Induction isn't the core of it. The idea is that you try to find counterexamples in science, and you don't have proofs in the general sense. Ever. The best you can say is that "If I guessed right, then I have math that worked out." Nothing more. Mathematics allows a definite "This is a solution that exists to this problem.". In Science, you can say that a solution exists to a problem, but that's not the tricky part -- the tricky part is figuring out the right problem to solve.

Science is effectively an attempt to reverse engineer the universe. Imagine that you were given a clock by a client. A very ornate clock that he doesn't want you to take apart. Then he tells you he wants you to make schematics for the clock so precise he could put together a precise functional duplicate. But you're not allowed to take it apart.

I would say that does not that count as mathematics, although mathematics would certainly have a use as a tool to prove that the behavior is correct in all observed respects. But you could probably find a new minute difference in the way the hand moves, and that would blast all your carefully crafted mathematics to bits.

I'm not slagging on mathematics, and I'm not trying to glorify science and say that one is a more powerful superset of the other.

They're both very useful and very tightly intertwined disciplines, but they're also very different. In a way, you could say that math is the study of solving problems, and science is the study of finding physically meaningful problems.