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.

To learn about Zeno, we must look to mostly secondary resources, as most of his primary work has been lost. We learn most about Zeno through Plato, Aristotle, Proclus, and Simplicius. However, from these four, we learn the most about Zeno through Aristotle, who comprehensively studied the innovative thinker. Our lack of primary sources have led many scholars to speculate on some of Zeno’s. In many cases, we only have the best educated guess of Zeno’s philosophy, based merely on highly researched, secondary sources.

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 differing interpretations muddy an appropriate exegesis of Zeno’s work, and the most fitting interpretation of his work should include more mathematics than I am willing to write, I will simply regard Zeno’s work through the traditional interpretation, originally put forward by Plato. Therefore, we shall now review the nine paradoxes of Zeno in light of their support of Parmenides where applicable.

The Achilles Paradox. To demonstrate that motion is an illusion, Zeno proposed the Achilles paradox. Imagine a runner who darts off and the obviously faster Achilles chases him after the runner takes a head-start. For Achilles to reach the runner, he must quickly move to a point at which the runner currently is. By the time he reaches that point, the runner has already moved to a new point. Then, Achilles must move to a new point, from which the runner has already moved, and Achilles again chases another point, ad infinitum. As we see, Zeno criticizes the concept of motion, in accordance with Parmenidean philosophy that rejects the idea of motion and change.

The Racetrack Paradox. This paradox is also named the “progressive dichotomy.” In this paradox, Zeno suggests that a runner prepares for a race. In this race, the runner begins at the starting line, a fixed point, and races toward the finish line, another fixed point. The runner must first run half of the distance between the start and finish lines. Once he has run half, he must then run half of the second half of the track, then half of that remainder, ad infinitum. Like Achilles won’t catch the runaway, the runner in this paradox will never reach the finish line. As a result, motion again seems to be paradoxical, and Zeno further supports the theories of his teacher.

The Arrow Paradox. Imagine that time exists as a sequence of “timeless” moments in space. In such a world, an archer shoots an arrow. The arrow, however, only takes up as much space as the arrow is long. So, in every moment, the arrow is taking up a space equal to its length. But in each moment, the arrow is not moving because there is no time for the arrow to move; it is stuck in a certain place (space) in each moment. Since places do not move, the arrow also never moves. We certainly see a trend here: motion is an illusion and does not exist.

The Stadium or Moving Rows Paradox. Perhaps his weakest paradox, it simply challenges a commonly held view of the time that considered passing bodies, although it unfortunately takes a number of paragraphs to explain. The general view held that if one body of fixed length moves at some constant speed past a stationary body of fixed length, then the moving body should be able to pass the stationary body again in the same amount of time.

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 a more thorough explanation of this paradox, including helpful diagrams, I encourage you to read this article on Zeno’s Moving Rows in the Stanford Encyclopedia of Philosophy.

Limited and Unlimited Paradox. Suppose there are many things in the world, but there is a fixed, or limited, amount, as opposed to just one thing in world, as Parmenides would say. If there are two things, they must be distinct from one another, but for them to be distinct, there must also be a third thing that separates them, or makes them distinct, namely a space or distance. Then for three things to exist, there must be a fourth thing… ad infinitum. So, for many things to exist, they would be both limited and unlimited, and this is impossible. Therefore, Zeno concludes, like Parmenides, there is only One Thing.

Stay tuned for the next segment, when I shall espouse the final four paradoxes and their significance to philosophical, mathematical, and scientific worlds.

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