  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.
  Keywords: arithmetized analysis; axiomatisation of geometry; axiomatisation
of physics; formalism; intuition; mathematical realism; modernism; Felix Klein;
David Hilbert; Karl Weierstrass
