Book Review: Fermat’s Last Theorem

Simon Singh has written on subjects ranging from a 17th century number theory problem and cryptography to cosmology and the Big bang theory. He may be a science writer, but at heart, he is a storyteller and in Fermat’s Last Theorem he shapes the three centuries long search for the solution of a mathematical problem into a gripping, at times poignant but always inspiring, tale. Singh brings a certain infectious enthusiasm to his subject that is contagious – once you get through the first few pages, you can be sure of not letting go until you turn over the very last page. There are certain dense, but truly satisfying, books on science and mathematics that, like a complex fugue, require a certain minimum amount of training and knowledge on the part of the reader to be truly appreciated. This is not one of them. With Singh’s writing flitting over the pages, light as a feather, Fermat’s Last Theorem is to books on mathematics what the Brandenburg Concertos are to Western classical music. With only a bare minimum of mathematical formalism, chockfull of entertaining anecdotes from the annals of mathematics and the world of mathematical puzzles and games, and filled with a cast of eccentric, but very much human mathematicians, Fermat’s Last Theorem is a very human story.

The book grabs the reader’s attention from the very first page, opening with mathematician Andrew Wiles scribbling away on the black board, in a tightly packed lecture hall at the Isaac Newton Institute in Cambridge, watched on by an audience holding their collective breath. Wiles jots down the last line – the statement of Fermat’s last theorem – and says “I think I’ll stop here”. The hall bursts into applause and cameras start flashing. But the foreword by John Lynch, producer of the BBC Horizon documentary film on which the book is based, prepares the reader not to be deceived by the euphoria. There’s a fateful twist in the tale. There’s a flaw in the proof.

Ahead of the euphoric scene that brings the first chapter to a close, the narrative digresses in time to Pythagorean Greece and into the clandestine Pythagorean brotherhood to trace the origins of the problem famously known as Fermat’s Last Theorem. Moving to 17th century France in the next chapter, the reader is introduced to the man who started it all: Pierre de Fermat. A civil servant by profession with an innate genius for number theory, Fermat, at a time when mathematics was almost a competitive sport than a professional pursuit, made significant contributions to number theory, the theory of probabilities, and laid down some of the basic precepts of the calculus. At the centre of the tale is a mischievous statement that Fermat wrote down in a margin of his copy of Diophantus’s Arithmetica, about the impossibility of solving a certain equation. He had, so Fermat claimed, “a truly marvellous demonstration of this proposition which this margin is too narrow to contain.” (!) This statement, by a master mathematician whose conjectures almost always turned out to be true, would go on to hold generations of mathematicians in thrall.

Andrew Wiles’s personal journey – his discovery of Fermat’s problem as a child, his training as a number theorist, his study of elliptic curves which inadvertently provided him with the tools needed to realise his childhood dream, and his eventual triumph – is the backbone of the story. Singh’s genius lies in seamlessly weaving the history and philosophy of mathematics, and most importantly, the lives of mathematicians into this main narrative fabric. There’s Fermat, the typical mischievous genius, “That damned Frenchman”; prolific Leonard Euler, the “Mathematical Cyclops” who would dash off a publication worthy calculation between the first and second calls for dinner; ‘Complex’ Augustin Cauchy, a religious bigot; young and rebellious Evariste Galois, was all of twenty years old when he was killed in a duel; paranoid Kurt Gödel, logician par excellence, proved that mathematics is inherently undecidable; the tragic Yutaka Taniyama, who committed suicide for no apparent reason other than having “lost confidence in” his “future”; and mild-mannered yet obsessive Andrew Wiles himself, who worked in secrecy for seven years to realise his childhood dream. My favourite moment in Fermat’s Last Theorem is that of two mathematicians, Barry Mazur and Ken Ribet, discussing a much-anticipated proof over cappuccino

Professor Mazur sipped his cappuccino and listened to Ribet’s idea. Then he stopped and stared at Ken in disbelief. ‘But don’t you see? You’ve already done it! All you have to do is add some gamma-zero of (M) structure and just run through your argument and it works. It gives you everything you need.’

Ribet looked at Mazur, looked at his cappuccino, and looked back at Mazur. It was the most important moment of Ribet’s career and he recalls it in loving detail. ‘I said you’re absolutely right – of course – how did I not see this? I was completely astonished because it had never occurred to me to add the extra gamma-zero of (M) structure, simple as it sounds.’

As Singh points out, “It should be noted that, although adding gamma-zero of (M) structure sounds simple to Ken Ribet, it is an esoteric step of logic which only a handful of the world’s mathematicians could have concocted over a casual cappuccino.”

This is a gem of a book that, like a shining beacon, strikes through the foreboding mists of both ancient and modern mathematics. I know this is a cliché, but I’ll still say it because it’s a cliché that fits this book like a dust jacket: if you decide to read just one book on mathematics, make it this one.

For those who are absolutely allergic to Xs, Ys, and Zs that even a small sprinkling of the dreaded letters of the alphabet over a few pages would trigger a hay fever reaction, here’s an embedded video of the BBC Horizon documentary film, Fermat’s Last Theorem, directed by Simon Singh:

 

For others, I would recommend reading the book before watching the film. The film is, without doubt, as gripping and moving as the book. If the embedded video doesn’t work use this link.

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