still don't think i understand the pythagorean theorem, all things considered
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I've been thinking about this for years
You can represent the hypotenuse as a series of steps, then just take the limit
All the vertical faces should add up the the vertical side, and same with the horizontal. So a+b=c
Surely someone else has come up with this 🤔
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the issue, in modern mathematical language, is a failure of continuity: in any topology in which that sequence of steps converges to the hypotenuse, the limit of the lengths is not the length of the limits, so length isn't continuous
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so if you try to write down a topology to make rigorous what it means for a sequence of curves to converge to a curve, the argument you've given is a proof that either
1) that sequence of steps won't converge to the hypotenuse, or
2) length won't be continuous
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well I'm banking on 2 bc 1 makes sense to me viscerally and thats all I want out of these infernal numbers
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