Published online by Cambridge University Press: 01 April 2022
In formal theories of measurement meaningfulness is usually formulated in terms of numerical statements that are invariant under admissible transformations of the numerical representation. This is equivalent to qualitative relations that are invariant under automorphisms of the measurement structure. This concept of meaningfulness, appropriately generalized, is studied in spaces constructed from a number of conjoint and extensive structures some of which are suitably interrelated by distribution laws. Such spaces model the dimensional structures of classical physics. It is shown that this qualitative concept corresponds exactly with the numerical concept of dimensionally invariant laws of physics.
A number of people have made helpful comments on an earlier draft of this paper. By far the most substantial contribution is that of William Craig who, on first listening to me discuss the problem in a seminar at Stanford University, strongly suggested that the best way to formulate the idea of meaningfulness is as invariance under automorphisms of the structure (Definition 3). He later gave me detailed criticisms, including pointing out an error, on a written draft. Others to whom I am also indebted are E. W. Adams, M. Bar-Hillel, R. L. Causey, F. Roberts, P. Suppes, and B. Wandell.
This work was supported in part by a National Science Foundation grant to the University of California at Irvine, Louis Narens principal investigator.