Why Beauty is Truth
By Ian Stewart – May 2nd, 2007
Soon after I was appointed to a lecturing position at the University of Warwick, in 1969, I was assigned a course on Galois Theory. This is a branch of algebra, mainly about how to solve equations, which came into being in the 19th century. This was the moment when mathematicians discovered how to make systematic use of symmetry. And the person who made the key discoveries, a young Frenchman called Évariste Galois, led a dramatic and rather tragic life, dying in a duel over a woman at the age of 21.
The idea of writing a popular science book, with Galois as the pivotal figure, took a while to emerge, but it goes right back to that lecture course, thirty years ago. Galois’s greatest achievement was to invent the concept of a ‘group’, which in due course gave rise to a huge and powerful theory, a kind of ‘calculus of symmetry’. Once mathematicians could specify symmetry as a precise concept, and work with it in a systematic and rigorous way, the topic really took off, and it now pervades the whole of mathematics, and many areas of science.
In fact, the story of symmetry is so huge and varied that no book on the subject could possibly cover everything. For example, the relation between symmetry and patterns in the natural world, such as the stripes on a tiger or ripples on a pond, would make a book in its own right. In fact, Martin Golubitsky and I wrote such a book 15 years ago: Fearful Symmetry. So this time I wanted to tackle something different.
The deepest use of symmetry occurs in fundamental physics. The two big areas, relativity and quantum mechanics, both rest on symmetry principles. Relativity is based on the symmetries of space and time; Quantum Mechanics is based on the rich and remarkable symmetries of fundamental particles. The currently fashionable (and controversial) topic of string theory tries to unify Relativity and Quantum Mechanics, essentially by unifying their symmetries in a suitable mathematical context.
The story that I finally settled on covered about four thousand years of human history. When the ancient Babylonians discovered how to solve quadratic equations, they set mathematics on a path that eventually led to Galois: what about more complicated equations? What about cubic, quartic, and quintic equations, for instance—ones that involved the third, fourth, or fifth powers of the unknown? Progress came from a variety of directions—Greek geometry, elegant work by the poet Omar Khayyam, and definitive algebraic solutions of cubic and quartic equations found by the mathematicians of Renaissance Italy. A key figure here was Girolamo Cardano, a polymath and a scoundrel, whose life reads like a sensationalist soap opera.
Galois disposed of the quintic equation, but not by solving it. Instead, he proved that no purely algebraic solution exists. This negative fact turned out to be far more important than any solution could ever have been, however, because of the method he created. Essentially, he showed that you can’t solve quintic equations because they have the wrong kind of symmetry. This beautiful and indirect way of thinking about algebra convinced mathematicians that group theory was worth developing. When Sophus Lie extended the ideas to the symmetries of ‘differential equations’, the favoured way to state laws of nature, symmetry concepts took centre stage in science as well as in pure mathematics.
The mathematics gets pretty complicated, but the key ideas are very simple, and those are all the story needs. And the people who contributed to the story are fascinating, their lives full of incident, with very human triumphs and tribulations. As the structure of the book became clearer, I realised that it had to be a ‘people book’, centred on the historical figures and their lives. Their mathematics, an integral part of those lives, then took care of itself.
Telling the story of symmetry that way leads to a fascinating question. For all but the final 150 years, practical considerations were almost completely absent. No one needed to solve cubics or quintics for practical reasons; what drove the research was the internal structure of mathematics itself, divorced from any considerations of the real world. And yet… what finally emerged was an immensely powerful technique for understanding the real world, at its deepest and most philosophical levels. The physicist Eugene Wigner, a significant player in the use of symmetry in quantum mechanics, called this mystery the ‘unreasonable effectiveness of mathematics’. What is the relation between a beautiful theory, and a true one? And that’s what my book Why Beauty Is Truth: A History of Symmetry is really about. I don’t claim to have an answer, but I do feel that the discovery of the underlying mathematics of symmetry has a lot to tell us about that relationship.