> how did people visualize/understand Fourier without all the neat abstractions, metaphors, and history ? The answer to that is, I think, exactly why so many p

> how did people visualize/understand Fourier without all the neat abstractions, metaphors, and history ? The answer to that is, I think, exactly why so many people don't like math; because, in math discovery, first comes the history, in the sense of the problem that motivates the nrew technique, then comes the metaphor comes, and only then is the math (which I suppose is what you mean by 'abstractions'?) built around it. Math professors, whether out of ignorance (we don't know the history), or preference (we learned this way and we like it), or even just time constraints (there's not time in a standard semester class to tell the full story of the concepts), will often teach only the math, or treat the intuition and metaphor as secondary; but they were, of course, the entire genesis of the subject. No-one retrofits intuition to mathematics; mathematics is rather a way for experienced practitioners to 'sanity-check' and formalise the intuition that they already have.