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Zeno's Paradoxes

by Miles Mathis

Atalanta and Achilles

Many readers of my paper on the calculus have assumed that I am opposed to the idea of infinities or limits. They imagine that I wrote my paper exposing the faults in the derivation of the calculus because I had philosophical problems with infinite progressions or with infinitely small quantities. But the fact is that I have no problem with the infinite as a concept, either mathematical or physical. Indeed, I find it much easier to believe that the universe is infinite in space and time. To believe the opposite is to beg so many questions. I also find it easier to believe in a real infinite progession in size, both large and small. That is (to give a concrete example) I find it more likely than not that particles smaller than photons will eventually be discovered or postulated.

As proof that I have no mathematical qualms about using limits, I will show that two of Zeno’s paradoxes have been solved simply and beautifully by the limit concept. I have the naïve opinion, based on nothing but instinct in this case (since Zeno’s life and work is almost entirely a mystery—most of what we know coming from the writings of Plato, which were not meant to be history) that Zeno posed his problems not as a serious attack upon commonsense notions, but as encouragement to others to discover some fairly esoteric mathematical notions that he had already discovered himself. In this way, his paradoxes were a teaching method somewhat like those of Socrates—study your simple prejudices and assumptions and you will uncover many other interesting facts. But Zeno did not teach by lecture or parable or dialectic. He taught by paradox. A paradox was for Zeno a sign that a subject was not fully understood. It was a signal of an important gap in knowledge. It was therefore a suggestion where students might learn more.
       If I am right, it would make Zeno a precursor of Archimedes. Zeno’s paradoxes point in the direction of Archimedes’ pre-calculus. Didactically, the two most famous paradoxes are useful mainly as signposts to the summing of infinite series, and I suspect that this was Zeno’s meaning from the beginning. That he was misunderstood is no surprise, since almost everyone who ever said or wrote anything has been mostly misunderstood. Besides, if I am wrong and Zeno was just a giant curmudgeon, it doesn’t really make any difference. The long-term usefulness of his paradoxes is the same no matter his intention.

Either the summing of the infinite series or the concept of the limit solves the most famous of Zeno’s paradoxes, since the limit and the sum are in many ways equivalent. I am not going to be terribly rigorous here. I am not trying to re-derive a proof. I am simply showing that I have a use for limits and infinite series.
       The two problems solved here are the problem of Atalanta and the problem of Achilles and the tortoise. The first problem concerns Atalanta running. Zeno says she never reaches her destination for this reason: she must first run half the distance, then 1/4 the distance, then 1/8 the distance, and so on. The second problem states that Achilles will never catch the tortoise, since while he is running to the point where the tortoise was, the tortoise has moved on. The tortoise is always one interval ahead.
       Both problems involve an infinite regression. They may therefore be solved by the summing of an infinite series. This is logically equivalent to solving by finding the limit, since you have to recognize a limit before you sum the series: if there is no limit there is no sum. Both methods make use of reasoning which I will put into common language here rather than mathematical symbolism. The methods work because they are both based on the realization that the infinite series ends at a definite time. The infinite series, even if it is taken to infinity, does not take an infinite time. There is a maximum time implied by the problem, and this time is the limit. The series therefore converges: the series is infinite but the sum is not. The sum is finite. Even if we imagine Atalanta and Achilles running every single interval, they will not take an infinite time to do so.
       Aristotle solved the first half of the problem by noticing that each of the intervals had a corresponding time. Each distance implied a time. As the distances got shorter so did the times. His solution was not accepted since it was argued that the total time also had an infinite number of intervals. How could an infinite number of intervals be traversed in a finite time? The problem was fully solved only with the later recognition that the infinite time intervals approached a limit. Given infinite calculations, you could reach the limit, but this limit was not infinity. It was actually the number 1. In other words, if we assume that it took Atalanta ½ a minute to reach the halfway mark, it is illogical to assume that the total time could take more than 1 minute. All the intervals added together, even if we allow an infinite regression, cannot be made to add up to anything greater than 1 minute. It therefore cannot possibly take an infinite time for her to reach her destination.
       Even if you argue that an infinite series can never be exhausted, that still leaves Atalanta somewhere short of the limit. Meaning that if she gives up because she is tired of crossing tiny intervals, she is still infinitesimally short of 1 minute. Therefore the time it took her to go from the beginning of her run (t0) to the time she quit in frustration (tf) is less than 1 minute. tf – t0 = Δt < 1. It is difficult to see why she is frustrated if she has been running for less than 1 minute.

Everything I have said so far is pretty standard. But I suspect that Zeno's point was even more subtle than that, and that we might look to an even more clever solution than the ones that have been offered historically. Summing the series certainly solves these problems, and Zeno probably meant to give the clue in that direction. But there is an even simpler answer, one that requires no limit and no summing. It does not even require looking at the series. This solution concerns noticing that Zeno never disallows us from assuming that Atalanta actually reaches the halfway mark. In fact it is a postulate of the problem. Zeno creates the paradox by assuming that Atalanta must reach the halfway mark. In order for her to reach the end, she must first go halfway, he says. In this way, the halfway mark becomes Zeno’s own postulate. He must assume it in order to create the paradox. Without a halfway mark, no infinite regression may be postulated. Without the infinite regression, no paradox is created. QED: the halfway mark may be taken as given. If so, then we may assume that it took her some time to reach the halfway mark. We must assume some velocity. Any given velocity will give us a time at the halfway point. With that time we may calculate the limit or the sum. But with that time, we may assume a finite total time, even without calculating any limit or sum. A finite halfway time must imply a finite total time; and a finite total time must be simply twice the halfway time (given a constant velocity).

