, Volume 194, Issue 12, pp 4839–4861 | Cite as

A match not made in heaven: on the applicability of mathematics in physics

  • Arezoo IslamiEmail author


In his seminal 1960 paper, the physicist Eugene Wigner formulated the question of the applicability of mathematics in physics in a way nobody had before. This formulation has been (almost) entirely overlooked due to an exclusive concern with (dis)solving Wigner’s problem and explaining the effectiveness of mathematics in the natural sciences, in one way or another. Many have attempted to attribute Wigner’s unjustified conclusion—that mathematics is unreasonably effective in the natural sciences—to his (dogmatic) formalist views on mathematics. My goal is to show that this reading misses out on Wigner’s highly original formulation of the problem which is presented throughout his body of work in physics as well as in philosophy. This formulation, as I will show, leads us in a new direction in solving the applicability problem.


Applicability of mathematics Wigner’s Puzzle Unreasonable effectiveness Laws of nature Invariance principles 



I thank Tom Donaldson, Krista Lawlor, Solomon Feferman, Michael Friedman, Mark Steiner, José Ferreirós and Jonathan Ettel for their insightful suggestions. My special thanks are to Thomas Ryckman for his enormously useful comments on several revisions of this paper.


  1. Bigelow, J., & Pargetter, R. (1991). Science and necessity. Cambridge: Cambridge University Press.CrossRefGoogle Scholar
  2. Burtt, E. A. (1923). The metaphysical foundations of modern physical science: A historical and critical essay. Trench, Trbner, London: London K. Paul, Trench, Trbner.Google Scholar
  3. Buzaglo, M. (2002). The logic of concept expansion. Cambridge: Cambridge University Press.Google Scholar
  4. Colyvan, M. (2001). The miracle of applied mathematics. Synthese, 127, 265–278.CrossRefGoogle Scholar
  5. Ferreirós, J. (forthcoming). Wigner’s ’unreasonable effectiveness’ in context. Mathematical Intelligencer.Google Scholar
  6. Grattan-Guinness, I. (2008). Solving wigner’s mystery: The reasonable (though perhaps limited) effectiveness of mathematics in the natural sciences. Mathematical Intelligencer, 30, 7–17.CrossRefGoogle Scholar
  7. Gross, D. J. (1995). Symmetry in physics: Wigner’s legacy. Physics Today, 48, 46–50.CrossRefGoogle Scholar
  8. Hamming, R. W. (1980). The unreasonable effectiveness of mathematics. American Mathematical Monthly, 87, 81–90.CrossRefGoogle Scholar
  9. Hardy, G. H. (1940). A mathematician’s apology. Cambridge: Cambridge University Press.Google Scholar
  10. Hawkins, T. (1974). The theory of matrices in 9th century. In Proceedings of the International Congress of Mathematicians Vancouver.Google Scholar
  11. Kahane, J.-P. (1991). Jacques hadamard. The Mathematical Intelligencer, 13, 23–29.CrossRefGoogle Scholar
  12. Longo, G., & Montévil, M. (2016). Comparing symmetries in models and simulations. In M. Dorato, L. Magnani, & T. Bertolotti (Eds.), Springer handbook of model-based science. Dordrecht: Springer.Google Scholar
  13. Longo, G., & Montévil, M. (2013). Extended criticality, phase spaces and enablement in biology. Chaos, Solition and Fractals, 55, 64–79.CrossRefGoogle Scholar
  14. Longo, G. (2005). The reasonable effectiveness of mathematics and its cognitive roots. In L. Boi (Ed.), Geometries of nature, living systems and human cognition (pp. 351–382). Singapore: World Scientific.CrossRefGoogle Scholar
  15. Lützen, J. (2011). The physical origin of physically useful mathematics. Interdisciplinary Science Reviews, 36, 229–243.CrossRefGoogle Scholar
  16. Mandic, D. P., & Lee Goh, V. S. (2009). Complex valued nonlinear adaptive filters: Noncircularity, widely linear and neural models. New York: Wiley.CrossRefGoogle Scholar
  17. Penrose, R. (1989). The emperor’s new mind: Concerning computers, minds and the laws of physics. New York: Oxford University Press.Google Scholar
  18. Pincock, C. (2014). Mathematics and scientific representation. New York: Oxford University Press.Google Scholar
  19. Roman, P. (2004). Why symmetry? some personal reflections. Symmetries in Science, 11, 1–12.Google Scholar
  20. Sarukkai, S. (2005). Revisiting the ’unreasonable effectiveness’ of mathematics. Current Science, 88, 415–423.Google Scholar
  21. Steiner, M. (1998). The applicability of mathematics as a philosophical problem. Cambridge, MA: Harvard University Press.Google Scholar
  22. Unger, R. M., & Smolin, L. (2014). The singular universe and the reality of time. Cambridge: Cambridge University Press.CrossRefGoogle Scholar
  23. Velupillai, K. V. (2005). The unreasonable ineffectiveness of mathematics in economics. Cambridge Journal of Economics, 29, 849–872.CrossRefGoogle Scholar
  24. Weinberg, S. (1992). Dreams of a final theory: The scientist’s search for the ultimate laws of nature. New York: Pantheon.Google Scholar
  25. Weyl, H. (1995). Topology and abstract algebra as two roads of mathematical comprehension. American Mathematical Monthly, 102, 453–460.CrossRefGoogle Scholar
  26. Wigner, E. (1949). Invariance in physical theory. Symmetries and Reflections, 93, 521–526.Google Scholar
  27. Wigner, E. (1960). The unreasonable effectiveness of mathematics in the natural sciences. Communications in Pure and Applied Mathematics, 13, 1–14.CrossRefGoogle Scholar
  28. Wigner, E. (1963). The role of invariance principles in natural philosophy. Symmetries and Reflections (pp. 28–37).Google Scholar
  29. Wigner, E. (1964a). Events, laws of nature, and invariance principles. Symmetries and Reflections (pp. 38–50).Google Scholar
  30. Wigner, E. (1964b). Symmetry and conservation laws. Symmetries and Reflections (pp. 14–27).Google Scholar
  31. Wigner, E. (1995). Symmetry in nature. In J.Mehra (Ed.), The Collected Works of Wigner (pp. 382–411). Berlin: Springer.Google Scholar
  32. Wilczek, F. (1999). Getting its from bits. Nature, 397, 303–306.CrossRefGoogle Scholar

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© Springer Science+Business Media Dordrecht 2016

Authors and Affiliations

  1. 1.Department of PhilosophyStanford UniversityStanfordUSA

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