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The way mathematics actually develops, the whole history of it, is a process of exploring that gap between human thought and abstract ideas: letting imaginary things surprise us, uncovering hidden assumptions, reworking proofs, sharpening definitions, finding new imaginary things to surprise us more.

And this is the point where I, like a broken record, tell you to read _Proofs and Refutations_, because it is profound and also delightful.

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