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Zeno's Paradox and Modern Physics

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By large-hammer - Posted on 02 September 2002

by Marcus Booker
The paradoxes of Zeno, a pre-Socratic Greek, have always posed a major problem for "continuous/infinite" mathematical theory. I have come to believe that the only satisfactory solution is to abandon the "continuity" paradigm assumed by these ancient paradoxes in favor of the "discrete" view. Thus my argument is for the discrete view, which I will expound upon further in this article. Moreover, this solution also makes sense of many of the "anomalies" of modern physics, including the "jumping of electrons" or "Plank's distance."The paradoxes of Zeno, a pre-Socratic Greek, have always posed a major problem for "continuous/infinite" mathematical theory. I have come to believe that the only satisfactory solution is to abandon the "continuity" paradigm assumed by these ancient paradoxes in favor of the "discrete" view. Thus my argument is for the discrete view, which I will expound upon further in this article. Moreover, this solution also makes sense of many of the "anomalies" of modern physics, including the "jumping of electrons" or "Plank's distance."Zeno had many paradoxes, but all of them posed the same problem for "continuity." Thus focusing on one paradox should suffice for the purpose of this article. I will focus on the paradox of Achilles and the Tortise.


Also, before I start, I must define what I mean by "continuous/infinite." An example might best define it: If I were to travel one straight mile (from "point A" to "point B"), I would be moving along a segment of a "line," and I would pass through an infinite number of "points" between A and B. My travel along that line is completely continuous and smooth, inasmuch as I pass through an infinite number of points. There are no jumps. This is the continuous model.

The mathematical model above is the model assumed by Zeno's paradoxes. Yet his paradoxes completely disprove this model; they reveal its impossibility. They especially reveal the inadequacy of that model in describing the real world on a micro-level. [Note: The "continuous/infinite" model is still useful on a macro-level, much like Newton's physics is useful when the speed of light approaches zero].


Zeno imagined a foot race between the world's fastest creature (Achilles) and the world's slowest creature (the Tortise). The tortise had a head start of one mile.

The race begins!

Before Achilles can pass the tortise, he must first travel one mile and make it to the tortise's original starting point. So Achilles does this. But by that time, the tortise has traveled ten feet. So the tortise is still ten feet ahead.

Achilles then must travel the ten feet, before he can pull ahead. But by that time, the tortise has traveled an additional inch. So the tortise remains an inch ahead.

Achilles next must travel the inch before he can catch up, but by that time, the tortise has moved another hair's width. So the tortise is a hair's width ahead of Achilles.


You can probably see the pattern here, how this progression would go on forever! And indeed, it would never end (as the paradox is modeled). Achilles would never surpass the tortise! The tortise would always be just a little bit ahead!

Actually, according to continuity (as shown by another of Zeno's paradoxes), neither one of the two competitors would have even been able to start the race at all, but I digress.

Of course, many people easily write-off or dismiss this paradox. They "know" or assume that the world is continuous. And they observe (correctly) that, in real life, Achilles would easily surpass the tortise.

Those who write-off this paradox assume that Zeno is some magician. They think that there is some sleight of hand to his words.

Yet I, for my part, do not think Zeno a trickster. I take his paradox seriously, and I consider the implications of it (and the assumptions that it makes). The solution is a different model--i.e. different assumptions.


To define these models plainly and easily, I will make an analogy. The model of continuity is much like analog. And the "discrete" model is like digital. Along this analogy, I claim that the world is digital rather than analog. Of course, it is digital with extremely good resolution (if you're following my analogy).

To continue with this analogy, let us transfer Zeno's paradox to a television set. We'll assume that there are 32-frames per second, which allows for the motion on the screen (as visible by the human eye) to *appear* completely smooth. Yet, we know that the motion is not *really* smooth. A character is in one position, then at the next moment, he is in another position. He is NEVER actually in between. He makes 32 small jumps each second!

So, if Achilles is allowed to make jumps like this, then he can indeed surpass the tortise, without having to pass through an infinitude of points. When the tortise is ahead by a hair's width, Achilles can simply jump ahead, without having to ever be "in between."

In the first ten-thousand frames, Achilles travels the mile; the tortise travels the ten feet.

In the next five-hundred frames, Achilles spans the ten feet; the tortise travels the inch.

In the next fifty frames, Achilles travels the inch; the tortise travels the hair's width.

But then, in the next single frame, Achilles travels that hair's width; the tortise moves some miniscule amount.

Yet in the frame after that one, Achilles "jumps" ahead of the tortise.


If something traveled one straight foot across the television, we know that it doesn't pass through an infinite number of points. Rather it makes many small jumps along the line. If that traveler can be said to pass through "points" at all, it passes through particular, distinct points (the pixels) that are finite in number.

