The value of learning mathematical proof

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.

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.)

Personally I think real analysis is an awkward way to learn mathematical proofs, and I agree discrete mathematics or elementary number theory is much better. I recommend picking up an Olympiad book for younger kids, like "Mathematical Circles, A Russian Experience."

This is also somewhat in reply to your elaboration in this comment. Just some data points:

In regards to this topic of proof, and more generally to the topic of formal science, I have found logic a very useful subject. For one, you can leverage your verbal reasoning ability, and begin by conceiving of it as a symbolization of natural language, which I find for myself and many others is far more convenient than, say, a formal science that requires more spatial reasoning or abstract pattern recognition. Later, the point that formal languages are languages in their own right is driven home, and you can do away with this conceptual bridge.

Logic also has helped me to conceive of formal problems as a continuum of difficulty of proof, rather than proofs and non-proofs. That is, when you read a math textbook, sometimes you are instructed to Solve, sometimes to Evaluate, sometimes to Graph; and then there is the dreaded Show That X or Prove That X! In a logic textbook, almost all exercises require a proof of validity, and you move up over time, deriving new inference rules from old, and moving onto metalogical theorems. Later returning to books about mathematical proof, I found things much less intimidating. I found that proof is not a realm forbidden to those lacking an innate ability to prove; you must work your way upwards as in all things.

Furthermore, in regards to this:

Even the math with simple foundations has surprising results with complicated proofs that require precise understanding.

In my opinion, very significant and complex results in logic are arrived at quite early in comparison to the significance of, and effort invested in, results in other fields of formal science.

And in regards to this:

I think this makes it all the more educational when a surprising result is proven, because there is less room for a beginner to wonder whether the result is an artifact of the funny formalish stuff.

I have found that in continuous mathematics I have walked away from proofs with a feeling best expressed as, "If you say so," as opposed to discrete mathematics and logic, where it's more like, "Why, of course!"

I agree with Epictetus' comment.

To my mind all of these things are better suited for learning the power of proof and the mathematical way of analyzing problems.

I think the main thrust of the article was less about the power of mathematics and more about the the habits of close reading and careful attention to detail required to do rigorous mathematics.

I'm not totally sure why, but I think a big part of it is that analysis has a pretty complicated technical foundation that already implicitly uses topology and/or logic (to define limits and stuff), even though you can sort of squint and usually kind of get away with using your intuitive notion of the continuum.

Seems like it's precisely because of the complicated technical foundation that real analysis was recommended. Theorems have to be read carefully, as even simple ones often have lots of hypotheses. Proofs have to be worked through carefully to make sure that no implicit assumptions are being introduced. Even great mathematicians ran into trouble playing fast and loose with the real numbers. It took them about two hundred years to finally lay rigorous foundations for calculus.

There's no single reason why my posts were misunderstood. You can think in terms of there being an underlying statistical model of whether or not a post will be understood. I acknowledge that my lack of communication skills have played a major role. But it's also true that people with very strong quantitative backgrounds (such as Paul Christiano) have understood my posts much more readily than most LWers have.

I think that this emphasis on explicit, built-from-scratch mathematical proofs runs counter to your previously expressed suggestion that learning via pattern matching is more efficient than learning via explicit reasoning.

My focus here was two-fold

1. Learning mathematical proof as a means of learning how to read and listen very carefully.
2. Learning the limits of rigorous reasoning by seeing how hard it is to give arguments that are actually rigorous, as opposed to just having the superficial appearance of rigor.

The second point is a part of my case for the first post that you linked.

I wouldn't recommend it for someone's first exposure to analysis. When you first meet a subject, you want to get a sense of how the bits fit together, and what the important concepts and theorems are supposed to "mean" (as opposed to their formal definitions). You learn this by slowly working through examples and thinking about special cases.

Unfortunately, Rudin has very few examples, his proofs are more elegant than enlightening (for the beginner anyway, his proofs are very enlightening if you already know the big picture and are want to know the answers to questions like "Do I really need this strong an assumption for this theorem?"), and develops his theories in a lot more generality than a typical introductory analysis course (Which again, isn't necessarily bad, but you do want to get a feel for how things work in R^n before diving into arbitrary metric spaces).

If you have three months, you might want to spend the first half or so on a more verbose book, and then go over the material again using Rudin. You'd get a deeper understanding, and it might even be faster than just going through Rudin once!

As a current Harvard math grad student I think you should read many different easy books to learn a subject whenever possible, especially if you can find them for free. When you say you are mathematically able it is unclear what level you are at. All of my favorite books for learning involve huge number of exercises, and I recommend you do all of them instead of reading ahead.

For basic real analysis, my favorite book is Rosenlicht's Introduction to Analysis but baby Rudin is pretty good too, and I recommend you flip back and forth between them both.

For learning math in general, I think real analysis is a poor place to start, but that may be personal preference because I have a more algebraic slant. I highly recommend books like Herstein's Abstract Algebra, Mathematical Circles: A Russian Experience, I.M. Gelfand's Trigonometry, and Robert Ash's Abstract Algebra: The Basic Graduate Year, mostly for the wealth of exercises. Some of these are books for small children and I think those are the best sort of books to first learn from.

I agree with SolveIt.

This should be it's own post, please do one! You're good enough and I'll love you for it!

Thanks. Which aspect of what I wrote jumped out at you as especially worthy of highlighting?

I think that complex analysis is much more intrinsically interesting (leading to Riemann surfaces, quasi conformal makings, etc,) but because the class of functions is so restricted, one is saved a lot of the nitty-gritty type work that one has to do in real analysis.