Page:Kant's Prolegomena etc (1883).djvu/137

will be found that the conception of the sum of 7 and 5 comprises nothing beyond the union of two numbers in a single one, and that therein nothing whatever is cogitated as to what this single number is, that comprehends both the others. The conception of twelve is by no means already cogitated, when I think merely of the union of seven and five, and I may dissect my conception of such a possible sum as long as I please, without discovering therein the number twelve. One must leave these conceptions, and call to one's aid an intuition corresponding to one or other of them, as for instance one's five fingers (or, like Segner in his Arithmetic, five points), and so gradually add the units of the five given in intuition to the conception of the seven. One's conception is therefore really enlarged by the proposition 7 + 5 = 12; to the first a new one being added, that was in nowise cogitated in the former; in other words, arithmetical propositions are always synthetic, a truth which is more apparent when we take rather larger numbers, for we must then be clearly convinced, that turn and twist our conceptions as we may, without calling intuition to our aid, we shall never find the sum required, by the mere dissection of them.

Just as little is any axiom of pure geometry analytic. That a straight line is the shortest between two points, is a synthetic proposition. For my conception of straight, has no reference to size, but only to quality. The conception of the "shortest" therefore is quite additional, and cannot be drawn from any analysis of the conception of a straight line. Intuition must therefore again be taken to our aid, by means of which alone the synthesis is possible.

Certain other axioms, postulated by geometricians, are indeed really analytic and rest on the principle of contradiction, but they only serve, like identical propositions, as links in the chain of method, and not themselves as principles; as for instance a = a, the whole is equal to itself, or (a + b) > a, i.e., the whole is greater than its part. But even these, although they are contained in mere conceptions, are only admitted in mathematics because they can be presented in intuition. What produces the common belief that the predicate of such apodictic judgments lies already in our conception, and that the judg-