Outer Limits of Reason (9 page)

Unfortunately, most of Zeno's original writings have been lost. Our knowledge of the paradoxes largely comes from people who wanted to prove him wrong. Aristotle briefly sets up some of Zeno's ideas before knocking them down. Because Zeno's ideas were given short shrift, it is not always clear what his original intentions were. This should not deter us since our central interest is not what Zeno actually said; rather, we are more interested to know if something is wrong with our intuition and how it can be adjusted. These ideas should not be taken lightly. They have bothered philosophers for almost 2,500 years. Regardless of whether one agrees with Zeno or not, he cannot be ignored.

The first and easiest of Zeno's paradoxes of motion is the
dichotomy paradox.
Imagine an intelligent slacker waking up in the morning. He tries to get from his bed to the door in his room (see
figure 3.2
).

To get the whole way to the door, he must reach the halfway point. Once he reaches that point, he still must go a quarter of the way more. From there he has an eighth of the way to go. At every point, he must still go halfway more. It seems that this slacker will never be able to reach the door. In other words, if he does want to get to the door, he will have to complete an infinite process. Since one cannot complete an infinite process in a finite amount of time, the slacker never gets to the door.

Our slacker can further justify his laziness with more logical reasoning. To reach the door, one has to go halfway. To reach the halfway point, he must first get to the quarter-way point, and before that the eighth-way point, etc. . . . Before any motion can be performed, he must perform half the motion. One needs to perform an infinite number of processes in order to get
anywhere
. An infinite number of processes demands an infinite amount of time. Who has an infinite amount of time? Why get out of bed at all?

Zeno's paradox is not only about movement but also about any task that has to be done. In order to complete a task, one must perform half the task first and go on from there. This shows that not only is movement an impossibility, but performing any task, indeed any change, within a time limit is unreasonable.

What are we to do with Zeno's little thought puzzle? After all, we do get to the end of our journey in a finite amount of time and when we do get out of bed in the morning, we can accomplish something. Following the theme of this book, Zeno's paradox has the form of a proof by contradiction. We are assuming something (that is wrong) and we are logically coming to a contradiction or an obvious falsehood. We came to the conclusion that there is no movement or change when, in fact, we see movement and change all the time. What exactly is our wrong assumption?

A mathematician might argue that there is no problem performing an infinite task. Look at the following infinite sum:

1/2 + 1/4 + 1/8 + 1/16 + 1/32 + . . . .

The uninitiated would say that the ellipsis means that the sum is going on forever, and so the sum total will be infinite. However, the sum total is the nice finite number 1.
⁷

There is a beautiful two-dimensional geometric way to see that this sum equals 1. Consider a square whose side length is 1, as in
figure 3.3
.

One can see this square as made up of half of the square plus a quarter plus an eighth plus . . . Every remaining part can be further split in half. It is obvious that the area of the entire square is 1.

However, a mathematician would be somewhat disingenuous to claim that this solves Zeno's paradox of performing an infinite process in a finite amount of time. After all, the mathematician is not adding each of the infinite terms to a running sum. She is simply displaying the first few terms and then indicating with an ellipsis that there are an infinite number of terms. She is doing a trick that shows what the sum of all of them would be if she summed them. If one were to sit down and add all infinite terms, it would indeed take an infinite amount of time.

A better solution is to say that the problem with Zeno's reasoning is that he assumes that space is continuous. That means that space looks like the real-number line and is infinitely divisible—that is, between every two points lies an infinite number of points. Only with this assumption can one describe the dichotomy paradox. In contrast, imagine that we are watching the slacker go to the door in an old-fashioned television made up of millions of little pixels. Then as he is moving, he is crossing the pixels. He crosses half of the pixels and then he crosses half of the rest of the pixels. Eventually the TV slacker will be one pixel away from the door and then he will be at the door. There are no half pixels to cross. A pixel is either crossed or not crossed. On the TV screen there is no problem with the slacker getting to his destination and Zeno's paradox evaporates. Maybe we can say the same thing with the real world. Perhaps space is made up of discrete points each separated from its neighbor and that between any two points there is at most a finite number of other points. In that case we would not have to worry about the dichotomy paradox. If we assume such a discrete space, then we can understand why our lazy slacker makes it to the door: he only has a finite number of points to cross. At a certain point, the intervals could no longer be split into two. Objects move in this type of space by going from one discrete point to the next without passing between them.

