## A new twist on Zeno’s Arrow Paradox

Zeno of Elea’s arrow paradox, like some of his other extant paradoxes, aims to show that our observations of change and becoming in the world are illusions. Our senses suggest that there are all kinds of change and motion of things around us, but our reason concludes otherwise. As good philosophers, we should listen to the voice of our reason, rather than the evidence of our senses, and reject the reality of motion and change in the world.

Here’s how the Arrow Paradox is supposed to help show the unreality of motion. (What follows is a common reconstruction of Zeno’s argument.) Consider an object like an arrow which our visual experience describes as moving in its trajectory in the air. Zeno claims that at every instant of its supposed flight, the arrow occupies a region of space exactly coinciding the size and shape of the arrow. But if an object occupies a region of space coinciding with the size and shape of the object, then the object must be at rest. The arrow at every instant during its supposed flight, therefore, is at rest; it is at no moment in that time interval in motion. So, contrary to the judgment of our senses, motion is impossible.

A popular solution to Zeno’s Arrow Paradox is Russell’s “at-at theory of motion.” According to Russell, an object cannot be in motion (nor can it be at rest) at an instant. To be in motion is to be at different locations at different times. (And to be at rest during an interval of time is to occupy the same location at every instant of that time interval.) Knowledge of the location of an object at a single instant does not tell us anything about its kinematic status.

Let us take, as Zeno seems to be doing, an instant or moment of time to be a time magnitude of zero duration. Call the space region occupied by the arrow at instant *t* the location *L* of the arrow at that instant. Notice that Zeno has no qualms about the arrow being at location *L* at time *t* ; he thinks the arrow is there, but is just not in motion there.

Let us take note of the following fact: The arrow is at location *L* for a* zero duration* of time. This is, by the way, the kind of claim that, not only Zeno, but all (Newtonian) physicists today would be apt to make. But what sense does it make to say that an object *is* at some place for an instant, that is, *for zero seconds*? Is there any difference between asserting, “The object was there for zero seconds” and “The object wasn’t there at all”? Imagine yourself saying to your friend, “I was in my office yesterday.” Your friend asks, “For how long were you there?” And you answer, “For zero seconds (or minutes or hours).” Aren’t you saying (in a weird way) that you were not in your office yesterday? It would seem that being present at some location for a zero duration of time is equivalent to being absent at that location. If that is so, then it follows that the arrow did not occupy *any* of the locations on its supposed trajectory! It’s not that the arrow was at rest at every point of the trajectory, as Zeno contends; the arrow was *not* at any point of the trajectory. Nevertheless, his and my conclusions are the same: The arrow could not possibly traverse that trajectory.

Now, the at-at theory of motion cannot solve the new paradox, for it presupposes that the arrow *can* occupy different locations on the trajectory at different times. But this is precisely what the “new twist” is rejecting.

To sum up: Whether Zeno’s argument for the motionlessness of the arrow is fallacious or not, our argument shows that the arrow is not to be found at any point of its presumed path. This means that the arrow simply didn’t budge; it stayed put the whole time at its initial rest position.

But things get more depressing: Being at rest during a certain interval of time involves an object’s occupying the same resting location at every instant of that time interval—for zero seconds. So, by a similar argument, we seem forced to conclude that the arrow is not to be found at its rest location, either, at any of the instants during the time period of its supposed rest.

Maybe Zeno’s conclusion was right after all—there is no such thing as motion. But, furthermore, there may be no such thing as rest either. Rest may not be any more possible than motion.

If you are not moved by my new twist on this old paradox, maybe you can walk me out of it…

Written by Erdinç Sayan

February 25, 2012 at 12:36 am

Posted in Metaphysics

### 77 Responses

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An “early Modernie” comment: the conclusion that there is neither any real motion nor any real rest is quite fair (qua conclusion), as the assumption here is actually that duration itself is not real. (How could time, as a “unity”, consist of “zero” durations?)

Arman BeslerFebruary 26, 2012 at 12:37 am

Thanks Arman. Two replies. One, I think neither my argument nor Zeno’s need to assume unreality of time and duration. He can assume (

pace, e.g. McTaggart) that there is real time (like the Newtonian “absolute time”) and deny the reality of motion. I can similarly be an absolutist about time—if only for the sake of my “new twist”—but deny that any trajectory can possibly be traversed by any object and also deny that there can be any location of rest occupied by an object. If someone were to ask, “Then where in the world is the object to be found, if not in motion or at rest?”, I would reply that I am not calling it a paradox for nothing. Second, how an infinity of zero-duration instants come together to constitute a nonzero length of time is an ancient mystery, but I don’t think I have to solve this mystery before I can set up my argument. Here’s an analogy. Think of the real number interval between zero and one. It’s a continuum, right? I can claim that numbertwo, for instance, is nowhere to be found in that interval without having to explain how the real numbers between zero and one can form a continuum.Erdinç SayanFebruary 27, 2012 at 2:39 am

I am not a philosopher, but this argument somehow depresses me deeply — gives a “new twist” to “The Wreck of the Hesperus” be it on the Bosphorus or not… No time, no motion, no rest, no instant, nothing… Does that mean no big-bang, no argument, no paradox, no problem?

BugsyMarch 3, 2012 at 8:58 pm

The guide for the depressed: Look at it from the brighter side. If there is none of those things, there is nothing to be depressed about. (To the third parties: Just kidding around with my old buddy Bugsy.)

Erdinç SayanMarch 4, 2012 at 1:25 am

Maybe we can expand your new paradox to a greater one.

If a physical object cannot be at rest at any time, then in the extended version we can say that it cannot be at rest in any spatial region at any time. This seems to be another way of saying, “Physical objects cannot exist in space.”

Since physical objects necessarily need to have spatial extensions in order to exist, we can conclude that physical objects cannot exist.

Tolga KurtMarch 12, 2012 at 10:27 pm

Thanks Tolga. Good suggestion. In my reply to Arman earlier I had something like that in mind when I said: “If someone were to ask, ‘Then where in the world is the object to be found, if not in motion or at rest?’, I would reply that I am not calling it a paradox for nothing.”

So a “side paradox” might go like this:

— A physical object can be neither at rest nor in motion in any region of space at any time. (As my argument in “New Twist” concludes.)

— Therefore, a physical object cannot be found in any region of space at any time, because a physical object can only be found either at rest or in motion in a region of space at any given time. (There is no third alternative.)

— It is a necessary condition for a physical object to exist that it is found in (occupies) some region of space at some time.

— Therefore, no physical object can exist!

Erdinç SayanMarch 14, 2012 at 9:28 pm

By the way, in an analogous way, we can form a new paradox.

If an object occupies a region in space, it means that that object also occupies all of the points within that region. Since a point in space has no extension, i.e. no volume, and if something is said to occupy that point, that “something” cannot be a physical object since physical objects necessarily are extended. In other words, something occupying a point occupies nowhere in space.

To conclude, no physical object can occupy any point, that is to say, no physical object can exist in space. Therefore, no physical object can exist.

Tolga KurtMarch 12, 2012 at 10:50 pm

You are right—an analogous reasoning to the one in the “New Twist” seems to yield another “side paradox.” As follows:

Consider any point in space. Can any physical object or part of a physical object occupy it? Suppose it can. How big could anything physical be that occupies a spatial point? Its size has to be zero. But a physical object of size zero is not an existing object. So, no existing physical object or part of a physical object can occupy a point in space. The same goes for any point of space. Therefore, no physical object can occupy space. But a physical object can only exist in space. Ergo, no physical object can exist!

Erdinç SayanMarch 14, 2012 at 10:38 pm

I think all the “follow-up arguments” suggested here are valid for Newtonian rigid bodies. No modern physicist will accept the condition that a physical object must have extension or that it must occupy certain space at a certain time.

First, there are many point particles like electron which do not have extension. They certainly are physical objects: they influence their surrounding regions with their spin, charge, etc. But to do so they do not need extension. I see no reason to insist that physical objects must have an extension. One needs to show that an object which does not have an extension cannot enter into causal relations. I do not see how one can do that.

Secondly, from Heisenberg uncertainty principle we know that the position of a particle whose velocity is known with certainty (“plane wave” particle) will be completely uncertain according to the Copenhagen interpretation of Quantum Mechanics. This uncertainty is ontological, that is, the particle does

nothave any position. Again, I see no reason to assume that an object which lacks position at certain moments cannot enter into causal relations.I think the main characteristic of a physical object is that it can enter into causal relations. So, these “follow up paradoxes” should be directed to Newtonian rigid bodies rather than quantum-physical objects.

