Abstract
In this paper I present two cases, taken from the history of science, in which mathematics and physics successfully interplay. These cases provide, respectively, an example of the successful application of mathematics in astronomy and an example of the successful application of mechanics in mathematics. I claim that an illustration of these cases has a twofold value in the context of the applicability debate. First, it enriches the debate with an historical perspective which is largely omitted in the contemporary discussion. Second, it reveals a component of the applicability problem that has received little attention. This component concerns the successful application of physical principles in mathematical practice. With the help of the two examples, in the final part of the paper I address the following question: are successful applications of mathematics to physics (direct applications) and successful applications of physics to mathematics (converse applications) two sides of the same problem?
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Notes
See, for instance, the accounts given in Steiner (1998), Pincock (2004), Bueno and Colyvan (2011), Rizza (2013), Bueno and French (2018) and McCullough-Benner (2020). In recent years, the most influential picture of applied mathematics has been the so-called ‘mapping account’ view, in which an explanation of the applicability of mathematics in science is given in terms of mappings (like homomorphisms, epimorphisms, and monomorphisms) that are established between mathematics and the empirical systems studied (e.g., see Bueno & Colyvan, 2011 and Bueno & French, 2018).
The converse applicability problem is acknowledged in Ginammi (2018). Ginammi considers different kinds of applicability involving mathematics and physics. Among these, he explicitly addresses applications of physics to mathematics, which he calls ‘physics-to-math applications’. In Sections 4 and 5, I shall bring up Ginammi’s analysis and discuss some of its aspects in the context of the present work.
In my presentation of Aristarchus’ proof I will follow Thomas Heath’s edition of On Sizes (Heath, 2004).
Aristarchus’ first result is correct, since the actual average distance of the Earth to the Sun in terms of its average distance to the Moon is 389. On the other hand, Aristarchus’ conclusion that the distance of the Sun from the Earth is less than 20 times the distance of the Moon from the Earth is, although obtained through a correct mathematical proof, false. The source of error lies in what Aristarchus assumes in hypothesis H4, namely that when the Moon is at quadrature the angle between the Moon and the Sun viewed from the Earth is 87∘ (\(\angle CBA\) in Fig. 1). The actual value of angle \(\angle CBA\) is about 89∘51′.
Angles \(\angle BAC\) and \(\angle EBD\) are congruent. In fact, \(\angle EBD=\angle EBA - \angle CBA\), where \(\angle EBA=90^{\circ }\) and \(\angle CBA=87^{\circ }\). Thus, \(\angle EBD=3^{\circ }\) and it is equal to \(\angle BAC\).
△(BEG) = △(BGI) because two sides and the included angle of △(BEG) are equal to two sides and the included angle of △(BGI): BE = BI, BG is common to the two triangles and \(\angle EBG=\angle GBI\).
The diagonal FG divides the square \(\square {(IGQF)}\) into two isosceles triangles: △(GIF) and △(GQF). If we apply the Pythagorean theorem to △(GIF), we get that FG2 = 2IG2, which is equivalent to say that \(\square {(FGKN)}\) is twice \(\square {(IGQF)}\).
Aristarchus remarks that FG2 : GE2 = 2 and 2 > 49 : 25. Therefore FG2 : GE2 > 49 : 25, which gives the inequality FG : GE > 7 : 5.
In this paper I am adopting Jan Dijksterhuis’ exposition of the Method (Dijksterhuis, 1987).
This result, concerning the property of the subtangent, is stated in the Quadrature of the Parabola, Proposition 2, where Archimedes mentions that the result was obtained by Aristaeus and Euclid in their treatises on conic sections (Heath, 2009, p. 235). The chord AC is parallel to the tangent to the parabola at B because D is the middle point of AC and DB is parallel to the axis of the parabola (this result is also stated in the Quadrature of the Parabola, Proposition 1).
Here Archimedes is implicitly applying three results obtained by Euclid in the Elements: Proposition 4 of Book VI and Propositions 11 and 9 of Book V.
The law of the lever, found by Archimedes in the treatise On the Equilibrium of Planes, states that bodies placed on opposite sides of the fulcrum are in equilibrium at distances reciprocally proportional to their weights. For instance, if two bodies of masses m1 and m2 are placed on the arms of a straight lever of fulcrum K, and if d1 and d2 are the distances of the bodies’ centers of mass from the fulcrum, then the two bodies will balance just in case m1/d2 = m2/d1.
