  This paper contains a case study of the work and self-definition of two
important mathematicians during the rise of modern mathematics: Felx Hausdorff
(1868--1942) and Hermann Weyl (1885--1955). The two had strongly diverging
positions with regard to basic questions of mathematical methodology, which is
reflected in the style and content of their mathematical research. Herbert
Mehrtens (1990) describes them as protagonists of what he sees as the two
opposing camps of ``modernists'' (Hilbert, Hausdorff et al.) and
``countermodernists'' (Brouwer, Weyl et al.). There is no doubt that Hausdorff
may be described as a mathematical ``modernist''', while the qualification of
Weyl as ``countermodern'' is rather off the track, once his work is taken into
account.
