Kant's Prolegomena Note I - Geometry being an objective representation of nature

I'm trying to understand this part of Kant's Prolegomena to Any Future Metaphysics, Note I to "How is pure mathematics possible?":

It would be completely different if the senses had to represent objects as they are in themselves. For then it absolutely would not follow from the representation of space, a representation that serves a priori, with all the various properties of space, as foundation for the geometer, that all of this, together with what is deduced from it, must be exactly so in nature.

Trying to organize that in more simple terms, this seems something like a reduction ad absurdum argument, like this:

• Premise 1: Senses represent things-in-themselves (which is actually false for Kant)
• Premise 2: Representation of space is a priori
• Conclusion: Representation of space and deductions from it may not be exactly so in nature (absurdum)

But, on a logical basis, why are P1, P2 and the negation of the conclusion incompatible? Are any other premises missing here? Wouldn't it be possible that our intuition a priori accurately represents (some) laws of nature as they are in themselves?

Side-note: I guess there could be possible modern objections based on theory of relativity, etc that could make the conclusion not an absurdum at all, but true. However that is not my question.

• Hint: Nature (and world) mean the sum of all possible experiences in Kant, ie. by definition the sum of all representations that are shaped by our pure intuitions (space and time) and our conceptual understanding. Commented Apr 30, 2022 at 5:25
• Ok, so combining that and P1, the senses could not represent nature, because things-in-themselves are disjointed from experience? If the senses represented things-in-themselves, wouldn't it be appropriate to also adjust the definition of "experiences" to include or equate things-in-themselves? Commented May 3, 2022 at 6:43

Kant recalls his basic discrimination, opposing

• D1: objects as they are in themselves

• D2: the external appearance of objects.

The external appearance of objects (D2) is the result of our intuition of objects in themselves. The latter intuition is provided by our sensibility, which is shaped by our a-priori intution of space. Therefore external appearance of objects relies on both the objects (D1) and on our sensibility of intuition.

According to Kant, the rules of our sensibility of intuition are the propositions of 3-dimensional Euclidean geometry. They are synthetic insights a-priori.

If these synthetic a-priori propositions would give us information about ‚objects as they are in themselves‘ (D1), this fact would seem inexplicable,

because we cannot see how things [D1] must of necessity agree with an image of them [D2], which we make spontaneously and previous to our acquaintance with them.

Kant concludes that the external appearance of objects (D2) does not give us information about objects as they are in themselves (D1).

• So if that was the argument used to get to "senses do not represent things-in-themselves", then the negation of this conclusion implies the negation of its premise which would go like "intuition of space and time are made after the objects, so they are a posteriori". This means P1 and P2 from the question contradict each other? If yes, I don't see how this was specifically used to talk about the relation between geometry and nature. Commented May 3, 2022 at 7:04

This is the best I could come up with so far, I'm not sure it is correct:

Kant seems to argue that, with P1 and P2, this is true:

• Representation of space and deductions from it may or may not be exactly so in nature

There is just not enough premises to assert that the representation of space does or does not objectively represent nature. So there would be skepticism about the reliability of geometry.

Conversely, further down in the same note, Kant writes this:

If, however, this image – or, better, this formal intuition – 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 very easy to (...) prove incontrovertibly: that all outer objects of our sensible world must necessarily agree (...), with the propositions of geometry, because sensibility itself, through its form of outer intuition (space), (...) makes those objects possible, as mere appearances.

So the key here appears to be that "this sensibility" or "this formal intuition" is the property by means of which alone objects are given to us. So this intuition has this monopoly over handing us objects. This common source enforces some kind of consistency between empirical (nature) and pure (geometrical) objects, which would presumably be more difficult if there were different sources of objects (things-in-themselves and our intuition).