Zeno of Elea (490-430 BCE) studied under and remained very loyal to his great teacher, Parmenides, who also was from Elea. Scholars credit Zeno as being the first person in Western history to evidence 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.
Before I being, I must elaborate on some interpretive problems with Zeno. Zeno wrote most extensively on what Philosophers now call "Zeno's Paradoxes." Plato, and most subsequent philosophers, interpret Zeno as a loyal student of Parmenides and see Zeno's work as an exposition and reinforcement of his teacher's work. Scholars have deemed this the most common, traditional interpretation. However, other interpretations do abide in academia today, seeing Zeno's work as contestations of commonly held views in his Greek world or reading Zeno's work as a minor critique of Parmenides in some cases. Some others have more recently interpreted Zeno's work as his criticism of 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. 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. 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. 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 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 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. 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 next installment, I will review the final four paradoxes of Zeno and his influence on not only the philosophical world but the scientific and mathematical worlds as well.
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.
Before I being, I must elaborate on some interpretive problems with Zeno. Zeno wrote most extensively on what Philosophers now call "Zeno's Paradoxes." Plato, and most subsequent philosophers, interpret Zeno as a loyal student of Parmenides and see Zeno's work as an exposition and reinforcement of his teacher's work. Scholars have deemed this the most common, traditional interpretation. However, other interpretations do abide in academia today, seeing Zeno's work as contestations of commonly held views in his Greek world or reading Zeno's work as a minor critique of Parmenides in some cases. Some others have more recently interpreted Zeno's work as his criticism of 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. 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. 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. 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 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 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. 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 next installment, I will review the final four paradoxes of Zeno and his influence on not only the philosophical world but the scientific and mathematical worlds as well.
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