Advertisement

The Applicability of Mathematics

  • Jane McDonnell
Chapter
  • 309 Downloads

Abstract

Since its publication in 1960, Wigner’s paper ‘The Unreasonable Effectiveness of Mathematics in the Natural Sciences’ has attracted comment from scientists, applied mathematicians and philosophers keen to give their take on what Wigner calls “the miracle of the appropriateness of the language of mathematics for the formulation of the laws of physics” (Wigner 1960: 14). There is much disagreement concerning both the nature of the miracle and, more fundamentally, whether there is a miracle, or, indeed, anything mysterious at all. Theoretical physicists such as David Gross tend to agree that it is “something of a miracle that we are able to devise theories that allow us to make incredibly precise predictions regarding physical phenomena” (Gross 1988: 8372). On the other hand, applied mathematicians such as Jack Schwartz speak of “the pernicious influence of mathematics on science” (Schwartz 2006: 231) and emphasise how tough it can be to find mathematical solutions for real-world problems. Biologists, economists and social scientists are of a similar mind to Schwartz and tend to see mathematics as a sometimes useful tool. Needless to say, philosophers are divided in their opinion. Ernst Nagel thinks that:

Keywords

Cellular Automaton Physical Theory Mathematical Concept Human Mind Mathematical Object 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

References

  1. Aaronson, S. 2002. Book Review: On “A New Kind of Science” by Stephen Wolfram. Quantum Computation and Information 5: 410–423.Google Scholar
  2. Atiyah, M., et al. 1994. Responses to “Theoretical Mathematics: Toward a Cultural Synthesis of Mathematics and Theoretical Physics”. Bulletin of the American Mathematical Society 30: 178–211.CrossRefGoogle Scholar
  3. Azzouni, J. 2000. Applying Mathematics: An Attempt to Design a Philosophical Problem. Monist 83: 209–227.CrossRefGoogle Scholar
  4. Bangu, S. 2006a. Pythagorean Heuristic in Physics. Perspectives on Science 14: 387–416.CrossRefGoogle Scholar
  5. ———. 2006b. Steiner on the Applicability of Mathematics and Naturalism. Philosophia Mathematica 14: 26–43.CrossRefGoogle Scholar
  6. Barrow, J. 1990. The World Within the World. Oxford: Oxford University Press.Google Scholar
  7. ———. 1992. Perché il Mondo é Matematico. Roma-Bari: Laterza.Google Scholar
  8. ———. 2007. New Theories of Everything. Oxford: Oxford University Press.Google Scholar
  9. Begley, S. 1998. Science Finds God, Newsweek, July 20.Google Scholar
  10. Berry, M., Ellis, J., and Deutsch, D. 2002. Review: A Revolution or Self Indulgent Hype? How Top Scientists View Wolfram. London: The Daily Telegraph, May 15.Google Scholar
  11. Bueno, O., and M. Colyvan. 2011. An Inferential Conception of the Application of Mathematics. Noûs 45(2): 345–374.CrossRefGoogle Scholar
  12. Carnap, R. 1956. Empiricism, Semantics and Ontology. In Meaning and Necessity, 205–221. Chicago: University of Chicago Press.Google Scholar
  13. Chaitin, G. 2007. Review: How Mathematicians Think. New Scientist 195: 49.CrossRefGoogle Scholar
  14. Courant, R. 1964. Mathematics in the Modern World. Scientific American 211(3): 41–49.CrossRefGoogle Scholar
  15. Dirac, P. 1931. Quantized Singularities in the Electromagnetic Field. Proceedings of the Royal Society of London A 133: 60–72.CrossRefGoogle Scholar
  16. ———. 1939. The Relation Between Mathematics and Physics. Proceedings of the Royal Society (Edinburgh) 59: 122–129.CrossRefGoogle Scholar
  17. ———. 1970. Can Equations of Motion be Used in High-Energy Physics? Physics Today 23: 29–31.CrossRefGoogle Scholar
  18. ———. 1977. History of Twentieth Century Physics. In Proceedings of the International School of Physics “Enrico Fermi”, Course 57, New York and London: Academic Press.Google Scholar
