Runner's Paradox: Finishing the Race in d Metres

[NeutronStar]

Hurkyl wrote:

The whole point of the thread is whether or not Zeno's paradox is or is not an actual flaw in the view point of modern science, is it not?

I've given up on trying to read everything that you guys are discussing, but this comment caught my attention.

I certainly didn't view this thread to be about any flaws in science. I am well aware that science cannot answer every question. Does someone think it can?

Zeno's question is almost philosophical in nature. Is it even important to science right now? I can imagine a time when it may actually become important. However for now I simply see it as interesting. And having value in the sense that it does spark food for thought. I see it as a worthy question to consider, but not to cry about if we can't answer it. That doesn't imply any flaw in science. Sheesh! There are a lot of questions science can't answer!

Hurkyl wrote:

The "solution" to Zeno's paradox (like almost every other paradox) is to state precisely what's going on.

As I tried to hint, the way it's been brought up here (as a question about tasks) is not well-formulated; e.g. what is a task and what does it mean to complete one, and why should one think a sequence of tasks should or should not be completable?

Seems to me that it's pretty straight-forward in this particular problem

What is a task?

To move 1/2 of the distance to finish line.

What does it mean to complete one?

You are 1/2 of the distance closer to the finish line than you were before.

Why should one think that this sequence of tasks cannot be completed?

Because these tasks are never ending. No matter how close you get to the finish line you can always move 1/2 of the distance closer.

You had suggested a "super task" a while back that would constitute reaching the final point. But who said that you could make up that task? That task isn't part of the problem. Your task is to step only 1/2 of the distance to the finish line with each step, and then explain how you can actually reach the finish line by completing only these tasks. Making up a new task to step directly to the finish line is cheating.

Your first reaction was to use calculus. But as has been pointed out, calculus will only guarantee that you will indeed reach the finish line if you manage to complete all of your tasks. Calculus in no way makes any claim that you should be able to actually complete your infinite many tasks. It only says that if you do manage to complete them you will end up at the finish line. But we already know that so we don’t even need calculus at all.

What we do need is a rigorous proof that an infinite number of tasks can actually be completed. That is impossible! This is why it's a paradox. Achilles can actually reach the finish line in the real world, but logically it isn't possible!

Even if you managed to prove that an infinite number of tasks could be completed, then all you would have proven is that infinity is finite. You aren't going to be able to prove that with rigor.

So going back to your original worry that this means that science is somehow flawed why not consider some alternatives.

1. Space and time cannot be divided up infinitely because they are quantized.

or,

2. Space and time are not absolute properties of the universe and so we can't even talk about dividing them up at all because the don't even exist!

I actually like both of these solutions to the paradox.

QM and String theory suggest solution number 1

SR and GR suggest solution number 2

It's interesting food for thought. Trying to prove that infinity can be finite seems to be futile to me. I'd much rather pick one of the other 2 possible solutions that I've offered.

Perhaps someone else has other possible solutions?

 

[Hurkyl]

What is a task?

To move 1/2 of the distance to finish line.

What does it mean to complete one?

You are 1/2 of the distance closer to the finish line than you were before.

If this is all one means by completing task, we should have been done on the first page. As I mentioned, one can clearly demonstrate that if Achilles follows the trajectory x(t) = t for t in [0, d], each task is completed.

Why should one think that this sequence of tasks cannot be completed?

Because these tasks are never ending. No matter how close you get to the finish line you can always move 1/2 of the distance closer.

But why does that imply they cannot be completed?


Given the meanings I attach to the words involved, that trajectory is rigorous proof that the tasks can be completed. (you just need to show it is smooth and to prove that the set of points {d - d/2ⁿ | n > 0} lie in [0, d]) That is sufficient to prove that geometry and infinite divisibility is not an obstacle to completion.

However you obviously don't accept this, which is why I've spent so much effort trying to get you guys to say what you mean by the words involved.

What we do need is a rigorous proof that an infinite number of tasks can actually be completed.

May I presume that if Achilles has completed the n-th task, he can complete the (n+1)-th task? The proof then follows by the standard calculus argument; it is then elementary to prove that each of the points in {d - d/2ⁿ | n >= 0} lie on Achilles's path, and continuity proves that d lies on it.


Alternatively we may use transfinite induction; if we add a final task "arrive at d" to Zeno's sequence, and agree that this final task can be completed if the entirety of Zeno's sequence can be completed, then transfinite induction guarantees that the "arrive at d" task is completed.

Even if you managed to prove that an infinite number of tasks could be completed, then all you would have proven is that infinity is finite. You aren't going to be able to prove that with rigor.

In the theory I am working, it is proven with rigor. Which is why I spend such an effort trying to figure out in what theory y'all are working.

So going back to your original worry that this means that science is somehow flawed why not consider some alternatives.

1. Space and time cannot be divided up infinitely because they are quantized.

I do consider alternatives; science is flawed, as demonstrated at the very least by the incompatability between the standard model and general relativity. I would be happy to see position quantized though at the moment there is no indication that will occur.

But my point is that Zeno's paradox is not sufficient to prove that an alternative must be considered.



And there is one comment I can't believe I missed because I was all but specifically requesting it!

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.

but I don't have time to respond to it at the moment, so I'll get it when I'm back from work... I'm quoting it so I'll remember it later.

 

[NeutronStar]

Hurkyl wrote:

May I presume that if Achilles has completed the n-th task, he can complete the (n+1)-th task? The proof then follows by the standard calculus argument; it is then elementary to prove that each of the points in {d - d/2n | n >= 0} lie on Achilles's path, and continuity proves that d lies on it.


Alternatively we may use transfinite induction; if we add a final task "arrive at d" to Zeno's sequence, and agree that this final task can be completed if the entirety of Zeno's sequence can be completed, then transfinite induction guarantees that the "arrive at d" task is completed.

