the second place, it may be admitted that the definition, like all definitions, is to a certain extent arbitrary. In the case of the small finite numbers, such as 2 and 3, it would be possible to frame definitions more nearly in accordance with our unanalysed feeling of what we mean; but the method of such definitions would lack uniformity, and would be found to fail sooner or later—at latest when we reached infinite numbers.
In the third place, the real desideratum about such a definition as that of number is not that it should represent as nearly as possible the ideas of those who have not gone through the analysis required in order to reach a definition, but that it should give us objects having the requisite properties. Numbers, in fact, must satisfy the formulae of arithmetic; any indubitable set of objects fulfilling this requirement may be called numbers. So far, the simplest set known to fulfil this requirement is the set introduced by the above definition. In comparison with this merit, the question whether the objects to which the definition applies are like or unlike the vague ideas of numbers entertained by those who cannot give a definition, is one of very little importance. All the important requirements are fulfilled by the above definition, and the sense of oddity which is at first unavoidable will be found to wear off very quickly with the growth of familiarity.
There is, however, a certain logical doctrine which may be thought to form an objection to the above definition of numbers as classes of classes—I mean the doctrine that there are no such objects as classes at all. It might be thought that this doctrine would make havoc of a theory which reduces numbers to classes, and of the many other theories in which we have made use of classes. This, however, would be a mistake: none of these theories
