Unravelling Xeno's Paradox

Jason Peterson
(Copyright 2005)

The ancient philosopher Xeno has received a lot of attention through the years for his take on the concept of infinite divisibility, and the paradox he arrived at in asserting the reality of such. While appearing to be a clearly thought-out and rational dilemma, I intend to show how this dilemma is not actually a paradox, nor does it live up to the title of a dilemma by any logical approach to the subject. A paradox by definition is that which is “an apparently sound argument leading to a contradiction”.

Xeno’s paradox is stated as follows:

1.For something to travel from point A to point B it must traverse a certain finite amount of space.

2.Before a finite amount of space can be traversed, a half of that space must first be traversed.

3.Any finite amount of space can be infinitely divided (as into halves).

4.Every fraction of space requires a finite amount of time to traverse.

5.A finite amount of time multiplied by an infinite amount of spacial fractions equals an infinite amount of time.

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Therefore: It would take an infinite amount of time to traverse a finite distance. Thus, motion is intelligibly impossible.

In evaluating the logical appeal and the soundness of this claim, there are five major critiques to consider: the Life-Span Argument, the Conceptual Rate Argument, the Conceptual Application Argument, the Conceptual Expansion Argument, and the Rule Of Number Argument.

The Life-Span Argument:

If it is true that space is infinitely divisible, and that it would actually take an infinite amount of time to traverse any distance, we are actually providing a proof that everything that undergoes motion is unmistakably existing for an infinite amount of time. Also, the only objects in existence that can be finite are those that never move. Since we all, as humans, continually undergo what qualifies for motion by any definition, it is fair to say then that we must all be existing for an infinite amount of time. However, we notice that having an infinite lifespan does not allow for the ability to discontinue the infinite nature of that span, and yet we also notice that people die everyday. This ability for mobile persons to stop living is in itself proof that we do not exist for an infinite amount of time. Thus, it must not require an infinite amount of time to traverse a finite distance.




The Conceptual Rate Argument:

If we are to say that space is infinitely divisible, before we can say that it would take an infinite amount of time to traverse a finite amount of space, we must first remember that motion is not time multiplied by distance, as Xeno seems to infer. Motion is a rate; it is distance divided by time. A rate is such that for every set portion of time, a set portion of space is traversed. In a finite amount of time, a finite amount of distance can be traveled, and likewise, in an infinite amount of time, an infinite amount of distance can be traveled. To divide a finite amount of space infinitely is to simultaneously infinitely divide the amount of time it takes to cross that finite space, so no numbers are changing here, the rate is not affected. For example, we will call the distance between point A and point B one “unit”. If it takes one unit of time to traverse one unit of space, it will not change anything to say that it takes ½ unit of time to traverse ½ unit of space, or ¼ unit of time to traverse ¼ unit of space, and so on.




The Conceptual Application Argument:

Reminiscent of Plato’s “Forms” it seems that concepts are infinite in nature. Whenever we think we have established knowledge of a concept, we have only in fact established a knowledge of a finite particular of that concept. In viewing a blue object, we have not discovered “blueness”, we have only stumbled upon a manifestation of that concept, as we well recognize that “blueness” is not quantifiable and so extends far beyond the blue object that we perceive. Infinity is a concept as well, so we have to realize that whenever we think we have established a knowledge of the infinite, we have only really established the knowledge of a finite manifestation of the infinite, or a particular finite point on the path towards the infinite. The catch here is that if we continue to apply the concept of infinity by adding/subtracting/multiplying/dividing on top of the first particular finite point to try to reach a full understanding, we are only establishing more and more finite manifestations of it. To demonstrate this is to say that if we begin with a finite distance between myself and the door, there is no longer a question as to if the space between is infinite, for only in having mentioned the door, we have applied the concept of the infinite and established a finite particular at that point in space, pushing the concept of the infinite to extend only beyond that point, so all the space before the door, or in between myself and the door, has already been established in a finite threshold. And even if we try to divide the space between to an infinite extent, we will only ever reach the finite particular divisions that we label or consider which will always be quantifiable. An infinite division does not exist in a finite world, only the application of infinity through finite conceived particulars, so there will never be an infinite amount of finite space, nor will it require an infinite amount of finite periods of time to traverse.




The Conceptual Expansion Argument:

In Xeno’s claim, he begins by establishing a finite distance between point A and point B. As the beginning of an argument, this is his base of expansion, which is to say that he has voluntarily given the quality of reality to this finite distance. Then, by way of his logic in incorporating finite values to an infinite amount of spacial fractions, he comes to a conclusion that this original finite distance is actually an infinite distance. By undermining his original base of expansion, he has destroyed all of the reference value and relevance of the rest of the argument. If the original finite distance does not exist as first stated, then his concept of the infinite has no basis for existence either. And if the finite distance is real, just the fact that it is the original real thing, and so the most direct influence, it should be given more rational credibility than any derivative of it. For example, picture a skyscraper. We start by building a foundation, and then the first floor is built upon the foundation, and the second floor upon the first, and so on. Imagine that you build up so many floors high that when you look down you can no longer see the foundation or the first few floors. If you were to, at that point, state without question that the foundation did not ever exist, or was actually built a few miles north of its original location, you would be making the same type of logical blunder as Xeno. If you were correct, then the building would never have been able to be built, and if you took away the foundation at the end of construction, it would tumble down to pieces. In this case, the distance between point A and point B in Xeno’s claim must necessarily be finite, so it cannot be an infinite collection of finite fractions, thus it cannot take an infinite amount of time to traverse.




The Rule Of Number Argument:

A sound argument is a valid argument with all true premises, and this critique is to show that Xeno’s argument includes a premise that does not ring true. The only way he is able to get from the point of incorporating infinite divisibility to requiring an infinite amount of time to traverse a finite distance is by the math involved in multiplying the finite distance by the infinite amount of spacial fractions. The rule of numbers is such that an infinite number is the largest possible number as it has no room to be enlarged in any way. To take an infinite number (spacial fractions) and multiply it by a finite number (time) would suggest that the infinite is becoming larger, and that is absolutely theoretically and physically impossible. As this type of mathematical relationship is the crux of his entire explanation of the space/time dilemma and/or paradox he addresses, and this relationship is one of impossibility, it should stand alone as proof that there is no clean logic in asserting that it takes an infinite amount of time to traverse a finite distance. Also, since that particular premise is not a working truth, the argument is not sound. As I stated before, a paradox requires a sound argument to lead to a contradiction, and since it is not sound, it should not be considered a paradox either.