Beauty of Numbers
New Statesman
January 2000

This week the trio of Tony Blair, Carol Vorderman and Johnny Ball launched Maths Year 2000, the government’s attempt to make Britain a more numerate nation. Most of the effort will be focussed on events that will either encourage children to realise the importance of basic maths or help adults to overcome poor numeracy skills. We are now bombarded on a daily basis with figures of all kinds, from mortgage rates to health statistics, and so a certain level of numeracy is essential.

There will also be a significant investment aimed at persuading teenagers that mathematics is a vital stepping stone towards lucrative careers in engineering, biotechnology and computing. In short, the government wants to persuade our youngsters to stick with maths so that Britain will have the brains needed to reap the rewards of the Information Age, e-commerce and the biotechnology revolution.

In addition to maths for day to day living and maths for profit, there is a third reason for studying mathematics, one that risks being overlooked during Maths Year 2000. Ever since Pythagoras came up with his theorem, mankind has studied mathematics for its own sake for the sheer joy of comprehending the deep truths that inhabit the abstract world of numbers. The motivation for so-called pure mathematicians is neither practical nor financial, but rather spiritual. Pure mathematicians are noble creatures, innocent and virtuous, driven by curiosity and passion.

As a non-mathematician who spent two years writing about pure mathematicians, I believe that they have much in common with poets, painters and musicians. However, mathematicians seem to suffer from an image problem, and instead of being viewed as creative thinkers of the highest order who tackle the toughest conundrums imaginable, society sees them as nerds and geeks who spend all day pushing buttons on calculators.

One of the first people who attempted to improve the image of mathematicians was the Cambridge number theorist G.H. Hardy. In 1940 he published “A Mathematician’s Apology”, a justification of his own life. Graham Greene hailed it alongside Henry James’s notebooks as ‘the best account of what it was like to be a creative artist.’ Sixty years later Hardy’s book is still available in most bookshops, is still one of the top twenty best-selling maths books and continues to outsell James’s notebooks.

In order to explain his work to non-experts, Hardy pointed out the similarities between mathematics and art:

   “A mathematician, like a painter or a poet, is a maker
   of patterns. If his patterns are more permanent than
   theirs it is because they are made with ideas.
   A painter makes patterns with shapes and colours,
   a poet with words... A mathematician, on the other
   hand, has no material to work with but ideas, and
   so his patterns are likely to last longer.”

Mathematicians attempt to prove conjectures, and achieve this by constructing proofs, which are patterns of logical arguments. Sometimes the proofs are simple and elegant, such as the proof that the square root of two is irrational, which is to say that it cannot be written as a fraction. This proof can be expressed in just half a dozen lines. In other cases, the proofs are enormously complicated, running to a hundred pages of dense equations, such as the proof of Fermat’s Last Theorem. Despite their obvious differences, both proofs have an immense mathematical beauty.

In the former case, the beauty emerges suddenly from the climax. The proof begins by assuming that the square root of two is rational, then goes on to demonstrate that this is impossible, therefore the opposite must be true. This approach is known as reductio ad absurdum. According to Hardy,

   "...reductio ad absurdum is one of a mathematician’s
   finest weapons. It is a far finer gambit than any chess
   gambit: a chess player may offer the sacrifice of a pawn
   or even a piece, but the mathematician offers the game.”

In the latter case, the beauty of the proof results from its intricacy. The proof twists and turns and has numerous parallel themes that collide at crucial moments, creating a vast braided edifice. Such a proof is similar to an epic drama with dramatic plot twists and characters who clash at pivotal moments. Alternatively, it is like a magnificent musical composition, whose various motifs converge to create a dazzling finale.

The aesthetic beauty of mathematics is not just wishful thinking by a community of academics desperate to obtain some glamour for their discipline, rather it is a crucial guide to mathematical truth. Although a mathematical proof consists of a series of logical steps, there is no logical way to determine these steps when first confronted with a problem. Creating a proof requires creativity, determination and intuition, and along the way a mathematician will rely on an aesthetic judgement in order to gauge whether or not he is on the right track. Hardy, who took a rather extreme view on this matter, stated:

   “Beauty is the first test: there is no permanent place
   in the world for ugly mathematics".

If mathematics can be beautiful, then why do most students fail to see it? Barry Lewis, the director of Maths Year 2000, recently tried to get to grips with the discrepancy by comparing mathematics and music. He pointed out the similarity between a page of mathematics and a sheet of music, stating that both contain strange symbols that are abstract, stylised and remote. However, the crucial difference is that the musical symbols can be expressed in a way that makes their beauty accessible, by which I mean the music itself can be appreciated without necessarily understanding musical notation or theory. However, there is no obvious equivalent representation for the majority of mathematics, and so there is no easy way for the non-mathematician to access the emotion of mathematics. The only way to understand the beauty of mathematics is to understand the symbols, which requires years of intense study.

Having tried to convey the intellectual and emotional value of mathematics, there may be those who still feel that the pursuit of mathematical proofs is a pointless waste of time. I would offer such people a second reason for encouraging research into these ethereal problems. Every so often pure ideas find an unexpected application, which then results in a practical benefit for society.

For example, pure mathematics is at the heart of encryption technology. For thousands of years mathematicians have studied prime numbers, those which have no divisors except 1 and the number itself. Modern mathematicians discovered in the 1970s an equation that used the properties of prime numbers as part of radically better system of guaranteeing privacy. This so-called public-key cryptography is now providing the infrastructure for e-commerce. When you enter your credit details on the Internet, they are encrypted using pure mathematics so that only the dealer can decrypt your message and complete the transaction. The entire boom in e-commerce would not have been possible without pure mathematics. Similarly, military and government encryption is also based on prime number theory. In fact, the world’s biggest employer of mathematicians is America’s National Security Agency, which is responsible for protecting U.S. communications and deciphering the foreign communications.

Hardy’s own research, however, has had no such direct application. The final paragraph of his book is a plea for recognition based on the inherent value of pure mathematics:

   “The case for my life, then, or for that of any one
   else who has been a mathematician in the same
   sense in which I have been one, is this: that I have
   added something to knowledge, and helped others
   to add more; and that these somethings have a value
   which differs in degree only, and not in kind, from
   that of the creations of the other great mathematicians,
   or of any of the other artists, great or small, who have
   left some kind of memorial behind them.”

Furthermore, Hardy could rest assured that his work would live on forever. Once established, a mathematical proof remains true for eternity, because it does not rely on a subjective opinion or imperfect measurement. Artistic values go in and out of fashion and scientific views of the world are continually overturned or revised, but mathematical truths are eternal. We have long since abandoned the cosmology and medicine of the Ancient Greeks, but the mathematical ideas of Euclid are still taught in today’s universities. As Hardy himself put it:

   “Archimedes will be remembered when Aeschylus
   is forgotten, because languages die and mathematical
   ideas do not. ‘Immortality’ may be a silly word, but
   probably a mathematician has the best chance of
   whatever it may mean.”