A Set of Independent Axioms for Extensive Quantities

Abstract

The modern viewpoint on quantities goes back at least to Newton’s Universal Arithmetick. Newton asserts that the relation between any two quantities of the same kind can be expressed by a real, positive number.² In 1901, O. Hoelder gave a set of ‘Axiome der Quantitaet’, which are sufficient to establish an isomorphism between any realization of his axioms and the additive semigroup of all positive real numbers. Related work of Hilbert, Veronese and others is indicative of a general interest in the subject of quantities in the abstract on the part of mathematicians of this period. During the last thirty years, from another direction, philosophers of science have become interested in the logical analysis of empirical procedures of measurement.³ The interests of these two groups overlap insofar as the philosophers have been concerned to state the formal conditions which must be satisfied by empirical operations measuring some characteristic of physical objects (or other entities). Philosophers have divided quantities (that is, entities or objects considered relatively to a given characteristic, such as mass, length or hardness) into two kinds. Intensive quantities are those which can merely be arranged in a serial order; extensive quantities are those for which a “natural” operation of addition or combination can also be specified. Another, more exact, way of making a distinction of this order is to say that intensive quantities are quantities to which numbers can be assigned uniquely up to a monotone transformation, and extensive quantities are quantities to which numbers can be assigned uniquely up to a similarity transformation (that is, multiplication by a positive constant).⁴ This last condition may be said to be the criterion of formal adequacy for a system of extensive quantities.

Reprinted from Portugaliae Mathematica 10 (1951), 163–172.

I am grateful to J. C. C. McKinsey for a number of helpful suggestions in connection with the present paper.