(3) Consequently it must base itself upon no experience at all, but upon something else, it must be derived from another source of knowledge!
Kant concludes (1) that there are some propositions which hold good only on one condition; (2) this condition is that they do not spring from experience, but from pure reason.
Thus, the question is, whence do we derive our reasons for believing in the truth of such propositions? No, whence does our belief get its cause? But the origin of a belief, of a strong conviction, is a psychological problem: and very limited and narrow experience frequently brings about such a belief! It already presupposes that there are not only "data a posteriori" but also "data a priori"— that is to say, "previous to experience." Necessary and universal truth cannot be given by experience: it is therefore quite clear that it has come to us without experience at all?
There is no such thing as an isolated judgment!
An isolated judgment is never "true," it is never knowledge; only in connection with, and when related to, many other judgments, is a guarantee of its truth forthcoming.
What is the difference between true and false belief? What is knowledge? He "knows" it, that is heavenly! Necessary and universal truth cannot be given by experience! It is therefore independent of experience, of all experience! The view which comes quite a priori, and therefore independent of all experience, merely out of reason, is "pure knowledge"!
"The principles of logic, the principle of identity and of contradiction, are examples of pure knowledge, because they precede all experience."—But these principles are not cognitions, but regulative articles of faith.
In order to establish the a priori character (the pure rationality) of mathematical axioms, space must be conceived as a form of pure reason.
Hume had declared that there were no a priori synthetic judgments. Kant says there are—the mathematical ones! And if there are such judgments, there may also be such things as metaphysics and a knowledge of things by means of pure reason!
Mathematics is possible under conditions which are not allowed to metaphysics. All human knowledge is either experience or mathematics.