for anything elsewhere.
Synthetic judgements demand a principle other than that of contradiction.
There are synthetic judgements a posteriori whose origin is empirical; but there are also others of an a priori certainty that spring from the Understanding and the Reason. But both are alike in this, that they can never have their source solely in the axiom of analysis, viz., the principle of contradiction; they require an altogether different principle, notwithstanding that whatever principle they may be deduced from, they must always conform to the principle of contradiction, for nothing can be opposed to this principle, although not everything can be deduced from it. I will first of all bring synthetic judgements under certain classes.
(1) Judgements of experience are always synthetic. It would be absurd to found an analytic judgement on experience, as it is unnecessary to go beyond my own conception in order to construct the judgement, and therefore the confirmation of experience is unnecessary to it. That a body is extended is a proposition possessing a priori certainty, and no judgement of experience. For before I go to experience I have all the conditions of my judgement already present in the conception, out of which I simply draw the predicate in accordance with the principle of contradiction, and thereby at the same time the necessity of the judgement may be known, a point which experience could never teach me.
(2) Mathematical judgements are in their entirety synthetic. This truth seems hitherto to have altogether escaped the analysts of human Reason; indeed, to be directly opposed to all their suppositions, although it is indisputably certain and very important in its consequences. For, because it was found that the conclusions of mathematicians all proceed according to the principle of contradiction (which the nature of every demonstrative certainty demands), it was concluded that the axioms were also known through the principle of contradiction, which was a great error; for though a synthetic proposition can be viewed in the light of the above principle, it can only be so by presupposing another synthetic proposition from which it is derived, but never by itself.
It must be first of all remarked that essentially mathematical propositions are always a priori, and never empirical, because they involve necessity, which cannot be inferred from experience. Should anyone be unwilling to admit this, I will limit my assertion to pure mathematics, the very conception of which itself brings with it the fact that it contains nothing empirical, but simply pure knowledge a priori.
At first sight, one might be disposed to think the proposition 7 + 5 = 12 merely analytic, resulting from the conception of a sum of seven and five, according to the principle of contradiction. But more closely considered it 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 thought as to what the single number is that combines both the others. The conception of twelve is by no means already thought, 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 … 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 no way thought 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 …
How is knowledge possible from pure reason?
We have already seen the important distinction between analytic and synthetic judgements. The possibility of analytic propositions can be very easily conceived, for they are based simply on the principle of contradiction. The possibility of synthetic propositions a posteriori, i.e., of such as are derived from experience, requires no particular explanation, for experience is nothing more than a continual adding together (synthesis) of perceptions. There remains, then, only synthetic propositions a priori, the possibility of which has yet to be sought for, or examined, because it must rest on other principles than that of contradiction.
But we do not require to search out the possibility of such propositions, that is, to ask whether they are possible, for there are enough of them, actually given, and with unquestionable certainty; and as the method we are here following is analytic, we shall assume at the outset that such synthetic but pure knowledge from Reason, is real; but thereupon we must investigate the ground of this possibility and proceed to ask – how is this knowledge possible? in order that, from the principles of its possibility, we may be in a position to determine the conditions, the scope, and limits of its use. The proper problem, on which everything turns, when expressed with scholastic precision, will accordingly stand thus: how are synthetic propositions a priori possible?
… Upon the solution of this problem, the standing or falling of metaphysics, in other words, its very existence, entirely depends. Let any one lay down assertions, however plausible, with regard to it, pile up conclusions upon conclusions to the point of overwhelming, if he has not been able first to answer satisfactorily the above question, I have a right to say: It is all vain, baseless philosophy, and false wisdom. You speak through pure Reason, and claim to create a priori cognitions, inasmuch as you pretend not merely to dissect given conceptions but new connections which do not rest on the principle of contradiction, and which you think you conceive quite independently of all experience. How do you arrive at them, and how will you justify yourself in such pretensions? …
How is pure mathematics possible?
Pure mathematics is only possible as synthetic knowledge a priori in so far as it refers simply to objects of sense, whose empirical intuition has for its foundation a pure intuition a priori (that of time and space). Such intuition is able to serve as a foundation because it is nothing more than the pure form of sensibility itself, that precedes the real appearance of objects, in that it makes them in the first place possible. This faculty of intuiting a priori does not concern the matter of the phenomenon … for that constitutes the empirical element therein, but only its form, space and time …
To contribute something to the explanation and confirmation of the above, we have only to consider the ordinary and necessary procedure of geometricians. All the proofs of complete likeness between two figures turn finally on the fact of their covering each other – in other words the possibility of substituting one, in every point, for the other; and this is obviously nothing else but a synthetic proposition, resting on immediate intuition. Now this intuition must be given pure and a priori, for otherwise the proposition in question could not count as demonstratively certain, but would possess only empirical certainty (in the latter case, we would only be able to say that it has always been so observed, or it is valid only in so far as our perception has hitherto extended) …
How is pure natural science possible?
Although all the judgements of experience are empirical (i.e. have their ground in the immediate perception of sense), on the other hand all empirical judgements are not judgements of experience. Beyond the empirical, and beyond the given sense-intuition generally, special conceptions must be added, which have their origin entirely a priori in the pure understanding. Every perception is primarily subsumed under these conceptions, and it is only by means of them that it can be transformed into experience …
Let us now attempt a solution of Hume’s problematical conception … namely the conception of Cause. First, there is given me a priori, by means of logic, the form of a conditioned judgement generally (one cognition as antecedent, the other as consequent). But it is possible that in
