Is Mathematics Unreasonably Effective?

ABSTRACT

Many mathematicians, physicists, and philosophers have suggested that the fact that mathematics—an a priori discipline informed substantially by aesthetic considerations—can be applied to natural science is mysterious. This paper sharpens and responds to a challenge to this effect. I argue that the aesthetic considerations used to evaluate and motivate mathematics are much more closely connected with the physical world than one might presume, and (with reference to case studies within Galois theory and probabilistic number theory) I show that they are correlated with generally recognised theoretical virtues, such as explanatory depth, unifying power, fruitfulness, and importance.

KEYWORDS:

• philosophy of mathematics
• applicability of mathematics
• Wigner
• unreasonable effectiveness
• aesthetics
• theoretical virtue

Disclosure Statement

No potential conflict of interest was reported by the author.

Notes

1 See, e.g., Quine [1960], Putnam [1975], Field [1980], and Colyvan [2001a] for a small sample of this literature.

2 Similar sentiments can be found in Feynman [1967: 171]: ‘I find it quite amazing that it is possible to predict what will happen by mathematics, which is simply following rules which really have nothing to do with the original thing.’

3 See Benacerraf [1973], Field [1989], and Schechter [2018]. See also Warren and Waxman [forthcoming] for a recent explanatory challenge concerning determinacy.

4 One option, inspired by Steiner [1998] and pursued in an earlier version of this paper, would be to direct the challenge at a species of naturalism, by construing it as an argument that mathematics is tacitly anthropocentric. However, helpful comments from a referee persuaded me that the relevant notion of anthropocentrism was neither entirely clear nor necessary for the dialectic.

5 See, e.g., the essays in Sinclair et al. [2006].

6 An immediate issue is that there is a difference between the boundary between mathematics/non-mathematics and the boundary between good/bad mathematics. It is surely a datum that demonstrably trivial or ugly or inelegant or uninteresting theories still count as mathematics, and so, if this view is to be made plausible, something must be said to finesse such counterexamples.

7 See, for instance, Pincock [2012]. Bueno and Colyvan [2011] emphasise the inferential role of mathematics, but are explicit [ibid.: 352] that they are building on, not repudiating, the representational account (their ‘mapping view’). For the purpose of raising the puzzle, we can be neutral on the precise details.

8 This is hinted at by Wigner [1960], but receives its clearest development by Steiner [1998] (a book that also contains illuminating discussion of many examples).

9 Field [1980] is the locus classicus.

10 For further arguments that the challenge does not rely on a particular view of mathematics, see Colyvan [2001b].

11 Steiner [1998] seems to believe that this subjectivist view is responsible for the force of the puzzle; Pincock [2012] attempts to disarm it by denying subjectivism. I believe that both are vulnerable to the point made in this paragraph.

12 Thanks to the Editor for pressing me to clarify here. The explanation offered in section 4 is in fact neutral between different conceptions of aesthetics: since it relies only on a reliable correlation between aesthetic properties in mathematics and certain non-aesthetic properties concerning theoretical virtues, nothing further needs to be assumed about the nature of the aesthetic properties in question.

13 Thanks to a referee for prompting clarification here.

14 See Maddy [2008] for a historically sophisticated and highly developed account along these lines.

15 The Langlands program is viewed as being of paramount importance within mathematics, wtih dozens of Abel Prizes and Fields Medals having been awarded for progress within it. See Gannon [2006] for a readable account.

16 For a more detailed exposition, see Stewart [2004], and see Kiernan [1971] for a comprehensive history.

17 A field is a set with recognisable analogues of addition, subtraction, multiplication, and division—a generalisation of the structure common to the rational numbers, the real numbers, and the complex numbers.

18 Technically: the automorphisms on $L$ that hold $K$ fixed.

19 In other words, we can compose or invert any of the permissible permutations of solutions and end up with what is still a permissible permutation.

20 Because $G_i$ is normal in $G_{i+1}$, we can form quotient groups.

21 Klainerman [2010: 279] reflects a prevalent view that PDEs are too disparate a class to admit of a general theory: ‘PDEs, in particular those that are nonlinear, are too subtle to fit into a too general scheme; on the contrary, each important PDE seems to be a world in itself.’

22 See, that is, Gilmore [2008].

23 That is, if $g\left(m\right)$ is the number of integers less than $m$ for which the inequality fails, then ${\displaystyle \frac{g\left(m\right)}{m}\to 0}$ as $m\to \mathrm{\infty }$.

24 This correlation might itself stand further explanation. One possible, relatively deflationary, approach might look for psychological reasons why we tend to find properties like simplicity, explanatory depth, importance, fruitfulness, to be beautiful or elegant when manifested within mathematics. But perhaps a deeper explanation is possible: for instance, perhaps mathematical beauty or elegance is grounded in properties like simplicity, explanatory depth, importance, fruitfulness, etc.; or perhaps aesthetic judgments serve as a way of expressing the presence of such properties. While these suggestions are intriguing, evaluating them would require a far more detailed discussion of aesthetic properties and judgments in mathematics than I am able to provide here. For the purpose of resolving the explanatory challenge, the fact of a reliable correlation is enough.

25 See, for instance, Psillos [2005]. ‘Epistemic’ here means something like ‘truth-directed’, as opposed to ‘merely pragmatic’: it is not meant to cover other notions of epistemic appraisal, such as ones involving epistemic responsibility or deontological considerations (although it is an open question how these relate to truth-directedness: for discussion, see Alston [2005]). Thanks to the Editor for pressing me for clarification on this point.

26 I am grateful to the Editor of AJP and two referees for helpful comments and suggestions which substantially improved the paper. Thanks to Michael Strevens for discussion on a (by now almost unrecognisable) first draft of this paper, to Rosa Cao for helpful comments, and especially to Jared Warren and Lavinia Picollo for comments and advice on several drafts.