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Offline McKay

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What is the Xeno motion paradox?
« on: 04/08/2012 10:13:43 »
So I found this video, in witch it was said and explained that all movement should be impossible .. it was explained somehow like this (too bad i can not find the video now) : imagine a arrow (or bullet.. or anything else) shooting/ moving from start to its target - in order to get to the target, the object has to reach the trajectories/ paths mid-point. To reach this mid-point, it has to reach the mid-point between start and first mid-point. To reach this new mid point, it has to reach the mid point between start and mid points mid point.. and so on to infinity.
So, according to this, all movement should be impossible - it would take infinite amount of time to traverse any infinitely small distance.
Even although the time to traverse each sub-mid-point would decrease, no matter how small the time gets, it would have to be multiplied by infinity, as there would be infinite .. steps.
That is .. weird.
It got me thinking and what I thought up was - space must be quantized. In that case, there would be a finite amount of steps and time required would be finite.

.. but that got me thinking even further - if space is quantized, then there is a actual limit of how round a circle can be - with the smoothness increasing with size! (basic geometry)


And so, to conclude, 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..
« Last Edit: 04/08/2012 11:02:48 by chris »


 

Offline RD

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Re: All movement should be impossible (?)
« Reply #1 on: 04/08/2012 10:50:55 »
So I found this video, in witch it was said and explained that all movement should be impossible ...

https://en.wikipedia.org/wiki/Zeno%27s_paradoxes#The_dichotomy_paradox

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Another proposed solution is to question one of the assumptions Zeno used in his paradoxes (particularly the Dichotomy), which is that between any two different points in space (or time), there is always another point. Without this assumption there are only a finite number of distances between two points, hence there is no infinite sequence of movements, and the paradox is resolved. The ideas of Planck length and Planck time in modern physics place a limit on the measurement of time and space
https://en.wikipedia.org/wiki/Zeno%27s_paradoxes#Proposed_solutions


... 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..

Pi is derived from mathematics which has ideal perfect shapes, not real-world objects made from lumpy atoms.
 

Offline evan_au

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Re: What is the Xeno motion paradox?
« Reply #2 on: 04/08/2012 11:56:57 »
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.

Just because it takes an infinite amount of time to write the equation in full does not mean that it takes forever for the arrow to get there! Of course, mathematicians get bored writing infinite things out in  full, and so have a notation to write this infinite series in 1 short line.

In the quantum version of Zeno's paradox, the Heisenberg uncertainty principle tells us that you cannot measure the precise position of any object, and the hypothesised Plank length is so short, that even macroscopic objects should be able to "quantum tunnel" over such a small distance. So motion to hit a target is still possible under quantum theory.
 

Offline Soul Surfer

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Re: What is the Xeno motion paradox?
« Reply #3 on: 05/08/2012 10:10:33 »
Zeno's paradox originates in ancient Greece and is essentially a verbal paradox demonstrating the limits of everyday language to describe some things.  It can in a simple sense become a mathematical paradox of infinity in everyday mathematics.  However as pointed out it can be clearly dealt with mathematically given the correct way of expressing the problem it can also be cleared up in everyday life by observing that things do move.

Zeno's original way of expressing the problem was in fact from the opposite direction because the no movement concept would have been refuted straight away.

Zeno described a race between Achilles and a tortoise and showed that even though Achilles was running faster than the tortoise he would never overtake it.
 

Offline David Cooper

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Re: What is the Xeno motion paradox?
« Reply #4 on: 05/08/2012 21:52:57 »
We have a clash of opinion here - one is that there is no problem and that there can be an infininte number of steps between two points in space, and the other is that there can't be - that the universe must be granular. Anyone who thinks that there can be an infinite number of consecutive steps simply hasn't understood the point of the Zeno's Arrow paradox: with an infinite number of consecutive steps where each step has a duration of any size, it will necessarily take forever to get from A to B. If the steps have no duration at all, then there can be no such thing as time - all events would play through in a single instant (of no length whatsoever), including the entire history and future of the universe.

With Zeno's Hare and Tortoise Paradox the solution is again to have a granular universe. Movements are all made in jumps and we have to focus on how long things are stationary. The wait-times for the hare are shorter than the wait-times for the tortoise, so it gets more turns to move and is able to catch the tortoise when the tortoise is not moving and leave it behind in the same manner.
 

Offline JP

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Re: What is the Xeno motion paradox?
« Reply #5 on: 06/08/2012 15:18:59 »
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.

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, 2nths, 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.
 

Offline David Cooper

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Re: What is the Xeno motion paradox?
« Reply #6 on: 08/08/2012 00:09:02 »
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.
 

Offline JP

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Re: What is the Xeno motion paradox?
« Reply #7 on: 08/08/2012 02:53:00 »
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.

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.
 

