Zeno's paradox: A notorious proof that all motion is impossible.
This paradox by Zeno of Elea, a Greek metaphysician of the 5th century B.C., is the oldest historically documented analytical investigation of the concept of infinity. Either Zeno himself did not publish anything or whatever he published has been lost. Hence, what we know of his paradoxes is from their mention in the works of other philosophers, particularly Aristotle.
A runner plans to travel a given distance D. However, to travel any distance he must first cover half of that distance D/2. Suppose he does that. He now has to cover half of the remaining route, D/4. And this continues ad infinitum. Though the runner is forever approaching his goal, he is always short of it, because there will always be an infinite number of minuscule distances in front of him. Since we can apply this result to any distance and any movement, it follows that all movements are impossible.
There is a slightly better-known version of this paradox, namely the imagined footrace between Achilles and the tortoise. Interestingly, Zeno was not bothered by the fact that his abstract results were in obvious contradiction to everyday experience. He shared with other Greek philosophers of his time the view that, in such a case, something had to be wrong with our experience. The fact that we seem to experience motion all the time is, by itself, a lame objection (although invariably the first one a person will make when the paradox is explained to him or her); if the proof conclusively establishes that motion is impossible, as it claims to, all that follows from the fact that we experience motion is that our experiences are illusions, a conclusion Zeno and his fellow Eleatics were happy to draw.
Over the centuries, Zeno's reasoning has enticed countless philosophers, mathematicians, and scientists to attempt to identify the hidden flaw nearly everyone instinctively thinks it most contain. The first and simplest objection was due to Diogenes: After listening to the story, he silently walked once all around his domicile. (As explained above, this objection is lame, even if the walking wasn't.) Much later, the arithmetic of number series developed by ►John Wallis and other mathematicians in the 17th century A.D. showed that we can cover even an infinite number of distances in a finite amount of time. The ►infinite sum D/2 + D/4 + D/8 + D/16 +... has the finite value D, so that Zeno's runner will reach his goal despite the infinite number of distances he has to cover. Finally, the paradox can also be challenged by denying that distances are in fact infinitely divisible. Indeed, the ►indeterminacy principle has shown that they are not.
Yet all these objections, with which some textbooks rashly dismiss Zeno's paradox, miss their target. For the paradox essentially is not a paradox of movement. Rather, it is a paradox concerning our concept of infinity: to reach the end of a route we have to cover the infinitely many subsections of which the route consists. But how can that be if there are "endless" subsections — if, in other words, after every subsection there is a new subsection?
Thus we see both Diogenes's empirical "rebuttal" and John Wallis's mathematical argument wrestling with our intuitive conception of infinity, rather than with Zeno's paradox. It was this contradiction that convinced Aristotle to reject actual infinity. We have already encountered the difference between potential and actual infinity in many contexts in this dictionary, but it is here that it originates. According to Aristotle, there is no actually endless magnitude. There is only potential infinity in the sense that we can always add something to something or that we can always continue dividing something up. But an actually infinite object such as a distance consisting of infinitely many actual subsections is not humanly conceivable. It results in unsolvable paradoxes. It must not exist. And with that Zeno's paradox is still at the basis of all the difficulties and contradictions that beset our concept of infinity.