The Unreasonable(?) Effectiveness of Mathematics in Science John Dougherty john.dougherty@lmu.de Munich Center for Mathematical Philosophy 20 July 2021 bit.ly/jdmcmpss John Dougherty (MCMP) Applicability 20 July 2021 1 / 23 Introduction Why are bubbles spherical? Brocken Inaglory. The image was edited by user:Alvesgaspar, CC BY-SA 3.0, via Wikimedia Commons John Dougherty (MCMP) Applicability 20 July 2021 2 / 23 Introduction Minimal surfaces Colin Bowring John Dougherty (MCMP) Applicability 20 July 2021 3 / 23 The Applicability Problem A first pass Observation #1 Mathematical facts can be used to predict and explain physical facts. Observation #2 Mathematical facts and physical facts are really different. The Applicability Problem How can mathematics be so effective in predicting and explaining physical phenomena? John Dougherty (MCMP) Applicability 20 July 2021 4 / 23 The Applicability Problem Differences Concrete/Abstract Experiment/Proof Necessary/Contingent Certain/Provisional Human-centric/human-independent John Dougherty (MCMP) Applicability 20 July 2021 5 / 23 The Applicability Problem Applicability Problems The Metaphysical Applicability Problem How can abstract math facts influence concrete physical facts? The Epistemic Applicability Problem How can a priori investigation of math facts tell us about the physical world? The Anthropocentric Applicability Problem How can math (a cultural artifact) be so useful in physics? John Dougherty (MCMP) Applicability 20 July 2021 6 / 23 The Applicability Problem Applicability Conflicts The Metaphysical Applicability Problem Mathematical objects have physical effects, which might be spooky. The Epistemic Applicability Problem We might have knowledge that goes beyond observation. The Anthropocentric Applicability Problem We might have a special place in the physical universe. John Dougherty (MCMP) Applicability 20 July 2021 7 / 23 The Mapping Account Analogy Paradigm Mathematics Describes the mathematical part of the world. Physics Describes the physical part of the world. Application Application involves drawing analogies between the mathy and physicsy parts. John Dougherty (MCMP) Applicability 20 July 2021 8 / 23 The Mapping Account Two big problems Quality Analogical reasoning isn’t especially good, but mathematical reasoning about physics is. (Right?) Nature of mathematics Need to say something about mathematics to study analogy. John Dougherty (MCMP) Applicability 20 July 2021 9 / 23 The Mapping Account Mapping account Mathematics Describes the part of the world consisting of structured sets. Physics Describes the physical part of the world. Application Application involves mapping parts of mathematical objects onto parts of physical objects. John Dougherty (MCMP) Applicability 20 July 2021 10 / 23 Alternatives Problems with the mapping account Unreasonableness Doesn’t explain why math is so effective in physics. Narrow Closely tied to features of structured sets. Inconsistency Inconsistent mathematics gets applied—and well! John Dougherty (MCMP) Applicability 20 July 2021 11 / 23 Alternatives Linguistic Paradigm Mathematics A language used in specialized discourses. Physics Describes the physical part of the world. Application Application of math to physics is like application of language to physics. John Dougherty (MCMP) Applicability 20 July 2021 12 / 23 Alternatives A basic mistake? Objection Mathematical language is used to talk about mathematics, and it’s mathematics that gets applied. (Beginning of) Response Careful not to assume the analogy view! What is mathematical language used for? John Dougherty (MCMP) Applicability 20 July 2021 13 / 23 Alternatives Quantum field theory (QFT) “Inconsistent” mathematics QFT is full of infinite quantities, even when you’re careful. Math without structured sets Some of the mathematical objects encountered in QFT do not admit (faithful) representation as structured sets. John Dougherty (MCMP) Applicability 20 July 2021 14 / 23 Conclusion Summary 1 The intuition that there’s something funny about the applicability of mathematics to the physical world can be distinguished into many applicability problems. 2 The most common framework for approaching this problem is the analogy paradigm in general, and the mapping account in particular. 3 There are problems with the mapping account and alternatives to it, but there’s work to be done! John Dougherty (MCMP) Applicability 20 July 2021 15 / 23 Conclusion Bibliography I Azzouni, J. (2000). Applying mathematics: An attempt to design a philosophical problem. The Monist, 83(2):209–227. Baker, A. (2005). Are there genuine mathematical explanations of physical phenomena? Mind, 114(454):223–238. Bangu, S. (2006). Steiner on the applicability of mathematics and naturalism. Philosophia Mathematica, 14(1):26–43. Bangu, S. (2012). The applicability of mathematics in science: Indispensability and ontology. Palgrave Macmillan Basingstoke. 