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Zeno's Paradox and Modern Physics
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.
DEFINITION OF "CONTINUOUS/INFINITE" MODEL
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].
ACHILLES AND THE TORTISE:
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.
IMPLICATIONS OF PARADOX
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.
THE DIFFERENT MODEL--DIGITAL-LIKE WORLD:
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.
FINITE JUMPS AS OPPOSED TO INFINITE CONTINUITY
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.
IMPLICATIONS FOR MODERN PHYSICS--PLANK'S DISTANCE
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.
A FRAME BY FRAME WORLD:
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.