5 - Qualitative axioms for random-variable representation of extensive quantities

Summary

In the standard theory of fundamental extensive measurement, qualitative axioms are formulated that lead to a numerical assignment unique up to a positive similarity transformation. The central idea of the theory of random quantities is to replace the numerical assignment by a random-variable assignment. This means that each object is assigned a random variable. In the case of extensive quantities, the expectation of the random variable replaces the usual numerical assignment, and the distribution of the random variable reflects the variability of the property in question, which could be intrinsic to the object or due to errors of observation. In any case, the existence of distributions with positive variances is almost universal in the actual practice of measurement in most domains of science.

It is a widespread complaint about the foundations of measurement that too little has been written that combines the qualitative structural analysis of measurement procedures and the analysis of variability in a quantity measured or in errors in the procedures used. In view of the extraordinarily large number of papers that have been written about the foundations of the theory of error, which go back to the eighteenth century with fundamental work already by Simpson, Lagrange, and Laplace, followed by the important contributions of Gauss, it is surprising that the two kinds of analysis have not received a more intensive consideration. Part of the reason is the fact that, in all of this long history, the literature on the theory of errors has been intrinsically quantitative in character.