Zeno's Paradoxes: Misunderstanding or Misconception?

[Mentat]

There are a couple of Zeno's paradoxes that I just don't agree with. There may be something to them that I haven't understood properly, but I just don't agree with them as I've understood them.

One is the paradox of motion. He posits that you can never get anywhere because you have to cover half of the remaining distance everytime. Since there are infinite halves, you can't ever get there. I disagree because you don't have to continue to cover half of each new distance, you can cover more than that, if you so desire. Also, once you have come into physical contact with something, I think it can be reasonably concluded that you have "gotten there".

I would very much like that someone should explain where my misunderstanding of this paradox is, before I go on to the next one please.

Any help would be appreciated.

 

[wuliheron]

There must be million websites devoted to this paradox. Here's the most mickey mouse plain english one I could find.


http://www.mathacademy.com/pr/prime/articles/zeno_tort/index.asp

 

[Mentat]

Ok, then that particular paradox is not considered authentic?

 

[Lifegazer]

Originally posted by Mentat
There are a couple of Zeno's paradoxes that I just don't agree with.

I hope you don't mind me hogging your posts tonight. I like supporting Zeno, when given a chance.

One is the paradox of motion. He posits that you can never get anywhere because you have to cover half of the remaining distance everytime. Since there are infinite halves, you can't ever get there. I disagree because you don't have to continue to cover half of each new distance, you can cover more than that, if you so desire.

That's not the point Mentat. A 'half' is used because it's the most readily-understandable fraction. Even if you were to traverse three-quarters of the distance every time, Zeno's ultimate conclusion would still apply: that if you travel 3/4 of a length, and then 4/7's of the remaining-length, and then 7/9's of the remaining-length - ad-infinitum - then you will never reach the end of that length.
I want to say more about this. But one of my favourite films is just starting...

 

[Mentat]

Originally posted by Lifegazer
I hope you don't mind me hogging your posts tonight. I like supporting Zeno, when given a chance.

That's not the point Mentat. A 'half' is used because it's the most readily-understandable fraction. Even if you were to traverse three-quarters of the distance every time, Zeno's ultimate conclusion would still apply: that if you travel 3/4 of a length, and then 4/7's of the remaining-length, and then 7/9's of the remaining-length - ad-infinitum - then you will never reach the end of that length.
I want to say more about this. But one of my favourite films is just starting...

Which film is that?

I understand that it could be any fraction, and that a point particle, with no real determined size, couldn't get to the end. However, I have mass, and so I would eventually be touching whatever was at the "end of the line".

 

[wuliheron]

Originally posted by Mentat
Ok, then that particular paradox is not considered authentic?

Nahhh... its only valid according to fundamentalist mathematicians in their ivory towers. Quantum Mechanics has its own versions of the paradox:

http://www.tcd.ie/Physics/Schools/what/atoms/quantum/uncertainty.html

No doubt it is easy, expedient, and comforting to tell little kids mom and dad know what life is all about and have a firm grasp on the situation, but the reality between adults is different. That's also why all these teachers and professors still teach outdated fundamentalist stuff first, and then get down and dirty with a more honest admission of the extent of their ignorance.

 

[quantumdude]

Originally posted by Mentat
One is the paradox of motion. He posits that you can never get anywhere because you have to cover half of the remaining distance everytime. Since there are infinite halves, you can't ever get there. I disagree because you don't have to continue to cover half of each new distance, you can cover more than that, if you so desire.

The fraction used really doesn't matter. The reason Zeno's paradox fails is that, in his argument, he assumes that the infinite series of powers of 1/2 diverges to infinity, when in fact it converges to 1.

It goes like this:

Assume you are traveling at speed v and must cross a distance L in time T.

Before you can get to L, you must get to L/2. Once you have gotten to L/2, you must then get to 3L/4, etc. This continuous halving of distances can be represented by the series:

Distance=L/2+L/4+L/8+...

