ABSTRACT
The purpose of this Part is to explain the kinds of applications of numbers which may be called measurement. For this purpose, we have first to consider generalizations of number. The numbers dealt with hitherto have been only integers (cardinal or ordinal); accordingly, in Section A, we consider positive and negative integers, ratios, and real numbers. (Complex numbers are dealt with later, under geometry, because they do not form a one-dimensional series.)
In Section B, we deal with what may be called ‘kinds’ of quantity: thus e.g. masses, spatial distances, velocities, each form one kind of quantity. We consider each kind of quantity as what may be called a ‘vector-family’, i.e. a class of one-one relations all having the same converse domain, and all having their domain contained in their converse domain. In such a case as spatial distances, the applicability of this view is obvious; in such a case as masses, the view becomes applicable by considering e.g. one gramme as + one gramme, i.e. as the relation of a mass m to a mass m′ when m exceeds m′ by one gramme. What is commonly called simply one gramme will then be the mass which has the relation + one gramme to the zero of mass. The reasons for treating quantities as vectors will be explained in Section B. Various different kinds of vector-families will be considered, the object being to obtain families whose members are capable of measurement either by means of ratios or by means of real numbers.
