Abstract
I present a reconstruction of Eugene Wigner’s argument for the claim that mathematics is ‘unreasonable effective’, together with six objections to its soundness. I show that these objections are weaker than usually thought, and I sketch a new objection.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Notes
- 1.
I deal with the puzzle in my (2009) and, more thoroughly, in my (2012, Chap. 7). Although there is some overlap between this paper and my treatment of the issue in my book, the current paper offers a different reconstruction of the puzzle. My conclusion, however, is the same—that Wigner’s riddle can be (dis)solved.
- 2.
Grattan-Guiness (2008) seems to me an example, among others, of conflating these separate issues: Wigner’s, who focused on the role of mathematics in describing nature, and others’ concerns with its role in theory-building.
- 3.
- 4.
In fact, Bourbaki, when referring to this issue, uses the word ‘preadaption’. Here is the entire quote: “Mathematics appears […] as a storehouse of abstract forms—the mathematical structures; and it so happens—without out knowing why—that certain aspects of empirical reality fit themselves into these forms, as if through a kind or pre-adaption.” (1950, p. 231) I found this quote in Ginammi (2014, p. 27).
- 5.
Wigner also writes that mathematical concepts “are defined with a view of permitting ingenious logical operations which appeal to our aesthetic sense … [they are chosen] for their amenability to clever manipulations and to striking, brilliant arguments.” (1960, p. 7).
- 6.
Jerome Cardan, who is credited with introducing them in the 16th century, remarked that “So progresses arithmetic subtlety the end of which, as is said, is as refined as is useless.” (Cited in Kline 1972, p. 253). According to Kline, neither did Newton regard complex numbers as significant, “most likely because in his day they lacked physical meaning.” (1972, p. 254).
- 7.
The selection of quotes is from Maddy (2007, p. 330).
- 8.
But, should one take into consideration their views on the matter, when they bothered to express them? My answer (for which I don’t have space to argue here) is ‘yes’, but not everybody agrees; see Azzouni (2000, p. 224).
- 9.
Hamming continues by saying that “we have tried to make mathematics a consistent, beautiful thing, and by doing so we have had an amazing number of successful applications to the physical world” (1980, p. 83)—yet another expression of the Wigner problem.
- 10.
- 11.
Wigner makes the point about the independence (hence objectivity) of the laws of a huge variety of particular circumstances on pp. 4–5 in his (1960).
- 12.
A well-known passage from Kepler reads: “Thus God himself was too kind to remain idle, and began to play the game of signatures, signing his likeness into the world; therefore I chance to think that all nature and the graceful sky are symbolized in the art of geometry.” Quoted in Dyson (1969, p. 9).
- 13.
As Kline (172, p. 1028) describes: “the Greeks, Descartes, Newton, Euler, and many others believed mathematics to be the accurate description of real phenomena (…) [T]hey regarded their work as the uncovering of the mathematical design of the universe.”
- 14.
A recent author seemingly embracing this idea is Plantinga (2011, pp. 284–91).
- 15.
See Wigner (1960, p. 2): “Furthermore, whereas it is unquestionably true that the concepts of elementary mathematics and particularly elementary geometry were formulated to describe entities which are directly suggested by the actual world, the same does not seem to be true of the more advanced concepts, in particular the concepts which play such an important role in physics.”
- 16.
I discuss a different kind of transitivity in my (2012, Chap. 7).
- 17.
Ivor Grattan-Guiness reasons along the same fallacious line: “Much mathematics, at all levels, was brought into being by worldly demands, so that its frequent effectiveness there is not so surprising.” (2008, p. 8; my emphasis).
- 18.
In the same passage quoted above from (1964, p. 429), Kline calls this kind of work “an ingenious bit of mathematical hocus-pocus”.
- 19.
In fact, the second passage in his 1854 masterpiece ‘On the Hypotheses Which Lie At The Bases Of Geometry’ contains the generalization point: “It will follow from this that a multiply extended magnitude is capable of different measure-relations, and consequently that space is only a particular case of a triply extended magnitude.” (Riemann 1854; reprinted in Hawking 2007, pp. 1031–2; translated by W.K. Clifford).
