Imre Lakatos's "Proofs and Refutations" is a wonderful book-length discussion of this and related points.https://twitter.com/AnalysisFact/status/752888067455852544 …
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Worst thing about teaching mathystuff (I'm including eg semantics) is showing defs up front rather than _designing_ them.
It's often possible to teach in such a way as to discover them. Slower, but deeper.
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Precisely. My pref. Trades quantity of material for depth of understanding. Defs are fascinating; books make them boring.
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It's such a strange deficiency in standard mathematical style that so little space is allocated to how defns are discovered
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