Going back to the roots in symmetrical fashion

- Why Beauty Is Truth

May 25, 2007

For many years, Ian Stewart has been the leading British writer on mathematics for a general audience. The title of his latest book, Why Beauty Is Truth, is poetic, but for a text on mathematics it is meaningless because beauty is known to be in the eye of the beholder, whereas truth in mathematics must be a matter of careful argument and common consent. The subtitle, A History of Symmetry, and the cover image of a (non-symmetric) butterfly are hardly more revealing: the simple geometric symmetry of an ideal butterfly is an almost insignificant part of Stewart's tale. The story that actually fills his pages runs as follows.

For at least 4,000 years (the earliest written records are from the area that is now Iraq), humans have challenged themselves to find numbers satisfying certain conditions, an activity that eventually became known as solving equations. Sometimes the unknown number appears with its own square, giving rise to the quadratic equations everyone learns to solve at school. If cubes (third powers) or fourth powers appear, the equations are more difficult, but by the 16th century these too could be handled. But all efforts to deal with fifth or higher powers failed and it was proved, most decisively by Abel in Norway in 1824, that such equations could not in general be solved in the form that was required.

In mathematics, however, there is rarely such a thing as a dead end, and work on equations led to new insights, one of which was that one can shuffle the roots of an equation around - mathematicians call this "permuting" them - yet certain properties arising from them will stay the same. The idea of working with permutations led in turn to what mathematicians call a group, and group theory eventually became a major branch of mathematics.

What is a group and why does it matter? Imagine first a counting system that goes 0, 1, 2, 3, 0, 1, 2, 3, 0, 1, 2, 3I. This may seem strange but we know that English clocks repeatedly run through the cycle I10, 11, 12, 1, 2, 3I and we have no difficulty doing simple arithmetic with them. In the 0, 1, 2, 3, 0I system it must be the case that 3 + 1 = 0, 3 + 2 = 1, and so on, and we can write out an addition table:

+ | 0 1 2 3
0 | 0 1 2 3
1 | 1 2 3 0
2 | 2 3 0 1
3 | 3 0 1 2

Now imagine a quite different situation: a square that can be rotated through a 1/4 turn, an action that we may call R1; or though a 1/2 turn (R2); or a 3/4 turn (R3); or a full turn, which is the same as doing nothing (R0). In every case the square ends up looking the same. Now we can follow one action by another and write, for example, R2.R3 for R2 followed by R3, and the rules turn out to be:

  .  | R0 R1 R2 R3
R0 | R0 R1 R2 R3
R1 | R1 R2 R3 R0
R2 | R2 R3 R0 R1
R3 | R3 R0 R1 R2

It is easy to see that the two tables are, in essence, identical: just swap + for . and 0 for R0, and so on. This is the kind of thing mathematicians get excited about because they can forget individual elements - numbers in the first table, rotations in the second - and just concentrate on the patterns or structures they generate. The structure here is a group of order 4, and there are infinitely many other groups, some with beautiful and intricate properties.

Groups were first discovered in the early 19th century by two French mathematicians, Évariste Galois and Augustin-Louis Cauchy. Galois died in a duel at the age of 20, leaving some badly written fragments containing astonishing new ideas. Cauchy spent most of his life in the Paris mathematical establishment and left hundreds of published papers but is a less romantic figure than Galois and is barely discussed by Stewart. Their independent contributions to group theory were brought together by others from the 1850s onwards. At about the same time, the wider applicability of groups was spotted by the English mathematician Arthur Cayley, but he does not figure in Stewart's book at all. Then in the early 20th century, physicists realised that groups gave them a language for describing the behaviour of quantum particles that, like roots of equations, have properties that remain the same when the pack is shuffled: this is the "symmetry" of Stewart's title. Thus Stewart's lengthy historical yarn spins eventually into a story about the nature of the universe and Theories of Everything.

The thread is so long and at times so thin that the second half of the book feels unrelated to the first and the whole at times lacks focus. There are good stories, but too many of them and too often superficially told. At one point, Stewart comments that mathematicians seldom make good historians, a remark he may come to regret for the book is riddled with errors. To give just two examples: the first arithmetic text to bring Hindu-Arabic numerals to Europe was not the Liber abaci (1202) of Leonardo Pisano but writings known in Latin as Algorismi (derived from a seminal text by al-Khwarizmi, 825), which reached Spain by the 10th century. And the diagrams that Stewart attributes to John Wallis in 1673, for illustrating imaginary numbers, bear no resemblance whatsoever to Wallis's ingenious constructions with conic sections; in any case he later negates his praise of Wallis by claiming that imaginary numbers were not understood until the 20th century. He would not allow such contradictions in a mathematical proof.

Factual errors are one thing, though in this book they come in multifarious forms. Errors of interpretation are another. Take, for instance, the claim that by the end of the 15th century Europe was getting its second wind. Are we to understand that the first wind was classical? In which case the centre of mathematical activity was more often Alexandria than Athens.

Should we suppose that Hellenistic Egypt was part of Europe while Islamic Spain was not? More insidiously, Stewart presents as a reason for this second wind not the recovery of Greek texts, or an infusion of new ideas from the Islamic world, but a cause internal to early modern Europe, the "struggling free of the embrace of the church of Rome and its fear of anything new". To suggest that the solution of cubic equations somehow became possible only when the cardinals took their eye off the ball is absurd.

Similarly unhelpful is the use of historical anecdote as padding. Euclid plays no useful role in the book, for instance, except to fill the long space between the Babylonians and Omar Khayyam. Endless details of childhoods and parentage become tedious, though not so irritating as the nicknames that make the book so difficult to navigate: epithets such as "the cunning fox" or "the weakly bookworm" do not make mathematicians more endearing, only more easily forgettable.

Despite these many criticisms, and especially towards the end of the book, an important story unfolds. Those who learnt from their newspapers in March 2007 that "Mathematicians have mapped the inner workings of one of the most complicated structures ever studied: the object known as the exceptional Lie group E8" will not understand the technicalities any better for reading Stewart's book, but they will at least have some sense of what the Lie group E8 is, and why it is something that even non-mathematicians should know about.

Why Beauty Is Truth: A History of Symmetry

Author - Ian Stewart
Publisher - Basic Books
Pages - 290
Price - £15.99
ISBN - 9780465082360

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