[gjm11]

Do you have any evidence that Dedekind's formalization of the real numbers was an essential step on the road to quantum physics and relativity? This seems very doubtful to me.

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".

[mathgradthrow]

My evidence is just the timeline. Mathematics blew up right before physics did.

[Sharlin]

Post hoc does not imply propter hoc.

[mathgradthrow]

Yeah, you're right, my evidence would be better if we did a randomized controlled trial.