So, in this world, there must be something like a frame and something like pixels. Whether these "frames" and "pixels" are uniform and regular, I cannot say--but they must be there somehow.


This view of the world also makes sense of the motion of electrons as observed by modern physicists. To those stuck on "continuity," it is counterintuitive. To those who see the "discrete," the following makes sense:

Scientists have noted that electrons move a small span, called "Plank's distance." Yet when the electrons move, they move from one spot to another--but they are never in between. They are in one location; then they are suddenly in another location, "Plank's distance" away. At least, that's how it was explained to me.

The interesting thing is that the friend who explained this phenomenon to me was quite puzzled by it. Before I had even heard these strange observations in modern physics, I had told him about my theory on the world being discrete (based upon Zeno's paradoxes), and how there must be little discrete jumps (with no "in-betweens)! Then he informed me about Plank's distance, which is an example of just such a little jump, with no in-betweens. Needless to say, he went away content.


So if time is an illusion. If the world moves from frame to frame, how does it seem so continuous?
Maybe it's like this:

The animator draws a cartoon. He is the one who writes the story, who assigns blame, who makes the "good guys." By His technique, He makes the story seem continuous, because He designs it that way. A character (and everything for that matter) is sustained from one frame to another by His hand. Of course, to the animator, the entire story is already written. The animator is not subject to the time in the animation. To Him the whole VHS tape exists--all before Him. All the frames are there at the same time in His presence. The characters are the ones who are stuck in their respective times; they are the ones who face trials and are rewarded according to their deserts. The animator is outside of all this, unless He somehow writes Himself into the script, making Himself a character like them.

Marcus Booker

JL's picture

Newton solved Zeno's Paradox with the invention of the calculus. Zeno's Paradox can be solved quite nicely in the continuum. Zeno's Paradox doesn't even make sense in a discrete space. It can't be expressed.

Plank's distance, L0 = (Gh/2pic^3)^(1/2) = 1.616 x 10^(-35) meters, has to do with the range over which quantum gravity occurs not electrons jumping.

Your whole analogy of electons jumping is not based on quantum mechanics and fails. An electron is not at a point. You've confused the problem of an electron's location with the measurement of that location.



JL Vaughn
Beyond Creation Science

large-hammer's picture

Are you sure?

Leibniz also invented calculus on a different model.

And yes, Zeno's paradox *doesn't* make sense in disrete space (but it would be a major problem for continuous space). That's exactly my point.

I have never heard it claimed that Newton's calculus solves Zeno's paradox--I've heard quite the contrary however. Did you hear that somewhere? Or are you deriving that yourself? If the latter, please expound.


JL's picture

I don't care whether you use Newton's infintesmals, Leibniz's indiscernables, or any of the other formulations of calculus that have been developed since. All you need is a workable concept of a limit.

I don't know if either Newton or Leibniz was aware of Zeno's paradox per se, but both of them routinely solved problems of this type. And before them no one did.

There were some problems in their concept of infinity that needed to wait until Cantor's work for a rigorus solution. But their answer was essentially unchanged.

Some people will still argue that Zeno's paradox has not been solved. However, the problem is purely deterministic and the math matches the measured answer. No discreteness need be invoked.



JL Vaughn
Beyond Creation Science

large-hammer's picture

I don't think that the calculus solves the paradox but works around it. The question of the paradox does concern the small side of things, the infinitely small or infintesimals. That is the level at which the paradox becomes a concern.

Yet there is a difference between reality and the useful approximations afforded to us by mathematical modeling.

The question then comes back, "How does Achilles *actually* surpass the tortise?" An "infintesimal" doesn't actually exist, except as a mathematical tool. How does Achilles pass that turtle without discreteness? How does it happen without a jump?

I agree that continuity is useful for mathematical modeling (that's why Newton's calculus is so useful). I may move from point A to point B, and someone may surmise that I pass through an infinite number of points. Yet I know that points don't exist, except as a useful tool to try to approximate reality. And there are no *real* lines. I always pass through *particular* places; and I'm always at a particular time. For the reality to be continuous is impossible. It's like trying to find the first positive real number after zero. What is it? 0.00000....0001
I would have to *actually* pass through this number if we live in a continuous world.


JL's picture

Quantum mechanics, which you appealed to, says that a particle's wave function exists in and travels through continuous space.

I can't measure it at every point because each measurement is a discrete act. You are equating the discreteness of the measurement process with the actual path traveled.

Why not ask if a tree falls in the woods and no one is there to here it, does it make a sound?

No one saw it travel from A to B. No one knows the path it took to get there. Why should I assume it just vanished from point A and suddenly appeared at point B?



JL Vaughn
Beyond Creation Science

large-hammer's picture

I don't know how electrons move. I am not a physicist.

