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:
[/color]
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[/color] 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[/color]

That's totally invalid. The definition of the limit is dependent on the fact that epsilon is greater than zero[/color]. 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[/color].

It does not prove this, nor can it be used to prove this.[/color]

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[/color]. 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.

 

[ahrkron]

Originally posted by NeutronStar
We did many series convergence problems.

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

Mathematicians use the same reasoning capabilities as you and I. Theorems are proven using such logic. "Points of view" (static, absolute or otherwise) need not enter in the discussion.

Well, if you learn that definition well, you can clearly see that mathematicians use it incorrectly all the time.

Either that, or you misunderstood the use they make of it.

Do you seriously sustain that all mathematicians since the formalization of calculus[/color] have used a definition incorrectly? I find that extremely hard to believe.

But it is not a matter of faith. I have gone over the resoning myself, as do quite a few high schools students (and college students and grad students and profesional mathematicians) every year, and find the to be quite clean.

The definition for the limit of f(x) at c clearly states: ...

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

There's no such "claim". The definition of limit allows you to consider situations in which:
1. the case epsilon=0 is not relevant for the discussion at hand,
2. the value at epsilon=0 is not defined
3. the value at epsilon=0 is different from the limit.
4. such value is defined and is equal to the limit.

When studying the behavior of a series, it is possible to use these tools. No dogmas or errors are involved; rather, further concepts are founded upon these studies.

calculus cannot make any statements whatsoever about what might happen should epsilon actually become zero[/color]. Such statements are outside of the scope of calculus. [/B]

No, they're not. Look up the definition of continuity. It is one of the basic concepts in calculus and it is concerned with the case epsilon=0. It is not an invalid extrapolation, but a case that has to be considered due to its enormously frequent occurrence on the formalism.

 

[NeutronStar]

Oops! I do owe you an apology. I have made a terrible error in my last post.

I was thinking in terms of the calculus limit[/color], but I was just typing the word calculus alone.

The reason that I did this is because our discussion is focusing on the convergence of a series, and so for all intents and purposes we are talking solely about a one-sided limit. We also must take note that we have no functional values for the value c in question (which would be the sum of the series). We are using a one-sided limit to come up with this value.

This is the context in which our discussion resides.

So if I may, I would like to rephrase my last comment by adding the following words in red[/color].

The calculus limit[/color] cannot make any statements whatsoever about what might happen should epsilon actually become zero. Such statements are outside of the scope of the calculus limit[/color].

Now, in response ahrkron gave the following comment.

Originally posted by ahrkron

No, they're not. Look up the definition of continuity. It is one of the basic concepts in calculus and it is concerned with the case epsilon=0. It is not an invalid extrapolation, but a case that has to be considered due to its enormously frequent occurrence on the formalism.

I am well aware of the definition (and limitations of) continuity.

If you carefully read all of the requirements for the definition of continuity you will first notice that we must have two limits. In the case of Zeno's paradox we have only one limit. (Also, in the case of a convergent series we have only one limit).

This in an of itself forbids any mention of continuity in this problem. The definition of continuity cannot be applied to a single-sided. This is a direct violation of its very own definition.

Secondly, even if we had two limits (one from the right, and one from the left), and they both existed and were the same value, we still can't conclude that a function is continuous unless we know the functional value at the point in question.

Why? Because in the very definition of continuity, it clearly states that continuity can only be said to exist if the functional value agrees with the values of the two limits. Therefore, it is not possible to conclude continuity without knowing the functional value at the point in question. This is clearly a restriction of the definition of continuity.

So even if we did have two limits here, we still could not conclude continuity even if the two limits were the same. Because we don’t' have a functional value for the point in question. The definition of continuity requires that we know this value!

So again, I apologize for my last post. The things that I said about calculus in general I meant to say about the calculus limit[/color].

It is true that we can make statements about when epsilon equals zero (if and only if) we know the functional value at the point in question.

However I must also point out here that this knowledge of the value of the point when epsilon equals zero does not come from calculus, it comes from the functional value! In other words, if we have a function and we know the value of the point in question we don’t' need calculus to find it! Not to mention the fact that calculus is completely useless to prove the existence of such a point if we don't know the functional value in the first place. The best calculus can do is say that the function has a limiting value as it approaches[/color] this point. Calculus in and of itself cannot make any direct statements about the existence of such values at the point where epsilon equals zero (without referring to the functional value at that point).

What we use can use calculus for, is to say that the entire function is continuous at a point. (Assuming of course that we know the functional value at that point.)

