  The first crisis in the geometry arose in the beginning of XIXth century,
when the mathematicians rejected the non-Euclidean geometry as a possible
geometry of the real world. Now we observe unreasonable rejection of the
non-Riemannian geometry by the official representatives of the contemporary
geometry. Class of the Riemannian geometries appears to be too narrow for
physical applications. The microcosm physics needs expansion of the class of
possible geometries appropriate for the role of space-time geometry. In the
framework of the non-Riemannian geometry one can construct the space-time
geometry, where the motion of free particles is primordially stochastic, and
this stochasticity depends on the particle mass. At the same time the geometry
in itself is not stochastic in the sense that the space-time intervals are
deterministic. Principles of quantum mechanics can be deduced from such a
space-time geometry. The crisis situation in geometry appears to be connected
with some preconceptions concerning the foundation of the geometry. The
preconceptions as well as the crisis generated by them are not purely
scientific phenomena. The human factor (social aspect) is rather strong in the
crisis phenomena. The preconceptions and the human factor appear to be so
strong, that usual logical arguments are not perceived, and the usual formal
mathematical language appears to be inappropriate for perception of an analysis
of the crisis origin and of a possibility of its overcoming. In the paper the
history and motives of the non-Riemannian geometry construction are presented.
There is a hope that such a less formal way of presentation helps to understand
and to overcome the existing preconceptions.
