> They do have insane pattern recognition skills when it comes to models. Good mathematicians also have this.
The skills physicists and mathematicians build aren't very comparable.
Physicists has the more complex equations and models, probably most complex of any field since physicists often invent new math to model things. Quantum field theory is insanely complicated math wise with infinite number of infinite integrals, mathematicians has still not figured out how to formalize that.
Mathematicians focuses much less on modeling and equations so they aren't as good at that, instead they are much better at formal proofs and theorems. The skills are very different and doesn't translate well.
Or in other words, physicists are experts at making complex math tools to model things. Mathematicians are experts at verifying tools that exists. Those two skills has less overlap than you might think.
I wouldn't be so dismissive of the overlap. A friend of mine has a math PhD and got a 32 on the Putnam each time he took it, which if you're familiar is proof he's no joke when it comes to math skills. He spent his career as a quant on wall street modeling economic relationships because it paid a lot more than staying pure math.
Interesting way to look at it. Your description of what physicists are experts at matches my math PhD pretty closely. I focused on mathematical modeling. I now work with a bunch of physicists, so I guess that checks out.
Yeah, if you work on the applied side of math it can be very similar to what people do in physics. But I was thinking more about pure math.
Edit: I think the main difference there is that in applied math they still prove that the models are mathematically correct. In physics they just show that the model align with experiments and skip math formalism.
I was with you on the generalities, but oh do theoretical physics suffer from math envy when it comes to formalism. Basically since the invention of quantum mechanics, has physics been dominated by “proofs” and “theorems” like the physical world is assumed to be axiomatically defined by Heisenberg’s and Pauli’s principles, and everything else is just maths.
No small part of the stagnation I sense in physics today stems from too deep a faith in the ultimate truth of the mathematical models we call theories. It doesn’t help the fact that we rely on Taylor expansions and perturbation methods for most experimental predictions.
The Higgs hunt and the passivity of the (experimental) physicists in challenging this stupid theory-driven search for new physics is emblematic of this era.
If only math was seen as a modeling language and not somehow truth/consistency itself, physics would be much better off.
Yeah, I saw that as well, met some professors in grad school that started talking about physics in terms of axioms and proofs instead of experimental results and models. At that point I lost interest and just went with math instead, if it is going to be math anyway why not go with the real thing.
> Physicists has the more complex equations and models
This isn't really the case. In general in any class of mathematical objects, the physically realizable ones are the most tame. Physical realizability is a huge constraint that typically dramatically simplifies problems.
> Quantum field theory is insanely complicated math wise with infinite number of infinite integrals, mathematicians has still not figured out how to formalize that.
My understanding was that the issue is that QFT as used by physicists is known to be mathematically inconsistent. For example, see Haag's theorem [0]. As Wikipedia puts it "there does not exist a well-defined interaction picture for QFT, which implies that perturbation theory of QFT, which underlies the entire Feynman diagram method, is fundamentally ill-defined."
So the problem is not that we don't know how to formalize it, it's that what physicists are doing is known to be wrong. So we need some better argument for why it sometimes seems to work under the conditions it does. And that's arguably more about getting the physics right than it is about math.
This is something that happens in general: physicists will often use tools or arguments that are known not to be correct despite the fact that they work well in practice.
I think in general, that's a good thing. Physicists need to be able to brainstorm ideas in a way that is unconstrained by needing to formally prove them. In this process they might find something that works 60% of the time. Then someone will refine it to work 80% of the time, etc. But at the end of the day, the tools do have to be mathematically consistent for them to actually describe the physics, unless we live in a mathematically inconsistent universe.
So the characterization I'd give is that physicists are generally better at modeling, but also less constrained by the need to be correct.
> Mathematicians are experts at verifying tools that exists. Those two skills has less overlap than you might think.
