Gold Member
Just curious.
Is everyone except me satisfied with the solutions to Zeno's paradoxes? see
It's been a while since I thought about these at any length, but I certainly remember never being satisfied with any solution to or refutation of them. If you are convinced they're rubbish, or quasiparadoxes, what convinced you?
Perhaps this belongs in a different forum, if so, please relocate :)
Happy thoughts
Rachel

EDIT- I didn't write what's on that site. I mentioned it because it's one of the more extensive. You can find brief explanations by just googling zeno's paradoxes.

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Honestrosewater:

You have done great work with the pirate problem.
I will study it tomorrow and replay to you there.

My belive it that everting is open even a mathematics true.

Thank you
Moshek

pig
That is a very long text and I am far too lazy to read it at the moment. Sorry about that :) But I will write something about one of the paradoxes mentioned - Achilles and the tortoise.

Let's consider a hippopotamus walking 2 m/s following an armadillo walking 1 m/s. The armadillo is 1 meter ahead of the hippopotamus.

What Zeno would say:

By the time the hippo reaches the dillo's original position d0, the dillo will have travelled to a position d1, half a meter ahead. By the time the hippo reaches d1, the dillo will be at d2, a quarter of a meter ahead, and so on. No matter how many times we do this, the dillo is always a little bit ahead. Therefore, the hippo will never reach him, and the dillo is safe from being crushed under its tremendous weight.

Where Zeno is right:

No matter how many times we do this, we will not reach the point where they are in the same spot.

What Zeno doesn't say:

No matter how many times we do this, we will not reach the point in time where 1 second has elapsed. Therefore, this doesn't prove that the hippo will never reach the dillo, since by Zeno's method, we never look far enough ahead in time. It only proves that at any moment in time before 1 second has elapsed, now matter how close, the dillo is ahead, which is true :)

That's why we say - As time tends to 1 second, the dillo tends to get stepped on. :) If we want to get to the 1 second mark Zeno's way, we must do it infinitely many times, and as we approach 1 second, the distance approaches zero. If you argue that it "doesn't make sense", and that the distance will "never actually reach zero" - then the time will also never actually reach one second.

I apologize if this point of view was mentioned in the text, or if the paradox still holds in some way in spite of it.

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Hurkyl
Staff Emeritus
Gold Member
To make an analogy... if you only look at red M&M's, does it make sense to conclude that no green M&M's exist?

That's the kind of impression I get from Zeno's paradoxes. You'll never see the end result in the sequence of events he studies, but why should one then conclude that the end result doesn't happen?

arildno
Homework Helper
Gold Member
Dearly Missed
If we set a distance travelled s equal to the product of velocity and time elapsed (s=v*t), I can't see that Zeno's observations amount to more than saying if the time elapsed is zero, then s is zero (which is true, since v*0=0 for any number v).
He then goes on to say this observation implies that v=0 (the so-called paradox).
I really can't see that implication..

Gold Member
To be fair to Zeno, we don't know exactly how he stated the paradoxes, so any criticism of him must bear that in mind.

pig, yes, nice effort :) So if you can define a (dimensionless) point on the hippo, for its position, the point the hippo must reach, for the limit, and measure infinitely small intervals of time, you have three less problems :)
How would you translate the absract mathematics to the physical world?

Hurkyl, my analogy- if you call a tail a leg, how many legs does a mule have? 4. Calling a tail a leg doesn't make it one. (before someone jumps on me, 4, assuming the obvious) That's the kind of impression I get from the solutions- they may be correct in themselves, but do not address the issue.

arildno, I think I missed something else in your argument, but can't put my finger on it yet.
Afraid I'm too tired now to get into a long post. Nothing moves until it moves. Case closed :) No, tomorrow...
Happy thoughts
Rachel

Funny thing about Zeno is that... You can actually see the paradox in the summation of Zeno's paradox... Its 1!!! Hehe...

pig
Rachel,

If you wish to avoid dimensionless points, let's define the distance between the two as the distance between the hippo's nose and the tip of the dillo's tail.

