The Problem of Motion

The Eleatic school of philosophers was founded by the religious thinker and poet Xenophanes (born c. 570 BC), whose main teaching was that the universe is singular, eternal, and unchanging. "The all is one." According to this view, as developed by later members of the Eleatic school, the appearances of multiplicity, change, and motion are mere illusions. ...

The greatest of the Eleatic philosophers was Parmenides (born c. 539 BC). In addition to developing the theme of unchanging oneness, he is also credited with originating the use of logical argument in philosophy. His habit was to accompany each statement of belief with some kind of logical argument for why it must be so. It's possible that this was a conscious innovation, but it seems more likely that the habitual rationalization was simply a peculiar aspect of his intellect. In any case, on this basis he is regarded as the father of metaphysics, and, as such, a key contributor to the evolution of scientific thought.

Parmenides's belief in the absolute unity and constancy of reality is quite radical and abstract, even by modern standards. He maintained that the universe is literally singular and unchangeable. However, his rationalism forced him to acknowledge that appearances are to the contrary, i.e., while he flatly denied the existence of plurality and change, he admitted the appearance of these things. Nevertheless, he insisted these were mere perceptions and opinions, not to be confused with "what is". Not surprisingly, Parmenides was ridiculed for his beliefs. One of Parmenides' students was Zeno, who is best remembered for a series of arguments in which he defends the intelligibility of the Eleatic philosophy by purporting to prove, by logical means, that change (motion) and plurality are impossible.

The four paradoxes

We can't be sure how the historical Zeno intended his arguments to be taken, since none of his writings have survived. We know his ideas only indirectly through the writings of Plato, Aristotle, Simplicus, and Proclus, none of whom was exactly sympathetic to Zeno's philosophical outlook. Furthermore, we're told that Zeno's arguments were a "youthful effort", and that they were made public without his prior knowledge or consent. Also, even if we accept that his purpose was to defend the Eleatic philosophy against charges of logical inconsistency, it doesn't follow that Zeno necessarily regarded his counter-charges as convincing. It's conceivable that he intended them as satires of (what he viewed as) the fallacious arguments that had been made against Parmenides' ideas. In any case, although we cannot know for sure how Zeno himself viewed his "paradoxes", we can nevertheless examine the arguments themselves, as they've come down to us, to see if they contain - or suggest - anything of interest.

Of the 40 arguments attributed to Zeno by later writers, the four most famous are on the subject of motion:

The Dichotomy: There is no motion, because that which is moved must arrive at the middle before it arrives at the end, and so on ad infinitum.

The Achilles: The slower will never be overtaken by the quicker, for that which is pursuing must first reach the point from which that which is fleeing started, so that the slower must always be some distance ahead.

The Arrow: If everything is either at rest or moving when it occupies a space equal to itself, while the object moved is always in the instant, a moving arrow is unmoved.

The Stadium: Consider two rows of bodies, each composed of an equal number of bodies of equal size. They pass each other as they travel with equal velocity in opposite directions. Thus, half a time is equal to the whole time.