Those who are confused by the paradox usually fail to distinguish between the time it would take to calculate an infinite series and the time it would take to traverse it, given some velocity. It would indeed take forever to calculate an infinite series, since there are an infinite number of terms. You could never truly exhaust the series. But motion is not the exhaustion of a mathematical series. Addition is the exhaustion of mathematical series.
       Beyond that, Zeno’s paradox relies on contradictions for its creation, as I have shown. In order to create his paradox, Zeno must assume that Atalanta reaches the halfway mark. How did she reach the halfway mark if she never moved? How do you create an infinite series without motion? An infinite series is only created by motion, real or abstract. Without change, there can be no progression or regression, mathematical or otherwise. Zeno assumes motion to disprove motion. The simplest solution requires you catch him at this. He says she can't ever reach the end. But all you have to do is say, "Zeno, my friend, you just assumed she could reach the halfway point! The halfway point is an end."
       If a halfway point can be reached, why should the reaching of an endpoint be paradoxical? If you cannot reach an endpoint, then you should not be able to reach a midpoint, since a midpoint is the endpoint of some smaller interval. Zeno’s assumption of a midpoint is enough, by itself, to prove the endpoint. If she reached the midpoint, as Zeno says, then I may show that that midpoint is the endpoint of a different interval, thereby proving motion.

As I said, I believe that it must have been Zeno’s intention to reveal these things to his students by forcing them to solve problems that seemed simple. He likely wished his students to discover that the limit approached was not at an infinite time. Furthermore, he wished his students to show the contradictions that created the paradox. A paradox is not created by a problem that is correctly stated. It is created by a problem that is incorrectly stated. He incorrectly stated a subtle problem in order the give practice in locating contradictions.

The Arrow

Another paradox of Zeno concerns an arrow flying through the air. Zeno states that at each instant the arrow must be imagined to be immobile—frozen in one spot. If it is frozen at each instant it must be frozen at all instants. If it is frozen at all instants it must not be moving. Therefore motion is an illusion.
       This is Zeno’s next lesson to the student who has solved the problems above by summing an infinite series. Here we have an infinite series each of whose terms appears to be zero. How do you sum a series of zeros and achieve a value? How do you sum a series of zero time intervals? Zero an infinite number of times is still zero. Limits and summations do not help us here.
       Historically this problem has been assumed to be linked to the other two problems above, and calculus is proposed as its solution. That is, calculus proposes that there is limit at every point in the line of motion, and that the solution is achieved by approaching an "everypoint." In my opinion this is a misuse of the limit concept and the concept of an infinite series. For calculus finds a velocity at every point. But there cannot be a limit at every point. An infinite series must approach a point from one direction or another. But if every point is a limit, then there is nowhere for numbers to converge from. You cannot postulate a line made up of an infinite number of limits. An infinite number of limits works exactly the way an infinite number of zeros would. You cannot solve the conundrum of an infinite number of zeros by postulating an infinite number of limits.
       I agree with Zeno that if the arrow were imagined to be "at an instant" its velocity must be imagined to be zero. Therefore the solution is that the arrow is never at an instant. A stopped arrow in motion is a mathematical abstraction. It is a physical impossibility. An arrow is either in motion or it is stopped. It cannot logically be imagined to be both. This is the contradiction in Zeno’s paradox. He imagines the arrow both stopped and in motion. But if it is stopped it is not in motion. If it is in motion, it is not stopped. One need only apply to the definition of motion. If an object is in motion, it is not stopped. Not over any possible interval. A zero interval is not a real interval. Motion is defined over a non-zero interval. Motion over a zero interval is a contradiction in terms. It is a mistake in logic.
       Time does not stop, both by definition and by experience. Time is change. If a change does not change it is not a change. Time stopping is therefore a contradiction in terms. Time cannot stop. If it did it would no longer be time.
       Math applied to a physical problem should express the mechanics of the problem. If it does not, it is bad math. Math that expresses motion as an infinite series of instants is bad math, since the instant does not exist in any physical problem. Only the interval exists. Math applied to motion should therefore express intervals, not instants. An instant is a non-physical, non-existential abstraction. It is useless even as a mathematical entity, since it has no referent except to the physical. Time divorced from physics is meaningless. It is just another zero-dimension variable, indistinguishable from any other naked variable, in which case the name "time" is arbitrary. In fact, time has meaning only in regard to motion.

