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)?

2If possible, please provide some reference (e.g. a link) to the question and answer you mention in your first sentence. – None – 2014-12-31T10:02:05.200

FWIW I was explicitly told in sci realism class that NO ONE SAYS THIS :) IIRC since then I found someone who did, but he wasn't very fashionable anyway haha – None – 2015-01-01T05:40:51.013

You are wrong that mathematics is not a phenomena itself. Mathematics is same thing as hamburger, or red shorts. You just have never (not so many times) "seen" it. That is why we regard mathematics and thinking as more

ideal. If the world will be filled with mathematical facts all over the place then in that world Pythagoras theorem or Atomism will be as dull as are red shorts in ours. Are red short actually dull? Hm. Ps: Comparison. – Asphir Dom – 2015-01-02T17:59:40.383