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A New Role for Mathematics in Empirical Sciences

Published online by Cambridge University Press:  01 January 2022

Abstract

Mathematics is often taken to play one of two roles in the empirical sciences: either it represents empirical phenomena or it explains these phenomena by imposing constraints on them. This article identifies a third and distinct role that has not been fully appreciated in the literature on applicability of mathematics and may be pervasive in scientific practice. I call this the “bridging” role of mathematics, according to which mathematics acts as a connecting scheme in our explanatory reasoning about why and how two different descriptions of an empirical phenomenon relate to each other. I discuss two bridging roles appearing in biological and physical explanations.

Type
Research Article
Copyright
Copyright 2021 by the Philosophy of Science Association. All rights reserved.

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Footnotes

I am very grateful to James Robert Brown, Laura Franklin-Hall, Franz Huber, Nicolas Fillion, Neil Dewar, Mario Günther, and three anonymous reviewers for very helpful feedback on earlier drafts of this article. I also thank audiences at the Canadian Society for the History and Philosophy of Science in Vancouver and the British Society for the Philosophy of Science in Durham for valuable discussions and suggestions.

References

Batterman, R. W. 2009. “On the Explanatory Role of Mathematics in Empirical Science.” British Journal for the Philosophy of Science 61 (1): 125.CrossRefGoogle Scholar
Bigelow, J. 1988. The Reality of Numbers: A Physicalist’s Philosophy of Mathematics. Oxford: Clarendon.Google Scholar
Born, M., Heisenberg, W., and Jordan, P. 1926. “Zur Quantenmechanik II.” Zeitschrift für Physik 35 (8–9): 557615.CrossRefGoogle Scholar
Bueno, O., and Colyvan, M. 2011. “An Inferential Conception of the Application of Mathematics.” Noûs 45 (2): 345–74.CrossRefGoogle Scholar
Bueno, O., and French, S. 2018. Applying Mathematics: Immersion, Inference, Interpretation. Oxford: Oxford University Press.CrossRefGoogle Scholar
Curiel, E. 2013. “Classical Mechanics Is Lagrangian: It Is Not Hamiltonian.” British Journal for the Philosophy of Science 65 (2): 269321.CrossRefGoogle Scholar
Durán, J. M. 2017. “Varying the Explanatory Span: Scientific Explanation for Computer Simulations.” International Studies in the Philosophy of Science 31 (1): 2745.CrossRefGoogle Scholar
Ermentrout, G. B., and Edelstein-Keshet, L. 1993. “Cellular Automata Approaches to Biological Modeling.” Journal of Theoretical Biology 160 (1): 97133.CrossRefGoogle ScholarPubMed
Franklin, J. 2014. An Aristotelian Realist Philosophy of Mathematics. London: Palgrave Macmillan.CrossRefGoogle Scholar
Heisenberg, W. 1925. “Über Quantentheoretische Umdeutung Kinematischer und Mechanischer Beziehungen.” Zeitschrift für Physik 33:879–93.CrossRefGoogle Scholar
Kitcher, P. 1999. “The Hegemony of Molecular Biology.” Biology and Philosophy 14 (2): 195210.CrossRefGoogle Scholar
Krantz, D., Luce, D., Suppes, P., and Tversky, A. 1971. Foundations of Measurement, Vol. 1, Additive and Polynomial Representations. San Diego, CA: Academic Press.Google Scholar
Lange, M. 2009. Laws and Lawmakers: Science, Metaphysics, and the Laws of Nature. New York: Oxford University Press.CrossRefGoogle Scholar
Lange, M. 2012. “What Makes a Scientific Explanation Distinctively Mathematical?British Journal for the Philosophy of Science 64 (3): 485511.CrossRefGoogle Scholar
Lange, M. 2017. Because without Cause: Non-causal Explanations in Science and Mathematics. New York: Oxford University Press.Google Scholar
Langton, C. G. 1986. “Studying Artificial Life with Cellular Automata.” Physica D 22 (1–3): 120–49.Google Scholar
Machamer, P., Darden, L., and Craver, C. F. 2000. “Thinking about Mechanisms.” Philosophy of Science 67 (1): 125.CrossRefGoogle Scholar
Manukyan, L., Montandon, S. A., Fofonjka, A., Smirnov, S., and Milinkovitch, M. C. 2017. “A Living Mesoscopic Cellular Automaton Made of Skin Scales.” Nature 544 (7649): 173–79.CrossRefGoogle ScholarPubMed
Muller, F. A. 1997. “The Equivalence Myth of Quantum Mechanics.” Pts. 1 and 2. Studies in History and Philosophy of Science B 28 (1): 3561.; 28 (2): 219–47.CrossRefGoogle Scholar
Nakamasu, A., Takahashi, G., Kanbe, A., and Kondo, S. 2009. “Interactions between Zebrafish Pigment Cells Responsible for the Generation of Turing Patterns.” Proceedings of the National Academy of Sciences 106 (21): 8429–34.CrossRefGoogle ScholarPubMed
North, J. 2009. “The Structure of Physics: A Case Study.” Journal of Philosophy 106 (2): 5788.CrossRefGoogle Scholar
Parker, W. S. 2017. “Computer Simulation, Measurement, and Data Assimilation.” British Journal for the Philosophy of Science 68 (1): 273304.CrossRefGoogle Scholar
Pincock, C. 2004. “A Revealing Flaw in Colyvan’s Indispensability Argument.” Philosophy of Science 71 (1): 6179.CrossRefGoogle Scholar
Pincock, C. 2007. “A Role for Mathematics in the Physical Sciences.” Noûs 41 (2): 253–75.CrossRefGoogle Scholar
Schrödinger, E. 1926a. “Quantisierung als Eigenwertproblem.” Annalen der physik 385 (13): 437–90.CrossRefGoogle Scholar
Schrödinger, E. 1926b. “Über das Verhältnis der Heisenberg-Born-Jordanschen Quantenmechanik zu der Meinem.” Annalen der Physik 384 (8): 734–56.CrossRefGoogle Scholar
Stewart, J. 2008. Calculus: Early Transcendentals. 6th ed. Belmont, CA: Thomson Learning.Google Scholar
Toffoli, T. 1984. “Cellular Automata as an Alternative to (Rather than an Approximation of) Differential Equations in Modeling Physics.” Physica D 10 (1–2): 117–27.Google Scholar
Turing, A. M. 1952. “The Chemical Basis of Morphogenesis.” Bulletin of Mathematical Biology 52 (1–2): 153–97.Google Scholar
von Neumann, J. 1951. “The General and Logical Theory of Automata.” In Cerebral Mechanisms in Behavior: The Hixon Symposium, 141. New York: Wiley.Google Scholar
von Neumann, J. 1955. Mathematical Foundations of Quantum Mechanics. Rev. ed. Princeton, NJ: Princeton University Press.Google Scholar
Wolfram, S. 1984. “Cellular Automata as Models of Complexity.” Nature 311 (5985): 419–24.CrossRefGoogle Scholar