Kant found, however, that the paradoxical characteristics of the causal principle also belong to mathematical and physical judgments. This fact makes it impossible to accept Hume’s sceptical conclusion. If even the assertion 7 + 5 = 12 is both synthetic and _a priori_, it is obviously impossible to question the validity of judgments that possess these characteristics. But they do not for that reason any the less urgently press for explanation. Such an enquiry might not, indeed, be necessary were we concerned only with scientific knowledge. For the natural sciences justify themselves by their practical successes and by their steady unbroken development. But metaphysical judgments are also of this type; and until the conditions which make _a priori_ synthetic judgment possible have been discovered, the question as to the legitimacy of metaphysical speculation cannot be decided. Such judgments are plainly mysterious, and urgently call for further enquiry. The problem to be solved concerns the ground of our ascription to the subject concept, as necessarily belonging to it, a predicate which seems to have no discoverable relation to it. What is the unknown _x_ on which the understanding rests in asserting the connection? It cannot be repeated experience; for the judgments in question claim necessity. Nor can such judgments be proved by means of a logical test, such as the inconceivability of the opposite. The absence of all apparent connection between subject and predicate removes that possibility. These, however, are the only two methods of proof hitherto recognised in science and philosophy. The problem demands for its solution nothing less than the discovery and formulation of an entirely novel method of proof.