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31 January 2000

A theorem is a joy for ever

The government wants us to improve our numeracy skills to help us cash in on e-commerce. But the rea

By Simon Singh

Mortgage rates. Health statistics. Sales prices. We are bombarded on a daily basis with figures of all kinds, and yet the majority of children and adults in Britain remain wary of maths and unembarrassed by their poor numeracy skills. International comparisons raise the alarming possibility of a nation so numerically challenged that it will fail to cash in on the e-commerce revolution, or reap the rewards of the Information Age. Enter Maths Year 2000, a government initiative – launched the other day by Tony Blair, Carol Vorderman and Johnny Ball – which aims to make Britons understand the importance of maths for day-to-day living and profit.

There is a third reason for studying mathematics, one that risks being overlooked during Maths Year 2000. Maths is beautiful. 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 found that they have much in common with poets, painters and musicians. However, mathematicians suffer from an image problem, and instead of being viewed as creative thinkers, they are regarded as 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, in which he 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.”

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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 and its 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, and therefore the opposite must be true. This approach is known as reductio ad absurdum.

Other proofs are enormously complicated, running to 100 pages of dense equations, such as the proof of Fermat’s Last Theorem. Here, 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.

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. 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 challenge our national blind spot 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. The crucial difference is that the musical symbols can be expressed in a way that makes their beauty accessible – the music itself can be appreciated without necessarily understanding musical notation or theory. There is no obvious equivalent representation for the majority of mathematics, however, 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.

The appeal of mathematics is not purely ethereal though. Every so often, pure ideas find an unexpected application, which results in a practical benefit for society.

For example, pure mathematics is at the heart of encryption technology. 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.

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 . . . 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.”

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 out of fashion and scientific views are continually overturned or revised, but mathematical truths are eternal.

Simon Singh is the author of “Fermat’ s Last Theorem” and “The Code Book – the science of secrecy from Ancient Egypt to quantum cryptography” . Details of Maths Year 2000 can be found at

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