- 20.
Maddy (2007, p. 337) accepts the (problematic) Kline picture, backing it up with a quote from Kline himself (1968, p. 234): “So Riemann’s motivation was not ‘purely aesthetic’ or in any sense ‘purely mathematical’; he was concerned, rather, with the needs of physical geometry and his efforts were successful.”
- 21.
Note that such an objector has no troubles to accept the first premise of WA, that these concepts are in the mathematical corpus because they are interesting and intriguing; if this objection from preparation and modeling is viable, the origin of concepts is just irrelevant.
- 22.
Recall that ‘many’ is relative; it means, ‘many among the truly important ones’, since these are the ones that matter, as we remember from the discussion of premise 3 above.
- 23.
See also Maddy (2007, p. 339): “As a mathematical analog, I suggest that we tend to notice those phenomena we have the tools to describe. There’s a saying: when all you’ve got is a hammer, everything looks like a nail. I propose a variant: if all you’ve got is a hammer, you tend to notice the nails.”
- 24.
Maddy (2007, p. 341) puts it as follows: “With the vast warehouses of mathematics so generously stocked, it’s perhaps less surprising that a bit of ready-made theory can sometimes be pulled down from the shelf and effectively applied.”
- 25.
I take Pincock (2012, pp. 184–5) to advance this line of thought: “Even an argument based on natural selection seems imaginable according to which our tendency to make aesthetic judgments is an adaptation precisely because these judgments track objective features of our environment.”
- 26.
Sometimes the physicists themselves try to develop the mathematics they need, but usually aren’t successful. Here is the story of Gell-Mann in Steiner (1998, p. 93), relying on Doncel et al. (1987, p. 489): “[In trying to generalize the Yang-Mills equations] [w]hat Gell-Mann did without knowing was to characterize isospin rotations as a ‘Lie Algebra’, a concept reinvented for the occasion, but known to mathematicians since the nineteenth century. He then (by trial and error) began looking for Lie Algebras extending isospin—unaware that the problem had already been solved by the mathematicians—but failed, not realizing that the first solution required eight components.”
- 27.
A point distantly related to the present one is that there are major scientific achievements (the theory of evolution, and other work in biology) in which mathematics doesn’t play any role (Wilczek 2007; Sarukkai 2005). However, Wigner’s problem centers on physics (despite the general title of his paper).
References
Azzouni, J.: Applying mathematics: an attempt to design a philosophical problem. Monist 83(2), 209–227 (2000)
Bangu, S.: Steiner on the applicability of mathematics and naturalism. Philosophia Math. (3) 14(1), 26–43 (2006)
Bangu, S.: Wigner’s puzzle for mathematical naturalism. Int. Stud. Philos. Sci. 23(3), 245–263 (2009)
Bangu, S.: The Applicability of Mathematics in Science: Indispensability and Ontology. Palgrave Macmillan, London (2012)
Bourbaki, N.: The architecture of mathematics. Am. Math. Monthly 57, 221–232 (1950)
Dirac, P.A.M.: The relation between mathematics and physics. In: Proceedings of the Royal Society (Edinburgh) (James Scott Prize Lecture), vol. 59, pp. 122–129 (1939)
Doncel, M., et al.: Symmetries in Physics (1600–1980). Servei de Publicacions, Universitat Autonoma de Barcelona, Barcelona (1987)
Dyson, F.J.: Mathematics in the physical sciences. In: The Mathematical Sciences (ed.) Committee on Support of Research in the Mathematical Sciences (COSRIMS) of the National Research Council, pp. 97–115. MIT Press, Cambridge (1969)
