Zeno's Paradoxes

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  • #26
ahrkron
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Originally posted by Lifegazer
He knew it was wrong, because...? Because he could see that motion was taking place? That just begs the question about perception being internal, or external.
I don't know about Zeno, but we know it is wrong because the argument is applied to whatever it is we call "motion" (be it internal, external or mixed), and it would imply that we would not even perceive it, which is evidently false.
 
  • #27
Lifegazer
Originally posted by ahrkron
I don't know about Zeno, but we know it is wrong because the argument is applied to whatever it is we call "motion" (be it internal, external or mixed), and it would imply that we would not even perceive it, which is evidently false.
The thing with internal-motion, is that it is conceptual. You don't really walk-around in your dreams, for example. It just appears that way. But it's not real motion.
Zeno's paradox fundamentally asks whether any form of perceived motion can be 'real'... or whether it's all conceptual.
 
  • #28
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Originally posted by Lifegazer
The thing with internal-motion, is that it is conceptual. You don't really walk-around in your dreams, for example. It just appears that way. But it's not real motion.
That is entirely irrelevant for Zeno's argument. Even in your dreams, in order to go to a different point, you need to have the perception of having passed through the intermediate points. You may argue otherwise in the case of dreams, but in the realm of perceptions, it is clearly the case.

Even if all reality was "in the Mind", Zeno's arguments can be equally applied. They do not tell anything about what is "behind perceptions", but only about our description of motion, and our assumptions about infinite decomposition.
 
  • #29
Lifegazer
Originally posted by ahrkron
That is entirely irrelevant for Zeno's argument. Even in your dreams, in order to go to a different point, you need to have the perception of having passed through the intermediate points.
Zeno's paradox asks the reader - even indirectly - to ponder the nature of reality. If motion (in an external reality) doesn't make sense, then one is forced to ponder the possibility that all motion is conceptual (in the mind). Of course, it may or may not be possible to show why Zeno's reasoning is incorrect. But I've never seen an argument to convince me that this is the case. Conceptual mathematics doesn't always apply to tangible (finite) reality. Especially when the term "infinite" is being used.
You may argue otherwise in the case of dreams, but in the realm of perceptions, it is clearly the case.
'Dreams' are evidence that motion can ~appear~ to occur where no motion has occured. But as for our concious perceptions, the discussion is still open to debate, as I see it.
Even if all reality was "in the Mind", Zeno's arguments can be equally applied.
But if motion was really a figment of the Mind's ability to fool itself somehow, then Zeno's arguments do make sense. It's when we apply Zeno's arguments to external-reality where the problems seem to arise.
 
  • #30
Tom Mattson
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Originally posted by Lifegazer
But does it actually converge to '1'?
Yes. I wouldn't have said it does if it does not.

That may well be a flaw. But I only find it to be a grammatical flaw.
The question remains whether the convergence to '1' can be achieved. If it cannot, then the underlying issue is still open to debate.
Grammatical flaw?! The flaw is mathematical, and it has been settled for quite some time.
 
  • #31
Lifegazer


Originally posted by Tom
Yes. I wouldn't have said it does if it does not.
Will you show us how? And can you do this without getting too fancy with the math?
 
  • #32
Tom Mattson
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Originally posted by Lifegazer
Of course, it may or may not be possible to show why Zeno's reasoning is incorrect.
We do know exactly why his reasoning is incorrect. It is because he assumed the divergence of an infinite series that actually converges.

But I've never seen an argument to convince me that this is the case. Conceptual mathematics doesn't always apply to tangible (finite) reality.
Then why listen to Zeno? He's the one who tried to show that motion is impossible using mathematics. You are willing to accept Zeno's use of "conceptual mathematics" with a glaring mistake and reject "conceptual mathematics" done correctly just because it supports your beliefs.

That makes no sense whatsoever.
 
  • #33
Tom Mattson
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Originally posted by Lifegazer
Will you show us how? And can you do this without getting too fancy with the math?
You and I interacted in 2 "Zeno" threads in PF v2.0, and I showed you how then. Rather than type it out again, I am going to refer you to this website:

http://mathworld.wolfram.com/GeometricSeries.html

Let r=1/2, and you've got Zeno's series.
 
  • #34
ahrkron
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Originally posted by Lifegazer
But if motion was really a figment of the Mind's ability to fool itself somehow, then Zeno's arguments do make sense. It's when we apply Zeno's arguments to external-reality where the problems seem to arise.
I think you did not get my point at all.

What I'm saying is that Zeno's arguments also apply to your theory, in which motion is just a projection within the Mind.

