Say you started in the middle of a room, and you walked to the wall. First you walked half way, and then half of that, and then half again, and so on. So how did you ever reach the wall? An ancient paradox of motion - after this on Earth and Sky.
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DB: This is Earth and Sky, on a paradox concerning motion posed by the ancient philosopher Zeno. Say you walked from the middle of a room to the wall. You can imagine your journey - as Zeno did -- as having been a series of steps. Dr. David Harbater of the University of Pennsylvania explains Zeno's paradox :
(Tape 0:12:43-0:13:10) Before you can get to the wall you have to walk halfway to the wall. Once you do that you're still not at the wall, so you have to go halfway again of what remains. Then you're still not at the wall so you have to get to the wall, but before you do that you have to go halfway of what remains again, and so forth. Which means that before you get there, there's infinitely many things that have to happen -- more things that have to happen than you have time for. It sounds like it would go on forever and as a result you would never reach the wall.
DB: Of course, you've already reached the wall, so you know it's possible. Here's Zeno's flaw:
(Tape 0:15:51-0:16:13) The flaw is that even though he can imagine that there are infinitely many things you have to do before you reach the wall, those things are shorter and shorter. And so what happens is that even though there are infinitely many of them, they add up to a finite amount, and that finite amount is how long it takes you to reach the wall. (Tape 0:17:03-0:17:18)
And in fact, when Newton was developing calculus in order to understand gravity -- how an apple falls and how the planets move -- he was in fact doing this very same thing of breaking things up into these small pieces, seeing how they fit together and then being able to predict the motion of objects.
DB: Thanks to Dr. David Harbater for speaking with us. And with thanks to the National Science Foundation, I'm Deborah Byrd, for Joel Block, for Earth and Sky.
Author: Beverly Wachtel
Thanks to the following individual who aided in the preparation of this script:
Dr. David Harbater
Department of Mathematics
University of Pennsylvania
Philadelphia, PA
If you enjoyed this program, you may be interested in the following:
Famous Paradoxes:
http://forum.swarthmore.edu/~isaac/problems/paradox.html
Zeno's Paradox:
http://www.seanet.com/~ksbrown/kmath440.htm
Mathematical Sciences Research Institute: