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

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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.

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Notes

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

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

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

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

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

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

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

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

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

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

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

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

  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.

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  1. Department of Philosophy, University of Missouri, 430 Strickland Hall, Columbia, MO, 65211, USA

    Kenneth Boyce

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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

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