Offline David Cooper

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Re: What is the Xeno motion paradox?
« Reply #8 on: 08/08/2012 20:53:22 »
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.
 

Offline JP

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Re: What is the Xeno motion paradox?
« Reply #9 on: 08/08/2012 20:59:14 »
No, David, you don't have to introduce infinity as a number in this limiting process.  That's the point of calculus. 

Talking about "granularity" of the universe is beside the point.  As I said, the quantum question is another thing entirely.  Calculus resolved the classical Zeno paradox.
 

Offline David Cooper

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Re: What is the Xeno motion paradox?
« Reply #10 on: 08/08/2012 23:39:04 »
Yes you do, JP - that's the whole point of the paradox. Calculus doesn't do anything to address the issue - all you're doing is making an assertion that it solves a problem which it manifestly doesn't. Anything less than going to the point where an infinity is brought in simply gives you a set of tiny chunks which can be cut in half over and over again forever, and that is the infinity coming in again whether you like it or not. If there's a finite number of chunks, it's still granular, and that's the case all the way down until you make the jump to infinity. The idea that calculus has anything new to say about the paradox is just wishful thinking.
 

Offline JP

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Re: What is the Xeno motion paradox?
« Reply #11 on: 09/08/2012 01:17:05 »
Calculus tells you why trying to shoehorn "infinity" into the problem as if it's a number as you're doing is wrong.
 

Offline William McCormick

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Re: What is the Xeno motion paradox?
« Reply #12 on: 09/08/2012 02:35:11 »
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.

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.

Infinity can be a real number, as a human with a designated life span, you will just never be able to count to it.  Or count a set number of objects, within that time frame. So any math performed on that un- knowable amount, within a humans lifetime, would be a guesstimate.

Infinity just means outside the realm of exacting mathematics. To say that by completing steps one two and three, and proving it, that step infinity will follow at a later time in some predefined order, is not scientific. No one has ever counted the particles of electricity that exist in any common object at any given time, so to say we have counted or weighed known objects, is a certain lie. It is a guesstimate that we live with.

Look at it this way, a particle of electricity is so small, moving so fast, that no one can ever see it.

Not that anyone sees any matter. Rather they see what other particles of light, bring to their eyes, about the object we claim to see. That is why no one will ever see a particle of electricity. Or ever know if the many particles of light needed to make our eyes or cameras work, are only representing one particle of electricity we think we are seeing or an infinite number. That is why for many,  many, years, a particle of electricity has been considered, infinitely small. And to me will stay that way.

My point being since we are not going to be able to prove something infinite as being something finite, we should probably not worry about complex formulas that may only be exaggerating the known error. Not really aiming this at you in particular.

                      Sincerely,

                            William McCormick

 

Offline David Cooper

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Re: What is the Xeno motion paradox?
« Reply #13 on: 09/08/2012 04:52:21 »
Calculus 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?

If you can go on counting up the items in a series forever, which in the case of Zeno's arrow paradox we can, then that necessarily gives us an infinite number of steps whether you're prepared to admit it or not, and that's the whole point of Zeno's arrow paradox - to ban us from considering the case where the infinity is involved, you aren't addressing the paradox at all, but merely avoiding the issue. The paradox is very specifically about the infinity, and calculus doesn't let you off the hook in that regard at all. If the infinite steps have a duration, the action can never complete, and if the infinite steps have no duration, the scale is lost and they don't add up to the required answer.

What is calculus? It's a mathematical trick get around the infinity problem for calculating exact values in awkward cases where you can go on working with smaller and smaller pieces forever, but that is all it is - it is not a mechanism to make the impossible possible.
 

Offline Geezer

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Re: What is the Xeno motion paradox?
« Reply #14 on: 09/08/2012 05:29:48 »

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


I don't think we can. The logic in the version of the Paradox of the Arrow that I know is faulty, so it's not really a paradox at all. I would not be too surprised to learn that Zeno understood that and decided to pull a fast one on us.
 

Offline Geezer

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Re: What is the Xeno motion paradox?
« Reply #15 on: 09/08/2012 06:15:20 »
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.

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.
 

Offline JP

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Re: What is the Xeno motion paradox?
« Reply #16 on: 09/08/2012 12:01:59 »
Calculus 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?

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.
 

Offline yor_on

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Re: What is the Xeno motion paradox?
« Reply #17 on: 09/08/2012 13:59:03 »
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.

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, 2nths, 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?
 

Offline David Cooper

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Re: What is the Xeno motion paradox?
« Reply #18 on: 09/08/2012 20:34:18 »
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.

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.

The time interval also decreases proportionally, but the whole point of the paradox is that when you take it to infinity in order to get rid of any possible granularity, the duration of the infinite chunks is zero, at which point the whole thing loses track of the total duration. If you don't take it that far, you're still dealing with granular solutions, just with smaller and smaller grain at every step.
 