The point of intersection of the medians of a triangle divides each median into segments with a 2 : 1 ratio. Thus, by taking point X on CK such that CK = 3XK, we find the point of intersection of the medians. But the point where the three medians of the triangle meet is also the center of gravity of the triangle, as Archimedes proves in his On the Equilibrium of Planes, Proposition 14 of Book I (Heath, 2009, p. 201).
The result is obtained through an application of Euclid’s Elements, Proposition 1 of Book VI.
The proof should be formally valid and contain no mistakes. Otherwise it would not count as proof.
I am grateful to a referee for pushing me to clarify this point.
In the radio-signal method, distance are calculated without trigonometry, using only the time-delay of the signal and the speed of light. For a description of the method see Webb(1999).
Archimedes’ result is also confirmed by modern calculus. Nevertheless, it is not necessary to resort to modern mathematics, since the geometrical proof given in the Quadrature of the Parabola is still regarded by mathematicians as a flawless demonstration.
Mark Steiner calls this the “metaphysical question concerning application: how can facts about numbers [thought of as abstract or nonphysical objects] be relevant to the empirical world?” (Steiner, 1998, pp. 1-2). Other philosophers of mathematics present the applicability problem in terms of the abstract/concrete dichotomy, as a metaphysical issue. See, for instance, Dummett, 1991, p. 301 and Nolan, 2015, p. 61.
Similar remarks on the importance to focus on (applied) mathematical theories, rather than (applied) mathematical objects, have been put forward in the debate on the enhanced indispensability argument (see section 3.4 of Panza & Sereni, 2016). Moreover, it can be noted that also mapping accounts of mathematical applicability focus on (applied) theories rather than (applied) entities. For instance, Otávio Bueno and Steven French observe how “In applying a mathematical theory to physics, we are often ‘bringing in’ structure from the mathematical level to the physical” (Bueno & French, 2018, p. 56).
By adopting this stance, I am not providing any argument ruling out the possibility that what is applied are objects (of theories). This possibility should be left open. And this especially because talking in terms of theories seems to leave a lot out (e.g., we may want to be more specific about what is applied to what, and talking about theories does not seem to provide this kind of specificity). I am indebted to a referee for bringing this consideration to my attention.
For a simple lever, if m1 and m2 are the masses of the two bodies on the arms and d1 and d2 are the distances of the bodies’ centers of mass from the fulcrum, the zero torque condition reads m1g ⋅ d1 = m2g ⋅ d2, which is Archimedes’ law of the lever m1/d2 = m2/d1.
For instance, Uspenskii (1961) also focuses on cases where a physical principle is used to establish a purely arithmetical result.
Conservation principles appear as the main ingredients of converse applications, even if sometimes the reference to such principles is not explicit. For instance, Mark Levi discusses a case of application of Kirchhoff’s second law (Levi, 2009, p. 76). Although not mentioned by Levi, we know that Kirchhoff’s second law is a consequence of charge conservation and conservation of energy.
Here I am specifically focusing on the import of principles, and not laws, in converse applications. The distinction between principles and laws is the object of a separate debate in philosophy of science, but for the point at stake here it is sufficient to add that I generally regard principles as more fundamental than laws. For instance, as I show in footnote 22, Archimedes’ law of the lever can be obtained from the conservation of angular momentum. And therefore the principle of conservation of angular momentum is more fundamental than the law of the lever. I do not exclude, however, that some physical laws may have such fundamental status. For instance, Newtons’ laws are sometimes regarded as fundamental principles of physics (Dilworth, 1994).
To overcome the problem with idealizations, some authors have proposed an extension of the mapping account in terms of partial structures (see, e.g., Bueno & French, 2018). Note, however, that this refinement does not provide an answer to the first criticism exposed here, since partial structures are still mathematical structures.
Interestingly, some positive reasons in favor of such unified perspective in terms of structural similarities and structural representation are given in Ginammi (2018). These considerations are surely important to address direct and converse applications in terms of a structuralist approach. Nevertheless, as observed by Ginammi himself, they still require further elaboration. In the present study I did not follow this route and I preferred to focus on an alternative direction of analysis.
References
Baker, A. (2009). Mathematical explanation in science. The British Journal for the Philosophy of Science, 60, 611–633.
Bangu, S. (2021). Mathematical explanations of physical phenomena. Australasian Journal of Philosophy, 99(4), 669–682.
Batterman, R. W. (2010). On the explanatory role of mathematics in empirical science. The British Journal for the Philosophy of Science, 61(1), 1–25.
Berggren, J. L., & Sidoli, N. (2007). Aristarchus’s on the sizes and distances of the sun and the moon: Greek and arabic texts. Archive for History of Exact Sciences, 61(3), 213–254.
