One Million Bucks for a 100% Solution

October 21, 2000

Philip J. Davis

Several months ago the world of research mathematicians was electrified by the announcement that the Clay Mathematics Institute, a private organization, had established a $7 million prize fund, $1 million each for the solution of seven famous mathematical problems.

These "Millennium Prize Problems," selected by a blue ribbon committee, are the P versus NP problem, the Riemann hypothesis, the Hodge conjecture, the Poincaré conjecture, the Yang-Mills existence and mass gap, Navier-Stokes existence and smoothness, and the Birch and Swinnerton-Dyer conjecture. These highly specialized problems are not easily explained to a general readership, although in its publicity release the Clay Institute took a stab at it.

The excitement seeped down to the general public, and within hours, several relevant e-mail addresses were jammed with thousands and thousands of inquiries-many more, in fact, than there are professional research mathematicians in the whole world. When I informed my friend Charles, who holds a PhD in applied mathematics and has worked for many years in the upper echelons of computer programming, he said,

Whee! Wonderful! A bit of pizzazz has finally entered the life of a discipline that strikes the average guy as about as interesting as a list of numbers from the Manhattan phone book. Wise up: Our profession deserves to have its Bobby Fischers and its Tiger Woodses.

Bobby and Tiger notwithstanding, my feelings are ambivalent. As a result, I undertook an informal, nonrandom, nonbinding poll of some of the mathematicians I know. I found that opinion within this group appears to be split. While I'm not prepared to extrapolate my percentages to the whole mathematical community (could I use small-sample theory?), I amassed enough individual justifications, pro and con, to put together this article.

A Bit of Background
There have always been prizes for mathematical accomplishment and there always will be. The first prize I've heard of goes back to c. 585 BC. It was given to Thales of Miletus, a somewhat fuzzy figure, who was reputed to have been the father of deductive geometry-a tremendous step in the history of mathematics. The prize had been offered for the wisest person around, and according to Diogenes Laertius, writing eight hundred years later, Thales won it twice. As to the prize he received, Diogenes gives us two versions. In the first, Thales won a tripod! In the second version, Thales won a kind of bowl. Surely neither is the ancient equivalent of a million dollars.

Somewhat later, the three famous problems of antiquity, squaring the circle, duplicating the cube, trisecting the angle-three very specific problems-were set forth, but not quite as a prize competition: According to the Delian Oracle, a plague could be avoided if their solutions were found. These three problems were settled negatively only in the last century and a half, by means of mathematical concepts and results that were hardly available at the time the oracle spoke.

The plague has persisted: Even today, mathematicians are plagued with submissions of "solutions" to these same problems, proposed by people who are unsophisticated mathematically and who understand neither the conditions of the problem nor the hallmarks of a valid solution. I have received my share of these solutions. I suspect that in the back of the minds of some of the submitters has been the thought that a large prize is offered for their solution. Another factor at work is the opportunity to show up those smart-aleck experts who said "no way."

In 1738 the French Académie des Sciences announced a prize for the best essay on the nature of fire. The prize was split between three people, one of whom was the great Swiss mathematician Leonhard Euler (1707-1783). We have in Euler's submission the beginnings of the trigonometric solutions of partial differential equations.

In 1889, Henri Poincaré won the King Oscar Prize for the solution of the n-body problem, though he did not give a 100% solution. (Very good man, Poincaré: Rules can always be stretched.)

Coming up to modern times, mathematician Paul Wolfskehl, a banker's son, offered in his will (1908) a prize of 100,000 German marks for the solution of the Fermat problem. On the brink of suicide because of an unsuccessful love affair, he had found psychological relief by devoting his thoughts to the problem. The prize is an indication of the compelling power of unsolved mathematical problems.

Among the large prizes now offered for mathematics---in addition to the Clay prizes---are the Goldbach Prize ($1,000,000), the Japan Prize (50 million yen), the Wolf Prize ($100,000), and the Ostrowski Prize (150,000 Swiss francs). Among this group, only the Clay and the Goldbach prizes are for the solution of specific problems.

The Nobel Prize (9,000,000 Swedish kroner) is, of course, the most prestigious prize in the world. But Alfred Nobel excluded mathematics from the prize areas. (Rumor has it that Nobel charged Swedish mathematician Mittag-Leffler with alienation of affections.)

