Why Beauty is Truth (62 page)

Why is mathematics so useful for purposes that its inventors never intended?

The Greek philosopher Plato said that “God ever geometrizes.” Galileo said much the same thing: “Nature's great book is written in mathematical language.” Johannes Kepler set out to find mathematical patterns in planetary orbits. Some of them led Newton to his law of gravitation; others were mystical nonsense.

Many modern physicists have commented on the astonishing power of mathematical thought. Wigner alluded to the “unreasonable effectiveness” of mathematics as a way to understand nature; the phrase appears in the title of an article he wrote in 1960. In the body of the article he said he would tackle two main points:

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.

And:

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.

Paul Dirac believed that in addition to being mathematical, nature's laws also had to be beautiful. In his mind, beauty and truth were two sides of the same coin, and mathematical beauty gave a strong clue to physical truth. He even went so far as to say he would prefer a beautiful theory to a correct one, and that he valued beauty above simplicity: “The research worker, in his efforts to express the fundamental laws of nature
in mathematical form, should strive mainly for mathematical beauty. He should still take simplicity into consideration in a subordinate way to beauty . . . where they clash the latter must take precedence.”

Interestingly, Dirac's concept of beauty in mathematics differed considerably from that of most mathematicians. It did not include logical rigor, and many steps in his work had logical gaps—the best-known example being his “delta function,” which has self-contradictory properties. Nevertheless, he made very effective use of this “function,” and eventually mathematicians reformulated the idea rigorously—at which point it was indeed a thing of beauty.

Still, as Dirac's biographer Helge Kragh has remarked, “All of [Dirac's] great discoveries were made before [the mid-1930s], and after 1935 he largely failed to produce physics of lasting value. It is not irrelevant to point out that the principle of mathematical beauty governed his thinking only during the later period.”

Not irrelevant, perhaps, but not correct either. Dirac may have made the principle explicit during his later period, but he was using it earlier.
All
of his best work is mathematically elegant, and he relied on elegance as a test of whether he was heading in a fruitful direction. What all this suggests is not that mathematical beauty is
the same as
physical truth but that it is
necessary
for physical truth. It is not sufficient. Many beautiful theories have turned out, once confronted with experiments, to be complete nonsense. As Thomas Huxley said, “Science is organized common sense, where many a beautiful theory was killed by an ugly fact.”

Yet there is much evidence that nature, at root, is beautiful. The mathematician Hermann Weyl, whose research linked group theory and physics, said, “My work has always tried to unite the true with the beautiful and when I had to choose one or the other, I usually chose the beautiful.” Werner Heisenberg, a founder of quantum mechanics, wrote to Einstein,

You may object that by speaking of simplicity and beauty I am introducing aesthetic criteria of truth, and I frankly admit that I am strongly attracted by the simplicity and beauty of the mathematical schemes which nature presents us. You must have felt this too: the almost frightening simplicity and wholeness of the relationship, which nature suddenly spreads out before us.

Einstein, in turn, felt that so many fundamental things are unknown—the nature of time, the sources of ordered behavior of matter, the shape of the universe—that we must remind ourselves how far we are from understanding anything “ultimate.” To the extent that it is useful, mathematical elegance provides us only local and temporary truths. Still, it is our best way forward.

Throughout history, mathematics has been enriched from two different sources. One is the natural world, the other the abstract world of logical thought. It is these two in combination that give mathematics its power to inform us about the universe. Dirac understood this relationship perfectly: “The mathematician plays a game in which he himself invents the rules while the physicist plays a game in which the rules are provided by nature, but as time goes on it becomes increasingly evident that the rules which the mathematician finds interesting are the same as those which nature has chosen.” Pure and applied mathematics complement each other. They are not poles apart, but the two ends of a connected spectrum of thought.

The story of symmetry demonstrates how even a negative answer to a good question (“can we solve the quintic?”) can lead to deep and fundamental mathematics. What counts is
why
the answer is negative. The methods that reveal this can be used to solve many other problems—among them, deep questions in physics. But our story also shows that the health of mathematics depends on the infusion of new life from the physical world.

The true strength of mathematics lies precisely in this remarkable fusion of the human sense of pattern (“beauty”) with the physical world, which acts both as a reality check (“truth”) and as an inexhaustible source of inspiration. We cannot solve the problems posed by science without new mathematical ideas. But new ideas for their own sake, if carried to extremes, can degenerate into meaningless games. The demands of science keep mathematics running along fruitful lines, and frequently suggest new ones.

If mathematics were entirely demand-driven, a slave of science, you would get the work you expect from a slave—sullen, grudging, and slow. If the subject were entirely driven by internal concerns, you would get a spoilt, selfish brat—pampered, self-centered, and full of its own importance. The best mathematics balances its own needs against those of the outside world.

This is what its unreasonable effectiveness derives from. A balanced personality learns from its experiences, and transfers that learning to new circumstances. The real world inspired great mathematics, but great mathematics can transcend its origins.

The unknown Babylonian who discovered how to solve a quadratic equation could never have realized, in his wildest dreams, what his legacy would be more than three thousand years later. No one could have predicted that questions about the solvability of equations would lead to one of the core concepts of mathematics, that of a group, or that groups would prove to be the language of symmetry. Even less could anyone have known that symmetry would unlock the secrets of the physical world.

Being able to solve a quadratic has very limited utility in physics. Being able to solve a quintic is even less useful, if only because any solution must be numerical, not symbolic, or else employ symbols specially invented for the purpose, which do little more than cover the question with a fig leaf. But understanding why quintics cannot be solved, appreciating the crucial role of symmetry, and pushing the underlying idea as far as it could go—that opened up entire new physical realms.

The process continues. The implications of symmetry for physics, indeed for the whole of science, are still relatively unexplored. There is much that we do not yet understand. But we do understand that symmetry groups are our path through the wilderness—at least until a still more powerful concept (perhaps already waiting in some obscure thesis) comes along.

In physics, beauty does not automatically ensure truth, but it helps.

In mathematics, beauty
must
be true—because anything false is ugly.

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WEBSITES

A. Hulpke, Determining the Galois group of a rational polynomial:
http://www.math.colosate.edu/hulpke/talks/galoistalk.pdf

The MacTutor History of Mathematics archive:
http://www-history.mcs.st-andrews.ac.uk/index.html