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be deduced from it. I will first of all bring synthetic judgments under certain classes.

(1) Judgments of experience are always synthetic. It would be absurd to found an analytic judgment on experience, as it is unnecessary to go beyond my own conception in order to construct the judgment, and therefore the confirmation of experience is unnecessary to it. That a body is extended is a proposition possessing à priori certainty, and no judgment of experience. For before I go to experience I have all the conditions of my judgment already present in the conception, out of which I simply draw the predicate in accordance with the principle of contradiction, and thereby at the same time the necessity of the judgment may be known, a point which experience could never teach me.

(2) Mathematical judgments are in their entirety synthetic. This truth seems hitherto to have altogether escaped the analysts of human Reason; indeed, to be directly opposed to all their suppositions, although it is indisputably certain and very important in its consequences. For, because it was found that the conclusions of mathematicians all proceed according to the principle of contradiction (which the nature of every apodictic certainty demands), it was concluded that the axioms were also known through the principle of contradiction, which was a great error; for though a synthetic proposition can be viewed in the light of the above principle, it can only be so by presupposing another synthetic proposition from which it is derived, but never by itself.

It must be first of all remarked that essentially mathematical propositions are always à priori, and never empirical, because they involve necessity, which cannot be inferred from experience. Should any one be unwilling to admit this, I will limit my assertion to pure mathematics, the very conception of which itself brings with it the fact that it contains nothing empirical, but simply pure knowledge à priori.

At first sight, one might be disposed to think the proposition 7 + 5 = 12 merely analytic, resulting from the conception of a sum of seven and five, according to the principle of contradiction. But more closely considered it