During a radio interview given by Philip Larkin in May 1974 to promote his High Windows collection, he pointed out that a good poem is like an onion. On the outside, both are pleasingly smooth and intriguing, and they become more and more so as their successive layers of meaning are revealed. His aim was to write the perfect onion.
The poetry of science is in some sense embodied in its great equations, and these equations can also be peeled. But their layers represent their attributes and consequences, not their meanings.
Despite the best efforts of poets and literary critics, no one has ever come up with an uncontroversial definition of a poem. No such problems beset mathematicians asked to define the term "equation". An equation is fundamentally an expression of perfect balance. For the pure mathematician, unconcerned with science, an equation is an abstract statement, having nothing to do with the real world. So when mathematicians see an equation such as y = x + 1, they think of y and x as abstract symbols, not as representing things that actually exist.
It is possible to imagine a universe in which mathematical equations have nothing to do with the workings of nature. Yet the marvellous thing is that they do. Scientists routinely cast their laws in the form of equations featuring symbols that represent quantities measurable by experimenters. It is through this symbolic representation that the equation has become one of the most powerful weapons in the scientists' armoury.
Best known of all the scientific equations is E = mc squared, first suggested by Einstein in 1905. Like most great equations, it asserts an equality between things that superficially appear to be quite different - energy, mass and the speed of light in a vacuum. Through this equation Einstein predicted that, for any mass (m), if you multiply it twice by the speed of light in a vacuum (c), the result is exactly equal to its corresponding energy (E). Like every other equation, E = mc squared balances two quantities, in the same way as a pair of weighing scales, with the = sign serving as the pivot.
This celebrated equation began its life as a confident speculation, and only decades later became part of the corpus of scientific knowledge, after experimenters had shown that it does indeed concur with nature. In common with all great equations, E = mc squared is in many ways similar to a great poem. Just as a perfect sonnet is spoiled if so much as a word or punctuation mark is changed, not a single detail of a great equation such as E = mc squared can be altered without rendering it useless. Great equations also share with fine poetry an extraordinary power - poetry is the most concise and highly charged form of language, just as equations are the most succinct form of understanding of the aspect of physical reality that they describe. E = mc squared is itself enormously powerful: its symbols encapsulate knowledge that can be applied to every energy conversion, from ones in every cell of every living thing on Earth, to the most distant cosmic explosion. Better yet, it seems to have held good since the beginning of time.
Great equations are just as rich a stimulus as poetry to the imagination. Shakespeare could no more have foreseen the multiple meanings that readers have perceived in "Shall I compare thee to a summer's day?" than Einstein could have predicted the myriad consequences of his equations of relativity.
None of this is to imply that poetry and scientific equations are the same. Every poem is written in a particular language and loses its magic in translation, whereas an equation is expressed in the universal language of mathematics: E = mc squared is the same in English as it is in Urdu. Also, poets seek multiple meanings and interactions between words and thoughts, whereas scientists intend their equations to convey a single, logical meaning.
The meaning with which a great scientific equation usually furnishes us is called a law of nature. An analogy popularised by the physicist Richard Feynman helps to clarify this relationship between equations and laws. Imagine people watching a game of chess. If they had never been taught the rules of chess, they could work them out simply by observing how the players moved the various pieces. Now imagine that the players are not playing ordinary chess, but are moving the pieces accord ing to a much more complicated set of rules on a hugely extended board. For the observers to be able to work out the rules of the game, they would have to watch parts of it extremely carefully, looking for patterns and any other clues they could muster. That, in essence, is the predicament of scientists. They closely observe nature - the movements of the pieces - and try to glean the underlying laws.
Armies of thinkers have been defeated by the enigma of why so many laws of nature can be written down conveniently as equations. Examples abound. In physics, Einstein's equation of general relativity gives a new theory of gravity by equating the curvature of space/time to the energy density of matter. In quantum theory, Schrödinger's equation describes the behaviour of matter in the micro-world, enabling an understanding of atoms and molecules that had proved impossible with older ideas. In ecology, the logistic map caricatures how the population of fish in a pond develops from one generation to the next (the map has many other applications). John Nash, the Nobel prize-winning mathematician who suffered from schizophrenia and is the subject of the forthcoming film A Beautiful Mind, came up with equations that determined how two people ought to behave in competitive games. Economists and biologists studying the theory of natural selection later found that his ideas were extremely relevant to their work.
