When Wigner claimed that the effectiveness of mathematics in the natural sciences was unreasonable it was due to a dogmatic formalist view of mathematics according to which higher mathematics is developed solely with a view to formal beauty. I shall argue that this philosophy is not in agreement with the actual practice of mathematics. Indeed, I shall briefly illustrate how physics has influenced the development of mathematics from antiquity up to the twentieth century. If this influence is taken into account, the effectiveness of mathematics is far more reasonable.
RobertoBonola. 1912. Non-Euclidean Geometry. Chicago: Open Court Publishing Co.
2.
LeoCorry. 1997. David Hilbert and the Axiomatization of Physics (1894–1905). Archive for History of Exact Sciences51: 89–197.
3.
LeoCorry. 2004. Hilbert and the Axiomatization of Physics (1898–1918): From ‘Grundlagen der Geometrie’ to ‘Grundlagen der Physik’. Dordrecht: Kluwer.
4.
MoritzEpple. 2003. The End of the Science of Quantity: Foundations of Analysis. In History of Analysis. Ch.10. Ed. , Hans NielsJahnke, ed. Providence, RI: American Mathematical Society and London Mathematical Society.
5.
ChristianGilain. 1991. Sur l'histoire du théorème fondamental de l'algèbre: théorie des équations et calcul intégral. Archive for History of Exact Sciences42(2): 91–136.
6.
KurtGödel. 1931, Uber formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme, I. Monatshefte für Mathematik und Physik38: 173–98.
7.
Grattan-Guinness, Ivor. 2008. Solving Wigner's mystery: The reasonable (though perhaps limited) effectiveness of mathematics in the natural sciences. Mathematical Intelligencer30(3): 7–17.
8.
JeremyGray. 1989. Ideas of Space: Euclidean, non-Euclidean, relativistic.New York: Oxford University Press.
9.
ThomasHankinsL. 1980. Sir William Rowan Hamilton. Baltimore, RI: The Johns Hopkins University Press.
10.
DavidHilbert. 1899. Grundlagen der Geometrie. Leipzig: Teubner.
11.
DavidHilbert. 1900. Mathematische Probleme. Nachrichten der Königlichen Gesellschaft der Wissenschaften zu Göttingen, mathematisch-physikalische Klasse3: 253–297.
12.
VictorKatzJ. 2009. A History of Mathematics. An Introduction. Boston: Addison-Wesley.
13.
LakatosImre. 1976. Proofs and Refutations. Cambridge: Cambridge University Press.
14.
LützenJesper. 1990. Joseph Liouville1809–1882: Master of Pure and Applied Mathematics. New York: Springer.
15.
LützenJesper. 1995. Interactions between mechanics and differential geometry in the nineteenth century.Archive for History of Exact Sciences49: 1–72.
16.
LützenJesper. 2003a. The Foundation of Analysis in the 19th century. In History of Analysis. Ed. , Hans NielsJahnke, ed. Providence, RI: American Mathematical Society and London Mathematical Society, 155–195.
17.
LützenJesper. 2003b. Between Rigor and Applications. Developments in the Concept of Function in Mathematical Analysis. In The Cambridge History of Science 5: The Modern Phys. and Math. Sciences, ed. , Marry JoNye, ed. Cambridge: Cambridge University Press, 468–487.
18.
LützenJesper. 2005. Mechanistic Images in Geometric Form: Heinrich Hertz’ Principles of Mechanics. Oxford: Oxford University Press.
19.
LützenJesper. 2007. The Mathematization of the Physical Sciences — Differential Equations of Nature. In History of Mathematics, in Encyclopedia of Life Support Systems (EOLSS), ed. , JeremyGray, Vagn LundsgaardHansen, ed. Developed under the Auspices of the UNESCO, http://www.eolss.net, Eolss Publishers, Oxford, UK. (22/4/11.)
20.
LützenJesper. 2011. The interaction of physics, mechanics and mathematics in Joseph Liouville's research. To appear. In The Dialectic Relation between Physics and Mathematics in the XIXth century, ed. , EvelyneBarbin, RaffaellePisano, ed. Springer.
21.
ShapiroStewart. 2000. Thinking about Mathematics. Oxford University Press, New York.
22.
ImreToth. 1969. Non-Euclidean Geometry before Euclid. Scientific American. November 1969: 87–101.
23.
EugeneWigner. 1960. The unreasonable effectiveness of mathematics in the natural sciences.Communications on Pure and Applied Mathematics13(1): 1–14. Page numbers refer to the reproduction in Mathematics: People, Problems, Results I–III, eds. , CampbellD M, HiggensJ C, ed. Belmont, California: Wadsworth International, 116–125.
