Mathematics, where nothing is ever as simple as it seems
By Robert Matthews and Nadejda Lobastova
Published: 12:01AM BST 20 Aug 2006
Grigory “Grisha” Perelman lives with his pensioner mother in a small apartment in St Petersburg, where the pair survives on her pension of £30 a month, plus some savings he made during his time as a lecturer.
Yet, on Tuesday, Perelman is expected to be hailed as one of the greatest minds on the planet, after solving a century-old mathematical riddle. His financial worries could also be at an end if – as many experts believe – he goes on to win the $1 million prize that has been offered to the first person to solve the mystery.
Professor Marcus du Sautoy, of Oxford University, is among the leading academics who hope that Perelman will turn up at the International Congress of Mathematicians, in Madrid, where the Russian is expected to named as the winner of the Fields Medal, regarded as “the Nobel Prize of mathematics”.
“Everyone is tipping him to get the prize – and everyone’s tipping him to turn it down,” says Prof du Sautoy. To judge by Perelman’s exclusive interview with The Sunday Telegraph, there is little hope of him breaking his self-imposed exile. “If anybody is interested in my way of solving the problem it’s all there, let them go and read about it,” he said. “I have published all my calculations. This is what I can offer the public. I do not believe anything that I say can be of the slightest public interest.”
There is nothing novel in geniuses rejecting mere baubles. Jean-Paul Sartre waved away the 1964 Nobel Prize for Literature and Marlon Brando rejected the 1973 Oscar for Best Actor. But Perelman is also in line to win $1 million from an American institute for solving one of the hardest problems in mathematics, known as the Poincaré conjecture. And no one thinks he will accept that, either – despite living in penury.
Welcome to the bizarre world of mathematics, where nothing is ever as simple as it seems. It is a world in which proving that one plus one really does equal two takes up a page of indecipherable symbols, and where people spend years pondering a doodle on a sheet of paper that divides it into two regions, called “inside and “outside”. In Perelman’s case, the challenge has been no less recherché: how to tell if an object is actually a sphere. The problem has been tantalising mathematicians since 1904, when Henri Poincaré, a brilliant Frenchman, suggested there was an infallible rule for showing that an object is a sphere (see box below).
In May 2000, the conjecture made world headlines after it was chosen as one of seven “Millennium Problems” whose solution carried the $1 million dollar reward. The money had been put up by the Clay Mathematics Institute, founded in 1998 by Landon Clay, an American millionaire with a fascination for mathematics. His aim was to attract more public interest in a subject notorious for its difficulty. But, even within academia, it is also notorious for the sheer unworldliness of many of its leading lights. Hints of just how strange mathematicians can be have emerged in such films as Good Will Hunting, which features Matt Damon as the troubled janitor-cum-genius, or in Russell Crowe’s portrayal of the schizophrenic Nobel Prize-winning theorist John Nash in A Beautiful Mind. But not even these Oscar-winning films could capture the reality.
Archimedes, the most brilliant mathematician of the ancient world, did his best work in the bath, and told the Roman soldier who killed him while he was pondering geometry: “Do not disturb my circles.” Isaac Newton wrote far more about the Book of Revelation than about mathematics, and worked out that Christ’s Second Coming would take place in AD 2060.
If Perelman does accept the Fields Medal, he will be joining the ranks of such brilliantly odd thinkers as Alexander Grothendieck, the French mathematician who won the prize in 1966. He became obsessed with the war in Vietnam and travelled to Hanoi, giving maths lectures in the forests while the US Air Force bombed the city. In 1988, he was awarded the Crafoord Prize – worth about £100,000 – but declined it in protest over what he claimed was the corrupting influence of such prizes. Three years later, the 63-year-old genius vanished from his home and is reputed to be living alone in a remote part of southern France.
Grothendieck shared his Fields Medal with Stephen Smale, an American mathematician who had his grant money cut after declaring that he did his best work while sitting on the beach at Rio de Janeiro. His work included major breakthroughs in the Poincaré conjecture – for which Perelman is now set to win the prize.
“Many mathematicians are a bit bonkers,” says Professor Keith Devlin, of Stanford University, California, and author of The Millennium Problems. “But so are many artists, business tycoons, movie stars and accountants – well, maybe not accountants.”
Prof du Sautoy blames the nature of the work, which requires years of concentrated effort. “You need a little bit of craziness to do mathematics, because it is such a weird world.”
