Achilles and the Tortoise

Through a random but fortuitous and perhaps inspired accident, I found myself the happy borrower of the following volume from the local library:


Mulling over the writings on the paradoxical nature of infinity has given me a great insight into my recent blue mood regarding my housework. Today I will give the first of a two-part explanation on how the concept of Infinity relates to my Laundry Basket.

The Greek philosopher Zeno of Elea had a thought. Perhaps he was in the shower at the time, perhaps he was "indisposed." ("In de what?" you may ask. "Indisposed. In de ... oh nevermind.") This is how it goes:

The Greek warrior Achilles was going to have a race with a tortoise. Because he was much faster than the tortoise, he decided to give the tortoise a head-start.

Zeno, through logical reasoning, proposed that Achilles was never able to catch up to the tortoise. The reasoning follows thus:

First, Achilles must reach the tortoise's starting point. But by the time he reaches it, the tortoise has moved to a point further on. And by the time Achilles has reached that point, the tortoise has moved even further on. By the time Achilles has reached that point, the tortoise has moved even further on, and by the time Achilles has reached that point, the tortoise has moved even further on.

Zeno said that it is obvious that the series is never-ending, and that there will always be a distance, however small, between the tortoise and Achilles.

Below is a nifty little fractal sometimes used to illustrate this paradox. Achilles has been replaced by a girl. The description on Wikipedia is:

The girl is assumed to walk three times as fast as the turtle, but whenever she turns a corner the turtle will, too. Even though she is faster, she will not see the turtle within a finite number of turns.

In order for Achilles (or the girl above) to catch up to the tortoise, they have to complete an infinite number of "travels" and Zeno went on to say that you can't complete an infinite list of Things To Do in a finite time. (Those of us who have ever looked after a husband, and/or children, and/or a house, and/or a dog, and/or paid employment didn't need a philosopher to tell us that.)

Hence, Achilles will never catch up to that darn tortoise.

Richard Morris, the author of my book, makes the point:

Of course, we all know that Achilles would catch the tortoise fairly quickly, but pointing this out does not refute Zeno's argument. Zeno is saying that Achilles must complete an infinite series of acts, and this cannot be done in a finite period of time. If we choose not to believe this, we must demonstrate where the fallacy lies.

I am certain that by now, you will have seen the obvious parallels between The Paradox of Achilles and The Tortoise and my Overflowing Laundry Basket, but for those of you who can't I will post all about it another day.

Next Up: Achilles and my Laundry Basket.