On memorizing versus understanding
I’ve been wondering a lot lately what is the difference between memorizing something versus understanding something. In this post I ask a lot of questions to which I don’t know the answers.
Say someone comes up to you today with a mathematical proof written on a piece of paper and tells you that tomorrow he’ll show up again and you’d have to recite that proof to him straight from your head. You have two choices: (1) you memorize everything (“word-to-word”) that is written on that piece of paper, (2) you understand that proof and you recite it by “figuring it out as you go” (with some small memorization used to remember how things are denoted or called).
Certainly we’ve all experienced the feeling of having understood something. It’s that “click” or “aha” moment in your brain associated with a pleasant sensation, perhaps even a sensation of relief. And perhaps some people would call the first option of memorizing word-to-word as the “dumb” way, especially when it comes to learning mathematics. But what is that feeling in your head after you’ve understood something? And how is that neurologically different from having memorized something?
I first started to ponder about this when I was trying to understand the proof to the following statement:
The set A of all sets that don’t contain themselves as their members cannot exist.
A way to prove it, is to show that we arrive at a contradiction in two excluding cases: assuming that A contains itself as its member and assuming that A does not contain itself as its member. Here’s the full proof:
That’s a reasoning that requires certain processing in your head before you “get convinced” that it is reasonable. It took me some time at the whiteboard before I had the “aha” moment when the whole reasoning started to make sense to me. Today I can recite it from my head, but I don’t feel that I am reciting it from memory. Rather, I feel that I’m inventing it on the spot from some underlying scheme that my brain generated during the “aha” moment. If I memorized everything without understanding it, to the outside world there would be no difference. There is only a difference in your brain that only you know about.
Is understanding a different form of memory that is more compressed? Is the pleasant feeling associated with understanding something our brain’s way of rewarding us for compressing information?