Abstract
Mathematical explanation is a topic of great contemporary interest in the philosophy of mathematics. The question of whether mathematics can play an explanatory role in empirical science is thought by many to be the key to making progress on the realism versus anti-realism debate in the philosophy of mathematics. Questions about explanation within mathematics are also interesting and are important for the development of a general account of explanation. In a series of groundbreaking papers from 1978 to 1983, Mark Steiner set the agenda for much of the modern debate about mathematical explanation. In the present paper we look at Steiner’s role in advancing mathematical explanation as a central topic in the philosophy of mathematics and his legacy for the modern debates about mathematical explanation.
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Notes
- 1.
Although the existence of an entity and its reality are often taken to amount to the same thing, a distinction can, and was, made between the two. Roughly, reality requires that the entities exist and that they exist independently of human activities. As we shall see, such a distinction was important for intuitionists such as Dummett and others.
- 2.
Throughout this paper we are assuming classical logic, standardly interpreted.
- 3.
One could argue that Steiner’s work here was an early example of philosophy of mathematical practice, which is a popular concern amongst philosophers of mathematics in recent years.
- 4.
- 5.
Maddy (2000) presents a case study of early atomic theory, in which the assumption of atoms was indispensable yet leading scientists Poincaré and Ostwald remained sceptical about the existence of atoms. This, she argues, is a problem for Quine, who either has to reject confirmational holism or naturalism—both core doctrines in Quine’s metaphysics and epistemology. See Colyvan (2001) for discussion of Maddy’s arguments and some Quinean replies.
- 6.
But emphasizing that scientists do not acknowledge the existence of all the objects they posit is not enough, it must be shown that they don’t commit themselves to mathematical objects, however cavalier their attitude to them may be. Maddy’s position, discussed above, in (Forthcoming) is one way of showing how scientists could avoid this commitment.
- 7.
One such approach is to invoke a causal test: an entity is real if and only if it has causal powers (Armstrong, 1978). Such a criterion has numerous problems (Colyvan, 1998; Oddie, 1982) and, in any case, in the current context, is question begging, since mathematical entities are usually take to be acausal.
- 8.
There was some work on mathematical explanation and non-causal explanation more generally (e.g. Maddy (1981), Oddie (1982), and Smart (1990)) before this turn in the debate, although such work did not seem a central issue until Maddy and Melia’s challenges for the indispensability argument were taken up.
- 9.
The explanatory version of the argument was suggested earlier, but not endorsed, by Hartry Field (1989, p. 14.).
- 10.
Recall that the intermediate-value theorem: Let f be a real-valued function, continuous on a closed interval [a,b] and let c be any number between f(a) and f(b) inclusive, then there is an x in [a,b] such that f(x)=c.
- 11.
Although one of the present authors also once went into print endorsing such a view about mathematical explanations requiring explanatory proofs (e.g. Colyvan, 2001), there is good reason to suspect that this is asking too much and explanatory proofs are not needed for the explanation of the physical phenomenon in question (Baker, 2012). This issue came up previously (in Sect. 1) in a slightly different context.
- 12.
- 13.
Indeed, the biologists interested in the issue surely would not be interested in any first-order logic reconstruction of their question.
- 14.
To be sure, this issue is far from settled, in part because (a) comparing theories or different formulations of theories with respect to explanatory power is notoriously difficult and (b) in most cases we don’t have the non-mathematical formulation to hand.
- 15.
Less in the sense that the set of objects required for explanation are a proper subset of the set required to do science. We’re not talking about cardinality of the sets in question because these may be the same.
- 16.
See Resnik (1981, 1982), Resnik, 1997) and Shapiro (1997) for such a realist structuralist view. Recently Resnik (2019) has defended a more ontologically deflationary version of structuralism (although there were similar nuances in his Resnik, 1997 as well). According to this latter view, structural properties do the explaining, in the sense that the objects referred to in the explanans having such and such structural properties is the reason for the phenomenon under investigation.
- 17.
When one of the present authors (Colyvan) started work on mathematical explanation in the mid 1990s, he was surprised to find that arguments for mathematical explanations of physical facts and the significance of these for debates about mathematical realism were already in place in Steiner’s (1978a) work. Indeed, in correspondence with Mark Steiner around this time Steiner was typically very generous and encouraging, despite the fact that Colyvan was initially just rehashing points made by Steiner some 17 years earlier.