Enis DokoMarch 28, 2012 at 10:25 pm

Enis said: “First, there are many point particles like electron which do not have extension. They certainly are physical objects: they influence their surrounding regions with their spin, charge, etc. But to do so they do not need extension.”

I want to remind Dr. Sayan’s approach in the text: “Is there any difference between asserting, ‘The object was there for zero seconds’ and ‘The object was not there at all’? “

In a similar way, I am asking: Is there any difference between saying “An object occupies zero extension in space” and “An object does not occupy anywhere in space”? The last step is to assert “Occupying nowhere in space is not being in space at all.”

If there

isa difference between these statements, then occupying somewhere with zero extension in space does not necessarily mean not being in space. So, your quantum particles can exist. But, then I will ask, which premise is wrong in which way.But if there is no difference, then neither Newtonian bodies, nor extensionless quantum particles can exist (if they necessarily need to exist in space).

Tolga KurtMarch 30, 2012 at 7:37 pm

Thanks for your comment Enis, my physics mentor.

Yes, I must confess that I had the Newtonian picture in mind when I supported Tolga’s new twist on my new twist.

So, you are saying: (1) a physical object like an electron can occupy zero extension in space (as it is literally a point-particle) and yet

isin space, as its causal interactions with its environment testify; (2) a particle whose velocity we determine with certainty is, by Heisenberg’s Uncertainty Principle, not locatable in space, or rather, has literally no location in space.I can see the relevance of your example of point-particles to Tolga’s argument in his second comment, but I don’t see how (2) undermines that argument.

Also, could you illuminate me about the following?

Re (1): Could it be the case that an electron does have a nonzero size, but it is too small for our measurement instruments to be able to measure it? (By the way, it seems to me that Heisenberg’s principle should be a barrier in measurements of size too, shouldn’t it?) If electrons and other “point-particles” in fact have nonzero sizes, so small that we can’t measure them, then Tolga’s argument will still have force.

Re (2): Wouldn’t it be more correct to say, for a particle whose velocity has been determined with full certainty, that it is located anywhere and everywhere in space, rather than being “outside of space”?

Erdinç SayanMarch 31, 2012 at 12:32 am

Thank you very much sir! I became interested in these topics after your Philosophy of Physics class. You are the efficient cause of this knowledge. :)

1. What we know certainly is that the electron has a radius smaller than 10^-13 m. Thus it may have extension after all. (But attributing extension to the electron would result in several theoretical problems, like why it does not seem to have internal structure, how the electron can stay in one piece without exploding due to its internal charge repulsion, etc.). But I do not think this will help you, since it is logically possible for an electron to be a point particle and be causally active. Since, as I said, it seems there is no direct relation between being causally effective and having extension.

2. What we know is that the particle’s position is infinitely uncertain in the ontological sense. I guess one may also interpret it as the particle being everywhere and anywhere.

Note: In Copenhagen interpretation Heisenberg uncertainty gives us ontological limits. A particle cannot have a definite velocity and position at the same time in an ontological sense. But of course one can use a different interpretation of QM such as the De Broglie-Bohm interpretation, in which Heisenberg Uncertainty Principle imposes an epistemological rather than ontological limit. But of course in this case, due to Bell’s theorem, we need to assume that we live in a non-local universe: the velocity and the position of any one particle depends on the value of the wave function, which in turn depends on the whole configuration of the universe. But all other laws of physics seem to be local, so most physicists prefer the Copenhagen interpretation of QM.

Enis DokoMarch 31, 2012 at 3:12 am

Assuming there is such a thing as causation, thanks.

I and probably Tolga will settle for mere nomological impossibility. If it is impossible for a point-object to exist in our good old actual world, and hence our “extended” paradox can do its dirty job in the actual world, that’s good enough for us.

Maybe logically speaking. But nomologically? I am not sure. And I am not sure if anyone can be sure, either.

Erdinç SayanMarch 31, 2012 at 2:10 pm

Yes, I agree that there can be nomological necessity between being causally effective and having extension. But since you are defending the argument, you need to show that there is nomological necessity. I think at this point no one can do it. Thus the follow-up argument is based on an unjustified assumption.

Let me complicate things a little more by including GR. Well, we know that space-time is causally active: it affects the mass distribution and trajectory of the particles. Space-time is not located anywhere, therefore is not extended. Thus unless you defend Machian relationism, GR shows that non-extended objects like space-time can be causally active. Thus , if you want to defend the follow-up argument, not only you have to show the said nomological necessity, but you also have to defend Machian relationism.

Enis DokoApril 1, 2012 at 3:24 am

Since you are the one who is attacking our argument by claiming “it is logically possible for an electron to be a point particle and be causally active,” maybe it is

youwho need to prove his claim. As I said, the extended paradox would still have force (i.e. apply to the actual world) unless you establish the following claim: It is nomologically possible for a point particle to have causal efficacy, and hence by your proposed principle, to exist in space. The example of an electron seems inconclusive, as an electron may in fact have spatial extension.Leaving aside the question of who has the burden of proof, how about the following argument, which more than meets your challenge?

(1) Tolga and I argued that a physical object that has zero extension in space cannot exist in space. This seems a

logical necessity, given our concepts of “space,” of “having no extension” and of “physical object.”(2) It seems to me

at least nomologically(if not logically)necessarythat a physical object that does not exist in space cannot have causal efficacy.Therefore: (3) It seems to me

logically impossibleto have a physical object with zero spatial extension having causal efficacy.To see that it is valid, compare the above argument with the following analogous argument:

(1’) It is

logically necessarythat a round cube cannot exist.(2’) It is

nomologically necessaryfor very large and highly condensed objects (like stars, planets, white dwarfs, etc.) to have a round shape.Therefore: (3’) It is

logically impossiblefor very large and highly condensed objects to have cubical shape.No, thanks. The extended paradox is about regular physical objects, not about weird things like space-time, Mr. GR.

Erdinç SayanApril 1, 2012 at 7:15 pm

I have been silently following this thread with excitement and –sometimes– with frustration.

I want to make a methodological point and then try to improve on my first attempted solution.

My methodological point may sound a bit crude and insensitive but it is really not my intention to anger or belittle anyone. I just find Dr. Sayan’s new spin on the paradox fascinating and want to get to the bottom of this issue with efficiency and with a clear head.

So, having made my heart-felt disclaimer, here is the methodological point: The original paradox and Dr. Sayan’s enhanced version engage the pre-theoretic concept of motion, which is not a part of, but a pre-requisit for all physical theories allowing for continuous trajectories in real space-time. So, quoting from and falling back to various interpretations of QM or GR can’t possibly help. After all, unless you reduce motion and rest to something else, you can’t meet the challenge. These theories and interpretations either assume motion and rest (not necessarily in an absolute sense) and therefore they beg the question against the enhanced version, or they deny motion (as in the block universe interpretation of GR) and therefore agree with the spirit of the original paradox. Either way they can’t possibly fit the bill for a resolution of the paradox. In this regard, it should not come as a surprise that the Russellian at-at-theory did not appeal to any fancy physics but just attempted to reduce motion to a special form of rest. Dr. Sayan’s point is that the Russellian reduction is not satisfactory because a similar difficulty arises with rest too. Point well-taken.

Now, it is easy to see that the methodological point yields a desideratum for any satisfactory resolution to the paradox:

Rest and motion should be reduced to or replaced with something pre-theoretic (e.g. geometric, algebraic, set-theoretic, phenomenologic, folk-physicsy, etc.)

In my original attempt involving line segments, I tried a geometric reduction but I did not manage to get rid of the assumption of rest. Now I will amend my original suggestion and try to reduce rest into some other geometric notion.

Originally I suggested something like this: “A particle is at B” means “for a non-zero amount of time the particle was at a line segment (i.e. [AC]) which includes B.” The difficulty with this suggestion is not that it presupposes motion, but presupposes rest. This is so because, the particle would be either in motion along [AC] or be at rest at some point in [AC]. Since we can’t admit motion, then we will have to admit rest.

Now, let’s reduce rest too.

I suggest: “A particle is at B” means “(1) The particle exists in real space and (2) you can make [AC] segment, which includes B, arbitrarily smaller around B in infinitely many different ways and still for a finite non-zero amount of time the particle will not be found outside [AC].”