In the language of
chapter 1
, we can say that this is a paradox because we are assuming that space is continuous:

Space is continuous ⇒ movement is impossible.

Since there is definitely movement in this world, and our assumption led us to a false fact, we conclude that space is not continuous. Rather, it is discrete, or separated into little “space atoms.”

Such ideas of discrete space are familiar to people who study quantum mechanics.
⁸
Physicists discuss something called
Planck's length
, which is equal to 1.6162×10
^−35
meters. Something smaller in length cannot be measured. To some extent, nothing smaller than that exists. Physicists assure us that objects go from one Planck's length to another. In high school chemistry it is taught that electrons fly in shells around a nucleus of an atom. When energy is added to an atom, the electrons make a “quantum leap” from one shell to the next. They do not pass in between the shells. Perhaps our lazy slacker also makes such quantum leaps and hence can finally reach the door.

Let us reconsider
figure 3.3
. The square is infinitely divvied up as illustrated. But this is only possible if we think of the square as a mathematical object. In mathematics every real number that represents a distance can be split into two, hence we can continue chopping forever. In contrast, let us think of the square as a piece of paper. We can start cutting paper into smaller and smaller pieces using finer and finer scissors. This will work for a while, but eventually we will reach the atomic level where no further cutting will be possible. This is true for any physical object made of atoms. We are forced to conclude that the square depicted in
figure 3.3
is not a good model for the physics associated with the paper square. The real numbers can be infinitely divided but the paper cannot be. What Zeno is forcing us to do is to ask the question of whether space (which is not made of atoms) can be infinitely divvied up. If it can be, the slacker will not reach his goal. If it cannot be, there must be discrete “space atoms,” and continuous real-number mathematics is not a proper model for space.
⁹

We cannot, however, be so flippant about asserting that space is discrete and not continuous. The world certainly does not look discrete. Movement has the feel of being continuous. Much of mathematical physics is based on calculus, which assumes that the real world is infinitely divisible. Outside of some quantum theory and Zeno, the continuous real numbers make a good model for the physical world. We build rockets and bridges using mathematics that assumes that the world is continuous. Let us not be so quick to abandon it.
¹⁰

 

Zeno's second paradox of motion is the story of
Achilles and the Tortoise
. Achilles was the ancient Greek version of the modern D.C. Comics character The Flash and was the fastest runner in town. One day he had a race with a slow Tortoise. To make the race more interesting (and because Achilles had a warm heart), Achilles gave the Tortoise a head start, as shown in the top line of
figure 3.4
.

The problem is, in order for Achilles to overtake the Tortoise, he must first pass the point where the Tortoise started (as in the second line in
figure 3.4
). At that point, the Tortoise has already moved further. Once again, in order for Achilles to overtake the Tortoise, he must get to the point where the Tortoise moved. At each point, Achilles is getting closer and closer to the pesky Tortoise, but he will never be able to reach him, let alone beat him.

Again there is a mathematical analogy to this. In calculus we say that the limit of 1
/x
as
x
goes to infinity is zero. That is, the larger
x
gets, the closer 1
/x
gets to zero. Since infinity is not a number,
x
can never get to infinity and 1
/x
can never get to zero. But the concept of a limit makes it meaningful. Similarly, the distance between Achilles and the Tortoise will never really be zero but the
limit
of the distance does get to zero. Again, we can find fault with this analogy. The concept of a mathematical limit is a type of trick. For no finite number will 1
/x
actually equal zero and at no time period will Achilles actually reach the Tortoise.

This paradox would also melt away if we assume that the racetrack is made out of discrete points. The fact that Achilles runs faster than the Tortoise simply means that he covers more of the discrete points in the same time. So eventually Achilles will overtake the Tortoise. Discrete space would answer the paradox, but again, we have to be careful. We should abandon the notion of continuous space with great trepidation since that mathematical model works so well in general physics.

In the third paradox, Zeno is not interested in determining whether a motion can be completed. Here he attacks the very idea of any motion whatsoever. In the
arrow paradox
, we are asked to think of an arrow flowing through space. At every instant in time, the arrow is in some particular position. If we think of time as a continuous sequence of “nows” that separates “pasts” from “futures,” then for each “now,” the arrow is in one particular position. At each point in time the arrow is in a definitive position and not moving. The question is, when does the arrow move? If it does not move at each of the “nows,” when does it?