## Abstract

In his seminal 1960 paper, the physicist Eugene Wigner formulated the question of the applicability of mathematics in physics in a way nobody had before. This formulation has been (almost) entirely overlooked due to an exclusive concern with (dis)solving Wigner’s problem and explaining the effectiveness of mathematics in the natural sciences, in one way or another. Many have attempted to attribute Wigner’s unjustified conclusion—that mathematics is unreasonably effective in the natural sciences—to his (dogmatic) formalist views on mathematics. My goal is to show that this reading misses out on Wigner’s highly original formulation of the problem which is presented throughout his body of work in physics as well as in philosophy. This formulation, as I will show, leads us in a new direction in solving the applicability problem.

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## Notes

Although for Aristotle astronomy, optics, mechanics, and harmonics were subordinate mathematical sciences, they were not considered to be parts of natural science. Moreover, applying mechanical principles to physical objects, which Galileo did, was nothing new. With the rise of modern physics, what is new is that one can give (natural causal) explanation for physical objects in terms of principles that are mathematical in nature. In other words, mathematical principles provide a basis for doing physical science. Along with E.A. Burtt, one might think this shift from understanding physical science as a theory of substantial forms to what involves causal explanations in terms of mathematical principles, involved a pure faith in the deep mathematical structure of the universe. In

*The Metaphysical Foundations of Modern Science*, Burtt argues that this trend in works of Galileo, Kepler, Copernicus must be understood based on the ‘pure faith’ (Burtt 1923).Physics especially in the last century has become increasingly mathematical to the extent that some parts of physics, string theory for instance, fit better in the description of mathematics. The status of string theory as a theory of physics is controversial. Moreover, here my focus is on characterizing mathematics and physics in the conventional way.

My paper, first and foremost, aims to contribute to the scholarship on Wigner’s problem of the “Unreasonable Effectiveness”. By extracting and highlighting what is interesting in Wigner’s project, which is almost entirely overlooked by other commentators, I hope to contribute to the scholarship on the question of the applicability of mathematics, in general. To fully answer this question, we need a much more detailed philosophical work about the

*relata*of this relationship, which is outside the capacity of this paper.I am convinced by Longo’s argument that Wigner’s removal of other natural sciences must be studied carefully. The comparison between biology and physics, in relation to the applicability of mathematics, is enlightening and deepens our understanding of the relationship between mathematics and physics. For an interesting and illuminating discussion see Longo and Montévil (2016) and Longo and Montévil (2013).

This question is often not distinguished from a connected question: what is it about

*mathematics*that makes it an appropriate language for modern physics? To provide an answer to the latter question, we need an elaborate account of mathematics. Following the format of Wigner’s paper and the trajectory of his work, and given the fact that other commentators in the field have mainly focused on the latter question, I will limit myself to making only some brief remarks about mathematics. The question of what mathematics is, given its applicability in modern physics, is at the center of another paper under construction. Of course to provide a full answer to the applicability problem we need to have an elaborate view of both*relata*of this relationship.Wigner, of course, wasn’t the first person concerned with the applicability of mathematics in natural sciences. In modern times, Kant most prominently raised this issue in

*Metaphysical Foundations of Natural Science*: “[I]n any special doctrine of nature there is only as much*proper*science as there is*mathematics*therein. For. proper science, and above all proper natural science requires a pure part lying at the basis of the empirical part, and resting on a priori cognition of natural things.” How is this relation possible? Or maybe in Kant’s jargon, how is mathematics possible? My paper, however, is focused on the current formulation of the problem, which is reinvigorated and reintroduced by Wigner’s lecture.My argument for this claim is in the following sections. Roughly put, while for Wigner advanced mathematics proceeds based on intra-mathematical criteria such as (formal) beauty and elegance, that is not the case for elementary branches of mathematics such as geometry and algebra. On Wigner’s view, the latter is suggested to us by our experience of nature. Moreover, while the puzzle for Courant is the relationship between mathematics and

*reality*, for Wigner the problem is the applicability of mathematics to (modern)*physics*, which is our study of inanimate nature. I will discuss these issues further in the course of this paper.Although these commentators advocate different views about mathematics and physics, they converge on their reading of Wigner. While for Wigner and Steiner physics is an empirical science, Hamming, for instance, argues that the whole of physics or at least “a surprising amount of it” can be derived deductively by armchair thinking only. It is in this way that physics seems to be almost an

*a priori*endeavor, much entangled with and similar to mathematics (Hamming 1980, pp. 88–89). Subtleties of this kind are very important and in my view, they ultimately amount to different formulations of the applicability problem. I have elaborated on these accounts in another work.Evaluating whether this solution is convincing or not is a separate matter which requires a space of its own. In this paper, I will focus on describing

*what it is*.While Wigner used invariance and symmetry principles interchangeably, the contemporary reader has a more nuanced notion. Symmetry is used to refer to symmetry groups (transformations), such as space-time symmetries, Lorentz and Poincare transformations. The invariance of certain quantities under symmetry transformations is linked to conservation laws (Noether’s theorems). For instance the invariance of energy of an isolated physical system under space translations leads to the conservation of linear momentum. Following Wigner, I use both symmetry and invariance principles in the same way.

Invariance principles were known to physicists prior to the twentieth century, but they were considered to be the consequences of and second to laws of nature. Einstein’s papers on special and especially general relativity reversed this trend, putting invariance principles first and deriving laws of nature on their basis. This reversal, however, didn’t have much of an influence on the work of theoretical physicists outside relativity theories. Wigner’s work on invariance principles in quantum mechanics had a profound effect on changing this situation. As Gross writes, “that today principles of symmetry are regarded as the most fundamental part of our description of nature, is in no small part due to the influence of Eugene Wigner.” (Gross 1995, p. 46).

With the exception of a few comments, mostly in this paper, one cannot find much evidence on Wigner’s views about mathematics. This is very telling, on my view, about the side of the applicability problem that Wigner found interesting. Unlike what most scholars in the field emphasize, the illuminating part of this paper on my view, is how Wigner understands the conditions for the possibility of mathematization of physics. That is, the aspects of modern theoretical physics that make for its mathematizability. Given the context of his intellectual environment, as Ferrerirós has recently argued, one can see the influence of the Hilbert school formalism on his account of mathematics. For an argument see Ferreirós (forthcoming).

Of course, Courant was not the only one. Wigner’s close friend and brother in law, Dirac had stated this puzzlement in more than one place. Mathematical beauty was also a concern for Walter Dubislav and Michael Polanyi, both of whom are mentioned in course of Wigner’s lecture.

Wigner, moreover, makes an effort to close his lecture by saying that lots of work remains to be done to solve this mystery ”a gift we neither understand, nor deserve”. As it were, he is saying, there is much more work left for the other Courant lecturers to do in this relation. See http://cims.nyu.edu/webapps/content/lectures/home for more information on these lectures.

It is quite remarkable that he calls this bafflement a

*cheerful*note, whereas it should have been a sad conclusion, an unsolved problem.This is a controversial position about pure mathematics. It is true that a big part of pure mathematics

*today*is not useful. Yet the situation might change in the future and in fact many mathematicians who develop highly abstract concepts hope that one day their mathematical work is useful in the natural sciences, technology and so on. Here I am not providing argument for the skeptical account on pure mathematics and its lack of applicability. I am only recounting it.Hamming, for instance, in his

*partial explanations*for Wigner’s puzzle suggests that: (1)“we see what we look for”, and (2) “We select the kind of mathematics to use. That is when the mathematical tools that we have chosen are not adequate in a particular case, we choose a different kind.” (Hamming 1980, pp. 87–89). These points haven’t gone unnoticed by Wigner as the following quote suggests.Suppose that A is a square matrix. An eigenvector of this matrix is a vector

*x*with the same direction as*Ax*. That is \(Ax=\lambda x\), where \(\lambda \) is a constant. Eigenvectors are these special vectors that when multiplied by a matrix, they are only stretched, or shrunk or reversed (or left unchanged). Now this constant \(\lambda \) is the eigenvalue of the matrix*A*. For instance if*A*is the identity matrix then all vectors are eigenvectors of it, since \(Ax=x\) with eigenvalue \(\lambda =1\). (In defining eigenvalues here, I am using the modern definition.)For an elaborate view of this history see Thomas Hawkins, “The Theory of Matrices in nineteenth Century” (Hawkins 1974).

