If math is just made up there ought to be more variability from one to the next mathematician. Physics, though, does not describe nature but tries to explain phenomena by mathematical analogy. Since analogy is not proof, that leaves physics not proving anything. If you want to describe nature you might start by saying what nature is, if possible.
There is a lot of variability between flat, spherical, and hyperbolic geometry results! They can't even agree on how many lines, if any, are parallel to a given line, that pass through a point not on the line!
What doesn't change is the use of logic, but what does change are the axioms on which each mathematical structure is based. In contrast, the axioms of physics are not for us to alter; they are what they are, and it is up to us to attempt to identify them. We can't even be sure that we've correctly identified any of them, e.g. that the speed of light cannot be exceeded by a material object, in any inertial frame. The most we can do is assert that no known facts or observations contradict what we believe to be true, but even this allows that some other conditions might exist, that we have never observed or conceived of, in which what we believe to be true would not be.