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So I found this video, in witch it was said and explained that all movement should be impossible ...

... if no circle can be a perfect circle and the roundness (for the lack of a better word) is limited, then the constant pi (3.14159...) does not have to be irrational and it can be finite..

In the original version of Zeno's paradox (pre-quantum theory), there was a flaw in the logic: It is possible to add up the infinite series of ^{1}/_{2}+^{1}/_{4}+^{1}/_{8}+....; the answer is finite and equals 1. This matches our experience that an arrow can reach the target.

Calculus didn't solve the paradox at all - it simply provided a way of calculating the exact answer which no amount of further chopping can take things beyond - it provides no mechanism for making infinity times zero equal to one.

Quote from: David Cooper on 08/08/2012 00:09:02Calculus didn't solve the paradox at all - it simply provided a way of calculating the exact answer which no amount of further chopping can take things beyond - it provides no mechanism for making infinity times zero equal to one.David, you're mistaking infinity for a number. You can't multiply by infinity--it isn't a real number. You can't add up an infinite number of things--infinity isn't a number. What you can do is to allow the number of steps to increase without limit, and the step size to decrease proportionally without limit. This is how calculus resolved the paradox.

Calculus tells you why trying to shoehorn "infinity" into the problem as if it's a number as you're doing is wrong.

which in the case of Zeno's arrow paradox we can,

Zeno was fully aware that you could increase the number of steps without limit and that the step size would decrease proportionally without limit, but he was also able to see that it did not resolve his paradox. You either have to stop at some point and accept that the universe is granular or you have to take it all the way to the point where an infinity is introduced, at which point the maths breaks down. Zeno's paradox tells us that the universe is granular.

Quote from: JP on 09/08/2012 01:17:05Calculus tells you why trying to shoehorn "infinity" into the problem as if it's a number as you're doing is wrong.Calculus does nothing of the kind. Zeno clearly understood this issue better than the calculus priests. Think about it carefully for a moment. How many numbers are there in the series 1, 2, 3, 4, 5, etc. if you count them all up. You get infinity: that's just well-established mathematics, so there's no dispute about that. What happens if you use a different series like 1, 2, 4, 8, 16, etc.? You get infinity again (and although it may be a different kind of infinity, it's still an infinity). Now do the same with the series 1/1, 1/2, 1/4, 1/8, etc. - how many of them are there? Infinity again. Dispute that if you like, but you're moving away from established mathematics if you try. So, what is it that makes you imagine that the idea of calculus can override that?

Quote from: evan_au on 04/08/2012 11:56:57In the original version of Zeno's paradox (pre-quantum theory), there was a flaw in the logic: It is possible to add up the infinite series of ^{1}/_{2}+^{1}/_{4}+^{1}/_{8}+....; the answer is finite and equals 1. This matches our experience that an arrow can reach the target.Put another way, let's say it takes 1 second to traverse the distance. If you chop it in half, it would take 1/2 second to traverse each half, which adds up to 1 second total. If you chop it in half again, you'd end up with 1/4 second to traverse each of 4 segments, which would add up to 1 second total. No matter how you chop it up, 8ths, 16ths, 2^{n}ths, it still takes 1 second because you're adding together many very small times, which can still add up to 1. Zeno didn't know the proper way to deal with this mathematically when you allow it to be chopped up endlessly, but modern calculus introduced the idea of a limit, which shows that no matter how small you chop these segments, the total time taken is still 1 second.The question of what happens in reality on the quantum scale is another thing altogether, but the original paradox was resolved by calculus.

Quote from: David Cooper on 08/08/2012 20:53:22Zeno was fully aware that you could increase the number of steps without limit and that the step size would decrease proportionally without limit, but he was also able to see that it did not resolve his paradox. You either have to stop at some point and accept that the universe is granular or you have to take it all the way to the point where an infinity is introduced, at which point the maths breaks down. Zeno's paradox tells us that the universe is granular.Yes, and the time interval also decreases proportionally without limit. You will end up with an arbitrarily large number divided by the same arbitrarily large number, which, if I'm not mistaken, is equal to one. Doesn't strike me as strong evidence that the Universe is granular.

