The theorems I’ve seen in analysis can be explained by analogies that invoke the macroscopic visible world, areas, volumes, life sized physical things in sets. Are there any theorems in math that if dumbed down into an analogy, can only be an analogy understandable with knowledge of quantum mechanics which deals with the smaller of the smaller? Or is it true that all intuition and analogies for explaining any theorems can be explained in both quantum and microscopic terms and macroscopic terms? The many theorems in geometry or topology can be explained using visual observable analogies. Are there any geometric or topological theorems that only make sense if you’re visualizing or understand the relativistic wave equations of the quantum mechanical world? I suppose not because not every mathematician understands QM and no one can see QM. The answer seems to also be no because Schrödinger managed to explain QM using macroscopic cats. And Einstein explained relativism using the touching a hot stove and woman example. But I’ve noticed that those trite analogies abstract way too much info from the real nature of QM.
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$\begingroup$ After sufficient reformulation, any theorem in all of mathematics can be explained in terms of apples without even resorting to analogy. $\endgroup$– MagmaCommented Feb 26, 2020 at 13:21
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$\begingroup$ Because everything boils down to axioms which can be explained using apples. $\endgroup$– user739402Commented Feb 26, 2020 at 13:22
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$\begingroup$ Sorta, yeah. But even in those cases where it's easier to use different parts of reality as we know it to produce an analogy for a mathematical statement, it's usually in a context that has barely anything to do with how the world works, but more how we perceive the world to work, in an idealized abstracted simplified model of our own experiences. I'm sure it would be possible to reformulate many mathematical statements in terms of quantum mechanics, but that would mostly defeat the point of having an analogy in the first place (which is to align your intuition with the abstract statements). $\endgroup$– MagmaCommented Feb 26, 2020 at 13:27
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$\begingroup$ So I’ve more or less asked whether physics knowledge makes one a better mathematician. And the answer is no, because physics knowledge doesn’t increase the number of useful analogies to describe mathematics. And more analogies to describe the same thing aren’t necessarily precise, non hand wavy or as good as a proof. $\endgroup$– user739402Commented Feb 26, 2020 at 13:30
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2$\begingroup$ Oh I'm sure that physics knowledge in general does increase the number of useful analogies, it's just that in the case of quantum mechanics, intuition ends up leading the other way: you gain intuitive understanding of quantum mechanics by applying your knowledge and experience in algebra and calculus. $\endgroup$– MagmaCommented Feb 26, 2020 at 13:35
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