The four Paradoxes of Zeno, which attempt to show that motion is impossible, are most conveniently treated as two pairs of paradoxes. The reasons for this will hopefully become clearer later. The first two paradoxes are as follows.
The Racecourse or Stadium argues that an athlete in a race will never be able to start. The reason for this is that before the runner can complete the whole course they have to complete half the course; and before they can complete half the course they have to complete a quarter; and before they can complete a quarter they have to complete an eighth; and so on. Therefore the runner has to complete an infinite amount of events in a finite amount of time - assuming that the race is to be run in a finite amount of time. As it is impossible to do an infinite amount of things in a finite amount of time, the race can never be started and so motion is impossible!
The second paradox is that of Achilles and the Tortoise, where in a race, Achilles gives the Tortoise a head start. The argument attempts to show that even though Achilles runs faster than the Tortoise, he will never catch her. The argument is as follows: when Achilles reaches the point at which the Tortoise started, the Tortoise is no longer there, having advanced some distance; when Achilles arrives at the point where the Tortoise was when Achilles arrived at the point where the Tortoise started, the Tortoise is no longer there, having advanced some distance; and so on. Hence Achilles can never catch the Tortoise, no matter how much faster he may run!
The diagram below may help to understand this argument.
The race starts at t0 with the Tortoise having a head start over Achilles. By time t1, when Achilles has reached the point at which the tortoise started, the tortoise has moved on; by t2 Achilles has reached the point where the tortoise was at t1 but the tortoise has moved on; by t3 Achilles has reached the point where the tortoise was at t2 but the tortoise has moved on; and so on. To be sure, the distance between Achilles and the tortoise is getting less and less each time but Achilles never catches up with - far less overtakes - the Tortoise.
Zeno, it seems, believed quite seriously that motion did not exist and that arguments such as these established it. What do we, who believe that races can be run and slow objects can be overtaken by faster moving ones, say in response?
One common reply is that Zeno has misunderstood the nature of infinity. Modern mathematics, it is said, has shown that the infinite sequences that Zeno generates do have a finite sum. In particular, to take the Racecourse example, the sequence 1/2 + 1/4 + 1/8 + 1/16 + . . . is equal to 1.
This reply, however, misunderstands what modern mathematics has shown. Mathematicians do use sequences such as 1/2 + 1/4 + 1/8 + 1/16 + . . . but they say that they have a limit of 1, or tend to 1. That is, we can get nearer and nearer towards 1 by adding on more and more members of the sequence, but not actually arrive at 1 - this would be impossible because we are considering an infinite sequence. So far from providing an argument against Zeno, mathematics is actually agreeing with him!
Further, this reply seems to miss the point of Zeno's argument: simply pointing out that there is a branch of mathematics that deals with the infinite does not reduce the puzzling aspects of the Paradoxes. We know that races can be run and that Achilles will overtake the Tortoise, what we want to know is what is wrong with the arguments that show that these things can't happen.
The first two Paradoxes of Zeno attempt to find contradictions in the idea that motion is continuous and space can be infinitely subdivided. But motion may not be continuous: space may be discrete and motion be a series of tiny jumps. On this view there would be a finite - but very large - number of steps between the beginning of the race and its end. So the Paradox of the Racecourse could be avoided by saying that there is some first, incredibly small, step that can be taken, where there is no step of half the size. Similarly, there is some small, and indivisible, last step that Achilles can take which will allow him to catch the Tortoise and then overtake her. This possibility is criticized by the other two Paradoxes of Zeno to be considered next time.