Note that it takes an infinite number of steps to travel the distance. Since it is not possible to perform an infinite number of steps in a finite time, it is not possible to cross any distance L in time T at speed V.

Zeno's mistake is in assuming that an infinite number of steps cannot be done in a finite amount of time. In other words, he tacitly assumes that it would require an infinite amount of time, and so could never be done.

Is that true? Let's see.

T=L/v=(1/v)Σ_n=0^oo(1/2)ⁿ

The series on the right is a geometric series. In order for Zeno's argument to hold up, it would have to blow up to infinity--but it actually converges to 1.

Also, once you have come into physical contact with something, I think it can be reasonably concluded that you have "gotten there".

That suggests the conclusion is wrong, but it doesn't say why it is wrong. The flaw in Zeno's argument is in assuming the divergence of a series that actually converges.

 

[ahrkron]

Originally posted by Mentat
One is the paradox of motion. He posits that you can never get anywhere because you have to cover half of the remaining distance everytime. Since there are infinite halves, you can't ever get there. I disagree because you don't have to continue to cover half of each new distance, you can cover more than that, if you so desire.

I think it is here's where you miss the point of the paradox. You cannot "cover more than that" without covering the menitoned half.

i.e., In order to cover 1m, you have to cover half a meter first, no matter how you do it, you will pass through the middle point. Similarly, on your way to cover the first half, you necessarily passed trhough the 25cm point. No matter what you are thinking during your movement (what you are "aiming for"), the matter is one of principle: you need to pass through an infinity of "middle-points".

Since all those partial movements, the paradox goes, must take some time, and you need an infinity of them, the final destination cannot be reached.

Does that help?

 

[Dissident Dan]

Yes, if you've ever studied Calculus, it would be more apparent to you.

The summation of a constanct divided by anything greater than -1 and less than 1 to the nth power is always a finite number, even when n goes to infinity.

[sum]1/xⁿ= c, when -1 < x < 1, regardless of bounds

This can be verified by comparing the integration of such a term from a finite number to infinity, because the integratin can be used as an upper bound for the summation.
for example,

[inte]1/2^x = -1/((ln2)2^x

so the integration of that from 1 to infinity is

0 - (-1/((ln2)2) = 1/((ln2)2)

 

[wuliheron]

Yeah, yeah, calculus suggests its not really a paradox. Philosophy suggests otherwise. I tried to keep it simple, but you asked for it. Zeno's paradox of motion is related to the Sorites Heap paradox.

http://plato.stanford.edu/entries/sorites-paradox/

This paradox and the Cretan Liar's paradox are both considered the most insoluable known. A central issue in the sorites paradox is vagueness.

http://plato.stanford.edu/entries/vagueness/#5

Something is vague if there are cases which do not fall clearly inside or outside the range of applicability of the term. That isn't to say vague words have no meaning. The word "heap" definitely has meaning, but its meaning is vague. Likewise the meaning of infinity is vague and it is debatable whether or not infinity is real.

 

[quantumdude]

OK, you're going to have to explain the connection between Sorites and Zeno. For one thing, I don't see the vagueness in Zeno's paradox. In fact, everything seems to be quite well defined.

As for the Cretan paradox, I don't see the vagueness there either, nor do I see the connection to Zeno. The Cretan paradox is an illustration of the breakdown of logic as a formal system, whereas Zeno attempted to prove the impossibility of motion. The former pertains to a mental construc (logic), while the latter pertains to something in the real world (motion).

 

[jammieg]

Here's another way of telling the story that takes into account Mentat's point of view:

A mathematician and an engineer were both standing 20 ft away from this pretty girl when they were asked by another Zeno guy that if they could only walk half the distance to the girl each time what would they do?
The mathematician exclaimed "Zeno, I will stand right here because I can conclusively prove that I'll never reach her", whereas the engineer said, "I agree but I can get close enough for practical purposes".