Perelman’s achievement is being compared with that of the English mathematician Professor Andrew Wiles. Prof Wiles, who is now at Princeton University, spent the best part of a decade proving Fermat’s Last Theorem, which centred on a mathematical formula scribbled in a textbook belonging to the eponymous 17th-century French mathematician. To make his breakthrough, Wiles had to combine ideas from apparently unconnected areas of mathematics – the same strategy that has now paid off for Perelman.
Wiles was awarded £30,000 for his achievement, the remnants of a huge fund established a century ago by Paul Wolfskehl, yet another eccentric mathematician. Wolfskehl, who was depressed after a failed romance, was contemplating suicide when he picked up a book that described Fermat’s Last Theorem and decided life was worth living after all. Wolfskehl celebrated by setting up a prize fund, the proceeds of which would go to the first mathematician to prove the theorem.
Its principal effect was to provoke vast numbers of cranks to send in purported proofs to university maths departments. So far, the Clay prizes do not seem to have had the same effect – not least because the seven Millennium Problems are all-but incomprehensible. “We do have one of our staff to handle the letters and emails,” says Dr James Carlson, the president of the Clay Millennial Institute. “And some of them have claimed to have solved all seven problems in one go.”
Anyone hoping to join the ranks of mathematical giants such as Perelman is best advised to have a crack at Goldbach’s conjecture, which is that every even number greater than two is the sum of two prime numbers. In 2000, the publisher Faber offered a prize of $1 million to the first person to prove it. Rather unsportingly, it put a two-year time limit on the challenge. So, for now at least, the only prize on offer is the one that means most to the likes of Perelman: a permanent place in mathematical history.
This is an old news article published in The Telegraph, which sums up the profession nicely (“Many mathematicians are a bit bonkers”). The article also draws out the reasons why I didn’t pursue a professional career in mathematics; I am too “normal” and my attention span is too short to give concerted effort on one thing for a long time, like all successful mathematicians do.
That being said, I have a huge sympathy for the profession. The people mentioned in the article are my personal heroes: Grisha Perelman, Alexander Grothendieck, Andrew Wiles. I am proud to tell people that I’ve met Andrew Wiles (who in 1995 proved the Fermat Last Theorem, the most famous “unsolved” problem in mathematics at the time) and John Nash (the subject of the film “A Beautiful Mind”; most of his work are in pure mathematics, though he is known today for his contributions in economics and game theory, for which he won the Nobel Prize).
In fact, I still maintain to this day that Grothendieck is the greatest mathematician, and perhaps the greatest scientific mind of our generation. I have read his handwritten Bangor manuscripts — a copy was shown to me by Dr. Alinor (UKM) in 2005 — and although the math is totally over my head (I know a little bit of algebraic topology and topos theory by self-study and some graduate classes I took in college) yet I knew what I held in my hand is a work of rare human genius that only comes out a few times every century.
Picture of A. Grothendieck, one of the greatest mathematicians of the 20th century. He won the Fields Medal in 1966, the highest honor that can be given to a mathematician (think the Nobel Prize for mathematics), but later became disillusioned with academia. He stopped publishing in journals, and would put down his groundbreaking ideas as handwritten notes to be distributed among close friends. He maintained a cult following among mathematicians, although his scientific output is low. In the 70s he started becoming weirder: he refused a prestigious award because he thought giving and receiving awards are detrimental to the mathematical profession, he gave advanced math lectures in deep jungle in Hanoi during the Vietnam War to protest the war, and suddenly in 1991, he just disappeared. Few people know for sure where he is now although some people believe he’s still alive and living in Andorra. Of course, he lives alone and does not entertain visitors.
This “weirdness” is on par with that of Grisha Perelman, who today lives in a small apartment in Moscow with his mother, surviving on her small pension. He refused the highest honor in mathematics (the Fields Medal) and a gift of one million dollars for solving the notoriously hard Poincare Conjecture. The reason given is a classic case of “pure mathematics” apologia: I am not doing this for the money, but only for my beloved mathematics. No wonder why people think mathematicians are bonkers.
LOL, sometimes I wonder what would happen if I chose to be a professional mathematician. Maybe I’d go bonkers too? Sometimes I really miss my beloved MIT Math Department and my math professors: Michael Artin, Steven Kleiman, Gilbert Strang, Sigurdur Helgason, Victor Kac, George Lusztig, Victor Guillemin, Michael Sipser, Richard Stanley, etc. These are some of the finest mathematical minds on the planet, and great teachers as well (except Lusztig).