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The Accidental Astrophysicists
An anonymous reader recommends a ScienceNews story that begins: "Dmitry Khavinson and Genevra Neumann didn't know anything about astrophysics. They were just doing mathematics, like they always do, following their curiosity. But five days after they posted one of their results on a preprint server, they got an email that said 'Congratulations! You've proven Sun Hong Rhie's astrophysics conjecture on gravitational lensing!'... Turns out that when gravity causes light rays to bend, it can make one star look like many. But until Khavinson and Nuemann's work, astrophysicists weren't sure just how many. Their proof in mathematics settled the question."
Math is not leading (and I don't think that xkcd strip claims it).
In fact in this case, Rhie found the limit first but just couldn't prove it.
As for proving stuff, Math can help prove some stuff, but not other stuff (it can't normally be used to prove what you had for lunch yesterday).