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
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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