by Eugene Wigner
The enormous usefulness of mathematics in the natural sciences is something bordering on the mysterious, and there is no rational explanation for it.
tags: mathematics, physics, philosophy, epistemology, beauty, science
"Mathematics, rightly viewed, possesses not only truth, but supreme beauty — cold and austere, like that of sculpture, without appeal to any part of our weaker nature, without the gorgeous trappings of painting or music, yet sublimely pure, and capable of a stern perfection such as only the greatest art can show."
There is a story about two friends, who were classmates in high school, talking about their jobs. One of them became a statistician and was working on population trends. He showed a reprint to his former classmate. The reprint started, as usual, with the Gaussian distribution and the statistician explained the meaning of the symbols for the actual population, for the average population, and so on. His classmate was a bit incredulous and was not quite sure whether the statistician was pulling his leg.
"How can you know that?" was his query. "And what is this symbol here?"
"Oh," said the statistician, "this is π."
"What is that?"
"The ratio of the circumference of the circle to its diameter."
"Well, now you are pushing your joke too far," said the classmate, "surely the population has nothing to do with the circumference of the circle."
Naturally, we are inclined to smile about the simplicity of the classmate's approach. Nevertheless, when I heard this story, I had to admit to an eerie feeling because, surely, the reaction of the classmate betrayed only plain common sense.
The preceding two stories illustrate the two main points which are the subjects of the present discourse.
Mathematical concepts turn up in entirely unexpected connections. Moreover, they often permit an unexpectedly close and accurate description of the phenomena in these connections.
Just because of this circumstance, and because we do not understand the reasons of their usefulness, we cannot know whether a theory formulated in terms of mathematical concepts is uniquely appropriate.
We are in a position similar to that of a man who was provided with a bunch of keys and who, having to open several doors in succession, always hit on the right key on the first or second trial. He became skeptical concerning the uniqueness of the coordination between keys and doors.
It is therefore surprising how readily the wonderful gift contained in the empirical law of epistemology was taken for granted. The ability of the human mind to form a string of 1000 conclusions and still remain "right," which was mentioned before, is a similar gift.
Wigner's central mystery: Mathematics is a tool invented by humans for human purposes — and yet it describes the physical universe with terrifying precision. Why? We have no answer. We have only the eerie, beautiful fact.
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.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 be united, 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.
The nightmare: If we were somewhat less intelligent, the false theories we have would appear to us perfectly true — their predictions match experiment. But they are in fact incompatible with each other. As more false theories are discovered, they are bound to prove to be in conflict. Similarly, it is possible that the theories we consider "proved" by numerical agreement are false because they conflict with a 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, this is the nightmare of the theorist.
Let us consider a few examples of "false" theories which give, in view of their falseness, alarmingly accurate descriptions of groups of phenomena.
The success of Bohr's early and pioneering ideas on the atom was always a rather narrow one and the same applies to Ptolemy's epicycles. Our present vantage point gives an accurate description of all phenomena which these more primitive theories can describe.
From the paper: "The same is not true any longer of the so-called free-electron theory, which gives a marvelously accurate picture of many, if not most, properties of metals, semiconductors, and insulators. In particular, it explains the fact, never properly understood on the basis of the 'real theory,' that insulators show a specific resistance to electricity which may be 10²⁶ times greater than that of metals."
The free-electron theory raises doubts as to how much we should trust numerical agreement between theory and experiment as evidence for the correctness of the theory. We are used to such doubts.
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.
✦ memory · ☽ night · ∞ loops · ❧ margins · ◆ proof
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