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> 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 o

> 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. [0] https://en.wikipedia.org/wiki/Haag%27s_theorem https://en.wikipedia.org/wiki/Haag%27s_theorem

Jensson 3y ago
> 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. https://en.wikipedia.org/wiki/Distribution_(mathematics) https://en.wikipedia.org/wiki/Distribution_(mathematics)
- ants_everywhere 3y ago
  > 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.

Jensson 3y ago

> 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. https://en.wikipedia.org/wiki/Distribution_(mathematics) https://en.wikipedia.org/wiki/Distribution_(mathematics)

ants_everywhere 3y ago
> 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.