  19. Dorato, M. 2005. The Laws of Nature and the Effectiveness of Mathematics. In The Role of Mathematics in Physical Sciences, edited by G. Boniolo et al., 131–144. Netherlands: Springer.Google Scholar
  20. Dummett, M. 1991. Frege: Philosophy of Mathematics. Cambridge, MA: Harvard University Press.Google Scholar
  21. Dyson, F. 1964. Mathematics in the Physical Sciences. Scientific American 211(3): 129–146.CrossRefGoogle Scholar
  22. ———. 1972. Missed Opportunities. Bulletin of the American Mathematical Society 78: 635–652.CrossRefGoogle Scholar
  23. ———. 1986. Paul A.M. Dirac. Obituary notice in American Philosophical Society Year Book, Philadelphia, American Physical Society, pp. 100–105.Google Scholar
  24. Einstein, A. 1954. On the Methods of Theoretical Physics. In Ideas and Opinions, edited by A. Einstein, 270–276. New York: Bonanza.Google Scholar
  25. Field, H. 1980. Science without Numbers: A Defense of Nominalism. Oxford: Blackwell.Google Scholar
  26. ———. 1989. Realism, Mathematics, and Modality. New York: Basil Blackwell.Google Scholar
  27. Frege, G. 1884. The Foundations of Arithmetic. Translated by J.L. Austin. Northwestern University Press, 1980.Google Scholar
  28. ———. 1962(1893/1903). Grundgesetze der Arithmetik, 2 vols., reprint in one vol. Hildesheim: Olms.Google Scholar
  29. Gabrielse, G. 2013. The Standard Model’s Greatest Triumph. Physics Today 66: 64–65.CrossRefGoogle Scholar
  30. Grattan-Guinness, I. 2008. Solving Wigner’s Mystery: The Reasonable (Though Perhaps Limited) Effectiveness of Mathematics in the Natural Sciences. The Mathematical Intelligencer 30: 7–17.CrossRefGoogle Scholar
  31. Gross, D. 1988. Physics and Mathematics at the Frontier. Proceedings of the National Academy of Sciences of the United States of America 85: 8371–8375.CrossRefGoogle Scholar
  32. Hale, B. 1987. Abstract Objects. Oxford: Blackwell.Google Scholar
  33. Hale, B., and C. Wright. 2001. The Reason’s Proper Study: Essays Towards a Neo-Fregean Philosophy of Mathematics. Oxford: Oxford University Press.CrossRefGoogle Scholar
  34. Hartle, J. 1993. The Quantum Mechanics of Closed Systems. In Directions in General Relativity, Volume 1: A Symposium and Collection of Essays in honor of Professor Charles W. Misner’s 60th Birthday, edited by B.-L. Hu, M.P. Ryan, and C.V. Vishveshwara. Cambridge: Cambridge University Press.Google Scholar
  35. Itzykson, I., and J. Zuber. 1985. Quantum Field Theory. New York: McGraw-Hill.Google Scholar
  36. Johnson, K. 2002. The Electromagnetic Field, downloaded from www-history.mcs.st-and.ac.uk/Projects/Johnson/Chapters/Ch4_4.htmlGoogle Scholar
  37. Lavers, C. 2002. How the Cheetah Got His Spots. London: The Guardian, August 3.Google Scholar
  38. Lepage, P. 1989. What Is Renormalization? In Proceedings of TASI’89: From Actions to Answers, edited by T. DeGrand and D. Toussaint. Singapore: World Scientific.Google Scholar
  39. Lesk, A. 2000. The Unreasonable Effectiveness of Mathematics in Molecular Biology. The Mathematical Intelligencer 22(2): 28–37.CrossRefGoogle Scholar
  40. Longo, G. 2005. The Reasonable Effectiveness of Mathematics and Its Cognitive Roots. In Geometries of Nature, Living Systems and Human Cognition series in “New Interactions of Mathematics with Natural Sciences and Humaties”, edited by L. Boi, 351–382. Singapore: World Scientific.Google Scholar
  41. McAllister, J. 1998. Is Beauty a Sign of Truth in Scientific Theories? American Scientist 86: 174–183.CrossRefGoogle Scholar
  42. ———. 2002. Recent Work on Aesthetics of Science. International Studies in the Philosophy of Science 16: 7–11.CrossRefGoogle Scholar
  43. Malament, D. 1982. Review of Field’s Science Without Numbers. Journal of Philosophy 79: 523–534.Google Scholar
  44. Mickens, R. 1990. Mathematics and Science. Singapore: World Scientific.CrossRefGoogle Scholar
  45. Nagel, E. 1979. Impossible Numbers. In Teleology Revisited, 166–194. New York: Columbia University Press.Google Scholar
