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The conventional method of a generalized geometry construction, based ondeduction of all propositions of the geometry from axioms, appears to beimperfect in the sense, that multivariant geometries cannot be constructed bymeans of this method. Multivariant geometry is such a geometry, where at thepoint P there are many vectors PP', PP'',... which are equivalent to the vectorQQ' at the point Q, but they are not equivalent between themselves. In theconventional (Euclidean) method the equivalence relation is transitive, whereasin a multivariant geometry the equivalence relation is intransitive, ingeneral. It is a reason, why the multivariant geometries cannot be deduced froma system of axioms. The space-time geometry in microcosm is multivariant.Multivariant geometry is a grainy geometry, i.e. the geometry, which is partlycontinuous and partly discrete. Multivariance is a mathematical method of thegraininess description. The graininess (and multivariance) of the space-timegeometry generates a multivariant (quantum) motion of particles in microcosm.Besides, the grainy space-time generates some discrimination mechanism,responsible for discrete parameters (mass, charge, spin) of elementaryparticles. Dynamics of particles appears to be determined completely byproperties of the grainy space-time geometry. The quantum principles appear tobe needless.

Multivariance as a crucial property of microcosm