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Fermat's Last Theorem - Simon Singh ***** 

CLASSICS REVISITED

This is the first popular maths book I ever read - and the one that persuaded me I wanted to be involved in the field of popular science. Just as the US publishers of Harry Potter and the Philosopher’s Stone reckoned the US public couldn’t cope with the word ‘philosopher’ and changed the title, this was originally called Fermat’s Enigma in the US, but such is its longstanding acclaim it's ended up with the correct name there too. Crazy assumptions from publishers apart, it’s the superb story of a bizarre little problem that no one could solve until the ever-wily mathematician Fermat scribbled in a margin that he had a wonderful solution, only there wasn’t room to write it down.

Fermat may well have been boasting, but his marginal claim threw down a gauntlet to hundreds of mathematicians who were to follow in his footsteps and fail, until it was finally achieved in the 20th century. Don’t worry if the maths itself isn't of great interest to you – the story will, both in its historical context and in the insight into the work and nature of modern mathematicians - at least, relatively modern given the book is a good 25 years old now.

In some ways the star of the book is Andrew Wiles, the British mathematician who pretty well single-handedly cracked the problem with an unusual level of secrecy, rather than the typical sharing approach of the profession. But equally it’s Fermat himself.

Whether or not Fermat actually had a solution is a moot point – but he certainly didn’t have Wiles’ complex approach and it's entirely possible it was a boast that he could have fulfilled. It seems so difficult to come up with a straightforward solution to this problem that Fermat has to be more than a little doubted. The nature of Wiles' solution is such that many mathematicians struggled with it, and Singh can only really give us an impression of what was involved. Any attempt to give meaning to the 100 plus page proof requires mathematics that is beyond the casual reader, and it's probably fair to say that bringing any clarity to the nature of the proof is the weakest part of this book.

Like all the best popular science books – and this certainly is one of the best – it brings in a whole range of extras historically and mathematically to add to the fascinating cast. Singh's writing style is fluid and genial. He came from a TV background, and the book combines the accessibility of that genre with the ability to go into far more depth than a TV documentary can. It's a milestone in the development of the popular science genre.

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Review by Brian Clegg - See all Brian's online articles or subscribe to a weekly email free here
Additional Review by Stephen Goldberg ****
Fermat’s last theorem was that a certain equation, under certain circumstances, had no possible solution. This theorem was finally proven in 1995 by mathematician Andrew Wiles. What made Fermat’s last theorem so intriguing to mathematicians was that Pierre de Fermat, in 1637, claimed to have proven it but left behind no written proof. Since that time and until 1995, mathematicians around the world have been trying to prove this theorem. It is not even known if Fermat himself actually proved it. The object of this book was to explain how this puzzle was finally solved. But the book is not just about Andrew Wiles. Author Simon Singh takes the reader through a fascinating tour of the history of mathematics before delivering the solution to us.
On his way to proving Fermat’s theorem, Wiles used a variety of techniques developed by earlier mathematicians. When Singh takes us though Wiles work and the use of earlier mathematical tools, he takes extensive detours to give significant biographical information on these earlier mathematicians. In this, Singh did a most admirable job. The book starts with Wiles’ presentation of his proof in 1993, but quickly detours to discuss the Greek mathematicians Pythagoras and Euclid. As Singh leads us through mathematical history he also pays significant attention to notable mathematicians Leonhard Euler (1707-1783), David Hilbert (1862-1943), and Alan Turing (1912-1954), among others. Particularly interesting was the chapter “A mathematical disgrace” where Singh discusses the difficulties faced by women mathematicians, most notably Sophie Germaine (1776-1831) and Emmy Nother (1882-1935). Also interesting was how Wiles worked in almost complete seclusion for a number of years. After Wiles presented his proof in 1993, errors were found, and he struggled for another two years before finally completing his work.
Where the book fails is in trying to actually explain number theory. There is a lot of math in this book, some of it relegated to appendices at the end. Very difficult to understand were E-series and M-series. Singh also failed to adequately explain mathematical techniques such as the method of Kolyvagin and Flach or the Taniyama–Shimura conjecture. If the objective of the book was to actually explain the proof of Fermat’s theorem then it fails as I understood it no better after having read the book than before. Where the book succeeds is in explaining how mathematicians build on other mathematician’s work and how a proof in mathematics, based on logical reasoning, is conceptually different than proof in other sciences that are based on experimentation and observation. The writing style was very accessible and easy to understand (aside from the math) and the biographies he writes are fascinating. Overall, this book was well worth reading for anyone interested in the history of science or mathematics.
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