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. Bueno, O. and French, S. (2018). Applying mathematics: Immersion, inference, interpretation. Oxford University Press. John Dougherty (MCMP) Applicability 20 July 2021 16 / 23 Conclusion Bibliography II Colyvan, M. (2001). The miracle of applied mathematics. Synthese, 127(3):265–278. Colyvan, M. (2014). The undeniable effectiveness of mathematics in the special sciences. In New directions in the philosophy of science, pages 63–73. Springer. Dirac, P. A. M. (1938). The relation between mathematics and physics. Proceedings of the Royal Society of Edinburgh, 59(Pt II):122. Dougherty, J. (2021). I ain’t afraid of no ghost. Studies in History and Philosophy of Science, 88:70–84. Dutilh Novaes, C. (2020). The dialogical roots of deduction. Cambridge University Press. Dyson, F. J. (1964). Mathematics in the physical sciences. Scientific American, 211(3):128–147. John Dougherty (MCMP) Applicability 20 July 2021 17 / 23 Conclusion Bibliography III Ferreirós, J. (2017). Wigner’s “unreasonable effectiveness” in context. The Mathematical Intelligencer, 39(2):64–71. Field, H. (2016). Science without numbers. Oxford University Press. Franklin, J. (1989). Mathematical necessity and reality. Australasian Journal of Philosophy, 67(3):286–294. Fraser, D. (2009). Quantum field theory: Underdetermination, inconsistency, and idealization. Philosophy of Science, 76(4):536–567. Fraser, D. (2011). How to take particle physics seriously: a further defence of axiomatic quantum field theory. Studies in History and Philosophy of Modern Physics, 42(2):126–135. Fraser, J. D. (2018). Renormalization and the formulation of scientific realism. Philosophy of Science, 85(5):1164–1175. John Dougherty (MCMP) Applicability 20 July 2021 18 / 23 Conclusion Bibliography IV Fraser, J. D. (2020a). The real problem with perturbative quantum field theory. The British Journal for the Philosophy of Science, 71(2):391–413. Fraser, J. D. (2020b). Towards a realist view of quantum field theory. In French and Saatsi (2020), pages 276–292. French, S. (2000). The reasonable effectiveness of mathematics: Partial structures and the application of group theory to physics. Synthese, 125(1):103–120. French, S. and Saatsi, J., editors (2020). Scientific Realism and the Quantum. Oxford University Press. Hamming, R. (1980). The unreasonable effectiveness of mathematics. The American Mathematical Monthly, 87(2):81–90. Islami, A. (2017). A match not made in heaven: On the applicability of mathematics in physics. Synthese, 194(12):4839–4861. John Dougherty (MCMP) Applicability 20 July 2021 19 / 23 Conclusion Bibliography V Islami, A. and Wiltsche, H. A. (2020). A match made on earth: On the applicability of mathematics in physics. In Phenomenological Approaches to Physics, pages 157–177. Springer. Lyon, A. and Colyvan, M. (2008). The explanatory power of phase spaces. Philosophia mathematica, 16(2):227–243. Maddy, P. (1992). Indispensability and practice. The Journal of Philosophy, 89(6):275–289. Morrison, M. (2015). Reconstructing reality: Models, mathematics, and simulations. Oxford Studies in Philosophy o. Pincock, C. (2011). Mathematics and scientific representation. Oxford University Press. Price, H. (2004). Naturalism without representationalism. Naturalism in question, pages 71–88. John Dougherty (MCMP) Applicability 20 July 2021 20 / 23 Conclusion Bibliography VI Rédei, M. (2020). On the tension between physics and mathematics. Journal for General Philosophy of Science, pages 1–15. Ruetsche, L. (2018). Renormalization group realism: The ascent of pessimism. Philosophy of Science, 85(5):1176–1189. Ruetsche, L. (2020). Perturbing realism. In French and Saatsi (2020), pages 293–314. Saatsi, J. (2011). The enhanced indispensability argument: Representational versus explanatory role of mathematics in science. The British Journal for the Philosophy of Science, 62(1):143–154. Saatsi, J. (2016). On the ‘indispensable explanatory role’ of mathematics. Mind, 125(500):1045–1070. Shabel, L. (2007). Apriority and application: Philosophy of mathematics in the moden period. In Shapiro (2007), pages 29–49. John Dougherty (MCMP) Applicability 20 July 2021 21 / 23 Conclusion Bibliography VII Shapiro, S., editor (2007). The Oxford Handbook of Philosophy of Mathematics and Logic. Oxford University Press. Simons, P. (2001). The applicability of mathematics as a philosophical problem. The British Journal for the Philosophy of Science, 52(1):181–184. Sklar, L. (1998). The language of nature is mathematics: But which mathematics? and what nature? In Proceedings of the Aristotelian Society, pages 241–261. JSTOR. Steiner, M. (1998). The applicability of mathematics as a philosophical problem. Harvard University Press. Stemeroff, N. (2021). Structuralism and the conformity of mathematics and nature. Studies in History and Philosophy of Science, 86:84–92. Stöltzner, M. (2004). On optimism and opportunism in applied mathematics. Erkenntnis, 60(1):121–145. John Dougherty (MCMP) Applicability 20 July 2021 22 / 23 Conclusion Bibliography VIII Thomasson, A. L. (2020). Norms and necessity. Oxford University Press, USA. Weinberg, S. (1986). Lecture on the applicability of mathematics. Notices of the American Mathematical Society, 33:725–733. Wigner, E. P. (1990). The unreasonable effectiveness of mathematics in the natural sciences. In Mathematics and Science, pages 291–306. World Scientific. Williams, P. (2019). Scientific realism made effective. The British Journal for the Philosophy of Science, 70(1):209–237. Wilson, M. (2000a). On the mathematics of spilt milk. In The Growth of Mathematical Knowledge, pages 143–152. Springer. Wilson, M. (2000b). The unreasonable uncooperativeness of mathematics in the natural sciences. The Monist, 83(2):296–314. John Dougherty (MCMP) Applicability 20 July 2021 23 / 23
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