The Principles of Mathematics (1903)

§ 153

(1) The kind of equality which consists in having the same number of parts has already been discussed in Part II. If this be indeed the meaning of quantitative equality, then quantity introduces no new idea. But it may be shown, I think, that greater and less have a wider field than whole and part, and an independent meaning. The arguments may be enumerated as follows: (α) We must admit indivisible quantities; (β) where the number of simple parts is infinite, there is no generalization of number which will give the recognized results as to inequality; (γ) some relations must be allowed to be quantitative, and relations are not even conceivably divisible; (δ) even where there is divisibility, the axiom that the whole is greater than the part must be allowed to be significant, and not a result of definition.(§ 153 ¶ 1)

(α) Some quantities are indivisible. For it is generally admitted that some psychical existents, such as pleasure and pain, are quantitative. If now equality means sameness in the number of indivisible parts, we shall have to regard a pleasure or a pain as consisting of a collection of units, all perfectly simple, and not, in any significant sense, equal inter se; for the equality of compound pleasures results on this hypothesis, solely from the number of simple ones entering into their composition, so that equality is formally inapplicable to indivisible pleasures. If, on the other hand, we allow pleasures to be infinitely divisible, so that no unit we can take is indivisible, then the number of units in any given pleasure is wholly arbitrary, and if there is to be any equality of pleasures, we shall have to admit that any two units may be significantly called equal or unequal[103]. Hence we shall require for equality some meaning other than sameness as to the number of parts. This latter theory, however, seems unavoidable. For there is not only no reason to regard pleasures as consisting of definite sums of indivisible units, but further—as a candid consideration will, I think, convince anyone—two pleasures can always be significantly judged equal or unequal. However small two pleasures may be, it must always be significant to say that they are equal. But on the theory I am combating, the judgment in question would suddenly cease to be significant when both pleasures were indivisible units. Such a view seems wholly unwarrantable, and I cannot believe that it has been consciously held by those[104] who have advocated the premisses from which it follows.(§ 153 ¶ 2)

(β) Some quantities are infinitely divisible, and in these, whatever definition we take of infinite number, equality is not coextensive with sameness in the number of parts. In the first place, equality or inequality must always be definite: concerning two quantities of the same kind, one answer must be right and the other wrong, though it is often not in our power to decide the alternative. From this it follows that, where quantities consist of an infinite number of parts, if equality or inequality is to be reduced to the number of parts at all, it must be reduced to number of simple parts; for the number of complex parts that may be taken to make up the whole is wholly arbitrary. But equality, for example in Geometry, is far narrower than sameness in the number of parts. The cardinal number of points in any two continuous portions of space is the same, as we know from Cantor; even the ordinal number or type is the same for any two lengths whatever. Hence if there is to be any spatial inequality of the kind to which Geometry and common-sense have accustomed us, we must seek some other meaning for equality than that obtained from the number of parts. At this point I shall be told that the meaning is very obvious: it is obtained from superposition. Without trenching too far on discussions which belong to a later part, I may observe (a) that superposition applies to matter, not to space, (b) that as a criterion of equality, it presupposes that the matter superposed is rigid, (c) that rigidity means constancy as regards metrical properties. This shows that we cannot, without a vicious circle, define spatial equality by superposition. Spatial magnitude is, in fact, as indefinable as every other kind; and number of parts, in this case as in all others where the number is infinite, is wholly inadequate even as a criterion.(§ 153 ¶ 3)

(γ) Some relations are quantities. This is suggested by the above discussion of spatial magnitudes, where it is very natural to base equality upon distances. Although this view, as we shall see hereafter, is not wholly adequate, it is yet partly true. There appear to be in certain spaces, and there certainly are in some series (for instance that of the rational numbers), quantitative relations of distance among the various terms. Also similarity and difference appear to be quantities. Consider for example two shades of colour. It seems undeniable that two shades of red are more similar to each other than either is to a shade of blue; yet there is no common property in the one case which is not found in the other also. Red is a mere collective name for a certain series of shades, and the only reason for giving a collective name to this series lies in the close resemblance between its terms. Hence red must not be regarded as a common property in virtue of which two shades of red resemble each other. And since relations are not even conceivably divisible, greater and less among relations cannot depend upon number of parts.(§ 153 ¶ 4)

(δ) Finally, it is well to consider directly the meanings of greater and less on the one hand, and of whole and part on the other. Euclid's axiom, that the whole is greater than the part, seems undeniably significant; but on the traditional view of quantity, this axiom would be a mere tautology. This point is again connected with the question whether superposition is to be taken as the meaning of equality, or as a mere criterion. On the latter view, the axiom must be significant, and we cannot identify magnitude with number of parts[105].(§ 153 ¶ 5)

§ 153 n. 1. I shall never use the word unequal to mean merely not equal, but always to mean greater or less, i.e. not equal, though of the same kind of quantities.

§ 153 n. 2. E.g. Mr Bradley, What do we mean by the Intensity of Psychical States? Mind, N. S. Vol. IV; see esp. p. 5.

§ 153 ¶ 6. Compare, with the above discussion, Meinong, Ueber die Bedeutung des Weberschen Gesetzes, Hamburg and Leipzig, 1896; especially Chap. I, § 3.