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
Gah! I hate this math. How do you ever get to 2? Because there's no fraction so small that half of it is zero. Are we saying there's some denominator so large that if it were x then 1/x = 0? Because that's the only way I can see this = 2. I am not suited to math like this.
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Replying to @TheCheekyGinger
Well, for many years we were trying to avoid saying that, and just define the idea of a "limit" without saying actual "infinite" and "infinitesimal" numbers exist But now there's non-standard analysis which says that those are okay
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Replying to @arthur_affect @TheCheekyGinger
But yeah ignoring that noise, the idea of the delta-epsilon definition of a limit is you don't ever need to "finish" the series (you can't, it takes infinitely long)
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Replying to @arthur_affect @TheCheekyGinger
You just need to ask the question "Every time I go one step further, is there a specific number I am getting closer to" And in this case yes, you can see that it's 2, so the answer to the question is 2
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Replying to @arthur_affect
I see that. But it feels wrong to say since it won't ever go over 2 it must *be* 2. Math isn't supposed to be horseshoes and hand grenades! This seems like saying "well ... close enough." Might as well give it a tolerance and call it machining.
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Replying to @TheCheekyGinger
Zeno of Elia would say if you can't accept this idea then if you think about it you can't map the real world onto numbers at all
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All of his paradoxes, when you strip them down, were just making this point - NOTHING in the real world is "exact", it CAN'T be
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Replying to @TheCheekyGinger
Yeah it's like Zeno's Paradox with the 1+1/2+1/4 stuff is just this complicated, mathy way of saying "How do you measure something like exactly one cup of water in real life You fill it up until it gets to the line and you stop? *Exactly when* do you stop"
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