With all due respect Hurkyl you are looking at the problem backwards. You are assuming that Achilles can obviously finish the race and reach the finish line. You therefore feel that can use that information as part of your proof.

Zeno's paradox is purely a thought problem. He was well aware that motion is possible, he could move himself! The whole point to Zeno's question is how can we explain motion without already assuming that it can be achieved.

Your first proof above just says what we have already allowed. That if Achilles can complete all of his tasks he will end up at d. So how does that prove that he has completed all his tasks? Simply because we know that he can actually reach the finish line in reality? That is not a permitted assumption! Secondly, even it we did permit that assumption that still doesn't prove that Achilles completed an infinite number of tasks, he may very well have cheated and moved though a space that is finitely divisible and have only completed a finite number of tasks. In other words, he may performed your supertask at the end where he ran out of half-way points and was forced to step completely to the last point because space is ultimately finitely divisible.

So your first so-called proof would prove nothing about completing an infinite number of tasks even if you were permitted to use the fact that Achilles actually finishes the race.

Your second proof is almost identical only in this case you are using induction. But you are still assuming two things. You are assuming that space can be infinitely divided, and you are assuming that Achilles can indeed reach the finish line. That second assumption is not permitted.

In Zeno's problem you are only permitted to use the first assumption and then logically prove that Achilles can finish the race. The mere fact that Achilles can finish the race in reality doesn't prove that space can be divided up infinitely. And this is really all that Zeno is saying with his paradox. He is saying, "Show me how Achilles can logically complete and infinite number of tasks without using the assumption that he can do it!"

In other words, prove that space is infinitely dividable without assuming it!

But every so-called proof that you have shown thus far contains both the assumption that space can be infinitely divisible and that Achilles can reach the finish line!

Therefore, I can only conclude that you don't fully appreciate Zeno's concern.

 

[NeutronStar]

Hurkyl,

I do want to agree with your proofs. Seriously.

If we allow both of your assumptions:

1. Space is infinitely divisible.
2. Achilles can obviously reach the finish line.

Then your induction proof would be valid. You would have indeed proven that infinity is finite!

And in reality we all know that assumption number 2 makes a whole lot of sense because we do this sort of thing everyday. (this is what I would call self-evident).

But we don't have any real reason to assume that number 1 is true. We have absolutely no reason to believe that space can be divided up infinitely. But do we have any reason to believe that can't be?

This is really where Zeno is at. Imagine this very wise old man asking you to PROVE you assumption number 1. To do so you cannot start by assuming it. You can't just assume that space is infinitely divisible, Zeno wants proof! And this is really what his paradox is all about. He is asking, "If space is infinitely divisible than how is it possible that we can move?".

I surrender to Zeno and openly admit that I cannot prove that space is infinitely divisible. Therefore Zeno has forced me to consider other alternatives.

This is the essence of Zeno's paradox.

It's a paradox because if anyone can prove it they will have proven that infinity is finite! It will always be a paradox.

It cannot be solved. Zeno offers it as food for thought. Any so-called solutions (like the ones that I have offered) are not really solutions at all. They are merely an acceptance that Zeno is right, and that we must reject the idea that space and/or time can be infinitely divisible.

I bow to Zeno's genius. He is my hero of the ancient Greeks.

Although, some of those other Greek dudes were pretty cool too!

 

[Hurkyl]

My goal is merely to observe that infinite divisibility can be internally consistent. Expecting to give a logical proof would be misguided; logical proofs are always relative to the hypotheses... and if the exceptional success science has had using models incorporating infinite divisibility isn't sufficient evidence that it is a practical idea, then I don't think it's worth spending additional effort on that angle.

I surrender to Zeno and openly admit that I cannot prove that space is infinitely divisible. Therefore Zeno has forced me to consider other alternatives.

Semantic correction; Zeno has forced you to consider that infinite divisibility is not the only option.

 

[NeutronStar]

Hurkly worte:

My goal is merely to observe that infinite divisibility can be internally consistent.

"merely observe"?

What happen to all the logical rigor that you claimed to be interested in?

That's Zeno's whole point! What do you mean "you merely observe"? What kind of a proof do you call that? Where's the logic behind your observation? What exactly is it that you have observed?

If your observation makes logical sense you should be able to state it in an unambiguous logical way with rigor.

It sounds to me like you are just saying, "My goal is merely to believe that infinite divisibility can be internally consistent.

And if this is what you choose to believe that's fine. But at least own up to the fact that you merely believe it and that you can't prove it at all!

If you could prove this with rigor, I'm sure that the entire world would love to see your proof!

You would win a Nobel Prize for such a proof to be sure!

 

[Hurkyl]

"merely observe"?

What happen to all the logical rigor that you claimed to be interested in?

Ala Godel, it's futile to use a logical theory to prove itself consistent. The best we can do is to seek potential contradictions and analyze them for flaws. Or, of course, we may pass the buck and appeal to a different theory within which we can prove the original theory consistent. (e.g. a relative consistency proof using set theory, but since I know you have issues with ZF, that's not really an option for this discussion)

 

[NeutronStar]

Originally posted by Hurkyl Ala Godel, it's futile to use a logical theory to prove itself consistent.

So in other words, when you are up against the wall and you can't make your case just cite Gödel's inconsistency proof.

I see.

I'll have to remember that one.

The best we can do is to seek potential contradictions and analyze them for flaws. Or, of course, we may pass the buck and appeal to a different theory within which we can prove the original theory consistent. (e.g. a relative consistency proof using set theory, but since I know you have issues with ZF, that's not really an option for this discussion)

Actually you do have a valid point here. I do have issues with ZF. So you're right that it isn't an option for this discussion.



From way back in this thread,...

drnihili worte:
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.

I never did get a chance to formally state that I agree with this. Even though it should be obvious.




Other than this I suppose I'm done on this thread. For real this time.