This gives us a negative definition of being at-at, which does away with the need for the particle to be at rest at any individual point.

Let me know what you think.

feylezofrizaApril 3, 2012 at 7:24 am

Thank you for your follow-up comment. Apologies for not responding sooner.

Interesting suggestion. If I understood it correctly, it doesn’t seem to yield the correct verdict under the following kind of situation: suppose a (point-)particle was at rest at B between 1:00pm and 2:00pm on a certain day, but after 2:00pm on that day it moved to somewhere else, going outside the initial spatial interval [A, C]. The problem seems to be that your definition does not specify

whenwe are to check if the particle is or is not found outside [A, C] for a finite non-zero amount of time. The following version would seem to take care of that problem:A (point-)particle is at point B during time interval [t1, t2] iff(1) The particle exists in real space and (2)for any spatial interval [A, C] which includes B, the particle is not found outside [A, C] during [t1, t2].This is a definition of “being at-during” (being at a point during an interval of time) rather than being at-at, and corresponds to our common sense (I would also say, Russellian and Newtonian) notion of being at rest at a point during an entire interval of time.

Does this definition of “being at-during” avoid my paradox? It seems to me no, because I am guessing it can mathematically, rigorously be shown that it entails the following:

A particle is at point B at every instant t within the time interval [t1, t2] iff(1) The particle exists in real space and (2)for any spatial interval [A, C] which includes B, the particle is not found outside [A, C] at t.This last is the at-at theory all over again, and if I am right, it is a logical consequence of the at-during theory. At this point my paradox shows its ugly head again—for how can a particle be anywhere for a zero instant of time? And this means that,

ipso facto, the at-during theory is also prone to that paradox.Erdinç SayanApril 8, 2012 at 3:06 pm

At first I was very impressed by your argument, but after a close inspection I think the argument is invalid. Before analyzing the argument let me briefly comment on the burden of proof issue. I think I did provide an argument for the claim “It is logically possible for an electron to be a point particle and be causally active.” The argument is simple: Most physicist believe that electron is a point particle and is causally active. Thus, even if they are wrong, they can conceive of the electron as a point particle being causally active. This conceivability is pretty good reason to assume that “it is logically possible for an electron to be a point particle and be causally active.” Besides, classical physics is based on point particles and the theory seems to be consistent, which I think provides a second argument for assuming that there is no logical contradiction in believing that electron is a point particle and is causally active.

If we return to your argument, I think the proper conclusion should be:

3. It is nomologically impossible to have a physical object with zero spatial extension having causal efficacy.

Similarly:

3. It is nomologically impossible for very large and highly condensed objects to have cubical shape.

Or:

3. It is not logically necessary for very large and highly condensed objects to have cubical shape.

I could not detect exactly the fallacy in your argument. But I think your argument can be reproduced like this (I did not have much time to think, so I may be misrepresenting your argument—in which case I am very sorry. It was not on purpose ):

Px: Object x is round.

Qx: Object x is cubical.

a: All stars.

Then your argument goes like this:

1. □(Px->~Qx) (Necessarily, if object x is round, then it cannot be cubical.)

2. Pa (All stars are round in the actual world.)

3. □(~Qa) (It is logically impossible for stars to be cubical.)

But this is clearly fallacious. You cannot apply modus ponens in this way. All you can conclude from above is ~Qa or ~(□Qa).

Of course you may be happy with the conclusion given as nomological necessity. But note that anyone who rejects the conclusion, most probably will reject the second premise too. (Given your definition, of course, which implies that point particles are not in space. But note that they can still be in time.) Thus I do not think that your argument would have much force for such person. But still, in my opinion the argument is very ingenious and impressive.

Lastly, let me briefly comment on feylezofriza’s methodological comment. You say: “Dr. Sayan’s enhanced version engage[s] the pre-theoretic concept of motion, which is not a part of, but a pre-requisit for all physical theories allowing for continuous trajectories in real space-time.” Let me remind you that particles in the Copenhagen interpretation of Quantum Mechanics do not follow continuous trajectories in space-time. Thus, you see, what you call pre-theoretic concepts can change from theory to theory. Thus in my opinion it is extremely important to be cautious about which physical theory you are using in analyzing the Zeno paradoxes. In fact, I am not the only person who thinks so. Hans Reichenbach, for example, was not impressed with the solution of Russell. He too, like me, believed that the paradox arises from assumption of Newtonian physics. According to Reichenbach, the Newtonian assumption of space and time being different entities leads to the paradox. Once we realize that we live in an Einsteinian rather than a Newtonian universe the paradox will disappear. Unlike Reichenbach, I believe that the solution lies in QM, not in relativity.

Enis DokoApril 3, 2012 at 2:31 pm

Yes, on the Cophenagen interpretation of QM particles do not follow continuous trajectories because they do not admit locations (even fuzzy locations) before the wave function collapses. But that is not explaining how motion is possible, it is just denying that motion ever happens. In that regard, it is not an adequate resolution to the paradox, and never can be.

feylezofrizaApril 3, 2012 at 5:18 pm

You are right, it was a blooper on my part. The conclusion “It is logically impossible for very large and highly condensed objects to have cubical shape” is so obviously false. What was I thinking…

OK, here’s my argument reinstated with a conclusion good enough for my purposes:

(1) It is

logically impossiblefor a physical object with zero spatial extension to exist in space (given our concepts of “space,” of “zero extension” and of “physical object”).(2) It is

nomologically impossiblefor a physical object that does not exist in space to have causal efficacy.Therefore: (3) It is

nomologically impossiblefor a physical object with zero spatial extension to have causal efficacy.(The parallel argument now is:

(1’) It is

logically impossiblefor a round cube to exist.(2’) It is

nomologically impossiblefor a star to have a round shape.Therefore: (3’) It is

nomologically impossiblefor a star to have a cubical shape.)I sure am.

If one believes in extra-spatial beings, like Descartes’s minds for example, one might attempt to reject (2). But (2) is talking about physical objects, not things like Cartesian minds (which are a headache even for Descartes because of the problem of interaction of them with spatial objects).

I doubt it. One could argue that a physical object that does not exist in space does not exist in spacetime, and hence does not exist in time either. I love the relativity theories as much as you do. :)

Small correction, if I may: Concepts that change from theory to theory are not

pre-theoretical.I thought that my paradox, (1) infects quantum objects after measurement, when they supposedly collapse into a state of motion or rest, (2) is a slap in the face of QM, as there can be no state of motion or rest that pre-measurement objects can collapse into—contra QM.

And here is a new idea that extends my original paradox to the quantum worlds: (3) To say that an object is in certain superposed states

for zero secondis just to say that the object is not in those superposed states (whatever “superposed states” means). Hence, objects cannot be in superposition of rest and motion (or, for that matter,anysuperposed states) either.This is an interesting issue altogether. It is a very common practice for scientists to make assumptions which they are totally sure to be false or even contradictory. (I think Nancy Cartright, for one, has examples.) They assume frictionless surfaces, ideal gases, uniform electric fields, they sometimes assume a huge star to have no mass or no extension. They make such idealizations and aproximations to simplify the mathematical treatment of the problem in their hand. In their “reflective moments” they may (or should) admit that a point particle cannot possibly exist, but they use the notion of point particle in their calculations as an aproximation or idealization, i.e. as an instrumental tool. This practice of scientists is not evidence that they can conceive of literally point particles. (You may remember that in the Appendix of my PhD dissertation I show that Newton’s statement of his 2nd law in terms of point particles leads to contradiction.)

Erdinç SayanApril 8, 2012 at 6:02 pm

I have an answer to your question “Is there any difference between asserting, “The object was there for zero seconds” and “The object wasn’t there at all”?”

Yes, there is and the at-at theory can account for it. To see the difference, imagine a line L which contains points A, B, C which are separated by finite non-zero distances and located in that order. Also imagine a point D outside L. Now consider a point particle moving through L. According to the at-at theory of motion, the particle was at B for zero seconds and it was not at D at all.

The difference between the two is that for a non-zero amount of time the particle was at a line segment (i.e. [AC]) which includes B, but there is no such line segment including point D.

This shows that the Russellian can account for the difference between being at some point for zero seconds vs not being there at all. Notice, this undercuts your inference to the conclusion “the arrow is not to be found at any point of its presumed path. This means that the arrow simply didn’t budge; it stayed put the whole time at its initial rest position.”

feylezofrizaMarch 15, 2012 at 9:45 pm

Thank you for your interesting reply.