Wigner gives a few examples of these theories such as Bohr’s early theory of atom, Ptolemy’s epicycles, and the free electron theory. In all of these cases there were amazing numerical agreements between the theory and experiment and were formulated in the language of mathematics (Wigner 1960, p. 236).

The relationship between mathematics and physics, as illustrated in these cases (where the constructions in mathematics, independent of their applicability, were successfully applied in physics) is far from trivial. We understand this by looking at the intricate history of the development of mathematics and physics. The skeptic’s skepticism as well as the mystic’s mysticism have their root in precisely ignoring this history, and hence cutting the philosophical investigation too short.

Formal beauty for Wigner is closely tied to rigor, simplicity and generality. As an example, in expanding the concept of ‘number’ to include negative numbers, mathematicians chose a definition that preserves many rules that hold between positive numbers such as associativity, commutativity, existence of a null member 0 (with addition), and associativity, commutativity, existence of a null member 1 (with multiplication).This expansion made definition of the set of integers \(\mathbb {Z}\) possible which is a superset of the set of natural numbers (\(\mathbb {N}\)) forming an algebraic group with the operation of \(+\) (unlike \(\mathbb {N}\)).The reason is that in \(\mathbb {Z}\) every number m has its inverse with respect to \(+\), −m in \(\mathbb {Z}\) (such that m+(−m)=0). Through such expansion, mathematicians were able to prove many theorems about integers, with the set of natural numbers as a spacial and limited case. This is what Wigner means by more rigor and generality. In addition, they made the proof of many theorems simpler and provided the solution to many unsolvable equations such as (\(x+1=0\)).

There is already a refinement that Wigner adds to Courant’s ideas: not

*all*mathematical concepts are developed playfully and aesthetically. There are some that were in a way “forced” on us by nature, by the actual world.Wigner’s definition of (advanced) mathematics is distinctly formalist, which is not so hard to understand given his ties to the Hilbert school. In this paper, he makes reference to Hilbert, Michael Polanyi and Walter Dubislav. In his recent paper, Josè Ferreiròs focuses on Walter Dubslav and his influence on Wigner’s views, which he characterizes as ‘strict formalism’ Ferreirós (forthcoming). On Wigner’s view (a) mathematical concepts -invented, not deduced- are governed by certain game-like rules, (b) the criteria for developing such concepts, isn’t their link to nature, or our experience of nature, but their formal characteristics which are internal to mathematics. Wigner states, “The great mathematician fully, almost ruthlessly, exploits the domain of permissible reasoning and skirts the impermissible. That his recklessness does not lead him to a morass of contradictions is a miracle itself” (Wigner 1960, p. 224).

All real numbers are thus complex numbers with \(y=0\).

The word “imaginary” was first coined by Descartes in 1630 to reflect his idea that For every equation of degree n, we can imagine n roots which do not correspond to any real quantity. See Magic of Complex numbers in Mandic and Lee Goh (2009) for more details.

The case of complex numbers in quantum mechanics is the focus of another paper, in which I argue that this way of characterizing their development is not historically accurate. Wigner, Penrose and many other commentators make a similar mistake in overlooking historical details regarding the development of these numbers, for instance, with regard to their geometrical representation. In this way, they turn the applicability of complex numbers in physics into a more puzzling and mysterious phenomenon.

