Abstract
Mathematics has a great variety ofapplications in the physical sciences.This simple, undeniable fact, however,gives rise to an interestingphilosophical problem:why should physical scientistsfind that they are unable to evenstate their theories without theresources of abstract mathematicaltheories? Moreover, theformulation of physical theories inthe language of mathematicsoften leads to new physical predictionswhich were quite unexpected onpurely physical grounds. It is thought by somethat the puzzles the applications of mathematicspresent are artefacts of out-dated philosophical theories about thenature of mathematics. In this paper I argue that this is not so.I outline two contemporary philosophical accounts of mathematics thatpay a great deal of attention to the applicability of mathematics and showthat even these leave a large part of the puzzles in question unexplained.
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REFERENCES
Baker, A. R.: (forthcoming) ‘Mathematics Indispensability and Scientific Progress’, Erkenntnis.
Balaguer, M.: 1998, Platonism and Anti-Platonism in Mathematics, New York, Oxford University Press.
Bigelow, J.: 1988, The Reality of Numbers: A Physicalist's Philosophy of Mathematics, Oxford, Clarendon.
Burgess, J. P.: 1983, ‘Why I Am Not a Nominalist’, Notre Dame Journal of Formal Logic 24, 93-105.
Chalmers, A. F.: 1973, ‘Maxwell's Methodology and His Application of it to Electromagnetism’, Studies in History and Philosophy of Science 4, 107-164.
Cheyne, C. and C. R. Pigden: 1996, ‘Pythagorean Powers or a Challenge to Platonism’, Australasian Journal of Philosophy 74, 639-645.
Colyvan, M.: 1998, ‘Is Platonism a Bad Bet?’, Australasian Journal of Philosophy 76, 115-119.
Colyvan, M.: 1999, ‘Confirmation Theory and Indispensability’, Philosophical Studies 96, 1-19.
Colyvan, M.: 2000, ‘Review of Steiner's, The Applicability of Mathematics as a Philosophical Problem,’ Mind 109, 390-394.
Colyvan, M.: 2001, The Indispensability of Mathematics, New York, Oxford University Press.
Davies, P.: 1992, The Mind of God, London, Penguin.
Dyson, F. J.: 1964, ‘Mathematics in the Physical Sciences’, Scientific American 211, 128-146.
Feynman, R. P.: 1965, The Character of Physical Law, London, BBC.
Field, H. H.: 1980, Science Without Numbers: A Defence of Nominalism, Oxford, Blackwell.
Field, H. H.: 1998, ‘Mathematical Objectivity and Mathematical Objects’, in S. Laurence and C. Macdonald (eds), Contemporary Readings in the Foundations of Metaphysics, Oxford, Blackwell, pp. 387-403.
Hamming, R. W.: 1980, ‘The Unreasonable Effectiveness of Mathematics’, American Mathematics Monthly 87, 81-90.
Hellman, G.: 1992, ‘The Boxer and His Fists: The Constructivist in the Arena of Quantum Physics’, Proceedings of the Aristotelian Society, Supplement LXVI, 61-77.
Hersh, R.: 1990, ‘Inner Vision Outer Truth’, in R. E. Mickens (ed.), Mathematics and Science, Singapore, World Scientific Press, pp. 64-72.
Hunt, B. J.: 1991, The Maxwellians, Ithaca, NY, Cornell University Press.
Kitcher, P.: 1984, The Nature of Mathematical Knowledge, New York, Oxford University Press.
Mac Lane, S.: 1990, ‘The Reasonable Effectiveness of Mathematics’, in R. E. Mickens (ed.), Mathematics and Science, Singapore, World Scientific Press, pp. 115-135.
Maddy, P.: 1990, Realism in Mathematics, Oxford, Clarendon.
Maddy, P.: 1992, ‘Indispensability and Practice’, Journal of Philosophy 89, 275-289.
Malament, D.: 1982, ‘Review of Field's Science Without Numbers’, Journal of Philosophy 79, 523-534.
Mickens, R. E. (ed.): 1990, Mathematics and Science, Singapore, World Scientific Press.
Penrose, R.: 1990, The Emperor's New Mind: Concerning Computers, Minds and the Laws of Physics, London, Vintage.
Putnam, H.: 1979, ‘Philosophy of Logic’, reprinted in Mathematics Matter and Method: Philosophical Papers Vol. I, 2nd edition, Cambridge, Cambridge University Press, pp. 323-357.
Quine, W. V.: 1980, ‘On What There Is’, reprinted in From a Logical Point of View, 2nd edition, Cambridge, MA, Harvard University Press, pp. 1-19.
Resnik, M. D.: 1985, ‘How Nominalist is Hartry Field's Nominalism?’, Philosophical Studies 47, 163-181.
Shapiro, S.: 1997, Philosophy of Mathematics: Structure and Ontology, New York, Oxford University Press.
Siegel, D. M.: 1991, Innovation in Maxwell's Electromagnetic Theory, Cambridge, Cambridge University Press.
Smart, J. J. C.: 1963, Philosophy and Scientific Realism, New York, Routledge and Kegan Paul.
Steiner, M.: 1989, ‘The Application of Mathematics to Natural Science’, Journal of Philosophy 86, 449-480.
Steiner, M.: 1995, ‘The Applicabilities of Mathematics’, Philosophia Mathematica (3) 3, 129-156.
Steiner, M.: 1998, The Applicability of Mathematics as a Philosophical Problem, Cambridge, MA, Harvard University Press.
Weinberg, S.: 1986, ‘Lecture on the Applicability of Mathematics’, Notices of the American Mathematical Society 33, 725-728.
Weinberg, S.: 1993, Dreams of a Final Theory, London, Vintage.
Wigner, E. P.: 1960, ‘The Unreasonable Effectiveness of Mathematics in the Natural Sciences’, Communications on Pure and Applied Mathematics 13, 1-14.
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Colyvan, M. The Miracle of Applied Mathematics. Synthese 127, 265–278 (2001). https://doi.org/10.1023/A:1010309227321
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DOI: https://doi.org/10.1023/A:1010309227321