I think your solution is crucially based on the assumption that a point particle

canmove—through L. This is the denial of my conclusion that there can be no motion. So, in your attempt to undermine my argument you are directly using the negation of my conclusion as one of your premises. It is like trying to undermine Zeno’s original arrow paradox by starting with the assumption that there is such a thing as motion. By contrast, Russell’s at-at theory simplyredefinesmotion, rather than setting up an argument against Zeno that starts with the denial of Zeno’s conclusion. You might say Russell’s redefinition threatens my paradox too. But then I think Russell (and you) need to directly attack my premise that says “To be at a point for zero duration is not to be there at all,” without presupposing that motion is possible in the first place.By the way, I don’t want to sound like I think “my paradox” is insoluble or anything…

Erdinç SayanMarch 17, 2012 at 1:21 am

I was worried about this too and I want to avoid making that assumption. Notice, the way I phrased the crucial bit of my reply was “for a non-zero amount of time the particle was at a line segment (i.e. [AC]) which includes B.” That does not seem to require motion to me.

Here is an argument why. I have been sitting in my chair motionless now for 5 minutes, and during this non-zero duration I have also been somewhere inside the line segment that starts at the floor and ends at the ceiling of my office. Does that require that I have been moving from the floor to the ceiling, or the other way around? No.

However, I would love to hear more if you think that I smuggled in the assumption of motion some other way.

feylezofrizaMarch 17, 2012 at 2:40 am

This time you may be smuggling in rest rather than motion. Recall that I argue that rest is as impossible as motion.

To say that the particle is

atthe line segment AC for a non-zero duration of time is to assert that the particle is either in motion or at rest at AC for that time period. I can’t think of a third alternative. But, given my major premise that says to be at a spatial point for a zero time is equivalent to not being there, rest is not any more possible than motion. So, contrary to what you may be thinking or feeling, you can’t be motionless in your office during that 5 minutes of “rest” from your work. See, the paradox really thickens…Erdinç SayanMarch 17, 2012 at 12:25 pm

I do not think that feylezofriza’s solution necessarily assumes that the particle is either at rest or moving. It is enough to claim that the particle was for a finite time interval between the points A and C. One may even use a weaker principle like “There is a finite probability for the particle being between A and C.” Of course you may object that “between” implicitly implies either being at rest between A and C or moving between these points. But I do not think this is necessarily true. If I use my old weapon, Quantum Mechanics, I can say that the particle is at some superposition of these two states.

Alternatively one can restate the principle as “The particle is or could be causally effective between points A and C for some finite time interval t”. I do not think this second principle is equivalent to the negation of your conclusion.

But I think you can attack the principles given above, as being too vague. Surely you can choose some big interval such that this point D at which the particle does not belong is also in this interval. Thus the condition is satisfied. Also the choice of time interval is vague. How big should that time interval be?

Thus one may need to modify the condition as, “For all space intervals [A,C] which include point B and all finite time intervals which include this ‘zero second instant’, the particle is (or is casually effective) between A and C.” I wonder what do you think about this modification—can it save the principle?

Enis DokoMarch 29, 2012 at 3:41 am

Thank you Enis, my friend and former student, a man who takes Quantum Mechanics and Relativity Theory everywhere he goes. :)

I love your comment. A cheap response on my part would be to admit that in advancing my paradox, I had in mind what Zeno too had in mind: the Newtonian-Mechanical world view. But why do I (and Zeno) have to be that old-fashioned?

Sure.

OK, I was being old-fashioned when I said I can’t think of a third alternative. The third alternative is, as you say, the superposition of both being at rest and in motion (add to that: with some probability). But what does it mean to say that the particle is at a superposition of the states of being at rest and of being in motion between the points A and C? I suppose something like this: (1) the particle is at rest between A and C, AND (2) the particle is in motion between A and C. But I have argued (on the basis of my big fat premise, “To be at a spatial point for a zero time is equivalent to not being there”) that neither (1) nor (2) holds.

Ergo 1: The particle

cannotbe at a superposition of motion and rest.Ergo 2: The Copenhagen Interpretation of QM which claims that (1) and (2) hold is wrong!

Ergo 3: Your old weapon backfires. :)

Gees, my paradox really really thickens…

Of course, one might object that being at the superposed states of being at rest and in motion is not quite (1) and (2) holding together. In that case, I don’t understand the notion of superposion of states, and if nobody understands it either, I don’t know how such an unintelligible notion can be used in a solution of my very intelligible paradox.

This is what I would assume here: To be (or disposed to be) causally effective between A and C, a necessary condition is that the particle be at rest, in motion, or in the superposed state of being at rest and in motion, between A and C. But if my reasoning above is correct, the particle cannot be in any of those states between A and C. I don’t find it intelligible to claim that the particle can be causally effective between A and C without possessing any of those three possible states between A and C.

Ergo 4: There can be no such thing as causation!

Gee whiz!!

Please somebody stop me before I go:

Ergo n: There can be no such things as

principles!Erdinç SayanMarch 30, 2012 at 2:47 am

Very interesting response, as I expected :).

I think your cheap solution will work; in a Newtonian universe you and Zeno are right. In my opinion the proper solution of all three paradoxes must involve QM.

Let me explain what I mean by superposition. When a particle is in the superposition of state X and Y, that does not mean that the particle is both at state X and Y. On the contrary, the particle is neither at state X nor at state Y. The particle will gain one of these states when we try to determine its state. Unless the particle is disturbed by measurement, it is neither at X nor at Y. Thus unless we measure either the particle’s position or velocity we can not say that it is at rest or moving. But this lack of information, in Copenhagen interpretation, is ontological in nature. The particle lacks position (thus is not at rest), and velocity (thus is not moving). This may sound counterintuitive, but I believe we have pretty good reasons to believe it. In my opinion Zeno and you provide additional arguments in support of Copenhagen interpretation. :)

But my second point may be promising too. Let me clarify it. Following Russell’s solution of the Arrow Paradox, I redefine the concept of occupying a certain position at some instant as follows:

“A particle occupies the position B at some time t, if the particle is (or maybe) causally effective for all space intervals [A,C] which include point B and all finite time intervals which include time t.”

I think this new definition does not imply being at rest in the sense you assume in your argument, i.e. occupying a certain point of space for some instant (zero time). After all, time may turn out to be discrete (in which case I guess your argument will fail again), so that there is no such thing as a zero-second instant. But my above definition will be valid even if time is discrete.

Enis DokoMarch 31, 2012 at 1:21 am

Thanks.

So, you agree that in a possible world governed by Newtonian physics motion, rest and all the rest will be impossible. Poor Newton wasted his time working out his theory of mechanics…

We’ll see about that…

Aha! Then my paradox is valid for

after the measurement, if not before. Your QM entails that after a measurement a particle will collapse into a state of motion or rest—welcome to the world of nonsuperposed rest and motion! My paradox picks it up from there. Moreover, since neither motion nor rest is possible for measured objects, there can be no state a pre-measurement superposed state can collapseinto.Ergo i (I lost count): There can be no superposed states and/or measurement. Or: No measurement of a particle’s kinematic status can yield any results!

Ergo i+1: QM (at least, its Copenhagen interpretation) is false!!

You see my new paradox (call it “The Newer Twist” if you like) devastates QM too. It is a paradox both for quantum worlds and Newtonian ones.

Gasp!

For the reasons I gave, I beg to disagree, Mr. QM.

Your principle is not going to save QM from collapsing into a state of annihilation. :)

Maybe one of these days I will come up with a paradox for staccato motion too. Maybe I’ll call it “A New Twist on Zeno’s Stadium Paradox.”

Erdinç SayanMarch 31, 2012 at 1:32 pm

t = 0: I am shifting the line [0, 1] to position [-0.5, 0.5]

t = 0.5: I am shifting the line from position [-0.5, 0.5] to position [-0.75, 0.25]

t = 0.75: I am shifting the line from position [-0.75, 0.25] to position [-0.875, 0.125]

…

This is a list of infinitely many steps. What is the position of the line after execution of all the steps on this list, at t = 1? Is there a step on this list shifting the line to position [-1, 0]?

Reinhard FischerAugust 11, 2012 at 12:05 pm

Thanks for the interesting puzzle.

