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> And your first quote is a trick on how to extend one's intuition, not abandon it! Can you elaborate on this? Do you mean that the point is 'to free up your i

• romwell 8y ago
  In a way, yes. But also: it is surprising how well building up the intuition in 3 dimensions works for N dimensions. You can build up a very working, visual intuition for Linear Algebra just by doing everything in 3D, for example. There's a textbook that takes this approach: "Practical Linear Algebra". But, of course, this comes with a caveat: it works until it doesn't. The volume of a unit sphere increases for dimensions 1-5, then starts to go down[1]. There exists an exotic R^4: a manifold that is homeomorphic, but not diffeomorphic to R^4[2]. And so on. But for a lot of (intro) differential geometry and topology, (e.g. concepts like covering spaces), I haven't found many examples where intuition based on 3 dimensions breaks. A friend of mine[3] once told me that geometry is the art of getting correct results from incorrect diagrams. I think that the quote we're discussing is in a similar spirit. [1]https://en.wikipedia.org/wiki/Volume_of_an_n-ball https://en.wikipedia.org/wiki/Volume_of_an_n-ball [2]https://en.wikipedia.org/wiki/Exotic_R4 https://en.wikipedia.org/wiki/Exotic_R4 [3]Dmytro Karabash