Runner's Paradox: Finishing the Race in d Metres

[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 .. ∞ of f(x) as n -> ∞

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(∞) = sum over x = 1 .. ∞ of f(x)

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

S(n) -> S(∞) as n -> ∞


Also, it is worth recalling that the ε-δ definition of limits is only used when the domain and range of the limit are finite real numbers. You may recall doing ε-M or N-δ 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?

 

[Hurkyl]

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

If a point X is not in a finite set S, the distance between S and X must be greater than zero (in nonpathological topological spaces). However, such a guarantee is not possible when S is infinite.

In particular, it is certainly possible for an event E to occur after every element of an infintie set S of events, yet there be no time elapsed between S and E.


Ordered finite sets have a well-defined first and last element. That is not always true of infinite sets.


Achilles covers the distance [0, d] in time [0, d]. If we consider everything but the endpoint, Achilles covers the distance [0, d) in time [0, d). The finishing time of a task is simply the first instant of time at which it can be said the task was completed. This time does not necessarily occur during the time interval in which the task was performed. Finishing times are what's called a "least upper bound" or a "supremum". The finishing time of each individual task occurs in [0, d). The finishing time of the entire sequence of tasks is d.

But the supremum of a finite set is an element of that set.


(all three of these comments are three different wordings of the same fact about infinite sets; unlike finite sets, infinite sets are not always topologically closed)

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

I was expecting the arrow paradox, but this one is nifty too.

Again the question is the same, what logical reason do we have to think this is or is not a valid scenario?

At least with Zeno's paradox, the fact that we can move from start to finish and that science allows infinite divisibility allowed us to prove that the scenario was a valid one. (You just failed to show that the consequences were impossible)

With this paradox, you've shown the consequences are impossible (actually, this part is still debatable, but irrelevant to my argument), but you've failed to prove the scenario is a valid one.

 

[drnihili]

Originally posted by Hurkyl

With this paradox, you've shown the consequences are impossible (actually, this part is still debatable, but irrelevant to my argument), but you've failed to prove the scenario is a valid one.

Valid in what sense?

 

[Hurkyl]

In the sense of it being anything more than an imaginative story.


I've gotten the impression throughout the thread that you aren't just presenting food for thought; you actually think that the paradoxes represent an actual inconsistency in the modern view of things. (If I'm wrong on this point, I apologize for dragging this thread out for so long)

In order for it to be an actual inconsistency, you have to both justify each step in the construction, and to prove that the consequences are contradictory with science.

With Zeno's paradox, we can rigorously justify the construction of the sequence, using the fact motion is possible and infinite divisibility. (I've taken the liberty of implicitly giving the tasks a precise meaning; to go from point A to point B means that at some time Achilles was at point A and at some later time Achilles was at point B, and at the time Achilles arrived at B, and any future time, it is correct to say the task has been completed)

We can't do that with the vows; at least I don't know of a way to rigorously prove it. Induction only tells that for any n, it is possible for there to be n gods with vows... it does not tell us the infinite case is possible.

 

[drnihili]

Originally posted by Hurkyl
In the sense of it being anything more than an imaginative story.


I've gotten the impression throughout the thread that you aren't just presenting food for thought; you actually think that the paradoxes represent an actual inconsistency in the modern view of things. (If I'm wrong on this point, I apologize for dragging this thread out for so long)

I think there are still unresolved problems with our formalizations of infinity. Those problems are not always inconsistencies in the formalizations themselves but are problems in the relationship between the formal models and the reality which they are taken to be models of.

In order for it to be an actual inconsistency, you have to both justify each step in the construction, and to prove that the consequences are contradictory with science.

With Zeno's paradox, we can rigorously justify the construction of the sequence, using the fact motion is possible and infinite divisibility. (I've taken the liberty of implicitly giving the tasks a precise meaning; to go from point A to point B means that at some time Achilles was at point A and at some later time Achilles was at point B, and at the time Achilles arrived at B, and any future time, it is correct to say the task has been completed)

We can't do that with the vows; at least I don't know of a way to rigorously prove it. Induction only tells that for any n, it is possible for there to be n gods with vows... it does not tell us the infinite case is possible.

It's actually a lot easier to validate the Gods example than the race example.

Let each vow be represented as a material conditional of the form "If Achilles reaches a point 1/2^n alive, then he will be instantly killed by God n." The conditionals are built by recursion on the counting numbers. Now let N be the conditional built at nth step of the recursion. If N is false, then each M>N must also be false. However, there is no contradiction in all of the Ns being true.

Either I don't understand your point about induction or you're confused. It is enough that we have a god for each n. We only need an omega sequence, not an omega+1 sequence.

 

[NeutronStar]

Hurkyl wrote:

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!

Hey! Wait a minute!

I came into this thread to offer a physical explanation of Zeno's paradox. This is a physics forum is it not?

My comments on the failure of pure abstract mathematical formalism are merely an observation. (not a complaint)

After having just completed four calculus refresher courses last fall and this spring I feel that I can safely say the following:

Calculus' answer to Zeno is simply this:

We are not concerned with whether or not the tasks can be completed. All we are concerned with is showing that the tasks will get closer and closer to the goal, and that they will continue to do so at an every increasing rate, and that this behavior will not change not matter how arbitrarily close we get.

Prove these things for convergence and you will get an A on a calculus exam!
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That's it. That is all that calculus can say about the matter. Calculus cannot be used to claim that an infinite number of tasks have ever been completed. That is not the purpose of calculus, nor is it the formalism of it.

So I am not complaining when I say that calculus cannot answer Zeno's question. I am stating this as a fact with absolute confidence based on my understanding of the rules, and definitions of the calculus used in this problem. And believe me, I've had long discussions with my mathematics professors about these very concepts. I am confident that they would agree with what I have said above in bold blue letters.