This being the case, it is clear that Zeno's paradox has no saying on whether motion is a projection within the Mind or an aspect of reality.

This is just natural, since the paradox does not contend with the nature of reality, but with the description of motion and infinite aggregates.
 
  • #35
Lifegazer
Originally posted by Tom
We do know exactly why his reasoning is incorrect. It is because he assumed the divergence of an infinite series that actually converges.
Yes... you've shown that it converges towards '1'. But you didn't explain why '1' is ever reached. That's why I asked at what point does "Distance=L/2+L/4+L/8+..." converge to 'L'? And even if Zeno is wrong with the terms which he uses, the fundamental-issue of real-motion is not resolved unless it can be shown how this happens. Because I don't understand how it can, to be honest. Not tangibly, anyway.
Then why listen to Zeno? He's the one who tried to show that motion is impossible using mathematics.
I happen to agree with his conclusion, even if he did make what I consider to be an error of language. I tried explaining why in my previous post to you.
"The length is singular. And the time to traverse it is also singular. It's not really surprising to see that '1' is at the heart of the debate. Zeno may have implied that the time to traverse a given length would be infinite. But what he means here is that the time to traverse a given-length cannot become singular in itself. I.e., that the oneness (completeness) of time to traverse a given singular-length cannot be achieved. [added note: basically, what he's saying is that it would take an eternity to achieve completeness/singularity of the given-length, if travelling in a manner which mirrors "L/2+L/4+L/8+...".]
I think he was really arguing (or he should have been arguing) that the time to traverse a given length can never become complete - and that therefore, time does not converge to a singular value. I.e., does not converge to '1' (which is the symbol of completeness, in this case)."
If it takes an eternity to converge towards '1', then '1' is not grasped.
You are willing to accept Zeno's use of "conceptual mathematics" with a glaring mistake and reject "conceptual mathematics" done correctly just because it supports your beliefs.
It is Zeno's use of language which is the issue. His 'paradox' can equally be applied to the eternal-convergence of singularity.
That makes no sense whatsoever.
I have no doubts that present-day mathematics are more advanced than in Zeno's day. But that's not the issue. The issue is whether those mathematics are conceptual, or whether they also reflect a tangible-reality (to which they are being applied). I think the issue is one of reason, rather than of mathematics.
 
  • #36
Lifegazer
Originally posted by ahrkron
What I'm saying is that Zeno's arguments also apply to your theory, in which motion is just a projection within the Mind.
Misunderstood you. Sorry.
However now that I understand your point, I have to disagree with it anyway. For Zeno's arguments are irrelevant to a Mind-reality. Such a reality is a physical-singularity. Motion doesn't really occur. And since Zeno's paradox seems to question the actuality of motion, my hypothesis fits in nicely with this challenge to real-motion. The motion of an observer, within his own mind (full of sensations), is the motion of the mind itself... within itself. It is a shift of perception which yields the appearance of motion - within the mind. The mind doesn't really cross lengths. It changes its perspective of that length, thus yielding the perception of motion. But the mind moves nowhere. Only sensations are changed.
 
  • #37
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Alright, I'd say that that particular paradox is pretty well covered. My thanks to everyone, especially those who explained it using math. I got up to basic Calculus, so I got the meaning out of the equations (possibly easier than I would have gotten it, if someone had tried to explain it in words - rather, it was much more solid/believable, when seen from a mathematical stand-point).

For some reason, I can't think of another of Zeno's paradoxes, at the present moment. I'll try and think of another one - I know I had some in mind, when I posted the thread. If anyone else can think of one to discuss, I'd be happy to discuss that one also.
 
  • #38
Tom Mattson
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Originally posted by Lifegazer
Yes... you've shown that it converges towards '1'. But you didn't explain why '1' is ever reached. That's why I asked at what point does "Distance=L/2+L/4+L/8+..." converge to 'L'?
The link I gave you explains how the series converges. As for "at what point" does it converge, I don't know what you mean.

"The length is singular. And the time to traverse it is also singular. It's not really surprising to see that '1' is at the heart of the debate. Zeno may have implied that the time to traverse a given length would be infinite. But what he means here is that the time to traverse a given-length cannot become singular in itself.
What do you mean by the last sentence? I know what Zeno meant, and it has nothing to do with 'singular', it has to do with 'infinite' (as in: "It will take an infinite amount of time to cross any distance").

I.e., that the oneness (completeness) of time to traverse a given singular-length cannot be achieved. [added note: basically, what he's saying is that it would take an eternity to achieve completeness/singularity of the given-length, if travelling in a manner which mirrors "L/2+L/4+L/8+...".]
What is "oneness(completeness) of time"? What is "completeness/singularity of the given-length"?

