The social justice movement espouses the notion that people who are privileged are often unfairly judgmental of those who were less privileged. Until recently, what they said didn't resonate with me. I knew that I had major advantages out of virtue of having been born a white, middle class male. But I recently realized that there were other privileges that I hadn't acknowledged as having benefited enormously from. In particular, I had the unusual experience of growing up with a very intellectually curious father, which gave me a huge head start in intellectual development.
I used to get annoyed when LWers misread my posts in ways that they wouldn't have if they had been reading more carefully. I conceptualized such commenters as being undisciplined, and being unwilling to do the work necessary to maintain high epistemic standards. I now see that my reading was in many cases uncharitable, analogous to many of my teachers having misread my learning disability as reflecting laziness. Many of my readers have probably never had the opportunity to learn how to read really carefully.
How did I myself learn? I don't remember in detail, but the one factor that seems most significant is my study of the mathematical subject of real analysis. A number of strongest thinkers who I know characterized the experience as a turning point in their development as well. It's the subject where one goes through rigorous proofs of the theorems of calculus.
Consider the extreme value theorem:
If a real-valued function f is continuous in the closed and bounded interval [a,b], then f must attain a maximum and a minimum, each at least once.
The theorem may seem obvious, but almost no undergraduate math majors would be able to come up with a logically impeccable proof from scratch. This ties in with why I almost never try to present rigorous arguments. If it's not clear to you that it might be very difficult to construct a rigorous proof of the extreme value theorem, you'd probably benefit intellectually from more exposure to mathematical 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.
If you've studied calculus, haven't yet had the experience of proving theorems from first principles beyond high school geometry, and would are interested, I would recommend:
- Abbott's Understanding Analysis
- Rosenlicht's Introduction to Analysis (as a less expensive second choice)
- Gelbaum and Olmsted's Counterexamples in Analysis
The last of these books is great for developing a sense for how superficially plausible statements are often false.
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!
That makes a lot of sense. Thank you.
Added: Much of the praise for baby Rudin suggests that trying to prove each theorem before having seen the proof, is one of the best ways to become a good mathematician. Can you comment on my thought that, after having read a more verbose book, I won't have that same experience? Or is that approach going to work with most real analysis books, so that I could still try to prove everything before it is explained?