Abstract
While there has been much discussion about what makes some mathematical proofs more explanatory than others, and what are mathematical coincidences, in this article I explore the distinct phenomenon of mathematical facts that call for explanation. The existence of mathematical facts that call for explanation stands in tension with virtually all existing accounts of “calling for explanation”, which imply that necessary facts cannot call for explanation. In this paper I explore what theoretical revisions are needed in order to accommodate this phenomenon. One of the important upshots is that, contrary to the current consensus, low prior probability is not a necessary condition for calling for explanation. In the final section I explain how the results of this inquiry help us make progress in assessing Hartry Field’s style of reliability argument against mathematical Platonism and against robust realism in other domains of necessary facts, such as ethics.
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Notes
Sometimes the explananda are events rather than facts, sometimes the relevant property is described as "surprise" or "puzzlement", rather than "calling for explanation". For a more nuanced discussion of Horwich see Baras (2019). For discussion of Schlesinger see Baras (unpublished ms.).
Prior to this paper, the possibility that necessary truths can call for explanation has been raised in blogposts by Pruss (2013) and Berry (2017), and in Baras (2017a, p. 203) and Baras (2017b, p. 485) in the context of Hartry Field’s (1989, pp. 25–30) style of argument against mathematical Platonism and parallel arguments against realisms in other abstract domains. As I completed this paper, Berry developed and published her view in Berry (2020). I engage with this literature and with her article in Sect. 5. In the context of philosophy of mind there is a debate about whether identities, such as mind–body identities, can call for explanation and some have made general claims about necessary facts in this context. Examples are Melnyk (2003, p. 52) and Papineau (2011, p. 9) who claim without much argument that necessary facts cannot call for explanation and Morris (2011, p. 378) who responds by citing mathematical explanations to show that they can call for explanation. These references are all to very brief comments. They do not explore the question in depth as I do here. (I thank Assaf Weksler for referring me to this debate). After completing this paper, I discovered that van Cleve (2018) explores at great length the idea that some necessary facts should be explained. Fortunately, there is not much overlap between the papers, so they nicely complement each other.
I thank an anonymous referee for this example.
For a formal argument (unfortunately, not very accessible to non-mathematicians) that mathematical claims can be translated to hypothetical claims in modal logic (S5) about what follows necessarily from certain axioms, see Hellman (1993, 1996). I remain neutral with regard to his further claim that this is all that mathematical claims amount to. (I thank Sharon Berry for discussion and the reference to Hellman).
For examples and some more elaboration, see Baras and Shenker (2020).
In ordinary conversation, “coincidence” is sometimes used as synonymous with “lacks an explanation”. Other times, people use “coincidence” and “explanation” differently, such that a coincidence is considered a legitimate type of explanation. My use of these terms here (and I’m following a core of the literature) is in accordance with the former, not the latter.
There is a theoretical question here, which I will explore elsewhere, whether the claim is just that E must have an explanation, or a stronger claim, that E must have a particular kind of explanation. For the purposes of this article, the distinction will not matter. For preliminary discussion, see Baras (2019) and Baras (2020).
I thank Assaf Weksler for this suggestion.
Perhaps there are other examples that do support the possibility of mathematical claims calling for explanation in sense (4). Some colleagues have suggested to me that there are arguments against certain mathematical axioms that can be understood as based on the claim that those axioms imply that a mathematical fact that calls for explanation could not be explained. Assessing such proposed examples would require further research.
Other adherents of epistemic interpretations add additional constraints for the priors because they believe that not every probability distribution that conforms to the axioms is rational. Such views are often called “logical” interpretations of probability.
I’m inspired here by Colyvan (2013), who distinguishes three types of motivations for idealization in normative domains. Colyvan lists mathematical simplicity as an independent type of motivation whereas I group it with other pragmatic considerations. (I thank Arnon Levy for acquainting me with this discussion).
I thank Jessica Wilson for raising a suggestion along these lines in conversation.
See Baras (2019, sec. 4) for some discussion.
References
Baras, D. (2017a). Our reliability is in principle explainable. Episteme, 14(02), 197–211. https://doi.org/10.1017/epi.2016.5.
Baras, D. (2017b). A reliability challenge to theistic Platonism. Analysis, 77(3), 479–487. https://doi.org/10.1093/analys/anx089.
Baras, D. (2019). Why do certain states of affairs call out for explanation? A critique of two Horwichian accounts. Philosophia, 47(5), 1405–1419. https://doi.org/10.1007/s11406-018-0047-x.
