Introduction to Mathematical Philosophy

of San Francisco; we cannot [Pg 40] say definitely that either of these two places is "between" England and New Zealand. But if a man chooses that route to go round the world, whichever way round he goes, his times in England and New Zealand are separated from each other by his times in Suez and San Francisco, and conversely. Generalising, if we take any four points on a circle, we can separate them into two couples, say and and and , such that, in order to get from to one must pass through either or , and in order to get from to one must pass through either or . Under these circumstances we say that the couple are "separated" by the couple . Out of this relation a cyclic order can be generated, in a way resembling that in which we generated an open order from "between," but somewhat more complicated.[12] [12]Cf. Principles of Mathematics, p. 205 (§ 194), and references there given. The purpose of the latter half of this chapter has been to suggest the subject which one may call "generation of serial relations." When such relations have been defined, the generation of them from other relations possessing only some of the properties required for series becomes very important, especially in the philosophy of geometry and physics. But we cannot, within the limits of the present volume, do more than make the reader aware that such a subject exists. [Pg 41]