New crisis in geometry??

Yuri A. Rylov

Institute for Problems in Mechanics, Russian Academy of Sciences
 101-1 ,Vernadskii Ave., Moscow, 119526, Russia
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Updated March 14, 2005


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,