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what does arnuald's paradox mean? does it just mean (-1)=1? or backwards? does it mean anything else?
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papernuke
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Can not really see a paradox in 1/-1=-1/1.
bo198214
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I think what it means is that the standard ordering for real numbers is not appropriate for making statements about ratios.

If, instead, you consider magnitudes, then 1 and -1 are the same size. So, it's unsurprising that swapping them doesn't change the ratio.
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I would rather say the standard intuition about (positive) quantities can not be transfered to the real numbers including negative numbers.
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To reach a destination, you must first reach the halfway point. To reach the halfway point, you must first reach the quarter-way point. This continues forever until there is an infinite number of points to reach before reaching a destination, therefore motion is an illusion.
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Nick
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pat
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What exactly is a paradox? and how can motion be an illusion if when you walk, you go forewards (or any direction)? it still happens.

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papernuke
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The paradox of having to walk infinite distance to go from point A to point B is, in itself, counterintuitive because we have to deal with an infinite number of points to cross but measure a finite time to get there (since it doesn't take us infinitely long to walk from A to B). The paradox arises because, back in the day of the paradox' conception, infinite sums and series were not properly understood so naively it led to an intuitive result that it should take an infinite time to reach B from A because there are infinitely many points between them and each point necessarily takes more than 0 instants of time to reach. It's paradoxical because intuitively (read: wrongly) it should take an infinity of time to get from A to B but the reality of it is we can walk from A to B in a finite time in the real world...so they disagree with each other...so it's a paradox.
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houserichichi wrote:The paradox arises because, back in the day of the paradox' conception, infinite sums and series were not properly understood

But thats not really an excuse, even the old greeks could consider the partial sums of 1/2^i and could see that they always are lower than 1, so it surely wouldnt lead to an infinite time.
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I wasn't aware the ancient Greeks had a working knowledge of limits or at least an intuitive concept of them. The fact that the infinite sum doesn't diverge to infinity wasn't provable until calculus, no? Perhaps it's the lack of mathematical proof that would tend it toward a paradox?
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nono probably they hadnt concept of a limit.
but they could add and compare, that suffice to deparadox the topic.
bo198214
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But is that not the paradox in and of itself? You add a half and then a half of that and then a half of that half, ad infinitum. You're taking a bit, adding a little more, then a little more, then a little more infinitely many times. It is by no means obvious (without the use of limit) that the sum converges to anything... But I do think I know what you're trying to say

The paradox, when I first read it, to me was that to go from A to B one has to pass through a continuum of (infinitely many) possible "points" between the two in a finite amount of time. More succinctly, to get from A to B one has to cross infinitely many places beforehand...and that's counterintuitive.
houserichichi
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houserichichi wrote:The paradox, when I first read it, to me was that to go from A to B one has to pass through a continuum of (infinitely many) possible "points" between the two in a finite amount of time. More succinctly, to get from A to B one has to cross infinitely many places beforehand...and that's counterintuitive.

The first time I read it, it was not a paradox for me, and I didnt thought about converging sums etc at all. The paradox shall be that Achilles or whoever never can pass the tortoise.
But this "never" means it takes infinitely many time.
And this is not the case as you can see already with elementary means (partial sums and estimates) that the time is limited.
Whether you can compute (by using limits and convergence) where and when exactly Achilles passes the tortoise does not matter so much for the paradox.

Hm what regards though infinitely many places ... already the greeks computed with ratios and the distance between A and B is full of infinitely many ratios... (or rational numbers or fractions, whatever you like).
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