Historian H. Mehrtens hypothesized an opposition between moderns and countermoderns in early 20th century mathematics, with the former led by Hilbert and the latter by 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 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. Rüdenberg 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 Göttingen [Publications of the Lower Saxony State and University Library in Göttingen], 19. Vandenhoeck & Ruprecht, Göttingen, 1985.

The question was posted at MO a month ago without generating much input.

Note 1. As Jan Peter Schäfermeyer pointed out, Klein not only published numerous 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.

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    $\begingroup$ The beginning of XX C was marked by different war lines in mathematical foundations, inspired in particular by the emergence of set theory with the controversial axiom of choice and the emergence of intuitionism $\endgroup$ – Vladimir Kanovei Mar 3 '17 at 7:02
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    $\begingroup$ Mikhail, "Mathematische Annalen" published not only one, but a series of six papers between 1879 and 1884 and later two more papers in 1895 and 1897. But this made Klein not necessarily a modern, but rather showed his liberal attitude, as opposed e.g. to Schwarz's. $\endgroup$ – Jan Peter Schäfermeyer Mar 12 '17 at 21:38

Mehrtens describes in his book "Moderne, Sprache, Mathematik" the trend towards abstraction and axiomatization in mathematics from the late 19th century onwards, whose main exponents to him were Hilbert, Cantor, Zermelo and Hausdorff and whose main adversaries were Kronecker, Klein, Poincaré and Brouwer.

While it is certainly true that there were controversies between Cantor and Kronecker and between Hilbert and Brouwer, which Poincaré participated in, I am not convinced that Klein's name belongs here. To begin with, Klein employed Cantor as a reviewer for the "Mathematische Annalen" (see Mehrtens, p. 207) and also published his main papers there. Then he brought Hilbert and Zermelo to Göttingen and never interfered in their scientific work. So, Mehrtens has to stretch historical facts a bit to make his point and he consequently calls Klein a proponent of the "kooperative Gegenmoderne".

He mainly bases his assertion on Klein's 1895 paper on The Arithmetizing of Mathematics where Klein criticizes Weierstrass and Kronecker who in turn considered him a charlatan. To Mehrtens, Klein's emphasis on geometric intuition, which not only shows up here, but also in a lecture he had given in Evanston and in his Erlangen program, makes him a counter-modern, while Gray's book features the latter as the first item in the chapter Mathematical modernism arrives.

In my opinion, Klein was - except for his Erlangen program - not a modern mathematician himself, but he was an enabler of modern mathematics, e.g. an early champion of Lie and Grassmann, and certainly not counter-modern.


I think it is safe to say that Klein was a more traditional mathematician than Hilbert. Yet Courant praises Klein's modern approach in teaching analysis in the preface to his calculus book.

Here are interviews with some people at Göttingen (Born, Courant, and von Karman) which show Klein's and Hilbert's shared deep interest in applying mathematics to physics. In this respect there was not much of a difference between the two.

As for the antagonism between Berlin and Göttingen, I think it had more to do with personal animosities, as can be seen from the Frobenius biography by Hawkins.

  • $\begingroup$ Jan, thanks for this information, which does tend to undermine Mehrtens' thesis in my opinion though only indirectly. According to Mehrtens as reported by Gray in his 2008 book, attributes of a "countermodern" include emphasis on applied mathematics, thought of as a throwback to the 19th century. In this sense the interviews you cited serve more to undermine Mehrten's thesis concerning Hilbert being a modern than his thesis concerning Klein being a countermodern :-) $\endgroup$ – Mikhail Katz Mar 6 '17 at 8:56
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    $\begingroup$ In his work on integral equations Hilbert does not appear very modern, does he? $\endgroup$ – Jan Peter Schäfermeyer Mar 10 '17 at 13:48
  • $\begingroup$ Jan, I am not familiar with Hilbert's work on integral equations, to my regret. Could you possibly elaborate? This might be an important part of the puzzle. $\endgroup$ – Mikhail Katz Mar 12 '17 at 9:03
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    $\begingroup$ You can conveniently read the main papers by Fredholm, Hilbert and Schmidt published between 1903 and 1905 in an English translation by G.W. Stewart and you will find that apart from Fredholm who uses a little bit of group theory, the authors stay within the boundaries of classical mathematics, i.e. analysis and the theory of infinite matrices. This would only change some years later due to the contributions by Schmidt and Riesz, who introduced the concept of a function space and an operator therein, respectively. $\endgroup$ – Jan Peter Schäfermeyer Mar 12 '17 at 16:14
  • $\begingroup$ Jan, would it be possible to get in touch with you via email? $\endgroup$ – Mikhail Katz Mar 13 '17 at 9:28

Our analysis of Mehrtens' claims appears at https://arxiv.org/abs/1803.02193 and is forthcoming at Mat. Stud.

Historian Herbert Mehrtens sought to portray the history of turn-of-the-century mathematics as a struggle of modern vs countermodern, led respectively by David Hilbert and Felix Klein. Some of Mehrtens' conclusions have been picked up by both historians (Jeremy Gray) and mathematicians (Frank Quinn). We argue that Klein and Hilbert, both at Goettingen, were not adversaries but rather modernist allies in a bid to broaden the scope of mathematics beyond a narrow focus on arithmetized analysis as practiced by the Berlin school. Klein's Goettingen lecture and other texts shed light on Klein's modernism. Hilbert's views on intuition are closer to Klein's views than Mehrtens is willing to allow. Klein and Hilbert were equally interested in the axiomatisation of physics. Among Klein's credits is helping launch the career of Abraham Fraenkel, and advancing the careers of Sophus Lie, Emmy Noether, and Ernst Zermelo, all four surely of impeccable modernist credentials. Mehrtens' unsourced claim that Hilbert was interested in production rather than meaning appears to stem from Mehrtens' marxist leanings. Mehrtens' claim that [the future SS-Brigadefuehrer] "Theodor Vahlen ... cited Klein's racist distinctions within mathematics, and sharpened them into open antisemitism" fabricates a spurious continuity between the two figures mentioned and is thus an odious misrepresentation of Klein's position.


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