Natural Isomorphisms in Group Theory

Abstract

VOL. 28, 1942 MA THEMA TICS: EILENBERG AND MacLANE then p, q, r, s form what may be called a pseudo-linear statistical quadruple, i.e., a quadruple which cannot be ordered by means of the between-relation though for each three of the four points one lies between the other two. If a statistical T-metric S metrized by means of jp contains more than four points, then by virtue of the properties of betweenness this relation can be used to order S. Moreover, the other ideas of metric geometry (convexity, geodesics, etc.) can be applied. The three principal applications of statistical metrics are to macroscopic, microscopic and physiological spatial measurements. Statistical metrics are designed to provide us (1) with a method removing conceptual difficulties from microscopic physics and transferring them into the underlying geometry, (2) with a treatment of thresholds of spatial sensation eliminating the intrinsic paradoxes of the classical theory. For a given point po the number II(0; Po, q) considered as a function of the point q indicates the probability that q cannot be distinguished from Po. The study of this function should replace the attempt to determine a definite set of points q which cannot be distinguished from po. This function could also be used advantageously instead of a relation of physical identity for which, as Poincare emphasized on several occasions, we always have triples p, q, r for which p = q, q = r, and p 0 r. Experiments indicate that q sometimes can and sometimes cannot be distinguished from po. Hence, the adequate description of the situation seems to arise from counting the relative frequency of these occurrences.