Those sets are the same kind of infinity in size. They're countably infinite. That also has nothing to do with what we're discussing.

It's completely different from saying that infinity is a real number that you can add or multiply by.

Look, you can argue that you're not making a mistake by treating infinity this way all you want, and throw around pejoratives like "calculus priests," but the fact is that you're wrong.

You argued above that if you can multiply infinity by zero which is wrong--that expression doesn't even make sense in math since infinity can't be multiplied: it's not a number.

Then you argue that adding infinite numbers of things together leads to an infinite result. That's also wrong, and you can prove that various infinite series converge using calculus/analysis.

Quote from: JP on 06/08/2012 15:18:59Quote from: evan_au on 04/08/2012 11:56:57In the original version of Zeno's paradox (pre-quantum theory), there was a flaw in the logic: It is possible to add up the infinite series of ^{1}/_{2}+^{1}/_{4}+^{1}/_{8}+....; the answer is finite and equals 1. This matches our experience that an arrow can reach the target.Put another way, let's say it takes 1 second to traverse the distance. If you chop it in half, it would take 1/2 second to traverse each half, which adds up to 1 second total. If you chop it in half again, you'd end up with 1/4 second to traverse each of 4 segments, which would add up to 1 second total. No matter how you chop it up, 8ths, 16ths, 2^{n}ths, it still takes 1 second because you're adding together many very small times, which can still add up to 1. Zeno didn't know the proper way to deal with this mathematically when you allow it to be chopped up endlessly, but modern calculus introduced the idea of a limit, which shows that no matter how small you chop these segments, the total time taken is still 1 second.The question of what happens in reality on the quantum scale is another thing altogether, but the original paradox was resolved by calculus.I think I see how you mean there JP, that it mathematically can't be a 'infinity', as adding it up again always will give you the same 'finite' answer. Or am I reading you wrong here? On the other hand? What stops me from splitting that second infinitely? One can argue that this second must be the finite answer to all splitting but there is still a 'infinity' represented in the 'instant' it takes, as it seems to me?

The problem is that you have to treat the idea of infinity in a mathematically correct way. It isn't a number like 1 or 10 or Pi or square root of 2. It's a concept, which in Zeno's paradox means that you divide the interval into halves without limit. But at the same time, you chop the times in half without limit. Treating this as a limiting process you can prove that the time is finite.

David, I'm not sure how to respond to you. Similar to your "Einstein was wrong" thread, your main arguments against the mathematical resolution of the paradox are insults aimed at mathematicians and error-filled proofs of your ideas.

I've pointed out where you're going wrong in your proofs and you respond with more insults at the "high priests of mathematics."

If you don't want to discuss mainstream mathematics or science, why are you posting on a mainstream science Q&A forum?

JP knows his math David I'm pretty sure on that one, doesn't mean that one can't have a different opinion though, just as I'm trying to see where that limit is thought as. To me the reason could be that a second is a limit of sorts whereas a infinity? On the other hand "It's a concept, which in Zeno's paradox means that you divide the interval into halves without limit. But at the same time, you chop the times in half without limit. Treating this as a limiting process you can prove that the time is finite."So i split a second in 2, then 2 again, then 2 again. then 2 ... ad infinitum, but how and where do I reach that limit? What in calculus demand that I can't just keep doing so for ever JP? There has to be some hidden logic to this reasoning that I'm missing.

