Tuesday, March 13, 2012

bottom-up and top-down thinking

I majored in math in college, which meant I spent a lot of time reading carefully through textbooks that started with very simple definitions and carefully built enormous beautiful structures out of those basic puzzle pieces, one step at a time. Then my task was to put some pieces together myself, by proving results on homework sets.

To prove something, you have to have a strong and thorough understanding of all of that foundational fodder that it's built on. Otherwise, there will always be black boxes in your understanding and in your proof.

If you're trying to understand something intuitively, though, black boxes are fine. If one black box seems to make sense, you can happily say you 'understand' that a higher level truth is true because the black box leads to it.

But if you've spent four years focusing on the agonizing details, it's really hard to switch to top-down thinking. For the longest time, every time I tried to look something math-related up on wikipedia, I would immediately get frustrated and discouraged, because nothing is ever presented in a bottom-up form. You get some discussion at the highest level, and have to click back on a dozen links to understand where it's coming from, and inevitably in those links you have to click on a dozen more each, and it never ends and it never meshes cleanly back together.

But as I've come to enjoy math as a tool and a hobby rather than as an end in itself, I've slowly gotten better at accepting black boxes. And slowly grown to love wikipedia.

And on that note, if you feel like learning something fun, go read about irrationality measures!

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