The Principles of Mathematics (1903)

§ 156

An objection to the above theory may be based on the relation of a magnitude to that whose magnitude it is. To fix our ideas, let us consider pleasure. A magnitude of pleasure is so much pleasure, such and such an intensity of pleasure. It seems difficult to regard this, as the absolute theory demands, as a simple idea: there seem to be two constituents, pleasure and intensity. Intensity need not be intensity of pleasure, and intensity of pleasure is distinct from abstract pleasure. But what we require for the constitution of a certain magnitude of pleasure is, not intensity in general, but a certain specific intensity; and a specific intensity cannot be indifferently of pleasure or of something else. We cannot first settle how much we will have, and then decide whether it is to be pleasure or mass. A specific intensity must be of a specific kind. Thus intensity and pleasure are not independent and coordinate elements in the definition of a given amount of pleasure. There are different kinds of intensity, and different magnitudes in each kind; but magnitudes in different kinds must be different. Thus it seems that the common element, indicated by the term intensity or magnitude, is not any thing intrinsic, that can be discovered by analysis of a single term, but is merely the fact of being one term in a relation of inequality. Magnitudes are defined by the fact that they have this relation, and they do not, so far as the definition shows, agree in anything else. The class to which they all belong, like the married portion of a community, is defined by mutual relations among its terms, not by a common relation to some outside term—unless, indeed, inequality itself were taken as such a term, which would be merely an unnecessary complication. It is necessary to consider what may be called the extension or field of a relation, as well as that of a class-concept: and magnitude is the class which forms the extension of inequality. Thus magnitude of pleasure is complex, because it combines magnitude and pleasure; but a particular magnitude of pleasure is not complex, for magnitude does not enter into its concept at all. It is only a magnitude because it is greater or less than certain other terms; it is only a magnitude of pleasure because of a certain relation which it has to pleasure. This is more easily understood where the particular magnitude has a special name. A yard, for instance, is a magnitude, because it is greater than a foot; it is a magnitude of length, because it is what is called a length. Thus all magnitudes are simple concepts, and are classified into kinds by their relation to some quality or relation. The quantities which are instances of a magnitude are particularized by spatio-temporal position or (in the case of relations which are quantities) by the terms between which the relation holds. Quantities are not properly greater or less, for the relations of greater and less hold between their magnitudes, which are distinct from the quantities.(§ 156 ¶ 1)

When this theory is applied in the enumeration of the necessary axioms, we find a very notable simplification. The axioms in which equality appears have all become demonstrable, and we require only the following (L, M, N being magnitudes of one kind):(§ 156 ¶ 2)

(a) No magnitude is greater or less than itself.(§ 156 ¶ 3)

(b) L is greater than M or L is less than M.(§ 156 ¶ 4)

(c) If L is greater than M, then M is less than L.(§ 156 ¶ 5)

(d) If L is greater than M and M is greater than N, then L is greater than N.(§ 156 ¶ 6)

The difficult axiom which we formerly called (b) is avoided, as are the other axioms concerning equality; and those that remain are simpler than our former set.(§ 156 ¶ 7)