Unreasonably Effective

In a much quoted academic paper, Eugene Wigner once wrote:

The miracle of the appropriateness of the language of mathematics to 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 worse, to our pleasure, even though perhaps also to our bafflement, to wide branches of learning. [1]

Wigner dubbed the amazing success of mathematics in accurately describing and predicting the physical world, “the unreasonable effectiveness of mathematics” [2]. It is quite astonishing. Why should it be that, for example, when Einstein was looking for some mathematical description of his theory of general relativity, he discovered that Riemann’s non-Euclidean geometry, invented about half a century earlier, was the perfect fit?

How is it that recent experiments to measure the magnetic moment of an electron should so precisely match the predictions of quantum electrodynamics, even when measured at a precision of eight parts in a trillion? One of the inventors of the theory of quantum electrodynamics, Freeman Dyson, said, “I’m amazed at how precisely Nature dances to the tune we scribbled so carelessly fifty-seven years ago, and at how the experimenters and the theorists can measure and calculate her dance to a part in a trillion.” [3]

Mario Livio writes:

In the late 1960s, physicists Steven Weinberg, Sheldon Glashow, and Abdus Salam developed a theory that treats the electromagnetic force and weak nuclear force in a unified manner. This theory, now known as the electroweak theory, predicted the existence of three particles (called the W+, W-, and Z bosons) that had never before been observed. The particles were unambiguously detected in 1983 in accelerator experiments (which smash one subatomic particle into another at very high energies) led by physicists Carlo Rubbia and Simon van der Meer. [4]

As Einstein put it, “How is it possible that mathematics, a product of human thought that is independent of experience, fits so excellently the objects of physical reality?” [5]

First, we might question the claim that mathematics is “independent of experience”. Sir Michael Atiyah wrote:

If one views the brain in its evolutionary context then the mysterious success of mathematics in the physical sciences is at least partially explained. The brain evolved in order to deal with the physical world, so it should not be too surprising that it has developed a language, mathematics, that is well suited for the purpose. [6]

This seems to accord with the latest research by cognitive scientists:

The cognitive scientists base their conclusions on what they regard as a compelling body of evidence from the results of numerous experiments. Some of these tests involved functional imaging studies of the brain during the performance of mathematical tasks. Others examined the math competence of infants, of hunter-gatherer groups such as the Mundurukú, who were never exposed to schooling, and of people with various degrees of brain damage. Most of the researchers agree that mathematical capacities appear to be innate. For instance, all humans are able to tell at a glance whether they are looking at one, two or three objects (an ability called subitizing). A very limited version of arithmetic, in the form of grouping, pairing, and very simple addition and subtraction, may also be innate, as is perhaps some very basic understanding of geometrical concepts (although this assertion is more controversial). Neuroscientists have also identified regions in the brain, such as the angular gyrus in the left hemisphere, that appear to be crucial for juggling numbers and mathematical computations, but which are not essential for language or the working memory. [7]

This explains only how it is that we come to have the ability to grasp the mathematical nature of the world: it does not explain why the world is, to such an impressive extent, mathematical. It is quite conceivable that the world should not exhibit regularities at all. It is even easier to conceive of a world which, although regular, exhibits a particular regularity rather than the universal regularity that we observe.

Why is it that the same law governs the motion of the heavens and the falling of an apple? As Sir Isaac Newton described it:

“Nature is very consonant and conformable to herself.” [8]

This was not always believed to be the case. The Scholastic philosophers, following Aristotle, believed that each thing behaves as it does because of its own nature. Thus it is the nature of the Moon that makes it orbit the Earth, it is the nature of the sea that causes it to rise and fall in tides, and it is the nature of an apple to fall toward the centre of the Earth. According to such a view, natural philosophy consists of determining and describing the various different natures of different entities. Newton’s great insight was that the universe is governed by universal laws: the orbit of the Moon, the rise and fall of the tides, and the falling of an apple are all the result of a universal law of gravitation. This was a revolutionary realisation:

This principle of nature being very remote from the conceptions of Philosophers, I forbore to describe it in that book, least I should be accounted an extravagant freak and so prejudice my Readers against all those things which were the main designe of the book. [9]

So not only is it possible to conceive of a world exhibiting local, particular regularity but no universal laws, that is indeed how the world was perceived for many centuries.

It also seems possible to conceive of a world that exhibits regularities which are nevertheless not mathematical: such a possibility is described by George Coyne and Michael Heller in their book A Comprehensible Universe [10]. However, if we accept the thesis that our mathematics is an evolutionary adaptation to the regularity of the world, then perhaps such a world could not exist in practice. If we found ourselves in a regular world then necessarily those regularities would be mathematical, as what we mean by mathematics would depend on the regularities whose nature our brains had evolved to accommodate.

