Ontology of geometry space selection in physical theories

Zoran Primorac, Ph.D. Professor
University of Mostar, Bosnia and Herzegovina

Relation between geometry of space selection and the space itself or extensiveness is given at the ontological level, i.e. between the space and geometry as a logical-mathematical form which draws its intrinsic value from that relation. The analysis of area concept in western philosophical tradition implies a dual appearance of the concept of space, namely as an immanent characteristic (of the body, field, etc.) and space as an independent entity. These two concepts are dichotomic, they define two different paradigms which are mutually irreducible. But, the fact is that both concepts will exist in some theories, but for a damage of theoretical coherence or an eclectic approach to physical reality. The full meaning of the term “space”, which gets its geometrical structure through analytical Euclidean geometry, appears with Newton and the concept of space as immanent characteristic gets its philosophical rounding in Descartes’ definitions of space. All modern philosophical attitudes or scientific attitudes on physical reality have derived their conceptual formulations from Newton and Descartes’ approaches. It is important to point out: both philosophers started with their constructions or explanations of the physical reality and their ontological approaches, as well as their concepts of geometry, heavily depended on their conception of the space. The fact is also that the concept of space or extensiveness appears in physical theories as paradigm both in implicit or explicit form. Accepting the concept of space as an independent entity, geometry appears as science of that entity structure. Geometrical objects such as point, direction, etc. represent some elements or parts of the space and do not belong to the set of physical objects. They can be approximated from physical objects but not identified. Then space enables spreading of physical objects, but it does not define them and that is the only difference between the pure spatial extension and the physical extension of material objects. In this case the selection of the geometry of space cannot be the consequence of experimental research. It has to be given in advance, or in such concept the intrinsic geometry of the physical space has to be given in the form of definition. In other words, the examination of the ontological truth of geometry is neither necessary nor possible, because we speak on the priory geometry of the space and not on the geometry of physical or material objects. Ontologically speaking it is the fact that the concept of space and its characteristics as well as its geometry are given in the form of definitions. Such status can bring to certain theoretical problematic situations which were manifested, e. g. in quantum mechanics. We may also ask the question; if Euclidean geometry is given as the inherent structure of space by definition, as it was done by Newton, is it possible to assign that inherence to any other geometrical system? The answer has to be affirmative, but Euclidean geometry has certain advantages over other systems. We could call them conditionally “generic” and “logic” advantages. The “generic” advantage could be in the fact that there is a generic connection of the Euclidean geometry and the concept of space. The concept of space has its historical genesis and the Euclidean geometry had an important place in that development. On the other side the abstract concept of space as such enabled the Euclidean geometry to leave the level of the phenomenal space. The “logical” advantage of the Euclidian geometry lies in certain mathematical “simplicity” in relation to other geometrical systems. However when we take the concept of space as an immanent characteristic of physical objects then we put the extensiveness at the level of physical properties. But, in order to make extensiveness experimentally applicable, all other physical conditions, which may change the dimensions of extensive objects, have to be eliminated. Then we hope that the behavior of the “clearly” extensive objects can give us the answer regarding the inherence of geometry and extensiveness. For that purpose some gauge of extensiveness has to be defined, for example: “solid” bodies which do not change its dimensions under the influence of some other physical factors. The second gauge of extensiveness which is often used in physics is the ray of light, which as a physical object which characteristics mostly correspond to the concept of geometrical direction. The problem of such concept of extensiveness and the geometry belonging to it is particularity, i.e. determination of a physical entity or a process as the carriers of extensiveness in relation to other qualitatively different physical entities. For example Einstein’s General theory of relativity favors the gravity field as the carrier of extensiveness, i.e. it considers extensiveness as the immanent characteristic of the gravity field. Empirically set Riemann’s type of geometry is considered as intrinsic to the gravity field, but there arises a question of relation to other physical realities, for example in the case of quantum objects and processes, etc. Here we have to point out that two concepts of the space have different approach to selection of intrinsic geometry, but they are put at the same level of conceptual formulation, which means that any of them doesn’t have any logical advantage in relation to the need of introducing the given definition in physics. As we already said, intrinsic geometry is given by definition in the concept of space as the independent entity, and in the concept of space as an intrinsic characteristic of physical entities we have the possibility of experimental fixing of the intrinsic geometry, but with the help of a gauge of extensiveness which has to be given by definition.