Abstract
The mathematical nature of modern physics suggests that mathematics is bound to play some role in explaining physical reality. Yet, there is an ongoing controversy about the prospects of mathematical explanations of physical facts and their nature. A common view has it that mathematics provides a rich and indispensable language for representing physical reality but that, ontologically, physical facts are not mathematical and, accordingly, mathematical facts cannot really explain physical facts. In what follows, I challenge this common view. I argue that, in modern natural science, mathematics is constitutive of the physical. Given the mathematical constitution of the physical, I propose an account of explanation in which mathematical frameworks, structures, and facts explain physical facts. In this account, mathematical explanations of physical facts are either species of physical explanations of physical facts in which the mathematical constitution of some physical facts in the explanans are highlighted, or simply explanations in which the mathematical constitution of physical facts are highlighted. In highlighting the mathematical constitution of physical facts, mathematical explanations of physical facts make the explained facts intelligible or deepen and expand the scope of our understanding of them. I argue that, unlike other accounts of mathematical explanations of physical facts, the proposed account is not subject to the objection that mathematics only represents the physical facts that actually do the explanation. I conclude by briefly considering the implications that the mathematical constitution of the physical has for the question of the unreasonable effectiveness of the use of mathematics in physics.
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Notes
- 1.
Steiner (1998, p. 9) argues that there are two problems with Wigner’s formulation of question of the “unreasonable effectiveness of mathematics in the natural sciences.” First, Wigner ignores the cases in which “scientists fail to find appropriate mathematical descriptions of natural phenomena” and the “mathematical concepts that never have found an application.” Second, Wigner focuses on individual cases of successful applications of mathematical concepts, and these successes might have nothing to do with the fact that a mathematical concept was applied. Steiner seeks to formulate the question of the astonishing success of the applicability of mathematics in the natural sciences in a way that escapes these objections. He argues that in their discoveries of new theories, scientists relied on mathematical analogies. Often these analogies were ‘Pythagorean’, meaning that they were inexpressible in any other language but that of pure mathematics. That is, often the strategy that physicists pursued to guess the laws of nature was Pythagorean: “they used the relations between the structures and even the notations of mathematics to frame analogies and guess according to those analogies” (ibid., pp. 4–5). Steiner argues that although not every guess, or even a large percentage of the guesses, was correct, this global strategy was astonishingly successful.
Steiner’s reasoning and examples are intriguing and deserve an in-depth study, which, for want of space, I need to postpone for another opportunity. I believe, though, that such a study will not change the main thrust of my analysis of the question of the mathematical constitution of the physical and its implications for the questions of mathematical explanations of physical facts and the “unreasonable effectiveness of mathematics in the natural sciences”.
- 2.
For a list of Pitowsky’s publications, see https://www.researchgate.net/scientific-contributions/72394418_Itamar_Pitowsky
- 3.
Bolzano’s example is believed to have been produced in the 1830s but the manuscript was only discovered in 1922 (Kowalewski 1923) and published in 1930 (Bolzano 1930). Neuenschwander (1978) notes that Weierstrass presented a continuous nowhere differentiable function before the Royal Academy of Sciences in Berlin on 18 July 1872 and that it was published first, with his assent, in Bois-Reymond (1875) and later in Weierstrass (1895).
- 4.
- 5.
Two comments: 1. The introduction of the calculus is often presented as a solution to Zeno’s arrow paradox. I believe that this presentation is somewhat misleading. The calculus does not really solve Zeno’s paradox. It just evades it by fiat, i. e. by redefining the concept of instantaneous velocity. 2. For a recent example of the role that the calculus plays in constituting physical facts, see Stemeroff’s (2018, Chap. 3) analysis of Norton’s Dome – the thought experiment that is purported to show that non-deterministic behaviour could exist within the scope of Newtonian mechanics. Stemeroff argues that: (i) this thought experiment overlooks the constraints that the calculus imposes on Newton’s theory; and that (ii) when these constraints are taken into account, Norton’s Dome is ruled out as impossible in Newtonian universes.
- 6.
- 7.
In Bohmian mechanics, a system’s momentum is not an intrinsic property. Rather, it is a relational property that is generally different from the system’s momentum ‘observable’. Thus, the momentum observable does not generally reflect the system’s momentum (Dürr et al. 2013, Chap. 7; Lazarovici et al. 2018).
- 8.
Since graph theory did not exist at the time, this is obviously an anachronistic account of Euler’s reasoning. That is not a problem for the current discussion, as the aim is not to reconstruct Euler’s own analysis but rather to consider Pincock’s account of mathematical explanation of physical facts. For Euler’s reasoning, see Euler (1736/1956), and for a reconstruction of it, see, for example, Hopkins and Wilson (2004).
- 9.
Vineberg (2018) suggests that the objection above does not apply to the structuralist understanding of mathematics. It is not clear, however, how an appeal to this conception of mathematics could help here. One may accept the structuralist rejection of mathematical objects yet argue that mathematical structures only represent the physical structures which actually do the explanation.
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Acknowledgments
I owe a great debt to Itamar Pitowsky. Itamar’s graduate course in the philosophy of probability stimulated my interest in the philosophical foundations of probability and quantum mechanics. Itamar supervised my course essay and MA thesis on the application of de Finetti’s theory of probability to the interpretation of quantum probabilities. During my work on this research project, I learned from Itamar a great deal about the curious nature of probabilities in quantum mechanics. Itamar was very generous with his time and the discussions with him were always enlightening. Our conversations continued for many years to come, and likewise they were always helpful in clarifying and developing my thoughts and ideas. Itamar’s untimely death has been a great loss. Whenever I have an idea I would like to test, I think of him and wish we could talk about it. I sorely miss the meetings with him and I wish I could discuss with him the questions and ideas considered above.
I am very grateful to the volume editors, Meir Hemmo and Orly Shenker, for inviting me to contribute, and to Meir for drawing my attention to Itamar’s commentary on the relationship between mathematics and physics. The main ideas of the proposed account of mathematical explanations of physical facts were first presented in the workshop on Mathematical and Geometrical Explanations at the Universitat Autònoma de Barcelona (March 2012), and I thank Laura Felline for inviting me to participate in the workshop. Earlier versions of this paper were also presented in the 39th, 40th, 43rd, and 46th Dubrovnik Philosophy of Science conferences, IHPST, Paris, IHPST, University of Toronto, Philosophy, Università degli Studi Roma Tre, Philosophy, Università degli Studi di Firenze, CSHPS, Victoria, CPNSS, LSE, Philosophy, Leibniz Universität Hannover, ISHPS, Jerusalem, Munich Center for Mathematical Philosophy, LMU, Faculty of Sciences, Universidade de Lisboa. I would like to thank the audiences in these venues for their helpful comments. For discussions and comments on earlier drafts of the paper, I am very grateful to Jim Brown, Donald Gillies, Laura Felline, Craig Fraser, Aaron Kenna, Flavia Padovani, Noah Stemeroff, and an anonymous referee. The research for this paper was supported by SSHRC Insight and SIG grants as well as Victoria College travel grants.
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Berkovitz, J. (2020). On the Mathematical Constitution and Explanation of Physical Facts. In: Hemmo, M., Shenker, O. (eds) Quantum, Probability, Logic. Jerusalem Studies in Philosophy and History of Science. Springer, Cham. https://doi.org/10.1007/978-3-030-34316-3_6
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