The ramifications of Euclidean geometry initiated by the invention of its non Euclidean alternatives, as well as the ramifications of Cantor's Set Theory derived by a recent proof of independency of the continuum hypothesis, gave rise to following question: Which of these ramifications is applicable to empirical (physical) reality? We say that a pure mathematical theory is applicable to empirical reality, if there is an isomorphy between its relations and the corresponding empirical data. This isomorphy is to be established by mesure of space and time, i.e. by experiment. In the case there is only one of the ramifications applicable to empirical reality, we speak about the uniqueness of this theory. To give a critical report on this issue is the aim of the present paper. On the past, prominent scholars maintained that concerning geometry the answer to the above question was a matter of convention or definition. So, according to H. Poincaré, there is not an experiment permitting to answer for the empirical status between Euclidean and non Euclidean geometries; because, we cannot maintain that certain phenomena which are possible in the Euclidean are no possible e.g. in the hyperbolic geometry. Therefore, it depends on convention to determine which of these geometries holds in the empirical world. This conventionalistic thesis is correct in case we restrict ourselves to geometries of constant curvature, i.e. to parabolic- (Euclidean), hyperbolic- and spherical geometry. As is well known, if we have two of these pure (postulational) geometries then no true interpretation can be given to both theories which interprets corresponding terms in the same manner: The two geometries are incompatible. But, if different interpretations are given to no corresponding terms, then the two theories are not always incompatible. So, any two of the three geometries of constant curvature are compatible if suitable different interpretations are given to their corresponding terms. The three geometries are formally intertranslatable. The situation is an other one in respect to Riemannian geometries of no constant curvature. According to General Theory of Relativity, the empirical applicability of Riemannian geometries depends on the distribution of mass. Because this distribution is determined on the experience, it follows that the question, which Riemannian geometry is empirically true, is uniquelly determined by the empirical (physical) reality. On the other hand, in the variation of the two factors, the curvature and the number of dimensions, there is a «formationlaw» for all possible manifolds, i.e. for all possible spaces. The uniqueness problem of geometry does not entails the behavior of the same problem for other mathematical fields. Special attention must be given, concerning uniqueness, as well for Arithmetic as even for Set Theory.
Academy of Athens
