Runner's Paradox: Finishing the Race in d Metres

[sage]

you have heard this before perhaps. its about a runner trying to run d metres. he covers d/2 in t1 second, then half of the distance that is left in t2 seconds, then half of the rest in t3 seconds and so on.as there is always a finite distance left, according to the paradox he can never cover d metres. so how does he do it?

 

[Hurkyl]

This infinite sequence of actions can be accomplished in finite time, so he does them all and then keeps going.

 

[AndersHermansson]

We had this over at sciforums recently. The short answer is that a sum of infinite series can be finite, which is where it might seem confusing. So that if you add an infinite amount of lengths the total length can still be finite. So the original question simply assumes it is not so.

 

[quartodeciman]

"there is always a finite distance left"

really means

"there is for any time before d/v (, with v being the speed of the runner) a finite distance left".

 

[sage]

yes this infinite sequence converges. but the point is if we go on adding the successive elements of the sequence one by one (as must be done here) we never reach the end of the sequence precisely because it is infinite. as we cannot reach the end of the sequence we cannot cover this finite distance in the calculated finite time. consider the finite time interval between n-th second and n+1-th second. first half a second passes by, then another 1/4-th, then another 1/8-th and so on. another infinite sequence converging at the limit, but that limit can never be attained. that is the problem.

 

[Hurkyl]

But why should one think that sequence of events cover the entire range of motion? Try this transfinite sequence:

Cover half the distance.
Cover half of what's left.
Cover half of what's left.
...
(countably finite repetitions)
...
Arrive at the destination.


Each step in the sequence picks up right there you left off if you perform all previous steps, includes the "Zeno sequence", and continues on afterwards to arrive at the destination.

 

[drnihili]

You have to be quite clear on what the question is. If you take Zeno to merely be asking how an infinite sequence can occupy a finite space, then calculus indeed answers the question. However, if you taking him to be asking the question of how one can complete and infinite sequence one member at a time, then calculus not only doesn't answer the puzzle but is entirely irrelevant to it. I think the latter question is the better way to understand the point of the paradox.

There are a host of related paradoxes which highlight the central issues. SOmetimes it helps to look at them instead of just the runner paradox.

 

[drnihili]

Originally posted by Hurkyl
But why should one think that sequence of events cover the entire range of motion? Try this transfinite sequence:

Cover half the distance.
Cover half of what's left.
Cover half of what's left.
...
(countably finite repetitions)
...
Arrive at the destination.


Each step in the sequence picks up right there you left off if you perform all previous steps, includes the "Zeno sequence", and continues on afterwards to arrive at the destination.

Ah, but this sequence can't be right. It presumes that after you've completed all the half distances you still have to do something further to arrive. If your sequence were correct, it would be possible to travel all the distances and yet still fail to arrive. But arriving cannot amount to traversing a distance or you give up the continuity of the reals. So on your account two runners could travers precisely the same distance and yet one of them would run d meters and the other wouldn't.

 

[Hurkyl]

It presumes that after you've completed all the half distances you still have to do something further to arrive.

Covering all of the half distances means covering the interval [0, d). If I run 1 meter per second, I cover all the half distances over the time interval [0, d).

You actually have to get to time d to have arrived at distance d. Zeno's paradox is a paradox because it presumes that you can't continue beyond the infinite sequence of covering half distances.

By continuity, any possible continuation of motion would have to include being at distance d at time d.

 

[drnihili]

Originally posted by Hurkyl
Covering all of the half distances means covering the interval [0, d). If I run 1 meter per second, I cover all the half distances over the time interval [0, d).

You actually have to get to time d to have arrived at distance d. Zeno's paradox is a paradox because it presumes that you can't continue beyond the infinite sequence of covering half distances.

By continuity, any possible continuation of motion would have to include being at distance d at time d.

The problem is that the open and closed intervals have the same distance. Closing the interval does not add any distance. Continuity comes in because the LUB of the two intervals is the same. If the runner really has completed all of the open intervals, he must have arrived at d.

Suppose otherwise, i.e that the runner has completed [0, d) but has not yet arrived at d. Call the runner's position r. r must be between the open interval and d. But this contradicts the fact that d is the least upper bound of the interval. So if r<d, then r must be in the open interval. But if r is in the open interval, then the runner has not yet completed the interval. This is because for every point in the interval there are infinitely many other points beyond it that are still in the interval. So r cannot be in the interval. thus the earliest point which can be r is d.

