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Vladimir_Nesov comments on How valuable is it to learn math deeply? - LessWrong
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(This discussion doesn't distinguish what could be called the rigorous and post-rigorous levels of skill, and so feels a little off (at least terminologically). At the rigorous level, which seems like what you are talking about, you know how the tools work, and can reassemble them to attack novel problems. At post-rigorous level, which seems like a better referent for "learning math deeply", you've sufficiently exercised intuitive mental models to offload most routine observations to System 1, freeing up conscious attention and allowing more ambitious intuitive inferences. Fluency as opposed to competence.)
Thanks Vladimir!
Why does my post give the impression of talking about the rigorous level?
You are opposing "learning math deeply" to rote memorization of brittle special cases, but the threshold of being able to work with standard tools (for e.g. understanding technical content of physics/statistics courses) is only rigorous level. Moving further requires additional practice/motivation, when you are already capable of using the tools, and that is not separately discussed in the post.
One (wo)man's brittle special case is another's generalization. There are many different levels of abstraction. One can be at the rigorous level on some dimensions and at the post-rigorous level on others. In the other direction, many things that once required post-rigorous thinking are now sufficiently codified so that they now require only rigorous thinking. There's not a well-defined body of "standard tools."