By they way, I would like to say that I did gain many ideas and food for thought from this discussion. I believe that is it always fruitful when great minds come together no matter how much they disagree. I am quite sure that everyone here has a very good understanding of what they understand.

We simply understand things differently.

Thanks to everyone who has contributed.

 

[Joy Division]

I've always been amused with Zeno's argument because even today people love to bring it up to argue various points. It's funny that an argument that Zeno used to show that motion and change were impossible is still brought up today.

Really though the logic in this argument can be very subtle and one has to tread carefully before saying what it really means. (I just hope I'll heed my own advice.)

Let's look at the perscribed motion of Achilles. Each of his steps covers half the distance of his last step and his first step covers half the overall distance. Let's look at what we've assumed here.

First we've assumed that motion is possible. So he can take his steps in the first place. Second we've assumed that space is infinitely divisible (unless we add some extra content to how he moves.) We get that because if space is not infinitly divisible then we get problems that we don't want when on one of his steps Achilles can no longer cross the minimum distance in our space and we get a contradiction.
It's not one we want though because all that contradiction proves is that unless we say what happens when to Achilles' motion when he reaches that point then the motion we've perscribed is not possible in a finitely divisible space.

Ok so if we want to consider that space may be finitely divisible then we have to either say achilles moves like we tell him to and stops when he can no longer make a step over the minium distance or that he stops taking smaller steps and continues covering the smallest distance each time. Notice that each one of these will end his movement in a finite number of steps, but in the latter he will actually reach his goal.

Ok the big thing to notice now is that Zeno's argument has not shed any light on whether or not space is infinitly divisible. In fact it's very consistant in all 3 cases we've considered. Neither does this shed any light on what would happen if you did move as Achilles does, we just really can't say.

Now let's consider the case when space is infinitly divisible. This one is the one in which he makes an infinite number of steps. Now our feelings and what is "intuitively obvious" isn't going to cut it anymore. We're dealing with infinities so mathematical formalism becomes key. The only sane thing to do is to use calculus. Really it is, we've got an ideal situation when we consider this argument and reality is what doesn't have any buisiness poking it head in here. Even if Calculus were completely wrong at describing the real world ,it is perfectly suited to the task of describing Achilles' motion.

Ok so what's the problem here. He takes an infinite number of steps! That must be crazy. Wrong because the series that describes his movement converges to a finite value there is no problem here. Really what's so wrong, the set of all his steps may be infinite but the distances he covers with each step, the time it takes for each step and even the work he does for each step approach zero.(That is in a limiting sense). So the amount of effort however you want to define it for each task ( or step as we've defined the tasks earlier) approaches zero too. That means because our long haired achean is an ideal runner he finishes those tasks like nobody's business.

Now this argument gets interesting when you actually ask, what if we tried to model Achilles' movement on a computer? It would take an infinite time for our digital Achilles to finish. That's the interesting part, because computers can't use the axiom of choice while our calculus running ideal Achilles can. What this means is that for the digital Achilles the two definitions of continuity (the one based on functions the other on sequences) aren't equivalent. You need the axiom of choice to prove they are.
That's where at least for me the interesting mathematics comes out from this argument.

 

[Ether]

The main point of contention here seems to be whether or not Achilles can complete an infinite number of tasks in a finite period of time. Hurkyl says he can. NeutronStar and drnihili say he cannot, and insist that Hurkyl give an explanation of how he can. This insistence, however, is based on the assumption that the completion of infinitely many tasks in finite time is (self-evidently) undoable.

My understanding of Hurkyl is that while the series of ever-shrinking segments Achilles covers converges to d, the time interval to complete each task converges to zero and the total time converges to some finite value. Calculus is thus telling us that the thing is indeed doable. Hurkyl's response as to how the thing is done, that Achilles merely runs from 0 to d, is thus sufficient, because performing an infinite number of tasks in finite time is no obstacle at all (in a sense it happens "automatically") and its undoability is not self-evident to everyone. Asking how it is done is like my asking how I will cross the river in front of me when in fact there is no river there.

Sorry if I misrepresented any of your points, Hurkyl.

 

[Hurkyl]

I think you have the semantics of the consistency argument right, but that point is not the one upon which neutron star and drnihili are focusing. To summarize the facts of the thread in two sentences:


One cannot logically prove an infinite divisibility model correctly describes the physical universe.

Zeno's paradox does not logically prove infinite divisibility models cannot correctly describe the physical universe.


neutron star and drnihili are focusing on the first fact, the rest have focused on the second fact.

 

[NeutronStar]

Hurkyl wrote:
One cannot logically prove an infinite divisibility model correctly describes the physical universe.

Zeno's paradox does not logically prove infinite divisibility models cannot correctly describe the physical universe.


neutron star and drnihili are focusing on the first fact, the rest have focused on the second fact.

Yes, I would agree that this is a good assessment of the conversation.

I just had some new thoughts about the infinite divisibility of space.

A while back Hurkyl mentioned that QM only states the bound particles are quantized, and it does not restrict free particles to this behavior.

As an important append to that I would like to add the following: (and please correct me if I am wrong, which I know you all will of course. )

QM does not precisely describe the behavior of free particles. In other words, it only describes them as a probability function. It does not assign to them any specific vector in space.

And as an additional prerequisite piece of information I would just like to remind everyone that the idea of an absolute space died with classical mechanics, and with Relativity came the idea that space, and time are merely relative notions.

With that I'd like to possibly spark a seminar on the discussion of the practicality of even thinking of an infinitely divisible space from a physical perspective.

I will begin the discussion with the following piece of information:

I just started studying classical analytical mechanics as a refresher course. (I've been trying to refresh this tired old brain lately. ) Just in case anyone is interested the book is Analytical Mechanics by Fowles & Cassiday ISBN 0-03-022317-2.

