Proofs as stories
If a proof is a story, then a memorable proof must tell a ripping yarn. What does that tell us about how to construct proofs? Not that we need a formal language in which every tiny detail can be checked algorithmically, but that the story line should come out clearly and strongly. It isn’t the syntax of the proof that needs improvement: it’s the semantics.
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Psychologists now tell us that without emotional underpinnings the rational part of our mind doesn’t work. It seems that we can only be rational about things if we have an emotional commitment to such a recently evolved technique as rationality … I don’t think I could get very emotional about a structured proof, however elegant. But when I can really feel the power of a mathematical story line, something happens in my mind that I can never forget … I’d rather we improved the storytelling of proofs, instead of dissecting them into bits that can be placed in stacks of file cars and sorted into order.