quite impossible through the fact that Kant did not distinguish the two kinds of mathematics in the fourth division of § 3, and that he is not at all conscious of the alleged double reference, as Vaihinger himself grants (p. 274), in asserting that this distinction had not been retained in consciousness. But it would be quite impossible to believe that Kant distinguished pure and applied mathematics in the first three divisions, and that in the fourth division he had again suddenly forgotten this distinction.
4) The same is true of § 8 (R Vb, pp. 64 if.), where, indeed, as I have already remarked in my edition of the Kritik, two trains of thought must be distinguished. But the first as well as the second of these is concerned with applied mathematics. The beginning of the second argument ("Läge nun, &c.") shows that Kant turned back to the beginning of the division; and, as so often, introduces by means of a particle which should really betoken an advance only a repetition of something already stated. The introduction of the word 'Vermögen' by no means signifies an advance. From his standpoint, Vaihinger can only explain (pp. 469 if.) the fact that the same examples are given in both arguments by the adoption of the greatest artificialities.
5) In note 1, § 13 of the Prolegomena, we do find a clear distinction between the problem of pure, and that of applied, mathematics. However, this is not made from Kant's standpoint, but from that of his opponents. For his opponents (Leibnizians, Hume), such a distinction was possible, and so Kant makes it in describing their theories. But when he propounds his own theory, he does not make the distinction. Why not?
6) The simple answer is, because he could not make it at all on his premises. For him there is only one problem: How is mathematics possible? while for his opponents this falls into two questions. The two most important unproven premises from which he sets out are the following: (a) The necessary geometrical axioms and propositions are given along with the necessary a priori perception of space; whether it is that they follow directly from the nature of space and its properties, or that they are derived a priori from its perception and discovered in it. (b) The a priori—that is, according to Kant, what precedes experience and makes experience possible, and is, therefore, valid for it and all its objects in general—space perception is identical with the a priori form of sensibility (Anschauung). The opponents of Kant answer the question regarding the possibility of pure mathematics by suggesting that mathematics is a mere science