If you wish to avoid limits and working with infinitely small numbers, then no matter how many times you divide, the distance to be travelled is always strictly greater than zero, but note that the elapsed time is then also always strictly less than 1 second, so the hippo doesn't reach the dillo within a second which is correct.

You get the paradox when you try to put infinity into this in an incomplete way - if you divide their movement into infinitely many steps, but don't divide the time proportionally. The possible reason is that it takes you a couple of seconds to imagine each step and would take forever to imagine the whole movement, but it doesn't work that way for them - in reality required time gets smaller and smaller for each step.

What Zeno does is basically divide a finite quantity into infinitely many parts, and say that because there are infinitely many parts, their sum is infinite. :)

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arildno said:
If we set a distance travelled s equal to the product of velocity and time elapsed (s=v*t), I can't see that Zeno's observations amount to more than saying if the time elapsed is zero, then s is zero (which is true, since v*0=0 for any number v).
He then goes on to say this observation implies that v=0 (the so-called paradox).
I really can't see that implication..
The implication is that you can't determine distance travelled by adding up the distance travelled at each individual instant in time. If you assume that you should be able to calculate distance in that manner than this is a paradox.

The problem with Zeno's paradoxes is that solving them generally involves changing definitions of terms like distance and velocity. That seems like "cheating" since you haven't really solved the paradox, just changed your definitions to avoid it; people are often very attached to their definitions of basic terms like distance.

pig
master_coda said:
The implication is that you can't determine distance travelled by adding up the distance travelled at each individual instant in time. If you assume that you should be able to calculate distance in that manner than this is a paradox.
By this same logic, you couldn't have divided a nonzero time interval into "each individual instant in time" in the first place:

If by summing the distances you get 0 meters because the travelled distance is 0 in each individual moment, then by summing the intervals you also get 0 seconds because each individual interval is 0 seconds, and since the original interval was nonzero, the division is obviously incorrect.

In both of these paradoxes, as Zeno divides distance, he also divides time, so he should treat them the same. He doesn't. You cannot ignore the concept of time when dealing with speed, because speed is defined as distance in time. In the hippo case, Zeno says "never" on the basis of infinite number of steps required, but that is irrelevant, what is relevant is the time all those steps take.

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pig said:
By this same logic, you couldn't have divided a nonzero time interval into "each individual instant in time" in the first place:

If by summing the distances you get 0 meters because the travelled distance is 0 in each individual moment, then by summing the intervals you also get 0 seconds because each individual interval is 0 seconds, and since the original interval was nonzero, the division is obviously incorrect.
Even more generally: take an interval of real numbers [a,b] with b>a. It is composed of points. Those points have zero length. Therefore the sum of all those lengths is zero. Therefore the interval has zero length. But the interval has length b-a>0 so this is a paradox.

Zeno's paradoxes rely on this paradox a great deal: take a concept that is well defined on intervals, extend that concept to points, then demonstrate a paradox.

Gold Member
pig said:
Rachel,

If you wish to avoid dimensionless points, let's define the distance between the two as the distance between the hippo's nose and the tip of the dillo's tail.
But exactly! Where exactly does the hippo's nose end and the "not hippo's nose" begin? You cannot avoid dimensionless points because the distance between the hippo's nose and the dillo's tail will eventually be smaller than any dimensional "point" you put on them. So you must then make your "point" smaller, only to eventually have to make it smaller again... this has nothing to do with time. It has to do with infinitely divisible space.

Even if you say: these two atoms, x and y, are touching; x is on the hippos nose and y is not on the hippos nose. And use the point where the two atoms touch as your "point". But then you have only changed the problem to the distance between any two "touching" atoms. Where do two atoms touch? You must now define where an atom ends and "not an atom" begins.

Happy thoughts
Rachel

Gold Member
master_coda said:
Even more generally: take an interval of real numbers [a,b] with b>a. It is composed of points. Those points have zero length. Therefore the sum of all those lengths is zero. Therefore the interval has zero length. But the interval has length b-a>0 so this is a paradox.

Zeno's paradoxes rely on this paradox a great deal: take a concept that is well defined on intervals, extend that concept to points, then demonstrate a paradox.
Thank you! There is no "next" real number! There is no "next" rational number either. The integers have "next" numbers, because they are constructed from the unit 1. You cannot define length until you define length. This is Zeno's Dichotomy paradox. There is no "next" position.