Baras, D. (2020). A strike against a striking principle. Philosophical Studies, 177(6), 1501–1514. https://doi.org/10.1007/s11098-019-01265-5.
Baras, D. (unpublished ms.). Calling for explanation: An extraordinary account. Retrieved August 7, 2020, from https://danbaras.com/papers/.
Baras, D., & Shenker, O. (2020). Calling for explanation: The case of the thermodynamic past state. European Journal for Philosophy of Science. https://doi.org/10.1007/s13194-020-00297-7.
Benacerraf, P. (1973). Mathematical truth. The Journal of Philosophy, 70(19), 661–679. https://doi.org/10.2307/2025075.
Berry, S. (2017). Trivializing Benacerraf and Monstrous Moonshine. http://philosophyinprogress.blogspot.co.il/2017/10/trivializing-benacerraf-and-monstrous.html.
Berry, S. (2020). Coincidence avoidance and formulating the access problem. Canadian Journal of Philosophy. https://doi.org/10.1017/can.2020.15.
Bjerring, J. C. (2013). Impossible worlds and logical omniscience: An impossibility result. Synthese, 190(13), 2505–2524. https://doi.org/10.1007/s11229-011-0038-y.
Burgess, J. P., & Rosen, G. (1999). A Subject with no object. Oxford: Oxford University Press. https://doi.org/10.1093/0198250126.001.0001.
Christensen, D. (2004). Putting logic in its place. Oxford: Oxford University Press. https://doi.org/10.1093/0199263256.001.0001.
Clarke-Doane, J. (2016). What is the Benacerraf problem? In F. Pataut (Ed.), New Perspectives on the philosophy of Paul Benacerraf: Truth, objects, infinity (pp. 17–43). Berlin: Springer. https://doi.org/10.1007/978-3-319-45980-6_2.
Clarke-Doane, J. (2020). Morality and mathematics. Oxford: Oxford University Press.
Colyvan, M. (2013). Idealisations in normative models. Synthese, 190(8), 1337–1350. https://doi.org/10.1007/s11229-012-0166-z.
D’Alessandro, W. (2020). Proving quadratic reciprocity: Explanation, disagreement, transparency and depth. Synthese. https://doi.org/10.1007/s11229-020-02591-6.
Enoch, D. (2011). Taking morality seriously. Oxford: Oxford University Press. https://doi.org/10.1093/acprof:oso/9780199579969.001.0001.
Faraci, D. (2019). Groundwork for an explanationist account of epistemic coincidence. Philosophers’ Imprint, 19(4), 1–26.
Field, H. (1989). Realism, mathematics, and modality. New York: Basil Blackwell.
Gaifman, H. (2004). Reasoning with limited resources and assigning probabilities to arithmetical statements. Synthese, 140(1–2), 97–119. https://doi.org/10.1023/B:SYNT.0000029944.99888.a7.
Gillies, D. (2000). Philosophical theories of probability. New York: Routledge.
Hájek, A. (2019). Interpretations of probability. In E. N. Zalta (Ed.), Stanford encyclopedia of philosophy (Fall 2019 Edition). https://plato.stanford.edu/archives/fall2019/entries/probability-interpret.
Harker, D. (2012). A surprise for Horwich (and some advocates of the fine-tuning argument (which does not include Horwich (as far as I know))). Philosophical Studies, 161(2), 247–261. https://doi.org/10.1007/s11098-011-9732-3.
Hellman, G. (1993). Mathematics without numbers. Oxford: Oxford University Press. https://doi.org/10.1093/0198240341.001.0001.
Hellman, G. (1996). Structuralism without structures. Philosophia Mathematica, 4(2), 100–123. https://doi.org/10.1093/philmat/4.2.100.
Horwich, P. (1982). Probability and evidence. Cambridge: Cambridge University Press.
Korman, D. Z., & Locke, D. (2020). Against minimalist responses to moral debunking arguments. In R. Shafer-Landau (Ed.), Oxford studies in metaethics, (Vol. 15). Oxford: Oxford University Press
Lange, M. (2016). Because without cause. Oxford: Oxford University Press. https://doi.org/10.1093/acprof:oso/9780190269487.001.0001.
Leslie, J. (1989). Universes. New York: Routledge.
Mancosu, P. (2018). Explanation in Mathematics. In E. N. Zalta (Ed.), Stanford encyclopedia of philosophy (Summer 2018 Edition). https://plato.stanford.edu/archives/sum2018/entries/mathematics-explanation
Manson, N. A., & Thrush, M. J. (2003). Fine-tuning, multiple universes, and the this universe objection. Pacific Philosophical Quarterly, 84, 67–83. https://doi.org/10.1111/1468-0114.00163.