I think he was really arguing (or he should have been arguing) that the time to traverse a given length can never become complete - and that therefore, time does not converge to a singular value. I.e., does not converge to '1' (which is the symbol of completeness, in this case)."
If it takes an eternity to converge towards '1', then '1' is not grasped.
Sure, that is what he was arguing. Basically it goes like this:

1. If it takes an eternity to traverse a distance L, then that distance cannot be traversed.
2. It takes an eternity to traverse a distance L.
3. Therefore, that distance cannot be traversed.

But Premise 2 is false.

It is Zeno's use of language which is the issue. His 'paradox' can equally be applied to the eternal-convergence of singularity.
No, it is not his use of language. It is his mathematical error which is the issue.

I have no doubts that present-day mathematics are more advanced than in Zeno's day. But that's not the issue. The issue is whether those mathematics are conceptual, or whether they also reflect a tangible-reality (to which they are being applied). I think the issue is one of reason, rather than of mathematics.
It is an issue of mathematics because that is how Zeno defined the problem from the start. Just read any account of the paradox, and you will see it.
 
  • #40
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Originally posted by Tom
Oooh, I t'ought you'd neva ask.

http://plato.stanford.edu/entries/paradox-zeno
I'd never heard of the "Plurality Paradox". It seems like utter nonsense to me (no offense to anyone who agrees with Zeno on this).

Would anyone like to try and defend it?
 
  • #41
Lifegazer
Originally posted by Tom
The link I gave you explains how the series converges. As for "at what point" does it converge, I don't know what you mean.
I'm not a mathematician, as you know. I was hoping you'd give a narrative of those math and explain what they say, in language. And I know we've spoken about it last year. But I honestly can't remember what your explanation was. But I do remember that I saw a problem with the tangibility of those math. I.e., I doubted that they could be applied to a tangible-reality (with real motion). This might not be important to you, but whether reality is 'tangible' (as opposed to conceptual... mind-ful) is the underlying issue raised by Zeno's paradox. And so I consider such a question to be worthy of discussion.

You say that Zeno was wrong because he dealt with the divergence of a series, as opposed to the convergence of a series. But the same 'paradox' exists with both possibilities. So it's not even relevant that he should make this error. The question still remains:-
How does "Distance=L/2+L/4+L/8+..." , converge to 'L'?
Or; how can there be an eternal progression towards '1', whereby '1' is finally yielded? Does this signify the end of eternity? Paradox abounds and Wuli should be all over you like a rash, at any minute.
What do you mean by the last sentence? I know what Zeno meant, and it has nothing to do with 'singular', it has to do with 'infinite' (as in: "It will take an infinite amount of time to cross any distance").
Again, it's not really relevant as to whether we deal with a diverging-series, or a converging-series. But you don't seem to grasp that "It will take an infinite amount of time to cross any distance"; can also be stated thus: It will take an eternity to converge towards any point (singularity). Or; it will take an eternity to converge towards '1'.
Zeno isn't really saying anything which contradicts a convergence towards '1'. I think he's been misread.
Sure, that is what he was arguing. Basically it goes like this:

1. If it takes an eternity to traverse a distance L, then that distance cannot be traversed.
2. It takes an eternity to traverse a distance L.
3. Therefore, that distance cannot be traversed.

But Premise 2 is false.
Given "Distance=L/2+L/4+L/8+..." , is a series that goes on forever, I do not see how a such a series can come to a convergence (an end) at 'L' (or '1'). If it comes to an end, then that series is not going-on forever. So how does mathematics overcome this problem?
It is an issue of mathematics because that is how Zeno defined the problem from the start. Just read any account of the paradox, and you will see it.
It is actually a discussion about concepts (motion; length; time.). Any mathematics which deals with these concepts, must obviously conform to the reasoning which distinguishes between 'tangible' and 'conceptual'. Because, if mathematicians trust the reason which has formulated math, then mathematicians should also take notice of any reason which makes those aforementioned distinctions.
Any philosophy which seeks to discredit Zeno, cannot do so merely with mathematics. For Zeno does not ask how mathematics manages to converge towards '1', conceptually. Zeno asks how tangible-things can achieve such events. And so, an argument of reason is required to discredit Zeno. Mathematics, when dealing with concepts such as 'infinity', has to take-note of the fact that 'infinity' is an intangible-concept.
 
  • #42
Tom Mattson
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Originally posted by Lifegazer
I'm not a mathematician, as you know. I was hoping you'd give a narrative of those math and explain what they say, in language.
I don't expect you to be a mathematician. That's why I presented a link that uses only high school math. I can't explain how to sum a series "in language".

