What is this passage from Proposition 4 of Kant's Metaphysical Foundations of Natural Science saying?

Question

In the following passage, Kant discusses the infinite divisibility of matter:

Matter is impenetrable, through its original expansive force…But this is only a consequence of the repulsive forces of each point in a space filled with matter. Now the space filled by matter is mathematically divisible to infinity, that is, its parts can be distinguished to infinity, although they cannot be moved, and thus cannot be divided (according to geometrical proofs). But in a space filled with matter, every part of it contains repulsive force, so as to counteract all the rest in all directions, and thus to repel them and to be repelled by them, that is, to be moved a distance from them.

Proposition 4, Metaphysical Foundations of Natural Science

What is Kant saying?

Answer 1

The point is to distinguish between

1. the mathematical (geometrical), infinite divisibility of space, where every point can be distinguished between either being occupied by matter or not, and
2. actual, physical divisibility of matter into infinity.

As a background, now, we need to know about two propositions:

1. Physical matter can only be divided so far (atomic theory) and
2. Space can only be occupied by a single object comprised of matter (Law of Excluded Middle, basically Aristotelian physics).

Thus, it follows that every infinitely small "geometrical" point occupied by matter "defends" its place against other matter (repelling force, forbidding infinitely close distance (=equity)), even if matter cannot actually be infinitely divided.

Answer 2

1. I understand the sentence quoted above as follows:

   Kant states that space, an object of geometry, is mathematically divisible without end. That’s correct, because any non zero volume of in space contains a proper volume which is less.

   But this division is only in thought, because one cannot cut physically space into pieces. Hence one cannot divide space physically and one cannot move the pieces apart.

   But I cannot follow Kant’s "proof" of proposition 4.

2. The whole attempt to derive fundamental properties of space from philosophical argumentation seems totally outdated to me. The investigation of space is an object of physics.

   It is to expect that below a certain scale (e.g., the Planck scale) space has a totally different structure than we register in the mesocosmos. The structure of space and time, and hence of spacetime, is currently an open question of science. E.g., see Carlo Rovelli: What is time? What is space.