Sums no more? - The changing role of additivity in measurement theory

Abstract

In the key writings on the foundations of measurement from the late 19th to the early 20th century (e.g. [1],[2],[3]), additivity was the central criterion for establishing appropriate axioms for measurement.This changed fairly abruptly around the middle of 20th century, and today the most comprehensive treatments of the foundations of measurement have relegated additivity to secondary status. My question in this paper is why and how this change came about. In what sense, if any, can this change in the foundations of measurement be regarded as a Kuhnian paradigm shift? I conclude that additivity fell out of favor as a necessary condition for measurement on the one hand because new scientific disciplines demanded a relaxation of apparently overly restrictive standards applicable to physical quantities, and on the other hand because new formal techniques were developed that allowed for a characterization of extensive quantities not based on the axioms of additivity. This episode in the history of measurement theory is philosophically interesting in many respects. It shows how a conceptual change—in our understanding of the concept of extensive quantity—is accomplished not through new data or empirical results, but arises from new demands in emerging scientific fields, and the development of new formal methods. In this way the switch from axioms of additivity as a basis for foundations of measurement meets Kuhnian conditions for paradigm shifts: a new paradigm, in this case new formal tools, has to be available before the shift can be completed. A result of this change, a sociological shift in science occurs as well: the foundations of measurement emerge as their own field of research, after having been regarded firmly in the domain of physics. In another respect the shift is remarkably un-Kuhnian, however: since measurement concerns not just a single discipline, it would be misleading to characterize the shift in understanding of measurement and quantities as a paradigm shift within a single field of research. Instead we see the emergence of an independent area of research out of the conflicting demands of different scientific disciplines.