 

[wuliheron]

Originally posted by Tom
OK, you're going to have to explain the connection between Sorites and Zeno. For one thing, I don't see the vagueness in Zeno's paradox. In fact, everything seems to be quite well defined.

As for the Cretan paradox, I don't see the vagueness there either, nor do I see the connection to Zeno. The Cretan paradox is an illustration of the breakdown of logic as a formal system, whereas Zeno attempted to prove the impossibility of motion. The former pertains to a mental construc (logic), while the latter pertains to something in the real world (motion).

Actually, the Cretan Liar's paradox is not widely considered vague. I mentioned it here just to put both paradoxes in perspective. That's the irony of it, the two most respected paradoxes cover both the vague and the well defined.

Zeno's paradox is vague because the concept of infinity is vague just as the concept of a heap is vague. The concept of time isn't that well defined either.

 

[dr-dock]

This ain't no paradox really!

if they both go with a constant speeds (V1 and V2) then:
dX1/V1=dt=dX2/V2 so if it was X1>X2 to become X2>X1 it has to be dX2>dX1 => dX2/dX1=V2/V1>1 => V2>V1 (it is enough).in fact it only seems like a paradox but it ain't cause as you shrenk dX you shrenk dt. at a infinite shrenking step you end up with 0/0=V=const.

just mantein your speed Achilles and you'll win.
wisdom is the might of the physically weaker.

 

[wuliheron]

Originally posted by dr-dock
if they both go with a constant speeds (V1 and V2) then:
dX1/V1=dt=dX2/V2 so if it was X1>X2 to become X2>X1 it has to be dX2>dX1 => dX2/dX1=V2/V1>1 => V2>V1 (it is enough).in fact it only seems like a paradox but it ain't cause as you shrenk dX you shrenk dt. at a infinite shrenking step you end up with 0/0=V=const.

just mantein your speed Achilles and you'll win.
wisdom is the might of the physically weaker.

He'll win at the end of eternity, what a paradox.

 

[Dissident Dan]

The thing to remember in Zeno's paradox is that if he continues at a constant speed, as the distance traveled halves, so does the time. So, with continual halving, the distance approaches 0, and so does the elapsed time. But time keeps on a'going...

 

[heusdens]

Pardox of motion

The essential problem, highlighted by Zeno’s paradoxes, is the inability of formal logic to grasp movement. Zeno’s paradox of the Arrow takes as an example of movement the parabola traced by an arrow in flight. At any given point in this trajectory, the arrow is considered to be still. But since, by definition, a line consists of a series of points, at each of which the arrow is still, movement is an illusion. The answer to this paradox was given by Hegel.

The notion of movement necessarily involves a contradiction. Consider the movement of a body, Zeno’s arrow for example, from one point to another. When it starts to move, it is no longer at point A. At the same time, it is not yet at point B. Where is it, then? To say that it is "in the middle" conveys nothing, for then it would still be at a point, and therefore at rest. "But," says Hegel, "movement means to be in this place and not to be in it, and thus to be in both alike; and this is the continuity of space and time which first makes motion possible." (Hegel, op. cit., Vol. 1, p. 273.) As Aristotle shrewdly observed, "It arises from the fact that it is taken for granted that time consists of the Now; for if this is not conceded, the conclusions will not follow." But what is this "now"? If we say the arrow is "here," "now," it has already gone.

Engels writes:

"Motion itself is a contradiction: even simple mechanical change of place can only come about through a body being both in one place and in another place at one and the same moment of time, being in one and the same place and also not in it. And the continual assertion and simultaneous solution of this contradiction is precisely what motion is." (Engels, Anti-Duhring, p. 152.)

 

[Lifegazer]

Originally posted by wuliheron
He'll win at the end of eternity, what a paradox.

No... he will win at the end of time/change. Which doesn't signify the end of existence.

 

[heusdens]

Originally posted by Lifegazer
No... he will win at the end of time/change. Which doesn't signify the end of existence.

