Achilles and the Tortoise

Rather than tackle Zeno head-on, let us pause to notice something remarkable. Suppose we take Zeno’s Paradox at face value for the moment, and agree with him that before I can walk a mile I must first walk a half-mile. And before I can walk the remaining half-mile I must first cover half of it, that is, a quarter-mile, and then an eighth-mile, and then a sixteenth-mile, and then a thirty-secondth-mile, and so on. Well, suppose I could cover all these infinite number of small distances, how far should I have walked? One mile! In other words,

1= $\frac{1}{2}$ + $\frac{1}{4}$ + $\frac{1}{8}$ + $\frac{1}{16}$ + $\frac{1}{32}$ +⋯ At first this may seem impossible: adding up an infinite number of positive distances should give an infinite distance for the sum. But it doesn’t—in this case it gives a finite sum; indeed, all these distances add up to 1! A little reflection will reveal that this isn’t so strange after all: if I can divide up a finite distance into an infinite number of small distances, then adding all those distances together should just give me back the finite distance I started with. (An infinite sum such as the one above is known in mathematics as an infinite series, and when such a sum adds up to a finite number we say that the series is summable.)

Now the resolution to Zeno’s Paradox is easy. Obviously, it will take me some fixed time to cross half the distance to the other side of the room, say 2 seconds. How long will it take to cross half the remaining distance? Half as long—only 1 second. Covering half of the remaining distance (an eighth of the total) will take only half a second. And so one. And once I have covered all the infinitely many sub-distances and added up all the time it took to traverse them? Only 4 seconds, and here I am, on the other side of the room after all.

And poor old Achilles would have won his race.