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#19
Nov2613, 09:29 PM

P: 139

Zeno's paradoxes seem like something that we only have secondhand accounts of, which are then repeated in distorted or incomplete versions in freshman calculus books.
My understanding is that the paradoxes, if we give the benefit of the doubt and call them that, are only paradoxes when taken together; Zeno was trying to argue that both infinitely divisible space and discrete space are impossible, which is why he came up with multiple scenarios. If Achilles can never catch the tortoise assuming infinitely divisible space, that's no paradox if space is actually discrete, is it? I never liked or could take seriously the "solutions" in those calculus books. They're just showing off and kind of missing the point, since if you're going to solve it using techniques the Greeks would clearly reject outright, you may as well make it easier on yourself, walk up to the wall and touch it, and say QED. Claiming that you need calculus to show that a faster runner will eventually overtake a slower one is just going to reinforce the common belief that mathematicians are crazy people who prove (column proofs of course...) obvious things. 


#20
Nov2613, 09:43 PM

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Positing that space is discrete would be unproven speculation. 


#21
Nov2613, 10:23 PM

P: 139

I'm not taking this too seriously and was partially joking with my physical proof, but then again, I don't think anyone can deny that touching a wall is a perfectly valid proof of the statement "I can touch a wall". I'd even say it's the proper proof :) *edit: I may have been thinking of the stadium paradox, but he may not have had discrete space in mind for that either. Also, I want to add that the calculus solutions seem to ultimately come down to accepting axioms about infininity that are at the very heart of the paradoxes. If Zeno was simply arguing that one couldn't reach a wall because it requires going halfway first, then we can just consider a wall twice as far away and go half that distance and we've reached the wall. He was arguing that motion itself was impossible, I guess because of some problems with "actual infinity" instead of "potential infinity". So even adding that first 1/2 of the infinite series is sort of begging the question, and if we somehow truly didn't know that motion was possible, I'd be torn between Zeno's arguments and the calculus ones. 


#22
Nov2613, 11:08 PM

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According to Simplicius, Diogenes the Cynic said nothing upon hearing Zeno's arguments, but stood up and walked, in order to demonstrate the falsity of Zeno's conclusions. To fully solve any of the paradoxes, however, one needs to show what is wrong with the argument, not just the conclusions. This is usually done by using convergent series and calculus, a tool at Zeno's time 2000 years in making. Zeno wasn't trying to prove motion or space or anything else was impossible but rather the paradoxes were meant IMO to show the limitations of math as of then. An interseting read: http://physics.bgsu.edu/~gcd/Spacetime3.html 


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