Yet given the theory of discreteness, I would *expect* that electrons move from one position to another without measurement between these positions being possible. I would expect that electrons are never in between (which is what I have heard to be the case).

Whether or not that is actually what has been observed, I cannot say.

If you assume continuity, then you might assume that the electron takes some unknown path between A and B (separated by Plank's distance). Physicists might be looking for this path. Yet under discreteness there need not be a path at all.


JL's picture


You are not a physicist. I am.

There is no theory of discreteness. Only some hypothesis you got somewhere and have no evidence for.

All theories of physics assume continuous time and space. Every experimental physicist wants to show that the standard model falls apart under certain conditions. They get Nobel prizes and great fame for such things. So far, no one has found any evidence of discreteness of the type you claim.

Measure the position of a free electron. Measure it again. It's no big deal to do. Take the two measurements closer together. Measure with less error. This is done all the time. All of this is well-known physics. All of these experiments always give results compatible the continuous space-time formulation of quantum mechanics.

Physicists give all sorts of vague descriptions to try to get a verbal handle on the dual particle-wave nature of matter. Don't take these too seriously. The wave packet exists and moves continuously through both time and space.

Whether the particle nature always exists or is just a manifestation of interference is a subject of light-hearted debate. It appears to not be answerable.

I've never seen a reference to Plank's distance in reference to particles except hypothetical gravitons. It has nothing to do with minimum observable displacement distances.

I've never understood why theologians who brag about their ignorance of a subject feel such a great need to make such absolute pronouncements concerning it. That's something you might be able to answer.

So unless you've got something more than emotional support for an illogical idea, I'll quit banging my head against the wall.



JL Vaughn
Beyond Creation Science

jowl's picture

"Measure the position of a free electron. Measure it again. It's no big deal to do. "

Heisenberg (sp?) uncertainty principle - anyone, anyone? Something to the effect you cannot measure the position of an electron or its velocity without observing it. Observing it requires light. Light will impact the electron and mess up your whole observation.

At least it is something like that.

Zeno's paradox cannot be solved because it tells you how you are supposed to solve it. It makes the rules for you. Using its rules, you can never determine the point of the "pass".

For example, say I tell you you can have a million dollars if you can walk ten feet. But there are a few rules, you will take a step every minute, but each step must be exactly half of your previous step, and your first step can be 5 feet.

Okay step 5 feet. ( you are excited now!)

Okay step 2.5 feet (whoo hoo - only 2.5 feet to go and you will be rich)

.... (100 minutes later) You are so close - but yet you will never, ever win, cause I will divide the amount of distance remaining each time in

In other words - I am declerating you to a point practically frozen in time.

The equation Distance solved for time:
d(t) = (10 - d(t-1))/2 where d(0) = 0;
d(1) = (10 - 0) / 2 = 5
d(2) = (10 - 5) / 2 = 2.5 etc

Since this is set up as the rules of the game, with slight of hand - I do not allow you to use the right equations to easily solve the problem.

Okay, Okay - is everything continuous or not? Well, I suppose all of Calculus fails if it is not, and certainly it has been demonstrated that time can be squished and bent by warping of space (relativity). So what the Achilles needs to do is simply warp space and then on the next step will pass the tortise.

Thus, the answer is - TO GET TO THE OTHER SIDE.



Virgil's picture


You are bringing up things that not many people think about on a daily basis, but they are of great importance, especially when it comes to our understanding of God and the universe in which we live. I have been thinking about how quantum mechanics relate to God and his attributes. Obviously, I don’t want to diminish God in any way, that is not my intent; I always long to know him better, and even try to understand a tiny part of who He is.

I have a big poster in my office with Eistein’s picture on it and his quote: “I want to know God’s thoughts…the rest are details”. I just wish more Christians today would long for that very thing…

Now, about the implications, theoretically, given enough computing power (and it should be safe to assume that God has the capacity for unlimited understanding and what we call “computing”), one can accurately measure, calculate and PREDETERMINE Plank’s distance and the location of the next jump of an electron. This has MAJOR implications on our understanding of God’s ability to know the universe completely, at any given time and in any given state (i.e. what you call frames). This is pretty fascinating, especially when you realize that only a being of infinite wisdom could actually achieve that kind of understanding and knowledge. Just by seeing how amazing the universe is, it only makes me marvel at how much more amazing God Himself is!

djelectrop's picture

That's the LAST time I read one of your articles @ 3:00AM. I was already having trouble getting to sleep...and NOW my head hurts. :)

Seriously though, regardless of whether or not the analogy holds any water in the real world of physics...I like the picture you drew in the closing paragraph which elevates God to a more Sovereign level than most modern Christians are willing to admit.

Brandon Scott

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