And the paradox isn't that you can't continue beyond the open interval, it's that you can't complete the interval at all.

 

[Hurkyl]

I'm aware the lengths of [0, d) and [0, d] are the same.


Anyways, a paradox is typically a contradiction that arises from an unfounded assumption. They usually get cleared up once you try to do everything rigorously.

So tell me, as precisely as possible, what you think the problem is.

 

[drnihili]

Well, I don't think I agree with your view of what a paradox is, but we'll leave the general theory of paradox for another thread.

The paradox in this case is that the runner, Achilles, must accomplish an infinite sequence of tasks. We know that he can complete them, we can even calculate precisely by when he will have completed them. The problem is in explaining how he completes them.

Achilles starts out with an infinite number of tasks to do. By the description of the problem, he must complete them one at a time. After he has accomplished his first task, there are an infinite number of tasks left. After he completes his second taks, there are an infinite number of tasks left. In fact after each task that he completes, there's always an infinite number left. As he moves down his list of tasks, he never gets any closer to the end of it. He always has just as many left to do as he started out with. As long as he is still working on the list, he has infinitely many left. The first point at which he has fewer than infinitely many tasks left is when he is all done, and at that point he has zero. He never decreases his list, he just suddenly finds that it is already done. So how is it that he manages to get to the end?

Geometry can predict the point at which Achilles will be done. Calculus can explain how it is that all the decreasing segments have a finite sum. But neither of them explains how it is that Achilles counts through the list, one task at a time - how he manages to complete an endless sequence.

 

[Hurkyl]

You still haven't answered the big question; why should an infinite sequence of tasks be impossible?


In particular (if I'm predicting your response correctly), why should every task in a sequence of tasks have a previous and a next task? (except, of course, for the first and last task, should they exist)

 

[drnihili]

Originally posted by Hurkyl
You still haven't answered the big question; why should an infinite sequence of tasks be impossible?


In particular (if I'm predicting your response correctly), why should every task in a sequence of tasks have a previous and a next task? (except, of course, for the first and last task, should they exist)

Because there's a function that given any task in the sequence returns the next task, and another function that returns the previous. If you take an ordering that lack that property it gets even more difficult. But Zeno's ordering does have the property.

 

[Hurkyl]

But why should an infinite series of tasks be impossible?


The resposne I was anticipating was something equivalent to saying that in my sequence of tasks, there is no task previous to "arrive at d". (it is eqiuvalent to say that there is no last task in Zeno's sequence)

 

[drnihili]

That response doesn't quite get it right. I've tried to explain it a couple times, but I'll have another go at it.

If Achilles accomplishes an infinite series of tasks, there must be some action of his which counts as completing all the tasks. But none of the tasks can be that action as each of the tasks leaves an infinite number remaining. So, if Achilles accomplishes all the tasks, then there must be something he does beyond the tasks themselves in virtue of which he can be said to have completed them all. By the description of the problem, there is no such action.

If there were such an action, then it would be theoretically possible for Achilles to accomplish each of the tasks and yet still fail to complete all of them. This is absurd. Hence there can be no such action.

 

[Hurkyl]

If Achilles accomplishes an infinite series of tasks, there must be some action of his which counts as completing all the tasks.

For the problem at hand, there must be some task which counts as the completion of all (previous) tasks, though this isn't always the case. But the question is why must that task be one of the infinite series of tasks?

Continuity (and completeness) guarantees that there must be a unique limiting event, but it does not guarantee that the unique limiting event must be one of the members of the infintie sequence.

In particular, the limiting task is the "arrive at destination" step I listed.

 

[drnihili]

Originally posted by Hurkyl
For the problem at hand, there must be some task which counts as the completion of all (previous) tasks, though this isn't always the case. But the question is why must that task be one of the infinite series of tasks?

Continuity (and completeness) guarantees that there must be a unique limiting event, but it does not guarantee that the unique limiting event must be one of the members of the infintie sequence.

In particular, the limiting task is the "arrive at destination" step I listed.

Obviously it can't be one of the listed tasks. But your proposal is no solution. What exactly does one do to arrive at the destination and when does one do it? Do you really mean to imply that one might complete each of the tasks and still not arrive at the destination?

 

[Hurkyl]

But your proposal is no solution. What exactly does one do to arrive at the destination and when does one do it?

