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.
Although we lack almost all of Zeno's work, we learn most about him through Plato, Aristotle, Proclus, and Simplicius. The majority of our sources, however, derive from Aristotle's writing on Zeno. In fact, because we have almost no primary sources, many scholars have filled in the missing gaps of Zeno's arguments with educated and very researched 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.
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. Imagine Achilles and another -- obviously slower -- runner. When the slower man starts running, Achilles then chases after him. However, by the time Achilles reaches the point where the other man presently is, the runner will have moved on to a new point. Then Achilles must run to a new point, from which the runner, again, has already moved, ad infinitum. From the traditional interpretation, Zeno wishes to discredit motion, or change, as a mere illusion in accordance with Parmenides' philosophy.
The Racetrack Paradox. Also known as the "progressive dichotomy," the racetrack paradox begins with a runner on a track with fixed starting and finish lines. Zeno argues that the runner will never reach a fixed point on the track. As the runner moves halfway towards the finish line, he must then run halfway through the second half, and he next runs half of that remainder, ad infinitum. This shows that a man may never move between fixed points and, again, supports Parmenides' view on motion and change.
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. 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 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.
Although we lack almost all of Zeno's work, we learn most about him through Plato, Aristotle, Proclus, and Simplicius. The majority of our sources, however, derive from Aristotle's writing on Zeno. In fact, because we have almost no primary sources, many scholars have filled in the missing gaps of Zeno's arguments with educated and very researched 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.
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. Imagine Achilles and another -- obviously slower -- runner. When the slower man starts running, Achilles then chases after him. However, by the time Achilles reaches the point where the other man presently is, the runner will have moved on to a new point. Then Achilles must run to a new point, from which the runner, again, has already moved, ad infinitum. From the traditional interpretation, Zeno wishes to discredit motion, or change, as a mere illusion in accordance with Parmenides' philosophy.
The Racetrack Paradox. Also known as the "progressive dichotomy," the racetrack paradox begins with a runner on a track with fixed starting and finish lines. Zeno argues that the runner will never reach a fixed point on the track. As the runner moves halfway towards the finish line, he must then run halfway through the second half, and he next runs half of that remainder, ad infinitum. This shows that a man may never move between fixed points and, again, supports Parmenides' view on motion and change.
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. 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 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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