Formalizing the method Ramanujan used here -- explicitly stating what rules actually define the things Ramanujan was doing so you can't just do anything and get 1=0 -- created the method known as "Ramanujan summation"
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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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Really cool exposition. Analytic continuation isn't magic. It comes from first extending functions on the real line to the complex plane. The nice ones with power series are "holomorphic" and have lots of nice properties as a result
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