Difficulty of math

I am interested in the difficulty of learning math, especially for “smart” people.

https://www.quora.com/Was-there-a-certain-age-when-abstract-physics-or-math-concepts-clicked-and-suddenly-made-sense-or-has-it-always-come-naturally-to-you/answer/Brian-Bi

https://www.quora.com/How-do-math-geniuses-understand-extremely-hard-math-concepts-so-quickly/answer/Satvik-Beri

From “Linear algebra: beware!”:

While linear algebra lacks any single compelling visual tool, it requires either considerable visuo-spatial skill or considerable abstract symbolic and verbal skill (or a suitable linear combination thereof). Note the gap here: the standard computational procedures require only arithmetic. But getting an understanding requires formidable visuo-spatial and/or symbolic manipulation skill. So one can become a maestro at manipulating matrices without understanding anything about the meaning or purpose thereof.

https://www.greaterwrong.com/posts/w5F4w8tNZc6LcBKRP/on-learning-difficult-things

https://www.facebook.com/vipulnaik.r/posts/10201718168211884

Christopher Olah, in quotes like the following:

Derivatives are cheaper than you think. That’s the main lesson to take away from this post. In fact, they’re unintuitively cheap, and us silly humans have had to repeatedly rediscover this fact. That’s an important thing to understand in deep learning. It’s also a really useful thing to know in other fields, and only more so if it isn’t common knowledge.

Michael Nielsen on Simpson’s paradox:

Now, I’ll confess that before learning about Simpson’s paradox, I would have unhesitatingly done just as I suggested a naive person would. Indeed, even though I’ve now spent quite a bit of time pondering Simpson’s paradox, I’m not entirely sure I wouldn’t still sometimes make the same kind of mistake. I find it more than a little mind-bending that my heuristics about how to behave on the basis of statistical evidence are obviously not just a little wrong, but utterly, horribly wrong.

https://www.facebook.com/qiaochu/posts/10155226053470811?comment_id=10155226054835811&reply_comment_id=10155226905650811&comment_tracking=%7B%22tn%22%3A%22R7%22%7D

https://en.wikipedia.org/wiki/Monty_Hall_problem#Confusion_and_criticism

https://mathoverflow.net/questions/38639/thinking-and-explaining

https://www.lesswrong.com/posts/2TPph4EGZ6trEbtku/explainers-shoot-high-aim-low

Terry Tao and Tim Gowers also talk about similar things.

Not about math, but related is http://johnsalvatier.org/blog/2017/reality-has-a-surprising-amount-of-detail. In thought, there are things we didn’t know that we don’t know as well as we think we do. I think this is related to ideas like cached thoughts and the illusion of consciousness, where we end up taking shortcuts or don’t think about things carefully enough, so we have an illusion of understanding something.

https://jonahsinick.com/the-truth-about-mathematical-ability/