Page:The World and the Individual, Second Series (1901).djvu/94

Rh find that between successive whole numbers, such as 2 and 3, 3 and 4, it is impossible to conceive other whole numbers inserted, so long as one takes the whole numbers in their natural order. And the same holds true, in the world of empirical things, regarding any simple series of objects whose type is that of the whole number series, where every object of the series is followed by a next one, with nothing between that belongs to the series. A series of this type we shall hereafter call, in accordance with recent mathematical usage, a Well-Ordered Series. But, on the other hand, as our Supplementary Essay showed at length, any collection of objects in the world is part of some infinite collection; and so the objects of any well-ordered series are themselves portions of the expression of a recurrent process, or of a well-ordered series of such processes. And in every such process, as was shown in the Supplementary Essay, an infinite number of discriminations are already implied. Hence, although it is indeed possible to find cases where we can no longer look for objects between a given pair of objects which have already been discriminated, it still appears that discriminations which are logically completed, merely by our distinguishing between two objects, are not to be found. Discrimination seems to be not merely of pairs, but of triads, or of larger systems of facts. For the sake of later use, it is proper to note here, regarding the definition of the Well-Ordered Series just given, that, while its type is that of the whole-number series, in so far forth as every term has a next following term, recent mathematical usage has extended the concept to include the so-called Transfinite Well-Ordered Series of Cantor, in which infinite series may follow infinite series without end, but so that every term has its own next following term.