If I understand your question correctly, the following would be a somewhat simplified version of it:

at time t = 0, the point P (which you can take to be the right end-point of the line, if you like) is at position x = 1,

at time t = 0.5, P is at position x = 0.5,

at time t = 0.75, P is at position x = 0.25,

at time t = 0.875, P is at position x = 0.125,

and so on.

So as t gets closer and closer to 1, x gets closer and closer to 0, in the manner of Zeno’s Dichotomy Paradox. Your question is what the position of P will be when t “finally reaches” 1, at the end of this infinite process.

This is not quite the same as Zeno’s Dichotomy Paradox, since you don’t seem to problematize the passage of time from t = 0 to t = 1. The Dichotomy Paradox would as much challenge the possibility of the passage of time as the possibility of motion.

Your question is also reminiscent of Thomson’s Lamp, or the problem of “supertasks,” which is a cousin of the Dichotomy Paradox. Thompson’s Lamp problem asks whether the lamp will end up on or off when t finally reaches 1. You instead ask whether P will be at position x = 0 when t reaches 1.

If that really is the puzzle you are posing, my answer would be “Yes”: P

willbe at x = 0, given that we assume that tcantake the exact value 1. The reason, I think, is that you set up a one-to-one correspondence between time and position points in your formulation of your question. So just as, for example, Pis atposition x = 0.25 at time t = 0.75, Pis atposition x = 0 at time t = 1.Your question strikes me as a hybrid between Zeno’s Dichotomy Paradox and Thompson’s Lamp. But it doesn’t pose a despairing choice for us the way Thompson’s Lamp does (i.e. between the on and off positions of the lamp), because I’d think the one-to-one correspondence between the time series and the position series must also extend to the

limitsof those two series. Maybe and hopefully arigorousproof of that extension can be given.Erdinç SayanAugust 13, 2012 at 12:38 am

Thank you for the response.

Yes, your simplified version is about the same as my version. And I can understand your argument, that the line reaches [-1, 0] as well as point P reaches 0.

What about my second question: Can we find an action in my list above, that enables the line to cover point -1? Does the list above contain all the actions necessary to accomplish that, ie. an action like “shifting the line to [-1, x]“? Or can we say, none of the steps in this list accomplishes that? If so, why don’t we call this a paradox nowadays?

Reinhard FischerAugust 13, 2012 at 7:03 pm

When t takes the exact value 1, P is at the position x = 0. Or, correspondingly, the

leftend-point of your line is at x = -1 when t = 1. I think we agree on that. I guess you are asking the following question: What is the penultimate step to the left end-point reaching x = -1? In other words, what was the step (or action) that took placeright beforethe left end-point reached x = -1 at t = 1?If I understood you correctly (sorry if I didn’t), your question is equivalent to this. OK, the limit of the series

SIGMA(1/2) to power nis 1. But what is the member of that series that is “closest” to the limit? In other words, what is the action right before we finally reach 1?If that is your question, I think the standard mathematical answer is that there is no such “last member” of that series. That is the peculiarity we find in infinite series and sequences; finite series and sequences don’t behave that way.

Perhaps your question can be put as: Can the series

SIGMA(1/2) to power never reach its limit? In other words, is the limit of the series (namely 1) a member of that series? The standard answer again would be “No.” The series gets closer and closer to its limit without reaching its limit—which is what gives us Zeno’s Dichotomy and Achilles Paradoxes, among other puzzles. If I am on the right track as far as understanding your question, yes itisa paradox, since at least the time of Zeno…Erdinç SayanAugust 15, 2012 at 1:13 am

If I am drawing a line from 0 to 1 continously, we know we will reach 1 at t = 1. And we will not lift the pencil before point 1 is drawn. Interestingly, we have drawn infinitely many points and we have even drawn a last point in this case.

If I am drawing the line in steps, like [0, 0.5) up to t = 0.5, [0.5, 0.75) up to t = 0.75, [0.75, 0.875) up to t = 0.875, … I will not draw point 1, because the limit is not element of this sequence. I cannot accomplish this line, even if I cannot prevent time from reaching 1. Thus, the existence of the line is not defined at t = 1, because the last point is not defined.

As soon as I am lifting the pencil, I have defined a last point – a last mark on the paper. I don’t get around defining a last point in the physical world to get something done.

My best guess is, the infinite series is not a sufficient definition of what happens in the real world. I don’t think, that there is something wrong with motion. For some reason the paradox tells us, that there is something wrong, how we are treating infinity, and our mind – which evolved in a finite world – is not trained to handle infinity correctly.

Reinhard FischerAugust 15, 2012 at 9:42 am

Your example about physically drawing a line with an end point, and the infinite sequence failing to produce an end point would be just another version of the Dichotomy Paradox. In the physical world, I can certainly draw a line complete with an end point, but if we analyze it as a series of half-way steps, I should be unable to do it. This is the kind of puzzle philosophers and mathematicians are grappling with since the time of Zeno.

Yes, there seems to be an inadequacy in our mathematical concepts and ways of thinking, since a discrepancy appears between what mathematics of infinity tells us and what happens in the physical world. (I am assuming that Zeno’s Dichotomy still awaits a satisfactory solution.) At the present, mathematics doesn’t look rich enough to deal with the physical world. But I still have hope. We will find some way to make mathematics square with the physical reality, I think.

Erdinç SayanAugust 16, 2012 at 12:45 pm

I am not sure, that logical thinking might get us there. Mathematics rich enough to deal with the physical world might not be understandable any more. We wouldn’t get around “understanding”, that the infinite can be completed.

It is like “understanding”, that uncountably many segments of size 0 added together are resulting in the segment [0, 1]. And it is like “understanding”, that infinitely many segments of size [0, 0.5], [0.5, 0.75], [0.75, 0.875], … added together are resulting in the segment [0, 1] – including the limit point 1.

I fear, logical thinking is rather hindering in this case.

Reinhard FischerAugust 17, 2012 at 9:56 am

“Mathematics rich enough to deal with the physical world might not be understandable any more.”

Interesting point Reinhard.

We hear it a lot, but I don’t know where we get the idea that a line consists of

infinitely manypoints ofzerosize which are “lined up.” It seems to me that if we divide a line segment infinitely many times, we don’t quite get zero-size elements, for reasons similar to the Dichotomy Paradox: we don’t reach zero-size elements just as we don’t reach the destination point (e.g. x = 1 in our examples) in the Dichotomy Paradox.Erdinç SayanAugust 17, 2012 at 9:09 pm

I am pretty sure the idea behind is: If you divide the line [0, 1] infinitely many times and you allow the resulting elements to have size after this process, reconstructing the line would result in a way larger line than [0, 1]. Because any size greater than zero multiplied by infinity results in infinity and not in [0, 1].

Obviously, whenever we have elements left which have size, we haven’t completed the process of infinite division, we simply divided the line by a very large natural number.

Reinhard FischerAugust 18, 2012 at 12:07 am

You are right of course about the impossibility of getting elements of finite (nonzero) size after “finishing” dividing the line infinitely many times. But if we get exactly zero size elements after the infinite division, that would mean we are reaching the limit; in other words, the limit becomes a member of the sequence (of the relevant kind). So, paradox once again…

Erdinç SayanAugust 18, 2012 at 12:14 pm

My approach to this is: If we can accept (somehow – not necessarily from a logical point of view), that there are infinitely many points in [0, 1], and there is a first point 0 and a last point 1, why not applying this to the segments [0, 0.5], [0.5, 0.75], [0.75, 0.875], …: There is a first segment [0, 0.5], and there is a last segment [x, 1].

Neither in the first case nor in the second a sequence is sufficient to define the points/segments.

Reinhard FischerAugust 22, 2012 at 10:24 am

I think there exists no segment of the form [x, 1], where x has any value. This x would have to be (if I understand your reasoning correctly) a real number which comes “right before” 1. But there is no real number right before (or “next to” or “closest to”) number 1. I think you can find a proof of this.

Erdinç SayanAugust 24, 2012 at 2:18 am

If I’d have to choose a real number I would assign x = 1, because the last segment [1, 1] is of zero size. Just like several segments left to this segment.

If we can accept, that there are infinitely many points in [0, 1], and if we accept, that there is only size zero left for each point (or segment) after infinite division, we can also accept a reverse operation, which enables us to regain size if we are adding infinitely many of these zero size segments. We know, that zero multiplied by any natural number is zero. What about multiplying zero by infinity?

Reinhard FischerAugust 24, 2012 at 12:41 pm

If your segments are obtained in accordance with the (1/2)-to-power-n rule, then I think you can’t ever obtain the segment [1, 1] by that rule. In fact you cannot obtain any “last segment” by that rule I believe.

