i did a lot of math while i was gone so i accumulated some math hot takes which i archived in roam and here they are for your convenience
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1. proofs in textbooks should be commented. the author should be commenting on which steps are routine, which steps are important to understand carefully, which steps are creative and surprising, etc. etc. sipser is the only textbook i've seen do anything like this and it rules
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2. students should be taught how to "debug" proofs. here is a basic technique that people don't get taught: if you proved something you know is wrong, you can figure out which step is wrong by stepping through the proof *with a counterexample*
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3. it's weird how we taught students to encode their understanding of math in their understanding of social permission (what they're "allowed" to do). permission has nothing to do with it. the question is what procedures turns true statements into true statements
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4. there should be two different words for "even prime" and "odd prime"
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5. there should be two different words for "simple abelian group" and "simple nonabelian group" (note that we already do this in lie theory)
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