Sunday, August 29, 2010

Down in the dumps

Success is failure turned inside out -
The silver tint in the clouds of doubt,
And you never can tell how close you are,
It might be near when it seems afar;
So stick to the fight when you're hardest hit -
It's when things seem worst that you must not quit
- Author unknown

Sunday, August 15, 2010

358 Years of Inspiration

Name of the Book: Fermat’s Last Theorem

Author : Simon Singh

Price : Rs.425

Personal Rating: *****

“The sum of the squares of the two sides of a triangle is equal to the square of the third side. To put mathematically, x^2 + y^2 = z^2.”

Replace the number 2 in above equation with n, where n is a whole number greater than 2, and lo, we have the toughest problem in mathematics that took close to four centuries to be solved. This theorem was called Fermat’s Last Theorem, named after the creator of this riddle Pierre de Fermat, and it was the last of Fermat’s riddles to be solved. Inspired by the Pythagoras theorem, Fermat discovered that the equation x^n + y^n = z^n, where n>2, has no solution. Fermat’s proof of it never saw the light of day, but he did mention in a corner of a page of Book of Arithmetica that he had a proof. That was enough for many eminent mathematicians to embark on a search for the holy grail of mathematics.

It was Euler who provided the first breakthrough after almost a century. He proved that the theorem holds good for the case n=3. It was only an evidence in favour of the theory, and not a solid, infallible proof. Mathematicians accept nothing but infallible proofs. But Euler’s reasoning provided mathematicians with a much need shot in the arm. It gave them hope that Fermat’s Last Theorem was not insurmountable. French Academy of Science offered prizes and medals to the mathematician who could solve Last Theorem. Gabriel Lame, who had proved the case n=7, proclaimed that he was on the verge of cracking it, and so did Cauchy, and the race was on. In the end, neither of them succeeded; Ernst Kummer pointed out a flaw in both their techniques.

For the next couple of centuries, many mathematicians attempted to put an end to this theory in vain. Even though they could not solve the problem, they developed several mathematical tools in the process, some of which would later be used to solve the problem. After many unsuccessful attempts, it boiled down to two theories - Fermat’s Last Problem and Taniyama-Shimura conjecture. It was discovered that both the theories were inextricably linked. Proving the latter would mean the former was true, and vice-versa.This is where a mathematician named Andrew Wiles took charge.

Andrew Wiles, who had a penchant for puzzles from a young age, first encountered the Last Theorem when he was ten and cracking it became his childhood dream. Many years later, he set out to prove the Taniyama-Shimura conjecture and worked for seven years in secrecy. By the time Wiles began working on the problem, mathematics had developed by leaps and bounds. Although he had the benefit of intricate tools and hindsight to learn from the mistakes of all those who worked before him, it was his ingenuity and stroke of genius that gave the finishing touch to the problem. After 358 years, the Fermat’s last theorem was conquered.

The narration of the author Simon Singh is extremely lucid and racy. It could have well been a thriller novel. Barring a couple of chapters that covers elliptic cures and modular forms, which would be difficult to grasp if these are alien terms to readers, he has outlined mathematical concepts in simple and layman terms that will make for an interesting read even to those who dread mathematics. Besides concepts and proofs, the author gives an insight into the lives of many mathematicians like Pythagoras, Euler, Euclid, Cauchy, Galois, Wiles and many others. I was particularly intrigued by the account of Galois’ life. Galois had prodigious talent for mathematics but he always found himself at the center of political controversies and was killed when he was only 21. The night before his death, being sure that he wouldn’t live beyond the following morning, he scribbled hurriedly his discoveries and mailed them to his friend, requesting that, should he be killed, those papers be sent to Jacobi or Gauss. Little did he know that his discovery was going to play a major role in cracking Fermat’s Last Theorem.

Fermat’s Last Theorem is an enthralling saga about a riddle that confounded mathematicians for three centuries. But what took my fancy was the man behind the riddle. I couldn’t but marvel at the brilliance of this french mathematician. Mathematics was in it’s nascent stages, modular forms were unheard of and high-speed computers existed only in fantasy during his time. Yet, he claimed that he had a proof and it remained unknown only because that was too large to fit in the margin. A rare genius!