Which in other words is infinity, but anyway that is not the answer, since the paradox is only a trick of the mind, and not a real paradox (since movement does occur).

 

[drag]

O.K. I believe that one is covered already.
What's next Mentat ?

Live long and prosper.

 

[DrChinese]

Zeno's Paradox is successful because it seems so reasonable.

When Einstein first published on Special Relativity, it was rejected by many classicists because the Newtonian model was so reasonable. Similarly, QM lacks some "reasonable" elements ("God does not play dice") and even today scientists still seek to restore the determinism implied by putative undiscovered hidden variables.

Science provides us with the means to compare one theory with another. Zeno's theory failed, and he knew it was completely wrong when it was proposed! He simply didn't understand how to make sense of an argument that appears so reasonable, yet is utterly wrong.

 

[Adam]

The reason it is NOT a paradox, and the reason it is basically not worth discussing, and the reason why Zeno was a dumb-arse, is because it relies on a faulty assumption. Ie. it relies on the premise that we travel some fraction (half, quarter, whatever) of the remaining distance, when in fact we travel a certain amount regardless of what the remaining distance is.

To clarify:

Zeno assumes that you are going 1000 metres. In a step you go 500 metres. next step 250 metres, then 125, and so on.

In reality, if you are traveling 1000 metres or 5000 metres, you will take steps of, for example, one metre, then another metre, then another metre, and so on.

Zeno's paradox only exists if you are dumb enough to consider the possibility that you would be stupid enough to reduce the size of your steps constantly.

You might say "Well, it's supposed to be more abstract than that". Whoopey-doo. Either you step 500 metres then step another 500 metres, or you do the Zeno Shuffle and take ever-shortening steps for no real purpose.

There's no actual paradox going on. It's very simple: if you choose to continually divide any number by anything other than 1, you get a fraction.

 

[wuliheron]

Originally posted by DrChinese
Zeno's Paradox is successful because it seems so reasonable.

When Einstein first published on Special Relativity, it was rejected by many classicists because the Newtonian model was so reasonable. Similarly, QM lacks some "reasonable" elements ("God does not play dice") and even today scientists still seek to restore the determinism implied by putative undiscovered hidden variables.

Science provides us with the means to compare one theory with another. Zeno's theory failed, and he knew it was completely wrong when it was proposed! He simply didn't understand how to make sense of an argument that appears so reasonable, yet is utterly wrong.

How do you know Zeno believed he was wrong when he proposed motion was impossible? He asserted that the universe is indivisible, eternal, immortal, and unchanging. Today, string theory proposes all distances are really the same one short distance. Who knows, maybe Zeno will have the last laugh after all. :0)

 

[Lifegazer]

Originally posted by Tom
Zeno's mistake is in assuming that an infinite number of steps cannot be done in a finite amount of time. In other words, he tacitly assumes that it would require an infinite amount of time, and so could never be done.

Is that true? Let's see.

T=L/v=(1/v)&Sigma;_n=0^oo(1/2)ⁿ

The series on the right is a geometric series. In order for Zeno's argument to hold up, it would have to blow up to infinity--but it actually converges to 1.

But does it actually converge to '1'?
At what point does "Distance=L/2+L/4+L/8+..." converge to 'L'?

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.
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).

The flaw in Zeno's argument is in assuming the divergence of a series that actually converges.

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.

 

[Lifegazer]

Originally posted by DrChinese
Zeno's theory failed, and he knew it was completely wrong when it was proposed! He simply didn't understand how to make sense of an argument that appears so reasonable, yet is utterly wrong.

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.

 

[ahrkron]

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.

 

[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.

 

[ahrkron]

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.

 

[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 conscious 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.

 

[quantumdude]

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.

 

[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?

 

[quantumdude]

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.

 

[quantumdude]

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.

 

[ahrkron]

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.

 

[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 traveling 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.