By these principles of the pure understanding the possibility of mathematics and of a pure science of nature may be fully and satis factorily explained. The matter of mathematics is the multiple object of space and time, which are given as the forms ofd priori intuition. This multiple matter is elaborated by the understanding according to the rules of logic, and as the phenomena must be in accordance with the conditions of space and time, or the forms under which they are intuitively viewed, that is, the relations of space and time must be discoverable in phenomena themselves. The possibility of mathematics therefore rests simply on this, that objects cannot be conceived of except in space and time, from which however it follows at the same time that mathematics do not admit of application beyond the sphere of sensible phenomena. The pure science of nature like wise cannot have any other object than the system of is priori laws. It is only under the forms of sensation that individual objects can be intuitively viewed, and their mutual connection cannot be thought of otherwise than under the forms of the understanding. If then the system of phenomena are to be an object of knowledge, they must correspond to the pure synthetical principles of the understanding, and it is only by these ct priori laws that a science of nature is possi ble. But the principles of this pure science of nature do not admit of being applied beyond the domain of experience.
The important result of the transcendental logic is that the ope rations of the understanding are only legitimate in reference to experience, and that consequently the use of the understanding is empirical, and not transcendental. It would be the latter if it could apply itself to objects not as phenomena merely, but as things abso lutely. But such a use of the understanding is obviously invalid, since the objective matter of a notion, or begriff, is given by intuition alone, and it is only by means of the empirical that the pure intuition itself comes to the object of which it is the form. These forms are simply representations of the object according as it conceived under them. To the subsunition of an object under a category, a schema, 'time,' is indispensable, aud, apart from all sensation, this schema itself does not subsist ; and the subeumtion, or arrangement of an object under the categories, is impossible. There may undoubtedly be a logical use of the categories beyond the domain of experience, but this, notwithstanding that it has its ground in the nature of human reason, is either altogether idle, or else involved in contra dictions (antinomie) which the transcendental dialectic investigates.
But besides phenomena there are other objects presented to the understanding, by a non-sensuous intuition of which consequently it can take cognisance. These Kant calls noumena (vooaeva). The dis tinction between noumena aud phenomena does not consist merely in a logical difference of the greater or less distinctness of their cognosei bility, but in a specific difference of the objects themselves. A nou
menon is not the thing in and by itself, for the thing in and by itself becomes evanescent for knowledge when conceived of independently of all sensuous forms. Nevertheless, as experience invariably refers back to something independent of and prior to sensation, the noume non may be considered as au object which is presented to the understanding by an uusensuous intuition. The general possibility of such a species of intuitiou is undeniable, notwithstanding that its objects are impossible to be known by man, whose knowledge is dependent on sensation. In a positive sense Kant applies the term of noumenon to the notion of God, and generally to all supra-sensible objects, which may be conceived of, but nevertheless cannot be an object of perception.
The criticism of the transcendental dialectic gives this reeult—that the ideas of the reason, as pure speculative Ideas, are nothing more than simple conmptions, for which no corresponding object can be scientifically shown to exist. Accordingly neither the existence of God, nor the Immortality of the soul, nor the freedom of the will, can be demonstratively established. Nevertheless, the reason is not merely a theoretical, but also a practical faculty, that is, it gives the law of human conduct and action. Now these laws present them selves with such an unconditional necessity (the categorical imperative) that no rational man endued with self-esteem can refuse obedience to them ; and, on the other Land, without the freedom of the will these laws could not be obeyed ; and without God and the soul's immor tality there would bo no final cause or motive for human conduct, which must be placed in a state of felicity, agreeable to morality, pro vided by and to be obtained through God, in another and a better life. Consequently every man who is conscious of his moral destination holds these practical ideas to be both true and objectively legitimate, notwithstanding that he is compelled or required to admit them merely by a subjective ground—the testimony of his own conscious nets, and of the moral wants resulting from its dictates. This Knot calls the postulate of the practical reason. The acceptance of this postulate as true and legitimate does not constitute a scientific certainty, or knowledge properly, which indeed does not exist for the supra-sensible ; it is merely a belief This faith, or belief, however, is thus distinguished from every other, that it is a moral or practical faith, and consequently possesses for the believer all the certainty requisite for the guidance and conduct of life, and consequently it enjoys a subjective certainty and authority. This faith is the proper foundation of religion, which is nothing else than a conscientious observance of all duties as divine commands, since God, as the moral law-giver, cannot bo worthily honoured otherwise than by obedience to the laws of morality.