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By Root 647 0

Greek philosopher Zeno of Elea, who lived in the fifth century bc. In one of his famous paradoxes, he described a theoretical race between Achilles and a tortoise. Achilles is faster than the tortoise, so the tortoise is given a head start. The famous warrior starts at a point A and his reptile challenger is ahead of him at a point B. Once the race starts, Achilles zooms forward and soon reaches point B, but by the time he gets there, the tortoise has already advanced to point C. Achilles then ploughs on to point C. But once again, when he reaches this point, the tortoise has shuffled forward to point D. Achilles must reach D, of course, but when he does, the tortoise is already at E. Zeno argued that the game of catch-up must carry on for ever and, therefore, that swift Achilles is never able to overtake his slower four-footed rival. The athlete is much faster than the tortoise, but he cannot beat him in a race.

Like this one, all of Zeno’s paradoxes draw apparently absurd conclusions by dissecting continuous motion into discrete events. Before Achilles can reach the tortoise, he must complete an infinite number of these discrete dashes. The paradox stems from the assumption that it is impossible to complete an infinite number of dashes in a finite amount of time.

The Greeks, though, didn’t have the depth of mathematical understanding of infinity to see that this assumption is a fallacy. It is possible to complete an infinite number of dashes in a finite amount of time. The crucial requirement is that the dashes are getting shorter and taking less time, and that both distance and time are approaching zero. Although this is a necessary condition, it’s not sufficient; the dashes also need to be shrinking at a fast enough rate.

Achilles and the tortoise.

This is what is happening with Achilles and the tortoise. For example, say that Achilles is running at twice the speed of the tortoise and that B is 1m ahead of A. When Achilles reaches B, the turtle has moved m to C. When Achilles reaches C, the turtle has moved another m to D. And so on. The total distance in metres that Achilles is running before he reaches the tortoise is:

If it takes Achilles one second to complete each of these intervals then it will take him for ever to complete the distance. But this is not the case. Assuming constant speed, it will take him a second to go a metre, it will take half a second to go half a metre, a quarter of a second to go quarter of a metre, and so on. So, the time in seconds it takes him to reach the tortoise is described by the same addition:

When both time and distance are described by the halving sequence they simultaneously converge at a fixed, finite value. In the above case, at 2 seconds and 2 metres. So, it turns out that Achilles can overtake the tortoise after all.

Not all of Zeno’s paradoxes, however, are solved by the maths of infinite series. In the ‘dichotomy paradox’ a runner is going from A to B. In this case, we’ll call the first point that the runner passes after leaving A point C. For the runner to get to C, however, he must have passed the point that is halfway to C. So C cannot be the first point he passes. It follows that there can be no ‘first point’ that the runner passes, since there will always be a point that he must pass before it. If there is no first point that the runner passes, Zeno argued, the runner cannot ever leave A.

According to lore, to refute this paradox Diogenes the Cynic silently stood up and walked from A to B, thereby demonstrating that such motion was possible. But Zeno’s dichotomy paradox cannot be dismissed so easily. In two and a half thousand years of scholarly head-scratching, no one has been able to solve the riddle totally. Part of the confusion is that a continuous line is not perfectly represented by a sequence of an infinite number of points, or an infinite number of smll intervals. Likewise, the unbroken passage of time is not perfectly represented by an infinite number of discrete moments. The concepts of continuity and discreteness are not entirely