Pure intuition: space and time

Part I proper (§§6–13) tackles the first keyhole: how is pure mathematics possible? We have the two clues already. Mathematics is synthetic (its judgments add content, not mere definitions) and it is a priori (necessary and universal). Kant now asks what could make both true at once. See Prolegomena to Any Future Metaphysics §6.

Start from the synthetic side. If mathematical judgments go beyond their concepts, what do they go to? Not to experience — that would make them a posteriori. Kant’s answer: to intuition (Anschauung) — immediate sensible presentation, the direct exhibiting of an individual. You don’t prove a theorem by staring at definitions; you construct the object: draw the triangle, lay out the units, count. Mathematics is the science that builds its concepts in intuition.

Now the tightening screw. The knowledge is a priori — necessary, universal, certain before you draw any particular triangle. But an intuition is normally of a single perceived thing, which is exactly the sort of a posteriori, contingent input a priori knowledge can’t rest on. So the intuition mathematics uses cannot be an empirical one. It must be a pure intuition: an intuition given a priori, before and independently of any actual object. See Prolegomena to Any Future Metaphysics §7.

How could there be an intuition before the object it presents? Only if it is not the object at all, but the form in which any object must be given to us. Kant’s thesis: what you intuit a priori is the mere form of your own sensibility — the shape every perception is bound to take, lying ready in the mind before anything fills it. See Prolegomena to Any Future Metaphysics §8.

And those forms are two: space is the a priori form of outer sense (everything given as outside us is given as spatial) and time is the a priori form of inner sense (every state of mind, and so every experience whatever, is given in temporal succession). This is the outline of what the larger Critique calls the Transcendental Aesthetic. See The Critique of Pure Reason Transcendental Aesthetic.

Put the pieces together and mathematics is explained. Geometry is the science of space, our pure form of outer intuition; arithmetic rests on pure succession in time, our form of counting. Because space and time are the very forms in which every object must be given to us, whatever geometry and arithmetic establish about those forms must hold, necessarily and without exception, of every object we can ever experience. That is why mathematics is synthetic (built in intuition) and a priori (holds of the forms themselves) and infallibly applicable to the world. See Prolegomena to Any Future Metaphysics §10.