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.
Unfortunately, we have been left with very little of Zeno’s original work. Plato, Aristotle, Proclus, and Simplicius wrote very much on Zeno’s work, and it is from these thinkers that we derive most of our information on him. Aristotle, however, wrote the most extensively on Zeno. Our lack of primary resources have forced scholars to interpret Zeno through secondary resources and speculate on some of his original arguments. In many cases, scholars leave us only educated 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. 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. 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. 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. Unfortunately, this is Zeno’s weakest, and perhaps seemingly his silliest, paradox. Even more unfortunately, it will take the longest to explain. With this paradox he wishes to discredit a commonly held belief in his day regarding motion and time. Consider one object of fixed length will pass another object of fixed length. Most believed that if the object were to turn around and traverse the latter object again, it would take the same amount of time to traverse the object on the second run as it did on the first.
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 better explanation with diagrams, you should visit the 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.
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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