Bird, A. (2007). Nature’s metaphysics: Laws and properties. Clarendon Press.
Bueno, O., & Colyvan, M. (2011). An inferential conception of the application of mathematics. Noûs, 45(2), 345–374.
Bueno, O., & French, S. (2018). Applying mathematics: Immersion inference interpretation. Oxford University Press.
Clarke-Doane, J. (2019). Modal objectivity1. Noûs, 53(2), 266–295.
Dijksterhuis, E. J. (1987). Archimedes. Princeton University Press.
Dilworth, C. (1994). Principles, laws, theories and the metaphysics of science. Synthese, 101, 223–247.
Dummett, M. (1991). Frege: Philosophy of mathematics. Harvard University Press.
Ellis, B. D. (2002). The philosophy of nature: A guide to the new essentialism. McGill-Queen’s University Press.
Ginammi, M. (2018). Applicability problems generalized. In M. Piazza G. Pulcini (Eds.) Truth Existence and Explanation. Boston Studies in the Philosophy and History of Science (pp. 209–224). Springer International Publishing.
Heath, T. L. (2004). Aristarchus of Samos, the ancient Copernicus. Dover.
Heath, T. L. (2009). The works of archimedes. Edited in modern notation with introductory chapters. Cambridge library collection - mathematics. Cambridge University Press.
Katz, V. (2009). A history of mathematics, 3rd edn. Pearson.
Kogan, Y. (1974). The application of mechanics to geometry. University of Chicago Press.
Lakatos, I. (1976). Proofs and refutations: The logic of mathematical discovery. Cambridge University Press.
Lange, M. (2017). Because without cause: Non-Causal explanations in science and mathematics. Oxford University Press.
Leeds, S. (2007). Physical and metaphysical necessity. Pacific Philosophical Quarterly, 88(4), 458–485.
Levi, M. (2009). The mathematical mechanic. Princeton University Press.
Linnemann, N. (2020). On metaphysically necessary laws from physics. European Journal for Philosophy of Science, 10, 23.
McCullough-Benner, C. (2020). Representing the world with inconsistent mathematics. The British Journal for the Philosophy of Science, 71(4), 1331–1358.
Netz, R., & Noel, W. (2007). The archimedes codex: How a medieval prayer book is revealing the true genius of antiquity’s greatest scientist. Da Capo Press.
Nolan, D. (2015). The unreasonable effectiveness of abstract metaphysics. Oxford Studies in Metaphysics, 9, 61–88.
Panza, M., & Sereni, A. (2016). The varieties of indispensability arguments. Synthese, 193(2), 469–516.
Pincock, C. (2004). A new perspective on the problem of applying mathematics. Philosophia Mathematica, 12(2), 135–161.
Pincock, C. (2015). Abstract explanations in science. The British Journal for the Philosophy of Science, 66(4), 857–882.
Pólya, G. (1981). Mathematics discovery: An understanding, learning and teaching problem solving (combined edition). Wiley.
Räz, T., & Sauer, T. (2015). Outline of a dynamical inferential conception of the application of mathematics. Studies in History and Philosophy of Science Part B: Studies in History and Philosophy of Modern Physics, 49, 57–72.
Rizza, D. (2013). The applicability of mathematics: Beyond mapping accounts. Philosophy of Science, 80(3), 398–412.
Shapiro, S. (1983). Mathematics and reality. Philosophy of Science, 50, 523–548.
Skow, B. (2015). Are there genuine physical explanations of mathematical phenomena? The British Journal for the Philosophy of Science, 66(1), 69–93.
Steiner, M. (1998). The applicability of mathematics as a philosophical problem. Harvard University Press.
Swoyer, C. (1982). The nature of natural laws. Australasian Journal of Philosophy, 60(3), 203–223.
Uspenskii, V. A. (1961). Some applications of mechanics to mathematics. Blaisdell Publishing Company.
Webb, S. (1999). Measuring the universe. The cosmological distance ladder. Springer Science & Business Media.
Wolff, J. (2013). Are conservation laws metaphysically necessary? Philosophy of Science, 80(5), 898–906.
Acknowledgements
I would like to thank two anonymous reviewers for their valuable comments.
Funding
This work was supported by FCiências.ID and the Portuguese Foundation for Science and Technology (FCT) through the project Exploring the Weak Objectivity of Mathematical Knowledge (grant no. CEECIND/01827/2018).
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Molinini, D. Direct and converse applications: Two sides of the same coin?. Euro Jnl Phil Sci 12, 8 (2022). https://doi.org/10.1007/s13194-021-00431-z
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DOI: https://doi.org/10.1007/s13194-021-00431-z