Some notable examples that come to mind are calculus, conservation laws, Dirac’s delta and bra-ket notation in quantum mechanics, path integrals, renormalization.
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.
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
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.
And it took a long time before the d/dx notation was properly formalized.
I think you mean “justified”; if there was one thing Leibniz was good at, it was formalizing!
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.
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.
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.