Wigner’s Puzzle for Mathematical Naturalism

Abstract

I argue that a recent version of the doctrine of mathematical naturalism faces difficulties arising in connection with Wigner’s old puzzle about the applicability of mathematics to natural science. I discuss the strategies to solve the puzzle and I show that they may not be available to the naturalist.

Acknowledgements

I thank Penelope Maddy, Susan Wineberg, and Patricia Marino for critical feedback on earlier drafts, and the two referees and the editor of this journal for suggestions.

Notes

[1] To the best of my knowledge, Colyvan (2001) is the only other attempt in the literature to relate these two themes. Maddy (2007) contains discussions of both themes as well.

[2] On the technical‐mathematical side, the debate concerns the supplementation of the axioms of standard set theory with new axioms. These new axioms are needed to settle the so‐called ‘open questions’ (‘open’ because they are shown to be neither provable nor disprovable, given the current axioms; among them, Cantor’s continuum hypothesis is the most notorious). On the philosophical side, Maddy is preoccupied to show that mathematical ontology should not be decided on the basis of the indispensable role played by (parts of) this ontology in physical theories (as Quine would have it), but rather on intra‐mathematical standards.

[3] Let me clarify a possible misunderstanding here. I take concepts like complex number, group, modular form, vector bundle, ramification point, etc., as instances of non‐set‐theoretic concepts. Thus, in not being set‐theoretical, these concepts are not foundational for all mathematics. Of course, some can be foundational for a particular mathematical field—algebraic geometry in this case. On the other hand, I take set‐theoretic concepts such as inner model, limit ordinal, constructible set, etc. as typical examples of foundational concepts. I also assume (trivially) that the potential reduction of the first series of concepts to concepts of set theory doesn’t render the first series of concepts foundational (set‐theoretic).

[4] See also the quotations about the ‘liberation’ of mathematics provided in the introduction above.

[5] In addition to disentangling Wigner’s argument, Steiner (1998) advances his own, new version of the puzzle, subtly different from Wigner’s. For further discussions of Steiner’s own version, see Colyvan (2001), Azzouni (2004, ch. 8), Bangu (2006).

[6] The emphasis is important, as Wigner grants that no applicability problem can occur for the elementary geometrical concepts in so far as they were designed to describe physical features in the first place—see the quotation in the introduction.

[7] This idea is very popular. The mathematician Hamming also wrote: ‘Artistic taste … plays … a large role in modern mathematics … 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’ (Hamming 1980, 83; 87). See also von Neumann’s opinion quoted in Section 4. Many other authors expressed similar views.

[8] Steiner (1989, 1998, 2005) offers an inventory of these strategies as well. Colyvan (2001) mentions them as well. Azzouni (2000) brings up some new considerations to the effect that Wigner failed to present us with a genuine mystery (I’ll discuss some of Azzouni’s points later on). The papers in Mickens (1990) span these strategies.

[9] See Mill (1947) for this early form of empiricism in the philosophy of mathematics, and Frege (1950, part II, sec. 23) for attacking him for confusing arithmetic with its application. Dummett (1991) gives an account of this exchange.

[10] See Colyvan (2006) for a similar discussion about the transitivity of indispensability (extending to applicability as well), with Colyvan claiming, against Rosen, that transitivity holds. As will be clear from what follows, I too believe transitivity holds but I’m sceptical that this will give the M‐naturalist any significant help eventually.

[11] Wigner (1967 [1960], 225) notes that ‘certainly, nothing in our experience suggests the introduction of these quantities’.

[12] As Kline (1972, 253) describes the episode, complex numbers first appeared in Cardan’s Ars Magna of 1545, chapter 37 (see Struik 1969, 67–69, for an English translation of a couple of relevant paragraphs). Cardan’s own view on these new mathematical objects is that they are only relevant for generalizing techniques of equation‐solving, where these equations have natural and rational coefficients. Commenting on their introduction, he notes that ‘So progresses arithmetic subtlety the end of which, as is said, is as refined as is useless’ (Kline 1972, 253). Kline also points out that even later on Newton did not regard complex numbers as significant, ‘most likely because in his day they lacked physical meaning’ (Kline 1972, 254, my emphasis).

