The Miraculous Effectiveness of Mathematics

From Nobel Prize Winner Promotes ID, Circa 1960

In 1960 Nobel prize winning physicist Eugene Wigner published a brief article entitled “The Unreasonable Effectiveness of Mathematics in the Natural Sciences.” See it here. In this article Wigner describes as “miraculous” (1) that “laws” of nature exist; and (2) that we should be able to discover those laws.

vjtorley has taken the time to give us a nice summary of and commentary on the article:


…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….

The depth of thought which goes into the formulation of the mathematical concepts is later justified by the skill with which these concepts are used. The great mathematician fully, almost ruthlessly, exploits the domain of permissible reasoning and skirts the impermissible. That his recklessness does not lead him into a morass of contradictions is a miracle in itself: 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. However, this is not our present subject….

The physicist is interested in discovering the laws of inanimate nature. In order to understand this statement, it is necessary to analyze the concept, “law of nature.”

The world around us is of baffling complexity and the most obvious fact about it is that we cannot predict the future. Although the joke attributes only to the optimist the view that the future is uncertain, the optimist is right in this case: the future is unpredictable. It is, as Schrodinger has remarked, a miracle that in spite of the baffling complexity of the world, certain regularities in the events could be discovered. One such regularity, discovered by Galileo, is that two rocks, dropped at the same time from the same height, reach the ground at the same time. The laws of nature are concerned with such regularities. Galileo’s regularity is a prototype of a large class of regularities. It is a surprising regularity for three reasons.

The first reason that it is surprising is that it is true not only in Pisa, and in Galileo’s time, it is true everywhere on the Earth, was always true, and will always be true. This property of the regularity is a recognized invariance property and, as I had occasion to point out some time ago, without invariance principles similar to those implied in the preceding generalization of Galileo’s observation, physics would not be possible. The second surprising feature is that the regularity which we are discussing is independent of so many conditions which could have an effect on it. It is valid no matter whether it rains or not, whether the experiment is carried out in a room or from the Leaning Tower, no matter whether the person who drops the rocks is a man or a woman…

The preceding two points, though highly significant from the point of view of the philosopher, are not the ones which surprised Galileo most, nor do they contain a specific law of nature. The law of nature is contained in the statement that the length of time which it takes for a heavy object to fall from a given height is independent of the size, material, and shape of the body which drops…

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. [Note 6: 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.] …

[T]he use of complex numbers is in this case [to describe the Hilbert space in quantum mechanics – VJT] not a calculational trick of applied mathematics but comes close to being a necessity in the formulation of the laws of quantum mechanics. Finally, it now begins to appear that not only complex numbers but so-called analytic functions are destined to play a decisive role in the formulation of quantum theory. I am referring to the rapidly developing theory of dispersion relations.

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

[I]t is possible that the theories, which we consider to be “proved” by a number of numerical agreements which appears to be large enough for us, are false because they are in conflict with a possible more encompassing theory which is beyond our means of discovery. If this were true, we would have to expect conflicts between our theories as soon as their number grows beyond a certain point and as soon as they cover a sufficiently large number of groups of phenomena. In contrast to the article of faith of the theoretical physicist mentioned before, this is the nightmare of the theorist.

It is even possible that some of the laws of nature will be in conflict with each other in their implications, but each convincing enough in its own domain so that we may not be willing to abandon any of them. We may resign ourselves to such a state of affairs or our interest in clearing up the conflict between the various theories may fade out. We may lose interest in the “ultimate truth,” that is, in a picture which is a consistent fusion into a single unit of the little pictures, formed on the various aspects of nature….

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… So far, the two theories could not be united, that is, no mathematical formulation exists to which both of these theories are approximations.

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. Mendel’s laws of inheritance and the subsequent work on genes may well form the beginning of such a theory as far as biology is concerned. Furthermore, it is quite possible that an abstract argument can be found which shows that there is a conflict between such a theory and the accepted principles of physics. The argument could be of such abstract nature that it might not be possible to resolve the conflict, in favor of one or of the other theory, by an experiment. Such a situation would 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.


Two things should be apparent from the foregoing quotes. First, the atheist’s faith in the enduring constancy of nature is pure superstition. Nature is not to be trusted; there is no reason to think that it will behave tomorrow as it did today. You can’t trust something mindless to keep behaving itself. You can only trust a Person to do something like that.

Second, the inadequacy of the Darwinian account of the origin of human intelligence should now be apparent. Our survival as a species does not require us to be able to discover laws of nature. Nor is it clear that human beings could only have emerged in a universe with mathematically interesting and science-friendly properties (such as universal laws that happen to be simple enough for us to comprehend). The only hypothesis that accounts for these things in a non-arbitrary fashion is the hypothesis that the universe was designed to be understood by its intelligent inhabitants.

Should this hypothesis be correct, then we would predict that the Designer would not want us to tie ourselves up in a mass of seeming contradictions – such as the apparent conflict between quantum mechanics and relativity. This the intelligent design hypothesis would predict that the current theoretical tensions in the fields of physics and cosmology will be successfully resolved, and that the best theories in each domain of science can all be fused into a single ultimate truth.

These are trying times for scientists, as this article shows (see especially articles 4, 8, 9, 10, 11, 12). Faint-hearted souls might be tempted to toss in the towel and admit defeat: “We’ll never understand it all.” Belief in God, far from being a science-stopper, is the only belief that can counter this defeatist frame of mind, and encourage scientists to keep doing more research. For God alone can guarantee that the universe is ultimately rational.

We can never be God, and understand it all; but it is honourable and fitting to put in the work to understand as much as we lawfully can, as adopted Sons of God.

After all, with greater understanding of God’s handiwork, comes greater praise for our Creator, Redeemer, Judge, and Rightful King!


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