Zeno's Paradox - in fractions?

One of Zeno's most famous paradoxes concerns Achilles and the tortoise.

Achilles and the tortoise are going to run a race. Achilles, being confident of victory, gives the tortoise a head start. Zeno supposedly proves that Achilles can never overtake the tortoise.

Before Achilles can overtake the tortoise, he must first run to point A, where the tortoise started. But then the tortoise has crawled to point B. Now Achilles must run to point B. But the tortoise has gone to point C, etc. Achilles is stuck in a situation in which he gets closer and closer to the tortoise, but never catches him. 




What Zeno is doing here, and in one of his other paradoxes, is to divide Achilles' journey into an infinite number of pieces. This is certainly permissible, as any line segment can be divided into an infinite number of points or line segments. This, in effect, divides Achilles' run into an infinite number of tasks. He must pass point A, then B, then C, etc. And what Zeno is arguing is that you can't do an infinite number of tasks in a finite amount of time.

But...why not?

Zeno says that you can divide a line into an infinite number of pieces. And then he says that you cannot divide a time interval into an infinite number of pieces. This is inconsistent. There is no paradox here. Zeno was just showing (pretending?) some ignorance of the nature of time. A time interval is just another line segment (when you graph it), that you can divide up in any way you want.

Read more...

(The paradox in its original form is as follows) 

The [second] argument was called “Achilles,” accordingly, from the fact that Achilles was taken [as a character] in it, and the argument says that it is impossible for him to overtake the tortoise when pursuing it. For in fact it is necessary that what is to overtake [something], before overtaking [it], first reach the limit from which what is fleeing set forth. In [the time in] which what is pursuing arrives at this, what is fleeing will advance a certain interval, even if it is less than that which what is pursuing advanced … . And in the time again in which what is pursuing will traverse this [interval] which what is fleeing advanced, in this time again what is fleeing will traverse some amount … . And thus in every time in which what is pursuing will traverse the [interval] which what is fleeing, being slower, has already advanced, what is fleeing will also advance some amount. (Simplicius(b) On Aristotle's Physics, 1014.10)

The Stanford Encyclopedia of Philosophy goes into greater detail...

There are plenty of sites - and blogs - that analyse Zeno's paradox using fractions.

By now you're probably asking whether this has got anything to do with the forthcoming plays being put on at The Dogstar, in Brixton, next week. (See poster at the side of this blog).

Bear with me.

Let me introduce you to Thompson's Lamp.

Consider a lamp, with a switch. Hit the switch once, it turns it on. Hit it again, it turns it off. Let us imagine there is a being with supernatural powers who likes to play with this lamp as follows. First, he turns it on. At the end of one minute, he turns it off. At the end of half a minute, he turns it on again. At the end of a quarter of a minute, he turns it off. In one eighth of a minute, he turns it on again. And so on, hitting the switch each time after waiting exactly one-half the time he waited before hitting it the last time. All these infinitely many time intervals add up to exactly two minutes.

In the play 'Wonder' (which is one of three plays that are being performed next week) there is a mention of Zeno's paradox. Certainly a lot less is said than is said about Fractional Reserve Banking. There is also a case of the lights being turned off.

To find out why you'll just have to turn up.

All tickets: £5 (of which 50p goes to The MS Society)
(www.mssociety.org.uk)

7.30pm on Tues 20^th, Wed 21^st and Thurs 22^nd November 2012.

At The Dogstar, 389 Coldharbour Lane, London, SW9 8LQ.
(Nearest tube: Brixton).