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Most physicists like to visualise stuff and have intuition on it, but they are not very bothered about the details. If their model matches experiment, it's perfect, even if it's mathematically invalid.
Mathematicians are focused on solving very specific super complex puzzle like enigmas, and they require themselves to adhere to all rules of the game they just made up, but which is autocoherent.
Good example is Feynman's integral vs Stochastic Calculus.