References
Armstrong, D. M. (1978). Universals and scientific realism. Cambridge University Press.
Baker, A. (2003). The indispensability argument and multiple foundations for mathematics. The Philosophical Quarterly, 53(210), 49–67.
Baker, A. (2005). Are there genuine mathematical explanations of physical phenomena? Mind, 114(454), 223–238.
Baker, A. (2009). Mathematical explanation in science. British Journal for the Philosophy of Science, 60(3), 611–633.
Baker, A. (2012). Science-driven mathematical explanation. Mind, 121(482), 243–267.
Bangu, S. I. (2008). Inference to the best explanation and mathematical realism. Synthese, 160(1), 13–20.
Baron, S. (2014). Optimization and mathematical explanation: Doing the Lévy walk. Synthese, 191(3), 459–479.
Baron, S. (2016). Explaining mathematical explanation. The Philosophical Quarterly, 66(264), 458–480.
Batterman, R. (2010). On the explanatory role of mathematics in empirical science. British Journal for the Philosophy of Science, 61(1), 1–25.
Benacerraf, P. (1965). What numbers could not be. Philosophical Review, 74(1), 47–73. Reprinted in Benacerraf, P. and Putnam, H. 1983. Philosophy of Mathematics, second edition, Cambridge: Cambridge University Press, pp. 272–294.
Cartwright, N. (1983). How the Laws of physics lie. Clarendon Press.
Colyvan, M. (1998). Can the Eleatic principle be justified? The Canadian Journal of Philosophy, 28(3), 313–336.
Colyvan, M. (1999). Confirmation theory and indispensability. Philosophical Studies, 96(1), 1–19.
Colyvan, M. (2001). The indispensability of mathematics. Oxford University Press.
Colyvan, M. (2002). Mathematics and aesthetic considerations in science. Mind, 111(441), 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). Springer.
Colyvan, M. (2010). There is no easy road to nominalism. Mind, 119(474), 285–306.
Colyvan, M. (2018). The ins and outs of mathematical explanation. Mathematical Intelligencer, 40(4), 26–29. Reprinted in M. Pitici (ed.), The Best Writing on Mathematics 2020, Princeton University Press, 2020, pp. 189–197.
Dummett, M. (1975). The philosophical basis of intuitionistic logic. In H. E. Rose, & J. C. Shepherdson (Eds.), Proceedings of the logic colloquium ‘73, special issue of Studies in logic and the foundations of mathematics (Vol. 80, pp. 5–40). North-Holland.
Dummett, M. (1978). Truth and other enigmas. Duckworth and Co.
Field, H. H. (1980). Science without numbers: A Defence of nominalism. Blackwell.
Field, H. H. (1989). Realism, mathematics and modality. Blackwell.
Hafner, J., & Mancosu, P. (2005). The varieties of mathematical explanation. In P. Mancosu, K. Jørgensen, & S. Pedersen (Eds.), Visualization, explanation and reasoning styles in mathematics (pp. 215–250). Springer.
Lange, M. (2018). Because without cause: Non-causal explanations in science and mathematics. Oxford University Press.
Lyon, A., & Colyvan, M. (2008). The explanatory power of phase spaces. Philosophia Mathematica, 16(2), 227–243.
Maddy, P. (1981). Sets and numbers. Noûs, 15(4), 495–591.
Maddy, P. (1992). Indispensability and practice. Journal of Philosophy, 89(6), 275–289.
Maddy, P. (1995). Naturalism and ontology. Philosophia Mathematica, 3(3), 248–270.
Maddy, P. (2000). Naturalism in mathematics. Oxford University Press.
Maddy, P. (2007). Second philosophy: A naturalistic method. Oxford University Press.
Maddy, P. (Forthcoming). Enhanced If-thenism. In P. Maddy (Ed.), A plea for natural philosophy and other essays. Oxford University Press.
Mancosu, P. (2008). Mathematical explanation: Why it matters. In P. Mancosu (Ed.), The philosophy of mathematical practice (pp. 134–149). Oxford University Press.
