What is Xeno's Paradox?

Xeno (or Zeno) was a disciple of the ancient Greek philosopher Parmenides. Parmenides believed that Reality was One: it was a single, homogeneous thing, indivisible and changeless. This seems, at first glance, to be an odd position to take as change seems to be happening around us all the time. Zeno (I prefer the "Z" spelling), a philosopher in his own right, was made famous for his rigorous defense of the Parmenidean belief system.

Zeno put forth four paradoxes concerning motion. The question addressed by the paradoxes was whether or not motion should be understood as continuous or discrete. For example, the first, and most famous paradox of his is this: suppose there is a runner, Achilles, who wishes to run from point A to point B. In order to get to point B, he must first run ½ the total distance. Then, to get from that point to B, he must run ½ the remaining distance, and from there, ½ that distance, and so on.

The end result is that he can never get to B: there is no end to the succession of half-distances, so the task before Achilles is an infinite task which, by definition, can never be completed. The paradox can also be reversed (I prefer this form). Supposes Achilles wishes to move from A to B. Before he can do that he must move ½ the total distance. But before he can do that, he must move ½ that half distance.

And so on, ad infinitum. In this situation, not only is Achilles unable to reach his final destination, but neither can he move at all because there is no first step. From this Zeno reasoned that space (and time) cannot be continuous. After that, he examined if space (and time) could be discrete and again found absurdities.

The reason Zeno put these paradoxes forward was to defend the semi-mystical views of his master, Parmenides. For us, though the mystical aspects may not be relevant, the paradoxes do serve a useful purpose. To me, at least, they demonstrate that there is something about the notion of movement we do not understand.