The Fields Medal, considered by many mathematicians as having the prestige of the Nobel, yields $15,000 (Canadian). The prestige of a prize is often thought to be more closely connected to the reputation of the winners than to the amount of the prize money. Think of the Olympics in classical Greece, where winners were crowned with a laurel wreath. Or do you suppose the athletes cashed in on tie-ins and residuals?

Schools, colleges, and societies give prizes-from pre-kindergarten, where every child gets a prize, to posthumous prizes, where only the honored dead do so. Among the lesser prizes as far as cash awards go are several administered by the American Mathematical Society. These yield approximately $4,000 each. As a result of recent bequests, SIAM'S top two prizes are $20,000 and $10,000.

Some years ago, an unsolicited prize went tax free. This is no longer the case. I've won a number of mathematical and national prizes. They didn't cause a blip in the family budget. And come to think of it, what is the distinction between a prize, an award, and a grant? They are all competitive processes. In the competition for an award to a department or to an institute, millions of dollars can be at stake; psychologically, nailing one of these awards often has the same effect on those who submitted the proposal as winning a prize. Thus, there always have been and always will be prizes. The ones mentioned here are just the tip of the iceberg.

The Swedes do it
The French do it
You can't be a mensch and not do it
Let's do it
Let's give a great big prize!
---Apologies to Cole Porter (1928)

Arguments in Favor of the Clay Prizes
A number of the following arguments, both pro and con, can be made for prizes quite generally; a few are specific to Clay.

It is appropriate to reward superior talent and excellence. Prizes, as we have seen, are one kind of reward. Prizes act as a spur to the creativity of others. Creative juices are stimulated thereby. Competition is good; competition is said to be the rule of nature, and it is occasionally built into law.

Large prizes provide excellent publicity for the mathematical profession; they give it higher visibility. They also provide excellent publicity, satisfaction, honors, and clout not only to recipients but also to donors, conceptualizers, and awards committees. "Who is honored? He who honors others." So it is written in the "Sayings of the Fathers."

Prizes show the public that mathematics is far from a closed book, that it is not "all out there simply to be picked up from books or from the Web." They make the public aware of the existence of important unsolved problems and hence of the need for public support and encouragement.

The Clay prizes call attention to certain specific, fundamental problems selected by the Scientific Advisory Board of the Clay Institute. The criteria for determining what is important are independent of those that may be set by contracting agencies or by granting foundations. In its selection of prize problems, with their thrust away from applications, the Clay Institute emphasizes that mathematics has been the Queen, and not merely the handmaiden, of the sciences-and that it ought to remain so.

The fact that the prizes are for very specific problems, and not for outstanding general accomplishment, lends objectivity and avoids subjectivity in the prize awards.

The ground rules for the award have been worked out thoughtfully. For example:

"Before consideration, a proposed solution must be published in a refereed mathematics journal of worldwide repute, and it must have general acceptance in the mathematics community two years after that publication."

If a solution passes these hurdles, further examinations will then be conducted by a board of experts in the field.

(These are pretty stringent conditions. They remind me of the manner in which the Vatican arrives at its saints. One difference is that the Vatican requires a very much longer waiting time before submission and final declaration. Also, in the mathematical case, no specific advocatus diaboli is named.)

Arguments Against the Prizes
Large prizes for notoriously difficult problems encourage a romantic view of a mathematical career. Of course, the world must always have its Shackletons, Lindbergs, and Hilarys. But it should be recognized that obsessive devotion to a single problem can destroy a professional career. An example, based on the experience of a real-life mathematician, can be found in mathematician and film maker Apostolos Doxiadis's novel Uncle Petros and Goldbach's Conjecture.

Large prizes feed right into the star system, which is now endemic and getting worse. They emphasize the individuals and not the accomplishments as such. The accomplishments become iconic acts: Although the mathematical results are held in high regard, not many people, not even professionals, will have read them or could understand without considerable study exactly what was accomplished or how it was accomplished.

Large prizes underline commercial values. Theorems can become commodities. The heros they bring forth will be one part Tiger Woods and one part Sammy Glick. Ambitious mathematicians are not going to ignore big problems just because there is no mega-money associated; on the other hand, prizes do not appear to accelerate the emergence of solutions to problems. The Fermat problem was solved only recently, almost a century after the announcement of Wolfskehl's prize.