This part is reversed IMO. Mathematicians created nearly every major tool used by physics since the history of humanity. Occasionally physicists will run ahead and come up with an idea that is lacking in tools, and they'll turn to mathematicians as the tool experts for help. Like, for example, Einstein did when trying to formulate relativity. More rarely, they'll come up with a novel tool like Feynman diagrams without knowing why they work, and then the mathematicians will be called in as the experts to sort things out.
And of course it is true that physicists will come up with ideas like the Ads/CFT correspondence that arise from physical considerations. It's also true that the problems of physics can inspire new mathematics simply because they're new and interesting problems.
> Mathematicians created nearly every major tool used by physics since the history of humanity
Calculus was created by a physicist who didn't even bother it since he just published the resulting physics, Fourier analysis was created by a physicist etc. Mathematicians first and foremost formalizes things after they were created, they don't create many useful tools to begin with. Very few fundamental tools were made first by mathematicians.
They often call physicists/polymaths mathematicians after it happened since they published a lot of math, but they did the physics/natural first at the time. The formalist mathematicians we see today weren't even a thing 150 years ago, the useful tools developed by formalist mathematicians is basically zero, the new useful tools physics used the past 100 years were done by themselves.
For example, generalized functions where done by physicists, mathematicians scoffed at it saying it is inconsistent as you said here, but then after over 100 years apparently it all worked out just and the physicists were right that you can work with functions that are defined by their properties when integrated instead of their values. The same will happen to QFT, it isn't wrong to work with infinities its just that mathematicians aren't good enough to figure out how to formalize such tools but such tools working with infinities works.
I have studied both theoretical physics and pure mathematics at a graduate level and published in both, I have a fair idea how the fields works. Rarely formalizing a tool can open up for more usages of it, but that is about it.
> Calculus was created by a physicist who didn't even bother it since he just published the resulting physics
Calculus was discovered by a number of people. I'm assuming you're thinking of Newton, who was a professor of mathematics and whose advisor was a mathematician. Remember the fundamental theorem of calculus had been proven by mathematicians Gregory and Barrow (Newton's advisor) before Newton, not to mention the work of people like Fibonacci. Calculus as we know it today was mainly developed by the mathematician Leibniz independently of Newton.
> Fourier analysis was created by a physicist etc
Joseph Fourier was a mathematician whose advisor was mathematician Joseph-Louis Lagrange whom he succeeded at ENS.
> Mathematicians first and foremost formalizes things after they were created, they don't create many useful tools to begin with. Very few fundamental tools were made first by mathematicians.
This is wildly wrong and contrary to history.
> They often call physicists/polymaths mathematicians after it happened since they published a lot of math
The people I'm calling mathematicians, like Newton and Fourier, are people who trained in mathematics and had mathematical advisors. What other definition do you want of a mathematician other than one who does math, studied math, is employed to do and teach math, and whose advisor was a mathematician?
For example, while Wikipedia calls Fourier a mathematician and physicist, Britannica just calls him a mathematician. Britannica does mention that he was also an Egyptologist but never calls him a physicist.
> generalized functions where done by physicists
Distributions have been used by mathematicians since the 19th century. I'm aware of no examples of physicists claiming priority. Physicists do often use them incorrectly, for example a lot of physicists view the Dirac delta "function" as a function despite the fact that no functions have those properties. They sometimes confuse that, as you seem to do, with thinking that mathematicians don't understand what generalized functions are about.
Footnote: Haag's theorem only invalidates naive perturbative approximation. But that's not what physicists actually use; they use it for motivation and then switch to a renormalized perturbative approximation, which is compatible with Haag's theorem.
Your point stands though: mathematicians are indeed able to cope with path integrals. But several of the models -- qed, Higgs, electroweak, basically everything but qcd -- used in particle theory suffer from a fatal flaw, the Landau pole. They can only exist as incomplete approximations to something else.
>So we need some better argument for why it sometimes seems to work under the conditions it does. And that's arguably more about getting the physics right than it is about math.
Douglas adams solved it though, he understands why this flawed physics seems to work -- 6x9 = 42
But the good ones are very hard working too, in order to build a big enough library of building blocks.