Front Page Titles (by Subject) Supplement V - Critique of Pure Reason
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Supplement V - Friedrich Max Müller, Critique of Pure Reason 
Immanuel Kant’s Critique of Pure Reason. In Commemoration of the Centenary of its First Publication. Translated into English by F. Max Mueller (2nd revised ed.) (New York: Macmillan, 1922).
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In all Theoretical Sciences of Reason Synthetical Judgments a priori are contained as Principles
1. 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. But, if we look more closely, we shall find that the concept of the sum of 7 and 5 contains nothing beyond the union of both sums into one, whereby nothing is told us as to what this single number may be which combines both. We by no means arrive at a concept of Twelve, by thinking that union of Seven and Five; and we may analyse our concept of such a possible sum as long as we will, still we shall never discover in it the concept of Twelve. We must go beyond these concepts, and call in the assistance of the intuition corresponding to one of the two, for instance, our five fingers, or, as Segner does in his arithmetic, five points, and so by degrees add the units of the Five, given in intuition, to the concept of the Seven. For I first take the number 7, and taking the intuition of the fingers of my hand, in order to form with it the concept of the 5, I gradually add the units, which I before took together, to make up the number 5, by means of the image of my hand, to the number 7, and I thus see the number 12 arising before me. That 5 should be added to 7 was no doubt implied in my concept of a sum 7 + 5, but not that that sum should be equal to 12. An arithmetical proposition is, therefore, always synthetical, which is seen more easily still by taking larger numbers, where we clearly perceive that, turn and twist our conceptions as we may, we could never, by means of the mere analysis of our concepts and without the help of intuition, arrive at the sum that is wanted.
Nor is any proposition of pure geometry analytical. That the straight line between two points is the shortest, is a synthetical proposition. For my concept of straight contains nothing of magnitude (quantity), but a quality only. The concept of the shortest is, therefore, purely adventitious, and cannot be deduced from the concept of the straight line by any analysis whatsoever. The aid of intuition, therefore, must be called in, by which alone the synthesis is possible.
[It is true that some few propositions, presupposed by the geometrician, are really analytical, and depend on the principle of contradiction: but then they serve only, like identical propositions, to form the chain of the method, and not as principles. Such are the propositions, a = a, the whole is equal to itself, or (a + b) > a, that the whole is greater than its part. And even these, though they are valid according to mere concepts, are only admitted in mathematics, because they can be represented in intuition.1 ] What often makes us believe that the predicate of such apodictic judgments is contained in our concept, and the judgment therefore analytical, is merely the ambiguous character of the expression. We are told that we ought to join in thought a certain predicate to a given concept, and this necessity is inherent in the concepts themselves. But the question is not what we ought to join to the given concept, but what we really think in it, though confusedly only, and then it becomes clear that the predicate is no doubt inherent in those concepts by necessity, not, however, as thought in the concept itself, but by means of an intuition, which must be added to the concept.
2. Natural science (physica) contains synthetical judgments a priori as principles. I shall adduce, as examples, a few propositions only, such as, that in all changes of the material world the quantity of matter always remains unchanged: or that in all communication of motion, action and reaction must always equal each other. It is clear not only that both convey necessity, and that, therefore, their origin is a priori, but also that they are synthetical propositions. For in the concept of matter I do not conceive its permanency, but only its presence in the space which it fills. I therefore go beyond the concept of matter in order to join something to it a priori, which I did not before conceive in it. The proposition is, therefore, not analytical, but synthetical, and yet a priori, and the same applies to the other propositions of the pure part of natural science.
3. Metaphysic, even if we look upon it as hitherto a tentative science only, which, however, is indispensable to us, owing to the very nature of human reason, is meant to contain synthetical knowledge a priori. Its object is not at all merely to analyse such concepts as we make to ourselves of things a priori, and thus to explain them analytically, but to expand our knowledge a priori. This we can only do by means of concepts which add something to a given concept that was not contained in it; nay, we even attempt, by means of synthetical judgments a priori, to go so far beyond a given concept that experience itself cannot follow us: as, for instance, in the proposition that the world must have a first beginning. Thus, according at least to its intentions, metaphysic consists merely of synthetical propositions a priori.
[1 ]This paragraph from It is true to intuition seems to have been a marginal note, as shown by Dr. Vaihinger. See Translator’s Preface, p. lii.