The Principles of Mathematics (1903)

§ 160

I have dwelt upon this case of colours, since it is one instance of a very important class. When any number of terms can be arranged in a series, it frequently happens that any two of them have a relation which may, in a generalized sense, be called a distance. This relation suffices to generate a serial arrangement, and is always necessarily a magnitude. In all such cases, if the terms of the series have names, and if these names have comparatives, the comparatives indicate, not more of the term in question, but more likeness to that term. Thus, if we suppose the time-series to be one in which there is distance, when an event is said to be more recent than another, what is meant is that its distance from the present was less than that of the other. Thus recentness is not itself a quality of the time or of the event. What are quantitatively compared in such cases are relations, not qualities. The case of colour is convenient for illustration, because colours have names, and the difference of two colours is generally admitted to be qualitative. But the principle is of very wide application. The importance of this class of magnitudes, and the absolute necessity of clear notions as to their nature, will appear more and more as we proceed. The whole philosophy of space and time, and the doctrine of so-called extensive magnitudes, depend throughout upon a clear understanding of series and distance.(§ 160 ¶ 1)

Distance must be distinguished from mere difference or unlikeness. It holds only between terms in a series. It is intimately connected with order, and implies that the terms between which it holds have an ultimate and simple difference, not one capable of analysis into constituents. It implies also that there is a more or less continuous passage, through other terms belonging to the same series, from one of the distant terms to the other. Mere difference per se appears to be the bare minimum of a relation, being in fact a precondition of almost all relations. It is always absolute, and is incapable of degrees. Moreover it holds between any two terms whatever, and is hardly to be distinguished from the assertion that they are two. But distance holds only between the members of certain series, and its existence is then the source of the series. It is a specific relation, and it has sense; we can distinguish the distance of A from B from that of B from A. This last mark alone suffices to distinguish distance from bare difference.(§ 160 ¶ 2)

It might perhaps be supposed that, in a series in which there is distance, although the distance AB must be greater than or less than AC, yet the distance BD need not be greater or less than AC. For example, there is obviously more difference between the pleasure derivable from £5 and that derivable from £100 than between that from £5 and that from £20. But need there be either equality or inequality between the difference for £1 and £20 and that for £5 and £100? This question must be answered affirmatively. For AC is greater or less than BC, and BC is greater or less than BD; hence AC, BC and also BC, BD are magnitudes of the same kind. Hence AC, BD are magnitudes of the same kind, and if not identical, one must be the greater and the other the less. Hence, when there is distance in a series, any two distances are quantitatively comparable.(§ 160 ¶ 3)

It should be observed that all the magnitudes of one kind form a series, and that their distances, therefore, if they have distances, are again magnitudes. But it must not be supposed that these can, in general, be obtained by subtraction, or are of the same kind as the magnitudes whose differences they express. Subtraction depends, as a rule, upon divisibility, and is therefore in general inapplicable to indivisible quantities. The point is important, and will be treated in detail in the following chapter.(§ 160 ¶ 4)

Thus nearness and distance are relations which have magnitude. Are there any other relations having magnitude? This may, I think, be doubted[112]. At least I am unaware of any other such relation, though I know no way of disproving their existence.(§ 160 ¶ 5)

§ 160 n. 1. Cf. Meinong, Ueber die Bedeutung des Weber'schen Gesetzes, Hamburg and Leipzig, 1896, p. 23.