Logical reloading as overcoming of crisis in geometry

The property of a geometry to be a science on mutual disposition of geometrical objects in the space or in the space-time will be called “geometricity”. This special term is neccessary, because the contemporary geometry does not possess the property of “geometricity”, in general. In other words, the contemporary geometry is not always is a science on mutual disposition of geometrical objects. Contemporary mathematics considers a geometry simply as a logical construction. For instance, the symplectic geometry is not a science on mutual dispositions of geometric objects.

It is a logical construction, whose form reminds the form of the Euclidean geometry. In applications of geometry to physics and to mechanics only the geometricity of the space-time geometry is important. It is of no importance, whether or not the geometry is a logical construction. If the real space-time geometry is nonaxiomatizable, it means, that it is not a logical construction. However, such a geometry may not possess the property of geometricity.

Contemporary mathematicians do not recognize nonaxiomatizable geometries, which have the property of geometicity, but which are not a logical construction. This situation should be qualified as a crisis in geometry [1], which reminds the crisis, when mathematicians did not recognize non-Euclidean geometries of Lobachevski -Bolyai.

Aforetime the geometry studied disposition of geometrical objects in usual space. The time was considered as an additional characteristic of the physical bodies description. After creation of the relativity theory the space and the time are considered as a united event space (or space-time). It is a more general approach to description of the event space. Any point of the event space is an event, which occurs at some place and at some time.

The geometry is described completely, if the distance ρ between any pair of points belonging to the space is given. The set Ω of points with a distance ρ, given on the set Ω, is known as a metric space M.

A use of the metric space in the physics and mechanics meets some problems. These problems lie in the definition of geometric objects in the metric space M. The distance ρ is supposed to satisfy the relations ρ : Ω × Ω → [0, ∞), ρ (Q, P ) = ρ (P, Q) , ∀P, Q ∈ Ω (1.1)

In the Euclidean geometry the distance has properties (1.1) -(1.3).

In the geometry of Minkowski the distance does not possess these properties. However, it would be very desirable to introduce a metric geometry (or some analog of metric geometry) for description of the space-time properties, because the metric geometry is free of such auxiliary concepts as coordinate system, dimension and such restriction as continuity. Metric geometry describes the geometric properties in terms of only distance, which is a true geometric concept.

After removal of the triangle axiom (1.3) the distance geometry arises [2]. Blumental failed to construct a straight line in terms of only distance. He was forced to introduce a straight as a continuous mapping of interval (0, 1) onto the space (a point set). Such an introduction of nonmetric concept of mapping in geometry seems to be undesirable, because an auxiliary concept is introduced and the distance geometry ceases to be a pure metric geometry.

In general, a construction of geometrical objects is the main problem of the metric geometry. One can easily construct sphere and ellipsoid, because in the Euclidean geometry these geometrical objects are constructed directly in terms of distance.

However, construction of other geometrical objects needs a use of some auxiliary means. For instance, a definition of a plane contains a reference to concept of linear independence of vectors. It is not quite clear, how to introduce this concept in terms of a distance.

A sphere Sp O,P with the center at the point O and the point P on the surface of the sphere is defined as a set of points R

An ellipsoid El F 1 F 2 P with focuses at the points F 1 , F 2 and a point P on the surface of the ellipsoid is defined as a set of points R

If the point P on the surface of ellipsoid coincides with the focus F 2 , the ellipsoid

In the proper Euclidean geometry the segment

has no thickness (it is onedimensional). However, if the triangle axiom (1.3) is not satisfied, the set

Criterion of one-dimensionality may be formulated in terms of distance. The section S P,

(1.7) The point P ∈ S P, T [F 1 F 2 ] in evident way. By definition the segment (1.6) is one-dimensional (has no thickness), if any section of T [F 1 F 2 ] consists of one p

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