Immanuel Kant's Critique of Pure Reason — trans. F. Max Müller

Not only in judgments, however, but even in certain concepts, can we show their origin a priori. Take away, for example, from the concept of a body, as supplied by experience, everything that is empirical, one by one ; such as colour, hardness or softness, weight, and even impenetrability, and there still remains the space which the body (now entirely vanished) occupied : that you cannot take away. And in the same manner, if you remove from your empirical concept of any object, corporeal or incorporeal, all properties which experience has taught you, you cannot take away from it that property by which you conceive it as a substance, or inherent in a substance (although such a concept contains more determinations than that of an object in general). Convinced, therefore, by the necessity with which that concept forces itself upon you, you will have to admit that it has its seat in your faculty of knowledge a priori.

SUPPLEMENT V

[See vol. ii. p. 7]

Empirical judgments, as such, are all synthetical ; for it would be absurd to found an analytical judgment on experience, because, in order to form such a judgment, I need not at all step out of my concept, or appeal to the testimony of experience. That a body is extended, is a proposition perfectly certain a priori, and not an empirical judgment. For, before I call in experience, I am already in possession of all the conditions of my judgment in the concept of body itself. I have only to draw out from it, according to the principle of contradiction, the required predicate, and I thus become conscious, at the same time, of the necessity of the judgment, which experience could never teach me. But, though I do not include the predicate of gravity in the general concept of body, that concept, nevertheless, indicates an object of experience through one of its parts : so that I may add other parts also of the same experience, besides those which belonged to the former concept. I may, first, by an analytical process, realise the concept of body, through the predicates of extension, impermeability, form, etc., all of which are contained in it. Afterwards I expand my knowledge, and looking back to the experience from which my concept of body was abstracted, I find gravity always connected with the before-mentioned predi" cates, and therefore I add it synthetically to that concept as a predicate. It is, therefore, experience on which the possibility of the synthesis of the predicate of gravity with the concept of body is founded : because both concepts, though neither of them is contained in the other, belong to each other, though accidentally only, as parts of a whole, namely, of experience, which is itself a synthetical connection of intuitions.

p. 719

SUPPLEMENT VI

[See vol. ii. p. 9]

In all Theoretical Sciences of Reason Synthetical Judgments a priori are contained as Principles

I . All mathematical judgments are synthetical. This proposition, though incontestably certain, and very important to us for the future, seems to have hitherto escaped the observation of those who are engaged in the anatomy of human reason : nay, to be directly opposed to all their conjectures. For as it was found that all mathematical conclusions proceed according to the principle of contradiction (which is required by the nature of all apodictic certainty), it was supposed that the fundamental principles of mathematics also rested on the authority of the same principle of contradiction. This, however, was a mistake : for though a synthetical proposition may be understood according to the principle of contradiction, this can only be if another synthetical proposition is presupposed, from which the latter is deduced, but never by itself. First of all, we ought to observe, that mathematical propositions, properly so called, are always judgments a priori, and not empirical, because they carry along with them necessity, which can never be deduced from experience. If people should object to this, I am quite willing to confine my statement to pure mathematics, the very concept of which implies that it does not contain empirical, but only pure knowledge a priori.

At first sight one might suppose indeed that the proposition 7 + 5 = 12 is merely analytical, following, according to the principle of contradiction, from the concept of a sum of 7 and 5.

p. 720

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