Fermat's Last Theorem

Title: Fermat’s Last Theorem

Author: Simon Singh

Publisher: Fourth Estate 2002 (First published: 1997)

ISBN: 1-84115-791-0

Pages: 323

Simon Singh is a writer, television producer, and great popularizer of science. His family emigrated from Punjab, India in 1950 and settled in England where Singh was born. He has a Ph.D in particle physics and worked for the BBC. His staunch opposition to quackery such as chiropractice caused him to be sued for libel, which he won. This book was his first, but the mastery of words and ideas doesn’t betray the novice in him. In fact, this book is a good example of how a scientific book should be written. Pierre de Fermat was a French mathematician in the 17^th century, who made a proposition that xⁿ + yⁿ = zⁿ is not true for n>2. He also noted down that he had a marvellous proof, which doesn’t fit in the margin of his book. This comment tantalized the world’s mathematicians for 350 years in search of the elusive proof. The theorem, which in fact was nothing more than a conjecture due to lack of proof, was called the ‘last theorem’ because all the others had been proved in less than a century of Fermat’s death. The last theorem stood tall, mocking the scholars who tried hard in vain when at last the professionals abandoned the quest altogether, even though hefty prizes were in store for the man who proved it.

Pythagoras’ theorem is a fundamental mathematical equation that is known to everybody who had had basic schooling. It linked the hypotenuse of a right-angled triangle to its sides by the equation x² + y² = z². Fermat formulated a variation to this equation, as shown above, and surmised that it won’t hold true for n>2. Proof of this theorem was not supplied and is assumed to be lost. Every mathematical concept need to be rigorously proved by logical arguments. In this respect, it differs from scientific proof. Every scientific theory is tested by experiments and its predictive power is verified by other set of experiments. If the volume of results supporting the theory is considerable, the theory is accepted, until it is proved wrong by another set of experiments. Even then, the original theory may still be used to explain a limited range of observations. Scientific proof is based on fallible human judgment in an experiment whereas mathematical proof is based on cold, infallible logic and is true till the end of time.

Pierre de Fermat (1601 - 1665) was an amateur mathematician,  being a civil servant and judge in an obscure province of France. He had a talented mind, and was a co-founder of the probability theory with Blaise Pascal, which grew from a collaboration between these two geniuses. Fermat was very secretive, reluctant to share results or proofs. The author claims that he was the inventor of calculus, though this is traditionally attributed to Newton. In 1934, Louis Trenchard Moore discovered a note from Newton saying that he developed his calculus based on Monsieur Fermat’s method of drawing tangents (p.47). Fermat got interested in number theory from the book ‘Arithmetica’, by the Greek mathematician, Diophantus, written about 14 centuries before. Fermat stated the theorem thus, Cubem autem in duos cubos, aut quadratoquadratum in duos quadratoquadratos, et generaliter nullam in infinitum ultra quadratum potestatem in duos eiusdem nominis fas est dividere, which means “It is impossible for a cube to be written as a sum of two cubes or a fourth power to be written as the sum of two fourth powers or, in general, for any number which is a power greater than the second to be written as a sum of two like powers” (p.66). In 1847, the French Academy of Sciences introduced a gold medal and a cash award of 3000 francs for the person who proves the theorem. Intense competition developed between Augustin Cauchy and Gabriel Lame, two famous French mathematicians, but both proofs were proved wrong by Ernst Kummer, a German mathematician who also proved by logic that a solution was not possible with the day’s mathematics. Work on it soon froze due to this, but was again kept warm by a curious incident. Paul Wolfskehl, a wealthy German industrialist gave a new life to it in 1908. Wolfskehl decided to commit suicide at a particular date and time due to a failed love affair. On that night, seeing that he had time, he spent some time in his personal library examining some books, Krummer’s treatise among them. Wolfskehl found an error in Krummer’s argument and was so engrossed in it that it dawned and he was invigorated with the joy of living. He decided to rewrite his will to offer 100,000 German marks (1 million pounds in today’s money) to be awarded to the person who proves the theorem before September 2007, 100 years hence!

