Having now briefly set forth the mathematical theory of cardinal numbers, it is time to turn our attention to the philosophical questions raised by this theory. I shall begin by a few preliminary remarks as to the distinction between philosophy and mathematics, and as to the function of philosophy in such a subject as the foundations of mathematics. The following observations are not necessarily to be regarded as applicable to other branches of philosophy, since they are derived specially from the consideration of the problems of logic.(§ 124 ¶ 1)

The distinction of philosophy and mathematics is broadly one of point of view: mathematics is constructive and deductive, philosophy is critical, and in a certain impersonal sense controversial. Wherever we have deductive reasoning, we have mathematics; but the principles of deduction, the recognition of indefinable entities, and the distinguishing between such entities, are the business of philosophy. Philosophy is, in fact, mainly a question of insight and perception. Entities which are perceived by the so-called senses, such as colours and sounds, are, for some reason, not commonly regarded as coming within the scope of philosophy, except as regards the more abstract of their relations; but it seems highly doubtful whether any such exclusion can be maintained. In any case, however, since the present work is essentially unconcerned with sensible objects, we may confine our remarks to entities which are not regarded as existing in space and time. Such entities, if we are to know anything about them, must be also in some sense perceived, and must be distinguished one from another; their relations also must be in part immediately apprehended. A certain body of indefinable entities and indemonstrable propositions must form the starting-point for any mathematical reasoning; and it is this starting-point that concerns the philosopher. When the philosopher's work has been perfectly accomplished, its results can be wholly embodied in premisses from which deduction may proceed. Now it follows from the very nature of such inquiries that results may be disproved, but can never be proved. The disproof will consist in pointing out contradictions and inconsistencies; but the absence of these can never amount to proof. All depends, in the end, upon immediate perception; and philosophical argument, strictly speaking, consists mainly of an endeavour to cause the reader to perceive what has been perceived by the author. The argument, in short, is not of the nature of proof, but of exhortation. Thus the question of the present chapter: Is there any indefinable set of entities commonly called numbers, and different from the set of entities above defined? is an essentially philosophical question, to be settled by inspection rather than by accurate chains of reasoning.(§ 124 ¶ 2)