Zeno of Elea, a philosopher who lived from 490 to 430 BCE, studied under the famous Parmenides, another philosopher also from Elea. Zeno contributed much to history of western philosophy in addition to also providing a lot of insight to our modern fields of mathematics and science. Zeno’s importance rests in his comprehensive writing on motion and change but particularly on the concept of infinity. While most philosophers before him presupposed a notion of infinity without any hesitation, Zeno was the first in Western history to really expose the problematic nature of infinity.

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.

I must also note that many debate Zeno’s intentions in writing so comprehensively on what is now known as “Zeno’s Paradoxes.” Traditionally, most agree that Zeno attempted to build upon Parmenides work. However, some suggest he sought to discredit Parmenides’ work; others claim he criticized the traditionally-held Greek views on motion; and more recently, interpreters propound that he was combatting Pythagorean thinkers.

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. Let us suppose that Achilles chases after another runner. As the runner starts out, Achilles then follows him shortly thereafter. First, Achilles runs toward a spot where the runner is. However, by the time Achilles reaches that spot, theoretically, the runner will have dashed to a new spot. Achilles naturally runs to the next spot, but the runner has spurred forward again… ad infinitum. Here, Zeno shows the deficiencies in the idea of motion, or change. This coincides with Parmenides’ philosophy in which motion is an illusion and does not exist.

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. My personal favorite is the Arrow paradox. Consider that times exists as a series of successive and “timeless” moments. If an archer were to shoot an arrow, the arrow would only take up as much space as it is long in each moment. The arrow is fixed to that position in each moment because to move in or out of the position would require time, or a new moment. Therefore, the arrow must always be contained in a particular place in each moment. And since places do not move, the arrow itself never moves. The arrow only “appears” to move, and as a result, motion is illusory yet again.

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 proposes a counter example. There exists a stadium that houses three parallel and linear tracks of equal length. Track A has a vehicle A that sits in the middle of the track; track B has vehicle B that starts from the very left of the track moving at constant speed X towards the right of the track; and track C has vehicle C that begins at the very right of the track moving toward the left of the track at constant speed X. As it turns out, vehicles B and C pass one another (or traverse each other’s fixed length bodies) in half the time that it takes either vehicle B or C to traverse vehicle A. Here, he considers the modern notion of relative velocity in Physics, and the scenario, in a twisted way, supports Aristotle’s description 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. This paradox challenges a metaphysical account of plurality, which Parmenides also opposed. Imagine there are many things that exist in the world but only a fixed number of things exist, or the number of thing in the world are limited to some number. If we start with two objects, then there must be something that separates these objects and makes them distinctive from one another. As a result, some third “thing” must exist to separate them, whether it be some space or quality. Then, if there are three things in the world, there must also be a fourth… ad infinitum. In this paradox, for a limited number of things to exist in the world, the number of things must also be unlimited, which is an apparent contradiction. Zeno thus supports Parmenides’ monistic metaphysics.

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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