If Atiyah is correct, then it is not ultimately the mathematical nature of the world that is surprising, but rather the universal regularity of the world. On the other hand, it is far from obvious that Atiyah is correct. Richard Hamming wrote:

If you pick 4,000 years for the age of science, generally, then you get an upper bound of 200 generations. Considering the effects of evolution we are looking for via selection of small chance variations, it does not seem to me that evolution can explain more than a small part of the unreasonable effectiveness of mathematics. [11]

The only answer to Hamming appears to be that, long before the age of science, we evolved, as the research of cognitive scientists mentioned above appears to tell us, an innate faculty for very rudimentary mathematics. We must then appeal to the nature of mathematics itself, what Atiyah calls the “abstract hierarchical nature of mathematics” [12], to explain how this rudimentary mathematics can be so remarkably successful when extended to every level from the subatomic to the galactic. Unfortunately, appealing to the nature of mathematics as we know it does seem to rather beg the question.

Nevertheless, whether we accept Atiyah’s evolutionary explanation or not, we are still left with the mystery of why it should be that the world exhibits universal regularities — regularities which, whether necessarily or contingently, we can express with astonishing accuracy in the language of mathematics. Coyne and Heller quote Einstein:

The very fact that the totality of our sense experiences is such that by means of thinking it can be put in order, this fact is one which … we shall never understand. One may say “the eternal mystery of the world is its comprehensibility”. [13]

Why is there order rather than chaos? Of course, this question cannot be answered rationally, for it is the validity of rationality that we are trying to explain. Coyne and Heller quote Sir Karl Popper:

Neither logical argument nor experience can establish the rationalist attitude; for only those who are ready to consider argument or experience, and who have therefore adopted this attitude already, will be impressed by them. That is to say, a rationalist attitude must be first adopted if any argument or experience is to be effective, and it cannot therefore be based upon argument or experience. So rationalism is necessarily far from comprehensive or self-contained. [14]

Coyne and Heller continue:

Why then should we not adopt irrationalism? Because when one confronts rationalism with irrationalism, one immediately sees that rationalism is a value. Therefore, “the choice before us is not simply an intellectual affair, or a matter of taste. It is a moral decision”. Indeed, the choice of value is the moral decision. Popper calls this kind of rationalism critical rationalism, the one “which recognizes the fact that the fundamental rationalist attitude results from an (at least tentative) act of faith — from faith in reason”. [15]

So there is a sense in which this mystery — this fundamental mystery about the world — cannot be answered, at least not rationally. As Einstein said, “we shall never understand” [16].

But the inquiring philosopher, who cannot help attempting to explain, is unlikely to be satisfied by this. As Coyne and Heller put it, “The principal tenet of rationality is that one is never allowed to cease asking questions if there remains something to be sought.” [17]

It seems clear to me that there is indeed something remaining to be sought.


[1] Wigner, E. P. 1960. Communications in Pure and Applied Mathematics, vol. 13, no. 1. Reprinted in Saatz, T. L., and Weyl, F. J., eds. 1969. The Spirit and the Uses of the Mathematical Sciences (New York: McGraw-Hill).

[2] Ibid.

[3] Livio, M. 2009. Is God a Mathematician? (New York: Simon & Schuster). p. 223.

[4] Ibid. p. 224

[5] Einstein, A. 1934. “Geometrie und Erfuhrung” in Mein Weltbild (Frankfurt am Main: Ullstein Materialien).

[6] Atiyah, M. 1995. Times Higher Education Supplement, 29th of September. Quoted in Livio, M. op. cit. p. 243.

[7] Livio, M. op. cit. p. 232

[8] Newton, I. 1704. Opticks (London: Smith & Walford)

[9] Ibid. The book to which he refers is his Philosophiae Naturalis Principia Mathematica, 1687 (London: Pepys)

[10] Coyne, G. and Heller, M. 2008. A Comprehensible Universe: The Interplay of Science and Theology (Springer)

[11] Hamming, R. W. 1980. The American Mathematical Monthly, 87(2), 81. Quoted in Livio, M. op. cit. p. 246.

[12] Atiyah, M. op. cit.

[13] Einstein, A. 1936. “Physics and Reality” in Journal of the Franklin Institute 221. pp. 349 – 382. Quoted in Coyne, G. and Heller, M. op. cit. p. 3

[14] Popper, K. P. 1974. The Open Society and Its Enemies: Volume 2 (London: Routledge & Kegan Paul) p. 224. Quoted in Coyne, G. and Heller, M. op. cit. p. 9

[15] Coyne, G. and Heller, M. op. cit. p. 9. In which they quote from Popper, K. P. op. cit. pp. 230 – 232

[16] Einstein, A. op. cit.

[17] Coyne, G. and Heller, M. op. cit. p. xiv

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Published in: on July 26, 2009 at 9:39 pm  Leave a Comment  

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