Why is it that so many laws can be expressed as an absolute imperative, that two apparently unrelated quantities (the equation's left and right sides) are exactly equal? Nor is it clear why fundamental laws exist at all. A popular, tongue-in-cheek explanation is that God is a mathematician, an idea that unhelpfully replaces profound questions with a doubly unverifiable proposition.
Even more contentious than the provenance of scientific equations is the question of whether they are invented or discovered. The Indian-American astrophysicist Subrahmanyan Chandrasekhar probably spoke for most great theoreticians when he remarked that when he found some new fact or insight, it appeared to him to be something "that had always been there and that I had chanced to pick up". According to this view, the equations that underlie the workings of the universe are in some sense "out there", independent of human existence, so that scientists are cosmic archaeologists, trying to unearth laws that have lain hidden since time began. The origin of the laws remains a mystery.
Of the hundreds of thousands of research scientists who have ever lived, very few have an important equation to their name. Two scientists who were adept at discovering fundamental equations and especially perceptive about the role of mathematics in science were Albert Einstein and the almost comparably brilliant theoretical physicist Paul Dirac. Neither was a mathematician per se, but both were remarkable in their ability to write down new equations that were as fecund as the greatest poetry. And both were captivated by the belief that the fundamental equations of physics must be beautiful.
This may sound strange. The concept of beauty is unwelcome in intellectual circles, and certainly has no place in academic critiques of high art. Yet it is a word that readily comes to all of us when we are moved by the sight of a smiling baby, a mountain vista, an exquisite orchid. What does it mean to say that an equation is beautiful? Fundamentally, it means that the equation can evoke the same rapture as other things that we describe as beautiful. Like a great work of art, a beautiful equation has among its attributes more than mere attractiveness - it has universality, simplicity, inevitability and elemental power. Think of masterpieces such as Cézanne's Apples and Pears, Buckminster Fuller's geodesic dome, Judi Dench's interpretation of Lady Macbeth, Ella Fitzgerald's recording of Manhattan. During my first experience of each of them, I realised that I was in the presence of something monumental in conception, fundamentally pure, free of excrescence and crafted so carefully that its power would be diminished if anything in it were changed.
An additional quality of a good scientific equation is that it has utilitarian beauty. It must tally with the results of every relevant experiment and, even better, make predictions that no one has made before. This aspect of an equation's effectiveness is akin to the beauty of a finely engineered machine of the kind we see in Stanley Kubrick's film Full Metal Jacket, when Gomer Pyle starts talking to his rifle. Pyle praises its meticulous construction, delighting in the qualities that make it fit for its lethal purpose. It would not be nearly so beautiful if it didn't work.
The concept of beauty was especially important to Einstein. According to his son Hans, "He had a character more like that of an artist than of a scientist as we usually think of them. For instance, the highest praise for a good theory was not that it was correct or exact, but that it was beautiful." He once went so far as to say that "the only physical theories that we are willing to accept are the beautiful ones", taking it for granted that a good theory must agree with experiment.
Dirac was even more emphatic than Einstein in his belief in mathematical beauty as a criterion for the quality of theories. In the latter part of his career, he spent much time touring the world, giving lectures on the origins of the equation that bears his name, stressing that the pursuit of beauty had always been a lodestar as well as an inspiration. During a seminar in Moscow in 1955, when asked to summarise his philosophy of physics, he wrote on the blackboard in capital letters, "Physical laws should have mathematical beauty."
Both he and Einstein were responsible for some of the seminal equations of modern science, which, through their concision, power and fundamental simplicity, can be regarded as some of the most beautiful poetry - the most exquisite onions - of the 20th century.