Notes on <Prolegomena to Any Future Metaphysics> - How is Pure Mathematics Possible
Section 7-13
All mathematical cognition has this distinguishing feature, that it must present its concept beforehand in intuition and indeed a priori, consequently in an intuition that is not empirical but pure, and can exemplify its apodictic teachings through intuition but can never derive them from it. Pure mathematics must be grounded in some pure intuition or other, in which it can present, or, as one calls it, construct all of its concepts in concreto yet a priori. Pure intuition is like empirical intuition since we can form concepts out of them, but with the difference that it is a priori, and is bound with the concept before all experience or individual perception.
Next question we need to ask is:
How is it possible to intuit something a priori?
There are concepts that we can think of a priori, like magnitude, cause, but in order to provide them with sense we need to use them in concreto.
If our intuition had to be of the kind that represented things as they are in themselves, then absolutely no intuition a priori would take place, but it would always be empirical. There is only one way possible for my intuition to precede the actuality of the object and occur as an a priori cognition, namely if it contains nothing else except the form of sensibility, which in me as subject precedes all actual impressions through which I am affected by objects. For I can know a priori that the objects of the senses can be intuited only in accordance with this form of sensibility. From this it follows: that propositions which relate merely to this form of sensory intuition will be possible and valid for objects of the senses; also, conversely, that intuitions which are possible a priori can never relate to things other than objects of our senses.
Therefore it is only by means of the form of sensory intuition that we can intuit things a priori, though by this means we can cognize objects only as they appear to us (to our senses), not as they may be in themselves.
Space and time are the intuitions upon which pure mathematics bases all its cognitions and judgments, which come forward as at once apodictic and necessary. Because if one eliminates from the empirical intuitions of bodies and their alterations (motion) everything empirical, that is, that which belongs to sensation, then space and time still remain.
Pure mathematics, as synthetic cognition a priori, is possible only because it refers to no other objects than mere objects of the senses, the empirical intuition of which is based on a pure and indeed a priori intuition (of space and time), and can be so based because this pure intuition is nothing but the mere form of sensibility, which precedes the actual appearance of objects.
The important point is that space and time are forms of sensibility, not actual qualities attaching to things in themselves. To prove that, Kant gives an example of comparing hand or ear with its image in mirror. If they are fully the same, then we can put one in place of the other in all cases, but we can not do that, so they are not the same. However, there are no inner differences between them that we can think of. They can only be distinguished in space. This means these objects are surely not representations of things as they are in themselves, and as the pure understanding would cognize them, rather, they are sensory intuitions, i.e., appearances, whose possibility rests on the relation of certain things, unknown in themselves, to something else, namely our sensibility. Space is the form of outer intuition of this sensibility, and the inner determination of any space is possible only through the determination of the outer relation to the whole space of which the space is a part, that is, the part is possible only through the whole, which never occurs with things in themselves as objects of the understanding alone, but does occur with mere appearances.
Note I - Objective Reality
Pure mathematics have objective reality only under the condition that it refers to objects of the senses. Sensibility, whose form lies at the foundation of geometry, is that upon which the possibility of outer appearances rests; these, therefore, can never contain anything other than what geometry prescribes to them. If this formal intuition of space is the essential property of our sensibility by means of which alone objects are given to us, and if this sensibility represents not things in themselves but only their appearances, then it is easy to prove that all outer objects of our sensible world must necessarily agree, in complete exactitude, with the propositions of geometry, because sensibility itself, through its form of outer intuition (space), with which the geometer deals, first makes those objects possible, as mere appearances.
Note II - Idealism
Everything that is to be given to us as object must be given to us in intuition. But all our intuition happens only by means of the senses. So all bodies together with the space in which they are found must be taken for nothing but mere representations in us, and exist nowhere else than merely in our thoughts. But it’s not idealism. Since idealism claim that there is no objective reality outside corresponding to our thought. But that is not the case here, we are saying there are things corresponds to our thought, which are things in themselves and are acquainted only with their appearances. Things in themselves is unknown to us but is nonetheless real. The existence of the thing that appears is not nullified, as with real idealism, but it is only shown that through the senses we cannot cognize it at all as it is in itself.
Note III - Transcendental Idealism
We can reject the claim that through the ideality of space and time the whole sensible world is an illusion. If an appearance is given to us, we are still completely free as to how we want to judge things from it. The difference between truth and dream, is not decided through the quality of the representations that are referred to objects, for they are the same in both, but through their connection according to the rules that determine the connection of representations in the concept of an object, and how far they can or cannot stand together in one experience. Thus if our cognition takes illusion for truth, it is the fault of understanding, not of sensibility. The illusion is not ascribed to the senses, but to the understanding. The theory described above is transcendental or critical idealism.