Andrew Wiles was born in 1953 in England, but later emigrated to U.S. As a young boy of 10, he was fascinated by the Fermat’s theorem which he discovered in a book in his public library. He was intrigued and interested by the centuries-old search for a proof of it and made it his mission for life. To do that, he must be in a position to handle the language of mathematics like an expert and a conventional education was the prime requirement. Wiles went on to become Professor at the Institute of Advanced Studies in Princeton, where Einstein spent his latter half of his career. His doctoral research was on a new branch of mathematics, called elliptic equations. This nascent branch was to find prime importance in the quest for proving Fermat right. In the 1950’s, two Japanese mathematicians, Yutaka Taniyama and Goro Shimura put forward a proposition, which was later called the Taniyama-Shimura conjecture. This stated that elliptic equations could be transformed into modular forms, which was another new branch developed in the twentieth century. Though the conjecture affirmed that the transformation was two-way, it couldn’t be proved. In 1984, Gerhard Frey proposed that Fermat’s last theorem, if proved false, lead to a peculiar elliptic equation which is so weird that it couldn’t be transformed to modular forms, thereby defying the Taniyama-Shimura conjecture. In other words, if the last theorem was false, Taniyama-Shimura was also false and vice versa, but the latter was bagging more and more adoption from mathematicians the world over. In 1986, Ken Ribet proved that Frey’s elliptic equation is indeed weird and can’t be transformed to modular form. It now remained to prove only the Taniyama-Shimura to confirm that Fermat was also true. Wiles continued his search for proof in isolation, lest anyone gets wind of it and steal the glory. He was really frightened when in 1988, Yoichi Miyaoka published a proof based on differential geometry, another new concept. However, to his utmost relief, the proof was later shown to be inconsistent. Wiles stepped up the tempo and succeeded in proving that every class of elliptic equations could be transformed to modular form with a new method developed by Kolyvagin and Flach.

Wiles announced his results in a seminar held at Cambridge in June 1993. The world lost no time in hailing him as the greatest mathematician of the century. The bouquets soon turned to brickbats when the judges appointed to referee his proof found some mistakes in it, which Wiles was unable to solve for 14 months. The mathematics community had almost relegated Wiles to another person who failed to come up with a convincing proof, than he supplemented his earlier work with a brilliant correction based on Iwasawa method, along with Richard Taylor. The proof finally appeared in the journal, Annals of Mathematics in May 1995, confirming Wiles stature for all eternity. With all this hindsight, we wonder whether Fermat really did have a proof as he claimed, as all of the concepts behind the proof was developed in the twentieth century.

The book is a pleasure to read and is an international best seller. Even people having only a fleeting experience with mathematics won’t find the book difficult to follow. Yes, you’d indeed find equations here, but those are analysed and made palatable to everybody’s taste and knowledge by the author’s masterful style. Singh’s sense of proportion in describing historical incidents which affected mathematics and determined its future flow is commendable. Every now and then he’d diverge from the main thread to illuminate a lateral point, but even those strays are delightful to read. A lot of interesting anecdotes are presented in the book, one of them concerning Pythagoras is noted here. The great Greek mathematician had a disciple, named Hippasus. The young talent soon discovered that irrational numbers (like square root of 2) existed, which contradicted with his master’s world view that all numbers are rational and defined the world. Instead of evaluating the new hypothesis to assess the truth in it, Pythagoras determined to stifle his opponent, at last ordering to drown his student! The concept of irrational numbers were kept a closely guarded secret for a few more years and came to light only after the death of the great thinker.

There are practically no drawbacks to counted against the book. A feeble, but politically significant one is that the author attaches too much importance to Greek learning and thought than they actually deserve. Mathematical development in Asia is proposed to be a continuation of Greek work when it sank into the dark ages. As he says, “While Europe abandoned the noble search for truth, India and Arabia were consolidating the knowledge which had been smuggled out of the embers of Alexandria and were reinterpreting it in a new and more eloquent language” (p.59). To claim that Asian knowledge was smuggled out of Egypt is simply outrageous, but we may pardon the author because he has given to us a very fine piece of work to be cherished by all serious readers.

The book is highly recommended.

Rating: 4 Star