38. Math Isn’t About Working Hard, It’s About Being Cleverly Lazy

(Epistemic status: a time-worn explanation I used to give my undergrads, fancied up a little.)

Math isn't about working hard to get the right answer. It's about being cleverly lazy. Let me explain.

For most of your life, you've probably been afraid of math, or burned out by it, or intimidated. There's a single right answer, and everything you need to do to get that answer is hard and weird and abstract. Surely as you keep going in math, as you learn harder math and are asked to do more difficult things, you'll also need to keep putting in more and more effort? Surely you need to work yourself to the bone doing it "the right way" just to keep up?

No! Absolutely not! If you try to pull that garbage you will end up doing a literally infinite amount of work! For a good example of this, consider something like the sum of negative powers of 2: 1, 1/2, 1/4, 1/8, and so on forever. You could spend a lifetime adding those up by hand and not get anywhere, and if you got anywhere with a computer instead, it'd be because of floating point errors, not because you finished. But if instead you spend a little time proving the telescoping series formula, then what would have taken you actual literal forever instead takes you seconds.

This, by the way, is why we like proofs so much in mathematics - it's not just because we get to know things for sure, but because every proof is a new tool we construct for ourselves in order to use it later. The trick is always to figure out the clever way to do as little total work as possible while still getting the right answer - to spend a bit of effort and a little bit of frame-shifting and tool-building, and leverage that into what's often literally infinite speedups. Math is about being cleverly lazy to get the right answer anyway, not working hard and grinding away.

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