by Marinus Jan Marijs

**The applicability of mathematics to the physical world**

It has been said that:

Basic researchers working in pure mathematics often develop fundamental laws, even entire branches of math, without any specific application in mind. Yet, many of these posited laws turn out—sometimes centuries later—to perfectly describe the behaviour of the real world with remarkable precision. This phenomenon was best articulated in the early 1900s by the Hungarian physicist Eugene Wigner as the “unreasonable effectiveness of mathematics.

In ”The Unreasonable Effectiveness of Mathematics in the Natural Sciences” by Eugene Wigner we find the following observations:

The first point is 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. Second, it is just this uncanny usefulness of mathematical concepts that raises the question of the uniqueness of our physical theories…..

I would say thatmathematicsis the science of skillful operations with concepts and rules invented just for this purpose. The principal emphasis is on theinvention of concepts.Mathematicswould soon run out of interesting theorems if these had to be formulated in terms of the concepts which already appear in the axioms…..

The preceding discussion is intended to remind us, first, that it is not at all natural that “__laws of nature__” exist, much less that man is able __to discover them (__E. Schrodinger, in his What Is Life? (Cambridge: Cambridge University Press, 1945), p. 31, says that this second miracle may well be beyond human understanding).

THE ROLE OF MATHEMATICS IN PHYSICAL THEORIES

The statement that the laws of nature are written in the language of mathematics was properly made three hundred years ago;[It is attributed to Galileo]

It is true, of course, that physics chooses certain mathematical concepts for the formulation of the laws of nature, and surely only a fraction of all mathematical concepts is used in physics. It is true also that the concepts which were chosen were not selected arbitrarily from a listing of mathematical terms but were developed, in many if not most cases, independently by the physicist and recognized then as having been conceived before by the mathematician.

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 itself into contradictions, or to the two miracles of the existence of laws of nature and of the human mind’s capacity to divine them. The observation which comes closest to an explanation for the mathematical concepts’ cropping up in physics which I know is Einstein’s statement that the only physical theories which we are willing to accept are the beautiful ones. It stands to argue that the concepts of mathematics, which invite the exercise of so much wit, have the quality of beauty. However, Einstein’s observation can at best explain properties of theories which we are willing to believe and has no reference to the intrinsic accuracy of the theory. We shall, therefore, turn to this latter question.

IS THE SUCCESS OF PHYSICAL THEORIES TRULY SURPRISING?

A possible explanation of the physicist’s use of mathematics to formulate his laws of nature is that he is a somewhat irresponsible person. As a result, when he finds a connection between two quantities which resembles a connection well-known from mathematics, he will jump at the conclusion that the connection is that discussed in mathematics simply because he does not know of any other similar connection. It is not the intention of the present discussion to refute the charge that the physicist is a somewhat irresponsible person. Perhaps he is. However, it is important to point out thatthe 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.This shows that the mathematical language has more to commend it than being the only language which we can speak; it shows that it is, in a very real sense, the correct language. Let us consider a few examples.

It may be useful to illustrate the alternatives by an example. We now have, in physics, two theories of great power and interest: the theory of quantum phenomena and the theory of relativity. These two theories have their roots in mutually exclusive groups of phenomena. Relativity theory applies to macroscopic bodies, such as stars. The event of coincidence, that is, in ultimate analysis of collision, is the primitive event in the theory of relativity and defines a point in space-time, or at least would define a point if the colliding particles were infinitely small. Quantum theory has its roots in the microscopic world and, from its point of view, the event of coincidence, or of collision, even if it takes place between particles of no spatial extent, is not primitive and not at all sharply isolated in space-time. The two theories operate with different mathematical concepts the four dimensional Riemann space and the infinite dimensional Hilbert space, respectively. So far, the two theories could not beunited, that is, no mathematical formulation exists to which both of these theories are approximations. All physicists believe that a union of the two theories is inherently possible and that we shall find it. Nevertheless, it is possible also to imagine that no union of the two theories can be found. This example illustrates the two possibilities, of union and ofconflict, mentioned before, both of which are conceivable.

Let me end on a more cheerful note. 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.(Eugene Wigner).

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How can it be that mathematics, being after all a product of human thought which is independent of experience, is so admirably appropriate to the objects of reality? —Albert Einstein

The most incomprehensible thing about the universe is that it is comprehensible.—Albert Einstein

Richard Hamming:

But it is hard for me to see how simple Darwinian survival of the fittest would select for the ability to do the long chains that mathematics and science seem to require.

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.

Eugene Wigner:

Certainly it is hard to believe that our reasoning power was brought, by Darwin’s process of natural selection, to the perfection which it seems to possess.

