Zeno's Paradoxes revisited

Anybody who has been in my ancient history class knows that I love paradoxes. This year, we did not spend as much time on philosophy as I had hoped, so I hope that you had time to think about the paradoxes on your own. I want to take a minute to go back over the "Achilles and the Tortoise" paradox, because I tried to explain another way to think about what the paradox is trying to say, but I felt a little rushed and am sure that I explained it well enough for people to catch on to what I was saying.

You remember that "Achilles and the Tortoise" involves a race between Achilles, the fastest runner, and a tortoise, who is less than the fastest runner. Since Achilles is so fast, he allows the tortoise to have a head start. The tortoise runs quite a ways before Achilles starts. Achilles is fast, but it still takes him some time to catch up to the tortoise. In that amount of time, the tortoise has moved on a bit. So now Achilles needs to make up the distance that the tortoise moved while Achilles was catching up. As soon as Achilles gets to where the tortoise had been, the tortoise has had time to move on a bit more. Now Achilles can keep getting closer and closer to the tortoise, but he is never going to completely catch up.

We also talked about a more recent retelling of this paradox, "The Frog and the Fly." There is a frog who wants a fly, but he can only jump halfway to where he is going. So he jumps half, half, half, but will never get there. In the ordinary world, is there any way to move somewhere without first going halfway?

What these little stories are supposed to help you imagine is a paradox of infinite divisibility. Say you have an object, a rubber ball, perhaps. You can cut it in half and discard one half of it. Then, take that half and cut it in half. How long can you keep this up? Forever? If you can do it forever, then any object is made up of infinite bits piled up together. How big are these infinite bits? They really cannot have any imaginable mass, since it takes infinity of them to make up something as small as a rubber ball.

If you can't cut it in half forever, what's stopping you? The ancient philosophers known as the "atomists" came up with the notion of "atoma," which translates to "without cutting." This is where we get our modern word "atom" - although they do not mean exactly the same thing. The atomists proposed that there was a point at which you reach a particle so small that it cannot be cut in half. According to them, this piece has some size and you can imagine half of it - it is "geometrically divisible" - but it cannot be physically divided. The atomists believe that this solves the paradox - at some point the frog moving towards the fly will reach a point where there is no such thing as "halfway" - it can either be on one side or the other.

I am not sure the atomists have solved the puzzle, though. Since an atom has a knowable size, then it is still divisible. It seems that in order to completely erase the paradox, an atom would need to be both physically and geometrically indivisible. In other words, it has to be special enough that you can pile up a finite number of them to make a rubber ball, but they are not made up of halves. This sort of thing is either very hard - or impossible - to imagine. It seems it is still a paradox.