The Principles of Mathematics (1903)

§ 152

Setting aside magnitudes for the moment, let us consider quantities. A quantity is anything which is capable of quantitative equality to something else. Quantitative equality is to be distinguished from other kinds, such as arithmetical or logical equality. All kinds of equality have in common the three properties of being reflexive, symmetrical, and transitive, i.e. a term which has this relation at all has this relation to itself; if A has the relation to B, B has it to A; if A has it to B, and B to C, A has it to C[102]. What it is that distinguishes quantitative equality from other kinds, and whether this kind of equality is analyzable, is a further and more difficult question, to which we must now proceed.(§ 152 ¶ 1)

There are, so far as I know, three main views of quantitative equality. There is (1) the traditional view, which denies quantity as an independent idea, and asserts that two terms are equal when, and only when, they have the same number of parts. (2) There is what may be called the relative view of quantity, according to which equal, greater and less are all direct relations between quantities. In this view we have no need of magnitude, since sameness of magnitude is replaced by the symmetrical and transitive relation of equality. (3) There is the absolute theory of quantity, in which equality is not a direct relation, but is to be analyzed into possession of a common magnitude, i.e. into sameness of relation to a third term. In this case there will be a special kind of relation of a term to its magnitude; between two magnitudes of the same kind there will be the relation of greater and less; while equal, greater and less will apply to quantities only in virtue of their relation to magnitudes. The difference between the second and third theories is exactly typical of a difference which arises in the case of many other series, and notably in regard to space and time. The decision is, therefore, a matter of very considerable importance.(§ 152 ¶ 2)

§ 152 n. 1. On the independence of these three properties, see Peano, Revue de Mathématique, VII, p. 22. The reflexive property is not strictly necessary; what is properly necessary and what is alone (at first sight at any rate) true of quantitative equality, is, that there exists at least one pair of terms having the relation in question. It follows then from the other two properties that each of these terms has to itself the relation in question.