And then there was the story of one the greatest mathematicians of the twentieth century Nobel prize winner Paul Dirac. Dirac, co-inventor of quantum mechanics, is now best known for his research on anti-matter. Paul Dirac was initially extremely hostile to religious views. Heisenberg recollected a conversation among young participants at the 1927 Solvay Conference about Einstein and Planck’s views on religion between Wolfgang Pauli, Heisenberg and Dirac. Dirac’s contribution was a criticism of the political purpose of religion, which was much appreciated for its lucidity by Bohr when Heisenberg reported it to him later. Among other things, Dirac said:

I cannot understand why we idle discussing religion. If we are honest—and scientists have to be—we must admit that religion is a jumble of false assertions, with no basis in reality. The very idea of God is a product of the human imagination. It is quite understandable why primitive people, who were so much more exposed to the overpowering forces of nature than we are today, should have personified these forces in fear and trembling. But nowadays, when we understand so many natural processes, we have no need for such solutions. I can’t for the life of me see how the postulate of an Almighty God helps us in any way. What I do see is that this assumption leads to such unproductive questions as why God allows so much misery and injustice, the exploitation of the poor by the rich and all the other horrors He might have prevented. If religion is still being taught, it is by no means because its ideas still convince us, but simply because some of us want to keep the lower classes quiet. Quiet people are much easier to govern than clamorous and dissatisfied ones. They are also much easier to exploit. Religion is a kind of opium that allows a nation to lull itself into wishful dreams and so forget the injustices that are being perpetrated against the people. Hence the close alliance between those two great political forces, the State and the Church. Both need the illusion that a kindly God rewards—in heaven if not on earth—all those who have not risen up against injustice, who have done their duty quietly and uncomplainingly. That is precisely why the honest assertion that God is a mere product of the human imagination is branded as the worst of all mortal sins.”

Heisenberg’s view was tolerant. Pauli, raised as a Catholic, had kept silent after some initial remarks, but when finally he was asked for his opinion, said: “Well, our friend Dirac has got a religion and its guiding principle is ‘There is no God and Paul Dirac is His prophet.'” Everybody, including Dirac, burst into laughter.

Remarks made during the Fifth Solvay International Conference (October 1927), as quoted in Physics and Beyond: Encounters and Conversations (1971) by Werner Heisenberg, pp. 85-86

In 1971, at a conference meeting, Dirac expressed his views on the existence of God. Dirac explained that the existence of God could only be justified if an improbable event were to have taken place in the past:

It could be that it is extremely difficult to start life. It might be that it is so difficult to start life that it has happened only once among all the planets. …Let us consider, just as a conjecture, that the chance life starting when we have got suitable physical conditions is 10^-100. I don’t have any logical reason for proposing this figure, I just want you to consider it as a possibility. Under those conditions…it is almost certain that life would not have started. And I feel that under those conditions it will be necessary to assume the existence of a god to start off life. I would like, therefore, to set up this connexion between the existence of a god and the physical laws: if physical laws are such that to start off life involves an excessively small chance, so that it will not be reasonable to suppose that life would have started just by blind chance, then there must be a god, and such a god would probably be showing his influence in the quantum jumps which are taking place later on. On the other hand, if life can start very easily and does not need any divine influence, then I will say that there is no god.

Later in life, Dirac’s views towards the idea of God were less hostile. As an author of an article appearing in the May 1963 edition of Scientific American, Dirac wrote:

It seems to be one of the fundamental features of nature that fundamental physical laws are described in terms of a mathematical theory of great beauty and power, needing quite a high standard of mathematics for one to understand it. You may wonder: Why is nature constructed along these lines? One can only answer that our present knowledge seems to show that nature is so constructed. We simply have to accept it. One could perhaps describe the situation by saying that God is a mathematician of a very high order, and He used very advanced mathematics in constructing the universe. Our feeble attempts at mathematics enable us to understand a bit of the universe, and as we proceed to develop higher and higher mathematics we can hope to understand the universe better.

Dirac was puzzled by the same fact that baffled Wigner and Einstein: that the human creation of mathematics enables us to understand the way the universe works. Based upon his mathematical view on the universe Paul Dirac came to the conclusion that the way the universe was created pointed in the direction of a creative transcendent intelligent force. His biographer physicist Graham Carmelo in ”Paul Dirac and the religion of mathematical beauty” says:

The words on his gravestone are ‘Because God made it that way…’.He (Dirac) almost certainly used those words.The Fifth Solvay International Conference (October 1927)

From back to front and from left to right :

- Piccard, E. Henriot, P. Ehrenfest, E. Herzen, Th. de Donder, E. Schrödinger, J.E. Verschaffelt, W. Pauli, W. Heisenberg, R.H. Fowler, L. Brillouin;
- Debye, M. Knudsen, W.L. Bragg, H.A. Kramers, P.A.M. Dirac, A.H. Compton, L. de Broglie, M. Born, N. Bohr;
- I. Langmuir, M. Planck, M. Curie, H.A. Lorentz, A. Einstein, P. Langevin, Ch.-E. Guye, C.T.R. Wilson, O.W. Richardson
Fifth conference participants, 1927. Institut International de Physique Solvay in Leopold Park