So people's intuitions about this are correct but they don't take it far enough Caught between the dueling intuitions between "Well the answer to this question can't be any ordinary number" and "It must HAVE an answer though"
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All new forms of math are built on that second part "Okay, I get that this is a stupid-ass question and Pythagoras or whoever would've just said 'shut the fuck up' if I asked him but WHAT IF you COULD add up numbers an infinite number of times"
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All summations of infinite series are based on making up new rules and saying "Okay let's pretend you can do this, what happens if you do, what new stuff do you discover if we just fuck around and act like this makes sense"
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So like, let's be clear This classic series: 1+1/2+1/4+1/8+1/16... Is, from a pure old-school POV, just as bad as the other one Even though this one looks like it has an answer (it adds up to 2 in the end)
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Replying to @arthur_affect
As a math grad student I think I get your point but I wouldn’t really put it that way. To the extent that an infinite sum has any precise meaning, it’s as a limit of partial sums, and by that definition makes perfect sense to say it equals 2
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Replying to @awildbread @arthur_affect
If you want to say that you can’t really add up an infinite number of things then sure. But when I see that infinite sum, what it means to me IS a limit, the precise thing that is equal to 2, not some more abstract idea of adding to infinity that can’t be described with math
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Replying to @awildbread @arthur_affect
There’s more of the thread now so I get your point more but I still don’t really agree with the framing of whether infinite series really ‘exist’. Why does the number 2 exist any more than the infinite series on the other side?
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Replying to @awildbread
*inhales* Have you ever even... like... SEEN a number, man
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Replying to @arthur_affect @awildbread
(As a humanities guy and not really a math guy, I will say that what I'm talking about here is Spengler's Decline of the West and his whole weird argument about whether or not classical civilization had a concept of "the Infinite")
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Replying to @arthur_affect @awildbread
(And placed very great importance on the idea that to the Greeks the idea of number itself was defined by ratio -- so much so that "irrational numbers" were a Lovecraftian horror)
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(While Christian Europe was obsessed with infinite space filled with infinite points -- the angels dancing on the head of a pin -- and could easily picture the reals along a "number line", their religion and their culture primed them for the concept of analysis)
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Replying to @arthur_affect @awildbread
So like -- here's this very stodgy early 20th-c German guy, a very old-fashioned historian, who nonetheless gets at the heart of all this PoMo woke SJW shit about math being culturally bound and contingent Everyone should read the book just for that
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Replying to @arthur_affect @awildbread
I mean as an old school essentialist who believed in "national spirits" and that sort of thing he was an antisemite in practice ("Jews don't do real analysis, they do symbolic logic") so, you know, grain of salt, although he did despise Hitler and the Nazis
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