# The Unreasonable Effectiveness of Mathematics in the Natural Sciences

"The Unreasonable Effectiveness of Mathematics in the Natural Sciences" is a 1960 article by the physicist Eugene Wigner, published in Communication in Pure and Applied Mathematics.[1][2] In it, Wigner observes that a physical theory's mathematical structure often points the way to further advances in that theory and even to empirical predictions. Mathematical theories often have predictive power in describing nature.

## Original paper and Wigner's observations

Wigner argues that mathematical concepts have applicability far beyond the context in which they were originally developed. He writes: "it is important to point out that the mathematical formulation of the physicist's often crude experience leads in an uncanny number of cases to an amazingly accurate description of a large class of phenomena".[3] He writes that the observation "the laws of nature are written in the language of mathematics was properly made three hundred years ago" (by Galileo) "is now more true than ever before."

Wigner's first example is Isaac Newton's law of gravitation. Originally used to model freely falling bodies on the surface of the earth, this law was extended on the basis of what Wigner terms "very scanty observations"[3] to describe the motion of the planets, where it "has proved accurate beyond all reasonable expectations".[4] Wigner: "It was Newton who then brought the law of freely falling objects into relation with the motion of the moon, noted that the parabola of the thrown rock's path on the earth and the circle of the moon's path in the sky are particular cases of the same mathematical object of an ellipse, and postulated the universal law of gravitation on the basis of a single, and at that time very approximate, numerical coincidence."

Wigner's second example comes from quantum mechanics: Max Born "noticed that some rules of computation, given by Heisenberg, were formally identical with the rules of computation with matrices, established a long time before by mathematicians. Born, Jordan, and Heisenberg then proposed to replace by matrices the position and momentum variables of the equations of classical mechanics. They applied the rules of matrix mechanics to a few highly idealized problems and the results were quite satisfactory. However, there was, at that time, no rational evidence that their matrix mechanics would prove correct under more realistic conditions." But Wolfgang Pauli found their work accurately described the hydrogen atom: "This application gave results in agreement with experience." The helium atom, with two electrons, is more complex, but "Nevertheless, the calculation of the lowest energy level of helium, as carried out a few months ago by Kinoshita at Cornell and by Bazley at the Bureau of Standards, agrees with the experimental data within the accuracy of the observations, which is one part in ten million. Surely in this case we 'got something out' of the equations that we did not put in." The same is true of the atomic spectra of heavier elements.

Wigner's last example comes from quantum electrodynamics: "Whereas Newton's theory of gravitation still had obvious connections with experience, experience entered the formulation of matrix mechanics only in the refined or sublimated form of Heisenberg's prescriptions. The quantum theory of the Lamb shift, as conceived by Bethe and established by Schwinger, is a purely mathematical theory and the only direct contribution of experiment was to show the existence of a measurable effect. The agreement with calculation is better than one part in a thousand."

Wigner concludes that "the enormous usefulness of mathematics in the natural sciences is something bordering on the mysterious and that there is no rational explanation for it". He ends his paper with the same spirit with which he began:

The miracle of the appropriateness of the language of mathematics for the formulation of the laws of physics is a wonderful gift which we neither understand nor deserve. We should be grateful for it and hope that it will remain valid in future research and that it will extend, for better or for worse, to our pleasure, even though perhaps also to our bafflement, to wide branches of learning.[5]

## Responses

There are examples beyond the ones mentioned by Wigner. Another oft-cited example is Maxwell's equations, derived to model the elementary electrical and magnetic phenomena known as of the mid-19th century. The equations also describe radio waves, discovered by David Edward Hughes in 1879, around the time of James Clerk Maxwell's death.

Wigner's original paper has inspired responses from researchers in many disciplines, among them Richard Hamming[6] in computer science, Arthur Lesk in molecular biology,[7] Peter Norvig in artificial intelligence,[8] Max Tegmark in physics,[9] Ivor Grattan-Guinness in mathematics[10] and Vela Velupillai in economics.[11]

Wigner's work provided insight into both physics and the philosophy of mathematics, and has been well cited in academic literature on the philosophy of physics and of mathematics. Wigner speculated on the relationship between the philosophy of science and the foundations of mathematics as follows:

It is difficult to avoid the impression that a miracle confronts us here, quite comparable in its striking nature to the miracle that the human mind can string a thousand arguments together without getting into contradictions, or to the two miracles of laws of nature and of the human mind's capacity to divine them.[12]

Later, Hilary Putnam (1975) explained the aforementioned two "miracles" as necessary consequences of a realist (but not Platonist) view of the philosophy of mathematics.[13] But in a passage discussing cognitive bias Wigner cautiously called[clarification needed] "not reliable", he went further:

The writer is convinced that it is useful, in epistemological discussions, to abandon the idealization that the level of human intelligence has a singular position on an absolute scale. In some cases it may even be useful to consider the attainment which is possible at the level of the intelligence of some other species.[14]

Whether humans checking the results of humans can be considered an objective basis for observation of the known universe is an interesting question, one followed up in both cosmology and the philosophy of mathematics.

