Sunday, 30 May 2004

Fermat’s Last Theorem [Fermat's Enigma] – Simon Singh *****

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 is calledFermat’s Enigma in the US (it could also be because of another book of the same name by Amir Aczel). But 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 it threw down a gauntlet to hundreds of mathematicians who were to follow until it was finally achieved in the 20th century. Don’t worry if the maths doesn’t interest you – the story will, both in its historical context and in the insight into the work and nature of modern mathematicians.
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. In fact it seems so difficult to come up with a straightforward solution to this problem that Fermat has to be more than a little doubted.
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. What can I say? Buy it!
Review by Brian Clegg
Community Review – Katy
I am 12 years old and I love maths. This book was amazing! It is the best book I have ever ever read. I really enjoyed all of it. I especially liked the bits with examples in because it helped me to understand more because I am doing my GCSE in maths in 4 months. It was fascinating to learn about the theorem and the story behind it. It tells you about lots of different people who have attempted to prove the theorem. It also talks about lots of different areas of mathematics associated with the theorem that do not even sound remotely similar. It is amazing to think that such a simple theorem has taken so long to solve – 358 years in fact. This book has been so inspirational to me. It has made my passion for mathematics stronger. I would strongly recommend it to anyone. You will not be able to put it down!!!!!!!!
Additional Review – 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.

Tuesday, 25 May 2004

The Demon Haunted World – Carl Sagan *****

An eloquent plea for reason and the scientific method when the media are pumping unexplained phenomena and X-Files fiction at us all the time. There’s nothing wrong at all with science fiction or fantasy, as long as we are aware that’s what it is – but Sagan points out just how easy it is for us to believe that the ‘truth is out there’ just because we want it to be.
Everything from presidents consulting horoscopes to witch burning and miracle cures come under Sagan’s logical but still human eye. Whether your demons are traditional or modern world alien abductors, spirit mediums or faith healers, he painstakingly shows that we are much more likely to invent the supernatural than to experience it for real.
Although Sagan goes on a bit, it’s a great counter to wide-eyed acceptance – as useful in business as it is in dealing with the unexplained. The cover asks ‘Are we on the brink of a new Dark Age of irrationality and superstition?’ and it’s a question anyone with an interest in science has to face up to.
Perhaps the only weakness Sagan doesn’t face is the scientists who have don’t take a scientific, rational view (‘let’s test this theory with experiment’) but instead refuse to even consider phenomena outside of their experience (‘There’s no point looking into it; it doesn’t exist.’) But we definitely need a few more sceptical works like this to put along the uncritical pap that TV companies regularly feed into our homes as ‘documentaries’.
Review by Brian Clegg

Wednesday, 19 May 2004

The Code Book – Simon Singh ****

Not in quite the same class as Singh’s definitive Fermat’s Last Theorem, but still a fascinating survey of the history of code making from the earliest days, through thewartime Enigma machines to the present day complexity of 128 bit encryption.
The great thing about the book is probably not the mathematical complexity of modern codes and ciphers but the very human studies of the use and need to transmit secret messages, from the ancient Greeks writing on a messengers bald head, then waiting for the hair to regrow, through the cipher that doomed Mary Queen of Scots to the race to crack the World War II Enigma machines.
One of Simon Singh’s great strengths is being able to get across complex principles in a way that the everyday reader doesn’t find intimidating. This shines through in The Code Book I don’t know if recent editions have the rather cringe-making ‘cipher challenge’ in the back – we can but hope this has disappeared by now – but this shouldn’t put anyone off.
Whatever your mathematical level or inclination, this book is likely to have something to offer.
Review by Brian Clegg

Friday, 14 May 2004

Chaos – James Gleick *****

The book on the most amazing development in mathematics since the introduction of the zero – chaos theory. Has a nice dramatic style that highlights the importance of the people involved in the maths, but can’t detract from the remarkable implications of this fundamentally new understanding of how almost everything really works, rather than the approximations we are used to in science…
… at least, that’s certainly the feeling you’ll get when you read the book. But then you take a step back and think, okay, what chaos done for the world since 1987 (or thereabouts) when the book first came out. And you have to say – very little. Chaos is fascinating, but usually turns out to be fundamentally impractical.
Does this detract from the book? Not at all. It’s still a fascinating read after all these years, and even if the best chaos can give us is Jeff Goldblum in Jurassic Park it doesn’t cease to intrigue.
Review by Peter Spitz