? _A priori_ judgements of the other
kind, viz. analytic judgements, offer no difficulty, since they
are at bottom tautologies, and consequently denial of them is
self-contradictory and meaningless. But there is difficulty where a
judgement asserts that a term B is connected with another term A, B
being neither identical with nor a part of A. In this case there is no
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.

The outline of the solution of this problem is contained in the
Preface to the Second Edition. There Kant urges that the key is to be
found by consideration of mathematics and physics. If the question be
raised as to what it is that has enabled these subjects to advance, in
both cases the answer will be found to lie in a change of method.
"Since the earliest times to which the history of human reason
reaches, mathematics has, amon