So at any rate, the concept of continuity cannot even be applied to Zeno's paradox (or the convergence of a series). Because in both of these cases we have only one limit, and we do not know the function value at the point in question. We would need to have both[/color] of these conditions present before we could even begin to apply the concept of continuity.

Continuity has no application in this problem. Pure and simple.

 

[Hurkyl]

I'll go off on this tangent to reply to your statements about calculus (though we are deviating quite a bit from Zeno's paradox):

If you read carefully the definition of a continuity, it says that f is continuous at a iff

f(x) -> f(a) as x -> a

(this means the limit of f(x) as x approaches a is f(a); this notation is much cleaner for text-based communication)

The definition of continuity says nothing about one-sided limits.


If you recall, the business about proving f(a) is equal to both the left-sided and the right-sided limits was based on a theorem about limits when the domain is the real numbers. In particular:

f(x) -> L as x -> a
iff
f(x) -> L as x -> a⁺ and f(x) -> L as x -> a^-

This theorem rests heavily on the fact that for any real number a, we can separate the real numbers into a left hand side and a right hand side. More generally, this theorem works for any interior point of an ordered space. However, when our domain does not permit such seperation, we can't even define one-sided limits, let alone prove this theorem.



Now, recall the definition of an infinite series;

Letting S(n) = sum over x = 1 .. n of f(x):

S(n) -> sum over x = 1 .. &infin; of f(x) as n -> &infin;

iow the infinite sum is equal to the limit of the sequence partial sums.

If we extend the domain of S(n) to the extended natural numbers by defining

S(&infin;) = sum over x = 1 .. &infin; of f(x)

then S(n) is a continuous function of n at &infin;! The definition of the infinite series is precisely the definition of continuity:

S(n) -> S(&infin;) as n -> &infin;


Also, it is worth recalling that the &epsilon;-&delta; definition of limits is only used when the domain and range of the limit are finite real numbers. You may recall doing &epsilon;-M or N-&delta; or even N-M limit proofs in your calc classes... alternatively one may use a topological definition of limits that is equally applicable to both the finite and infinite cases.

Secondly, even if we had two limits (one from the right, and one from the left), and they both existed and were the same value, we still can't conclude that a function is continuous unless we know the functional value at the point in question.

Quite often, we're told from the outset that the function in qusetion is continuous. In such cases, finding the limit of the f(x) as x approaches a yields the value of f(a).

and we do not know the function value at the point in question.

But we do; we know the limit of the sequence of positions is the destination point.


Also, one should note that the term "continuous" when applied to topological spaces is not the same term as "continuous" when applied to functions.

 

[NeutronStar]

(P.S. for the sake of correctness, I should point out you have the if statement in your definition of continuity backwards)

Yes, I did type that in backwards. Thanks for pointing that out.

Some of my math books print definitions as "whenever" statements, and some of them print them as "if-then" statements.

I prefer to use the "if-then" style. Unfortunately I was looking at a book that uses the "whenever" style when I typed the information into my post. During the translation to the "if-then" style I forgot to swap things around.

Sorry for the confusion. I've corrected that post for future readers.

 

[drnihili]

Originally posted by 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"?

The problem with the task is that it is comprised simply of all the other tasks. So you have a sequence of discrete tasks, and a further task which is doing all of them.

Since the original problem is to say how it is that Achilles can complete the entire series, saying that he does is by completing all the tasks is no help - it just begs the question.

The "all tasks" task is different in that it requires no action on Achilles' part that is not already required by some other task on the list. It can be dropped from the list of required task with absolutely no effect on Achilles. In this sense it is an empty task.

Since the task is empty, and since it's proposal merely begs the question, it cannot constitute an answer to the paradox.

Achilles must complete an infinite series of tasks. We know that completing them only requires a finite period of time. But we don't know how he can complete them at all.

 

[Hurkyl]

But we also don't have a logical reason why he can't complete them.

 

[drnihili]

Originally posted by Hurkyl
But we also don't have a logical reason why he can't complete them.

Yes, we do. I've given it above, but here it is again. He can't complete them because there is nothing he could do which would count as completing them.

The list of tasks is unbounded (in the sense of not containing it's own bound.) So there is no task in the list the doing of which would count as completing the list. But neither is their any task not on the list the doing of which would count as completing the list. If there were, then Achilles could do each of the tasks on his list and still not arrive at d - which is absurd. Thus there is no task either on or off the list the doing of which would count as completing the list. Hence there is nothing Achilles can do which would count as completing the list, since if there were it would be describable as a task. Thus Achilles cannot complete the list.