Quoted in Jean-Pierre Kahane’s 1991 article

*Jacques Hadamard*(Kahane 1991, p. 26).Buzaglo writes, “Eventually it became clear that without this “nonsense” there would be no mathematics- or at least no modern mathematics. Moreover, without expansions it is hard to see how we could progress in physics” (Buzaglo 2002, p. 11).

I am convinced by Ferrerirós’ argument in his recent paper, where he traces Wigner’s formalism to his connections to the Hilbert school. Not only does Wigner think that advanced mathematical concepts are invented, but also he emphasizes that they are invented

*just*for the purpose of their manipulability. It is this emphasis on ‘just’ that reveals his formalist tendencies about mathematics. See Ferreirós (forthcoming) and Lützen (2011) for a clear characterization of Wigner’s formalism.See Ferrerirós’ recent paper on the historical context of Wigner’s ’strict formalism’ and his relation to Walter Dubislav (Ferreirós forthcoming).

While it might seem that either formalists or Platonists have more difficulty explaining the applicability of mathematics, Colyvan argues that the question of applicability of mathematics has its full force for different schools of philosophy of mathematics from nominalists to Platonists. For an interesting discussion see Colyvan (2001).

P. Roman in

*Why Symmetry*also notes: “Nature, be it even objective reality, just*is*. It cannot be ‘explained’, at least not as far as science is concerned. Existence is a primary category, including, by the way, ourselves, too. (Which would imply that the explainer himself must be explained.) And certainly science is much more than ‘the description of Nature’ ...We have gone far beyond such a casual phenomenology and even empiricism. We want to ‘understand’, and we have in part succeeded. ...And the twentieth century German writer Hermann Hesse tells us: ‘Every science is ...a kind of ordering, simplifying; an attempt to make digestible for the spirit that which is indigestible.’ Indeed, we are safe to say:*Science is the attempt to correlate individual phenomena and events into a coherent framework (or systems of such frameworks)*” (Roman 2004, p. 2).Wigner writes, “If we look a little deeper into the situation we realize that we would not live in the same sense that we do if the events around us had no structure. .. There would be no way our volition could manifest itself and there would be no such thing as that which we call life, This does not mean, of course, that life depends on the existence of the unbelievable precision and accuracy of the correlation between events which our laws of nature express and indeed the precision of these laws has all elements of a miracle that we can think of.”

By virtue of characterizing laws in this way, Wigner takes a route different from a more metaphysically inclined way of understanding laws, as statements of a necessary relation between events. He rather offers an anti-realist account of laws as statements of correlations between events, as opposed to a more strict causal conception, where causality is tightly linked to metaphysical necessity. Using an anti-realist undertone, he defines physics not as concerned with nature but with regularities that we

*experience*among events.Wigner writes, “The surprising discovery of Newton’s age is just the clear separation of laws of nature on the one hand and initial conditions on the other. The former are precise beyond anything reasonable; we know virtually nothing about the latter” (Wigner 1964a, p. 40).

As another example take the case of Halley’s comet. Bigelow and Pargetter write: “If we want to explain why, say, Halley’s comet returns about every seventy-six years, we will use the laws of mechanics, but we will also have recourse to matters of particular fact. For instance, we will have to feed in details on the mass of the comet, its position and velocity at some given moment, the paths of the planets, and so forth. These facts are clearly not laws of nature - they are particular, not general, and they not only could have been otherwise, but have been otherwise, and will be otherwise again. They are what we call initial conditions (Bigelow and Pargetter 1991, pp. 300–301).”

The commas in these formula can be replaced by the logical conjunction \('\wedge '\).

This formulation doesn’t make the universal form of these laws explicit. A better form could be \(\forall x (A(x),B(x),\ldots \rightarrow X_1(x),X_2(x),\ldots )\).

This is a necessary condition for any causal analysis of nature that enables one to make predictions.

