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 the perception, a rule of the relation may be met with which will say that on the occurrence of a given phenomenon another always follows (though not conversely). This would be a case of making use of the hypothetical judgement – to say, for instance, if a body is illuminated long enough by the sun it will become warm. There is certainly no necessity of connection here, in other words, no conception of cause. But to continue: if the above proposition, which is merely a subjective connection of perception, is to be a proposition of experience, it must be regarded as necessary and universally valid. Such a proposition would run: sun is, through its light, the cause of heat. The above empirical rule is now looked upon as law, and indeed not just as valid of phenomena, but valid of them in relation to a possible experience – which requires thoroughly, and therefore necessarily, valid rules. I perfectly understand, then, the conception of cause as a conception necessarily belonging to the mere form of experience, and its possibility as a synthetic union of perceptions, in a consciousness in general. But the possibility of a thing in general as a cause I do not understand, because the conception of cause does not refer at all to things, but only indicates the condition attaching to experience: that this can only be an objectively valid knowledge of phenomena, and their sequence in time, in so far as their antecedent can be united to the consequent according to the rule of hypothetical judgements.
Hence, the pure conceptions of the understanding have no meaning whatever, when they quit the objects of experience and refer to things in themselves (noumena). They serve, as it were, to spell out phenomena, that these may be able to be read as experience. The axioms arising from their relation to the world of sense, only serve our understanding for use in experience. Beyond this, are only arbitrary combinations, destitute of objective reality, and the possibility of which can neither be known a priori, nor their reference to objects be confirmed, or even made intelligible by an example, because all examples are borrowed from some possible experience, and consequently the objects of those conceptions are nothing but what may be met with in a possible experience.
This complete solution of Hume’s problem, although it turns out to be contrary to the opinion of its originator, preserves for the pure conceptions of the understanding their origin a priori, and for the universal laws of Nature their validity as laws of the understanding, but in such a manner that their use is limited to experience, because their possibility has its basis, solely, in the reference of the understanding to experience; not because they are derived from experience, but because experience is derived from them, which completely reversed mode of connection