Remember Russell's paradox? I think it is related. If you want to define a set by listing its elements, you cannot define a set that contains itself. After every definition, you must extend the definition.

Happy thoughts
Rachel

Gold Member
Perhaps Zeno saw the difficulties in dividing space infinitely and in time infinitely and decided to combine them, using motion. He also has plurality paradoxes which treat only space.

Happy thoughts
Rachel

honestrosewater said:
Perhaps Zeno saw the difficulties in dividing space infinitely and in time infinitely and decided to combine them, using motion. He also has plurality paradoxes which treat only space.

Happy thoughts
Rachel
But expressing the paradoxes in terms of space and time attaches useless baggage to them. These aren't paradoxes within space and time, only our attempts to define them with mathematics.

honestrosewater said:
Remember Russell's paradox? I think it is related. If you want to define a set by listing its elements, you cannot define a set that contains itself. After every definition, you must extend the definition.
Russell's paradox isn't about defining sets as lists of their elements. Nor is it a problem with sets that contain themselves. It's a problem with unrestricted comprehension...the idea that all we need to define a set is to provide a rule for determining what is contained in that set.

honestrosewater said:
Thank you! There is no "next" real number! There is no "next" rational number either. The integers have "next" numbers, because they are constructed from the unit 1. You cannot define length until you define length. This is Zeno's Dichotomy paradox. There is no "next" position.
I don't see how the fact that there is no next position is interesting. It's rather trivial to demonstrate without Zeno's paradoxes.

Gold Member
master_coda said:
But expressing the paradoxes in terms of space and time attaches useless baggage to them. These aren't paradoxes within space and time, only our attempts to define them with mathematics.
But Zeno couldn't express them by saying "take an interval of real numbers [a,b] with b>a..." I don't see what you mean.

master_coda said:
Russell's paradox isn't about defining sets as lists of their elements. Nor is it a problem with sets that contain themselves. It's a problem with unrestricted comprehension...the idea that all we need to define a set is to provide a rule for determining what is contained in that set.

What about the idea that all we need to define a path is to provide a rule for determining what is contained in that path?
You must also specify the range, what can be contained in that set, or to what the rule will apply?
The path is a set of points. But is the path not also a set of intervals? Think of a point as an element (an element is not a set). Then an interval is a set (an interval contains points). So how can you switch from points to intervals in the same rule? Wouldn't you need two different rules, one that applies to points, and one that applies to intervals? And even to intervals that contain intervals? Not to mention applying the same rules to physical objects and mathematical objects. Perhaps I am not saying this correctly. Or maybe I'm just wrong.

master_coda said:
I don't see how the fact that there is no next position is interesting. It's rather trivial to demonstrate without Zeno's paradoxes.
Same as above, Zeno didn't have modern mathematics. I think he is expressing the same concept that is expressed using mathematics. But the mathematical solution or explanation does not make sense for space and time, for the physical world.

Happy thoughts
Rachel

honestrosewater said:
The path is a set of points. But is the path not also a set of intervals? Think of a point as an element (an element is not a set). Then an interval is a set (an interval contains points). So how can you switch from points to intervals in the same rule? Wouldn't you need two different rules, one that applies to points, and one that applies to intervals? And even to intervals that contain intervals? Not to mention applying the same rules to physical objects and mathematical objects. Perhaps I am not saying this correctly. Or maybe I'm just wrong.
Yes, this is the idea...we have to be careful not to extend definitions beyond the point where they make sense. We can find the distance travelled over two intervals of time by adding together the distance travelled over each interval. That doesn't mean that we can find the distance travelled over an interval by summing the distance travelled at each point.

Zeno's paradoxes aren't really paradoxes in nature...they're paradoxes in mathematics. When we try and use that inconsistent math to define nature then the illusion is created that there are paradoxes in nature itself. The solution is to use math which is not inconsistent.

