Zeno's Paradoxes: Commentary

Zeno of Elea

Commentary on the text

1. We have joined Aristotle in midstream, engaged in an extended discussion on the nature of motion and change. It is important to note that Aristotle, the empiricist, believes strongly that "Motion is eternal" and that change is a fundamental ingredient of nature. He is in fundamental disagreement with Zeno and his Parmenidean philosophy of the denial of motion. In the following segment of his treatise, he "resolves" Zeno's paradoxes by dismissing them altogether as the result of fallacious reasoning, but in fact he does not identify any flaws in the application of the rules of logic by the Elean. Rather, he disputes with Zeno's underlying assumptions, as we will see.

2. Aristotle alludes here to the first of the paradoxes, the Dichotomy.

In this paradox, Zeno claims that no object can move from point A to point B for that would entail having to first reach the halfway point C. But even before getting to C, the object must first reach the point D halfway between A and C; indeed, prior to this, the point E halfway between A and D must be attained. As this argument can be repeated ad infinitum, it follows that infinitely many points must be traversed beforethe object can get from A to B. In fact, infinitely many points must be reached as soon as the motion from A begins. Since it is clearly impossible to accomplish infinitely many tasks (each being the passing from one point to another) in a finite time, no motion at all is possible!
Aristotle disputes Zeno's claim that the object cannot pass over infinitely many points between A and B in a finite time by asserting that the space between the points, as well as the duration of time elapsed as the motion occurs, is a continuum. That is, he takes the view that there are always infinitely many points in space between any two given ones, that if we bisect any distance AB, no matter how small, there will always be a point C available in between. Similarly, between any two moments of time there is an intermediary moment. Another way to understand Aristotle's perspective is to note the diagram drawn above, one that is implicitly provided by the text. To see space (or time) as a continuum is to have the mental picture of a line segment as representing this phenomenon: the line is made up of infinitely many points without any gaps. This geometric representation of nature is a hallmark of Greek mathematics. (Read the beginning of Euclid's Elements for a further discussion of the geometric world-view of Greek mathematics.)

3. That is, there are two kinds of infinity: the "divisible" infinity of points between any two given points, and the infinity of extent or "extremes" exhibited by a line having no endpoints.

4. It takes an infinitely long time for a point to travel along an infinitely long line.

5. It is this latter assertion, that "the passage over the finite [the space from A to B] cannot occupy an infinite time," that forms the content of Zeno's argument. For Zeno, the paradox comes from contemplating the consequence of traversing the infinitely many intermediary points between A and B. We will analyze this argument more fully below.

6. Here is the picture Aristotle has in mind:

Apparently, he envisions the object as moving right to left from B to E, while time is measured from left to right starting at C.

7. If BE is an "exact measure" of AB, or as is also said, if BEmeasures AB, then the length of AB is an integer multiple of the length of BE. Another term used to describe this is to say that segments AB and BE are commensurable. Aristotle affirms that BE need not be an exact measure of AB; this is not important to the argument. However, it is an important geometric consideration. Geometric arguments were often made much easier to explain if the segments involved were commensurable. (See the article on Hippasus for further discussion on this point.)

8. Evidently, Aristotle is assuming that the object moves at a constant rate of speed. In modern terms, since

distance = rate ¥time

under the condition that the rate is constant throughout, equal distances must take equal times to traverse because then

time = (distance)/(rate).

9. Aristotle sees the important distinction between continuous and discrete phenomena. A continuous substance is one that can be modeled by a line segment, as we discussed above (note 2): between any two points of the line, no matter how close together, there can always be found another, and no gaps exist between the extremities. On the other hand, a discrete substance is one that breaks down into indivisible "atoms". These atoms can be counted one by one, like grains of sand. It is this discrete view of nature that Aristotle imputes to Zeno, and rejects as false.
However, since Zeno says that "the passage over the finite cannot occupy an infinite time," he must be allowing space to be infinitely divisible, otherwise it would not be possible for him to continually bisect the initial segments as is explained in the argument.

What is more, it is incorrect to claim that he is assuming here that time is discrete. To see this, let us analyze the paradox more fully.
One way to describe Zeno's argument is to time the passage of the object as it passes between the marked points. Suppose we believe that it takes a certain amount of time for the object to pass from A to B. Then (assuming travel at a constant speed), it will take 1/2 of the time to pass from C to B, and 1/4 = (1/2)(1/2) = (1/2)^2 of the time to pass from D to C, and 1/8 = (1/2)(1/4) = (1/2)^3 of the time to pass from E to D, and so on. In particular, the time periods that elapse between successive pairs of points must be shrinking in size. This does not conform to a discrete view of time, for if time were made up of indivisible instants, it would be impossible to consider a time period any shorter than that indivisible instant. As the time periods in the argument will eventually become shorter than any given indivisible instant, no such indivisible could exist.
The total amount of time required to pass from A to B is given by adding up the time periods between the successive points; this gives the sum 1/2 + 1/4 + 1/8 + ... . Assuming the infinite divisibility of the space between A and B and of the time elapsed, there will be infinitely many terms in this sum. This is where Zeno's crisis lies: how is it possible to sum these infinitely many numbers? Surely an infinite sum of nonzero time durations gives an infinite value for the total time; but supposedly, this sum represents a finite time. Hence the paradox. This interpretation of Zeno's argument is somewhat more sophisticated than the one Aristotle gives. In addition, it specifies that among Zeno's hypotheses are that space and time are continuous, not discrete.

