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 formulæ 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 are any the worse for the doctrine that classes are fictions. What the doctrine is, and why it is not destructive, I will try briefly to explain. On account of certain rather complicated difficulties, culminating in definite contradictions, I was led to the view that nothing that can be said significantly about things, _i.e._ particulars, can be said significantly (_i.e._ either truly or falsely) about classes of things. That is to say, if, in any sentence in which a thing is mentioned, you substitute a class for the thing, you no longer have a sentence that has any meaning: the sentence is no longer either true or false, but a meaningless collection of words. Appearances to the contrary can be dispelled by a moment's reflection. For example, in the sentence, "Adam is fond of apples," you may substitute _mankind_, and say, "Mankind is fond of apples." But obviously you do not mean that there is one individual, called "mankind," which munches apples: you mean that the separate individuals who compose mankind are each severally fond of apples.