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gjm comments on The value of learning mathematical proof - Less Wrong
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Sorry to harp on it again, but to enjoy real analysis one does require a fair amount of math aptitude, not just being white middle class male with a very intellectually curious parent. I had all those, and a good math instructor in the 2nd year, and a good TA, and a group of friends I would explain the stuff I learned to, and I got a high mark in the course, but I never really enjoyed it the way I enjoyed physics and some computer courses. I could do rigorous proofs when required, I just never had a thing for it. I would get a kick of figuring out a fancy integral, but not out of figuring out a fancy proof.
I agree that it is a humbling experience to learn "how difficult it can be to offer rigorous proofs of even relatively simple statements," and I felt suitably humbled, and it has value for the armchair AI researchers frequenting this site, but if your goal is to teach humility, I suspect there are better ways.
What the books you are suggesting are good for is to find people who think they are bad at math, but aren't. I have seen an occasional case of a person being taken in by the beauty of mathematical proofs.
That seems too heavy, You ought to learn that in your first programming course, where a program that looks perfectly correct inevitably contains multiple bugs.
I don't think humility is what Jonah is trying to teach -- rather, it's something more like the habit of working really hard at understanding things. (Though I worry that there's a roughly opposite error: thinking that skill in other domains is like skill in pure mathematics and requires the same kind of intellectual work. The same amount, maybe -- though actually I suspect it varies -- but not necessarily the same kind.)
I was addressing the specific skill of reading carefully and not making assumptions that the author hasn't stated, which is highly relevant to learning in general. I agree that the work that goes into understanding things outside of pure math isn't necessarily of the same type as within pure math.