Quote from: yor_on on 09/08/2012 22:52:11JP knows his math David I'm pretty sure on that one, doesn't mean that one can't have a different opinion though, just as I'm trying to see where that limit is thought as. To me the reason could be that a second is a limit of sorts whereas a infinity? On the other hand "It's a concept, which in Zeno's paradox means that you divide the interval into halves without limit. But at the same time, you chop the times in half without limit. Treating this as a limiting process you can prove that the time is finite."So i split a second in 2, then 2 again, then 2 again. then 2 ... ad infinitum, but how and where do I reach that limit? What in calculus demand that I can't just keep doing so for ever JP? There has to be some hidden logic to this reasoning that I'm missing.In calculus, you can. That's why I was careful to say above that I'm not talking about quantum effects. In classical physics and calculus, there is no issue in chopping up time and distance into as small a chunk that you want. There is a problem with saying you have an infinitely small segment unless you're precise about what infinity means. Infinity is not a number like 1, sqrt(-1) or Pi. Infinity has to be used with care as a concept for very large things. It does have meaning, but since it's not a number, it doesn't make sense if you try to add, divide or multiply by it. You can (and physicsts often do) use it similarly to a number to mean things can increase forever or decrease forever without stopping, but that's a concept, not a single number.Again, the Zeno-like paradox is to say that if it takes you 1 second to walk a distance, you can chop that distance up into as many segments as you want. If you do that, each segment is of finite time. But if you keep chopping, eventually you're adding up an infinite number of finite times which must be infinite. The problem is that if you really have an infinite (meaning increasing without limit) number of steps, then each step takes an infinitely short (decreasing without limit) time to cross. For example, if you chop it in half, each segment takes 1/2th of a second to cross. 2*1/2=1 second. If you chop it into 3rds, each takes 1/3rd of a segment, and 3*1/3rd=1 second. If you shop it into n segments, each one takes 1/nth of a second, and n*1/n=1. Calculus basically says that the proper way to treat this is to take n*1/n seconds and then to let n get bigger without limit. Since n*1/n=1 for all possible real values of n, then the limit has to be 1 no matter how big n gets, even if it's allowed to increase forever without limit.Zeno's paradox is a bit more sophisticated, since it's basically chopping the segment in half, then that half in half, then the next half in half, etc., so it's not as obvious that it has to sum to 1, but it's a similar proof in calculus. You can write the sum of a finite series and calculate the sum as you chop it up without limit. (The answer was actually found before calculus--thanks to Matthew for pointing that out to me--but calculus really made it rigorous and showed how to treat it within a firm mathematical framework.)

Due to the reality that exists, isn't the paradox then as stated, just a mathemtical exercise, admittedly or obviously not taking into consideration any actuality or reality?

William, to me it's logic and it is correct, at least for calculus. And as we use that kind of math for all sort of daily needs and constructions, and find it working? But ask JP of what he didn't discuss instead

Again, the Zeno-like paradox is to say that if it takes you 1 second to walk a distance, you can chop that distance up into as many segments as you want. If you do that, each segment is of finite time. But if you keep chopping, eventually you're adding up an infinite number of finite times which must be infinite. The problem is that if you really have an infinite (meaning increasing without limit) number of steps, then each step takes an infinitely short (decreasing without limit) time to cross. For example, if you chop it in half, each segment takes 1/2th of a second to cross. 2*1/2=1 second. If you chop it into 3rds, each takes 1/3rd of a segment, and 3*1/3rd=1 second. If you shop it into n segments, each one takes 1/nth of a second, and n*1/n=1. Calculus basically says that the proper way to treat this is to take n*1/n seconds and then to let n get bigger without limit. Since n*1/n=1 for all possible real values of n, then the limit has to be 1 no matter how big n gets, even if it's allowed to increase forever without limit.

David, you might be interested in Georg Cantor (...sorry, you cannot view external links. To see them, please REGISTER or LOGIN. You will learn many things such as the nice theorem "there are an infinite amount of sizes of infinity".

...And of course if it grows without bound, it naturally represents an infinite family of numbers, which is precisely why you can't treat the symbol for infinity, itself, as a number: the symbol takes on meaning relative to expression in which it is used. Do correct me if I'm wrong, that's as far as I understand at the moment.

the first step, which is half the distance,can also be considered a distance in and of itself?