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Hilbert's lecture at the ICM in Paris in 1900 presented 10 of the famous 23 open problems. It is well known that the idea of the lecture came from Hermann Minkowski. Hilbert was at Göttingen at the time where he was hired through untiring efforts of Felix Klein. As detailed by historian David Rowe and others, both Hilbert and Klein were involved in a battle against the Berliners at the time. The Berlin school dominated by followers of Kummer, Weierstrass, and Kronecker was known for its focus on arithmetized analysis. Hilbert's 23 open problems sought to broaden the scope of mathematics beyond such narrow focus. It seems as though it would have been natural for Hilbert to have discussed the 23 problems with Klein. Is there any evidence of such discussions in published work or private correspondence?

Here is what Minkowski wrote in a letter to Hilbert:

"Most alluring would be the attempt to look into the future, in other words, a characterization of the problems to which the mathematicians should turn in the future. With this, you might conceivably have people talking about your speech even decades from now. Of course, prophecy is indeed a difficult thing" (Minkowski 1973, 5 January 1900; see German original).

The reference is

Hermann Minkowski, Briefe an David Hilbert, Hg. L. Ru¨denberg und H. Zassenhaus, New York: Springer-Verlag, 1973.

This information comes from page 16 of Rowe's article

Rowe, D. "Mathematics made in Germany: on the background to Hilbert's Paris lecture." Math. Intelligencer 35 (2013), no. 3, 9--20.

Beyond the issue of possible correspondence concerning Hilbert's Paris lecture, Frei's book on the Klein-Hilbert correspondence may contain further evidence that Klein and Hilbert were, first of all, allied against the Berliners, and second of all both moderns contrary to the thrust of the Mehrtens hypothesis on Klein being allegedly countermodern:

Der Briefwechsel David Hilbert-Felix Klein (1886-1918). [The correspondence between David Hilbert and Felix Klein (1886-1918)] Edited, with comments, by Guether Frei. Arbeiten aus der Niedersachsischen Staats- und Universitätsbibliothek Göttingen [Publications of the Lower Saxony State and University Library in Göttingen], 19. Vandenhoeck & Ruprecht, Göttingen, 1985.

Note 1. As Jan Peter Schäfermeyer pointed out here, Klein not only published six papers by Cantor in Mathematische Annalen but also used Cantor as a referee for the journal. From the modern perspective this would indicate an attitude of a modern on Klein's part. Any further details would be appreciated.

Note 2. (added in 2024) This issue is analyzed in this recent publication: Heinig, P.; Katz, M.; Kuhlemann, K.; Schaefermeyer, J.P.; Sherry, D. "Exploring Felix Klein's contested modernism." Antiquitates Mathematicae 17 (2023), 29–65. https://arxiv.org/abs/2402.00122, coming soon to https://dx.doi.org/10.14708/am.v17i1.7245

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    $\begingroup$ Could you please add a reference about your claim It is well-known that the idea of the lecture came from Hermann Minkowski? $\endgroup$
    – Francesco Polizzi
    Commented Feb 1, 2017 at 15:54
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    $\begingroup$ Here's a link (click on pg 120) to the page with the letter: books.google.com/… $\endgroup$
    – Suvrit
    Commented Feb 1, 2017 at 17:09
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    $\begingroup$ Thiele's article also mentions Hurwitz as another person from whom Hilbert sought advice concerning his ICM lecture. Klein is, however, not mentioned. Constance Reid's book Hilbert also mentions both Minkowski and Hurwitz (but not Klein), and given the extensive discussion of Klein's role in Hilbert's life in that book, one may expect that if any evidence of discussion between Klein and Hilbert concerning the ICM lecture is known, it would have been included. $\endgroup$
    – Willie Wong
    Commented Feb 1, 2017 at 18:06
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    $\begingroup$ On the other hand, both Hurwitz and Minkowski were in Zurich at the time, and Klein in Goettingen, I don't know much about 1900 era German custom, but why would Hilbert write Klein a letter to talk about the lecture when he could just walk down the hall? (This is in regards to your asking about "private correspondence".) $\endgroup$
    – Willie Wong
    Commented Feb 1, 2017 at 18:11
  • $\begingroup$ They probably would have talked. Why do you ask? $\endgroup$
    – user44143
    Commented Feb 2, 2017 at 8:44

2 Answers 2

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As Constance Reid points out, Klein had lost his interest in pure mathematics around 1900 and had devoted himself to projects in applied mathematics and teaching, which Hilbert had scarcely any interest for.

To Runge, whom he would make the first full professor in applied mathematics in Germany in 1904, he had already written in 1894 that he thought that mathematicians were too often occupied with artificial problems that were bred in university rooms, a view that was shared by Runge, who had already in the 1880s defected from pure mathematics.

