Mathematical Structures and Psychological Measurements

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Abstract
The nature of psychological measurements in relation to mathematical structures and representations is examined. Some very general notions concerning algebras and systems are introduced and applied to physical and number systems, and to measurement theory. It is shown that the classical intensive and extensive dimensions of measurements with their respective ordinal and additive scales are not adequate to describe physical events without the introduction of the notions of dimensional units and of dimensional homogeneity. It is also shown that in the absence of these notions, the resulting systems of magnitudes have only a very restricted kind of isomorphism with the real number system, and hence have little or no mathematical representations. An alternative in the form of an extended theory of measurements is developed. A third dimension of measurement, the supra-extensive dimension, is introduced; and a new scale, the multiplicative scale, is associated with it. It is shown that supra-extensive magnitudes do constitute systems isomorphic with the system of real numbers and that they alone can be given mathematical representations. Physical quantities are supra-extensive magnitudes. In contrast, to date, psychological quantities are either intensive or extensive, but never of the third kind. This, it is felt, is the reason why mathematical representations have been few and without success in psychology as contrasted to the physical sciences. In particular, the Weber-Fechner relation is examined and shown to be invalid in two respects. It is concluded that the construction of multiplicative scales in psychology, or the equivalent use of dimensional analysis, alone will enable the development of fruitful mathematical theories in this area of investigation.

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