In any case, the book starts off with some rather interesting historical information about the definitions of particular physical quantities. These definitions are really all we have to go on as physicists. Length (which is really the same thing as distance) was originally quantified as a unit by the length of a particular physical object. By that physical definition the notion of length (and therefore distance) would necessarily be dependent on the bound state of matter. Thus the very notion of distance would indeed be subject to the quantized nature of QM.

As time progressed we moved on to redefine length as the distance that light travels over the duration of a specific time. At first this may sound like we have freed the definition of distance from the grasp of QM. However, further analysis reveals otherwise.

We have defined time as the oscillations between particular states of the cesium-133 atom, a quantum object to be sure. Our concept of time is based on this quantum jumping little fellow, so these quantum ticks of time are necessarily pushed onto our idea of distance as defined by the distance that light travels during these ticks. In other words, we can only time the light in an intermittent fashion (with each jump of the quantum atomic clock).

So by our very definitions of space and time we have no choice but to own up to the fact that space and time is indeed finitely quantified as we have defined them to exist. The finite nature of it all stems from the quantum jumpiness of our universe. It is an unavoidable part of the universe that we live in.

Now to speak abstractly about some concept of absolute space being infinitely divisible it purely a fabrication of our minds. We have no reason to believe that such an infinitely divisible absolute space exists. In fact, Albert Einstein would be giving us all the tsk, tsk if he knew we were still attempting to hang onto such an ideal.

So with this in mind let me quote Hurkyl once again:

Hurkyl wrote:
One cannot logically prove an infinite divisibility model correctly describes the physical universe.

Zeno's paradox does not logically prove infinite divisibility models cannot correctly describe the physical universe.


neutron star and drnihili are focusing on the first fact, the rest have focused on the second fact.

Based on everything that I've just said here I would have to side with neutron star and drnihili.

Hey! I'm allowed to take my own side on this!

Of course whether or not Zeno has proven that infinite divisibility models cannot correctly describe the physical universe is certainly arguable.

Whether Zeno's Paradox proves it or not, it sure looks like he hit the nail on the head to me.

I yield the floor to whomever wishes to jump in next.

 

[Joy Division]

While it is interesting that quantum mechanics shows us that we can't observe the universe in an infinitly divisible way (as far as I know), whether or not that means that space itself is actually discrete is an interesting question.

I don't have an answer. But the idea that space may be discrete does not invalidate the fact that calculus is still internally consitant.

I know some people think that any math not related to the real world is uninteresting (or worse nonexistant ). I just don't share their opinions. This is why there is in fact a separate study of mathematics and physics.

Of course calculus alone can't describe our universe at all times.

 

[Ether]

Originally posted by NeutronStar
Now to speak abstractly about some concept of absolute space being infinitely divisible it purely a fabrication of our minds.

How does an assumption of infinite divisibilty imply that space is absolute?

The mathematics of both Special and General Relativity is a continuum math. I'm not sure Einstein wouldn't take you to task.

 

[NeutronStar]

Ether
The mathematics of both Special and General Relativity is a continuum math. I'm not sure Einstein wouldn't take you to task.

I wish Einstein were alive today. I would most certainly like to ask him how he justifies quantifying space when he himself claims that there is no such thing to quantify?

Please understand that I am in no way suggesting that Einstein would be stumped by this question. I am simply saying that I would genuinely like to hear his response to it.

Einstein's genius is unquestionable. However, his tendency to stubbornly hang onto preconceived notions was also historically reordered as being notorious. He totally could not accept the idea the quantum mechanics predicts that god plays dice with the universe. He was a determinist through and through. So while he was willing to reject the idea of an absolute space, he was not prepared to reject the idea of an absolutely determined universe. Very ironic I think.

Einstein's General Relativity describes gravity as warped space-time. That doesn't amount to a very comprehensible idea coming from someone who has denounced the very idea of an absolute space-time that can be warped.

However, we do know that QM and GR are not compatible. So something's got to give. Will it be QM, or GR? OR some combination of the two? We don't yet know. All we do know is that QM hasn't failed yet. And while the relativity of space and time have been experimentally verified their infinite divisibility has not.

Einstein was not even a very good mathematician actually. He relied on other mathematicians to formulate his ideas. Minkowski was a big player in this among others. In fact it was actually Minkowski who first suggested that time should be a dimension. Einstein actually opposed that idea in the very beginning.

Einstein's mathematical descriptions are necessarily continuous because he is using a continuous mathematics. He had no choice. I have reason to believe that mathematical formalism is actually incorrect in this regard. (See the thread entitled "What Makes Mathematics?")

If we correct the basis of mathematical formalism, then Einstein's GR would automatically become discrete because all of mathematics would become discrete.

Who knows? That alone may automatically make it compatible with QM! I'm not far enough into these topics to really say.

We may actually have a complete theory of everything and just don't know it!

(Disclaimer) This particular post was an extreme instantaneous ramble. I just happened to be passing by my computer and noticed the response on this thread, so I sat down and typed in my thoughts.

 

[Hurkyl]

QM does not precisely describe the behavior of free particles. In other words, it only describes them as a probability function. It does not assign to them any specific vector in space.

The classical notion of particles is incorrect. The "particles" of quantum mechanics are particular states of a field that is approximately like a classical particle. (the very nature of how this approximation works yields the uncertainty principle)

By that physical definition the notion of length (and therefore distance) would necessarily be dependent on the bound state of matter. Thus the very notion of distance would indeed be subject to the quantized nature of QM.

Recall that it is the generally the energy of the state that is quantized in a bound state; not position. (e.g. the modern electron cloud model of the atom has replaced the Bohr model)

We have no reason to believe that such an infinitely divisible absolute space exists.

That doesn't amount to a very comprehensible idea coming from someone who has denounced the very idea of an absolute space-time that can be warped.

