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komponisto comments on Scientist vs. philosopher on conceptual analysis - Less Wrong
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Calculus is not a fair example, because the disciplines of mathematics and physics were not separated at the time; the work of Newton and Leibniz was the best "mathematics" (as well as "physics") of the era. Noether's theorem is not an example at all but a counterexample: a mathematical theorem, proved by a mathematician, that provided new insights into physics.
Dirac's delta and Feynman path integrals are fair examples of your point.
I concede the calculus point on a technicality :). Certainly there was no clear math/physics separation at the time (and the term physics as currently understood didn't even exist), but the drive to develop the math necessary to solve real-life problems was certainly there, and separate from the drive to do pure mathematics. And it took a long time before the d/dx notation was properly formalized.
As for the Noether's theorem, it was inspired by Einstein proving the energy-momentum tensor conservation in General Relativity, without realizing that it was a special case of a very general principle.
Except that the problems in question (explaining the motion of the planets and so on) would not have been considered "real-life problems" back then; rather, they would have been considered "abstract philosophical speculations" that would have carried the same kind of stigma among "practical men" that "pure mathematics" does today.
I think you mean "justified"; if there was one thing Leibniz was good at, it was formalizing!
According to your Wikipedia link, it represented the solution to a physical problem in its own right: a paradox wherein conservation appeared to be violated in GR.