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New crisis in geometry??

#### Yuri A. Rylov

*Institute for Problems in Mechanics, Russian
Academy of Sciences*

* 101-1 ,Vernadskii Ave., Moscow, 119526, Russia*

* email: rylov@ipmnet.ru*

Web site: http://rsfq1.physics.sunysb.edu/~rylov/yrylov.htm

or mirror Web site: http://gas-dyn.ipmnet.ru/~rylov/yrylov.htm

Updated March 14, 2005

#### abstract

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.

There is
text of the paper in English and in Russian,