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Section Twelve: The First Part, Continued

25 September, 2015 - 14:22

To contribute something to the explanation and confirmation of the above, we have only to consider the ordinary and necessary procedure of geometricians. All the proofs of complete likeness between two given figures, turn at last upon the fact of their covering each other; in other words, of the possibility of substituting one, in every point, for the other, which is obviously nothing else but a synthetic proposition resting on immediate intuition. Now this intuition must be given pure and apriori, for otherwise the proposition in question could not count as apodictically certain, but would possess only empirical certainty. We could only say in that case, it has been always so observed, or it is valid so far as our perception has hitherto extended. That complete space, itself no boundary of a further space, has three dimensions, and that no space can have more than this number, is founded on the proposition that not more than three lines can bisect each other at right angles in a single point. But this proposition cannot be presented from conceptions, but rests immediately on intuition, and indeed on pure aprioriintuition, because it is apodictically certain that we can require a line to be drawn out to infinity (inindefinitum), or that a series of changes (e.g., spaces passed through by motion) shall be continued to infinity, and this presupposes a presentation of space and time, merely dependent on intuition, namely, so far as in itself, it is bounded by nothing, for from conceptions it could never be concluded. Pure intuitions apriori, then, really lie at the foundation of mathematics, and these make its synthetic and apodictically valid propositions pos- sible, and hence our transcendental deduction of conceptions in space and time explains at the same time the possibility of pure mathematics, which without such a deduction, and without our assuming that “all which can be given to our senses (the outer in space, the inner in time) is only intuited by us, as it appears to us, and not as it is in itself,” might indeed be conceded, but could in nowise be understood.