Section 12

§ 12.

To add something by way of elucidation and confirmation, we need only consider the usual and indispensably necessary procedure of geometers. All proofs of the complete equality of two given figures—where one can be substituted for the other in every respect—ultimately come down to their coinciding with one another. This is plainly nothing but a synthetic proposition resting on immediate intuition; and this intuition must be pure and given a priori, for otherwise that proposition could not count as apodictically certain, but would have only empirical certainty. It would mean only that this is always observed to be so, and would hold only as far as our perception had hitherto extended. That complete space—which is itself no longer the boundary of another space—has three dimensions, and that space in general can have no more, is based on the proposition that no more than three lines can intersect at right angles at a single point. But this proposition cannot be demonstrated from concepts at all: it rests immediately on intuition, indeed pure intuition a priori, because it is apodictically certain. That we can demand that a line be drawn on indefinitely (in indefinitum), or that a series of alterations—for example, spaces traversed by motion—be continued indefinitely, presupposes a representation of space and time that can attach only to intuition, namely insofar as intuition is in itself bounded by nothing; for it could never be inferred from concepts. Pure intuitions a priori therefore really do underlie mathematics and make possible its synthetic propositions with their apodictic validity. Hence our transcendental deduction of the concepts in space and time also explains the possibility of a pure mathematics. Without such a deduction, and without our assuming that “everything that may be given to our senses—to the outer senses in space, to the inner sense in time—is intuited by us only as it appears to us, not as it is in itself,” that possibility could indeed be conceded, but by no means understood.

§ 12.

To add some clarification and confirmation, we need only look at the usual and absolutely necessary procedure of geometers. All proofs of the complete equality of two given figures, where one can replace the other in every respect, ultimately come down to their coinciding with one another. This is plainly nothing but a synthetic proposition resting on immediate intuition. That intuition must be pure and given a priori. Otherwise the proposition could not count as apodictically certain, but would have only empirical certainty. It would say only that this is how we always observe it, and would hold only as far as our perception had so far extended. The claim that complete space—which is no longer itself a boundary of another space—has three dimensions, and that space in general cannot have more, is based on the proposition that no more than three lines can intersect at right angles at one point. But this proposition cannot be demonstrated from concepts at all. It rests directly on intuition, specifically pure a priori intuition, because it is apodictically certain. That we can require a line to be drawn on indefinitely (in indefinitum), or a series of changes—for example, spaces traversed through motion—to be continued indefinitely, presupposes a representation of space and time that can belong only to intuition, insofar as intuition is in itself limited by nothing. For it could never be inferred from concepts. So pure a priori intuitions really do underlie mathematics and make its synthetic, apodictically valid propositions possible. Our transcendental deduction of concepts in space and time therefore also explains the possibility of pure mathematics. Without such a deduction, and without assuming that “everything that may be given to our senses—to the outer senses in space, to the inner sense in time—is intuited by us only as it appears to us, not as it is in itself,” that possibility could be admitted, but could in no way be understood.