Was Zeno unfair toward Achilles in his paradox?
Last week I was reading the Croatian edition of Bryan Magee’s “The Story of Philosophy” and he reminded me of Zeno’s famous “Achilles and the tortoise” paradox.
Here is how the paradox goes (taken from Wikipedia):
“In the paradox of Achilles and the Tortoise, Achilles is in a footrace with the tortoise. Achilles allows the tortoise a head start of 100 meters. If we suppose that each racer starts running at some constant speed (here instead of ‘one very fast and one very slow’ I would stick to the original: Achilles is twice time faster than the Tortoise), then after some finite time, Achilles will have run 100 meters, bringing him to the tortoise's starting point. During this time, the tortoise has run a much shorter distance, 50 meters. It will then take Achilles some further time to run that distance, by which time the tortoise will have advanced farther; and then more time still to reach this third point, while the tortoise moves ahead. Thus, whenever Achilles reaches somewhere the tortoise has been, he still has farther to go. Therefore, because there are an infinite number of points Achilles must reach where the tortoise has already been, he can never overtake the tortoise.”
Now, I think that the paradox exists only because Zeno is manipulating with the time-component of his paradox.
First let’s clarify what Zeno means when he says that Achilles is twice time faster than the Tortoise. If, for example, Achilles runs 100 meter in 10 seconds the Tortoise who is twice time slower than Achilles will run only 50 meters. It is cleat that we cannot talk about the speed without including the time component (speed = distance/time).
Zeno says that when Achilles reaches the Tortoise's starting point, the Tortoise will be run only 50 meters. This all has happened during certain amount of time (with proposed figurers, after 10 seconds).
Next, Zeno says “It will then take Achilles some further time to run that distance, by which time the tortoise will have advanced farther”. With the proposed figures that mean that when Achilles runs 150 meters the Tortoise will only reach 75 meters (but it is steal 25 meters ahead because of the head start of 100 meters). However, the second sequence lasts only 5 seconds – if it would last full 10 seconds than Achilles and the Tortoise would be side by side.
In the third sequence Zeno executes another unmentioned cutting of time – this time he “stops” the race when Achilles reaches 175 meters, while the Tortoise only reaches 87,5 meters. That sequence lasts just 2,5 seconds.
Just for the fun let’s go further:
4th sequence: A= 187,5 meters / T= 93,75 meters / time= 1,25 seconds
5th sequence: A= 193,75 meters / T = 96,875 meters / time= 0,625 seconds
In the last sequence it is like Zeno gives the signal to start the race – and then instantly (after just 0,655 seconds) shouts: ”Stop! Stop! Achilles, please come back. I now you couldn’t stop on time, but the next sequence starts from this point here (with his finger he shows a spot 193,75 meters from the starting point). Now, we can start the 6th sequence…”, which will last only 0,3125 if we wants to prevent Achilles to win the race. [It sounds like Monty Python’s comedy, isn’t it :)]
So, my conclusion is that Achilles was never given enough time to win the race, or that Zeno was (consciously or not) manipulating “behind the curtains” with the time component of his paradox.
I am aware that this paradox is not the only one, but at least here we do not have to use complex mathematics to prove that the sum of 1/2 + 1/4 + 1/8 + 1/16... is still 1.
What do you think?
Slavonski Brod, CROATIA
P.S. Sorry for my clumsy English