Derived Measurement, Dimensions, and Dimensional Analysis

Abstract

This paper presents a representational theory of derived physical measurements. The theory proceeds from a formal definition of a class of similar systems. It is shown that such a class of systems possesses a natural proportionality structure. A derived measure of a class of systems is defined to be a proportionality-preserving representation whose values are n-tuples of positive real numbers. Therefore, the derived measures are measures of entire physical systems. The theory provides an interpretation of the dimensional parameters in a large class of physical laws, and it accounts for the monomial dimensions of these parameters. It is also shown that a class of similar systems obeys a dimensionally invariant law, which one may safely subject to a dimensional analysis.