[13] A problem that might occur is that this indirectness sometimes causes surprise. One can still wonder how complex numbers can have any relevance for the empirical realm, in spite of their being connected with it via natural numbers. But this is a rather minor problem, not a genuine mystery, since this surprise is a mere psychological phenomenon due to ‘implicational opacity’, as Azzouni (2000, p. 211) aptly calls it. The not‐immediately recognizable (psychologically speaking) inferential relations between mathematical concepts (complex numbers and natural numbers in this case) generate a kind of inferential opacity, which would explain the feeling of surprise.

[14] ‘Significant’ is of course patently vague. Moreover, as it has been noted, ‘significant’ is problematic in two senses. First, it is not clear what a ‘significant’ number of mathematical concepts is, and second, what it is for these concepts to be ‘significantly’ effective in applications.

[15] Wilson (2000) comes closest to this position in recent literature. Wilson documents in detail the ‘uncooperativeness’ of mathematics in the natural sciences.

[16] Graphically, this is represented in the diagram as ‘M* ⊃ P*’.

[17] Some of this work was done by Wigner himself; yet, out of modesty, he barely discussed it in his paper. The central idea of this field of work is to map elementary particles onto irreducible group representations, where the gauge group is chosen depending on what symmetries the interaction in question obeys; for instance, the symmetries of the strong interaction are characterized by the SU(3) group (unitary symmetry of dimension 3).

[18] Hadamard (1949, 129–130) presents a couple of other examples, such as Bernoulli’s introduction of the functional.

[19] Straightforward avowals of aestheticism are quite frequent among mathematicians; see, for instance, Hardy (1961, 2027) and Hadamard (1949, sec. 9).

[20] The converse is also true; in the particle physics community it is notorious how Gell‐Mann embarked upon the hopeless task of reinventing the theory of group representations when realizing he might need it. Fortunately for him, he soon found out that the mathematical apparatus had been fully developed years before. See Gell‐Mann (1964).

[21] Famous physicists find the occurrence of these correlations genuinely surprising. Weinberg’s oft‐cited remark that ‘It is positively spooky how the physicist finds the mathematician has been there before him or her’ (Weinberg 1986, 725) is an instance of such recent bafflement. (His ‘there’ refers in the scheme to the physical territory described by P*.)

[22] It was precisely this kind of match that deeply impressed Dirac and prompted him to write: ‘One may describe this situation by saying that the mathematician plays a game in which he himself invents the rules while the physicist plays a game in which the rules are provided by Nature, but as time goes on it becomes increasingly evident that the rules which the mathematician finds interesting are the same as those which Nature has chosen’ (Dirac 1939, 124).

[23] As Steiner (1998) points out, our concepts of simplicity and convenience are a reflection of the computational power of our brains.

[24] The version of anthropocentrism I describe here is articulated by Steiner (1998).

[25] As noted, one strand of the line of this Wignerian argument involving anthropocentrism is developed recently in Steiner (1998). More exactly, unlike Wigner (who discussed the role of mathematics in describing nature), Steiner argues that a similar (yet more robust) puzzle occurs with regard to the role of mathematics in discovering physical laws. He advances the subtle point that the discovery of some new physics P* is in many cases guided by mathematical generalizations. In fact, a version of the scheme I employed here can be found in a footnote of one of his discussions of concept invention in physics and mathematics. See Steiner (1998, 53n10). For criticisms of this argument, see Liston (2000), Simons (2001), Bangu (2006), Maddy (2007, IV.2).

[26] I thank Mark Janoff for drawing my attention to this point, discussed in his MPhil dissertation.

[27] For more on how a ‘caricature’ of the physical system retains the ‘essential physics’, see Batterman (2009).