The value of learning mathematical proof

by JonahS1 min read2nd Jun 201542 comments


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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:

The last of these books is great for developing a sense for how superficially plausible statements are often false. 

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