So in the spirit of physics let me offer the following

There mere fact that the tasks are getting smaller as they continue on indefinitely is not good enough. There must be a 'physical' explanation as to how they can be completed.

I am satisfy that an infinite number of tasks can never be completed. No explanation possible.

So my answer is that one of our assumptions are wrong. And the assumption that I take to be wrong is that space can be divided up into infinitely many parts. Take away that assumption and the paradox goes away.

Space can only be divided up into a finitely small amount. This does not imply an absolute space by the way! It simply implies that whatever we mean by space, that concept is going to necessarily be quantized.

I think that QM, and even to a much greater extent, String Theory, both indicate that this is most likely the case anyhow. So unless we know differently why do we keep assuming that space can be divided up indefinitely?

Who told us that we could do that?

 

[ahrkron]

Hurkyl is right on the money when he says that

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!

Zeno's paradox hides a self-inconsistency in the treatment of infinities that is neatly cleaned up by calculus.

The basic structure of the paradox is:

1. Think of a trajectory (i.e., our name for something that necessarily has a non-zero extension in two quantities: space and time).
2. Divide the space extension infinitely (i.e., "assume we can infinitely divide the distance to be traversed").
3. Think of the times needed for each section defined in step 2.
4. The total time should be the sum of all times we found on step 3.
5. But summing those times is impossible, since there is an infinity of them.

There is a blatant contradiction between the assumption hidden in step 5 (an infinite sum cannot add up to a finite value), and the explicit division put by hand into step 2 (lets divide this distance into an equivalent infinite number of subintervals)!

The resolution of this has nothing to do with the physics of motion. It is just a matter of self-consistency.

What calculus has to say about this is that, doing things carefully, continuous magnitudes and infinite sums (as those assumed to exist in the formulation of the paradox) can be solidly dealt with.

 

[drnihili]

Originally posted by ahrkron
Hurkyl is right on the money when he says that



Zeno's paradox hides a self-inconsistency in the treatment of infinities that is neatly cleaned up by calculus.

The basic structure of the paradox is:

1. Think of a trajectory (i.e., our name for something that necessarily has a non-zero extension in two quantities: space and time).
2. Divide the space extension infinitely (i.e., "assume we can infinitely divide the distance to be traversed").
3. Think of the times needed for each section defined in step 2.
4. The total time should be the sum of all times we found on step 3.
5. But summing those times is impossible, since there is an infinity of them.

There is a blatant contradiction between the assumption hidden in step 5 (an infinite sum cannot add up to a finite value), and the explicit division put by hand into step 2 (lets divide this distance into an equivalent infinite number of subintervals)!

The resolution of this has nothing to do with the physics of motion. It is just a matter of self-consistency.

What calculus has to say about this is that, doing things carefully, continuous magnitudes and infinite sums (as those assumed to exist in the formulation of the paradox) can be solidly dealt with.

If that were the problem, you would be correct that calculus solves it. But since you don't understand the problem, it's not surprising that your answer is irrelevant to it. Yes, calculus is a wonderful mathematical theory. Bully for it. It just doesn't have anything to do with the problem at hand.

The problem is not finding a finite sum of an infinite series. The problem is saying what it would be to complete an infinite series one step at a time. Calculus doesn't even begin to adress this issue, let alone provide an answer for it.

 

[NeutronStar]

ahrkron wrote:

5. But summing those times is impossible, since there is an infinity of them.

…(snip)…

What calculus has to say about this is that, doing things carefully, continuous magnitudes and infinite sums (as those assumed to exist in the formulation of the paradox) can be solidly dealt with.

I agree that calculus solidly deals with these concepts from a purely mathematical point of view. I love calculus!

I'm not out to put down calculus.

But I also understand the definitions and the methods of how calculus deals with these problems. And I would like to state some 'facts' regarding this issue.


Fact 1. There is nothing in the formalism of calculus that says that an infinite number of tasks can be completed.


If you can find a calculus book that states otherwise I would be glad to read it. But to my knowledge calculus does not make that claim anywhere in its formalism.


Fact 2. Calculus handles the problem of infinite sums by basically 'cornering' the problem


I'll grant you that this statement is purely off-the-cuff, but I feel that it genuinely describes the situation.

Proving that a series converges in calculus is all about proving the behavior of the quantities that are being added together. That they have certain limit properties. And if all of these properties are satisfied then we can ignore the fact that the sums can never be completed. We then say with confidence that the sum converges to a particular value, and for all intents and purposes equals that value.

This is not at all the same as proving why Achilles can complete an infinite number of tasks.

In fact, all calculus is really saying is that if Achilles can manage to complete all his tasks he will definitely end up at the finish line because this is where he is converging to.

The same thing goes for an infinite sum. If you could possibly complete it then you most certainly will end up with the precise value that calculus has predicted because this is where the sum is going to converge.

I mean, calculus is perfectly correct in both cases. It calculates the correct point where Achilles will end up, and it calculates the correct sums for infinite series. But it does NOT explain how an infinite number of task can be completed. Neither does it claim to have completed the infinite additions. It simply gives what the results would be if they could be completely.

I think calculus is great! But it does what it does. And it doesn't explain how an infinite number of tasks can be completed. Nope, it doesn't do that, nor does it claim to do that anywhere in the formalism that I am aware of. It just corners the convergence.

But it does deal with that solidly, I do agree with that!

 

[Hurkyl]

Fact 3:

Nobody here (but me) has made any attempt at making a rigorous notion of what a "task" is, and what it means to "complete" a task.

 

[HallsofIvy]

I would agree that "There is nothing in the formalism of calculus that says that an infinite number of tasks can be completed."

But you will have to agree that calculus then gives ways of arriving at the same result AS IF an infinite number of tasks were completed!