Zeno’s paradox of motion – part 1 (Achilles and the tortoise)
The most famous of Zeno’s paradoxes, and also the one with amusing historical examples: Zeno’s paradox of motion. In one version of the paradox Zeno proposes that there is no such thing as motion. There are many variations, and Aristotle recounts four of them, though essentially one can call them variations of two paradoxes of motion. One concerning time and the other space. For today, let us focus on space and recount the Achilles and tortoise paradox:
“in a race the quickest runner can never overtake the slowest, since the pursuer must first reach the point whence the pursued started, so that the slower must always hold a lead”; and “the non-existence of motion on the ground that that which is in locomotion must arrive at the half-way stage before it arrives at the goal” (Aristotle’s Physics, Book VI.9).
Though intuitively illogical, there is some sense in this. Right?
The description is more complicated than the paradox actually is. Zeno’s point is simply that space is divisible, and because it is divisible one cannot reach a specific point in space when another has moved from that point further. Let us take an example:
The distance is 1000x (where x is the measure unit for distance – mile, meter, whatever).
Let us also say that Achilles give 100x head start to the tortoise.
Whenever Achilles reaches 100x in time t1, the tortoise would have moved further (for instance, 150x).
Thus, when Achilles reaches 150x in time t2, the tortoise would have moved even further to 175x.
Zeno’s point is that given these conditions, Achilles cannot catch up with the tortoise because space can be infinitely divided into smaller units still – where the tortoise will always be a fraction of space ahead.
There are by now numerous solutions to this ‘paradox’, (some have written a 272 pages long [amazon text=book on Zeno’s paradox&asin=0452289173]). Then again, as Aristotle pointed out already, this is not really a paradox, but poor physics. Anyone with high-school level of physics will see the problem: both Achilles and tortoise will have stopped moving as such at some point in time.
Perhaps by today’s standards we can say that this paradox is a challenge to conventional physics. But what if that is Zeno’s point with the paradox? – A challenge to all of Ancient Greek thought that everything is in motion, always – a challenge to Heraclitus ‘everything is flux’ view pointed out in the Ship of Theseus paradox (fragment DK B12).
For this, it is best to look at Zeno’s point on motion in relation to time.
This is a 4-part series on Zeno’s paradox of motion.
- Part 1: Zeno’s paradox of motion in relation to bisection of space (Achilles and the tortoise)
- Part 2: Zeno’s paradox of motion in relation to bisection of time (Flying arrow is at rest)
- Part 3: Further applications of Zeno’s paradox: The Ross-Littlewood paradox
- Part 4: Further applications of Zeno’s paradox: Thomson’s lamp