contradiction in asserting that A is not B, and it would seem that only experience can determine whether all A is or is not B. Otherwise we are presupposing that things must conform to our ideas about them. Now metaphysics claims to make _a priori_ synthetic judgements, for it does not base its results on any appeal to experience. Hence, before we enter upon metaphysics, we really ought to investigate our right to make _a priori_ synthetic judgements at all. Therein, in fact, lies the importance to metaphysics of the existence of such judgements in mathematics and physics. For it shows that the difficulty is not peculiar to metaphysics, but is a general one shared by other subjects; and the existence of such judgements in mathematics is specially important because there their validity or certainty has never been questioned.[11] The success of mathematics shows that at any rate under certain conditions _a priori_ synthetic judgements are valid, and if we can determine these conditions, we shall be able to decide whether such judgements are possible in metaphysics. In this way we shall be able to settle a disputed case of their validity by examination of an undisputed case. The general problem, however, is simply to show what it is which makes _a priori_ synthetic judgements as such possible; and there will be three cases, those of mathematics, of physics, and of metaphysics. [11] Kant points out that this certainty has usually been attributed to the analytic character of mathematical judgements, and it is of course vital to his argument that he should be successful in showing that they are really synthetic.