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thought experiments that demonstrate that motion is an illusion. Since motion occurs within space and time, Zeno’s paradoxes will challenge our intuition of these obvious concepts.
    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 ).
    Figure 3.2
    Zeno’s dichotomy paradox
    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 .
    Figure 3.3
    A two-dimensional infinite sum
    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