Grigori Yakovlevich Perelman

In August 2006, Perelman was offered the Fields Medal for “his contributions to geometry and his revolutionary insights into the analytical and geometric structure of the Ricci flow⁹”, but he declined the award, stating: “I’m not interested in money or fame; I don’t want to be on display like an animal in a zoo.” On 22 December 2006, the scientific journal Science recognized Perelman’s proof of the Poincaré conjecture as the scientific “Breakthrough of the Year”, the first such recognition in the area of mathematics.

On 18 March 2010, it was announced that he had met the criteria to receive the first Clay Millennium Prize (Millennium Prize Problems¹⁰) for resolution of the Poincaré conjecture. On 1 July 2010, he rejected the prize of one million dollars, saying that he considered the decision of the board of the Clay Institute to be unfair, in that his contribution to solving the Poincaré conjecture was no greater than that of Richard S. Hamilton¹¹, the mathematician who pioneered the Ricci flow partly with the aim of attacking the conjecture. He had previously rejected the prestigious prize of the European Mathematical Society in 1996.

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V. A thing that nonmathematical readers want to know, a question that is always asked when mathematicians address lay audiences, is, What use is it? Suppose the RH [Riemann Hypothesis] were proved true, of false. What practical consequences would follow? Would our heal, our convenience, our safety be improved? Would new devices be invented? Would we travel faster? Have more devastating weapons? Colonize Mars?

I had better unmask myself at this point as a pure mathematician sans mélange, having no interest in such questions at all. Most mathematicians–and most theoretical physicists, too–are motivated not by any thought of advancing the health or convenience of the human race, but by the sheer joy of discovery and the challenge of tackling difficult problems. Mathematicians are generally pleased when their work turns out to have some practical results (at any rate if the result is peaceful), but they rarely think about such things in their working lives. At the Courant conference I sat through four days of solid lectures and discussions on topics related to the RH (Riemann Hypothesis), from 9:30 A.M. to 6:00 P.M. every day, without ever hearing a mathematician mention practical consequences.

Here is what Jacques Hadamard had to say on this point in the The Psychology of Invention in the Mathematical Field.

“The answer appears to us before the question … Practical application is found by not looking for it, and one can say that the whole progress of civilization rests on that principle … Practical questions are most often solved by means of existing theories … It seldom happens that important mathematical researches are directly undertaken in view of a given practical use: they are inspired by the desire which is the common motive of every scientific work, the desire to know and to understand.”

G. H. Hardy, in the concluding pages of his strange little Apology¹⁶, was more blunt and more personal about it.

“I have never done anything “useful.” No discovery of mine has made, or is likely to make, directly or indirectly, for good or ill, the least difference to amenity of the world … Judged by all practical standards, the value of mathematical life is nil. “

In the case of prime number theory, Hadamard’s “the answer appears to us before the questions” applies, and Hardy’s claim is no longer true. Beginning in the late 1970s, prime numbers began to attain great importance in the design of encryption methods for both military and civilian use. Ways to test a large number for primality, ways to resolve large numbers into their prime factors, ways to manufacture gigantic primes; these all became very practical matters indeed in the last two decades of the twentieth century. Theoretical results, including some of Hardy’s, were essential in these developments, which, among other things, allow you to use your credit card to order goods over the internet. A resolution of the RH would undoubtedly have further consequences in this field, validating all those countless theorems about primes that begin, “Assuming the truth of the RH …” and acting as a spur to further discoveries.

And of course, if the physicists really do succeed in identifying a “Riemann dynamics,” our understanding of the physical world will be transformed thereby.

Unfortunately, it is impossible to predict what things will follow from that transformation. Not even the cleverest people can make such predictions, and those who do should not be trusted. Here is a mathematician at work, not quite 100 years ago.

“Every morning I would sit down before a blank sheet of paper. Throughout the day, with a belief interval for lunch, I would stare at the blank sheet. Often when evening came it was still empty … The two summers of 1903 and 1904 remain in my mind as a period of complete intellectual deadlock … It seemed quite likely that the whole of the rest of my life might be consumed in looking at that blank sheet of paper.”

That is from Bertrand Russell’s autobiography¹⁷. What was stumping him was the attempt to find a definition of “number” in terms of pure logic. What does “three,” for example, actually mean? The German logician Gottlob Frege¹⁸ had come up with an answer; but Russell had found a flaw in Frege’s reasoning and was searching for a way to plug the break.

If you had asked Russell, during those summers of frustration, whether his perplexities were likely to lead to any practical application, he would have hooted with laughter. This was the purest of pure intellection, to the degree that even Russell, a pure mathematician by training, found himself wondering what the point was. “It seemed unworthy of a grown man to spend his time on such trivialities …,” he remarked. In fact, Russell’s work eventually brought forth Principia Mathematica¹⁹, a key development in the modern study of the foundations of mathematics. Among the fruits of that study have been, so far, victory in World War II (or at any rate, victory at a lower cost than would otherwise have been possible) and machines like the one on which I am writing writing this book.

The RH should therefore be approached in the spirit of Hadamard and Hardy, though preferably without the overlay of melancholy Hardy put on his disclaimer. As Andrew Odlyzko told me, “Either it is true, or else it isn’t.” One day we shall know. I have no idea what the consequences will be, and I don’t believe anyone else has, either. I am certain, though, that they will be tremendous. At the end of the hunt, our understanding will be transformed. Until then, the joy and fascination is in the hunt itself, and–for those of us not equipped to ride–in observing the energy, resolution, and ingenuity of the hunters. Wir müssen wissen, wir werden wissen.²⁰