One traverses the position interval [0,d) over the time interval [0, d). That is sufficient to be at position d at time d. (I'm assuming the traversal is in the manner being discussed)

Do you really mean to imply that one might complete each of the tasks and still not arrive at the destination?

I mean to imply that one does not reach the destination during the time interval in which one is performing Zeno's tasks. In this case, the time interval [0, d). One arrives at the destination at time d, after all of Zeno's tasks have been completed.

 

[drnihili]

Originally posted by Hurkyl
One traverses the position interval [0,d) over the time interval [0, d). That is sufficient to be at position d at time d. (I'm assuming the traversal is in the manner being discussed)

Here you've essentially said that completing all the tasks is sufficient for arrival. But you haven't said how that is accomplished. I agree that it's sufficient, that's not the issue. The issue is saying how it is done.

I mean to imply that one does not reach the destination during the time interval in which one is performing Zeno's tasks. In this case, the time interval [0, d). One arrives at the destination at time d, after all of Zeno's tasks have been completed.

This can't be right. One doesn't first complete the tasks and then arrive. If that were the case then there would have to be a moment in between finishing the tasks and arriving. (given infinite divisibility.) But that would contradict what you said above about completing the tasks being sufficient for arriving. Arriving can't be separate from completing all the tasks. It can't occur after completing them, nor can it occur before completing them. It has to occur simultaneously with completing them. But this still leaves the problem of saying what it means to complete and endless sequence.

 

[Hurkyl]

The issue is saying how it is done.

You do it by crossing the entire path between you and the destination. What's wrong with that?

If that were the case then there would have to be a moment in between finishing the tasks and arriving. (given infinite divisibility.)

Why must there be a moment between finishing the tasks and arriving? There is zero time between finishing the tasks and arriving at the destination; no matter how you infinitely divide zero, all of the pieces must be zero.

 

[drnihili]

Originally posted by Hurkyl
You do it by crossing the entire path between you and the destination. What's wrong with that?

That's just begging the question.

Why must there be a moment between finishing the tasks and arriving? There is zero time between finishing the tasks and arriving at the destination; no matter how you infinitely divide zero, all of the pieces must be zero.

If there is zero time between the two events, then they are simultaneous. You stated one was after the other.

 

[Hurkyl]

If there is zero time between the two events, then they are simultaneous. You stated one was after the other.

But we're not talking about the time between two individual events, are we?

 

[drnihili]

Originally posted by Hurkyl
But we're not talking about the time between two individual events, are we?

If we are not, then there must be just one event. In that case, please say what that event is, and what specific action of Achilles' brings it to pass.

Also, if it is just one event, then I'm puzzled why you said it occurred after itself.

 

[Hurkyl]

We are talking about the time between a single event and an infinite sequence of events. There is a big difference there, and this time may be zero, even if the single event occurs strictly later than every event in the infinite sequence.

 

[drnihili]

Originally posted by Hurkyl
We are talking about the time between a single event and an infinite sequence of events. There is a big difference there, and this time may be zero, even if the single event occurs strictly later than every event in the infinite sequence.

Again, without begging the question this time, what is this single event, and what action does Achilles take to bring it about?

 

[Hurkyl]

The single event is arriving at the destination. He does this by covering all the ground between his starting point and this destination.

 

[drnihili]

Originally posted by Hurkyl
The single event is arriving at the destination. He does this by covering all the ground between his starting point and this destination.

You'll recall I asked for a non-question begging answer.

 

[Hurkyl]

I don't see what's wrong with this task.

But instead of debating the merits of this description, I'll ask how this is any different from any of the other tasks in Zeno's sequence? How is "Go from here to there" any different from "Go from here to half way between here and there"?

 

[NeutronStar]

Originally posted by sage
you have heard this before perhaps. its about a runner trying to run d metres. he covers d/2 in t1 second, then half of the distance that is left in t2 seconds, then half of the rest in t3 seconds and so on.as there is always a finite distance left, according to the paradox he can never cover d metres. so how does he do it?

Sage, I haven't read every detail of this thread because it appears to have deteriorated into a mathematics argument that can never be fully justified in a comprehensible intuitive way. The inevitable conclusion of any such mathematical arguments can only be had by accepting the abstract axioms of mathematics and forfeiting an intuitive comprehension of any physical explanation of quantity.

For a physical explanation I would like to offer the following:

In physics we have discovered that the nature of the universe is indeed quantized. For this reason it is not physically possible to continue to divide up time and distance in half indefinitely. There comes a point when we reach a length of distance that has no half distance. In other words, it makes no sense to talk about space between these points. Therefore Zeno's paradox is not a paradox at all.

