[roenxi]

Even if a lot of mathematicians think math is about studying nature they are demonstrably wrong. One can do maths that has no known relation to physical reality, has no basis in physical reality or sits in direct opposition to how physical reality actually works. That is precisely the opposite of physics to say that "well, if I can imagine it it must be nature". So by counterexample, since the field of maths contains things that are anathema to the science of physics maths cannot be a branch of physics.

Math is the study of abstract patterns and those cannot be escaped. But just because individual mathematicians dedicate their lives to finding abstract patterns inspired by physics doesn't mean that either physics or math are branches of the other.

[spacedome]

You are making a lot of ontological and epistemological assumptions that are contentious in the philosophy of math. Not saying you are wrong in thinking this, metaphysical questions don't necessarily have answers, but many would not agree with you.