysis, 55, 223–229.

    Google Scholar 

  41. Melia, J. (2000). Weaseling away the indispensability argument. Mind, 109, 455–479.

    Google Scholar 

  42. Melia, J. (2002). Response to Colyvan. Mind, 111, 75–79.

    Google Scholar 

  43. Melia, J. (2008). A world of concrete particulars. In D. W. Zimmerman (Ed.), Oxford studies in metaphysics (Vol. 4, pp. 99–124). NY: Oxford University Press.

    Google Scholar 

  44. Morrison, J. (2010). Just how controversial is evidential holism? Synthese, 173, 335–352.

    Google Scholar 

  45. Morrison, J. (2012). Evidential holism and indispensability arguments. Erkenntnis, 76, 263–278.

    Google Scholar 

  46. Parsons, C. (1983). Quine on the philosophy of mathematics. In Mathematics in philosophy (pp. 176–205). Ithaca, NY: Cornell University Press.

  47. Psillos, S., & Ruttkamp-Bloem, E. (2017). Scientific realism: Quo Vadis? introduction: New thinking about scientific realism. Synthese, 194, 3187–3201.

    Google Scholar 

  48. Putnam, H. (1971). Philosophy of logic (NY: Harper & Row). Reprinted in Mathematics, matter and method by Hilary Putnam (NY: Cambridge University Press, 1979) (pp. 60–78).

  49. Putnam, H. (1975). “What is mathematical truth” Historia Mathematica, 2, pp. 529–543. Reprinted in Mathematics, matter and method by Hilary Putnam (NY: Cambridge University Press, 1979), (pp. 323–357).

  50. Quine, W. V. (1948). On what there is. Review of Metaphysics, 2, 21–38.

    Google Scholar 

  51. Quine, W. V. (1951). Two dogmas of empiricism. The Philosophical Review, 60, 20–43.

    Google Scholar 

  52. Quine, W. V. (Ed.). (1981a). Success and limits of mathematization. In Theories and things (pp. 148–155). Cambridge MA: Harvard University Press.

    Google Scholar 

  53. Quine, W. V. (Ed.). (1981b). Things and their place in theories. In Theories and things (pp. 1–23). Cambridge MA: Harvard University Press.

    Google Scholar 

  54. Saatsi, J. (2011). The enhanced indispensability argument: Representational versus explanatory role of mathematics in science. British Journal for the Philosophy of Science, 62, 143–154.

    Google Scholar 

  55. Sober, E. (1993). Mathematics and indispensability. The Philosophical Review, 102, 35–57.

    Google Scholar 

  56. Sober, E. (2011). Evolution without naturalism. In J. Kvanvig (Ed.), Oxford studies in philosophy of religion (Vol. 3, pp. 187–221). New York: Oxford University Press.

    Google Scholar 

  57. Vineberg, S. (1996). Confirmation and the indispensability of mathematics to science. Philosophy of Science, 63(Supplement), S256–S263.

    Google Scholar 

  58. Yablo, S. (2005). The myth of seven. In M. E. Kalderon (Ed.), Fictionalism in metaphysics (pp. 88–115). NY: Oxford University Press.

    Google Scholar 

  59. Yablo, S. (2012). Explanation, extrapolation, and existence. Mind, 121, 1007–1029.

    Google Scholar 

Download references

Author information

Affiliations

Authors

Corresponding author

Correspondence to Kenneth Boyce.

Ethics declarations

Conflict of interest

The author declares that they have no conflict of interest.

Human and animal rights

The author did not conduct research involving human participants or animals.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Boyce, K. Why inference to the best explanation doesn’t secure empirical grounds for mathematical platonism. Synthese 198, 583–595 (2021). https://doi.org/10.1007/s11229-018-02043-2

Download citation

Keywords

  • Indispensability arguments
  • Mathematical platonism
  • Nominalism
  • Inference to the best explanation
  • Scientific realism
  • Confirmation