As I show in my paper on time, time is actually one distance measured against another distance. This is the operational fact that grounds the entire problem of motion. Therefore the interval in question is always finally a distance interval. A stopped arrow supplies us with no distance interval, and it velocity is thereby incalculable. This should not be a surprise since a stopped arrow is not in motion. To calculate a velocity, we must be given two distances: the distance passed by the arrow and the distance passed simultaneously by the pendulum or second hand or cesium atom. The velocity is then the ratio of these distances. Any other analysis of the problem of motion is an unnecessary abstraction. Most other analyses of the problem are not only unnecessary abstractions; they are unnecessary obstructions. They invariably introduce contradictions, paradoxes, or other fine confusion.

In my paper on the calculus I show that points on a mathematical curve or line are not physical points or instants. I show that a physical point or instant cannot be graphed. Here I have shown that physical points and instants do not exist in the mechanics of motion anyway, so that graphing them would have not been helpful. Mind you, this in no way compensates for the error of the calculus. In fact, it doubles the error. Calculus postulates points in space, then proposes to graph them and take them to limits. It therefore uses a false method to graph things that do not exist. A point on a mathematical curve thereby becomes the ghost of a ghost, a faux pas gilded with a faux pas.

Paradox as Signal

I would like to make one final point concerning the paradox in mathematics. The paradox is now seen as a sign of distinction. From transfinite math to Special Relativity to QED, the paradox is seen not as a fault but as an esoteric badge, a hint that the transcendent is being approached. This belief could not be more false or deluded. A paradox is not a good sign. It is a very bad sign. It is a sign of disease. It is a very loud noise warning of a coming train. It is usually a sign that the question has been badly posed, as with Zeno. It is the outcome of a contradiction.
       Whether we are speaking of the twin paradox, Cantor’s or Hilbert’s paradoxes, or the two-slit paradox, we are speaking of a basic misunderstanding. We are being clued by the paradox to the existence of an error in our operations or axioms. With Einstein, Cantor and Hilbert, we are being shown a simple logical error. With the wave-particle duality, we are being clued by the paradox to the partial state of our knowledge. We are being told in no uncertain terms, "You need to know more about the structure and mechanics of the photon and the 'light packet' before you build a final theory of propagation." The propagation of light is not beyond understanding, as Bohr or Feynman would have it. It is simply beyond current understanding. It is not understood yet. All the mysticism and paradox is created by scientists who want a final theory now. They are so unbelievably arrogant that they cannot imagine the possibility of something they do not understand. They imply that if we do not understand it at the present time, it is a permanent mystery. "I do not know, therefore it is unknowable." The hubris is so transparent it is difficult to believe that these scientists have been taken seriously—that they have become famous while mouthing such absurdities.
       I have disproved the twin paradox simply by doing the rather simple math on an approaching object, showing that the time transforms are the inverse of a receding object. They therefore cancel out in any roundtrip. Why no one else thought to do this is beyond me. Maybe Mileva Einstein was the only one who even partially understood relativity, and once Albert left her she left the men to fend for themselves. Alone, they could not even do the algebra to disprove the twin paradox. Or maybe they liked the twin paradox and did not care to disprove it. It has been the single biggest selling point of the theory to the science fiction crowd and the readers of Scientific American. If we could no longer dream about time travel then relativity would revert to the dusty shelves of science along with Maxwell and Archimedes.
       As for Cantor’s and Hilbert’s various paradoxes, they all rest on the assumption that you can add 1 to infinity. This mistake is even more difficult to understand than the mistake of Einstein. Any mathematician who would even entertain the idea of adding 1 to infinity should be disbarred from the field as a nuisance. As I have pointed out in another paper, adding 1 to infinity is a contradiction in terms. An infinite set contains all terms of the set. There is therefore no entity or set of entities that can be added to the set. For instance, if you have an infinite set of people, all people are in the set. You cannot even think of adding another person. The same applies to numbers. Cantor tries to separate the infinite set of numbers into various subsets: integers, rationals, irrationals, etc. But this is just equation finessing. If you are counting one infinite subset with another, for example irrationals with integers, the only operational fact that is to the point is that you will always have a term to count with. No matter how many irrationals you want to count, you will always have an integer to count with. That is what infinite means. If the set of integers ends before the set of irrationals, then it was not infinite. The infinite cannot end. This falsifies everything else Cantor has to say.
       I can only suppose that Cantor was taken seriously because mathematicians were bored. They needed some new games to play to make themselves appear interesting. Different sized infinities are nearly as sexy as time travel. David Foster Wallace, the current prince of pseudo-intellectualism, can even get published in the mainstream talking about the transfinite. Imagine getting a mainstream publisher to look seriously at my algebraic corrections to Einstein or Newton. Algebra only reminds pseudo-intellectual readers of hated highschool classes. But the transfinite leads to Buddhism and the Dalai Llama and Star Trek and whatever else is in fashion. Another leader in science publishing, Paul Davies, proves this in every article, peppering his paste-ups with asides to pop culture. As if Hollywood scriptwriters or religious gurus had anything to add to the history of physics or mathematics.
       Zeno remains a hot topic in web postings, due to his status as first paradoxer. But Zeno would not have had any patience with contemporary mathematics. A true mathematician does not use logic or equations to create paradoxes; he uses logic and equations to solve paradoxes. This, I believe, is the intended legacy of Zeno.

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