Zeno's paradox would only be a paradox if we lived in a purely continuous universe. But we don’t. We live in a quantized universe. Therefore there is no paradox.

It may very well be impossible to move if the universe is indeed continuous.

 

[ahrkron]

The resolution of the paradox lies within well understood math (convergence of series). The universe may well have continuous quantities, and they would still be able to change. What we need to understand is how our description of such changes has to be used.

 

[Hurkyl]

The gross mischaracterization of mathematics aside...

Zeno's paradox is only a paradox when one makes unfettered attempts at extrapolating the properties of finite sets to those of infinite sets without any attempt at proof. (though at this point, it's merely boiled down to drnihili complaining that I don't have acceptable semantics for describing motion)


As for the physical theory, only bound systems have been shown to be quantized. Free particles are not quantized. And no aspect of the geometry of space-time has yet been shown to be quantized. Even in theoretical physics, no theory implies that distance is quantized (though LQG implies area and volume are quantized).

 

[NeutronStar]

Hurkyl wrote:
As for the physical theory, only bound systems have been shown to be quantized. Free particles are not quantized. And no aspect of the geometry of space-time has yet been shown to be quantized. Even in theoretical physics, no theory implies that distance is quantized (though LQG implies area and volume are quantized).

It would seem to me that Zeno being a macro object would be made up of bound particles. He could hardly be thought of as an unbound system. The same would go for his start and finish lines that would necessarily be locations on a macro scale. (After all, if he isn't moving relative to some other macro object can he even be said to be moving at all?)

As far as distance being quantized goes, it follows from Planck's constant of energy. After all, if energy is quantized then so must be time. And of course if time is quantized then so must be distance. I'm sure that I've seen references to the Planck length of quantized distance. In fact I believe that Brian Green refers to the Planck distance in his book on string theory called "The Elegant Universe". I think it was on the order of ten to the -33 centimeters or something like that. A distance that below which it is absurd to talk about space as a dimension.

In fact, I quite sure that he also referred to the concept of the unit of Planck time in that book too. A time duration below which time no longer holds meaning.

It would seem to me that these concepts would need to apply to free particles as well as bound ones. But maybe not, since free particles are really nothing more than free waves of probability while bound ones are restricted to standing waves of probability.

Alright, so I have no idea what I'm talking about. Big deal. It still makes more sense to me than the abstract mathematical converging series. (See my next post in response to the capabilities of calculus)

 

[Hurkyl]

It still makes more sense to me than the abstract mathematical converging series.

One does not need to consider infinite series to describe motion through a connected space.

 

[NeutronStar]

Ahrkron wrote

The resolution of the paradox lies within well understood math (convergence of series). The universe may well have continuous quantities, and they would still be able to change. What we need to understand is how our description of such changes has to be used.

"well understood math"?

I remember studying the series convergences in calculus II. In fact, I just had a refresher course in calculus II this last spring. We did many series convergence problems.

I disagreed with those conclusions. I mean from a static or absolute point of view.

In other words, all of calculus is based on the idea of the limit. This is the foundation of calculus. Everything in calculus rests upon the definition of the limit.

Well, if you learn that definition well, you can clearly see that mathematicians use it incorrectly all the time. The definition for the limit of f(x) at c clearly states:


For every epsilon greater than zero there exists a delta greater than zero such that,
If the distance between x and c exists and is less than delta.
Then the distance between f(x) and L exists and is less than epsilon,


Yet mathematicians never fail to claim that calculus can prove something about when epsilon equals zero

That's totally invalid. The definition of the limit is dependent on the fact that epsilon is greater than zero. As soon as epsilon actually becomes zero the formalism of calculus is no longer applicable. Yet mathematicians continually claim that calculus proves that something like a converging series actually converges when epsilon equals zero.

It does not prove this, nor can it be used to prove this.

All it can possibly prove is that as the series converges it gets close to some number. Period amen. To claim that it actually converges is to totally ignore the definition of the limit upon which calculus rests.

All of calculus rest on the definition of the limit. And anyone who truly understands that definition should clearly undestand that calculus cannot make any statements whatsoever about what might happen should epsilon actually become zero. Such statements are outside of the scope of calculus.

Edited to correct the order of the if-then statement in the definition of the limit. Sorry about that.