Wigner also laid out the challenge of a cognitive approach to integrating the sciences:

A much more difficult and confusing situation would arise if we could, some day, establish a theory of the phenomena of consciousness, or of biology, which would be as coherent and convincing as our present theories of the inanimate world.[15]

He further proposed that arguments could be found that might

put a heavy strain on our faith in our theories and on our belief in the reality of the concepts which we form. It would give us a deep sense of frustration in our search for what I called 'the ultimate truth'. The reason that such a situation is conceivable is that, fundamentally, we do not know why our theories work so well. Hence, their accuracy may not prove their truth and consistency. Indeed, it is this writer's belief that something rather akin to the situation which was described above exists if the present laws of heredity and of physics are confronted.[5]

### Richard Hamming

Mathematician and Turing Award laureate Richard Hamming reflected on and extended Wigner's Unreasonable Effectiveness in 1980, discussing four "partial explanations" for it,[6] and concluding that they were unsatisfactory. They were:

1. Humans see what they look for. The belief that science is experimentally grounded is only partially true. Hamming gives four examples of nontrivial physical phenomena he believes arose from the mathematical tools employed and not from the intrinsic properties of physical reality.

• Hamming proposes that Galileo discovered the law of falling bodies not by experimenting, but by simple, though careful, thinking. Hamming imagines Galileo as having engaged in the following thought experiment (the experiment, which Hamming calls "scholastic reasoning", is described in Galileo's book On Motion.):

Suppose that a falling body broke into two pieces. Of course, the two pieces would immediately slow down to their appropriate speeds. But suppose further that one piece happened to touch the other one. Would they now be one piece and both speed up? Suppose I tie the two pieces together. How tightly must I do it to make them one piece? A light string? A rope? Glue? When are two pieces one?[16]

There is simply no way a falling body can "answer" such hypothetical "questions." Hence Galileo would have concluded that "falling bodies need not know anything if they all fall with the same velocity, unless interfered with by another force." After coming up with this argument, Hamming found a related discussion in Pólya (1963: 83-85).[17] Hamming's account does not reveal an awareness of the 20th-century scholarly debate over just what Galileo did.[clarification needed]

2. Humans create and select the mathematics that fit a situation. The mathematics at hand does not always work. For example, when mere scalars proved awkward for understanding forces, first vectors, then tensors, were invented.

3. Mathematics addresses only a part of human experience. Much of human experience does not fall under science or mathematics but under the philosophy of value, including ethics, aesthetics, and political philosophy. To assert that the world can be explained via mathematics amounts to an act of faith.

4. Evolution has primed humans to think mathematically. The earliest lifeforms must have contained the seeds of the human ability to create and follow long chains of close reasoning.

### Max Tegmark

A different response, advocated by physicist Max Tegmark, is that physics is so successfully described by mathematics because the physical world is completely mathematical, isomorphic to a mathematical structure, and that we are simply uncovering this bit by bit.[9][19] The same interpretation had been advanced some years previously by Peter Atkins.[20] In this interpretation, the various approximations that constitute our current physics theories are successful because simple mathematical structures can provide good approximations of certain aspects of more complex mathematical structures. In other words, our successful theories are not mathematics approximating physics, but simple mathematics approximating more complex mathematics.

### Ivor Grattan-Guinness

Ivor Grattan-Guinness found the effectiveness in question eminently reasonable and explicable in terms of concepts such as analogy, generalization, and metaphor.[10][clarification needed]

### Michael Atiyah

The tables were turned by Michael Atiyah with his essay "The unreasonable effectiveness of physics in mathematics". He argued that the toolbox of physics enables a practitioner like Edward Witten to go beyond standard mathematics, in particular the geometry of 4-manifolds. The tools of a physicist are cited as quantum field theory, special relativity, non-abelian gauge theory, spin, chirality, supersymmetry , and the electromagnetic duality.[21]

