This article is a translation of Les problèmes de Hilbert, written in 2010 by Étienne Ghys for the French mathematics magazine Images des Mathématiques, and is published here with permission.
In August 1900, David Hilbert delivered a lecture which remains popular among mathematicians. Entitled “Mathematical Problems”, it presented a list of twenty-three open problems that Hilbert considered as important for the future of mathematics. To discuss all of these problems, and review their current status one hundred and ten years later, would require twenty-three separate articles. Here, I will only quote a few excerpts from the introduction to this lecture.
First, what is a good problem according to Hilbert?
The role of problems in the development of mathematics
A good problem must be crystal clear
Hilbert alludes to a comment by Joseph Diaz Gergonne, the French geometer who, in 1810, founded the first real mathematical journal: the Annales de mathématiques pures et appliquées, also known as Annales de Gergonne.1 Anyway, we should meditate on this sentence sometimes: “What is clear and easily comprehended attracts,whereas the complicated repels us.”
A problem must be difficult, but approachable
I think everyone will agree with the following comment: if it is too easy, a problem is not interesting, but if it is too difficult, it is useless. How does one know in advance, though, that a problem is not inaccessible?
Where do good problems come from?
Non-mathematicians often think that mathematics should be applied and should be useful to solve concrete problems. In front of problems of “pure mathematics”, they ask: “what is it useful for?” Hilbert explains that things are more subtle.
The first problems came from the real world.
However, these concrete problems generate new problems within mathematics, which a priori do not promise any direct “utility”.
These pure problems, sometimes, reflect on the concrete in return.
Rigour before everything!
Today, the following message can hardly be read in national curricula—
Besides, it is an error to believe that rigour in the proof is the enemy of simplicity. On the contrary we find it confirmed by numerous examples that the rigorous method is at the same time the simpler and the more easily comprehended. The very effort for rigour forces us to find out simpler methods of proof. It also frequently leads the way to methods which are more capable of development than the old methods of less rigour.
Generalise to understand better but understand particular cases
Does one go from the general to the particular? Or from the particular to the general? These are two different but complementary approaches and are two ways to “do math”.
In dealing with certain mathematical problems, specialization plays, as I believe, a still more important part than generalization. Perhaps in most cases where we seek in vain the answer to a question, the cause of the failure lies in the fact that problems which are simpler and easier than the one in hand have themselves been either not at all, or incompletely solved. All depends, then, on finding these easier problems,and on solving them by means of devices as perfect as possible, and also via concepts capable of generalization. This rule is one of the most important levers for overcoming mathematical difficulties, and it seems to me that it is used almost always, though perhaps unconsciously.
This reminds me of this beautiful quotation of Grothendieck, who explains in Récoltes et semailles2 his way of solving problems:
Sometimes, one can show that a solution to a problem is impossible
For centuries, there were attempts to square the circle, until it was proved to be impossible towards the end of the nineteenth century.
The “conviction” of the mathematician
This conviction of the solvability of every mathematical problem is a powerful incentive to the worker. We hear within us the perpetual call: There is this problem. Seek its solution. You can find it by pure reason, for in mathematics there is no ignorabimus.
Thirty years later, Hilbert was forced to revise his “conviction which every mathematician shares, but which no one till then had yet shown, by providing a proof”. Gödel’s theorem, in 1931, would indeed show that in every formal system containing arithmetic, there exists statements which are impossible to prove and to refute, with their negation being equally impossible to prove as well. This is liable to test our “convictions”. This was a deep change in our vision of right and wrong. However, it must be said that even though, nearly all mathematicians are aware of this theorem, they do not believe it too much in their everyday practice, and they stick to the conviction mentioned by Hilbert. Sometimes, we have convictions which we also know, deep inside, are not completely justifiable…
On 8 September 1930, a year before Gödel’s theorem, Hilbert was lecturing about “The knowledge of nature and logic”. The conclusion of his lecture was broadcast on the radio, where Hilbert claimed:
We must know,
We will know.
This beautiful conclusion, “Wir müssen wissen, Wir werden wissen”, was engraved on Hilbert’s grave (he died in 1943), which could imply that he upheld it all his life.
An optimistic conclusion
One hundred and ten years later, his conclusion still stands!3
- In an article, (page 88, footnote), Quételet quotes a letter from Gergonne: “For a long time I have repeated to my students that we do not have the final say about a problem in science as long as one cannot bring it to the point where it can be told to a man met by chance in the street.” Thanks to Karine Chemla for the reference, which is: des lettres et des beaux-arts de Belgique. Académie royale des sciences. Nouveaux Mémoires de L’Académie Royale des Sciences et Belles-Lettres de Bruxelles. Volume t.4. 1820. ↩
- Alexandre Grothendieck. Récoltes et semailles. Réflexions et témoignage sur un passé de mathématicien, 1985. ↩
- Images des Mathématiques carries articles on three of the twenty-three problems–-the third problem, about the volume of polyhedrons (Daniel Perrin. Aires et volumes: découpage et recollement (I). 24 Nov 2010. Online; Michéle Audin. Une « vie brève » de Max Dehn. 8 Sep 2010. Online), the tenth one on Diophantine equations (Pierre de la Harpe. Le nombre d’or en mathématique. 14 Jan 2009. Online) and the eighteenth about space tilings (Pierre de la Harpe. Ornements et cristaux, pavages et groupes, III. 10 May 2009. Online). The following book gives a current state of the art about the problems: Jeremy J. Gray. The Hilbert Challenge. Oxford University Press, 2000.↩