But I do remember that I saw a problem with the tangibility of those math. I.e., I doubted that they could be applied to a tangible-reality (with real motion). This might not be important to you, but whether reality is 'tangible' (as opposed to conceptual... mind-ful) is the underlying issue raised by Zeno's paradox. And so I consider such a question to be worthy of discussion.
You still don't get it: Zeno is the one who brought the math into it. Any problems you have with the 'tangibility' of math, and you'll have to take it up with Zeno. I'm only telling you why he's wrong.

You say that Zeno was wrong because he dealt with the divergence of a series, as opposed to the convergence of a series. But the same 'paradox' exists with both possibilities. So it's not even relevant that he should make this error.
I don't know what you're thinking, but the paradox disappears completely if you correct the mistake.

The question still remains:-
How does "Distance=L/2+L/4+L/8+..." , converge to 'L'?
As I said, the link explains it. Ask me again, and I'll tell ya the same.

Originally posted by Lifegazer
Again, it's not really relevant as to whether we deal with a diverging-series, or a converging-series.
Of course it is--Zeno relies on the divergence of the series to say what you are about to say...

But you don't seem to grasp that "It will take an infinite amount of time to cross any distance"; can also be stated thus: It will take an eternity to converge towards any point (singularity). Or; it will take an eternity to converge towards '1'.
...and of course, it won't take an eternity. The series I gave you is the time that it will take.

Zeno isn't really saying anything which contradicts a convergence towards '1'. I think he's been misread.
Certainly by you, it seems, because the infinite series is part of the problem, and he gets it wrong.

Given "Distance=L/2+L/4+L/8+..." , is a series that goes on forever, I do not see how a such a series can come to a convergence (an end) at 'L' (or '1'). If it comes to an end, then that series is not going-on forever. So how does mathematics overcome this problem?
It does converge, and the link explains why.

It is actually a discussion about concepts (motion; length; time.). Any mathematics which deals with these concepts, must obviously conform to the reasoning which distinguishes between 'tangible' and 'conceptual'. Because, if mathematicians trust the reason which has formulated math, then mathematicians should also take notice of any reason which makes those aforementioned distinctions.
No--really--it is a discussion about mathematics. Just read the paradox on any one of a zillion webpages.

Any philosophy which seeks to discredit Zeno, cannot do so merely with mathematics.
Of course it can, because Zeno brought it into the arena of mathematics. Again, just read the paradox.

For Zeno does not ask how mathematics manages to converge towards '1', conceptually.
Of course he doesn't ask how the series converges--He asserts that it diverges. Again, just read the paradox.

You've obviously got some studying to do on this issue. Read the paradox, see where Zeno puts the infinite series into it, and see where he gets it wrong. It's not that difficult.
 
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  • #43
Lifegazer
Originally posted by Tom
...and of course, it won't take an eternity. The series I gave you is the time that it will take.
Does L/2 + L/4 + L/8 + L/16 + L/32... ad-infinitum... converge to 'L'?
And if the above series is considered to be infinite, then how can it ever stop at 'L'?

An infinite-convergence (towards 'L' or '1') cannot stop - simply by definition. The fact that it stops means that it isn't an infinite-series.
There's more to this than math. But if you're not interested, then never mind.
 
  • #44
Tom Mattson
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Originally posted by Lifegazer
Does L/2 + L/4 + L/8 + L/16 + L/32... ad-infinitum... converge to 'L'?
Yes

You can stop asking me the same question. Trust me, I'm not going to change my answer.

And if the above series is considered to be infinite, then how can it ever stop at 'L'?
The "how" is in the link I gave you. All the deductions are there.

An infinite-convergence (towards 'L' or '1') cannot stop - simply by definition. The fact that it stops means that it isn't an infinite-series.
There's more to this than math. But if you're not interested, then never mind.
No, there really isn't more to it than math. Again, read the paradox. I can't stress that enough. The solution is found in the calculus of infinite series. If you can't be bothered taking it upon yourself to read the problem and learn the math that solves it, then why bother with these discussions?
 
