Challenge 569 Response

Space and or Time

with Don Webb

“Living in the Moment” appears in this issue.

How does the story illustrate Zeno’s paradox of Achilles and the Tortoise?
What is the logical flaw in Zeno’s paradox?

“Living in the Moment” relies on Zeno’s paradox of Achilles and the Tortoise by assuming that time is static rather than dynamic. The narrator “burrows” into time, as he calls it, by shaving each moment into increasingly smaller intervals. As long as he does this, he can live forever even while being run over by a car.

Zeno’s “Achilles” paradox assumes that Achilles chases a moving tortoise by traversing an infinite series of increasingly smaller intervals in space approaching a limit of zero. We know where Achilles is at each interval, but only because he’s not moving; he’s right where Zeno says he is. No wonder poor Achilles can’t catch the tortoise!

The paradox is based on an attempt to reconcile data of two different orders: position and velocity. Zeno implicitly assumes that we can determine both Achilles’ average velocity or speed and his position in space at the same time. No, sorry, Zeno, Werner Heisenberg says it can’t be done.

Let’s assume that Achilles is running at an average speed faster than that of the tortoise. We see him outrun the tortoise, but we can’t say exactly where he is at any given moment.

Of course, we could film the “race” and stop the film whenever we want. Then we would know — like Zeno — exactly where Achilles and the tortoise are at that moment in the film. But we could not tell from the scene itself how fast either is moving. An uninformed casual spectator could logically speculate that the isolated frame in the film shows that the tortoise is speeding away from Achilles.

The narrator in “Living in the Moment” treats time in the same way as Zeno treats space in his famous paradox: as an infinite series of increasingly smaller intervals approaching zero as a limit.

But does the narrator live forever? No, he doesn’t, for the same reason that Achilles outruns the tortoise. The narrator is not frozen in time any more than Achilles is frozen in space. The only way the narrator can have his infinity is to stop all motion: his own, the car’s, and that of the entire universe.

In the arts, time is determined by motion in the medium. For example, musical and dramatic performances impose duration; they proceed at their own pace and last only so long. In contrast, museums, galleries and books leave duration to the viewer or reader; you can ponder over them as little or as long as you like.

Of course, just as one can reread a page in a book or revisit a painting or sculpture, one can replay a segment of a film or audio recording. You can even stop the replay at a single frame of a film or at a single set of musical notes. But what would be the point? For all practical purposes, the work would disappear, because motion — namely time — is of the essence.

We can’t live solely in space, because nothing would change; we wouldn’t even have time to get started. We can’t live solely in time, either, because we’d have no space to do it in.

Fortunately, we can outrun tortoises and keep an eye on traffic. Einstein’s discovery of the space-time continuum lets us off the hook. We might even say it came at just the right time and in the right place.