This might seem unsatisfactory. After all, we can't "redefine nature". But the abstractions that we use to describe nature can be redefined.

honestrosewater said:
Same as above, Zeno didn't have modern mathematics.
I certainly don't fault Zeno for not being able to apply the knowledge accumulated in the many years after he died. But that doesn't mean we can't apply that knowledge now.

When we express the problems Zeno brought up in terms of space and time, we complicate the issue. Everyone has preconceived notions of what space and time are, and those notions get in the way of discussing the core issue "how do we measure infinite sets?" We can discuss the physical world after we fix up the math.

Gold Member
So how do we measure infinite sets?

I also remember running into problems when "learning" about air resistance on a falling body- elementary, right? As I recall, an object can encounter n meters/second of resistance without ever reaching a speed of n m/s (unless it was travelling n m/s while at rest, before it was dropped).
i.e., it slowed down this much because it would have encountered this much resistance if it hadn't slowed down ;) Maybe I just didn't get it. Seems like a lot of information being exchanged instantaneously. Guess it boils down to instantaneous velocity which I never liked. Maybe I just don't get lots of stuff. Maybe it's because I saw this strange French film last night and am feeling rather pessimistic and broody.

Happy thoughts (if it matters)
Rachel

pig
honestrosewater said:
But exactly! Where exactly does the hippo's nose end and the "not hippo's nose" begin? You cannot avoid dimensionless points because the distance between the hippo's nose and the dillo's tail will eventually be smaller than any dimensional "point" you put on them. So you must then make your "point" smaller, only to eventually have to make it smaller again... this has nothing to do with time. It has to do with infinitely divisible space.
I now understand what you are talking about. From my experience, most people conclude that mathematically, one object will not reach another no matter how much time passes - they keep slowing down the hippo in their "thought experiment", but not time. :) And that's not where the paradox is. I hope you understand what I'm trying to say. I originally assumed you were making that mistake.

I don't really have anything smart to add here since master coda explained it better than I could. The problem is how we deal with infinity, not in nature itself. :)

[edited to remove something stupid and irrelevant.. note to self - read, comprehend, post]

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arildno
Homework Helper
Gold Member
Dearly Missed
Excellent discussion of the meaning of Zeno's paradoxes, master_coda and honestrosewater!
Clearly, I've been much too dismissive of his thoughts.
Thx a lot!

honestrosewater said:
So how do we measure infinite sets?

I also remember running into problems when "learning" about air resistance on a falling body- elementary, right? As I recall, an object can encounter n meters/second of resistance without ever reaching a speed of n m/s (unless it was travelling n m/s while at rest, before it was dropped).
i.e., it slowed down this much because it would have encountered this much resistance if it hadn't slowed down ;) Maybe I just didn't get it. Seems like a lot of information being exchanged instantaneously. Guess it boils down to instantaneous velocity which I never liked. Maybe I just don't get lots of stuff. Maybe it's because I saw this strange French film last night and am feeling rather pessimistic and broody.

Happy thoughts (if it matters)
Rachel
I prepared a nice reply to this, but my computer ate it. Now I'm pessimistic and broody.

Tom Mattson
Staff Emeritus
Gold Member
master_coda said:
I prepared a nice reply to this, but my computer ate it. Now I'm pessimistic and broody.
Confucius say, when writing a long post, do it on MS Word, cut, and paste.

Happy thoughts,

Tom Mattson said:
Confucius say, when writing a long post, do it on MS Word, cut, and paste.

Happy thoughts,
That doesn't sound like Confucius. It's more of a Zen saying. Telling people to protect themselves from computer crashes by writing in MS Word is like telling people to protect themselves from car crashes by driving into oncoming traffic.

Gold Member
Who said the second mouse gets the cheese? heehee

Great, I feel like I've actually learned something :) Clean up the math. Then go count the sheep. Thanks everyone.
Happy thoughts
Rachel

Oh dear Zenon

Honestrosewater , great Thread you posed here!

Zenon paradox will live forever !

No one can really solved it and also be happy for that.
It show in amassing way that the way we see realty
is a function of the language we use to describe it.

Mathemattion today should not be afraid from any Paradox in mathematics on the contrary it is the only place were mathematics live today when the binary logic way of thinking have no more real significant.

So think really as quantum mechanic is working !

Moshek