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МОСКОВСКИЙ ГОСУДАРСТВЕННЫЙ УНИВЕРСИТЕТ
ИМЕНИ М.В. ЛОМОНОСОВА
МАТЕМАТИКА И РЕАЛЬНОСТЬ
Тезисы Третьей всероссийской научной конференции
27-28 сентября 2013 г.
МОСКВА Центр стратегической конъюнктуры
УДК 5:1 ББК 22.1:87
проф. В.А. Бажанов, проф. А.Н. Кричевец, доц. Е.В. Косилова, проф. В.Я. Перминов, доц. В.А. Шапошников
Проведение конференции поддержано Российским фондом фундаментальных исследований, проект № 13-06-06076.
Философия математики: актуальные проблемы. Математика и реальность. Тезисы Третьей всероссийской научной конференции; 27-28 сентября 2013 г. / Редкол.: Бажанов В.А. и др. – Москва: Центр стратегической конъюнктуры, 2013. – 270 с.
Конференция по философии математики – традиционная встреча специалистов в этой области и смежных с ней областях. В ее работе приняли участие профессиональные математики, преподаватели математики в системе высшего образования, философы, логики, психологи, историки математики. Тезисы в сборнике сгруппированы в разделы, соответствующие секциям конференции. Приоритетная тема конференции 2013 года – «Математика и реальность».
© Авторы тезисов, 2013
МАТЕМАТИКА И РЕАЛЬНОСТЬ
Darvas György, director, Symmetrion, Budapest
THREE GENERATIONS OF NON-EUCLIDEAN GEOMETRIES IN THEIR RELATION TO
APPLICATIONS IN PHYSICS Our geometrical world view is usually held to be based on
Euclidean geometry. It is a plausible view since someone standing on the Earth sees the surroundings as approximately flat. But that view, propagated through our schools, is mistaken.
Many pupils have been convinced that Euclidean geometry is absolute. Many students are then surprised when first meet a non- Euclidean geometry. Many of them are sceptical about the reality of such a geometry. And many of them question the reality of those physical phenomena, which inspired and demanded the elaboration of non- Euclidean geometries.
Geometry is an abstraction. Physical phenomena represent reality. The geometries they demand are those correctly describing the spatial environment in which these phenomena appear. Flat (that means, Euclidean) geometry is an abstract approximation, which disregards the material environment in our current theories. Physical reality assumes the presence of matter which, in turn, determines the structure of space, and which structure appears different from the idealised „empty” world’s geometry.
From Euclidean to curved geometries
The first non-Euclidean geometries were elaborated in the middle of the nineteenth century (Lobachevsky, Bolyai, Gauss, ...). Those geometries assumed (different) constant-curvature spaces (hyperbolic, elliptic, spherical, etc.). The curvature also determines the metric of the given space. However, those geometries were still far from physical reality, for the curvature around a physical source is not constant. It depends, among other factors, on the distance from the material source, which causes the space to be curved.
From constant curvature geometries to space-time dependent curvatures
The second generation of the non-Euclidean geometries took into consideration the dependence of the curvature (and the metric) on the displacement of a given point (event) in space and time. This is a family of Riemannian geometries. The curvature (and metric) in such geometries is a function of space-time co-ordinates. Without Riemann geometry, the theory of gravitation, better known as the general theory of relativity, could not have been elaborated. This was the first and up to now probably the most successful application of a non-Euclidean geometry in physics. From space-time dependent geometries to many-parameter dependent curvatures
More sophisticated mathematical theories were developed that allowed the curvature to depend on other variables. One can mention first the two theorems elaborated by Emmy Noether just with the aim to prove the invariances (a phenomenon associated with symmetries since the 18th c.) evoked by the general theory of relativity. These two theorems establised a correspondence between conservation laws (that means, invariance under certain actions, such as a translation in time) and symmetries, which are so fundamental in most discoveries concerning the physical structure of matter in the recent nearly hundred years. These two mathematical theorems allow an indefinite number of parameters as variables of functions. These parameters can be endowed with physical properties. The possibilities of the variations of applicable physical parameters in the second theorem of Noether are still not fully exploited.
Riemann geometry assigns a fixed curvature to each space-time point. This means that curvature (and the metric of the space) depends on four variables. If one increases the number of variables (in our case: physical parameters) on which the curvature of the space depends, one can assign a variety of curvatures to any space-time point, according to these additional parameters. Application of such additional parameters opened the way to extending the geometries from the characterization of space-time to physical fields.
This potential program was predicted, at least in mathematical terms, by Paul Finsler. Finsler geometries treat spaces whose curvature,
and accordingly metric, depend not only on their location (like in Riemann geometry), but also on directions assigned to each spatial point. This explains, why the applications of quaternions and octonions in physical theories interpreted on Finsler spaces are so widespread. The original idea was that the curvature changed according to the possible spatial directions where a (bundle of) vector(s) could be attached to each point at each time. From many-parameter dependent curvature spaces to physical fields
The idea was plausible, for many physical quantites behave like vectors, and their value really depends on spatial (spatio-temporal) direction. Later the concept of Finsler geometry was extended to generalised additional parameters as well. Finsler geometries represent a third generation of non-Euclidean geometries.
Physical fields often depend on vector quantities. Physics describes nature in permanent change. Physical nature can be characterised by quantities which may change both their place and their direction at every moment. Many phenomena call for a description in a direction-dependent reference frame. Further, physics establishes laws that represent constancy in the continuously changing world. These laws are based on more or less stable physical principles. Fruition of symmetry is one of the most fundamental principles of physics. It embodies constancy in change (cf., Symmetry: Culture and Science, Vols. 14-15, 2003-4). Symmetry means that while certain properties are changing at least one other property is conserved. Derivation of conservation theorems in field theories comply in most cases with the second Noether theorem mentioned above.
It is interesting to mention that this geometry was formed by P. Finsler (in his PhD dissertation) during the same year and at the same university (1918, Göttingen) when and where E. Noether published her theorems. And yet, his geometry was acknowledged only much later. Moreover, it started to be widely applied in physics only in the 21st century. The idea was not far from physicists. Similar ideas appeared in field theories many decades ago. In general terms, most gauge fields follow Finsler geometry, although are not reflected in explicite form in earlier publications. Despite much evidence, most physicists did not refer
to Finsler and did not consciously apply the geometrical tools developed by him. Only in the recent decade has Finsler geometry become part of tools used by physicists. This geometry became part of the description of physical fields, not only the spaces on which these fields are interpreted.
The more parameters appear on which a curvature and metric of a space depends, the less symmetric is that space. However, there appear new symmetries in the fields which are interpreted on these spaces. Finsler geomet