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Volume 10, No. 5 - July/August 2000
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Is It Wise to Bet On Mathematical Progress?

ALTHOUGH I, for one, would be reluctant to wager real money on whether a unified theory of physics will be discovered before the decade is out, or on whether the mind-body problem will be dispatched by the end of the century, others are more daring. A member of the Lloyd's of London insurance syndicate, for example, has recently bet that a certain mathematical conjecture will not be proved before March 15, 2002.

The proposition -- Goldbach's conjecture -- is one of the oldest famous unsolved problems in mathematics. It was first stated in a letter that Christian Goldbach (1690-1764) -- a Prussian math enthusiast who served as tutor to Peter II, the teenage czar of Russia -- sent in 1742 to the great mathematician Leonhard Euler. Goldbach had noticed that any even number he examined could be written as the sum of two prime numbers. (A number is prime if it has no divisors other than itself and 1.) Thus, for example, the even number 24 can be written as the sum of 11 and 13, which are both primes. Perhaps, Goldbach idly guessed, this is true for all even numbers. Euler responded that it probably was but he had no idea how to go about proving it.

Although it is not of central importance to mathematics, the conjecture has captured the imagination of many mathematicians, professional and amateur, because of its seeming simplicity. Recently, it caught the eye of Apostolos Doxiadis, a Greek film and theater director who studied mathe matics at Columbia University and the Ecole Pratique des Hautes Etudes. His novel Uncle Petros and Goldbach's Conjecture was published in Greece in 1992 -- the 250th anniversary of Goldbach's conjecture. Earlier this year it was brought out in England by Faber and Faber and in the United States by Blooms bury USA. To promote the novel, these two publishing houses came up with a good gimmick: They would give a prize of one million dollars to anyone who could achieve the feat that obsesses the novel's protagonist -- prove Goldbach's conjecture. They stipulated that the proof must be submitted by March 2002 and published in a "reputable" mathematics journal by March 2004.

But what if someone out there actually succeeded in doing this? To indemnify themselves against that ruinous possibility -- a million dollars is still a lot of money in the book trade -- the publishers went to Lloyd's of London. There they found an underwriter who was willing to take on this contingent liability -- for a fee, of course. Although the editors I spoke with at Faber and Faber would not divulge the underwriter's identity, they did say that the cost of the insurance policy was "in the tens of thousands of dollars." So in effect the underwriter made a bet against the solution of Goldbach's conjecture: If it goes unsolved by 2002, his profit will be in the "tens of thousands of dollars"; if it is solved by the deadline, his loss will be a million dollars minus that five-figure sum.

To repeat my opening question: Is such a bet wise? How does one go about computing the odds of an intellectual breakthrough? Well, if Lloyd's of London is comfortable writing policies on everything from shipping accidents to (re portedly) Jennifer Lopez's derriere, there is no reason why Goldbach's conjecture should daunt them. Judging from the policy's terms, the underwriter must have put the odds against Goldbach being proved at around 100 to 1. Would a mathematician agree? I called Joe Buhler, a distinguished number theorist who is currently deputy director of the Mathematical Sciences Research Institute at Berkeley. "We're just waiting for a new idea, not a whole new branch of mathematics," he told me. "I'd say there's at least a fifty-fifty chance that Goldbach's conjecture will be proved in the next fifty years." These odds, a little arithmetic shows, roughly jibe with those arrived at by the Lloyd's underwriter.

Even though Goldbach's conjecture has been recalcitrant to proof over the last two and a half centuries, most mathematicians none theless believe that the proposition is true -- that each of the infinity of even numbers can indeed be expressed as the sum of two primes. Still, not everything that is true in mathematics can be proved. From Gödel's incompleteness theorem we know that in any formal system of arithmetic there are infinitely many propositions that are neither provable nor disprovable. Could Goldbach's conjecture be one of them? That is what the dispirited protagonist of Uncle Petros and Goldbach's Conjecture begins to suspect. Had he (or his novelist-creator) been cleverer, he might have seen a silver lining here. If Goldbach's conjecture could be shown to be undecidable -- neither provable nor disprovable -- then this would be tantamount to proving it true! For if it is false, there must be some counterexample to it; that is, some even number that could not be written as the sum of two prime numbers. But such a counterexample would constitute a disproof of the conjecture -- thereby contradicting its undecidability.

Would such a "meta-proof" of Goldbach's conjecture qualify for the million- dollar prize? The folks at Faber and Faber were a little nonplussed when I raised the possibility. They did, however, warm to my suggestion that, in the event the money did have to be paid out, it should go to the winner as a pair of checks, one for $499,943, the other for $500,057 -- two prime numbers that, à la Goldbach, sum to an even million.

In addition to "Hypotheses," Jim Holt writes a regular column for the Wall Street Journal covering books on science and philosophy. His most recent book is Worlds Within Worlds: How the Infitesimal Revolutionized Thought (Four Walls/Eight Windows).

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