It doesn't map onto the real world in a familiar way very well, at all But it wasn't meant to Counting things up infinitely isn't something you can do in the real world AT ALL either, remember?
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And yet, bizarrely, this result is meaningful in an applied context It comes up in quantum physics, where the real physical world gets extremely weird and we need to use weird math to talk about it
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It's part of the mathematical description of the Casimir effect, which has been experimentally verifiedhttps://en.wikipedia.org/wiki/Casimir_effect …
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The extremely broad metaphor the guy gives us for trying to imagine this is -- the actual positive numbers that we add up in 1+2+3+4... can't give us a real sum The number keeps getting bigger, it leads us only toward ∞, and ∞ isn't a number, so the sum just doesn't exist
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Analytic continuation is about saying "Okay, so if you take out everything that you're not allowed to do that absolutely by the rules cannot give you an answer, can you find something about it that *does* give you an answer"
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Paradoxical Zen koan shit Like Death in Discworld saying that the universe by definition is everything that exists and everywhere is inside it so you can't be outside it and standing outside it is a contradiction in terms But if you could, then from the outside it'd be blue
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-1/12 is a "residue", it's a "leftover piece" of 1+2+3+4..., the one bit that can be made to act like a number
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And that, in its way, is connected to the paradoxical thing the Casimir effect is, this thing where in a total vacuum where nothing should be exerting any force on anything, two metal plates very close together will attract each other
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I.e. -- getting really sloppy and vague from my layman's POV again, in real life in quantum physics a "vacuum" is actually filled with "virtual particles", which kinda almost do exist but then don't
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It is obviously invoked constantly in silly and meaningless ways on the Internet but it is also obviously a real thing that works, as evidenced by the fact that we have an Internet
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Replying to @arthur_affect
Is this an "all models are wrong, some are useful" moment?
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Replying to @damonkord
No it's just saying that the fact that quantum physics is really weird obviously means that some people like to go "Well if it's weird then there's NO RULES AT ALL and anything can be true like the fact that if you pay me $500 I can cure your cancer via energy healing over Skype"
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