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I'm reading Wittgenstein's On Certainty, and he mentioned that no one feels surprise when mathematics proves itself consistent. Except I do.
Basically every time I do some mental arithmetic, I do the problem multiple times, coming at it from different directions. For example, maybe I need to find half of 47. I'd immediately take half of 46 and add 0.5, getting 23.5. But then I'd also take half of 50 and subtract 1.5, also getting 23.5.
(Sidenote: I know not everyone does this, as demonstrated by how outraged people are over Common Core. It just makes plain good sense to me. Mathematics shouldn't be the blind application of fixed algorithms -- you need to choose the approach that works best for you. And to do that, you need to see the different options and really understand how they're all the same thing, fundamentally. But most people don't really understand that. They can only solve problems in a single way they memorized 30 years ago. Then they feel dumb when their kid asks for help with their homework, and lash out.)
In part I do this to provide to a checksum on the original answer, but also because I always feel a small thrill of surprise and delight. Math is internally consistent, and every problem has an infinite number of ways you can solve it. It's just so neat -- and also staggeringly impressive. Imagine writing an operating system with no bugs. Imagine being able to design a legal system without any need for judges, because there was a single, obvious, undeniable verdict for any case. Imagine a taxonomy with no edge cases, no "miscellaneous" categories.
Math is quite literally inhuman in its perfection.
Take that, Wittgenstein.
Basically every time I do some mental arithmetic, I do the problem multiple times, coming at it from different directions. For example, maybe I need to find half of 47. I'd immediately take half of 46 and add 0.5, getting 23.5. But then I'd also take half of 50 and subtract 1.5, also getting 23.5.
(Sidenote: I know not everyone does this, as demonstrated by how outraged people are over Common Core. It just makes plain good sense to me. Mathematics shouldn't be the blind application of fixed algorithms -- you need to choose the approach that works best for you. And to do that, you need to see the different options and really understand how they're all the same thing, fundamentally. But most people don't really understand that. They can only solve problems in a single way they memorized 30 years ago. Then they feel dumb when their kid asks for help with their homework, and lash out.)
In part I do this to provide to a checksum on the original answer, but also because I always feel a small thrill of surprise and delight. Math is internally consistent, and every problem has an infinite number of ways you can solve it. It's just so neat -- and also staggeringly impressive. Imagine writing an operating system with no bugs. Imagine being able to design a legal system without any need for judges, because there was a single, obvious, undeniable verdict for any case. Imagine a taxonomy with no edge cases, no "miscellaneous" categories.
Math is quite literally inhuman in its perfection.
Take that, Wittgenstein.