Humbling index: Where do you stand?(indexed.blogspot.com)
indexed.blogspot.com
Humbling index: Where do you stand?
http://indexed.blogspot.com/2008/08/humbling.html
3 comments
Reminds me of the unconsciously-incompetent, consciously-incompetent, consciously-competent and finally the unconsciously-competent concepts...
These graphs are somewhat misleading. As education (A) increases, knowledge (B) and "things you know you don't know" (C) increase on approximately the same curve. I'd say that A:C grows slowly at first, then gets steep, while A:B is roughly linear.
There is one other comment with same opinion on the site itself.
But the idea is not to get into such gory details. Its a fun thing. They essentially get the highly insightful statement in a very creative way. :)
I was definitely impressed to be reminded of Humility and what makes the best people this way.
But the idea is not to get into such gory details. Its a fun thing. They essentially get the highly insightful statement in a very creative way. :)
I was definitely impressed to be reminded of Humility and what makes the best people this way.
I present an example from personal experience. It's not so much that I didn't know the concept being discussed, but rather the example I will present was used to kick off my real analysis class three days ago.
Consider a number, B (B for bad). Let B be the limit of the alternating harmonic series as the number of terms approaches infinity. The professor showed by merely switching the location of parentheses and rearranging terms that this same series can be shown to sum to 1.5B. Naturally, the number B, the limit, approaches ln 2. But by rearranging the terms, he showed otherwise. The point of the lecture was to show that the human mind, unless trained in the concept of limits at infinity, falls short. Now I know the reason why he was able to prove this was due to a result of Riemann, specifically that conditionally convergent series can be made to sum to anything you want by rearranging the terms in a sufficient manner (see Rudin's Principles of Mathematical Analysis for a proof), much of the class was stumped as to why this occurred. Just when you think you have mastered something as beautiful as the calculus, when you start to dig deeper into the theoretical underpinnings of it, you realize that there is so much more to learn.