Section 7
Original German
§ 7.
Wir finden aber, daß alle mathematische Erkenntniß dieses Eigenthümliche habe, daß sie ihren Begriff vorher in der Anschauung und zwar a priori, mithin einer solchen, die nicht empirisch, sondern reine Anschauung ist, darstellen müsse, ohne welches Mittel sie nicht einen einzigen Schritt thun kann; daher ihre Urtheile jederzeit intuitiv sind, an statt daß Philosophie sich mit discursiven Urtheilen, aus bloßen Begriffen, begnügen muß und ihre apodiktische Lehren wohl durch Anschauung erläutern, niemals aber daher ableiten kann. Diese Beobachtung in Ansehung der Natur der Mathematik giebt uns nun schon eine Leitung auf die erste und oberste Bedingung ihrer Möglichkeit: nämlich es muß ihr irgend eine reine Anschauung zum Grunde liegen, in welcher sie alle ihre Begriffe in concreto und dennoch a priori darstellen oder, wie man es nennt, sie construiren kann.281.1 Können wir diese reine Anschauung und die Möglichkeit einer solchen ausfinden, so erklärt sich daraus leicht, wie synthetische Sätze a priori in der reinen Mathematik, und mithin auch, wie diese Wissenschaft selbst möglich sei; denn so wie die empirische Anschauung es ohne Schwierigkeit möglich macht, daß wir unseren Begriff, den wir uns von einem Object der Anschauung machen, durch neue Prädicate, die die Anschauung selbst darbietet, in der Erfahrung synthetisch erweitern, so wird es auch die reine Anschauung thun; nur mit dem Unterschiede: daß im letztern Falle das synthetische Urtheil a priori gewiß und apodiktisch, im ersteren aber nur a posteriori und empirisch gewiß sein wird, weil diese nur das enthält, was in der zufälligen empirischen Anschauung angetroffen wird, jene aber, was in der reinen nothwendig angetroffen werden muß, indem sie als Anschauung a priori mit dem Begriffe vor aller Erfahrung oder einzelnen Wahrnehmung unzertrennlich verbunden ist.
281.1. Siehe Kritik S. 713.
Musean translation
Mouseia’s complete machine-assisted Musean translation, made directly from the complete German of the Prolegomena (AA IV 253–383, all §§1–60 and the Appendix), for fidelity, the steps of Kant’s argument, clarity and structure.
§ 7.
We find, however, that all mathematical cognition has this distinctive feature: it must first exhibit its concept in intuition, and do so a priori, hence in an intuition that is not empirical but pure; without this means it cannot take a single step. Its judgments are therefore always intuitive, whereas philosophy must content itself with discursive judgments, from mere concepts, and may indeed elucidate its apodictic teachings through intuition, but can never derive them from it. This observation about the nature of mathematics already guides us to the first and highest condition of its possibility: some pure intuition must underlie it, in which it can exhibit all its concepts in concreto and yet a priori—or, as it is called, construct them.281.1 If we can discover this pure intuition and the possibility of such an intuition, it will be easy to explain how synthetic propositions a priori are possible in pure mathematics, and thus how this science itself is possible. For just as empirical intuition makes it possible without difficulty for us synthetically to enlarge, in experience, the concept we form of an object of intuition by adding new predicates supplied by intuition itself, so pure intuition will do the same. There is only this difference: in the latter case the synthetic judgment will be certain a priori and apodictic, whereas in the former it will be certain only a posteriori and empirically. For the empirically certain judgment contains only what is encountered in contingent empirical intuition, whereas the a priori judgment contains what must necessarily be encountered in pure intuition, since this intuition, as intuition a priori, is inseparably connected with the concept before all experience or particular perception.
281.1. See Critique, page 713.
Plain English translation
Mouseia’s complete Plain English edition, made independently and directly from the same complete German text.
§ 7.
We find that all mathematical cognition has this distinctive feature: it must first present its concept in intuition, and must do so a priori, hence in an intuition that is not empirical but pure. Without this means it cannot take a single step. Its judgments are therefore always intuitive. Philosophy, by contrast, must be content with discursive judgments, from mere concepts. It can certainly clarify its apodictic teachings through intuition, but can never derive them from intuition. This observation about the nature of mathematics already points us toward the first and highest condition of its possibility. Some pure intuition must underlie it, in which it can present all its concepts concretely yet a priori—or, as it is called, construct them.281.1 If we can discover this pure intuition and how it is possible, we can easily explain from it how synthetic a priori propositions are possible in pure mathematics, and therefore how this science itself is possible. Empirical intuition makes it possible without difficulty for us to extend synthetically, in experience, the concept we form of an object of intuition, by adding new predicates that the intuition itself supplies. Pure intuition will do the same. There is only this difference: in the latter case the synthetic judgment will be certain a priori and apodictically; in the former, only a posteriori and empirically. The empirically certain judgment contains only what is found in contingent empirical intuition. The a priori judgment contains what must necessarily be found in pure intuition. For this pure intuition, as an a priori intuition, is inseparably connected with the concept before all experience or individual perception.
281.1. See Kritik S. 713.
About this section
Section 7 is part of Prolegomena to Any Future Metaphysics by Immanuel Kant. This page sets the German original beside 2 direct-source translations, so any line can be checked against the source. It is section 12 of 72 in the work, which is kept in the European tradition.
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