Offline David Cooper

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Re: What is the Xeno motion paradox?
« Reply #19 on: 09/08/2012 20:47:33 »
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.

The idea that they might count as different kinds of infinities was just mentioned in passing, and I suspect you're wrong about them being the same kind - maybe you aren't up to speed with the latest developments in mathematics in that area.

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

It has everything to do with what we're discussing - it's the entire point of the paradox, and your proposed solution (which isn't actually yours, but is being pushed by the priests and must therefore be correct no matter how wrong it is) is to ban the issue from being addressed and claim that avoiding the issue equates to a solution.

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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.

Well, obviously you're right because you're simply right, and never mind what correctly applied reason has to say on the matter.

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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.

If you don't take it to the point where there are an infinite number of chunks, you're actually pushing a granular solution, so you're not succeeding in solving the paradox with a non-granular solution. If you attempt to consider an infinite number of chunks, the maths breaks down and you fail. That's the whole point! You are pushing an infinite number of granular solutions and claiming they aren't granular.

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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.

No, I point out that any attempt to add up the infinite number of chunks goes wrong because if they're all zero in duration, there's nothing to add up, and if they're all greater than zero you can only get an infinite result (and this is not incorrect maths) - neither are mathematically useful as the whole scale has been lost. The only solutions are granular, and the tricks of calculus have nothing profound to say on the matter whatsoever.

If when you count up the items in a series of numbers such as 1, 2, 3, 4, etc. you get an infinity, what sense does it make to say that if you add up 1 + 1 + 1 + 1 + etc. forever you don't also get an infinity? That is identical to counting the items in the series. You are simply not entitled to pick and choose in such a way as to decide that one of these is valid and the other invalid when they are identical processes.
« Last Edit: 09/08/2012 21:30:51 by David Cooper »
 

Offline JP

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Re: What is the Xeno motion paradox?
« Reply #20 on: 09/08/2012 21:25:52 »
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.

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, 2nths, 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 I'm pointing out is that you can't treat "infinity" as a number.  You have to be very careful when saying you're taking an infinite number of things or making things infinitely large or small.  For example, infinity cannot be treated as a real number.  Zeno's paradox essentially argues that if you divide the interval into an infinite number of steps, each one still takes a tiny amount of time to cross, call this time t.  The argument goes that the total time to cross N of these steps is N times t therefore the total time to cross an infinite number of these steps is infinity times t, which must be infinity!

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.
 

Offline David Cooper

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Re: What is the Xeno motion paradox?
« Reply #21 on: 09/08/2012 21:38:57 »
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.

The time is finite, but the amount of chopping you can do is infinite. Every solution short of that infinity is a granular solution. You cannot escape from a granular solution without directly addressing the infinity, and any attempt to handle that infinity will self-destruct.
 

Offline JP

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Re: What is the Xeno motion paradox?
« Reply #22 on: 09/08/2012 21:41:26 »
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?
 

Offline David Cooper

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Re: What is the Xeno motion paradox?
« Reply #23 on: 09/08/2012 22:23:33 »
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.

Everything I said in the thread about Einstein's philosophical interpretation of relativity was correct - I've just started a new thread about it because nothing's been done about the previous one which was wrongly hacked out of a thread and shunted into "new theories" where it clearly doesn't belong.

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

No, you've simply made claims about mathematics which don't stack up. The priests I refer to are not all mathematicians, but to specific ones who push faulty philosopy as fact and misinform the public, shutting down their thinking in the process. Calculus does not solve Zeno's arrow paradox - claims that it does are simply false.

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

When I see people pushing ideas that are wrong on the basis of authority, that is sufficient justification for putting across an alternative view. You push other people's assertions while I push mine, and the people who read the arguments put forward to back the assertions can make up their own minds. History is littered with cases of false beliefs holding sway, and when they're eventually overturned everyone looks back and says how silly people must have been to believe whatever it was - the same will happen again over many things that are mainstream and wrong. My objective here is simply to open people's minds to the possibility that certain beliefs are wrong, or at the very least go beyond the competence of the people who came up with them.

In this argument here, we have had a claim that 1 + 1 + 1 + 1 + etc. doesn't add up to infinity, whereas if you count all the numbers in the series 1, 2, 3, 4, etc. you get an infinite number. The former is the very process by which the latter is done. You can't just ban things from being infinities on an arbitrary basis - that is neither mathematics nor science.
 

Offline yor_on

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Re: What is the Xeno motion paradox?
« Reply #24 on: 09/08/2012 22:52:11 »
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.
 

The Naked Scientists Forum

Re: What is the Xeno motion paradox?
« Reply #24 on: 09/08/2012 22:52:11 »

 

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