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.
The experience of seeing how difficult it can be to offer rigorous proofs of even relatively simple statements trains one to read very carefully, and not make any unwarranted assumptions.
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.
The last of these books is great for developing a sense for how superficially plausible statements are often false.
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.
Sorry to harp on it again, but to enjoy real analysis one does require a fair amount of math aptitude,
Is there a reason why you keep bringing up this subject? I’m not complaining – I just want to know whether there’s a point that you’ve been trying to make that I’ve been missing.
In my present post I was advocating learning real analysis for the sake of getting into the habit of reasoning carefully, not for enjoyment.I think that a large fraction of LWers have the mathematical aptitude required to find it tolerable, even if not exciting. I usually don’t advocate people learning things that they don’t find especially interesting, but in this particular case, the skill is so important that I think it might be worth it – I see it as analogous to literacy.
Naturally it depends on the other alternatives on the table, and how far one wants to go. But I do know several people who report that learning the subject changed how they think in general, not only in the context of math.
I would have to add another point on the anecdotal side for this. I made it through Real Analysis (barely!) when I was a math major—and it made a significant difference on the thought process I go through when I consider things. If nothing else, it was very instrumental in breaking the “good rhetoric = good argument” connection I’d been operating under up until that point. And this was long before I’d any notion that places like CFAR or LW even existed.
(I will disclaimer that it also made certain kinds of communication more difficult—because most folks don’t like it when you try to make their opinions rigorous—but that could as easily be from how I implemented those ideas as from the change in thinking itself. )
but if your goal is to teach humility, I suspect there are better ways
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.)
(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.
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.
Is there a reason why you keep bringing up this subject? I’m not complaining – I just want to know whether there’s a point that you’ve been trying to make that I’ve been missing.
In my present post I was advocating learning real analysis for the sake of getting into the habit of reasoning carefully, not for enjoyment.I think that a large fraction of LWers have the mathematical aptitude required to find it tolerable, even if not exciting. I usually don’t advocate people learning things that they don’t find especially interesting, but in this particular case, the skill is so important that I think it might be worth it – I see it as analogous to literacy.
I am not sure the cost-benefit analysis is favorable, in particular for people who do not intend to become professional mathematicians.
Naturally it depends on the other alternatives on the table, and how far one wants to go. But I do know several people who report that learning the subject changed how they think in general, not only in the context of math.
I would have to add another point on the anecdotal side for this. I made it through Real Analysis (barely!) when I was a math major—and it made a significant difference on the thought process I go through when I consider things. If nothing else, it was very instrumental in breaking the “good rhetoric = good argument” connection I’d been operating under up until that point. And this was long before I’d any notion that places like CFAR or LW even existed.
(I will disclaimer that it also made certain kinds of communication more difficult—because most folks don’t like it when you try to make their opinions rigorous—but that could as easily be from how I implemented those ideas as from the change in thinking itself. )
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.