Mancosu, P. (2018). Explanation in mathematics. In E. N. Zalta (Ed.), The Stanford encyclopedia of philosophy (Summer 2018 Edition). https://plato.stanford.edu/archives/sum2018/entries/mathematics-explanation/
Melia, J. (2000). Weaseling away the indispensability argument. Mind, 109(435), 455–479.
Melia, J. (2002). Response to Colyvan. Mind, 111(441), 75–79.
Oddie, G. (1982). Armstrong on the Eleatic principle and abstract entities. Philosophical Studies, 41(2), 285–295.
Odom, B., Hanneke, D., D’Urso, B., & Gabrielse, G. (2006). New measurement of the electron magnetic moment using a one-electron quantum cyclotron. Physical Review Letters, 97(3), 030801. (First published 17 July 2006; Erratum published Physical Review Letters 99, 039902 (2007)).
Pincock, C. (2011). Mathematical explanations of the rainbow. Studies in History and Philosophy of Modern Physics, 42(1), 13–22.
Putnam, H. (1979). Mathematics matter and method: Philosophical papers (Vol. 1, 2nd ed., pp. 60–78). Cambridge University Press.
Quine, W. V. O. (1953). On what there is. In From a logical point of view (pp. 1–19). Harvard University Press. (Originally published in 1948).
Resnik, M. D. (1981). Mathematics as a science of patterns: Ontology and reference. Noûs, 15(4), 529–550.
Resnik, M. D. (1982). Mathematics as a science of patterns: Epistemology. Noûs, 16(1), 95–105.
Resnik, M. D. (1997). Mathematics as a science of patterns. Clarendon Press.
Resnik, M. D. (2019). Non-ontological structuralism. Philosophia Mathematica, 27(3), 303–315.
Resnik, M. D., & Kushner, D. (1987). Explanation, Independence and realism in mathematics. British Journal for the Philosophy of Science, 38(2), 141–158.
Reutlinger, A., & Saatsi, J. (Eds.). (2018). Explanation beyond causation. Oxford University Press.
Shapiro, S. (1997). Philosophy of mathematics: Structure and ontology. Oxford University Press.
Smart, J. J. C. (1990). Explanation—Opening address. In D. Knowles (Ed.), Explanation and its limits (pp. 1–19). Cambridge University Press.
Steiner, M. (1975). Mathematical knowledge. Cornell University Press.
Steiner, M. (1978a). Mathematics, explanation, and scientific knowledge. Noûs, 12(1), 17–28.
Steiner, M. (1978b). Mathematical explanation. Philosophical Studies, 34(2), 135–151.
Steiner, M. (1983). Mathematical realism. Noûs, 17(3), 383–395.
Steiner, M. (1998). The applicability of mathematics as a philosophical problem. Harvard University Press.
van Fraassen, B. C. (1980). The scientific image. Clarendon.
Yablo, S. (2012). Explanation, extrapolation, and existence. Mind, 121(484), 1007–1029.
Acknowledgements
First and foremost, both authors are deeply indebted to Mark Steiner for his fascinating, and often provocative, work in the philosophy of mathematics. We are also grateful for many conversations with Mark over the years on topics related to this paper. We have learned a great deal from him and have been inspired to pursue some of the topics he opened up for discussion. We have not always agreed with Mark but this made our conversations and exchanges all the more interesting—at least for us but we’re pretty sure for Mark as well. We both counted Mark as a friend as well as a colleague and we miss him, his sense of humor, and his generous nature. We are also indebted to Carl Posy and Mark Rothery for several helpful comments on an earlier draft of this paper, and to Jody Azzouni, Steve Cole, Thomas Hofweber, Marc Lange, and Pen Maddy for conversations or correspondence concerned the topics treated here. Mark Colyvan’s work on this paper was funded by an ARC Discover Grant (grant number: DP170104924).
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Colyvan, M., Resnik, M.D. (2023). Explanation and Realism: Interwoven Themes in the Philosophy of Mathematics. In: Posy, C., Ben-Menahem, Y. (eds) Mathematical Knowledge, Objects and Applications. Jerusalem Studies in Philosophy and History of Science. Springer, Cham. https://doi.org/10.1007/978-3-031-21655-8_4
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