Studying under the great Parmenides, Zeno of Elea lived from 490 to 430 BCE, writing on topics ranging from mathematics, science, and philosophy. From all academic perspectives, Zeno's significance to intellectual history lies in his contribution to and development of the concept of infinity. In fact, most consider Zeno to be the first thinker in the West to demonstrate the problems with infinity in practical application.
Although we lack almost all of Zeno's work, we learn most about him through Plato, Aristotle, Proclus, and Simplicius. The majority of our sources, however, derive from Aristotle's writing on Zeno. In fact, because we have almost no primary sources, many scholars have filled in the missing gaps of Zeno's arguments with educated and very researched guesses.
It is important that I also highlight some interpretive issues. Zeno spent a majority of his time on what is currently referred to as his "Paradoxes." Most philosophers traditionally interpret Zeno's paradoxes as supporting arguments to the monistic metaphysics of his teacher, Parmenides. Others interpreters say he meant to oppose Parmenides, while some contend he merely meant to contest the ideas of motion that were commonly held in his time. Still yet, recent researchers claim his paradoxes responded to Pythagorean philosophy.
Since scholarship finds Zeno's philosophy very problematic to interpret, and thorough contemplation of Zeno's work requires more mathematics than I am willing to write about, I will espouse here the nine paradoxes that scholars have extrapolated from Zeno's philosophy by means of the traditional interpretation when applicable.
The Achilles Paradox. Imagine Achilles and another -- obviously slower -- runner. When the slower man starts running, Achilles then chases after him. However, by the time Achilles reaches the point where the other man presently is, the runner will have moved on to a new point. Then Achilles must run to a new point, from which the runner, again, has already moved, ad infinitum. From the traditional interpretation, Zeno wishes to discredit motion, or change, as a mere illusion in accordance with Parmenides' philosophy.
The Racetrack Paradox. Scholars also refer to this as the progressive dichotomy. The paradox supposes a runner that begins a race at a fixed point, the starting line, and quickly moves to another fixed point, the finish line. However, according to Zeno, by the time he traverses half the distance of the track, the distance between start and finish, he must again traverse half the distance of the remainder, then half of the next remainder, ad infinitum. We see in yet another way how Zeno suggests motion and change is an illusion, or better yet, an impossible goal.
The Arrow Paradox. We see here another paradox on the illusion of movement. If we assume that time exists as a succession of "timeless" moments, Zeno suggests the idea of an archer. The archer will shoot an arrow, but this arrow can only take up a distance in space that equals the length of the bow. Since in every "moment" the arrow cannot move in or out of this space, because that would require time, or a new moment, then the arrow stays perpetually in some place. Since a place cannot move, the arrow itself never moves but stays stuck in a particular place. Motion is only illusory.
The Stadium or Moving Rows Paradox. Zeno here proposes a very weak paradox, at least in its assumption, but highlights a very important concept in Physics. However, this paradox will take several sentences to explain. In this paradox, he wishes to refute a commonly held belief of the time. The belief held said that a body of fixed length that traverses the fixed distance of another body will do so in the same amount of time if the former body were to traverse the second distance (or body) again.
Zeno contests this theory, proposing another paradox. Imagine a stadium where there are three equal, parallel, horizontal, and linear tracks. On track A, there is a stationary vehicle A, that rests in the center of the track; on track B, there is a vehicle B that starts from the very left of the track and moves at a constant speed, X, toward the right of the track; and on track C, there is a vehicle C that starts from the very right of the track and moves at a constant speed, X, toward the left of the track. It turns out that vehicles B and C pass one another in half the time that it takes for either vehicle B or C to pass A. He merely points out what we now consider relative velocity, but in this scenario, he stretches the analogy in attempt to state the following point that Aristotle rephrases in his Physica: "it turns out that half the time is equal to its double."
For diagrams and a similar, yet longer explanation, read this article on Zeno's Moving Rows in the Stanford Encyclopedia of Philosophy.
Limited and Unlimited Paradox. Should there exist many things in the world but only in a limited amount, in contrast to a world in which only one thing exists, we may suppose at first two things existing. Zeno would state, that for these two objects to exist, they must have distinctive characteristics that separate them, but for the objects to be separated, there must also exist a third thing, whether it be a generic thing, a space of separation, or a quality of separation. For three things to exist, then there must be a fourth... ad infinitum. In order that many things could exist in a limited amount, they must actually be unlimited as well, and this is an obvious contradiction. Zeno, as a result, concludes with Parmenides' thesis that the world is One.
In the second and final segment, I shall continue with the final four paradoxes of Zeno and consider their importance to the intellectual history of philosophy, mathematics, and science.
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