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I would like to start off by saying this is not a philosophical question. I have a specific question pertaining to physics after the following explanation and background information, which I felt was necessary to properly formulate my question.

I have done some research on Zeno's Paradoxes as well as some of their modern day mathematical and physical "solutions." I understand that Zeno's dichotomy paradox can be explained mathematically by constructing an argument that makes use of the following infinite series: $$ \sum_{n=1}^\infty {1 \over 2^n} = 1$$ Because this series converges to $1$, then that implies that if physically traveling from some point A to another point B truly does require an infinite amount of actions in which an object travels ${1 \over 2}$ of the distance, then ${1 \over 4}$ then ${1 \over 8}$ then ${1 \over 16}$ and so on, the sum of those actions would eventually result in traveling the full distance. Mathematically, this makes perfect sense. However, knowing that the mathematics we use to describe the physical world is only a model, adopted when there is sufficient evidence to support it, could it be that the solution gained by the above argument comes to the correct conclusion using the wrong path? Using the infinite series provides a solution to the dichotomy paradox if it does indeed require an infinite amount of actions to move from point A to point B. So, then the question becomes, does it truly take an infinite amount of actions to physically move from one point to another (in which case, I believe the mathematical solution is a perfect model), or are there other phenomena at play that cannot be described in this fashion?

I have one idea pertaining to a quantum mechanical description of the situation that does not disprove, but provides an alternative to the reasoning that it requires an infinite number of actions to move from point A to point B.

  • Suppose we have a box of some macroscopic size (say $1m^3$), and it is in the process of moving with a constant velocity from some point A to another point B. Obviously, the box is made up of atoms, and if we zoom in far enough on the leading face of the box we could see that the motion of said box is really the motion of a very large number of atoms moving in the same direction. According to the uncertainty principle, it is impossible to determine both a particle's momentum and position simultaneously. According to the physics professor who first taught me about this, the reason for this uncertainty does not lie within our measurement methods, but rather, it is an impossibility inherent in the fact that all particles have wavelike characteristics. Thus, their position and momentum are not even clearly defined at a specific moment in time. Could it not be possible then, that if we zoomed in far enough on the leading face of our box, that we would find the face of the box does not have a definite position? If this is true, then wouldn't there be a certain point very close to the destination point, B, where the idea of cutting the remainder of the distance to the destination in half makes no difference to the particles in question? We may say that to complete the trip, we must travel another ${1 \over 2^x}$ portion of the distance we started with, but if the uncertainty in position is larger than this remaining distance, could the particle (and by the same argument, all of the particles that make up the leading face of the box), traverse the final distance to point B without physically having to travel the "infinite" amount of points remaining?

Also, I have a second idea that I want to present as an aside, something that I don't believe can be proven as of now, but I wonder if, according to current understanding, it is possible.

  • What if, instead of analyzing the situation at hand through the distance traveled, it was analyzed by analyzing the passage of time. Specifically, if time is quantized, would it resolve the dichotomy paradox in a physical sense? I realize that whether or not time is quantized is up for debate (there is even this question here at stack exchange that explores the idea), so I am not going to ask if time is quantized. Based off of my understanding, however, there is nothing known as of now that says time cannot be quantized (please correct me if I am wrong). Therefore, if time were quantized then could we not say that for a given constant velocity there is a minimum distance that can be traveled? A distance that corresponds to $\Delta t_{min} * v$? Would this not imply that as our moving box reached a certain distance, $d$ away from point B, and if $d < \Delta t_{min} * v$, the box would physically not be able to travel such a small distance, and in the next moment that the box moved, it would effectively be across the destination point B?

All of that being said, here is my specific question: In my above arguments, is any of my logic faulty? Is there any known law that would disprove either of these explanations for Zeno's dichotomy paradox? If so, is there a better way of physically (not mathematically) resolving the paradox?

