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.
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.
Differing interpretive opinions, as we see on many other texts, do persist today as to how we may appropriately read Zeno. In a similar vein, the most fair interpretation would include much more mathematical examination than I am willing to provide. In light of these two statements, I will simply lay out Zeno's nine paradoxes according to the traditional interpretation put forward by Plato.
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. 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 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.
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.
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.
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.
Differing interpretive opinions, as we see on many other texts, do persist today as to how we may appropriately read Zeno. In a similar vein, the most fair interpretation would include much more mathematical examination than I am willing to provide. In light of these two statements, I will simply lay out Zeno's nine paradoxes according to the traditional interpretation put forward by Plato.
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. 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 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.
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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