In 1963, a yet-to-be-proven Mathematics problem caught the attention of a 10 year old shy, quiet kid in Cambridge, England. The kid was Andrew Wiles, and the problem, which would become his obsession in adulthood, was Fermat’s theorem – the most famous problem in Maths which had confounded the world for the last 300 years.
The theorem was proposed by Fermat, a 17th century councilor, who devoted all his energy to Mathematics in his free time. Fermat was a mischievous figure, who would often invent new problems, and refuse to publish the solution; fame and recognition did not matter to him at all. What certainly interested him was finding new riddles, and then teasing his colleagues by writing letters about these problems, without providing any proofs. One such problem – actually discovered by his son after his death – was scribbled in the margins of a book, along with a note attached in Latin which stated,
“I have a truly marvelous demonstration of this proposition which this margin is too narrow to contain.”
For 358 years, this ‘marvelous’ solution escaped the fate of every serious Mathematician, who tried their hands at the problem, only to be left defeated by it. When it was eventually solved in the late 20th century, the solution ran into more than 100 pages, using techniques developed long after Fermat’s death. Certainly Fermat did not have this way of solution to his mind. Was the ‘Prince of amateurs’ bluffing then? But then, how was he so confident of his proposition?
Simon Singh takes from here, and presents an interesting narrative covering generations of Mathematicians struggling, and at the same time advancing and developing new techniques, getting closer to the problem, but not quite getting it. This is definitely not a book about Maths equations, but of people behind those equations, and of death and deception, curiosity that breeds innovation, obsession that kills, and of dreams that keep you alive. Thus for all non-mathematicians (like me), it provides a fascinating account of what drove and inspired these people, whose equations we studied in school, beginning with the famous Pythagoras. Fermat, in fact, created his renowned puzzle while studying the Pythagoras’ theorem in 1637.
I was really excited to know about Pythagoras, the man whose eponymous theorem had come to my rescue on countless occasions while solving geometry problems – if nothing works out, just drop a perpendicular, join some lines, and see the magic. Pythagoras, of Somas, in ancient Greece, was an influential figure of 6th century BC : one who fought an intolerant and conservative society to establish a highly secretive, egalitarian school to practice Maths and philosophy. He truly believed that the law of nature is governed by rational numbers; legend has it that when one of his pupils discovered irrational numbers, he got him killed. Irrational numbers reeked of evil – these were numbers with no pattern and no end – and thus did not fit, ironically, in the ideals of a man who worshipped logic.
Pythagoras was also one of the first mathematicians who laid the foundation for ‘absolute proof’, something which is at the heart of Maths, and is different from ‘scientific proof’. Mathematicians rely on logic rather than hypothesis and evidences, such that once something is proven, it’s proven forever with no room for change. The simple example is Pythagoras’ theorem which states that for any right angled triangle, square of hypotenuse must equal to the sum of the square of the other two sides : x2+y2 = z2 is true for all right-angled triangles, and has infinite combinations of x, y and z satisfying this equation.
Fermat must have toyed with the idea of changing powers in the Pythagoras’ theorem, and to his astonishment, he found that the equations, x3+y3 = z3, x4+y4 = z4, and so on to infinity, did not have any solution i.e. xn+yn = zn has no positive integer solution for n>2. The Frenchman, true to his character, did not bother to leave any proof behind this general equation. Instead, for the next three and a half centuries he seemed to have the last laugh from his grave, looking at so many brilliant minds failing to prove his proposition.
On this journey of discovery, Simon recounts the history of number theory, and puts in your arsenal the knowledge of prime, whole, rational, irrational, and imaginary numbers to make some sense of the proofs attempted. Along this route, you meet Pascal creating laws of probability; Euclid writing the highest selling text-book of all time, Elements; Euler making the first breakthrough by proving the theorem for a special case, n=3; Sophie Germain defying traditions becoming the first woman to attend lectures at the Academy of Sciences, Paris and; Taniyama and Shimura, an unlikely duo, collaborating over the same problem in 1950s Japan, and discovering new Maths on the way.
Among these prominent individuals, a story which fascinated me was of an amateur mathematician – Paul Wolfskehl. A rich German industrialist, Paul got depressed over a failed romance, and meticulously planned for his suicide. Few hours from shooting himself, with all his planned pre-suicide work completed, he started studying the latest development on the Fermat’s theorem. He found a flaw in the logic, and got so engrossed in it that he forgot about his suicide. The most difficult problem of Maths had given him a new life, and so when he actually died, he left behind a fortune of £1,000,000 (in today’s money) for the one who would finally solve it.
As the world now knows it, Andrew Wiles won the Wolfskehl’s award, conquering the giant beast of a problem that was Fermat’s theorem, in 1995. His is a story of unimaginable determination and immense perseverance, and a seven year ordeal in secrecy to pursue his childhood dream. There is frustration and despair too, and high-tension drama such that when he presented his argument in 1993, a flaw was found in his solution in the process of peer-review. The world had to wait for two more years for the worthy winner of the prize, and the fame & recognition that came with it.
For me, Andrew’s story, and multiple others in the book, were about the power of dreams; dreams which sometimes seem naive, sometimes crazy, sometimes invincible, and sometimes enigmatic, but at the same time give you courage and hope. How many of us still remember our childhood dreams? How many of us are actually pursuing them, or like so many, have you kept them aside for practicality and rationality? As I finished the book, I could not stop humming a song I love from the movie ‘La La Land’ which raises a toast for people who engage in seemingly foolish, romantic pursuit of dreams.
Here’s to the ones who dream, Foolish as they seem Here’s to the hearts that ache, Here’s to the mess we make