| Half of the work of mathematics is in correct definitions. Groethendieck referred to the division between mathematical labors as hunting and farming. This is not my most popular opinion, but probably the most consequential invention of the last 400 years was the set. Suddenly all mathematical knowledge could be verified in one framework. Physicists had a target in which to state their models. If you could state your hypothesis in the language of mathematics, "everyone" knew exactly what you meant by it, and how to go about testing your claims, or proving them, if they happened to be about mathematics itself. Calculus was invented in 1690ish, physicists like to claim that this was the most important advance in physics, but quantum mechanics and relativity didn't happen until dedekind invented the real numbers, 200 years later. It turns out that knowing what you're talking about matters. |
A more plausible claim: the general move towards greater rigour in mathematics, one of whose expressions was Dedekind's formalization of the real numbers, improved the state of mathematical understanding in ways that were necessary for the arrival of quantum physics and relativity. E.g., to do quantum physics you want the notion of "vector space"; to do general relativity you want the notion of "Riemannian manifold"; to do special relativity maybe you want to have encountered the "Erlangen programme".
But I'm not 100% convinced. It's not unusual for physicists to make use of mathematical notions that they don't have precise definitions of. E.g., I'm not sure anyone has an entirely satisfactory formal account of "path integrals"; string theory may or may not turn out to have anything to do with how the universe actually works, but if it doesn't it probably won't be because we don't have a complete account of what it actually is. Newton managed to do pretty impressive things with calculus before anyone had a really convincing definition of such advanced notions as, er, "derivative".