[math_dandy]

Historically, mathematicians have spent a huge amount of time and effort formulating optimal axioms and foundations so that theorems would follow naturally from structure. Theorems following “trivially” from a theoretical framework that took years to develop isn’t an indictment of the theorem, but an endorsement of the incredible effort expended to develop an optimal context for expressing and understanding the theorem.

[mathgradthrow]

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.

[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.

[chermi]

Lol the authority you speak with is so hard to reconcile with the absurdity of your claims.

[mathgradthrow]

I didn't clain to have a popular opinion.

[chermi]

I know. But you should realize that sometimes opinions can be wrong.

[mathgradthrow]

the best I can do is assure you that it is informed by experience. Calculus could not progress without analysis. einstein needed poincare, who needed topology, which needed sets.