The competitive aspects of prizes can lead to publicly expressed acrimony. I know of three, perhaps four, serious instances of it during the years of my professional career. Recall that the Trojan war was begun because of the decision of a prize contest. Large prizes can increase secrecy, bickering, and envy. They could disrupt the sense of mathematical community. Norbert Wiener (1894-1964), one of the leading mathematicians of his generation, was strongly of this opinion. It was Wiener who coined the term "cybernetics," which has now entered the common language in a hundred different forms. In a letter (1941) of resignation from the National Academy of Sciences, Wiener wrote:

"As to medals, prizes, and the like, the less said of them the better. The heartbreak to the unsuccessful competitors is only equalled by the injury which their receipt can wreak on a weak or vain personality, or the irony of their reception by an aging scholar long after all good they can do is gone. I say, justly or unjustly administered, they are an abomination and should be abolished without exception."

A million dollars is considerably greater than previously established mathematical prizes, and, according to a well-known dialectic that Marxists love to cite, quantity changes quality. A huge prize demeans lesser prizes; it embarrasses those who sponsor the lesser ones. To some, the Clay prizes spell a vulgarization of mathematics.

There are better ways to use the money to foster mathematical research. The Clay Institute has allocated a portion of its income toward a variety of lesser prizes, grants, workshops, and so forth. These activities are certainly welcome, but will hardly cause a stir in the media.

Though the conditions of the prize allow for the awards committee to judge the relevance of previous sustaining results, so that the whole prize doesn't necessarily go to the person who drove in the final nail, how far back does one have to go in mathematical history to identify sustaining results? I hear through the grapevine that the first problem on the Clay list is close to being solved: The P versus NP problem is rumored to be undecidable, on the basis of the ZFC (Zermelo-Fraenkel-Choice) axioms. In this litigious age, shall the heirs of Zermelo and Fraenkel be factored into the award? The "C" designates the axiom of choice, formulated by Zermelo and, independently, by Erhardt Schmidt, so should Schmidt's descendents also harbor "great expectations"? As regards the "thousands of workers in the field," do "they also serve who only stand and wait"? Could the whole complex nexus of mathematics have been developed by a few prize winners?

There are many important unsolved problems in mathematics. The International Mathematical Union recently sponsored a book (Mathematics: Frontiers and Perspectives) that lists at least a hundred of them. The Clay Institute's designation has produced an imaginary dichotomy: the "really important problems" and other problems. Subjective judgment decides which problems are more likely to be fruitful in the long run. No one at the beginning of the 20th century would have predicted that the most significant mathematical development of the century would be the digital computer, insofar as it has affected the lives of everyone on earth.

Then there is the question of what constitutes a mathematical proof. What is a solution or a complete solution? Formalists, logicians think they know. But they don't, really. Consider, for example, the famous four color problem. When the computer-assisted solution was made public, the result was rejected by a certain subset of mathematicians. Or consider proofs that are extremely long---up to five thousand pages, I've heard. What committee can go through so many pages without falling asleep? It's worse than the mass of evidence presented by a Special Counsel in Washington.

A solution, therefore, is what the educated mathematical public declares is a solution. I've been plugging this view all along, but I think it may shock the Scientific Advisory Board when they realize that their ground rules are antiplatonic. The history of mathematics has surprises for us.

My own principal objection to the Clay prizes is that they distort the nature of mathematical research, which is to proceed, by freedom of the imagination, from established starting points toward results that in some way seem fruitful. Initial goals may not be relevant to the final work. The Clay prizes distort public understanding of mathematics by emphasizing problem solving at the expense of concept building, elaboration, and application. Yes, mathematicians start by specifying specific goals, but at the end of the day, there are no long-range goals for mathematics other than to be fruitful and grow, and to merit the support of the community. In this connection, I quote a response I received from a historian of science:

"It is a vast relief that my field, history, has no such prizes. My sense, especially after reading Elisabeth Crawford's work on the history of the Nobel Prizes [which are not for preassigned problems], is that their effect is to distort (and narrow) the range of possible research, as well as the range of the appreciation of same."

A Final Thought
If, as in Greek antiquity, some foundation were to offer a prize for "the wisest of us all," then as a result of the competition, would we all wise up? Would evolution thereby take one giant step forward?

Philip J. Davis, professor emeritus of applied mathematics at Brown University, is an independent writer, scholar, and lecturer. He lives in Providence, Rhode Island.

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