## References

1. ^ Wigner, E. P. (1960). "The unreasonable effectiveness of mathematics in the natural sciences. Richard Courant lecture in mathematical sciences delivered at New York University, May 11, 1959". Communications on Pure and Applied Mathematics. 13 (1): 1–14. Bibcode:1960CPAM...13....1W. doi:10.1002/cpa.3160130102. S2CID 6112252. Archived from the original on 2021-02-12.
2. ^ Note: Wigner's mention of Kellner and Hilleraas "... Jordan felt that we would have been, at least temporarily, helpless had an unexpected disagreement occurred in the theory of the helium atom. This was, at that time, developed by Kellner and by Hilleraas ..." refers to Georg W. Kellner (Kellner, Georg W. (1927). "Die Ionisierungsspannung des Heliums nach der Schrödingerschen Theorie". Zeitschrift für Physik. 44 (1–2): 91–109. Bibcode:1927ZPhy...44...91K. doi:10.1007/BF01391720. S2CID 122213875.) and to Egil Hylleraas.
3. ^ a b Wigner 1960, §Is the Success of Physical Theories Truly Surprising? p. 8
4. ^ Wigner 1960, p. 9
5. ^ a b Wigner 1960, p. 14
6. ^ a b Hamming, R. W. (1980). "The Unreasonable Effectiveness of Mathematics". The American Mathematical Monthly. 87 (2): 81–90. doi:10.2307/2321982. hdl:10945/55827. JSTOR 2321982. Archived from the original on 2022-06-22. Retrieved 2021-07-30.
7. ^ Lesk, A. M. (2000). "The unreasonable effectiveness of mathematics in molecular biology". The Mathematical Intelligencer. 22 (2): 28–37. doi:10.1007/BF03025372. S2CID 120102813.
8. ^ Halevy, A.; Norvig, P.; Pereira, F. (2009). "The Unreasonable Effectiveness of Data" (PDF). IEEE Intelligent Systems. 24 (2): 8–12. doi:10.1109/MIS.2009.36. S2CID 14300215. Archived (PDF) from the original on 2022-08-09. Retrieved 2015-09-04.
9. ^ a b Tegmark, Max (2008). "The Mathematical Universe". Foundations of Physics. 38 (2): 101–150. arXiv:0704.0646. Bibcode:2008FoPh...38..101T. doi:10.1007/s10701-007-9186-9. S2CID 9890455.
10. ^ a b Grattan-Guinness, I. (2008). "Solving Wigner's mystery: The reasonable (though perhaps limited) effectiveness of mathematics in the natural sciences". The Mathematical Intelligencer. 30 (3): 7–17. doi:10.1007/BF02985373. S2CID 123174309.
11. ^ Velupillai, K. V. (2005). "The unreasonable ineffectiveness of mathematics in economics". Cambridge Journal of Economics. 29 (6): 849–872. CiteSeerX 10.1.1.194.6586. doi:10.1093/cje/bei084.
12. ^ Wigner 1960, p. 7
13. ^ Putnam, Hilary (1975). "What is Mathematical Truth?". Historia Mathematica. 2 (4): 529–543. doi:10.1016/0315-0860(75)90116-0.
Reprinted in Putnam, Hilary (1975). Mathematics, Matter and Method: Philosophical Papers. Vol. 1. Cambridge University Press. pp. 60–78. ISBN 978-0-521-20665-5.
14. ^ Wigner 1960, p. 12 Footnote 11
15. ^ Wigner 1960, p. 13
16. ^ Van Helden, Albert (1995). "On Motion". The Galileo Project. Archived from the original on 21 December 2017. Retrieved 16 October 2013.
17. ^ Pólya, George; Bowden, Leon; School Mathematics Study Group (1963). Mathematical methods in science; a course of lectures. Studies in mathematics. Vol. 11. Stanford: School Mathematics Study Group. OCLC 227871299.
18. ^ Folland, Gerald B.; Sitaram, Alladi (1997). "The Uncertainty Principle: A Mathematical Survey". Journal of Fourier Analysis and Applications. 3 (3): 207–238. doi:10.1007/BF02649110. S2CID 121355943.
19. ^ Tegmark, Max (2014). Our Mathematical Universe. Knopf. ISBN 978-0-307-59980-3.
20. ^ Atkins, Peter (1992). Creation Revisited. W.H.Freeman. ISBN 978-0-7167-4500-6.
21. ^ Atiyah, Michael (2002). "The unreasonable effectiveness of physics in mathematics". In Fokas, A.S. (ed.). Highlights of Mathematical Physics. American Mathematical Society. pp. 25–38. ISBN 0-8218-3223-9. OCLC 50164838.