[youoy]

Yes! I agree with you, mathematics is not a natural science. You can study the physical reality or the intellectual reality, mathematics would be closer to the intellectual reality.

I guess this points back to the eternal debate about if mathematical objects are invented or discovered. If they are discovered, then that would mean that there is an underlying truth/semantics behind the mathematical formalisms. So in that case mathematics studies and interacts with reality. Take for example Calculus and Newton and Leibniz [0] who invented/discovered calculus independently.

I will let you decide on what side of the debate you feel more comfortable :)

[0] https://en.m.wikipedia.org/wiki/Leibniz%E2%80%93Newton_calcu...