Did Hilbert discuss his 23 problems with Felix Klein? 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 Gottingen 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 Universitatsbibliothek Gottingen [Publications of the Lower Saxony State and University Library in Gottingen], 19. Vandenhoeck & Ruprecht, Gottingen, 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 a progressive attitude on Klein's part. Any further details would be appreciated.
 A: 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:


*

*Hilbert's full text in English translation

*Klein on intuition and the Berliners

*Minkowksi with a letter to Hilbert using "dear" and "yours"

*Moritz Epple summarizing Mehrtens on Klein as countermodern

*Ivor Grattan-Guinness on topics missing from the address 

*Parshall and Rowe on Klein's enthusiasm for international fairs

A: 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.