It's unclear how the notion of "relative" and "absolute" measurement bears any relation to these ideas.

All we do know is that QM hasn't failed yet.

IIRC, QM fails spectacularly when we push it too far out of its flat space assumption.

If we correct the basis of mathematical formalism, then Einstein's GR would automatically become discrete because all of mathematics would become discrete.

GR depends crutially on continuity; it simply does not work if you use a discrete space-time. People have studied such models, but no acceptable model has been discovered.

 

[drnihili]

Originally posted by Hurkyl
May I presume that if Achilles has completed the n-th task, he can complete the (n+1)-th task? The proof then follows by the standard calculus argument; it is then elementary to prove that each of the points in {d - d/2ⁿ | n >= 0} lie on Achilles's path, and continuity proves that d lies on it.

Alternatively we may use transfinite induction; if we add a final task "arrive at d" to Zeno's sequence, and agree that this final task can be completed if the entirety of Zeno's sequence can be completed, then transfinite induction guarantees that the "arrive at d" task is completed.

Let's start with the case of finite induction above. Yes, if Achilles can complete task n, then he can complete task n+1. I grant you that there is no task such that Achilles cannot complete it. However, it does not follow that Achilles can complete the list.

Your argument assumes that if each task on a list is completable, then the list as a whole is completable. This is an example the fallacy of composition. You need to explicitly show that the completability of each of the tasks implies the completability of the collection. It's easy, and perhaps instructive, to come up with examples where this sort of reasoning leads us astray.

If the nth task preceds infinitely many tasks, then the n+1th task also precedes infinitely many tasks. Thus each task precedes infinitely many tasks. However, it does not follow that the list as a whole precedes infinitely many tasks. "Precedes infinitely many tasks" is not an composable property.

Not all numeric properties are composable. For example, for any natural number n, if there is there is a natural number bigger than n, then there is a natural number bigger than n+1. However, there is no natural number bigger than all natural numbers.

Consider an omega sequence (a set ordered as the natural numbers) of sentences where sentence S_n is given by
"There is an m>n such that S_m is false."
Now for any n, the initial segment whose last member is S_n is consistent. However, the entire seequence of sentences is inconsistent. (This, btw, is Yablo's paradox. Perhaps it's inclusion here is a hint that we may really be dealing with omega-inconsistency. I'll have a think on that.)

What your proof above really does is to show that if Achilles completes the list, he will have completed each task and arrived at d. But that has already been granted. What has yet to be shown is that Achilles can complete the list as a whole.

Transfinite induction won't help either as you would need to justify the limit step in order to not merely beg the question.

[I should be clear that NeutronStar and I are grinding entirely different axes. I have no special problem with continuous space. It does have some oddities, but as near as I can tell, discrete space does not fare any better.

I do not know how to resolve the paradox. Thus I am not arguing for any particular resolution of it.]

 

[NeutronStar]

Hurkyl wrote:
Recall that it is the generally the energy of the state that is quantized in a bound state; not position. (e.g. the modern electron cloud model of the atom has replaced the Bohr model)

Actually that is totally irrelevant. When using an atomic clock we are only concerned with the oscillations between energy states. We are not tracking any imagined position that an electron might be in an imaginary electron cloud.

To think of the oscillations of a cesium-133 atom as a sine wave would be a totally incorrect picture. QM tells us that an atom is either in one energy state, or it is in the other. There are no partial states in between. This is the famous quantum leaping. This is what makes quantum mechanics so mysterious to us. We simply can't comprehend how this can be so.

So if we were to graphically draw the frequency of a cesium-133 atom as it changes states that graphic would necessarily have to be a square wave with zero rise and fall times. (Not a sine wave that we would be tempted to intuitively imagine it to be). QM tells us that this must be so. Therefore our clock is necessarily quantized. And since our definition of distance is based on time then it too must be quantized.

Now that I have had time to think about this a little bit I now realize that Einstein would be forced to accept this.

When Einstein started to postulate relativity he said the following:

Distance (or space) is what we measure with a measuring rod

Time is what we measure with a clock

He was a very practical physicist. He didn't want to deal with any erroneous made-up definitions, he wanted to conceptualize the problem at hand.

Based on these conceptions for space and time we have no choice but to conclude that both space and time are indeed quantized by these definitions.

Therefore any claim that that space can indeed be infinitely divisible is not supported by this model. To make such a claim would require a new definition for both space and time.

Whether anyone else is buying this or not doesn't really concern me. I'm selling it to myself here big time.

I'm glad that I came to this forum. This is very useful for me to talk these things out.

Hurkyl wrote:
GR depends crutially on continuity; it simply does not work if you use a discrete space-time. People have studied such models, but no acceptable model has been discovered.

If what you say here is true then GR might ultimately be incorrect.

Gee, I've been studying QM because I thought it was weird. But if GR demands a continuous space then it has its own sort of interesting weirdness.

However, when you say that it depends on continuity exactly what does that mean? I mean, if space is quantized and you move between the only points available isn't that a form of continuity? I mean, there's nowhere else to move to!

We might just be misinterpreting the meaning of infinity as it has to do with continuity there. In fact, I'm willing to bet that this is all there is to that.

That just goes back to what I keep saying about the problems associated with how we have incorrectly defined the concept of One (and therefore infinity) in mathematics.

 

[drnihili]

Originally posted by NeutronStar
[argument that our measurement of time must be discrete rather than continuous]

Based on these conceptions for space and time we have no choice but to conclude that both space and time are indeed quantized by these definitions.

Therefore any claim that that space can indeed be infinitely divisible is not supported by this model. To make such a claim would require a new definition for both space and time.

You can't get rid of infinite divisibility so easily. It may be the case that any way of dividing time yields discrete units. However, all that is needed for infinite divisibility is that given any way of dividing time into discrete units, there is another way which divides it into a greater number of discrete units. What you need to show is that this process has an end. For example, you could show that there is no way to divide time more finely than by the motions of a cesium atom.

