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The Applicability of Mathematics

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The Pythagorean World

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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:

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Notes

  1. 1.

    See, for example, Lesk (2000) and Velupillai (2005).

  2. 2.

    It is not clear whether or not quantum mechanics can be explained in nominalistic terms as required by Field: Malament (1982) suggests not, but Balaguer (1996a) makes an attempt.

  3. 3.

    The fact that it is renormalisable merely means that the coupling constants are calculable from theory rather than measured experimentally.

  4. 4.

    See, for example, Grattan-Guinness (2008), Azzouni (2000) and Wilczek (2006).

  5. 5.

    For example, traditional mathematical tools could only scratch the surface of fluid dynamics problems involving turbulence or systems far from equilibrium (common features in the real world), so new tools were invented to explore chaotic and non-equilibrium systems.

  6. 6.

    Of particular interest is the rule 110 cellular automaton which emulates a universal Turing machine and so can do any computation that can be done by any computer.

  7. 7.

    See, for example, Aaronson (2002), Barrow (2007), Berry et al. (2002), Lavers (2002), Rucker (2003), Weinberg (2002).

  8. 8.

    See, for example, Shapiro (2000). If all else fails, the applicability of mathematics can be taken as a brute fact.

  9. 9.

    See, for example, Dummett (1991) and Steiner (1995: 132–138).

  10. 10.

    See, for example, Hale and Wright (2001).

  11. 11.

    At the moment, only a subset of mathematics is applied, but the true extent of the applicability of mathematics is not known, so nothing can be inferred from this.

  12. 12.

    For example, singularities in General Relativity leading to the study of black holes; infinities in black-body radiation leading to the postulation of quanta; singularities in quantum field theory leading to the development of renormalisation group methods.

  13. 13.

    I think that, already, the Standard Model of Particle Physics and General Relativity taken together are close to a TOE in the sense that we haven’t been able to break them yet and we have been trying with incredibly sophisticated experiments at very high energies for many years. Of course, as matters stand, we know that they cannot be unified to form a candidate TOE because of inconsistencies which arise at the Planck scale. Also, they require up to 37 free parameters to be put in by hand to account for phenomena which elude our conceptual understanding (e.g. the origins of mass and dark energy). That is why there is so much interest in String Theory, because it is a consistent quantum field theory which naturally unifies gravity and gauge theories and doesn’t require extrinsic parameters. Despite recent travails in the development of String Theory and the resort of some physicists to a multiverse explanation, I still count it as a candidate for TOE.

  14. 14.

    To use Barrow’s terminology, see Sect. 2.3.5.

  15. 15.

    See, for example, Resnik (1990), Hale (1987), Zalta (1983).

References

  • Aaronson, S. 2002. Book Review: On “A New Kind of Science” by Stephen Wolfram. Quantum Computation and Information 5: 410–423.

    Google Scholar 

  • 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.

    Article  Google Scholar 

  • Azzouni, J. 2000. Applying Mathematics: An Attempt to Design a Philosophical Problem. Monist 83: 209–227.

    Article  Google Scholar 

  • Bangu, S. 2006a. Pythagorean Heuristic in Physics. Perspectives on Science 14: 387–416.

    Article  Google Scholar 

  • ———. 2006b. Steiner on the Applicability of Mathematics and Naturalism. Philosophia Mathematica 14: 26–43.

    Article  Google Scholar 

  • Barrow, J. 1990. The World Within the World. Oxford: Oxford University Press.

    Google Scholar 

  • ———. 1992. Perché il Mondo é Matematico. Roma-Bari: Laterza.

    Google Scholar 

  • ———. 2007. New Theories of Everything. Oxford: Oxford University Press.

    Google Scholar 

  • Begley, S. 1998. Science Finds God, Newsweek, July 20.

    Google Scholar 

  • 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 

  • Bueno, O., and M. Colyvan. 2011. An Inferential Conception of the Application of Mathematics. Noûs 45(2): 345–374.

    Article  Google Scholar 

  • Carnap, R. 1956. Empiricism, Semantics and Ontology. In Meaning and Necessity, 205–221. Chicago: University of Chicago Press.

    Google Scholar 

  • Chaitin, G. 2007. Review: How Mathematicians Think. New Scientist 195: 49.

    Article  Google Scholar 

  • Courant, R. 1964. Mathematics in the Modern World. Scientific American 211(3): 41–49.

    Article  Google Scholar 

  • Dirac, P. 1931. Quantized Singularities in the Electromagnetic Field. Proceedings of the Royal Society of London A 133: 60–72.

    Article  Google Scholar 

  • ———. 1939. The Relation Between Mathematics and Physics. Proceedings of the Royal Society (Edinburgh) 59: 122–129.

    Article  Google Scholar 

  • ———. 1970. Can Equations of Motion be Used in High-Energy Physics? Physics Today 23: 29–31.

    Article  Google Scholar 

  • ———. 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 

  • 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 

  • Dummett, M. 1991. Frege: Philosophy of Mathematics. Cambridge, MA: Harvard University Press.

    Google Scholar 

  • Dyson, F. 1964. Mathematics in the Physical Sciences. Scientific American 211(3): 129–146.

    Article  Google Scholar 

  • ———. 1972. Missed Opportunities. Bulletin of the American Mathematical Society 78: 635–652.

    Article  Google Scholar 

  • ———. 1986. Paul A.M. Dirac. Obituary notice in American Philosophical Society Year Book, Philadelphia, American Physical Society, pp. 100–105.

    Google Scholar 

  • Einstein, A. 1954. On the Methods of Theoretical Physics. In Ideas and Opinions, edited by A. Einstein, 270–276. New York: Bonanza.