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wgrenard
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  • More on Zeno's paradox: http://physics.stackexchange.com/search?q=+is%3Aq+Zeno%27s – Qmechanic Feb 02 '14 at 21:12
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    Frankly you've got it backwards. The artificiality is not is summing the series, the artificiality is in Zeno's formulation. You don't move half way and then move half of the rest and then ... You move forward continuously and after a finite time you have arrived. You don't need quantum mechanics because classical mechanics is all over it. You can use quantum mechanics and be all over it that way, too, but focusing on the HEP just confuses the issue. – dmckee --- ex-moderator kitten Feb 02 '14 at 21:19
  • But doesn't the very idea of moving "continuously" imply that you must move half before moving the next half? That is, if motion is truly continuous, there would be no splitting it into smaller "chunks." You couldn't move from one place to the other without traveling the distance in between. – wgrenard Feb 02 '14 at 21:35
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    I'm not even sure that $\sum_{n=1}^\infty {1 \over 2^n} = 1$ solves the problem. This just says that the limit according to some axioms is $1$, the definition of limit implies that if a number is infinitely close to $L$, then the limit is $L$. So it's circular reasoning. – jinawee Feb 02 '14 at 21:39
  • @wgrenard It implies that you pass through the halfway point, but there is not an infinite sequence of things that we have to count through to get anywhere. Zeno's argument would imply that we can't get to the halfway point either. Speed is distance/time so the the time required is distance/speed and that is finite for any positive speed. Zeno uses sequencing words to make it seem like a infinite number of discrete events must happen and thus imply that an infinite time is required. Math is the language of physics and word pictures are the tool of obsfucation. – dmckee --- ex-moderator kitten Feb 02 '14 at 21:48
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    Zeno's "paradox" is sort of meant to be a joke, I mean, it's not like a grapefruit thrown against a wall cares that you could potentially decompose its path into an infinite sequence of subpaths whose union is the classical path, it still does what it does. Anyone who would tell you that Zeno's "paradox" is still an unsolved problem in classical physics is either incompetent or a crank. That's not to say that the topic isn't entertaining to discuss or that this isn't a good question to ask, but I think it should be accompanied with a grain of humor. – DumpsterDoofus Feb 02 '14 at 21:48
  • @dmckee Okay I think I understand what you are saying; I'll have to ponder it awhile. – wgrenard Feb 02 '14 at 22:04
  • @DumpsterDoofus Simply because we can observe that the grapefruit hits the wall doesn't resolve the paradox. Since, obviously all things can make it from one point to another its obvious there is faulty reasoning in Zeno's argument. The question is, what part is faulty. – wgrenard Feb 02 '14 at 22:07
  • @wgrenard: Because it's wrong. It's not a rigorous proof. There's an entire branch of recreational mathematics devoted to "joke proofs" which look and sound like proofs and which "prove" false things by using loose, nonrigorous reasoning and hoping that the nonrigor is not noticed by the reader. Some of them are really good, and hard to catch. The point is, Zeno's argument is not a legitimate proof; it doesn't actually prove what it's claiming to show in the first place, so it makes no sense to ask what goes wrong, when it doesn't actually prove anything in the first place. – DumpsterDoofus Feb 02 '14 at 22:44
  • It's similar to the joke proof by induction that "all horses are white", or the book "Mathematics Made Difficult" by Carl Linderholm (which, by the way, is a fantastic book, but unfortunately out of print). – DumpsterDoofus Feb 02 '14 at 22:45
  • @DumpsterDoofus Okay, you make a fair point. When formulating my original question I was focused on solving the outcome of the problem when I should have been trying to solve the proposal. – wgrenard Feb 02 '14 at 23:04
  • @dmckee After thinking about what you said for quite awhile, I'm still not convinced the answer satisfies my question. You bring up the point that time = distance/speed, which obviously gives us a finite time. But this goes back to my original problem with a mathematical solution. That equation works and gives a finite time because it was devised based off of what is observed. Therefore, of course it will yield a finite answer. This does not show physically, however, that we cannot divide the distance into infinite points, which presents a problem that can't be resolved by classical mechanics – wgrenard Feb 03 '14 at 17:17
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    No. You are assuming that a infinite number of points to be passed means a infinite number of finite periods of time. Passing from one point to another takes infinitesimal time. In other words, calculus is the answer and calculus is rigorous and exact: there are no approximations or things left unchecked. Again, just because it would take you infinite time to talk through the process in Zeno formulation does not mean that it takes infinite time to happen. – dmckee --- ex-moderator kitten Feb 03 '14 at 19:34

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There are a lot of comments. The end result is that your first proposition is correct as far as the physics models we have validated with experimental evidence go.

Even classically, when one has an extended body whose trajectory is assumed by the center of mass, the paradox becomes trivial, even if the center of mass never reaches the goal, half of the solid body minus an infinitesimal essentially does.

The Heisenberg uncertainty principle is involved at the elementary particle state, and yes the classical motion is no longer relevant and does not describe the behavior of nature at the microscopic scale.

As for your second proposition, what disallows discreteness of space and time is not lack of imagination. It is at the moment incompatibility of known and validated physics and mathematical models proposed with such discreteness incorporated. At the moment as far as I know, locality for Lorenz invariance, which has been validated an innumerable number of times, is the main obstacle for such theories, but also I am not aware if there exist proposals that can also incorporate the standard model of particle physics, which is an encapsulation of all the experimental measurements we have up to now. Thus discreteness in time exists in some models but is not validated by any data and is not a proposition accepted by the mainstream physics community. In any case even if you take time as a variable the argument with the Heisenberg uncertainty principle is sufficient as there also exist a delta(E)delta(t)>h_bar form of it.

anna v
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  • Thank you. Though the extended comments were not bad, I accepted because you actually addressed the question I posed. – wgrenard Feb 03 '14 at 07:31