Of course that example is impossible to prove since we can divide time more finely by using two out of synch cesium atoms. So if time is quantized, it smallest unit can be no bigger than the period of oscillation of a cesium atom divided by the number of out of synch cesium atoms within a light cone. If this were the case, it would have the interesting side effect that size of a quanta was relative to one's location in spacetime.

 

[Hurkyl]

So if we were to graphically draw the frequency of a cesium-133 atom as it changes states that graphic would necessarily have to be a square wave with zero rise and fall times. (Not a sine wave that we would be tempted to intuitively imagine it to be). QM tells us that this must be so. Therefore our clock is necessarily quantized

First off, only the energy is quantized. The times at which QM allows the atom to change states is not.

Secondly, even if the times at which the atom can change states were quantized, that provides no reason to suppose that the quantization would have the same grain across all possible clocks.

Thirdly, even if there were reason to suppose that the quantization was the same grain across all possible clocks, that provides no reason to think that the allowed transitions would be synchronized across all possible clocks... in fact, because of relativity of simultaneity, such a possibility is forbidden by special relativity.


In summary, this doesn't even provide a mildly compelling reason to think time comes in discrete steps.

This is very useful for me to talk these things out.

I agree with the sentiment; even when arguing against the most hopeless crackpot (I'm not referring to you, incidentally), I find the practice of expressing ideas and concepts useful. (and I gain a greater precision of thought and expression every time)

However, when you say that it depends on continuity exactly what does that mean?

The mathematics of GR use the idea of a "4 dimensional differentiable manifold", which has as part of its definition that sufficiently small open sets on a manifold can be mapped to R⁴ with an invertible continuous function. This let's us lift the infintie divisibility of R to that of a differentiable manifold.


At a more fundamental level, the topology of "discrete" spacesis radically different from those of "continuous" spaces. The EEP is an axiom about local properties which "discrete" sets don't really have; as you shrink your perspective on a discrete set, it looks different.

(From here on, by a "discrete" topological space I mean a space such that neighborhoods contain finitely many points)

The fact that the neighborhoods of a discrete space have only finitely many points allows you to prove, for each pointX , one of the two following things must be true:

(a) X is topologically "well-seperated" from the rest of the space; the only sets which the point is near are those that it is a part of. It has a neighborhood consisting of only itself.

(b) the point is not seperable from some other point. There is some point Y distinct from such that whenever X is part of an open set, Y is also part of that set.

 

[Hurkyl]

Your argument assumes that if each task on a list is completable, then the list as a whole is completable.

I did misspeak; I was saying "can complete" when I was thinking "completes", which makes a huge difference.

 

[drnihili]

Originally posted by Hurkyl
I did misspeak; I was saying "can complete" when I was thinking "completes", which makes a huge difference.

Yes it does. Now you've brought tense into play. As a result your assumption is either incorrect or begs the question depending on where you set the present moment.

 

[Hurkyl]

"can" connotes something vastly different from temporal concerns. (at least depending on your presumptions!)

But that's a little misleading. The current hypothesis with "can complete" gets formalized to:

P(n) := the set of the first n tasks can be completed. (whatever "can be completed" means)

while the "completes" version gets formalized to

P(n) := Achilles completes the n-th task (meaning he is at the specified destination at some point in time)


The present moment has nothing to do with it.

 

[drnihili]

Originally posted by Hurkyl
"can" connotes something vastly different from temporal concerns. (at least depending on your presumptions!)

But that's a little misleading. The current hypothesis with "can complete" gets formalized to:

P(n) := the set of the first n tasks can be completed. (whatever "can be completed" means)

while the "completes" version gets formalized to

P(n) := Achilles completes the n-th task (meaning he is at the specified destination at some point in time)


The present moment has nothing to do with it.

"completes" introduces temporal concerns. "can" is just an alethic modality.

The present moment has a lot to do with it. If you take "completes" to be present tense, then your assumption that if Achilles completes task n, then he completes task n+1 is simply wrong. If you are using "completes" atemporally or with a specious present, then the assumption begs the question as it tacitly assumes the completion of the list just as the use of a future tense would.

In any case you still would not have addressed the issue of composition.

 

[Hurkyl]

And we go back to why I was so insistent on getting precise definitions of terms. The english language is not only horribly imprecise, but highly subjective.

 

[drnihili]

Unfortunately you're stuck with the messiness of natural language any time you try to make mathematics connect to the real world. Take something as simple as 1+1=2. Is it true? Of course it is mathematically speaking. But once you try to tell me that if I put one apple in a bag and then another that I will have two apples and you give as your reason the fact that 1+1=2, then you've gone and botched it up.

Since the original question deals with the physical rather than the mathematical world, we have to show that the math correctly depicts the real world. Doing that invariably requires some messines. In particular, our original problem is inherently temporal. The question isn't whether the sequence of ever decreasing segments has a finite sum. The question is whether it can be completed when done one task at a time.

So yes, it's bogged down in the messiness of the real world. But unless you want to simply be a Hilbertian formalist, you're kind of stuck with that.

 

[NeutronStar]

drnihili wrote:
Not all numeric properties are composable. For example, for any natural number n, if there is there is a natural number bigger than n, then there is a natural number bigger than n+1. However, there is no natural number bigger than all natural numbers.

Actually if you are happy with this condition then you should have no problem accepting that even an imaginary space can only be divided up into finite many distances.

My whole basis for the argument that a finite line can only be said to contain an infinite number of points is precisely the same logic used here. Just because the set of natural numbers itself is infinite doesn't force that property of infinity onto any of its elements. In other words, no natural number itself is infinite.

Imagine starting with a line defined only by its two endpoints. (That line can represent a distance or space) We can now place a point between the two endpoints. We now have 3 points. We can place points halfway between those existing points. We now have 5 points. Toss in more points between those, we have 9 points. Do it again we get a line with 17 points and so on.

