Through perfect balance we find universal beauty

February 22, 2002

Although scientific equations are not easily accessible, like painting and poetry, they possess a unique aesthetic value, argues Graham Farmelo.

Westminster Abbey is not the sort of place you would expect to find scientific equations. But it does feature one, named after the great British theoretical physicist and aesthete Paul Dirac, born a century ago this year. His master equation, one of the great triumphs of modern science, was the fruit of his quasi-religious conviction that the fundamental equations of the universe are beautiful.

Dirac's scientific method was based on the assumption that the beauty of these equations is mathematical, based on symbols and the logic that binds them. For Dirac, mathematical beauty was more profound than the concept of beauty in art, literature and music. In his view, whereas we can disagree about the aesthetic merits of, say, Rembrandt and Warhol, mathematical beauty transcends the personal. Such beauty is universal, even if its appreciation is not. To those without mathematical training who inquired about his aesthetic credo, the famously taciturn Dirac would reply firmly but courteously: "You could not possibly understand."

This sort of comment, along with Dirac's championing of what he saw as the objective beauty of mathematics and his indifference to the arts, would not find favour among today's cultural commentators or in a world in which scientists are expected as a matter of principle to engage with people about their work. But does Dirac have a point? Should scientists be more assertive in promoting mathematical beauty above the subjective aesthetics of the arts?

Anyone who works with mathematics will confirm that it has a beauty of its own, able to induce the same rapture as a great work of art. But not all mathematicians agree on what this beauty entails. In 1988, The Mathematical Intelligencer published a questionnaire inviting readers to score a selection of theorems for their beauty on a scale of one to ten. The magazine received more than 70 replies. There was a fair amount of agreement about which theorems were most beautiful, but the fact that all the responses were not the same indicated that mathematicians do not share the same notion of what constitutes beauty.

Dirac believed that all fundamental equations are beautiful. From this, he evolved a new technique of "playing with pretty mathematics", regardless of any application the work may have. It was "good luck" if it then turned out that the work did apply to the way the universe worked. He even insisted that it was more important for an equation to be beautiful than to agree with experimental data.

Few scientists could work successfully this way, but for Dirac it was a productive credo, at least when he was young. It enabled him to write down his magical equation for the behaviour of the electron, which he later used to predict successfully the existence of antimatter. We now know that antimatter at one time made up almost half the material contents of the universe during the early moments of the big bang, so it is true to say that the Dirac equation enabled him to foresee half of all material existence.

As beauty was taking centre stage in fundamental science, it was taking a back seat in the arts. While Dirac, Albert Einstein and other great scientists were becoming obsessed by beauty, the word was becoming infra-dig in the arts. "What is beauty, anyway?" scoffed Picasso, adding that he hated the aesthetic game played by "the mandarins who 'appreciate'

beauty". He had a point, of course. The classical notion of beauty - an attribute of something that cannot be even slightly altered without destroying its power- certainly does not apply to all works of art, especially to those of modernism and postmodernism.

Not that beauty has ever been completely abandoned as an aspiration. Henry Moore once said that the greatest sculptures, for example those by Michaelangelo, can be viewed - indeed, should be viewed - from all distances since new aspects of beauty will be revealed in every detail. He made that remark to the great Indian-American astrophysicist Subrahmanyan Chandrasekhar, who noted that the beauty Moore described is precisely the kind possessed by Einstein's general theory of relativity. Whether one admires the surface shape of the theory or one immerses oneself in its fine details, its inevitability is entrancing - nothing in it can be changed without ensuring its destruction. It was the fragile beauty of the theory that convinced physicists of its worth long before it had any serious support from experiment.

But of all the art forms, it is poetry whose beauty is most akin to that of mathematics. Just as poetry is the most concise and highly charged form of language, the great equations of science are the most succinct form of understanding of the aspect of physical reality they describe. Each draws their power from concision, from the perfect positioning of its constituent symbols. To change anything in Shakespeare's sonnet Farewell, thou art too dear for my possessing would be as deleterious to its impact as marking the slightest alteration to the Dirac equation would be.

Much modern poetry is, in its way, as difficult as theoretical physics. Although almost all equations are written in mathematical language inaccessible to the majority, the same is true of many of the finest modern poems. The work of the revered poet Geoffrey Hill, for example, might be regarded as the string theory of modern literature.

One of the tasks of the literary critic is to broaden (as well as deepen) popular appreciation of the art they are studying. The task of today's science populariser is the same. Dirac was not alone in believing it impossible to communicate the idea of mathematical beauty to someone who has little or no mathematics. The great intuitive physicist and populariser Richard Feynman agreed. He went so far as to say that C. P. Snow's "two cultures" refers not to the arts and the physical sciences, but to those who have the experience of understanding mathematics well enough to appreciate nature, and those who do not.

Even if Feynman was right about the difficulty of discussing mathematical beauty to those who are not mathematically trained, one could argue that we must never accept the two cultures notion. If the slippery word culture has any useful meaning these days, there is only one culture, and it includes both science and art. Scientists probably have a tougher job than artists to explain the aesthetic value of their work to the public. But in the end, most fundamental science is publicly funded and whatever beauty means, most people have a sense that its pursuit is well worth supporting.

Graham Farmelo is head of science communication at the Science Museum, London, associate professor of physics at Northeastern University, United States, and editor of It Must Be Beautiful : Great Equations of Modern Science , published next week by Granta, priced £20.00.

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