But we find that all mathematical knowledge has this speciality, that it must present its conception previously in intuition, and indeed a priori, that is, in an intuition that is not empirical but pure, without which means it cannot make a single step; its judgments therefore are always intuitive, whereas philosophy must be satisfied with discursive judgments out of mere conceptions; for though it can explain its apodictic doctrines by intuition, these can never be derived from such a source. This observation respecting the nature of mathematics, itself furnishes us with a guide as to the first and foremost condition of its possibility, namely, that some pure intuition must be at its foundation, wherein it can present all its conceptions in concreto and a priori at the same time, or as it is termed, construct them. If we can find out this pure intuition together with its possibility, it will be readily explicable how synthetic propositions a priori are possible in pure mathematics, and therefore, also, how this science is itself possible. For just as empirical intuition enables us, without difficulty, to extend synthetically in experience the conception we form of an object of intuition, by new predicates, themselves afforded us by intuition, so will the pure intuition, only with this difference: that in the last case the synthetic judgment a priori is certain and apodictic, while in the first case it is no more than a posteriori and empirically certain, because the latter only contains what is met with in chance empirical intuition, but the former what is necessarily met with in the pure intuition, inasmuch as being intuition a priori, it is indissolubly bound up with the conception before all experience or perception of individual things.
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