| Geometry is a science which determines the properties of space synthetically, and yet a priorI. What, then, must be our representation of space, in order that such a cognition of it may be possible? |
| It must be originally intuition, for from a mere conception, no propositions can be deduced which go out beyond the conception, and yet this happens in geometry. |
| (Introd. V.) But this intuition must be found in the mind a priori, that is, before any perception of objects, consequently must be pure, not empirical, intuition. |
| For geometrical principles are always apodeictic, that is, united with the consciousness of their necessity, as; "Space has only three dimensions." But propositions of this kind cannot be empirical judgements, nor conclusions from them. |
| (Introd. II.) Now, how can an external intuition anterior to objects themselves, and in which our conception of objects can be determined a priori, exist in the human mind? |
| Obviously not otherwise than in so far as it has its seat in the subject only, as the formal capacity of the subject's being affected by objects, and thereby of obtaining immediate representation, that is, intuition; consequently, only as the form of the external sense in general. |
| Thus it is only by means of our explanation that the possibility of geometry, as a synthetical science a priori, becomes comprehensible. |
| Every mode of explanation which does not show us this possibility, although in appearance it may be similar to ours, can with the utmost certainty be distinguished from it by these marks. |
| SS 4. Conclusions from the foregoing Conceptions. |
| (a) Space does not represent any property of objects as things in themselves, nor does it represent them in their relations to each other; in other words, space does not represent to us any determination of objects such as attaches to the objects themselves, and would remain, even though all subjective conditions of the intuition were abstracted. |
| For neither absolute nor relative determinations of objects can be intuited prior to the existence of the things to which they belong, and therefore not a priorI. (b) Space is nothing else than the form of all phenomena of the external sense, that is, the subjective condition of the sensibility, under which alone external intuition is possible. |
| Now, because the receptivity or capacity of the subject to be affected by objects necessarily antecedes all intuitions of these objects, it is easily understood how the form of all phenomena can be given in the mind previous to all actual perceptions, therefore a priori, and how it, as a pure intuition, in which all objects must be determined, can contain principles of the relations of these objects prior to all experience. |
| It is therefore from the human point of view only that we can speak of space, extended objects, etc. |
| If we depart from the subjective condition, under which alone we can obtain external intuition, or, in other words, by means of which we are affected by objects, the representation of space has no meaning whatsoever. |
| This predicate is only applicable to things in so far as they appear to us, that is, are objects of sensibility. |
| The constant form of this receptivity, which we call sensibility, is a necessary condition of all relations in which objects can be intuited as existing without us, and when abstraction of these objects is made, is a pure intuition, to which we give the name of space. |
| It is clear that we cannot make the special conditions of sensibility into conditions of the possibility of things, but only of the possibility of their existence as far as they are phenomena. |
| And so we may correctly say that space contains all which can appear to us externally, but not all things considered as things in themselves, be they intuited or not, or by whatsoever subject one will. |
| As to the intuitions of other thinking beings, we cannot judge whether they are or are not bound by the same conditions which limit our own intuition, and which for us are universally valid. |
| If we join the limitation of a judgement to the conception of the subject, then the judgement will possess unconditioned validity. |
| For example, the proposition, "All objects are beside each other in space," is valid only under the limitation that these things are taken as objects of our sensuous intuition. |