American Journal of Applied Mathematics and Statistics
ISSN (Print): 2328-7306 ISSN (Online): 2328-7292 Website: Editor-in-chief: Mohamed Seddeek
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American Journal of Applied Mathematics and Statistics. 2017, 5(3), 95-100
DOI: 10.12691/ajams-5-3-2
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Studying the Winger’s “Enigma” about the Unreasonable Effectiveness of Mathematics in the Natural Sciences

Michael Gr. Voskoglou1,

1Mathematical Sciences, School of Technological Applications, Graduate Technological Educational Institute of Western Greece, Patras, Greece

Pub. Date: August 07, 2017

Cite this paper:
Michael Gr. Voskoglou. Studying the Winger’s “Enigma” about the Unreasonable Effectiveness of Mathematics in the Natural Sciences. American Journal of Applied Mathematics and Statistics. 2017; 5(3):95-100. doi: 10.12691/ajams-5-3-2


The effectiveness of mathematics in the natural sciences was characterized by the famous Nobel prize holder E. P. Winger as being unreasonable. It is not difficult for one to understand that this characterization is related to a question that has occupied the interest of philosophers, mathematicians and other scientists at least from the Plato’s era in ancient , until today: “Is mathematics discovered or invented by humans”? In the present work in an effort to obtain a convincing explanation of the above Winger’s “enigma”, the existing philosophical views about the above question are critically examined and discussed in connection with the advances in the history of mathematics that affected the human beliefs about them.

philosophy of mathematics platonism mathematical realism non euclidean geometries set theory continuum hypothesis axiom of choice incompleteness theorems canonical distribution metaphysics of quality

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[1]  Voskoglou, Μ.Gr., A Contribution to the Study of Rings, Ph. D. Thesis, Department of Mathematics, University of Patras, Greece (in Greek language), 1982.
[2]  Majid, S., “What is a Quantum Group?”, Notices of the American Math. Soc., 53, 30-31, 2006.
[3]  Lopez-Permouth, S., “Matrix Representations of Skew Polynomial Rings with Semisimple Coefficient Rings”, Contemporary Mathematics, 480, 289-295, 2009.
[4]  Winger, E.P. , “The unreasonable effectiveness of mathematics in the natural sciences”, Communications on Pure and Applied Mathematics, 13, 1-14, 1960) (Richard Courant Lecture, New York University, 11-5-1959).
[5]  Putnam, H., “What is Mathematical Truth?” Mathematics, Matter and Method, Cambridge University Press, Cambridge, 2nd Edition, pp.60-78, 1975.
[6]  Hamming, R., “The unreasonable effectiveness of mathematics”, American Mathematical Monthly, 87(2), 81-90, 1980.
[7]  Livio, M., Is God a Mathematician? Simon & Schuster, London, 2009.
[8]  Hardy, G.H., The Apology of a Mathematician, Cambridge University Press, Cambridge, UK, 1940.
[9]  Shapiro, S., Thinking about Mathematics, Oxford University Press, Oxford, 2000.
[10]  Tegmark, M., Our Mathematical Universe: My Quest to the Ultimate Reality, A. Knopf, New York, 2014.
[11]  Changeux, J.-P.& Connes, A. , Conversations on Mind, Matter and Mathematics, Princeton University Press, Princeton, 1995.
[12]  Muller, F.M., Immanuel Kant’s Critique of Pure Reason, Macmillan, London, 1881 (translation of the original Kant’s work, 1781).
[13]  Grassmann, G., Die Lineale Ausdehnungsiehre, Wiegand, Leipzig, English, 1844, translation by L. Kannenberg, 1885. Linear Extension: A New Branch of Mathematics, Open Court, Chicago.
[14]  Pesic, P., Beyond Geometry: Classic Papers from Riemann to Einstein, Dover Publications, Mineola, New York, 2007.
[15]  Gödel, C., The Consistency of the Axiom of Choice and of the Generalized Continuum Hypothesis with the Axioms of Set Theory, Princeton University Press, Princeton, USA, 1940.
[16]  Cohen, P.J., Set Theory and the Continuum Hypothesis, Benjamin, New York, 1966.
[17]  Franzen, T., Gödel’s Theorem: An Incomplete Guide to its Use and Abuse, A.K. Peters, Wellesley, Mass.., USA, 2005.
[18]  Kasner, E. & Newman, J.R., Mathematics and the Imagination, Tempus Books, Reelmond, Washington, 1989.
[19]  Lakoff, G. & Nunez, R. E., Where Mathematics Comes From, Basic Books, New York, 2000.
[20]  Atiyah, M.F., “Book review: Conversations on Mind, Matter and Mathematics ( J.-P. Changeux & A. Connes)”, Times Higher Education Supplement ,September, 29, 1995.
[21]  Wiles, A., “Modular Elliptic Curves and Fermat’s Last Theorem”, Annals of Mathematics, 142, 443-551, 1995.
[22]  Shing, S.L., Fermat’s Last Theorem, Fourth Estate, London, 1997.
[23]  Nicholson, J.S.. “A Perspective on Winger’s “Unreasonable Effectiveness of Mathematics’, Notices of the American Math. Soc., 59, 38-42, 2012.
[24]  Pirsig, R.M., Zen and the Art of Motorcycle Maintenance, William Morrow and Company, New York, 1974,.
[25]  Pirsig, R.M., Lila, Bantam Books, New York, 1991.
[26]  Davis, P.J. & Hersh, R., The Mathematical Experience, Birkhauser, Boston, 1981.
[27]  Russell, B. , The Problems of Philosophy, Home University Library, London, 1912 (retyped by Oxford University Press, Oxford, 1997).
[28]  Ma, Li, “Towards a Yin-Yang Balance in Mathematics Education”, Proceedings 4th Mediterranean Conference on Mathematics Education, Palermo, Italy, pp.685-689, 2005.