10. This is the second paradox of Zeno, the so-called Arrow paradox. It goes like this: Consider an arrow in flight. At any given moment in time, the arrow takes up a particular position in space; previously, it was in a different place, and later it will also be elsewhere, but now it happens to be here. But if it is here, and not somewhere else, then it isn't really moving. That is, no motion can be taking place. Notice that argument in this paradox comes to the same conclusion as in the Dichotomy: motion is nonexistent.

11. Here is Aristotle's criticism. He again accuses Zeno of the erroneous (to him) assumption that time is discrete. Now, however, Aristotle gets it right. Evidence of this occurs when Zeno partitions the flight of the arrow into separate moments of time: the moment before, the present moment, and the moment after. Still, the argument comes to the same conclusion as in the Dichotomy: motion is nonexistent.

12. This is the Dichotomy paradox (note 2). It is reported by the early historian Proclus that Zeno had devised forty paradoxes, but these are the only ones that survive in the literature.

13. The Achilles paradox considers successive moments in a foot race between "the fleet-footed Achilles," the legendary warrior from Homer's Iliad, and some other runner, who must necessarily be the slower of the two to be true to the legend. Let's call him Bob. Somewhat like the foolish Hare in Aesop's fable of the Tortoise and the Hare, Achilles is so confident of his abilities that he gives Bob a headstart. Then the race begins.

While Achilles obviously runs much faster than his opponent, it takes him some time to reach the position from which Bob started the race, by which time Bob has moved forward a bit to a second landmark. At the next stage of the race, Achilles must reach this second landmark, but by then Bob, slow as he is, has moved forward some to a third landmark. This process can be carried on forever, since no matter how fast Achilles runs, Bob will always be able to advance during the time it takes Achilles to reach the next landmark. Consequently, Achilles can never overtake his opponent, despite being the faster runner.

14. Aristotle dismisses this paradox on the basis of its similarity to the Dichotomy, the only difference being "the division of the space in a certain way". Our interpretation of the Achilles is also similar to the one we gave of the Dichotomy (but different from Aristotle's): the distance between the runners in successive stages of the race is decreasing in size, indicating the continuous divisibility of space, but then also the durations of time between successive stages is decreasing, indicating the continuous divisibility of time.

15. The fourth of the paradoxes is called the Stadium. Aristotle sketches his understanding of the paradox, why he believes it can be dismissed, and then gives an explanation of the details of the Zeno's argument.

16. The picture brought to mind is that of three trains of carriages, two of which are passing each other and a third stationary train by moving in opposite directions at the center of the field:

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Trains B and C are moving at equal speeds, so that at some later time the situation is this one:

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Next, three inferences about what has transpired in the race are made.

17. The first conclusion is straightforward, but the second is confusing. Unfortunately, it is at the heart of the paradox, so we must work to understand what is being said. Implicit here is that Zeno is assuming that time can be measured discretely; in fact, the trains are supposed to be moving as fast as they possibly can--so fast that one indivisible instant of time passes as a carriage of train B (or likewise, of train C) passes one train A. But then we note that in the same time, a carriage of train B passes two carriages of train C. Therefore, it takes half an instant for the B and C to pass each other. Here is the paradox: there is no such thing as half an instant, since an instant is by definition indivisible. Another way of conlcuding this is to argue that an instant from the perspective of a moving train passing the stationary train is twice as long as an instant from the perspective of a moving train passing the other moving train; so, as Aristotle puts it, "half a given time is equal to double that time", a clear impossibility.

18. The third conclusion is equally confusing, but the idea is the same: if one carriage passes another in one instant of time, then an instant measured with respect to one pair of trains is half as long (or twice as long, depending on your choice of trains) as an instant with respect to another pair of trains.
    Aristotle's criticism of the paradox, given above in the text, is that Zeno makes the erroneous "assumption that a body occupies an equal time in passing with equal velocity a body that is in motion and a body of equal size that is at rest." But this is rather a consequence of the more primitive assumption that the time it takes a carriage to pass another is one indivisible instant of time. Aristotle took Zeno to task earlier for taking a discrete view of time, but makes no mention of this here.
    Let us now step back and reconsider the four paradoxes and try to filter out Aristotle's perspective on the matter. In the Dichotomy and Achilles, Zeno argues that motion is impossible from the hypothesis that time is a continuous phenomenon. In the Arrow and the Stadium, he argues that motion is impossible from the hypothesis that time is a discrete phenomenon. Cobbling these together into one meta-argument gives us an even more powerful conclusion: regardless what your stand is on the nature of time, continuous or discrete, the conclusion is that motion is impossible. Has Zeno convinced you of this?
    Even if, like Aristotle, you would rather believe your own senses than Zeno's deductions, even if you are convinced that motion, transformation, and change are present everywhere in nature, Zeno has left some fascinating puzzles for us to ponder. What they say about the mathematical universe is even more important for us than what they say about the physical universe.

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