The Paradoxes of Zeno of Elea
A Philosophical Examination of the Infinite
Zeno of Elea may have been an iconoclast, but he posed several interesting problems that would baffle thinkers for generations to come.
Very little is known of the actual life of Zeno of Elea (not to be confused with the Stoic philosopher Zeno of Citium), apart from his famously frustrating paradoxes. He was a Greek Philosopher – Pre-Socrates, even – and seemed to delight in demonstrating that the world seemed to be full of intrinsic inconsistencies.
Zeno was a contemporary of Aristotle, which puts his lifetime somewhere in the 5th century BC, and it is from Aristotle that we get most of our relevant information about this character, whose personal writings have fortunately been long since lost, somewhere in the annals of time.
An Examination of the Paradoxes
Zeno’s paradoxes, in essence, were a not-so-subtle attempt to disprove the very existence of motion. According to these rather simple little explanations, the very idea that something can move from one place to another seems to be impossible.
While these paradoxes may not be entirely effective today in convincing one of the non-existence of motion, they may be at least effective in serving to liven up a dying conversation, or to ponder while passing time waiting in long lines.
Eight of Zeno’s paradoxes have survived the passage of time in one form or another, though most of them are rather uninteresting and repetitive. These are paraphrased versions of three of the most famous of them:
1) The Paradox of Achilles and the Tortoise
Simply put, the paradox goes like this: Suppose a quick person (Zeno used Achilles as his example) and a slow person (a tortoise) joined in a footrace. Achilles, knowing that he was much faster, gave the tortoise a head start of 20 feet. As Zeno figured it, when the race begins, it would take Achilles only seconds to reach the tortoise's starting point, but by this time the tortoise would have already moved a few feet forward himself. It would take still another second or two for Achilles to catch up to the tortoise’s new location, but by that point the tortoise would have already moved still again, and so on, forever. In Zeno's mind, this argument proves that the quick runner can logically never catch up with the slow runner, as the argument can continue on like this infinitely. And therefore, logically speaking, motion can't possibly exist, except in one's own mind.
2) The Dichotomy Paradox
The most scientifically titled of Zeno's paradoxes, this one is much simpler than the first, yet even harder to refute. Imagine a person trying to run from one place to another. Perhaps they are getting up off the couch to answer your phone, which is sitting 20 feet away on a table, ringing loudly. Zeno would argue that one cannot possibly make it this entire distance, and here is why: In order to travel the 20 feet to answer the phone, one would first have to travel half that distance; that is, 10 feet. In order to travel those 10 feet, however, one would first have to travel half of that distance, which is five feet. The only possible way one can ever travel five feet, however, is by first traveling half that distance... etc. Once again, the argument continues on into infinity, causing one to need some serious headache medication if they think of it for too long. In the end, though, Zeno simply summarizes by stating once again that no distance can actually ever be traveled, except in our own minds.
3) The Arrow Paradox
Perhaps the cleverest of Zeno’s paradoxes, the Arrow Paradox postulates that an arrow in flight is the perfect example of the non-existence of motion. According to Zeno, at any point during the arrow's flight, there are only two possible locations: Where the arrow is, and where the arrow isn't. At any given moment, the arrow's location can only be where the arrow is, and there is no possibility of it ever moving to where the arrow isn't, and thus, the arrow never moves to a different place, therefore it is always perfectly still, therefore motion as a whole is impossible, therefore philosophers everywhere are given something to discuss amongst themselves for the next few centuries.
Consequences of the Paradoxes
The paradoxes of Zeno have truly given generation upon generation of philosophers, scientists and mathematicians something to think about. After all, it was clear just from looking at the world that they could not possibly be true (though Zeno might have argued that what we see is all in our minds), but actually figuring out why is another story altogether.
At this point in history, however, mathematicians have found both very interesting and very complicated ways of finally disproving each of these paradoxes using calculus and other mathematical tools. They remain, however, interesting ways to look at difficult concepts as that of “infinity” in an entirely new light.
References:
“Modernity of Zeno’s Paradoxes.”
“Zeno’s Paradoxes.” Stanford Encyclopedia of Philosophy.
