The Principles of Mathematics (1903)

§ 155

(3) In the absolute theory, there is, belonging to a set of equal quantities, one definite concept, namely a certain magnitude. Magnitudes are distinguished among concepts by the fact that they have the relations of greater and less (or at least one of them) to other terms, which are therefore also magnitudes. Two magnitudes cannot be equal, for equality belongs to quantities, and is defined as possession of the same magnitude. Every magnitude is a simple and indefinable concept. Not any two magnitudes are one greater and the other less; on the contrary, given any magnitude, those which are greater or less than that magnitude form a certain definite class, within which any two are one greater and the other less. Such a class is called a kind of magnitude. A kind of magnitude may, however, be also defined in another way, which has to be connected with the above by an axiom. Every magnitude is a magnitude of something—pleasure, distance, area, etc.—and has thus a certain specific relation to the something of which it is a magnitude. This relation is very peculiar, and appears to be incapable of further definition. All magnitudes which have this relation to one and the same something (e.g. pleasure) are magnitudes of one kind; and with this definition, it becomes an axiom to say that, of two magnitudes of the same kind, one is greater and the other less.(§ 155 ¶ 1)