    Google Scholar 

  • Field, H. 1980. Science without Numbers: A Defense of Nominalism. Oxford: Blackwell.

    Google Scholar 

  • ———. 1989. Realism, Mathematics, and Modality. New York: Basil Blackwell.

    Google Scholar 

  • Frege, G. 1884. The Foundations of Arithmetic. Translated by J.L. Austin. Northwestern University Press, 1980.

    Google Scholar 

  • ———. 1962(1893/1903). Grundgesetze der Arithmetik, 2 vols., reprint in one vol. Hildesheim: Olms.

    Google Scholar 

  • Gabrielse, G. 2013. The Standard Model’s Greatest Triumph. Physics Today 66: 64–65.

    Article  Google Scholar 

  • 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.

    Article  Google Scholar 

  • 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.

    Article  Google Scholar 

  • Hale, B. 1987. Abstract Objects. Oxford: Blackwell.

    Google Scholar 

  • Hale, B., and C. Wright. 2001. The Reason’s Proper Study: Essays Towards a Neo-Fregean Philosophy of Mathematics. Oxford: Oxford University Press.

    Book  Google Scholar 

  • 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 

  • Itzykson, I., and J. Zuber. 1985. Quantum Field Theory. New York: McGraw-Hill.

    Google Scholar 

  • Johnson, K. 2002. The Electromagnetic Field, downloaded from www-history.mcs.st-and.ac.uk/Projects/Johnson/Chapters/Ch4_4.html

    Google Scholar 

  • Lavers, C. 2002. How the Cheetah Got His Spots. London: The Guardian, August 3.

    Google Scholar 

  • 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 

  • Lesk, A. 2000. The Unreasonable Effectiveness of Mathematics in Molecular Biology. The Mathematical Intelligencer 22(2): 28–37.

    Article  Google Scholar 

  • 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 

  • McAllister, J. 1998. Is Beauty a Sign of Truth in Scientific Theories? American Scientist 86: 174–183.

    Article  Google Scholar 

  • ———. 2002. Recent Work on Aesthetics of Science. International Studies in the Philosophy of Science 16: 7–11.

    Article  Google Scholar 

  • Malament, D. 1982. Review of Field’s Science Without Numbers. Journal of Philosophy 79: 523–534.

    Google Scholar 

  • Mickens, R. 1990. Mathematics and Science. Singapore: World Scientific.

    Book  Google Scholar 

  • Nagel, E. 1979. Impossible Numbers. In Teleology Revisited, 166–194. New York: Columbia University Press.

    Google Scholar 

  • Priest, G. 2013. Mathematical Pluralism. Logic Journal of IGPL 21: 4–13.

    Article  Google Scholar 

  • 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 

  • 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 

  • Rucker, R. 2003. Review: A New Kind of Science. American Mathematical Monthly 110: 851–861.

    Article  Google Scholar 

  • Ryckman, T. 2005. The Reign of Relativity: Philosophy in Physics 1915–1925. Oxford: Oxford University Press.

    Book  Google Scholar 

  • 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.

    Chapter  Google Scholar 

  • Shapiro, S. 2000. Thinking about Mathematics: The Philosophy of Mathematics. Oxford: Oxford University Press.

    Google Scholar 

  • Steiner, M. 1989. The Application of Mathematics to Natural Science. The Journal of Philosophy 86: 449–480.

    Article  Google Scholar 

  • ———. 1995. The Applicabilities of Mathematics. Philosophia Mathematica 3: 129–156.

    Article  Google Scholar 

  • ———. 1998. The Applicability of Mathematics as a Philosophical Problem. Cambridge, MA: Harvard University Press.

    Google Scholar 

  • 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 

  • Suppes, P. 1960. A Comparison of the Meaning and Uses of Models in Mathematics and the Empirical Sciences. Synthese 12(2–3): 287–301.

    Article  Google Scholar 

  • Tegmark, M. 2008. The Mathematical Universe. Foundations of Physics 38: 101–150.

    Article  Google Scholar 

  • Velupillai, K. 2005. The Unreasonable Ineffectiveness of Mathematics in Economics. Cambridge Journal of Economics 29: 849–872.

    Article  Google Scholar 

  • 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 

  • Wang, H. 1996. A Logical Journey: From Gödel to Philosophy. Cambridge, MA: MIT Press.

    Google Scholar 

  • Weinberg, S. 1994. Dreams of a Final Theory. New York: Random House.

    Google Scholar 

  • ———. 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 

  • ———. 2002. Is the Universe a Computer? New York Times Review of Books 49: 1–2.

    Google Scholar 

  • Wigner, E. 1960. The Unreasonable Effectiveness of Mathematics in the Natural Sciences. Communications of Pure and Applied Mathematics 13: 1–14.

    Article  Google Scholar 

  • Wilczek, F. 2006. Reasonably Effective: I Deconstructing a Miracle. Physics Today 59: 8–9.

    Article  Google Scholar 

  • Wilson, M. 2000. The Unreasonable Uncooperativeness of Mathematics in the Natural Sciences. Monist 83: 296–315.

    Article  Google Scholar 

  • Wolfram, S. 2002. A New Kind of Science. Champaign, IL: Wolfram Media.

    Google Scholar 

  • Zalta, E. 1983. Abstract Objects: An Introduction to Axiomatic Metaphysics. Dordrecht, Holland: Reidel.

    Book  Google Scholar 

  • ———. 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/

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  1. Philosophy Department Clayton, Victoria, Australia

    Jane McDonnell

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McDonnell, J. (2017). The Applicability of Mathematics. In: The Pythagorean World. Palgrave Macmillan, Cham. https://doi.org/10.1007/978-3-319-40976-4_2

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