## 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.
- 3.
- 4.
- 5.
See (Field 1980) for an influential challenge to this claim.

- 6.
- 7.
- 8.
- 9.
For a similar point see (Morrison 2012, pp. 265–266).

- 10.
- 11.
- 12.
- 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.
- 17.
- 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.
- 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

Azzouni, J. (2004).

*Deflating existential consequence*. NY: Oxford University Press.Azzouni, J. (2012). Taking the easy road out of dodge.

*Mind,**121,*951–965.Baker, A. (2005). Are there genuine mathematical explanations of physical phenomena?

*Mind,**114,*223–238.Baker, A. (2009). mathematical explanation in science.

*British Journal for the Philosophy of Science,**60,*611–633.Baker, A. (2016). Parsimony and inference to the best mathematical explanation.

*Synthese,**193,*333–350.Balaguer, M. (1998).

*Platonism and anti-platonism in mathematics*. NY: Oxford University Press.Balaguer, M. (2009). Fictionalism, theft, and the story of mathematics.

*Philosophia Mathematica,**17,*131–162.Bangu, S. I. (2008). Inference to the best explanation and mathematical realism.

*Synthese,**160,*13–20.Batterman, R. W. (2002).

*The devil in the details: Asymptotic reasoning in explanation, reduction, and emergence*. NY: Oxford University Press.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.Busch, J. (2011). Is the indispensability argument dispensable.

*Theoria,**77,*139–158.Busch, J. (2012). Can the new indispensability argument be saved from euclidean rescues.

*Synthese,**187,*489–508.Busch, J., & Morrison, J. (2016). Should scientific realists be platonists?

*Synthese,**193,*435–449.Chakravartty, A. (2017). Scientific realism. In E. N. Zalta (Ed.)

*The Stanford encyclopedia of philosophy*. https://plato.stanford.edu/entries/scientific-realism/.Colyvan, M. (1999). Confirmation and indispensability.

*Philosophical Studies,**96,*1–19.Colyvan, M. (2001).

*The indispensability of mathematics*. NY: Oxford University Press.Colyvan, M. (2002). Mathematics and aesthetic considerations in science.

*Mind,**111,*69–74.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.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.Colyvan, M. (2010). There is no easy road to nominalism.

*Mind,**119,*285–306.Colyvan, M. (2012). Road work ahead: Heavy machinery on the easy road.

*Mind,**121,*1031–1046.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/.Daly, C., & Langford, S. (2009). Mathematical explanation and indispensability arguments.

*The Philosophical Quarterly,**39,*641–658.Field, H. (1980).

*Science without numbers: A defense of nominalism*. Princeton, NJ: Princeton University Press.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.Hale, B., & Wright, C. (2001).

*The reason’s proper study: Essays towards a Neo-Fregean philosophy of mathematics*. NY: Oxford University Press.Hale, B., & Wright, C. (2002). Benacerraf’s dilemma revisited.

*European Journal of Philosophy,**10,*101–129.Hempel, C. G. (1945). Studies in the logic of confirmation (II.).

*Mind,**54,*97–121.Leng, M. (2005a). Mathematical explanation. In C. Cellucci & D. Gillies (Eds.),

*Mathematical reasoning and heuristics*(pp. 167–189). London: King’s College Publications.Leng, M. (2005b). Platonism and anti-platonism: Why worry?

*International Studies in the Philosophy of Science,**19,*65–84.Leng, M. (2010).

*Mathematics and reality*. NY: Oxford University Press.Liggins, D. (2008). Quine, Putnam, and the ‘Quine-Putnam’ indispensability argument.

*Erkenntnis,**68,*113–127.Lipton, P. (2004).

*Inference to the best explanation*(2nd ed.). NY: Routledge.Maddy, P. (1992). Indispensability and practice.

*The Journal of Philosophy,**89,*275–289.Maddy, P. (1995). Naturalism and ontology.

*Philosophia Mathematica,**3,*248–270.Maddy, P. (1997).

*Naturalism in mathematics*. NY: Oxford University Press.Marcus, R. (2014). The holistic presumptions of the indispensability argument.

*Synthese,**191,*3575–3594.Marcus, R. (2015).

*Autonomy platonism and the indispensability argument*. NY: Lexington Books.McCain, K. (2016).

*The nature of scientific knowledge: An explanatory approach*. Berlin: Springer.Melia, J. (1995). On what there’s not.

*Analysis,**55,*223–229.Melia, J. (2000). Weaseling away the indispensability argument.

*Mind,**109,*455–479.Melia, J. (2002). Response to Colyvan.

*Mind,**111,*75–79.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.Morrison, J. (2010). Just how controversial is evidential holism?

*Synthese,**173,*335–352.Morrison, J. (2012). Evidential holism and indispensability arguments.

*Erkenntnis,**76,*263–278.Parsons, C. (1983). Quine on the philosophy of mathematics. In

*Mathematics in philosophy*(pp. 176–205). Ithaca, NY: Cornell University Press.Psillos, S., & Ruttkamp-Bloem, E. (2017). Scientific realism: Quo Vadis? introduction: New thinking about scientific realism.

*Synthese,**194,*3187–3201.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).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).Quine, W. V. (1948). On what there is.

*Review of Metaphysics,**2,*21–38.Quine, W. V. (1951). Two dogmas of empiricism.

*The Philosophical Review,**60,*20–43.Quine, W. V. (Ed.). (1981a). Success and limits of mathematization. In

*Theories and things*(pp. 148–155). Cambridge MA: Harvard University Press.Quine, W. V. (Ed.). (1981b). Things and their place in theories. In

*Theories and things*(pp. 1–23). Cambridge MA: Harvard University Press.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.Sober, E. (1993). Mathematics and indispensability.

*The Philosophical Review,**102,*35–57.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.Vineberg, S. (1996). Confirmation and the indispensability of mathematics to science.

*Philosophy of Science,**63*(Supplement), S256–S263.Yablo, S. (2005). The myth of seven. In M. E. Kalderon (Ed.),

*Fictionalism in metaphysics*(pp. 88–115). NY: Oxford University Press.Yablo, S. (2012). Explanation, extrapolation, and existence.

*Mind,**121,*1007–1029.

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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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### Keywords

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