We are building a set of the finite elements, {2, 3, 5, 9, 17,…}

There is no end to how many times we can do this. In other words we can add more elements to this set endlessly by simply placing more points between existing points. However, it is quite clear (to me anyway) that just because there is no end to the number of times we can do this, it does not follow that we can make the line contain an infinite number of points. On the contrary this example shows that no matter how hard we try we can never force the property of infinity onto any of the elements within the set. So while the number of points that we can add to a line appears to be unbounded, it must also always retain the property of being finite.

I take Zeno's Paradox to simply be saying that if you attempt to actually add an infinite number of points to a line you can never succeed. Not even by using pure logic! Logic itself is telling us that this is an impossible task. Set theory only serves to verify this conclusion.

drnihili wrote:
Consider an omega sequence (a set ordered as the natural numbers) of sentences where sentence S_n is given by
"There is an m>n such that S_m is false."
Now for any n, the initial segment whose last member is S_n is consistent. However, the entire seequence of sentences is inconsistent. (This, btw, is Yablo's paradox. Perhaps it's inclusion here is a hint that we may really be dealing with omega-inconsistency. I'll have a think on that.)

I don’t see the problem of a finite number of points in a finite line, or the finite divisibility of a finite distance, being a problem of self-reference.

However you might have a point concerning the completion of an infinite number of tasks. If so, all this would do is move you from Zeno's Paradox into Yablo's Paradox which may very well be a similar situation. (i.e. Something that isn't solvable: a Paradox)

If that’s the case, then you will have just added yet another way of saying that it isn't possible to infinitely divide up a finite space.

 

[NeutronStar]

Hurkyl wrote:
First off, only the energy is quantized. The times at which QM allows the atom to change states is not.

Alright, I confess I jumped the gun on that one.

I'm starting to just post my first thoughts now. That's not good on a forum board.

This whole conversation has me thinking along those lines though, and I must confess to already have a strong belief that time and space must both be quantized.

I'm still convinced that they are, and I still believe that I'm on the right track thinking in terms of what Einstein said that distance is what we measure with a rod, and time is what we measure with a clock.

I do like that way of thinking, and this does imply that time and space are related to material objects rather than to some imaginary idea of an absolute space.

So while I confess to being a complete idiot here, I still claim to be an idiot who's on the right track!

And this was still good food for thought too!

 

[drnihili]

Originally posted by NeutronStar
Actually if you are happy with this condition then you should have no problem accepting that even an imaginary space can only be divided up into finite many distances.

That depends on what you mean by "be divided". If you mean that any activity of dividing yields only finitely many divisions, then I agree. If you mean that there is some inconsistency in supposing that the line is constituted by infinitely many points, then I disagree.

I take Zeno's Paradox to simply be saying that if you attempt to actually add an infinite number of points to a line you can never succeed. Not even by using pure logic! Logic itself is telling us that this is an impossible task. Set theory only serves to verify this conclusion.

Simlarly here. I'm with you until you switch to set theory. Set theory is not about tasks or actions. When one talks of constructing the hierarchy of sets, there is no pretense that one actually builds the entire hierarchy. Rather one is setting out rules which define the hierarchy. Those definitions cover an infinite number of cases. However no one can actually carry out the construction for an infinite number of cases.

I don’t see the problem of a finite number of points in a finite line, or the finite divisibility of a finite distance, being a problem of self-reference.

Yablo's paradox does not contain any self reference, so I'm not sure where you're headed with this one.

 

[NeutronStar]

drnihili wrote:[/]
Simlarly here. I'm with you until you switch to set theory. Set theory is not about tasks or actions. When one talks of constructing the hierarchy of sets, there is no pretense that one actually builds the entire hierarchy. Rather one is setting out rules which define the hierarchy. Those definitions cover an infinite number of cases. However no one can actually carry out the construction for an infinite number of cases.

It was in fact the idea of the rules that I was referring to with the set theory example.

There is no rule (or axiom) in set theory that states that the quantitative property of a set must necessarily be transferred onto any of it's elements. There mere fact that we can continue to constructs elements without bound does not in itself imply that any of the elements within the set must necessarily become infinite.

All I'm saying here is that there is no rule in set theory that allows us to do that, or make that conclusion in any way. If we jump to any such conclusion then we are doing that intuitively on our own, and we really have no logical reason to do it.

On the contrary the rules of addition tell us just the opposite. We have started out with a finite number of point (the two end points). We then begin to add points half way between these two end points, etc. Well each time we do this we are adding a finite number of points to a finite number of points, and we have an axiom somewhere that says that the result of any such operations on sets must also produce a finite set. So from the operation of addition we have proven that all individual elements in our infinite set must indeed be finite.

It seems to me that putting all of this together set theory has indeed shown that any finite line must contain a finite number of points. Even though there is no bound on the finite number that we may choose to express this quantity.

The Points Salemans:

There is a points salesman who has cards that each contain a finite number of points on them. He has an infinite number of cards. Before you begin the race you much purchase a card to describe the number of points in your line. Then you may begin your race. No matter which card you chose from his infinite inventory you will end up with a card that contains a finite number of points. Then when you run the race, you will only need to complete a finite number of tasks.

Zeno's Paradox:

Zeno comes along and says, "Hey wait! Try doing this instead,…". And then he tells you to try stepping half way to the next point each time instead of using the points on a card from the set that defines the possible number of points in a line.

All that Zeno has done was to force you to use the last card in an infinite set of cards. But that is impossible because an infinite set has no last card. Zeno has set you up to fail. He has transferred the infinite property of the main set onto the elements.

