contradiction, this is possible only when another synthetical
proposition precedes, from which the latter is deduced, but never
of itself.

Before all, be it observed, that proper mathematical propositions
are always judgements a priori, and not empirical, because they
carry along with them the conception of necessity, which cannot be
given by experience. If this be demurred to, it matters not; I will
then limit my assertion to pure mathematics, the very conception of
which implies that it consists of knowledge altogether non-empirical
and a priori.

We might, indeed at first suppose that the proposition 7 + 5 = 12 is
a merely analytical proposition, following (according to the principle
of contradiction) from the conception of a sum of seven and five.
But if we regard it more narrowly, we find that our conception of
the sum of seven and five contains nothing more than the uniting of
both sums into one, whereby it cannot at all be cogitated what this
single number is which embraces both. The conception of twelve is by
no means obtained by merely cogitating the union of seven and five;
and we may analyse our conception of such a possible sum as long as
we will, still we shall never discover in it the notion of twelve.
We must go beyond these conceptions, and have recourse to an intuition
which corresponds to one of the two--our five fingers, for example,
or like Segner in his Arithmetic five points, and so by degrees, add
the units contained in the five given in the intuition, to the
conception of seven. For I first take the number 7, and, for the
conception of 5 calling in the aid of the fingers of my hand as
objects of intuition, I add the units, which I before took together
to make up the number 5, gradually now by means of the material image