You cannot do anything an infinite number of times The answer to "1+2+3+4+5+... = x" is that x can't be anything because you will never actually finish adding up numbers and you will never get an answer "Infinite" is another way of saying "nonexistent"
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You can't say x="infinity" or ∞ because in ordinary arithmetic that's not a number, it's a meaningless word
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The *rules of arithmetic* say this -- numbers are *defined* by the fact that if you add 1 or subtract 1 from a number, you get a different number, if "∞" breaks this rule, it's not a number at all and you can't use it
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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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*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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