As opposed to this group of invariance principles, with which we are on

*terra cognita*, newer principles of invariance aren’t necessarily continuous, global or geometrical. With them we are in*terra incognita*. Symmetries of general theory of relativity, symmetries of quantum mechanics such as permutation symmetry are among the latter group. These principles, unlike the first group, have no roots in the history of science, hence with them we enter*terra incognita.*In Newton’s

*Principia*the third invariance principle is recognized as a corollary to the laws of motion, that is, as a consequence of the laws and not one of their underlying assumptions. Huygens, unlike Newton, considered this principle as a basic postulate from which to derive laws of motion. But as it stands, and presented initially in Newton’s manuscripts, it is an independent assumption.These three symmetry principles, for Wigner, are symmetries of both Newtonian mechanics and special relativity (Wigner 1964a, p. 45).

A linear operator is an operator that preserves addition and multiplication. An operator

*U*on a Hilbert space*H*, \(U:H\rightarrow H\) is called unitary if \(UU^*=U^*U=I\) where \(U^*\) is the adjoint of*U*and*I*is the identity transformation. A unitary operator acting on a vector changes its direction but preserves the length.For a more detailed discussion see Gross (1995).

More precisely, mathematics and physics are the result of a similar abstraction where relations between objects are pushed to their limit, abstracted from their particularity. Of course the subject of mathematics is more idealized and hence farther from the physical world. A closer study of this abstraction and idealization is subject of another paper under construction.

Wigner devotes the bulk of his paper to three examples: Planetary Motion, Heisenberg’s Matrix Mechanics, and Quantum Electrodynamics, in order to show the appropriateness of the language of mathematics for theories of physics.

As an example take the case of the principle of parity which was rejected due to Wu’s experiment. Wigner formalized the principle of conservation of parity (mirror-symmetry) in 1927 as a principle according to which our world and its mirror image behave in the same way, with only difference that left and right are reversed. Wu’s experiment, carrie din 1956, was an experiment for checking the truth of parity conservation for beta decay. To all physicists surprise, the result of the experiment was the parity is violated (in fact its violation was maximal) . It was surprising because the physicists believed the conservation of parity holds everywhere. Wigner writes, “Only very elementary theory was necessary to see that Wu’s experiment was in conflict with parity principle” (Wigner 1963, p. 31) which led ultimately to the refutation of parity conservation in this case.

Note that the situation is not quite the same for testing the validity of every older invariance principle. Given the gravitational field of the earth, on which we do our experiments, for instance, it might be extremely hard (if not impossible) to compare experiments on earth to check the rotational invariance. For an interesting discussion on possible differences see (Wigner 1995, 390–392).

Steiner for instance gives examples of this attitude among physicists in

*Applicability of Mathematics as a Philosophical Problem*.I am not sure why Wigner said that the free electron theory is considered false.The free electron theory in condensed matter is not considered false. Its a pretty good working model that is still taught in condensed matter course. Bohrs early atomic theory, on the other hand, is not correct because it fails to work for any atom except hydrogen. The latter suffices to make Wigner’s point about the relationship between mathematical formulation and truth.

Beauty as a guide to truth, a slogan of theoretical physics and a regulative principle in its practice, gives the theorist emotional encouragement that she needs, echoing Hardy’s famous statement: “Beauty is the first test: there is no permanent place in this world for ugly mathematics.” But this emotional encouragement, taken as a true statement about the fundamental structure of the universe, can be very misleading.

Lützen 2011 argues that the origin of physically useful mathematical concepts is physical (Lützen 2011).

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## Acknowledgments

I thank Tom Donaldson, Krista Lawlor, Solomon Feferman, Michael Friedman, Mark Steiner, José Ferreirós and Jonathan Ettel for their insightful suggestions. My special thanks are to Thomas Ryckman for his enormously useful comments on several revisions of this paper.

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Islami, A. A match not made in heaven: on the applicability of mathematics in physics.
*Synthese* **194**, 4839–4861 (2017). https://doi.org/10.1007/s11229-016-1171-4

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DOI: https://doi.org/10.1007/s11229-016-1171-4