Abstract
Anyone interested in understanding the nature of modern physics will at some point encounter a problem that was popularized in the 1960s by the physicist Eugene Wigner: Why is it that mathematics is so effective and useful for describing, explaining and predicting the kinds of phenomena we are concerned with in the sciences? In this chapter, we will propose a phenomenological solution for this “problem” of the seemingly unreasonable effectiveness of mathematics in the physical sciences. In our view, the “problem” can only be solved—or made to evaporate—if we shift our attention away from the why-question—Why can mathematics play the role it does in physics?—, and focus on the how-question instead. Our question, then, is this: How is mathematics actually used in the practice of modern physics?
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Notes
- 1.
To be sure, it could be pointed out that criteria other than empirical adequacy do have their place in physics, especially when physicists invoke super-empirical virtues to break underdetermination on the level of empirically equivalent theories. However, apart from the fact that it is unclear if such super-empirical values should qualify as genuinely epistemic, the role of virtues like rigor, elegance, simplicity, manipulability or formal beauty seem much more fundamental in mathematical research. In mathematics, these virtues are not merely the means to decide between otherwise indistinguishable theories; they are rather the guiding principles for the development and assessment of theories.
- 2.
Of course, one could question whether such a “empirical paradigm of theory assessment” is adequate in light of more recent developments in contemporary physics. For instance, since the scale of string theory is roughly of the order of the Planck length, the chances of finding direct empirical confirmation of the theory’s core claims seem rather remote. Given this lack of empirical backing, it is not surprising that string theorists are guided to a much stronger extent by considerations resembling those used in pure mathematics. However, since the jury is still out on whether string theory is in fact too far detached from the binding norms of experimental science or, alternatively, on how the research practice of string theorists would change if empirical tests were possible, we will ignore this case here (cf., for further discussions, Penrose 2004, Smolin 2006, Dawid 2013 and Hossenfelder 2018).
- 3.
It should be noted for the sake of completeness that there have been skeptical voices as well, effectively denying the distinctness thesis and, consequently, the existence of the applicability problem. The argument, in a nutshell, is this: If mathematics is really just a game-like invention, and if, furthermore, its inventors had genuinely physical purposes in mind, then there is nothing mysterious about the usefulness of mathematics in physical research. This view can easily be substantiated by several well-known examples from science history: Leibniz and Newton invented differential and integral calculus for the explicit purpose to describe systems with trajectories through space and time with forces acting on them. Given this practical background and given the ingenuity of its inventors, the applicability of differential and integral calculus is no more surprising than the fact that hammers are well-suited to drive nails. However, such a deflationary stance toward the applicability problem faces several problems: First, it would be a serious mistake to reduce the role of mathematics to that of a convenient tool for the successful framing of physical descriptions. Quite the opposite: In the face of lacking empirical data, physicists quite often turn to mathematics itself in order to discover novel theories or even previously unknown physical phenomena (cf., e.g., Steiner 1989, 1998 and Colyvan 2001). Second, and closely related, it is also not true that the mathematical tools that proved useful in physics were always developed with genuinely physical purposes in mind. Some of the most productive mathematical innovations such as complex numbers, non-Euclidean geometries or spinors were regarded as purely theoretical first and went on to demonstrate their high practical relevance decades, sometimes even centuries later.
- 4.
- 5.
Galileo’s argument, in a nutshell, is that rigorous mathematical proofs allow us to participate in the perfection of God’s knowledge. Hence, when an empirical problem can be dealt with mathematically, Galileo feels warranted to regard the geometrical model of the empirical target system as a truthful representation of how God perceives reality (cf., e.g., Galilei 1967, 103; McTighe 1967; Redondi 1998).
- 6.
It should be noted that there are actually two distinct metaphysical arguments that operate in the background of Galilean physics. First, there is the doctrine of primary and secondary qualities that Galileo introduced in Il Saggiatore and that became common currency in philosophical circles through the works of Descartes, Locke, Hume and others. The second argumentative strategy, which seems to play a particularly prominent role in Galileo’s scientific practice, is based on the distinction between natural occurrences and phenomena (cf. Koertge 1977 and McAllister 1996): Natural occurrences are the physical processes, exactly as they occur under normal, lifeworldly conditions. Phenomena, on the other hand, are the abstract invariant forms that allegedly underlie natural occurrences. According to Galileo, a natural occurrence is always the result of one or more phenomena and great number of accidents. And although Galileo acknowledges that the accidents are responsible for the huge variety of observable natural occurrences, he claims that they must be systematically excluded from physics through the method of geometrical idealization.
- 7.
Although it is not an easy task to determine the empirical adequacy of Galilean science from a contemporary perspective, the following episode shows how hard it was to apply Galilean mechanics successfully in the seventeenth century: Four years after Galileo’s death the gunner Giovanni Ranieri attempted to apply Galileo’s theory of projectile motion to his craft. However, as Ranieri reports in a letter to Evangelista Torricelli—Galileo’s successor at the University of Pisa—the experimental results did not even come close to matching the theoretical predictions. Ranieri replicated one of Galileo’s geometrical “proofs” by using an elevated gun to perform a number of point-blank shots. While the theory predicted a range of approximately 96 paces, Ranieri achieved ranges of 400 paces and more (cf. Segre 1991, 94–97). Particularly interesting is how Torricelli reacted to Ranieri’s complaint: Torricelli explained the empirical inadequacy of Galileo’s theory by pointing out “that Galileo [speaks] the language of geometry and [is] not bound by any empirical result” (Segre, 1991, 44). Even more interesting is the fact that Galileo himself was perfectly aware of the practical insufficiencies of his own theory. Shortly after he has presented his “proof” that projectiles describe a semi-parabolic path, he freely admits that the “conclusions proved in the abstract will be different when applied in the concrete and will be fallacious to this extent, that neither will the horizontal motion be uniform, nor the path of the projectile a parabola” (Galilei, 1954, 251).
- 8.
- 9.
The suggestion to read Koyré from a phenomenological perspective is by no means far-fetched: Not only was Koyré a student of Husserl in Göttingen; Koyré had plans to write his dissertation on the antinomies of set theory under Husserl’s supervision. What is more, as Parker has argued in detail, there are good reasons to believe that Koyré’s later interpretation of Galilean physics was heavily influenced by Husserl’s take on the issue (cf. Parker 2017). Although the phenomenological traces in Koyré’s oeuvre have been overlooked by many, there are, of course, exceptions. For instance, Michel Foucault remarks that “we run across phenomenology in someone like Koyré […] who […] developed a historical analysis of the forms of rationality and knowledge in a phenomenological perspective” (Foucault 1998, 438).
- 10.
It should also be noted that space constraints will prevent us from commenting on two recent solutions to the applicability problem that are in some ways similar to our own approach, namely da Silva’s transcendental-phenomenological account (da Silva 2017) and the account that has recently been developed by Bueno and French (2018).
- 11.
This is the subject of our future work.
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Islami, A., Wiltsche, H.A. (2020). A Match Made on Earth: On the Applicability of Mathematics in Physics. In: Wiltsche, H.A., Berghofer, P. (eds) Phenomenological Approaches to Physics. Synthese Library, vol 429. Springer, Cham. https://doi.org/10.1007/978-3-030-46973-3_8
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