Interesting but somewhat confusing.
You want a differentiation between the mathematical description, and actual, physical space.
You have stated we have the mathematical description, but haven't yet tackled what we mean by actual, physical space.
IOW, we have the model, but what is the actual reality ?
( and Dad says we don't know, which you agree is valid )
I would assume, as a scientist, you know how science works.
We build models ( mathematical models, not toys ) and test them against reality to find their limited areas of applicability.
The only model that is 100% applicable is the reality itself.
I have to ask, how else would you model actual, physical space, if not mathematically ??