  46. Priest, G. 2013. Mathematical Pluralism. Logic Journal of IGPL 21: 4–13.CrossRefGoogle Scholar
  47. Quine, W. 1976. Whither Physical Objects? In Essays in Memory of Imre Lakatos: Boston Studies in the Philosophy of Science 39; Synthese Library volume 99, edited by Robert S. Cohen and Marx W. Wartofsky, 497–504.Google Scholar
  48. Resnik, M. 1990. Between Mathematics and Physics. PSA: Proceedings of the Biennial Meeting of the Philosophy of Science Association, Vol. 1990, Volume Two: Symposia and Invited Papers (1990): 369–378.Google Scholar
  49. Rucker, R. 2003. Review: A New Kind of Science. American Mathematical Monthly 110: 851–861.CrossRefGoogle Scholar
  50. Ryckman, T. 2005. The Reign of Relativity: Philosophy in Physics 1915–1925. Oxford: Oxford University Press.CrossRefGoogle Scholar
  51. Schwartz, J. 2006. The Pernicious Influence of Mathematics on Science. In Unconventional Essays on the Nature of Mathematics, ed. Reuben Hersh, 231–235. New York: Springer.CrossRefGoogle Scholar
  52. Shapiro, S. 2000. Thinking about Mathematics: The Philosophy of Mathematics. Oxford: Oxford University Press.Google Scholar
  53. Steiner, M. 1989. The Application of Mathematics to Natural Science. The Journal of Philosophy 86: 449–480.CrossRefGoogle Scholar
  54. ———. 1995. The Applicabilities of Mathematics. Philosophia Mathematica 3: 129–156.CrossRefGoogle Scholar
  55. ———. 1998. The Applicability of Mathematics as a Philosophical Problem. Cambridge, MA: Harvard University Press.Google Scholar
  56. Stoltzner, M. 2005. Theoretical Mathematics: On the Philosophical Significance of the Jaffe-Quinn Debate. In The Role of Mathematics in Physical Sciences, edited by G. Boniolo et al., 197–222. Netherlands: Springer.Google Scholar
  57. Suppes, P. 1960. A Comparison of the Meaning and Uses of Models in Mathematics and the Empirical Sciences. Synthese 12(2–3): 287–301.CrossRefGoogle Scholar
  58. Tegmark, M. 2008. The Mathematical Universe. Foundations of Physics 38: 101–150.CrossRefGoogle Scholar
  59. Velupillai, K. 2005. The Unreasonable Ineffectiveness of Mathematics in Economics. Cambridge Journal of Economics 29: 849–872.CrossRefGoogle Scholar
  60. von Neumann, J. 1947. The Mathematician. In The Works of the Mind, edited by R.B. Heywood, 180–196. Chicago: University of Chicago Press.Google Scholar
  61. Wang, H. 1996. A Logical Journey: From Gödel to Philosophy. Cambridge, MA: MIT Press.Google Scholar
  62. Weinberg, S. 1994. Dreams of a Final Theory. New York: Random House.Google Scholar
  63. ———. 1997. What Is Quantum Field Theory, and What Did We Think It Is? In Proceedings of the Conference on “Historical and Philosophical Reflections on the Foundations of Quantum Field Theory”. Boston University, March 1996, arXiv:hep-th/9702027.Google Scholar
  64. ———. 2002. Is the Universe a Computer? New York Times Review of Books 49: 1–2.Google Scholar
  65. Wigner, E. 1960. The Unreasonable Effectiveness of Mathematics in the Natural Sciences. Communications of Pure and Applied Mathematics 13: 1–14.CrossRefGoogle Scholar
  66. Wilczek, F. 2006. Reasonably Effective: I Deconstructing a Miracle. Physics Today 59: 8–9.CrossRefGoogle Scholar
  67. Wilson, M. 2000. The Unreasonable Uncooperativeness of Mathematics in the Natural Sciences. Monist 83: 296–315.CrossRefGoogle Scholar
  68. Wolfram, S. 2002. A New Kind of Science. Champaign, IL: Wolfram Media.Google Scholar
  69. Zalta, E. 1983. Abstract Objects: An Introduction to Axiomatic Metaphysics. Dordrecht, Holland: Reidel.CrossRefGoogle Scholar
  70. ———. 2013. Gottlob Frege. In The Stanford Encyclopedia of Philosophy (Spring ed.), edited by Edward N. Zalta, downloaded from http://plato.stanford.edu/archives/spr2013/entries/frege/

Copyright information

© The Author(s) 2017

Authors and Affiliations

  • Jane McDonnell
    • 1
  1. 1.Philosophy Department ClaytonVictoriaAustralia

Personalised recommendations