of concepts, and constitutes, therefore, a necessary system of propositions which follow from one another according to the principle of contradiction. The question whether this pure mathematics may be at once applied to objects, would be answered in the negative, at least in part. Kant regarded this solution of the problem as false, because mathematical propositions depend, according to him, on per- ception. From this newly gained standpoint, if one takes account of the two premises given above, there can no longer be any difference between pure and applied mathematics. For the same a priori perception which gives necessity to the mathematical judgments, in opposition to the individual cases which experience furnishes, imparts to it also objective validity, i.e., applicability to all objects which are given in that perception. One may, therefore, neither assert with Paulsen that Kant only treated the problem of applied mathematics; nor with Fischer that he was only concerned with the problem of pure mathematics; nor with Vaihinger that both problems receive attention, sometimes confused, sometimes (at least for a knowing eye) in separation. We must rather say that Kant only recognized one problem: the possibility of mathematics in general. Whenever mathematics is founded on perception, there is no longer any distinction between necessity, and validity for objects; both are identical. They can only be distinguished by one who derives mathematics from concepts. It is at once evident that the three detached Reflexionen from B. Erdmann's collection which Vaihinger quotes (p. 282, note), cannot be used in contradiction of my view. Vaihinger himself brings together (p. 273, note 1) the confusion of the mathematical problems, and of pure perception with the form of perception, without once recognizing the necessity of both of these from Kant's standpoint.
7) As a matter of fact—that is, if one disregards all unproved premises—there also exists for Kant a problem of pure mathematics peculiar to himself which I think he cannot solve. It consists in the question: How do the different geometrical axioms and propositions follow from the nature of space and its properties? Are they deduced from its ready made a priori perception, or discovered in it? Whence arises, then, their necessity, since it is always single experiences which inform us regarding the relations of that a priori mathematical space, and since we can accordingly never be certain whether or not those experiences may be repeated, and whether they reveal to us essential and permanent relations and not rather those which are accidental and transitory?