How did he do that? It should be obvious. Imagine that you begin with a line that contains only two points. To step half way to the end point you must trade in your card with the 2 on it and request a card with a 3 on it. (in other words, you must dynamically redefine your line and move to the next card in the set). But after you make that step you again run out of points and must trade in your 3 card for a 5 card. Only then can you make the next step. This will continue forever because you are not actually completing the number of points of any particular card, instead what you are being forced to do is to continually choose a new card from the infinite deck with each step. In other words, you are being forced to redefine your line with every step you take!''

To me Zeno's Paradox simply proves that we can't construct an axiom that would allow us to force the quantitative property of a set onto its elements, because to do so would cause a logical contradiction.

Anyhow, this is how I think of the problem from the point of view of set theory. I'm sure that there are other ways to think of it as well.

drnihili wrote:[/]
Yablo's paradox does not contain any self reference, so I'm not sure where you're headed with this one.

I don’t see how you can say that. The truth value of each sentence is dependent on the very set that it is a member of. Or should I say, the membership of each element in the proposed set is dependent on the truth value of a statement about the structure of the set itself. If that isn't self-reference I have no idea what is!

 

[Hurkyl]

This whole conversation has me thinking along those lines though, and I must confess to already have a strong belief that time and space must both be quantized.

I would too like to see a discrete model of space-time (marginally different than "quantized", but I think you mean discrete anyways)... but I don't go promoting the idea because there's no proof.

I still believe that I'm on the right track thinking in terms of what Einstein said that distance is what we measure with a rod, and time is what we measure with a clock.

Right. In fact, in my casual reading of both GR and QM, the issue of what constitutes a "rod" and "clock" is one of the more interesting and still active research topics. One interesting fact is that, according to QM, any physical clock has a finite probability of giving the same reading twice!

So yes, it's bogged down in the messiness of the real world. But unless you want to simply be a Hilbertian formalist, you're kind of stuck with that.

I am a formalist! (at least to some extent)

I put a very clear demarcation here; it is the purpose of mathematics to tell us what are and are not the consequences of a set of axioms, and it is the purpose of science to determine which axioms the subject of interest models.

(but the two are not disjoint)

I take Zeno's Paradox to simply be saying that if you attempt to actually add an infinite number of points to a line you can never succeed. Not even by using pure logic! Logic itself is telling us that this is an impossible task. Set theory only serves to verify this conclusion.

Logic does not tell us this is an impossible task; logic merely tells us that induction is not sufficient to prove it possible.


Incidentally, the question about lines is not whether we can place a set of points on a line, but a question about the set of points that are on the line. So while induction cannot (by itself) prove that we can place infinite points on a line, it can prove that there are an infinite number of points on a line.

(given the reponse written while I wrote this, I think this warrants proof, so I'm writing it now)

 

[Hurkyl]

(the context is Euclidean geometry)

Theorem: Any line has an infinite number of points on it.

Proof:

Let L be any line.
Choose P on L.


Lemma: For n > 1, there is a set of n points on L, S_n, such that there is a point Q_n in S_n such that for any other point R in S_n, R*Q_n*P. (That is, Q_n is between R and P)

Base case:
An incidence axiom guarantees that any line has at least two distinct points. One of those points has to be different than P; call it A.

A betweenness axiom guarantees a point Q such that A*Q*P. Define S₂ := {A, Q}. This satisfies the statement of the lemma for n = 2.

Inductive step:
Let S_n satisfy the statement of the lemma. The same betweenness axiom guarantees the existence of a point Q_n+1 such that Q_n*Q_n+1*P.

For every point R in S_n different from Q_n, it is true that R*Q_n*P. We also have Q_n*Q_n+1*P, thus we can conclude R*Q_n+1*P. (I don't know wehre my text is, I can't remember if this is an axiom or a theorem) This allows us to conclude that R is distinct from Q_n+1

Define S_n+1 := S_n U {Q_n+1}. This set satisfies the statement f the lemma.

Thus, the lemma has been proven for all natural numbers n > 1.


Let T be the union of all of the S_n for natural numbers n > 1

Every point in T lies on L (because it is an element of one of the S_n's)

|T| >= |S| = n for all natural numbers n > 1, so T cannot be finite.

Therefore T is an infinite set of points on L, and this concludes the theorem.

 

[drnihili]

Originally posted by NeutronStar
I don’t see how you can say that. The truth value of each sentence is dependent on the very set that it is a member of. Or should I say, the membership of each element in the proposed set is dependent on the truth value of a statement about the structure of the set itself. If that isn't self-reference I have no idea what is!

Self reference is when a sentence refers to itself as in "This sentence is false". If you want to prohibit reference to any set of which the sentence is a part, then you won't be able to say anything about sentences at all. Not even innocuous things like "the first sentence of this post begins with the letter 'S'."

But whatever, this isn't a thread about self-reference. I was merely pointing out that there are lots of cases where we can find things such that if they are true of n, they are true of n+1 and yet they are not true of the natural numbers taken as a set.

 

[drnihili]

Originally posted by Hurkyl
II am a formalist! (at least to some extent)

I put a very clear demarcation here; it is the purpose of mathematics to tell us what are and are not the consequences of a set of axioms, and it is the purpose of science to determine which axioms the subject of interest models.

(but the two are not disjoint)

That's fine. In that case you have to agree that calculus (nor any branch of mathematics) cannot by itself resolve Zeno's paradox. Any resolution would have to consist not only of a set of axioms and cosequences but also an interpretation of the axioms in terms of the real world and also an argument that the axioms correctly model the world. The first of these tasks might, perhaps, be mathematical, but the second clearly isn't.

So from your revised formalist perspective, you still owe an argument that you have an accurate model of what's going on. At this stage it's entirely unclear how such an argument might go.

 

[Hurkyl]

Zeno's paradox is a pseudoparadox; it derives no contradiction. There's nothing to resolve.


As to whether an infinite divisbility model correctly models the world, that's a completely separate question.