 

[Hurkyl]

If there were, then Achilles could do each of the tasks on his list and still not arrive at d

During the time interval when Achilles is completing the tasks, [0, d), he does not arrive at position d.

But any single point in time at which it can be said "Achilles finished the tasks", he has (had) arrived at d, precisely at time d.

Hence there is nothing Achilles can do which would count as completing the list, since if there were it would be describable as a task.

The task is to complete the list. (actually, I've heard this type of task called a supertask since it's a task to complete tasks)



Anyways, the individual tasks in Zeno's sequence are of the form "Go from point a to point b". If these are acceptable tasks, then so must "go from 0 to d". And in the process of performing this task, every task in Zeno's sequence is completed.

 

[drnihili]

Originally posted by Hurkyl
During the time interval when Achilles is completing the tasks, [0, d), he does not arrive at position d.

But any single point in time at which it can be said "Achilles finished the tasks", he has (had) arrived at d, precisely at time d.

This is true, but not especially relevant to the question at hand

The task is to complete the list. (actually, I've heard this type of task called a supertask since it's a task to complete tasks)

Anyways, the individual tasks in Zeno's sequence are of the form "Go from point a to point b". If these are acceptable tasks, then so must "go from 0 to d". And in the process of performing this task, every task in Zeno's sequence is completed.

Every task requires doing something. The problem with your supertask isn't that it requires going from point a to point b. The problem is that it either doesn't require anything at all, or begs the question.

I can, for example, specify a task that requires that Achilles complete his first two tasks. But the task is empty since it doesn't require Achilles to do anything that he's not already required to do by his other tasks. We can also add the task that Achilles should exist while he runs. There are an infinite number of empty tasks that can be added to the list. But these aren't real tasks since they don't require anything more of achilles than is already required by the tasks on the list. These sorts of "tasks" don't really specify something that must be done, they aren't additions to the list.

Furthermore, even if I were to grant that your supertask was nonempty, it doesn't help answer the original question. I ask "How can Achilles complete all the tasks on his list?" You answer "by ccompleting the super-task." I ask "And what must Achilles do to complete the super task?" You answer "He must complete all the tasks on his list." We've just come full circle. I want to know how he completes all the tasks, you answer that he does it by completing all of them. Perhaps we can grant that your answer is true, but it is entirely vacuous.

 

[NeutronStar]

Hurkyl wrote:

But any single point in time at which it can be said "Achilles finished the tasks", he has (had) arrived at d, precisely at time d.

Tsk, tsk, Hurkyl. All you are saying here is that it is obvious that we can move so it therefore it must also be obvious that we have completed all of the tasks.

Zeno was well aware that we can move. His whole point is that there is no logical explanation for it. So the argument that we can obviously move, and therefore we must be able to move, is actually quite silly don't you think?

Zeno wants an explanation of how it can be done, and so do we.

As drnihili has suggested, the task of actually finishing is unimportant, it's the question of how the infinite many tasks in between are completed in a finite way. What constitutes the completion of these tasks, save for the trivial response that we can obviously move.

As I've already posted my answer is to conclude there simply aren't an infinite number of tasks. The concept of distance cannot be broken up continuously and indefinitely. The whole paradox suggests to me that the universe (including motion) must necessarily be quantized.

Mathematical tricks of limits and or convergent series just don't satisfy my quest for an explanation. Those definition are mathematical abstractions and kind of miss the point of physical reality. Just because we say that a mathematical series can converge doesn't mean that it actually can. We are just satisfied that as the additions become smaller and small they don't add anything significant because they are continually decreasing in the amount that they add. So we ignore them. They are never really *completed*. And calculus doesn't claim that they ever do get completed. It only shows that in the limit, the additions tend toward zero so we can ignore them. But we can never have claim to actually have stopped the process! If we did we would have an finite number of additions and not an infinite sum.

This problem actually reminds me a lot about the number of points in a finite line.

Anyone who accepts that a finite line can contain an infinite number of points should have absolutely no problem with Zeno's paradox. Or, maybe better said, if Zeno's paradox bothers a person, then the idea of an infinite number of points in a line should also bother them.

After all, to say that you have an infinite number of points in a finite line is to also accept the idea that a finite line contains an infinite number of finite distances. Anyone who can accept that should have no problem imagining an infinite number of distances being completed in a finite amount of time. Just imagine the line as being a timeline.

Personally I don't accept the idea that a finite line can contain an infinite number of points, and so Zeno's paradox holds interest for me. But I solve it in the same way that I solve the finite line problem. Real physical distance cannot be divided up into infinitely many parts, and therefore it does not require an infinite number of tasks to move. No paradox here.

