To a physicist like me, the assumption that mathematicians are the source of mathematical revelations is the same as assuming that the best novels are produced by people who studied english.
Group theory.
Laplace transforms were considered an irrelevant curiosity until somebody figured out they were isomorphic to the wildly effective Heaviside D operator. Then everyone did their best to bury Heaviside's demonstrations of their utility, and pretend that Laplace had done all the heavy lifting.