Melnyk, A. (2003). A physicalist manifesto. Cambridge: Cambridge University Press. https://doi.org/10.1017/CBO9780511498817.
Mogensen, A. L. (unpublished ms.). Ethics, evolution, and the coincidence problem: A skeptical appraisal. Retrieved May 20, 2020, from https://www.andreasmogensen.com/publications-all.
Morris, K. (2011). Theoretical Identities as Explanantia and Explananda. American Philosophical Quarterly, 48(4), 373–385.
Nummela, E. (1987). No coincidence. The Mathematical Gazette, 71(456), 147. https://doi.org/10.2307/3616509.
Papineau, D. (2011). What exactly is the explanatory gap? Philosophia, 39(1), 5–19. https://doi.org/10.1007/s11406-010-9273-6.
Price, H. (2002). Boltzmann’s time bomb. The British Journal for the Philosophy of Science, 53(1), 83–119. https://doi.org/10.1093/bjps/53.1.83.
Pruss, A. R. (2013). Necessary coincidences. http://alexanderpruss.blogspot.co.il/2013/11/necessary-coincidences.html.
Pust, J. (2004). On explaining knowledge of necessity. Dialectica, 58(1), 71–87. https://doi.org/10.1111/j.1746-8361.2004.tb00292.x.
Schechter, J. (2010). The reliability challenge and the epistemology of logic. Philosophical Perspectives, 24(1), 437–464. https://doi.org/10.1111/j.1520-8583.2010.00199.x.
Schechter, J. (2013). Rational self-doubt and the failure of closure. Philosophical Studies, 163(2), 429–452. https://doi.org/10.1007/s11098-011-9823-1.
Schechter, J. (2018). Is there a reliability challenge for logic? Philosophical Issues, 28(1), 325–347. https://doi.org/10.1111/phis.12128.
Schlesinger, G. N. (1991). The sweep of probability. Notre Dame: University of Notre Dame Press.
Smithies, D. (2015). Ideal rationality and logical omniscience. Synthese, 192(9), 2769–2793. https://doi.org/10.1007/s11229-015-0735-z.
Smolin, L. (1997). The life of the cosmos. Oxford: Oxford University Press.
Steiner, M. (1978). Mathematical explanation. Philosophical Studies, 34(2), 135–151. https://doi.org/10.1007/BF00354494.
Street, S. (2016). Objectivity and truth: You’d better rethink it. In R. Shafer-Landau (Ed.), Oxford studies in metaethics (Vol. 11, pp. 293–334). Oxford: Oxford University Press. https://doi.org/10.1093/acprof:oso/9780198784647.003.0012.
Strevens, M. (2017). Notes on Bayesian confirmation theory. http://www.nyu.edu/classes/strevens/BCT/BCT.pdf
Talbott, W. J. (2016). Bayesian epistemology. In E. N. Zalta (Ed.), Stanford encyclopedia of philosophy (Winter 2016 Edition). https://plato.stanford.edu/archives/win2016/entries/epistemology-bayesian.
Van Cleve, J. (2018). Brute necessity. Philosophy Compass, 13(9), e12516. https://doi.org/10.1111/phc3.12516.
White, R. (2005). Explanation as a guide to induction. Philosophers’ Imprint, 5(2), 1–29.
White, R. (2018). The argument from cosmological fine-tuning. In G. Rosen, A. Byrne, J. Cohen, S. V. Shiffrin, & E. Harman (Eds.), The Norton introduction to philosophy (2nd ed., pp. 29–35). New York: Norton.
Wielenberg, E. J. (2010). On the evolutionary debunking of morality. Ethics, 120(3), 441–464. https://doi.org/10.1086/652292.
Yli-Vakkuri, J., & Hawthorne, J. (2018). The necessity of mathematics. Noûs. https://doi.org/10.1111/nous.12268.
Acknowledgements
I am thankful to Sharon Berry, David Enoch, Arnon Levy, Alexander Pruss, Joshua Schechter, Assaf Weksler, two anonymous reviewers and to the audiences at my presentations at the 22nd Israeli Philosophical Association conference (2019) and the 93rd Joint Session of the Aristotelean Society and the Mind Association (2019) for very helpful comments and discussion. This article was written with the support of the Martin Buber Society of Fellows at the Hebrew University of Jerusalem.
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Baras, D. How can necessary facts call for explanation?. Synthese 198, 11607–11624 (2021). https://doi.org/10.1007/s11229-020-02817-7
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DOI: https://doi.org/10.1007/s11229-020-02817-7