I’ve been reading up on Hilbert’s problems recently. They’re the 23 problems presented by German mathematician David Hilbert at the International Congress of Mathematicians held in Paris in 1900. (To be more precise, Hilbert presented 10 of them in a talk at the congress. The full list was published later. Mathematicians don’t make a distinction between the 10 presented in person and the rest.)
Hilbert was one of the giants of mathematics at the turn of the twentieth century, and his assessment of the most tantalizing open problems in mathematics has influenced the course of mathematics research in the time since.
I started looking into the Hilbert problems because of a project I was working on that touched on the Millennium Prize Problems. This is a list of seven problems put forth in 2000 by the Clay Mathematics Institute as seven of the most important unsolved problems in mathematics at the start of the new millennium. I had thought of the millennium prize problems as a sort of update of Hilbert’s list, but I was less familiar with his questions, so I wanted to learn more.
Each Millennium Prize Problem has a $1 million prize attached. Solve any one, and you could be a millionaire. (Or, like the only person to solve one of them so far, you could reject the prize and leave the research world entirely.*) Probably because of the lucrative purse attached to each problem, the Clay Institute has detailed guidelines for how prizes are decided and awarded and official write-ups of each problem by experts in the field. The status of each problem is well-defined. (The Poincaré conjecture is solved, and the other six are unsolved.)
Compared to the Millennium Prize Problems, Hilbert’s list looks a bit like a term paper that got thrown together at the last minute by a student who forgot about the deadline. Some problems are too easy: the third problem had actually been solved before Hilbert stated it, though the solution was not published at the time; Max Dehn, who actually published the solution, solved it very quickly after Hilbert stated it. Some problems are ill-posed: the fourth problem is sometimes described as “too vague to have a solution,” and the 23rd problem is just “Further development of the methods of the calculus of variations.” Sorting the problems into “solved” and “unsolved” lists is a bit of a mess. The 10th problem (pdf) has clearly been solved. The eighth problem, which includes the Riemann hypothesis, is clearly still open. But mathematicians have varying perspectives about the extent to which, for example, the second problem should be considered solved. In my research for the article, I consulted several books and websites, all of which had slightly different points of view on which questions were solved and which were still open. I could not help but think that in the most charitable interpretation, the amateurish Hilbert problems had paved the way for the smooth, professional Millennium Prize Problems. (Hence the title of this post.)
But maybe that description is unfair to Hilbert. Why should a list of unsolved problems in mathematics be tidy? After all, the problems are unsolved, which means we don’t know whether or how they will be solved or if their solutions will show us that we were asking the wrong questions in the first place. Hilbert himself wrote, “It is difficult and often impossible to judge the value of a problem correctly in advance; for the final award depends upon the gain which science obtains from the problem.” From another perspective, the Millennium Problems are corporate and sanitized compared to Hilbert’s more organic, sprawling list.
The money clouds things as well. An open-ended question like “further development in the calculus of variations” is not reasonable when there’s a lucrative prize for coming to some defined endpoint, but the calculus of variations has certainly developed a lot since 1900, in part because of Hilbert’s challenge.
After I got past seeing Hilbert’s list as sloppy and nebulous, I began to appreciate the way these problems—either in spite or because they are a little messy—fostered several fruitful lines of inquiry in 20th century mathematics. Sometimes the most important part was that they inspired mathematicians to figure out what the “right” questions were in the first place. Which combinations of axioms, sets of numbers or shapes, and quantifiers lead to the most interesting questions to look at? Even resolved problems, like the 10th, have opened up areas of research that continue to enthrall mathematicians.
Nothing says the Millennium Problems are not having the same effect. Only one of them has been solved, but partial results for others continue to come in. Will this smaller collection of more tightly-focused problems inspire as much innovation as Hilbert’s did? There’s no way to know now.
In any human endeavor, necessity is the mother of invention. In math, problems are often the source of that necessity. Hilbert said it as well:
The deep significance of certain problems for the advance of mathematical science in general and the important role which they play in the work of the individual investigator are not to be denied. As long as a branch of science offers an abundance of problems, so long is it alive; a lack of problems foreshadows extinction or the cessation of independent development. Just as every human undertaking pursues certain objects, so also mathematical research requires its problems. It is by the solution of problems that the investigator tests the temper of [their] steel; [they find] new methods and new outlooks, and [gain] a wider and freer horizon.
*Grigori Perelman, who proved the Poincaré conjecture, refused both the Clay Institute prize money and a Fields Medal. The money was used to establish the Poincaré Chair, which funds 6- and 12-month visiting positions at the Institut Henri Poincaré in Paris for promising researchers. My spouse received one of these positions a few years ago, so my assessment should be considered very biased, but funding several short-term positions seems like a better way to encourage math research than giving a big pile of money to one person.