Bruce Schechter – New York Times
In the Life of Pure Reason, Prizes Have Their Place
by Bruce Schechter
For those who want to be millionaires but shudder to contemplate facing a smirking Regis Philbin as he asks, “Is that your final answer?” the publishers of “Uncle Petros and Goldbach’s Conjecture,” a novel by Apostolos Doxiadis offer a challenging alternative.
The first person to submit a correct proof of Goldbach’s conjecture — a well-known mathematical poser that has resisted solution for more than 250 years — before March 15, 2002, will receive a check for $1 million.
But the money is the least of it. As the mathematician G. H. Hardy wrote in “A Mathematician’s Apology,” his unnecessary but eloquent justification of a life devoted to pure mathematics, “Mathematical fame, if you have the cash to pay for it, is one of the soundest and steadiest of investments.” Whoever manages to prove Goldbach’s conjecture within two years will have earned the glory and the cash to pay for it too.
Goldbach’s conjecture is the type of thing that moved Hardy to write his apology, for it is a problem (at least before the prize was established) of no practical use whatsoever. It is simply a beautiful conjecture about how numbers fit together, a guess about the infinite.
In 1742, a little-known mathematician named Christian Goldbach wrote a letter to Leonhard Euler in which he speculated that every even number greater than two could be expressed as the sum of two prime numbers. Primes, of course, are whole numbers that are evenly divisible by only one and themselves — numbers like 2, 3, 5, 7, 11 and 13.
The number 26 is not a prime because it is divisible by both 2 and 13. Goldbach’s conjecture is easy to verify for small numbers: 24=19+5 and 72=19+53. Using computers and clever algorithms, mathematicians have verified Goldbach’s conjecture to about 400 trillion.
No reasonable mathematician seriously doubts the validity of Goldbach’s conjecture.
But mathematical proof is not a matter of what seems reasonable, but a product of pure reason. And so far, Goldbach’s little observation has resisted the reasoning power of the greatest minds in mathematics. In the novel, Uncle Petros is driven to despair and ultimately madness by his failure to prove Goldbach’s conjecture.
The publisher Faber & Faber’s offer of a prize for solving Goldbach’s conjecture is both a clever publicity stunt and in the finest traditions of mathematics. When a problem bothered Dr. Paul Erdos, the eccentric, itinerant mathematician, who died in 1996, he often “put a price on its head,” offering a cash prize roughly calibrated to its assumed difficulty.
Dr. Erdos, who never had a permanent home or job was not a rich man, so the prizes he offered were usually small, ranging from less than a dollar to $10,000 for one problem he was convinced would never be solved, and that still never has been. Even if someone had managed to solve it, Dr. Erdos joked that it would probably take so long that the solver would have been working for far less than minimum wage. Mathematicians did not seem to care. Most of the checks Dr. Erdos wrote for solved problems have been framed, not cashed.
Mathematicians have often been wildly wrong in their evaluation of the difficulty of problems. Knowing this, Dr. Erdos limited himself to offering prizes that he could somehow pay in the event of an unforeseen brainstorm. Before offering their prize, Faber & Faber and Bloomsbury Books, the British and American publishers of “Uncle Petros,” secured insurance from Lloyd’s.
“The insurers had to take a punt,” said Tony Faber, managing director of Faber & Faber. “Although the chance of someone solving it is low, we don’t think it is zero.” He would not reveal the premium Lloyd’s demanded, but would only say it was “in the five figures.”
With insurance in hand, Mr. Faber would be thrilled if someone found the proof, though his happiness would have little to do with the advancement of mathematics. “I’d love it,” Mr. Faber said. “It would be great publicity.”
For many years, the mathematical puzzle that had the biggest price on its head was one that was scribbled around 1637 by the French mathematician Pierre Fermat in the margins of one of his math books.
“I have a truly marvelous demonstration of this proposition which this margin is too narrow to contain,” Fermat wrote, a sentence that has driven and inspired generations of mathematicians. For hundreds of years nobody had been able to find a proof of Fermat’s conjecture and had begun to suspect that he was either mistaken or mischievous.
In 1908, a German industrialist and amateur mathematician named Paul Wolfskehl established a prize of 10,000 marks — roughly $1 million — for a proof of Fermat’s last theorem. Unfortunately, by the time the Princeton mathematician Andrew Wiles had collected the prize in 1997, inflation had eroded Wolfskehl’s prize to $50,000.
Dr. Wiles had worked on the problem in secrecy, much as, in the book, Mr. Doxiadis’s Petros had worked on Goldbach’s conjecture, for more than seven years. He said he would wake up with Fermat’s theorem first thing in the morning and think about it all day. “And,” Dr. Wiles added, “I would be thinking about it when I went to sleep.” In other words, true to Dr. Erdos’s estimate, Dr. Wiles earned less than minimum wage.
Fortunately, Dr. Wiles was not trying to become a millionaire by pursuing Fermat’s theorem. Neither was the fictional Uncle Petros, who as a youth was drawn to a seemingly simple problem that has proven to be far more difficult to solve than Fermat’s conjecture.
Uncle Petros’s despair was due not only to his having wasted his creative years — for a mathematician, this means until about the age of 30 — on a proof, but also to the possibility that indeed no proof of Goldbach’s theorem was possible.
Mathematicians once believed that the truth or falsity of any mathematical proposition could be decided by applying the rules of mathematical logic. A proof may be very difficult, like that of Fermat’s theorem, but in principle, given enough genius and dedication, it could be found.
But in 1931 Kurt Gödel proved that this belief was nothing more than wishful thinking. Some propositions, Dr. Gödel showed, can be neither proved nor disproved within the confines of mathematical logic. Uncle Petros, after devoting his life to Goldbach’s conjecture, finally lost hope when he believed that Goldbach’s conjecture might be undecidable. Perhaps he gave up too soon.
Working mathematicians generally ignore the possibility that proofs they are pursuing are impossible. For them, the techniques invented and principles illuminated by pursuing the proof of a hard problem are the real rewards. Fame is nice too. And $1 million always comes in handy.
April 25, 2000: Bruce Schechter – New York Times