[Zeno’s] second [paradox] is the so-called “Achilles”, and it amounts to this, that 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. (Aristotle, Physics VI:9)
The second argument was called “Achilles,” from the fact that Achilles was a character in it, and the argument says that it is impossible for him to overtake the tortoise when pursuing it. For it is necessary that what is to overtake, before overtaking must 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, On Aristotle’s Physics 1014:10)
Simplicius? He really didn’t live up to his name then, did he? After reading that explanation, I don’t really know much more about Zeno’s paradox, and I think one of my eyeballs may well have overtaken the other. So, here’s an attempt to work out what they were trying to say…
The godlike Achilles, Warrior-Hero and nifty runner, wakes one morning and decides to race a tortoise. (Admittedly, this is already sounding slightly unlikely). Apparently being ‘godlike’ just wasn’t enough, he needed that little bit of extra affirmation.
The tradition is not completely clear as to where he procured the tortoise (or indeed whether the tortoise was anything other than a shared sense of ‘tortoiseness’ later remembered by its associates). What is clear is that he gave it a head start. Now, if you can imagine anything more narcissistic than being semi-divine, deciding to race a tortoise, and then generously giving it a head start, well… hat’s off to you) The Testudo Graeca was to begin the race some distance ahead of Achilles, although they both set off at the same time. To complete the picture don’t forget that Achilles would have been attired after the Hellenic athletic fashion – Naked. (Totally. Nood.)
That may have a lot more to do with the outcome of this race than has been previously recognised.
Socrates gave the signal for the race to commence, and our two doughty competitors set off: Achilles somewhat faster than the tortoise. Now Achilles’ plan is to overtake the tortoise and finish the race ahead of it (thereby winning resounding glory?!?!). But in order to overtake the tortoise he must run past the point at which the tortoise started the race. Even though Achilles is moving faster than the tortoise, he finds that once he has reached the tortoise’s starting point, the crafty reptile has already moved on (a bit). So he sets out again to overtake the tortoise. Once again he needs to pass through the new point at which the tortoise set out. But again he finds that when he gets there, the tortoise has moved.
In fact, this keeps happening. What was supposed to be a quick ego boost for an insecure Greek playboy, ends up with Achilles running around in the buff all day chasing a very alarmed tortoise.
That’s Zeno’s paradox.
It’s actually not very hard to solve. But what makes it a useful thought-experiment? (Other than a spectacularly bizarre excuse for not chasing turtles)