Trying to build a summation method this way for a diverging series 1+2+3+4... does not work, the way 1+2+3+4... diverges makes it neither "linear" nor "stable" I.e. if I stick an extra zero in there ("0+1+2+3+4..."), I can use his method to prove 1=0
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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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The vacuum energy is adding up all the particles that *could* be interacting with those metal plates, an *infinite number* of ways those plates could be pushing against each other and exerting force via an electromagnetic field
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But infinite force can't exist, doesn't mean anything, and what this actually adds up to instead is the "residue", the -1/12, i.e. a very small negative number -- a force that goes the opposite direction and attracts things together, but only at a very small distance
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Yes, actual physicists, I don't know what I'm talking about The point here is just me dreaming of one of the many things in heaven and earth I hadn't previously had in my philosophy
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The knowledge that a summation of 1+2+3+4...=-1/12 exists doesn't affect my life -- if I am ever asked to add up all of the natural numbers I will probably stop at 1+2+3=6, honestly -- nor is the proof that they gave the real, final proof
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But damn if it isn't a beautiful, powerful demonstration that you can fuck around with seemingly simple things and open a Pandora's Box of wonders
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Like the motto of Stephen Fry's show QI, "If you can't be right, be interesting" At every step along this process someone could have, like Dostoyevsky's coxcomb, blocked the path and waved their arms and said "No you can't do that turn back"
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(Hell, the ancient Greeks, who believed you could not in fact do that, were still pretty lax traffic cops and didn't *enforce* not doing that because they didn't have the rigor, which is why Archimedes was allowed to discover a limit-based method for approximating pi)
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But it's always more interesting to just ignore him and keep walking anyway I mean what's the worst that can happen? You try to say something makes sense but everyone just looks at it a while and goes "Nah this is stupid" and moves on? You gotta at least try
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End of conversation
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