I was looking at arguments about mathematics being a science (or not), here for example, but it seems that these arguments are more about some metaphysical idea of mathematics rather than the subject itself. Mathematics as it exists is a study of the most schematic properties of physical objects, it has both theoretical and empirical aspects. If our world consisted of unstable shapes constantly morphing into each other we would have a very different kind of geometry, if the objects we encounter sprouted new ones when put together we would have a very different kind of arithmetic and set theory.
Mathematics doesn't study some "arbitrary" systems of axioms, it never did and it never will. Which systems are "good" or "interesting" is discovered by trial and error, just as theoretical parts in all sciences. Sure, mathematics is unfalsifiable in the sense that if you accept the axioms (and logic) you have to accept the theorems, but Newtonian mechanics is unfalsifiable in exactly the same sense. And in the sense that it is falsifiable so is Euclidean geometry. I suppose, one can imagine a being doing mathematics without sensory input, but one can also imagine a being that creates new worlds as it pleases with their own laws of physics.
Scientific theories are not verified or falsified by experiments alone either, as Einstein remarked "only theory decides what one observes". String theory gives a stark example of just how indirect a relation to observations and experiments can be for a vibrant physical theory. Mathematics is the most theoretical of sciences, but there are more commonalities between it and physics than between physics and say psychology, so it seems to be well within the spectrum.
Is there a separation in modern philosophy between mathematics as practiced and "ideal mathematics" that "has no bearing on the physical world and cannot be affected by it"? What is the philosophical basis for such separation, is there similar separation for other sciences? Is the controversial status of mathematics as a science due to historical tradition of Plato, Kant and Husserl who treated its source as metaphysically different?
EDIT: Euclidean and non-Euclidean geometry are not "arbitrary" at all, one summarizes our basic space experience, the other modifies only one aspect of it which may not even be noticable at small distances. Grothendieck invented etale cohomology to prove Weil conjectures about algebraic varieties, following up on that quickly leads to tangible things as well. Even the purest of areas like logic, set theory or algebra can be easily traced to their roots, including Godel's theorems with their reliance on arithmetic and "effective axiomatization". There is leeway in every science, no one expects some direct correspondence with reality or applicability to it from every theory. There is more leeway in mathematics, but where is the qualitative difference?