The Principles of Mathematics (1903)

§ 159

The questions to be discussed in the present chapter are these: What kinds of terms are there which, by their common relation to a number of magnitudes, constitute a class of quantities of one kind? Have all such terms anything else in common? Is there any mark which will ensure that a term is thus related to a set of magnitudes? What sorts of terms are capable of degree, or intensity, or greater and less?(§ 159 ¶ 1)

The traditional view regards divisibility as a common mark of all terms having magnitude. We have already seen that there is no à priori ground for this view. We are now to examine the question inductively, to find as many undoubted instances of quantities as possible, and to inquire whether they all have divisibility or any other common mark.(§ 159 ¶ 2)

Any term of which a greater or less degree is possible contains under it a collection of magnitudes of one kind. Hence the comparative form in grammar is primâ facie evidence of quantity. If this evidence were conclusive, we should have to admit that all, or almost all, qualities are susceptible of magnitude. The praises and reproaches addressed by poets to their mistresses would afford comparatives and superlatives of most known adjectives. But some circumspection is required in using evidence of this grammatical nature. There is always, I think, some quantitative comparison wherever a comparative or superlative occurs, but it is often not a comparison as regards the quality indicated by grammar.(§ 159 ¶ 3)

O ruddier than the cherry,
O sweeter than the berry,
O nymph more bright
Than moonshine light,

are lines containing three comparatives. As regards sweetness and brightness, we have, I think, a genuine quantitative comparison; but as regards ruddiness, this may be doubted. The comparative here—and generally where colours are concerned—indicates, I think, not more of a given colour, but more likeness to a standard colour. Various shades of colour are supposed to be arranged in a series, such that the difference of quality is greater or less according as the distance in the series is greater or less. One of these shades is the ideal ruddiness, and others are called more or less ruddy according as they are nearer to or further from this shade in the series. The same explanation applies, I think, to such terms as whiter, blacker, redder. The true quantity involved seems to be, in all these cases, a relation, namely the relation of similarity. The difference between two shades of colour is certainly a difference of quality, not merely of magnitude; and when we say that one thing is redder than another, we do not imply that the two are of the same shade. If there were no difference of shade, we should probably say one was brighter than the other, which is quite a different kind of comparison. But though the difference of two shades is a difference of quality, yet, as the possibility of serial arrangement shows, this difference of quality is itself susceptible of degrees. Each shade of colour seems to be simple and unanalyzable; but neighbouring colours in the spectrum are certainly more similar than remote colours. It is this that gives continuity to colours. Between two shades of colour, A and B, we should say, there is always a third colour C; and this means that C resembles A or B more than B or A does. But for such relations of immediate resemblance, we should not be able to arrange colours in series. The resemblance must be immediate, since all shades of colour are unanalyzable, as appears from any attempt at description or definition[111]. Thus we have an indubitable case of relations which have magnitude. The difference or resemblance of two colours is a relation, and is a magnitude; for it is greater or less than other differences or resemblances.

§ 159 n. 1. On the subject of the resemblances of colours, see Meinong, Abstrahiren und Vergleichen, Zeitschrift f. Psych. u. Phys. d. Sinnesorgane, Vol. XXIV, p. 72 ff. I am not sure that I agree with the whole of Meinong's argument, but his general conclusion, dass die Umfangscollective des Aehnlichen Allgemeinheiten darstellen, an denen die Abstraction wenigstens unmittelbar keinen Antheil hat (p. 78), appears to me to be a correct and important logical principle.