  • #45
Lifegazer
Originally posted by Tom
No, there really isn't more to it than math. Again, read the paradox. I can't stress that enough. The solution is found in the calculus of infinite series. If you can't be bothered taking it upon yourself to read the problem and learn the math that solves it, then why bother with these discussions?
I consider the discussion to be one of reason. Fundamentally, the discussion is about the nature of reality, and discusses the possible 'substance' of particular concepts. Whether they can exist as tangible-entities, outside of perception.
You don't seem to want to debate any issue concerned along these lines. Fair enough. But to think that all such matters are solved because 'mathematics' (which is conceptual) can make an infinite-series come to a stop, is an incorrect attitude, imo.
I could probably come to understand the mathematics you pointed-out. But after I came to understand these mathematics, I would make the exact-same points I've been making all along. I do not contend that the mathematics are wrong. I contend that they don't make sense in relation to existence. A series which is 'infinite', by definition, simply cannot stop.
 
  • #46
Tom Mattson
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Originally posted by Lifegazer
I consider the discussion to be one of reason. Fundamentally, the discussion is about the nature of reality, and discusses the possible 'substance' of particular concepts. Whether they can exist as tangible-entities, outside of perception.
Everyone involved--you, me, Zeno--considers the discussion to be one of reason. The question is, "How did Zeno reason his argument, and where did he go wrong?"

You don't seem to want to debate any issue concerned along these lines. Fair enough. But to think that all such matters are solved because 'mathematics' (which is conceptual) can make an infinite-series come to a stop, is an incorrect attitude, imo.
First, I never said that "all such matters" were solved with mathematics. I said that Zeno's paradox was solved with mathematics, and that is because Zeno based his argument on mathematics. Again, you would know that if you had read the problem.

Second, the infinite series does not "come to a stop". It converges, meaning that all its terms add up to a finite number.

I could probably come to understand the mathematics you pointed-out. But after I came to understand these mathematics, I would make the exact-same points I've been making all along.
I doubt it. For one thing, you would not be making the point that it takes an eternity to traverse a finite length at constant speed. That is precisely where Zeno went wrong.

I do not contend that the mathematics are wrong. I contend that they don't make sense in relation to existence.
Well then tell Zeno! I am only analyzing his argument.

A series which is 'infinite', by definition, simply cannot stop.
And it doesn't. It's just that the infinite number of parts add up to a finite number. I know what your next question is going to be:

"But how...?"

The "how" is contained in the link I gave you. If you go through it--and understand it--you will see that some infinite series can be summed, despite the fact that they do not "stop".
 
  • #47
In other words, you can keep adding up 1/2n all day long, but you will never get past 1. If you feel really bored, try it...
 
  • #48
ahrkron
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Originally posted by Lifegazer
The motion of an observer, within his own mind (full of sensations), is the motion of the mind itself... within itself. It is a shift of perception which yields the appearance of motion - within the mind. The mind doesn't really cross lengths. It changes its perspective of that length, thus yielding the perception of motion. But the mind moves nowhere. Only sensations are changed.
Phrase it as you will. The fact is, the perspective/perception/appearance or anything you want to use for the description, gives as good a basis for the paradox as realism.

For instance:

In order for the mind to "change perspective", it needs to do it in a way that (according to its own perspective) sweeps the intermediate perspective-states.

In order for it to "change sensations" from here to 1m, it first has to "change perspectives" up to 0.5m, and then to half that "perspective", and then half that.

i.e., if you don't accept that an infinite sum can converge to a finite number, then you have to accept that even in your mind, motion cannot be completed ever.

In any case, you can see that Zeno's argument does not favor either view (with or without the "Mind").
 
  • #49
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Second, the infinite series does not "come to a stop". It converges, meaning that all its terms add up to a finite number.
I'm no mathematician but I don't think that's strictly true. The infinite series sums to what is, to all intents and purposes, a finite number given that .9999999....... is equal to 1.

Taking Uncertainty into account, the 'smearing' of space time as one approaches infinitesimal quantities might logically translate to numbers in that any conceptualy finite number n, is in 'reality' or practical use, n+-1/[oo]. The problem being that using this concept, 1 is not a strictly finite number. If there was a symbol for 'infinitesimal' that would be better.

Applying this to Zeno, in the case of Achilles, depending on what infinitesimal amount short of the total distance is deemed to be equal to the distance, he reaches the finish line long before a infinite amount of time has passed. In the case of the arrow, it always in motion an never stationary.

Raavin

my head hurts
 
  • #50
Lifegazer
Originally posted by ahrkron
In order for it to "change sensations" from here to 1m, it first has to "change perspectives" up to 0.5m, and then to half that "perspective", and then half that.

i.e., if you don't accept that an infinite sum can converge to a finite number, then you have to accept that even in your mind, motion cannot be completed ever.

In any case, you can see that Zeno's argument does not favor either view (with or without the "Mind").
If the mind changes perception like the frames of a movie (incrementally, rather than smoothly), then would motion be achieved without such considerations?
 

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