The Principles of Mathematics (1903)

§ 154

(2) There is therefore in quantity something over and above the ideas which we have hitherto discussed. It remains to decide between the relative and absolute theories of magnitude.(§ 154 ¶ 1)

The relative theory regards equal quantities as not possessed of any common property over and above that of unequal quantities, but as distinguished merely by the natural relation of equality. There is no such thing as a magnitude, shared by euqal quantities. We must not say: This and that are both a yard long; we must say: This and that are equal, or are both equal to the standard yard in the Exchequer. Inequality is also a direct relation between quantities, not between magnitudes. There is nothing by which a set of equal quantities are distinguished from one which is not equal to them, except the relation of equality itself. The course of definition is, therefore, as follows: We have first a quality or relation, say pleasure, of which there are various instances, specialized, in the case of a quality, by temporal or spatiotemporal position, and in the case of a relation, by the terms between which it holds. Let us, to fix ideas, consider quantities of pleasure. Quantities of pleasure consist merely of the complexes pleasure at such a time, and pleasure at such another time (to which place may be added, if it be thought that pleasures have position in space). In the analysis of a particular pleasure, there is, according to the relational theory, no other element to be found. But on comparing these particular pleasures, we find that any two have one and only one of three relations, equal, greater, and less. Why some have one relation, some another, is a question to which it is theoretically and strictly impossible to give an answer; for there is, ex hypothesi, no point of difference except temporal or spatio-temporal position, which is obviously irrelevant. Equal quantities of pleasure do not agree in any respect in which unequal ones differ: it merely happens that some have on relation and some another.(§ 154 ¶ 2)

This state of things, it must be admitted, is curious, and it becomes still more so when we examine the indemonstrable axioms which the relational theory obliges us to assume. They are the following (A, B, C being all quantities of one kind):(§ 154 ¶ 3)

(a) A=B, or A is greater than B, or A is less than B.(§ 154 ¶ 4)

(b) A being given, there is always a B, which may be identical with A, such that A=B.(§ 154 ¶ 5)

(c) If A=B, then B=A.(§ 154 ¶ 6)

(d) If A=B and B=C, then A=C.(§ 154 ¶ 7)

(e) If A is greater than B, then B is less than A.(§ 154 ¶ 8)

(f) If A is greater than B, and B is greater than C, then A is greater than C.(§ 154 ¶ 9)

(g) If A is greater than B, and B=C, then A is greater than C.(§ 154 ¶ 10)

(h) If A=B, and B is greater than C, then A is greater than C.(§ 154 ¶ 11)

From (b), (c), and (d) it follows that A=A[106]. From (e) and (f) it follows that, if A is less than B, and B is less than C, then A is less than C; from (c), (e), and (h) it follows that, if A is less than B, and B=C, then A is less than C; from (c), (e), and (g) it follows that, if A=B, and B is less than C, then A is less than C. (In the place of (b) we may put the axiom: If A be a quantity, then A=A.) These axioms, it will be observed, lead to the conclusion that, in any proposition asserting equality, excess, or defect, an equal quantity may be substituted anywhere without affecting the truth or falsehood of the proposition. Further, the proposition A=A is an essential part of the theory. Now the first of these facts strongly suggests that what is relevant in quantitative propositions is not the actual quantity, but some property which it shares with other equal quantities. And this suggestion is almost demonstrated by the second fact, A=A. For it may be laid down that the only unanalyzable symmetrical and transitive relation which a term can have to itself is identity, if this be indeed a relation. Hence the relation of equality should be analyzable. Now to say that a relation is analyzable is to say either that it consists of two or more relations between its terms, which is plainly not the case here, or that, when it is said to hold between two terms, there is some third term to which both are related in ways which, when compounded, give the original relation. Thus to assert that A is B's grandparent is to assert that there is some third person C, who is A's son or daughter and B's father or mother. Hence if equality be unanalyzable, two equal terms must both be related to some third term; and since a term may be equal to itself, any two equal terms must have the same relation to the third term in question. But to admit this is to admit the absolute theory of magnitude.(§ 154 ¶ 12)

A direct inspection of what we mean when we say that two terms are equal or unequal will reinforce the objections to the relational theory. It seems preposterous to maintain that equal quantities have absolutely nothing in common beyond what is shared by unequal quantities. Moreover unequal quantities are not merely different: they are different in the specific manner expressed by saying that one is greater, the other less. Such a difference seems quite unintelligible unless there is some point of difference, where unequal quantities are concerned, which is absent where quantities are equal. Thus the relational theory, though apparently not absolutely self-contradictory, is complicated and paradoxical. Both the complication and the paradox, we shall find, are entirely absent in the absolute theory.(§ 154 ¶ 13)

§ 156 n. 1. This does not follow from (c) and (d) alone, since they do not assert that A is ever equal to B. See Peano, loc. cit.