In the fifth century B.C.E., Zeno of Elea offered arguments that led to conclusions contradicting what we all know from our physical experience—that runners run, that arrows fly, and that there are many different things in the world. The arguments were paradoxes for the ancient Greek philosophers. Because many of the arguments turn crucially on the notion that space and time are infinitely divisible, Zeno was the first person to show that the concept of infinity is problematical.
In the Achilles Paradox, Achilles races to catch a slower runner—for example, a tortoise that is crawling in a line away from him. The tortoise has a head start, so if Achilles hopes to overtake it, he must run at least as far as the place where the tortoise presently is, but by the time he arrives there, it will have crawled to a new place, so then Achilles must run at least to this new place, but the tortoise meanwhile will have crawled on, and so forth. Achilles will never catch the tortoise, says Zeno. Therefore, good reasoning shows that fast runners never can catch slow ones. So much the worse for the claim that any kind of motion really occurs, Zeno says in defense of his mentor Parmenides who had argued that motion is an illusion.
Although practically no scholars today would agree with Zeno’s conclusion, we cannot escape the paradox by jumping up from our seat and chasing down a tortoise, nor by saying Zeno should have constructed a new argument in which Achilles takes better aim and runs to some other target place ahead of where the tortoise is. Because Zeno was correct in saying Achilles needs to run at least to all those places where the tortoise once was, what is required is an analysis of Zeno's own argument.
There are ten known paradoxes. In the Achilles Paradox, Zeno assumed distances and durations can be endlessly divided into (what modern mathematicians call a transfinite infinity of indivisible) parts, and he assumed there are too many of these parts for the runner to complete. Aristotle's treatment said Zeno should have assumed instead that there are only potential infinities, so that at any time the hypothetical division into parts produces only a finite number of parts, and the runner has time to complete all these parts. Aristotle's treatment became the generally accepted solution until the late 19th century. The current standard treatment or so-called "Standard Solution" implies Zeno was correct to conclude that a runner's path contains an actual infinity of parts at any time during the motion, but he was mistaken to assume this is too many parts. This treatment employs the mathematical apparatus of calculus which has proved its indispensability for the development of modern science. The article ends by exploring newer treatments of the paradoxes—and related paradoxes such as Thomson's Lamp Paradox—that were developed since the 1950s.
There are ten known paradoxes. In the Achilles Paradox, Zeno assumed distances and durations can be endlessly divided into (what modern mathematicians call a transfinite infinity of indivisible) parts, and he assumed there are too many of these parts for the runner to complete. Aristotle's treatment said Zeno should have assumed instead that there are only potential infinities, so that at any time the hypothetical division into parts produces only a finite number of parts, and the runner has time to complete all these parts. Aristotle's treatment became the generally accepted solution until the late 19th century. The current standard treatment or so-called "Standard Solution" implies Zeno was correct to conclude that a runner's path contains an actual infinity of parts at any time during the motion, but he was mistaken to assume this is too many parts. This treatment employs the mathematical apparatus of calculus which has proved its indispensability for the development of modern science. The article ends by exploring newer treatments of the paradoxes—and related paradoxes such as Thomson's Lamp Paradox—that were developed since the 1950s.
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