To my knowledge, the result of zero multiplied by infinity is undefined in standard arithmetic. And strictly speaking, “infinity” is not really a number, so it cannot be multiplied by any number. (It is a different story in transfinite arithmetic, though.)

Erdinç SayanAugust 25, 2012 at 1:50 am

A supertask – as defined today – wouldn’t enable us to divide [0, 1] into zero size segments. That’s true. I am not really doing math here.

All I am doing is, suggesting that the infinite division of [0, 1] can be completed. A mathematician would agree, that infinite division can be completed, but he would not agree, that this ends up in zero size segments. This seems kind of paradox to me for the reason mentioned: Not ending up in zero size segments means not really completing the infinite division from a geometrical point of view.

As far as the (1/2)-to-power-n rule is concerned: It is actually today’s math, that the union of the segments [0, 0.5], [0.5, 0.75], [0.75, 0.875], … is [0, 1). The only point not covered by these segments is 1, the zero size segment [1, 1].

Reinhard FischerAugust 25, 2012 at 4:55 am

Here’s another, similar scenario. Suppose we have a 1cm line segment.

We divide it into two equal portions; each of the 2 portions is 0.5cm long.

We divide each portion into two equal portions; each of the 4 portions is 0.25cm long.

We divide each portion into two equal portions; each of the 8 portions is 0.125cm long.

and so on,

ad infinitum.All that mathematics tells us is that as the number of steps of division “goes to infinity,” the size of each of the portions “goes to zero.” Mathematics won’t ask what happens to the size of the portions when this infinitely long process of division is “complete.” For ‘infinitely long process of division’ means “

noncompleteableprocess of division.” So, mathematics relies on the notion of limit in such matters and refuses to talk about completed infinities. Presumably, the reason mathematics refrains from talking about completed infinite processes is to stay clear of paradoxes.I am not certain whether you and I are in agreement about all this.

It maybe that resolving the paradoxes involved in such matters requires that we go beyond current mathematics. But I don’t know “how much beyond” we need to go.

Erdinç SayanAugust 25, 2012 at 10:50 pm

As far as I know mathematics refrains from talking about processes at all. They refrain from using time (as used in processes). But I am pretty sure, mathematics is treating infinite sets as complete. The infinite set { [0, 0.5], [0.5, 0.75], [0.75, 0.875], … } is complete at the very moment. And it doesn’t take time to sum up 0.5 + 0.25 + 0.125 + …, the sum exists already.

The set { [0, 0.5], [0.5, 0.75], [0.75, 0.875], … } is bijectable to the set of natural numbers, which is also complete and exists already. The set of natural numbers is the notion transfinite mathematics is build on. I am questioning the notion of a “set” of natural numbers.

According to set theory, omega is the number bigger than every natural number. If we are indexing the segments: [0, 0.5] is 0, [0.5, 0.75] is 1, [0.75, 0.875] is 2, and so on, then it would be a natural choice to use omega as an index for the last segment [1, 1]. And according to set theory all segments to the left of [1, 1] can be indexed by natural numbers.

If I am comparing any natural number to omega, the natural number – no matter how large the number is – does not even reach 0.000001% of omega. So, what about this undefined infinitely large distance between omega an any natural number? If we have an answer for this question, we might be able to explain, what is really needed to “reach” point 1 and to “reach” omega. And what elements an infinite set has to be made of.

Reinhard FischerAugust 26, 2012 at 10:16 am

Yes, the members of many (actually, infinitely many) different infinite sequences are already wholly contained in the interval [0, 1]. In that sense there is no division process going on within the interval [0, 1] itself. But we sometimes find it useful to imagine an abstract division process. Notice that you are using the notion of division too. And a division process can be written as a mathematical function, anyway. So we are not talking about anything physical, involving time, etc. When we do that, we realize/prove that the “end result” of such a division process cannot be segments of zero size, and hence [1, 1] is not obtainable by the kind of process you mentioned earlier. In other words, the limit is not a member of the sequence.

“So, what about this undefined infinitely large distance between omega an any natural number? If we have an answer for this question, we might be able to explain, what is really needed to ‘reach’ point 1 and to ‘reach’ omega.”

You may be right, but I personally have no idea how that can be accomplished.

Erdinç SayanAugust 26, 2012 at 6:15 pm

Ending up in zero size segments makes sense even from this point of view:

t = 0: I am drawing the line segment [0, 0.5]

t = 0.5: I am drawing the line segment [0.5, 0.75]

t = 0.75: I am drawing the line segment [0.75, 0.875]

.

.

.

t = 1: I am drawing the line segment [1, 1]

t = 1: I am drawing the line segment [1, 1]

t = 1: I am drawing the line segment [1, 1]

.

.

.

Neither additional line segments nor additional time segments will be added.

Even if we didn’t define a last step of this process I can lift the pencil, since there is no difference between executing this endless loop and not executing it. So, we at least don’t get a logical problem to define a last point in an infinite process.

Reinhard FischerAugust 27, 2012 at 6:50 pm

Here’s a sequence same in form to the initial part of yours:

at t = 0 the light bulb is on,

at t = 0.5 the light bulb loses half of its brightness,

at t = 0.75 the light bulb loses 75% of its brightness,

at t = 0.875 the light bulb loses 87.5% of its brightness,

and so on

ad infinitum.What is the brightness of the bulb when t = 1? The answer I think is that the bulb is off.

This is a benign (unproblematic) version of Thomson’s Lamp, because unlike the original version of the Thomson’s Lamp puzzle, there

isa limit to the sequence. Moreover, we assume that t = 1isreached, unlike Zeno’s Dichotomy. But with the sequence I gave above (or the analogous one you gave) we solve neither the Thomson’s Lamp puzzle nor Zeno’s Dichotomy I think.Erdinç SayanAugust 27, 2012 at 8:46 pm

The difference I can see: In your last process and in all other processes which we discussed before we didn’t define t = 1. We only defined states or actions for t < 1. I am not sure if I am cheating here, but in my previous task I am defining an endless loop right at t = 1. Problematic is, the endless loop does do nothing in time and space. Am I allowed to add "nothing" to the previous process? And does it mean, I have defined t = 1 or not?

Reinhard FischerAugust 27, 2012 at 10:22 pm

“And does it mean, I have defined t = 1 or not?”

While judging this take into account, that I am looking for the definition of a process that is not just going to the limit but actually reaching the limit (i.e. zero size segments) at t = 1.

Feinhard FischerAugust 28, 2012 at 5:47 am

“And does it mean, I have defined t = 1 or not?”

While judging this take into account, that I am looking for the definition of a process that is not just going to the limit but actually reaching the limit (i.e. zero size segments) at t = 1.

Reinhard FischerAugust 28, 2012 at 5:50 am

Your sequence which includes the endless repetition at and after t=1 is really a function you are defining. One can define any function one wants. But what we need to solve the Dichotomy is not a mere definition of a function, but a

proofthat: (i) t=1 is reachable, and (ii) x=1 is reachable.Erdinç SayanAugust 29, 2012 at 12:41 am

I know. What I am trying here is some kind of reverse engineering, i.e. I am defining what I think is needed and look for a process that fits the needs. I have started a thread in Google Groups sci.math to see what the mathematicians think about it (thread “Dichotomy paradox reverse engineering”).

Reinhard FischerAugust 29, 2012 at 11:53 pm

I hope you can get it to work. I am afraid I have no ideas or suggestions to make to help your project.

Erdinç SayanAugust 30, 2012 at 1:26 am

What do you think about this:

We know we can biject the segments of size 0.5, 0.25, 0.125, … to the segments of size 0.25, 0.125, 0.0625, …, furthermore we can biject these segments to the segments of size 0.125, 0.0625, 0.03125, …, and there is no end to it. So, we can use index #1 for segment 0.5. We can as well use index #1 for segment 0.25 or segment 0.125 and so on, since we know there are always infinitely many more segments following. If we create the task:

t = 0: move pen from 0 to 0.5 (taking 1/2 second), naming it segment #1

t = 0.5: move pen from 0.5 to 0.75 (taking 1/4 second), naming it segment #1

t = 0.75: move pen from 0.75 to 0.875 (taking 1/8 second), naming it segment #1

t = 0.875: move pen from 0.875 to 0.9375 (taking 1/16 second), naming it segment #1

…

At which step should we stop using index #1 for the segment to have enough segments of non-zero size left to assign the rest of the natural numbers?

