Abstract
Wigner famously referred to the `unreasonable effectiveness' of mathematics in its application to science. Using Wigner's own application of group theory to nuclear physics, I hope to indicate that this effectiveness can be seen to be not so unreasonable if attention is paid to the various idealising moves undertaken. The overall framework for analysing this relationship between mathematics and physics is that of da Costa's partial structures programme.
Similar content being viewed by others
References
Bueno, O.: (forthcoming), 'structures, Applicability of Mathematics and Quasi-Truth'
Bueno, O.: 1997, 'Empirical Adequacy: A Partial Structures Approach', Studies in History and Philosophy of Science 28, 585–610.
Bueno, O., S. French, and J. Ladyman: forthcoming, 'On Representing the Relationship Between the Mathematical and the Empirical', preprint.
Coleman, A. J.: 1997, 'Groups and Physics – Dogmatic Opinions of a Senior Citizen', Notices of the American Mathematical Society 44, 8–17.
da Costa, N. C. A.: 1986, 'Pragmatic Probability', Erkenntnis 25, 141–162.
da Costa, N. C. A. and S. French: 1989, 'Pragmatic Truth and the Logic of Induction', British Journal for the Philosophy of Science 40, 333–356.
da Costa, N. C. A. and S. French: 1990, 'The Model-Theoretic Approach in the Philosophy of Science', Philosophy of Science 57, 248–265.
da Costa, N. C. A. and S. French: 1993, 'Towards an Acceptable Theory of Acceptance: Partial Structures and the General Correspondence Principle', in S. French and Kamminga (eds), Correspondence, Invariance and Heuristics: Essays in Honour of Heinz Post, D. Reidel, Dordrecht, pp. 137–158.
Donini, E.: 1987, 'Einstein and a Realistic Conception of Light-Matter Symmetry, 1905–1925', in M. A. Doncel, et al. (eds), pp. 105–112.
Doncel, M. A. et al.: 1987, 'Symmetries in Physics (1600–1980)', Proceedings of the 1st International Meeting of the History of Scientific Ideas, (Bellaterra: Barcelona).
Fermi, E.: 1926, 'Zur Quantelung des idealen einatomigen Gases', Zeitschrift für Physik 36, 902–912.
French, S.: 1997, 'Partiality, Pursuit and Practice', in M. L. Dalla Chiara et al. (eds), Proceedings of the 10th International Congress on Logic, Methodology and Philosophy of Science, Kluwer, Dordrecht, pp. 35–52.
French, S.: 1999, 'Models and Mathematics in Physics: The Role of Group Theory', in J. Butterfield and C. Pagonis (eds), From Physics to Philosophy, Cambridge University Press, Cambridge.
French, S. and J. Ladyman: 1998, 'A Semantic Perspective on Idealisation in Quantum Mechanics', in N. Shanks (ed.), Idealization IX: Idealization in Contemporary Physics: Poznán Studies in the Philosophy of the Sciences and the Humanities, Rodopi, pp. 51–73.
French, S. and J. Ladyman: 1999, 'Reinflating the Semantic Approach', International Studies in the Philosophy of Science 13, 103–121.
Gell-Mann, M.: 1987, 'Particle Theory from S-Matrix to Quarks', in M. A. Doncel, et al. (eds), pp. 473–497.
Giere, R.: 1988, Explaining Science: A Cognitive Approach, University of Chicago Press, Chicago.
Grattan-Guiness, I.: 1992, 'Structure-Similarity as a Cornerstone of the Philosophy of Mathematics', in J. Echeverria, A. Ibarra, and T. Mormann (eds), The Space of Mathematics, de Gruyter, Berlin, pp. 91–111.
Heisenberg, W.: 1932, 'Ñber den Bau der Atomkerne, I', Zeitschrift für Physik 77, 1–11.
Judd, B. R.: 1993, 'Applied Group Theory 1926–1935', in A. S. Wightman (ed.), The Collected Works of Eugene Paul Wigner, Part A, The Scientific Papers, Vol. 1, Springer-Verlag, Berlin, pp. 17–33.
Ladyman, J.: 1998, 'Structural Realism and the Semantic Approach to Theories', Studies in History and Philosophy of Science 29, 409–424.
Lloyd, E.: 1988, The Structure and Confirmation of Evolutionary Theory, Greenwood, Westport, CT.
Mackey, G. W.: 1993, 'The Mathematical Papers', in A. S. Wightman (ed.), The Collected Works of Eugene Paul Wigner, Part A, The Scientific Papers, Vol. 1, Springer-Verlag, Berlin, pp. 241–290.
Mikenberg, I., N. C. A. da Costa, and R. Chuaqui: 1986, 'Pragmatic Truth and Approximation to Truth', Journal of Symbolic Logic 51, 201–221.
Miller, A.: 1987, 'Symmetry and Imagery in the Physics of Bohr, Einstein and Heisenberg', in M. A. Doncel et al., pp. 299–325.
Newman, M. H. A.: 1957, 'HermannWeyl', Biographical Memoirs of Fellows of the Royal Society 3, 305–328.
Newton, T. D. and Wigner, E. P.: 1949, 'Localized States for Elementary Systems', Reviews of Modern Physics 21, 400–406.
Ne'eman, Y.: 1987, 'Hadron Symmetry, Classification and Compositeness', in M. A. Doncel et al., pp. 499–540.
Redhead, M. L. G.: 1975, 'Symmetry in Intertheory Relations', Synthese 32, 77–112.
Slater, J. C.: 1929, 'The Theory of Complex Spectra', Physical Review 34, 1293–1322.
Speiser, D.: 1987, 'The Principle of Relativity in Euler'sWork', in M. A. Doncel et al., pp. 31–47.
Suppes, P.: 1957, Introduction to Logic, Van Nostrand, New York.
Suppes, P.: 1967, 'Set-Theoretical Structures in Science', mimeograph, Stanford University.
von Meyenn, K.: 1987, 'Pauli's Belief in Exact Symmetries', in M. A. Doncel et al. (eds), pp. 331–58.
Weyl, H.: 1925, 'Theorie der Darstellung kontinuierlicher halb-einfacher Gruppen durch lineare Transformationen, I', Math. Zeitschrift 23, 271–309, II, ibid. 24, 328–76, (1926); III, ibid., 24, 377–95, (1926); rep. in H. Weyl (1968) Vol. II, pp. 543–647.
Weyl, H.: 1926, 'Quantenmechanik und Gruppentheorie', Zeitschrift für Physik 46, 1–46; also in H. Weyl (1968) Vol. III, pp. 90–135.
Weyl, H.: 1931, The Theory of Groups and Quantum Mechanics, Methuen and Co.; English trans. 2nd ed., 1928.
Weyl, H.: 1968, Gesammelte Abhandlungen, Springer-Verlag, Berlin.
Wigner, E. P.: 1926, 'Ñber nicht kombinierende Terme in der neueren Quantentheorie', Zeitschrift für Physik 40, 492–500.
Wigner, E. P.: 1927, 'Ñber nicht kombinierende Terme in der neueren Quantentheorie', Zeitschrift für Physik 40, 883–892.
Wigner, E. P.: 1959, Group Theory and Its Application to the Quantum Mechanics of Atomic Spectra, Academic Press; English trans., 1931.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
French, S. The Reasonable Effectiveness of Mathematics: Partial Structures and the Application of Group Theory to Physics. Synthese 125, 103–120 (2000). https://doi.org/10.1023/A:1005246608001
Issue Date:
DOI: https://doi.org/10.1023/A:1005246608001