Current axiomatizations for extensive measurement postulate the existence of infinitely small objects. This assumption is neither necessary nor reasonable. This paper develops this theme and presents a more acceptable axiom system. A representation theorem is stated and proved in detail. This work improves some previous results of the author.

Science Without Numbers caused a stir in 1980, with its bold nominalist approach to the philosophy of mathematics and science. It has been unavailable for twenty years and is now reissued in a revised edition with a substantial new preface presenting the author's current views and responses to the issues raised in subsequent debate.

Unlike the standard representational theory of measurement, which takes the real numbers as a pregiven numerical domain, the approach presented in this paper is based on an abstract concept of a procedure of measurement, and ‘values of measurement’ are understood in terms of such procedures. The resulting ‘type approach’ makes use of elementary model-theoretic notions and emphasizes the constructibility of scales. It provides a natural starting point for a systematic discussion of issues that tend to be neglected in the standard (...) framework (such as the relation between measurement and computation). At the same time it is perfectly compatible with the modern representational theory of measurement and helps elucidate a number of issues central to that theory (e.g. the role of Archimedean axioms). (shrink)