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.