number of birds on yonder bough). Taking the whole numbers as the abstract numbers, i.e. as the members of a certain ideal series, arithmetically defined, the mathematician can, therefore, view them all as given by means of their universal definition, and their consequent clear distinction from all other objects of thought.[1] Taking them thus as given, the numbers become entities of the type contemplated by our Third Conception of Being; and as such entities we can admit them here for the moment, not now asking whether or no they have, or can win, a reality of our Fourth type.
Now the numbers form, in infinitely numerous ways, a self-representative system of the type here in question. That is, as has repeatedly been remarked, by all the recent authors who have dealt with this aspect of the matter, the number-system, taken in its conceived totality, can be put in a one-to-one correspondence with one of its own constituent portions in any one of an endless number of ways. For the numbers, if once regarded as a given whole, form an endless ordered series, having a first term, a second term, and so on. But just so the even numbers, 2, 4, 6, etc., form an endless ordered series, having a first, second, third term, and so on. In the same way, too, the prime numbers form a demonstrably endless series, whereof there is a first member, a second member, and so forth. Or, again, the numbers that are perfect squares, those that are perfect cubes, and those, in general, that are of the form ɑn, where n is any one whole number, while a takes successively the value of every whole number, — all such derived systems of whole numbers, form similarly ordered series, wherein each member of each system has its determined place as first, second, third, or later member of its own system, while the system forms a series without end. Take, then, any
- ↑ How they are to be defined is of course itself a significant logical problem, whereof we shall soon hear more. Cantor’s account of the well-defined multitude, Menge, or ensemble, is found in French translation in the Acta Mathematica, torn. II, p. 363. On the general sense in which any multitude can be viewed as given for purposes of mathematical discussion, see Borel’s Leçons (cited above), p. 2.