Reinhard FischerSeptember 4, 2012 at 10:42 pm

If I understand your question right (and I am not sure I do), here’s my reply:

I think it can be proven that for any finite natural number k, such that we mark all of the first k members in the sequence (1/2)n (by ‘#1’ or some other marker), there will always be left countably infinitely many members of the sequence. So, in your terms, there is no last step where we must stop using index #1 for the segment to have enough segments left to assign the rest of the natural numbers to them. But in your question you require that there be left as many segments of

non-zero size as there are natural numbers. That would make the length of the line we are segmenting infinite. So your requirement cannot be met.Erdinç SayanSeptember 9, 2012 at 1:00 pm

That’s what I mean. If we don’t stop using #1 all segments in [0, 1) will be named #1 at t = 1. What about the size of all other segments (#2, #3, …) at t = 1 then?

I guess, we don’t get around ending up in zero size segments by the definition of the task already. I mean this task:

t = 0: move pen from 0 to 0.5, naming it segment #1, [0.5, 0.75] is #2, [0.75, 0.875] is #3, …

t = 0.5: move pen from 0.5 to 0.75, naming it segment #1, [0.75, 0.875] is #2, [0.875, 0.9375] is #3, …

t = 0.75: move pen from 0.75 to 0.875, naming it segment #1, [0.875, 0.9375] is #2, [0.9375, 0.96875] is #3, …

and so on.

Reinhard FischerSeptember 9, 2012 at 2:32 pm

Suppose as time advances from t=0 to t=1, we mark with ‘#1’ (or any other marker) the segments created in accordance with the (½)n function. I think all we are allowed to say are the following: (i) For every initial k segments all of which are marked #1

beforet=1, there will remain countably infinitely many segments not marked yet. (ii) When t=1, all of the segments in [0,1) have been marked #1. So no unmarked segment will remain. I don’t think (or at least I don’t know whether) we can prove that x=1 is part of any segment marked #1, if that’s what we are trying to do.Erdinç SayanSeptember 17, 2012 at 2:32 pm

x = 1 cannot be part of the segments marked #1. But that’s not what I mean. Every step of this task has a segment named #2 and #3 and so on. The size of these segments is converging towards 0. See:

t = 0: move pen from 0 to 0.5, naming it segment #1, [0.5, 0.75] is #2, [0.75, 0.875] is #3, …

t = 0.5: move pen from 0.5 to 0.75, naming it segment #1, [0.75, 0.875] is #2, [0.875, 0.9375] is #3, …

t = 0.75: move pen from 0.75 to 0.875, naming it segment #1, [0.875, 0.9375] is #2, [0.9375, 0.96875] is #3, …

What is the limit case for these segments (#2, #3, …)? Doesn’t it make sense to assume a limit for these segments? All of them being 0 at t = 1? These would be the zero size segments needed for the endless loop I am looking for.

If we don’t allow any zero size segment (#2, #3 …) to exist in the limit case, then the questions remains, at which step of this task should I stop naming the segments #1 to have enough non-zero segments left existing in the limit case to assign the rest of the natural numbers (#2, #3, …)? And here I think we agree, that defining such a step is not possible.

Reinhard FischerSeptember 18, 2012 at 9:16 pm

“x = 1 cannot be part of the segments marked #1.”

At least not if we allow only non-zero size segments to be marked and not also the zero size segments which I am suggesting that are also elements of the set of segments in [0, 1].

Reinhard FischerSeptember 18, 2012 at 10:31 pm

“What is the limit case for these segments (#2, #3, …)? Doesn’t it make sense to assume a limit for these segments? All of them being 0 at t = 1? These would be the zero size segments…”

Yes, as t approaches to 1, the size of those other segments will approach to 0, I think. And if we assume t can reach 1 (which of course Zeno would deny), the total length of the segments marked #1 will be 1, and all the other segments would disappear, i.e. their lengths will be zero I guess.

“If we don’t allow any zero size segment (#2, #3 …) to exist in the limit case, then the question remains, at which step of this task should I stop naming the segments #1 to have enough non-zero segments left existing in the limit case to assign the rest of the natural numbers (#2, #3, …)? And here I think we agree, that defining such a step is not possible.”

I guess you are right. By not allowing the total length of the segments marked #1 to reach 1, it seems we are not allowing time to reach 1, which would be fine with Zeno.

Erdinç SayanSeptember 20, 2012 at 3:08 am

“And if we assume t can reach 1 (which of course Zeno would deny), the total length of the segments marked #1 will be 1, …”

I still think Zeno is right as far as it concerns reaching 1 by means of non-zero size segments. He didn’t take into account the zero size segments whose existence I am trying to investigate here.

“… and all the other segments would disappear, i.e. their lengths will be zero I guess.”

That’s the point. What enables us to say, that the segments would disappear (i.e. no longer exist) just because their length is zero at t = 1? Don’t we also accept, that the line [0, 1] consists of infinitely many points, each of size 0? If none of these points exists just because they are of size 0, then the line wouldn’t exist. But each point (e.g. x = 1, which is also the zero size segment [1, 1]) is an existing element of the set of points in [0, 1].

If we allow only the non-zero size segments to exist in the limit case of our task, we get [0, 1). If we allow also the zero size segments to exist, we get [0, 1].

Reinhard FischerSeptember 20, 2012 at 8:44 am

(1) I think that if we are trying to solve the Dichotomy Paradox, we should simply not rely on time reaching 1.

(2) I think saying that a line segment consists of infinitely many “segments of zero size” which are “lined up” to build the line segment is mistaken. For one thing, zero times infinity is undefined in mathematics.

(3) As I understand, you want to say that the total length of the segments marked ‘#1’ in the procedure you described (which relies on passage of time and the time’s reaching 1, which Zeno would reject) is [1, 0), but when we add all the infinitely many remaining segments (all of which are of zero size) marked ‘#2’, ‘#3’, ‘#4’, …, we get the distance [0, 1]. It seems to me that this argument still does not show that the pen

cantraverse the whole distance [0, 1]. Zeno would probably insist that your pen can only traverse [0, 1), i.e. only those segments marked ‘#1’.Erdinç SayanSeptember 23, 2012 at 1:23 pm

I am not sure if we should call it ‘trying to solve the Dichotomy Paradox’. I am questioning, that the paradox is set up right already. I am trying to figure out if it makes any sense to assume (like Zeno did), that there is enough space for infinitely many segments of non-zero size in [0, 1] – no matter if we are talking about infinitely many segments of equal size or if we are talking about infinitely many non-zero size segments created in accordance with the (½)n function.

Reinhard FischerSeptember 23, 2012 at 9:07 pm

If we allow only finitely many copies of a line segment s in [0, 1], where the size of s is an element of (0, 0.5], how can we allow infinitely many line segments { [0, 0.5], [0.5, 0.75], [0.75, 0.875], … } in [0, 1], where the size of each line segment is an element of (0, 0.5]? We wouldn’t allow an infinite number of copies of any element of { [0, 0.5], [0.5, 0.75], [0.75, 0.875], … } in [0, 1], right?

Reinhard FischerOctober 2, 2012 at 8:20 am

I am sorry, I don’t follow it. What do you mean by allowing “only finitely many copies of a line segment s in [0, 1], where the size of s is an element of (0, 0.5]”?

Erdinç SayanOctober 8, 2012 at 2:55 am

For example, we allow four copies of the segment [0.5, 0.75] in [0, 1], which are [0, 0.25], [0.25, 0.5], [0.5, 0.75], [0.75, 1]. For each segment of the set { [0, 0.5], [0.5, 0.75], [0.75, 0.875], … } is true, that the number of copies in [0, 1] is finite. Zeno assumes, that there is enough space for infinitely many segments in [0, 1], even if there is not even one segment in the set { [0, 0.5], [0.5, 0.75], [0.75, 0.875], … } that could be copied infinitely many times into [0, 1].

Reinhard FischerOctober 8, 2012 at 9:50 pm

Interesting thought. As I understand, your question is: Although every individual segment in the (1/2)-to-power-n sequence can fit in the [0, 1] interval only finitely many times, how come there are infinitely many segments in [0, 1], as Zeno claims?