Now I go back to work fellas! These forum boards can be addictive!

 

[Hurkyl]

This is true, but not especially relevant to the question at hand

I bring it up because I get the impression that you are confusing the properties of individual events with the properties of infinite sets of events.

Every task requires doing something...

Then why are you saying there must be a task that counts as completing the list? Such a task is as "empty" as my supertask.

 

[NeutronStar]

drnihili wrote:

I ask "How can Achilles complete all the tasks on his list?"

Be careful drnihili. If you are willing to accept that Achilles has completed the very first task then you are done. He moved.

All that is needed now is to move the finish line up to where he moved to by completing his first task and the race is over. Motion is possible and there is no paradox.

But Achilles can not even complete his first task, because in order to do this he must first complete the task of moving halfway toward that first destination.

It works in both directions. Not only can Achilles never finish the race, he can't even start it!

He cannot move at all.

Why? Because the assumption is that space can be divided up into an infinite many parts. So in order to move to any point we must first move to the point halfway between that point and where we are, etc, etc, etc. We can't even make our first move, or complete our first task, because to do so would assume that we had already completed an infinite number of tasks just to get to that first halfway point!

At this point I can't help but bring in Sherlock Holmes.

We have a mystery. The mystery of how motion can be achieved in an infinitely divisible space. After investigating all of the evidence, and removing all of the impossibilities only one explanation remains.

And in the spirit of Sherlock Holmes whatever remains must be the truth.

"Space must be finitely divisible my dear Watson. For this is the only suspect that we cannot eliminate."

 

[Hurkyl]

Tsk, tsk, Hurkyl. All you are saying here is that it is obvious that we can move so it therefore it must also be obvious that we have completed all of the tasks.

...

Zeno wants an explanation of how it can be done, and so do we.

You're not asking for an explanation of how motion can be done, you're asking how to reconsile apparent paradoxes in the ability to describe motion in terms of an infinite sequence of tasks. My response is that the paradoxes only arise from incorrectly extrapolating properties of the finite to the infinite.


As to why we can move at all is somewhat more fundamental than any questions about infinite sequences.

Mathematical tricks of limits and or convergent series just don't satisfy my quest for an explanation.

As I mentioned before, mathematical "tricks" of limits and convergence (iow ordinary tools for working with infinite sequences) only arise when you insist on describing things with an infinite sequence.

If you don't like dealing with infinite sequences... then don't go seeking them and then complain you don't like dealing with them! Incidentally, differential and integral calculus can be introduced without any use of limits and infinite sequences.


Edit: I was wondering when someone was going to get around to Zeno's other paradoxes.

 

[drnihili]

Originally posted by Hurkyl
I bring it up because I get the impression that you are confusing the properties of individual events with the properties of infinite sets of events.

Fine, let's try this. Specifically what properties do you think I am confusing?

Then why are you saying there must be a task that counts as completing the list? Such a task is as "empty" as my supertask. [/B]

That is the point. There cannot be any task that completes the list. But absent such a task, there is no reason to think that the list can be completed and indeed there is every reason to think that it cannot.

 

[drnihili]

Originally posted by NeutronStar
Be careful drnihili. If you are willing to accept that Achilles has completed the very first task then you are done. He moved.

Yes I know. However this version grants the completion of each task and shows that even with that assumption Achilles cannot get to d.

I actually prefer the form in which he cannot start. My favorite rendition is the following.

Imagine that Achilles wants to proceed from point A to point B, but that an infinite number of Gods have vowed to prevent him. God 1 vows that if Achilles makes it 1/2 way, he will kill him with a Lightning bolt. God 2 vows that if Achilles makes it even as far as 1/4 of the way he will kill him with a lightning bolt. God 3 vows that if he makes it 1/8 of the way ... In general, God N vows that if Achilles makes it 1/2^Nth of the way, he will be killed with a lightning bolt before proceeding further.

It turns out, that Achilles can't start across the road. For suppose that he travels some distance. In that case he must have already traveled 1/N^2 of the distance for some N. But that means that God N+1 would have already killed him. In turn that means that he wouldn't have made it 1/N^2 of the distance. Since the assumption that Achilles travels some distance across the road yields a contradiction, it can't be right. So Achilles doesn't travel any portion of the distance.

Now here's the clincher. Since Achilles doesn't travel any portion of the distance, none of the God's have to fulfill their vow. Their vows alone make it logically impossible for him to cross the road. But it seems absurd that Achilles can be prevented from crossing the road without any of the gods having to carry out their vow. WHat, after all, is stopping him?