Dyson, F.: ‘Paul A. M. Dirac’ American Philosophical Society Year Book 1986 (1986)
French, S.: The reasonable effectiveness of mathematics: partial structures and the application of group theory to physics. Synthese 125, 103–120 (2000)
Ginammi, M.: Structure and Applicability. An Analysis of the Problem of the Applicability of Mathematics. PhD Dissertation, Scuola Normale Superiore, Pisa (2014)
Grattan-Guiness, I.: Solving Wigner’s mystery: the reasonable (though perhaps limited) effectiveness of’ mathematics in the natural sciences. Math. Intelligencer 30(3), 7–17 (2008)
Hamming, R.: The unreasonable effectiveness of mathematics. Am. Math. Monthly 87(2), 81–90 (1980)
Hawking, S. (ed.): God Created the Integers. Running Press, Philadelphia & London, The Mathematical Breakthroughs that Changed History (2007)
Kline, M.: Mathematics in Western Culture. Oxford University Press, New York (1964)
Kline, M.: Mathematics in the Modern World. W. H. Freeman and Company, San Francisco (1968)
Kline, M.: Mathematical Thought from Ancient to Modern Times. Oxford University Press, New York (1972). (Vol. 3)
Kline, M.: Mathematics: The Loss of Certainty. Oxford University Press, New York (1980)
Maddy, P.: Second Philosophy. Oxford University Press, New York (2007)
Nagel, E.: ‘Impossible Numbers’ in Teleology Revisited. Columbia University Press, New York (1979)
Pincock, C.: Mathematics and Scientific Representation. Oxford University Press, New York (2012)
Plantinga, A.: Where the Conflict Really Lies: Science, Religion, and Naturalism. Oxford University Press, New York (2011)
Riemann, B.: On the Hypotheses Which Lie At The Bases Of Geometry. Reprinted in Hawking (2007) (1854)
Sarukkai, S.: Revisiting the ‘unreasonable effectiveness’ of mathematics. Curr. Sci. 88(3), 415–423 (2005)
Schwartz, J.T.: The pernicious influence of mathematics on science. In: Kac, M., Rota, G.C., Schwartz, J.T. (eds.) Discrete Thoughts. Essays on Mathematics, Science and Philosophy. Birkhauser, Boston (1986)
Steiner, M.: The Applicability of Mathematics as a Philosophical Problem. Harvard University Press, Cambridge (1998)
Von Neumann, J.: The Mathematician. In: Newman, J.R. (ed.) The World of Mathematics, vol. 4, pp. 2053–2063. George Allen and Unwin, London (1961)
Weinberg, S.: Lecture on the applicability of mathematics. Not. Am. Math. Soc. 33, 725–733 (1986)
Weinberg, S.: Dreams of a Final Theory. Vintage, London (1993)
Wigner, E.: The unreasonable effectiveness of mathematics in the natural sciences. Commun. Pure Appl. Math. 13(1), 1–14 (1960)
Wilczek, F.: ‘Reasonably effective: I. Deconstructing a Miracle’, Physics Today, pp. 8–9 (2006)
Wilczek, F.: ‘Reasonably effective: II. Devil’s advocate’, Physics Today, pp. 8–9 (2007)
Acknowledgements
Thanks are due to the stimulating audience of the Models and Inferences in Science conference, in particular to the editors of this volume, Emiliano Ippoliti, Fabio Sterpetti, and Thomas Nickles. The responsibility for the final form of the text is entirely mine.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2016 Springer International Publishing Switzerland
About this chapter
Cite this chapter
Bangu, S. (2016). On ‘The Unreasonable Effectiveness of Mathematics in the Natural Sciences’. In: Ippoliti, E., Sterpetti, F., Nickles, T. (eds) Models and Inferences in Science. Studies in Applied Philosophy, Epistemology and Rational Ethics, vol 25. Springer, Cham. https://doi.org/10.1007/978-3-319-28163-6_2
Download citation
DOI: https://doi.org/10.1007/978-3-319-28163-6_2
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-28162-9
Online ISBN: 978-3-319-28163-6
eBook Packages: Religion and PhilosophyPhilosophy and Religion (R0)