As an example, take the 3. segment, i.e. [0.75, 0.875], whose size is 0.125. We can fit it 8 times in [0, 1]. But since the previous segments [0, 0.5] and [0.5, 0.75] are “already taken,” actually there can be 4 segments that fill [0, 1], namely [0, 0.5], [0.5, 0.75] (the ones already taken), and two segments of size 0.125 filling the remaining space. Another example: In the case of the 4. segment, i.e. [0.875, 0.9375], whose size is 0.0625, the number of segments that can fill in [0, 1] is 5 (the ones already taken, namely the first 3 segments and two segments of size 0.0625 that fill the remainder of [0, 1]).

So the general formula seems to be that, for the n-th segment, there are n+1 segments that can fill in [0, 1]: n-1 of them are the segments already taken before the n-th segment, and 2 more segments with the size of the n-th segment to fill the remainder of [0, 1]. Now, of course for any n-th segment, n+1 is a finite number. I would turn your question into: Although for any n, the number of segments up to the n-th segment plus the number of times the n-th segment can be fitted into [0, 1] total n+1 (where n+1 is a finite number), how come there are infinitely many segments in [0, 1], as Zeno claims?

I think the short reply to your puzzle is that your question is similar to the following question: Although

everynatural number is finite, how come there are infinitely many natural numbers? The answer to this last question is of course that there are infinitely many natural numbers, because for any natural number n, there is a next natural number (a number bigger than n). Similarly, we can answer your question by saying that for any n-th segment, for which the total number of the previous segments plus the times the n-th segment can fit in the [0, 1] interval is n+1, there is a next (smaller) segment for which the total number is n+2 times. To simply put it: the sequence n+1, which expresses the number of times a segment plus the previous segments that can be fitted into our interval, goes to infinity.QED, I think…

Erdinç SayanOctober 13, 2012 at 3:36 am

What you say makes perfect sense to me – from a numerical point of view. And for each natural number n there is not just a next natural number n+1, there are infinitely many next natural numbers.

But – from a geometrical point of view – this means, for every finite segment we need infinitely many segments to follow. So, no matter how close we get to 1 in the interval [0, 1), there are always infinitely many segments between this point and 1. If this is true for every segment in [0, 1) and if it is true, that every segment in [0, 1) is a finite segment, is there always enough space for infinitely many more segments of non-zero size – no matter where we are pointing to in [0, 1)?

Reinhard FischerOctober 13, 2012 at 4:38 pm

“What you say makes perfect sense to me – from a numerical point of view. And for each natural number n there is not just a next natural number n+1, there are infinitely many next natural numbers.”

The

reason whythere are infinitely many natural numbers is because for every number n, there is a next number, n+1. That’s the definition of infinity for this kind of case. That’s why I said for every natural number there isanext number.“But – from a geometrical point of view – this means, for every finite segment we need infinitely many segments to follow. So, no matter how close we get to 1 in the interval [0, 1), there are always infinitely many segments between this point and 1. If this is true for every segment in [0, 1) and if it is true, that every segment in [0, 1) is a finite segment, is there always enough space for infinitely many more segments of non-zero size – no matter where we are pointing to in [0, 1)?”

My (informal) proof in my previous comment is well aware of the geometrical nature of the problem. The answer to your question is “Yes”: there is always enough space for infinitely many more segments to fit in the remaining space of [0, 1], because the number of segments that can fit in the remaining space of [0, 1] goes to infinity. Notice that I am not saying “infinitely many more segments of

non-zero size.” All we can legitimately say is that as the number of segments increases without end (i.e. for every n-th segment there is an (n+1)-th segment), the size of the segments correspondingly come close to zero without end (i.e. for each segment of a certain size there is a next segment with a smaller size, without the size of any segment reaching zero ). You can say, if you like, that the sizes of the segments come “infinitely close to zero”. I am not sure if “infinitely close to zero” can easily be described as “non-zero.”Erdinç SayanOctober 15, 2012 at 2:48 am

Then the question arises, how do we distinguish between “finitely close to zero” and “infinitely close to zero”. I think, “finitely close to zero” can be described as “non-zero”. Is there a difference between these two phrases at all (i.e. “finitely close” and “infinitely close” to zero)?

There are infinitesimals in mathematics. But as far as I know this is not standard mathematics. Even if, what is the physical representation of an infinitesimal? An object which is neither of non-zero size nor of zero size?

Reinhard FischerOctober 28, 2012 at 11:22 pm

“… no matter how close we get to 1 in the interval [0, 1), there are always infinitely many segments between this point and 1. If this is true for every segment in [0, 1) and if it is true, that every segment in [0, 1) is a finite segment,

is there always enough space for infinitely many more segments of non-zero size– no matter where we are pointing to in [0, 1)?”We can take “finitely close to zero” to be simply “non-zero,” as you suggest. In the context of our discussion, we could roughly define a number to be ‘non-zero’ iff it is equal to 1/r, where r is a (finite) real number larger than 1. Given this definition of ‘non-zero,’ the answer to your question is “No.” But if we don’t insist on segments being of non-zero size, and ask the question as, “

Is there always enough space for infinitely many more segments?”, the answer is “Yes,” as I said in my comment of Oct. 15.Now, since there

areinfinitely many segments and they cannotallbe of non-zero size (given our definition of ‘non-zero’ above), wouldn’t it follow that some of the segments have to be such that they arenotof non-zero sizes (again, given our definition of ‘non-zero)? Have we discovered the existence of infinitesimals by that argument?Erdinç SayanNovember 3, 2012 at 10:42 pm

“Have we discovered the existence of infinitesimals by that argument?”

I doubt it, because every segment turns out to be of size

finitelyclose to zero according to my definition in my previous comment. The reason is that for every segment whose size is equal to 1/r, there is a next segment whose size is equal to 1/s, where s is also a (finite) real number larger than 1 (s being larger than r). So, by mathematical induction, all of the segments are finitely close to zero.Then how can infinitely many segments with sizes finitely close to zero fit in [0, 1]?

Consider the following case which is similar to the Dichotomy Paradox. We divide the interval [0, 1] by the (1/2)-to-power-n formula. First we divide it into 2 (equal) segments; then each of these 2 segments we divide into 2 segments, thus producing 4 segments; then each of these 4 segments we divide into 2 segments again, thus producing 8 segments; and so on

ad infinitum. We are supposed to never get zero-size segments (that is, segments with the size of a point) by this process. At each step of the process the segments obtained have size finitely close to zero, i.e. they have size equal to 1/r, where r is a real number larger than 1 (in our example r=2 over n), but there is nosmallest-sizesegment in the sequence, because there is no final step of the division process. So,for nor can we talk about infinitely many segments of size 1/r, thereby getting an infinitely long length—which would have contradicted [0, 1].Erdinç SayanNovember 4, 2012 at 2:47 am

So, I can see we are sharing the same thoughts. For now my conclusion is, that Zeno’s paradoxes teach us, that our logical understanding of (actual) infinity is not sufficient. Since mathematics is not necessarily bound to our logical understanding “mathematics rich enough to deal with the physical world” should accept (axiomize), that we are ending up in zero size segments after (actual) infinite division. I would even accept the notion of infinitesimally small segments, because the properties of these segments are pretty much the same as the properties of zero size segments, I think.

Reinhard FischerNovember 4, 2012 at 1:05 pm

I too have the hunch that the (1/2)-to-power-n series does reach the destination point 1. I would love to see a proof of it one day.

Erdinç SayanNovember 6, 2012 at 2:12 am

I am not sure, that it can be proven at all. I think, all we can show is, that our understanding, if things can be completed in countably many steps or not, is not sufficient. Another example: We are creating this list in countably many steps:

step 1:

0

1

step 2:

00

10

01

11

step 3:

000

100

010

110

001

101

011

111

…

Wouldn’t it make some kind of sense to assume, that the resulting list (after infinitely many steps) contains ‘000…’ at the top of the list and ‘111…’ at the bottom of the list? I think so. But according to our current understanding of processes of countably many steps: No. You can simply replace the binary sequences above by their decimal representation (11 = 3, 111 = 7, …), and you will simply get the list of natural numbers after infinitely many steps. You might find ‘000…’ on a list of natural numbers (which simply is 0), but there is no natural number that could be represented by ‘111…’.

Reinhard FischerNovember 6, 2012 at 7:39 am

This reminds me of the following question I sometimes ask my students:

Isn’t the greatest natural number 999999… (infinite concatenation of 9’s)?

The answer is that 999999… is not a natural number, because every natural number is finite.

Erdinç SayanNovember 10, 2012 at 12:53 am