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  • $\begingroup$ Thanks, Jan Peter. I wouldn't rely on Constance Reid for accurate historical scholarship. She mainly had a flair for writing about mathematics and was not particularly known for serious scholarship. In particular, I would take the claim that Klein limited himself to applied mathematics starting in 1900. You may have noticed that Bieberbach got his degree under Klein and Bieberbach was not working in applied mathematics. Around 1918 Klein wrote a paper in relativity theory, which is not what is usually understood by "applied mathematics". $\endgroup$
    – Mikhail Katz
    Commented Mar 15, 2017 at 12:29
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    $\begingroup$ Mikhail, if you look at the list of Klein's Ph.D. students you will notice that after the period of 1881-1899 where he supervised a staggering number of 43 theses almost exclusively in pure mathematics, not only his pace slowed down considerably, but also that the theses he supevised were mostly in applied mathematics, with the exception of Bieberbach's and Freundlich's in complex analysis and the one by Ihlenburg which covered a problem in plane geometry. $\endgroup$
    – Jan Peter Schäfermeyer
    Commented Mar 15, 2017 at 13:33
  • $\begingroup$ Yes , as far as pure math is concerned, "slowed down" is better than "lost interest". This applies both to research and student supervision. As you know, applied mathematics is still alive and kicking today, so it is hard to understand by what magic wand interest in applied mathematics (not exclusive, as we have discussed) would make Klein a "countermodern". $\endgroup$
    – Mikhail Katz
    Commented Mar 15, 2017 at 13:37
  • $\begingroup$ As for Klein's last two papers of 1918, they are of course remarkable, because they contain the first application of the then still unpublished, now famous Noether's theorem, but they too, belong to the field of mathematical physics. $\endgroup$
    – Jan Peter Schäfermeyer
    Commented Mar 15, 2017 at 13:40
  • $\begingroup$ Jan Peter, not everybody accepts the Mehrtens-Gray-Quinn take on mathematical physics as being beyond the pale of pure mathematics as you know :-) Hilbert's famous list had a few problems in mathematical physics as you have pointed out. Some people consider Atiyah to be a pure mathematician :-) $\endgroup$
    – Mikhail Katz
    Commented Mar 15, 2017 at 14:12
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We can imagine what Klein might have said:

Dear Hilbert,

Thanks for showing me the draft of your address for Paris before I left for the Exposition Universelle. I appreciate your kind words on me in the text.

In the areas of mathematics that you have covered, you are closer to the frontier of knowledge than I am, and I can only admire your selection of questions in those areas.

However, I would revise the emphasis of your problems. I would put the set theory and axiomatics towards the end or only in the written version, since I find it poor material for lively conversation.

I would also include some of the mathematical problems arising from Maxwell's physics, or Poincare's mechanics, or Italian algebraic geometry, which are absent from your questions now.

I see that you have both praise and criticism for the Berliners. You talk about rigor as much as they do, and I wish you would say more about the intuition that is also necessary for mathematical progress.

Finally, you speak frequently of mathematical knowledge, and I hope you will also include mathematical practice as applied in both science and industry, which is what I am seeing here at the Exposition before we gather for the conference.

Yours, F. Klein

Some references:

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    $\begingroup$ :-) Very imaginative. It is hard to criticize a work of fiction but I do detect certain misconceptions concerning Klein. The comment on set theory seems to indicate that Klein was less than enthusiastic about it. Do you have any evidence for this? As indicated in this comment, Klein published not one but six papers by Cantor in Mathematische Annalen, and moreover used Cantor as a referee for the journal. This indicates more enthusiasm for set theory than your passage. $\endgroup$
    – Mikhail Katz
    Commented Mar 14, 2017 at 17:10
  • $\begingroup$ Concerning intuition: your passage seems to suggest that Klein was more enthusiastic about it than Hilbert. This is similarly a misconception (perhaps a common one), as Hilbert frequently talked about intuition and specifically emphasized it in his text with Cohn-Vossen. This misconception (as the one above) fits like a glove with Mehrtens' take on Klein as a countermodern and a partisan of aryan intuition, but this perspective on Klein appears to be incorrect. $\endgroup$
    – Mikhail Katz
    Commented Mar 14, 2017 at 17:13
  • $\begingroup$ Similarly, with regard to applications to science, your comment appears to emphasize that Klein was more interested in these than Hilbert. This may also be a common misconception, given Hilbert's deep interest in physics; see in particular this question. $\endgroup$
    – Mikhail Katz
    Commented Mar 14, 2017 at 17:15
  • $\begingroup$ P.S. If you read German I would much appreciate if you could take a look at Frei's volume of Klein-Hilbert correspondence and see if it indicates how modern Klein may have been. $\endgroup$
    – Mikhail Katz
    Commented Mar 14, 2017 at 17:19
  • $\begingroup$ @MikhailKatz, on your third comment, I agree Hilbert got interested in physics, but see Grattan-Guinness on the notable omission of physics from the 1900 talk. $\endgroup$
    – user44143
    Commented Mar 14, 2017 at 17:20

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