

Humbling index: Where do you stand? - jyothi
http://indexed.blogspot.com/2008/08/humbling.html

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brentr
I think the point is: the more you learn, the more you realize how much you
don't know.

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.

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raju
Reminds me of the unconsciously-incompetent, consciously-incompetent,
consciously-competent and finally the unconsciously-competent concepts...

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time_management
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.

~~~
jtj
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.

