Reasoning with external material formal notation (squiggles on paper) accomplishes abstraction in two ways discussed by @cdutilhnovaes: de-semantification and ease of calculation.
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Yes, I’m unconvinced that blackboard lecturing is anything more than a small (and mostly unimportant) fraction of “learning about” and “thinking about” mathematics.
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How people behave when they read a textbook/paper and hit something they want to understand seems to me far more important. And this seems to be very idiosyncratic. I do it eyes closed, talking to myself. But others are v different. (Some via diagrams, some via algebra, ...)
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I think blackboard/paper work is important, possibly central, but also _legible_ in a way other kinds of work aren't, so likely to dominate any ethnomethodological study.
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I bet varying levels of a/phantasia matter, too - different capabilities, different ways of making ideas legible to oneself (ethnomethodological inquiry aside)
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Absolutely. My own rhythm was a sort of dialectic between kinesthetic work in my head and symbolic work on paper. But I've known mathematicians who didn't seem to do anything but grind chains of symbols.
End of conversation
New conversation -
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Yeah I "wrote" most of my dissertation walking the dog
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. Banned in Sweden. SubGenius, Zhuangist, white-hat troll. Defrocked mathematician. Brain problems.