In answer to someone's question regarding Kant's idealist view of space vs. modern science, someone referred to the dichotomy between a cup as normally perceived and a cup as a collection of atoms. If we also admit the notion of the cup itself we then have a "trichotomy". I wonder if anyone holds that the cup-in- itself is just a collection of atoms. Could we say that atoms are not phenomena but mathematical entities; that things themselves are fundamentally mathematical? I imagine though that Kant sees mathematics as a human construct.
Regarding the thing itself...
I'm sure there are people who hold the thing in itself is the fundamental construct of matter (probably subatomic particles rather than atoms). In fact, isn't this the materialist position?
However, if we look at the cup-in-itself as noumenon in the Kantian sense, then I don't think atoms qualify. Noumenon are unknowable. Atoms on the other hand are (at best) indirectly knowable through their effects, or are (at worst) mathematical abstractions that help us model certain things.
Regarding whether the thing itself is mathematical...
Are things fundamentally mathematical? Well, cosmologist Max Tegmark has put forward his Mathematical Universe Hypothesis and wrote a book called Our Mathematical Universe in which those views are presented.
I also recall an author who made a half-facetious argument that we all believe mathematics is the fundamental ontology. His argument went something like this. Put 1 ball into an empty box. Now put another ball into that same box. Now count the number of balls in the box. If there aren't 2 balls, then we'd look for holes in the box, physical anomalies, and even doubt our very senses. Yet the one thing we would not do is doubt that 1+1 = 2.
This raises all sorts of fascinating questions. What does it mean for something to be a mathematical structure, as opposed to a physical one? Are mathematical structures things that lack any properties and consist purely of relations? If we go down to the smallest substrate of matter, would there be any properties? Would properties imply parts? At that point, would we be in an area where our traditional thinking simply cannot cope (although some would argue that we got there with Quantum Mechanics)?
A collection of atoms is something that can be captured by our intellect. According to Kant, the thing in itself is the thing as it exists before being captured by our intellect. The collection of atoms won't do, nor any scientific theory.
Sellars' distinction between the 'manifest image' and the 'scientific image' of the world better express the dichotomy you are referring at.
For Kant, mathematics correspond to the prerequisite of understanding. Mathematics is the tool which allow us to put phenomena into shape so our intellect can capture it. Saying that the 'things in itself' is mathematical is thus far from Kant's philosophy.
I have just looked at Max Tegmark's book. He is widely quoted online as saying that the ding an sich is unnowable. Actually this position is one that he categorizes as meaningless. It seems I have to accept his view that reality is ultimately mathematical. Sad, because I like mystery. Anyway we still have the mystery of affective consciousness.Apologies for not being able to direct my comments to exactly the right correspondants. Sorry Quentin.