| I can cognize beforehand by analysis the conception of body through the characteristics of extension, impenetrability, shape, etc., all which are cogitated in this conception. |
| But now I extend my knowledge, and looking back on experience from which I had derived this conception of body, I find weight at all times connected with the above characteristics, and therefore I synthetically add to my conceptions this as a predicate, and say, "All bodies are heavy." Thus it is experience upon which rests the possibility of the synthesis of the predicate of weight with the conception of body, because both conceptions, although the one is not contained in the other, still belong to one another (only contingently, however), as parts of a whole, namely, of experience, which is itself a synthesis of intuitions. |
| But to synthetical judgements a priori, such aid is entirely wanting. |
| If I go out of and beyond the conception A, in order to recognize another B as connected with it, what foundation have I to rest on, whereby to render the synthesis possible? |
| I have here no longer the advantage of looking out in the sphere of experience for what I want. |
| Let us take, for example, the proposition, "Everything that happens has a cause." In the conception of "something that happens," I indeed think an existence which a certain time antecedes, and from this I can derive analytical judgements. |
| But the conception of a cause lies quite out of the above conception, and indicates something entirely different from "that which happens," and is consequently not contained in that conception. |
| How then am I able to assert concerning the general conception--"that which happens"--something entirely different from that conception, and to recognize the conception of cause although not contained in it, yet as belonging to it, and even necessarily? |
| what is here the unknown = X, upon which the understanding rests when it believes it has found, out of the conception A a foreign predicate B, which it nevertheless considers to be connected with it? |
| It cannot be experience, because the principle adduced annexes the two representations, cause and effect, to the representation existence, not only with universality, which experience cannot give, but also with the expression of necessity, therefore completely a priori and from pure conceptions. |
| Upon such synthetical, that is augmentative propositions, depends the whole aim of our speculative knowledge a priori; for although analytical judgements are indeed highly important and necessary, they are so, only to arrive at that clearness of conceptions which is requisite for a sure and extended synthesis, and this alone is a real acquisition. |
| V. In all Theoretical Sciences of Reason, Synthetical Judgements "a priori" are contained as Principles. |
| 1. Mathematical judgements are always synthetical. |
| Hitherto this fact, though incontestably true and very important in its consequences, seems to have escaped the analysts of the human mind, nay, to be in complete opposition to all their conjectures. |
| For as it was found that mathematical conclusions all proceed according to the principle of contradiction (which the nature of every apodeictic certainty requires), people became persuaded that the fundamental principles of the science also were recognized and admitted in the same way. |
| But the notion is fallacious; for although a synthetical proposition can certainly be discerned by means of the principle of contradiction, this is possible only when another synthetical proposition precedes, from which the latter is deduced, but never of itself which |
| Before all, be it observed, that proper mathematical propositions are always judgements a priori, and not empirical, because they carry along with them the conception of necessity, which cannot be given by experience. |
| If this be demurred to, it matters not; I will then limit my assertion to pure mathematics, the very conception of which implies that it consists of knowledge altogether non-empirical and a priorI. We might, indeed at first suppose that the proposition 7 + 5 = 12 is a merely analytical proposition, following (according to the principle of contradiction) from the conception of a sum of seven and five. |
| But if we regard it more narrowly, we find that our conception of the sum of seven and five contains nothing more than the uniting of both sums into one, whereby it cannot at all be cogitated what this single number is which embraces both. |
| The conception of twelve is by no means obtained by merely cogitating the union of seven and five; and we may analyse our conception of such a possible sum as long as we will, still we shall never discover in it the notion of twelve. |
| We must go beyond these conceptions, and have recourse to an intuition which corresponds to one of the two--our five fingers, for example, or like Segner in his Arithmetic five points, and so by degrees, add the units contained in the five given in the intuition, to the conception of seven. |