Abstract
In his influential 1960 paper ‘The Unreasonable Effectiveness of Mathematics in the Natural Sciences’, Eugene P. Wigner raises the question of why something that was developed without concern for empirical facts—mathematics—should turn out to be so powerful in explaining facts about the natural world. Recent philosophy of science has developed ‘Wigner’s puzzle’ in two different directions: First, in relation to the supposed indispensability of mathematical facts to particular scientific explanations and, secondly, in connection with the idea that aesthetic criteria track theoretical desiderata such as empirical success. An important aspect of Wigner’s article has, however, been overlooked in these debates: his worries about the underdetermination of physical theories by mathematical frameworks. The present paper argues that, by restoring this aspect of Wigner’s argument to its proper place, Wigner’s puzzle may become an instructive case study for the teaching of core issues in the philosophy of science and its history.
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Notes
The indispensability thesis, promulgated by W.V.O. Quine and Hilary Putnam, is motivated by a realist commitment to those entities—including mathematical entities—that our best scientific theories quantify over.
While previous commentators on Wigner’s paper have noted that his ‘puzzle’ is really several problems rolled into one, they have tended to highlight the role of mathematics in the very discovery of scientific theories. Sometimes this departure from Wigner’s original concern is made explicit (e.g. by Steiner 1995, p. 153, fn. 68); sometimes it leads to a conflation of several distinct problems, as when Colyvan treats Steiner’s approach as a ‘version of Wigner’s thesis’ (Colyvan 2001, p. 267), thereby explicitly assimilating the two accounts into a single ‘Wigner/Steiner problem’ (ibid.).
Feynman expresses a similarly optimistic view when he likens the multiplicity of mathematical approaches to ‘a bridge with lots of members, [which] is over-connected; if pieces have dropped out you can reconnect it another way’ (Feynman 1985: 47). (I am grateful to an anonymous reviewer for pointing me towards Feynman’s text; see also the discussion in Sect. 3 of this paper).
One of the few exceptions is Wigner’s colleague and sometime co-author Henry Margenau, who seems to have Wigner’s underdetermination in mind when he insists that ‘historically, uniqueness of mathematical description certainly does not exist, and methodologically it is not necessary for the success of science’ (Margenau 1966, p. 354).
This point has only recently started to receive attention; see (Gelfert 2011, esp. pp. 284–285) and references therein.
What follows is only a rough sketch of structural realism and its relation to Wigner’s puzzle; a fuller discussion would require extensive engagement with the literature on structural realism, for which this is not the place.
Ontic (as opposed to epistemic) structural realists go one step further in arguing that not only is structure all there is to know, but that structure is all there is. (For a review, including a discussion of structural realism’s historical influences, see Massimi 2010).
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Acknowledgments
I am grateful to Sorin Bangu (Bergen), Grant Fisher (Daejeon), and Ricardo Karam (Hamburg) for comments on an earlier version of this paper, as well as to four anonymous reviewers for exceptionally detailed and helpful comments.
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Gelfert, A. Applicability, Indispensability, and Underdetermination: Puzzling Over Wigner’s ‘Unreasonable Effectiveness of Mathematics’. Sci & Educ 23, 997–1009 (2014). https://doi.org/10.1007/s11191-013-9606-5
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DOI: https://doi.org/10.1007/s11191-013-9606-5