*You can't do things infinite times* If we imagine the addition as a real-world activity that takes time to do -- a second to write the new sum on a piece of paper, a fraction of a microsecond for a processor to encode it in memory -- then the Sun will go out before you get 2
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Ancient Greek philosophers knew about this shit This is, famously, Zeno's paradox Getting around this and saying "Well let's just pretend you CAN do it infinite times" is not *answering* Zeno, it's just telling him to shut the fuck up
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Specifically, saying Zeno was wrong, and I *can* just wave a magic wand and say "skip to the end of something I just said was by definition endless", is formally inventing the idea of a "limit" Which has made many people very angry and been widely regarded as a bad move
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The fact that being able to say "1+1/2+1/4+1/8...=2" is very useful, and all of calculus is based on it, doesn't actually mean we were *right* Whether "infinite converging series" are, like, a real thing that exists in the world of atoms is this big heavy question (probably not)
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There's a famously sexist quote apocryphally attributed to Shaw that I'll paraphrase as "If you wouldn't sell out for $10, but you will sell out for $10 million, then you are a sellout and you're just haggling over the price"
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When you agreed you could just shrug off the paradoxical-ness of doing anything an infinite number of times because pretending like you can is useful, you left the original rules of arithmetic behind Now we're just arguing over how weird we want to get
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This is, in fact, a *whole field of mathematics*, and the summation of infinite series can be done according to any number of different methods, which mathematicians invent at their pleasure, designed according to different criteria
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(I'm verging into stuff I only half-remember here, full disclosure) 1+1/2+1/4+1/8+... is one of the easiest series to just up and say has a solution, that it's 2 Because you're only adding, not subtracting, so you can move around all those numbers at will
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This is called "absolute convergence" If the numbers change sign, you have "conditional convergence" Paradoxically, for people used to doing things a finite number of times, it suddenly matters how the numbers are arranged
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Replying to @arthur_affect
This isn't quite right. A series like 1/2-1/4+1/8-1/16 is still absolutely convergent. A conditionally convergent series is one where the positive terms sum to infinity and the negative terms sum to negative infinity. Which, yeah, requires negative terms.
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Yeah, I'm not a real math major, I've just played one on TV
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Replying to @arthur_affect @eggynack
You know I should look up what you did before you slapped terfs on twitter on the regular.
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