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.
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 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. 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. 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. Perhaps his weakest paradox, it simply challenges a commonly held view of the time that considered passing bodies, although it unfortunately takes a number of paragraphs to explain. The general view held that if one body of fixed length moves at some constant speed past a stationary body of fixed length, then the moving body should be able to pass the stationary body again in the same amount of time.
As a result, Zeno gives us another racing scenario. Imagine three equal, parallel, and linear racetracks, where the A track has a stationary vehicle placed in the middle, a B track has a vehicle moving left from the right end of the track, and the C track has a vehicle moving right from the left end of the track. The A vehicle is stationary while the B and C vehicles move toward one another at a constant and equal speed. The B and C vehicles pass one another in half the time it takes for either of them to pass the A vehicle. So, in a very convoluted sense, "it turns out that half the time is equal to its double, Aristotle notes in his Physica.
For diagrams and a similar, yet longer explanation, 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.
Stay tuned for the next segment, when I shall espouse the final four paradoxes and their significance to philosophical, mathematical, and scientific worlds.
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 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. 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. 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. Perhaps his weakest paradox, it simply challenges a commonly held view of the time that considered passing bodies, although it unfortunately takes a number of paragraphs to explain. The general view held that if one body of fixed length moves at some constant speed past a stationary body of fixed length, then the moving body should be able to pass the stationary body again in the same amount of time.
As a result, Zeno gives us another racing scenario. Imagine three equal, parallel, and linear racetracks, where the A track has a stationary vehicle placed in the middle, a B track has a vehicle moving left from the right end of the track, and the C track has a vehicle moving right from the left end of the track. The A vehicle is stationary while the B and C vehicles move toward one another at a constant and equal speed. The B and C vehicles pass one another in half the time it takes for either of them to pass the A vehicle. So, in a very convoluted sense, "it turns out that half the time is equal to its double, Aristotle notes in his Physica.
For diagrams and a similar, yet longer explanation, 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.
Stay tuned for the next segment, when I shall espouse the final four paradoxes and their significance to philosophical, mathematical, and scientific worlds.
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I look forward to reading the subsequent posts!
ReplyDeleteI wrote something about Zeno a while ago, it can be seen here http://bit.ly/i1Hia6 (and the Review of Metaphysics paper downloaded).
Also more broadly I have written about a general paradox (http://bit.ly/dFQ0IG) which is a sort of challenge to all those who can reason beyond the prejudices of scientific assumptions and presumptions about the observed reality.
When I formulated this argument about the nature and limits of our knowledge of "reality" in 1996, whilst I was working on Zeno's paradoxes of movement (see above), I called it the "paradox of phenomenal observation". Like most of the so called paradoxes in our philosophical tradition, including those of movement and of measurement in quantum physics, it is born out of the structure of our language and logic, a language that can only name what is identical with itself and thus gives self-identical names to processes that are supposed to be dynamic and as such descriptions of changing phenomena.
It can be considered the father of all paradoxes. If you understand it, it will give you a sort of dizziness, like that experienced when you look at an image which you believe to be just what it seems at a first glance, but that suddenly reveals itself, as you transfix your glare on it, to be a completely different picture (e.g. a hidden apparently 3D image), a picture that you feel you are about to lose very easily as you shift from the old picture to the new one.
Applied to our traditional picture of reality such experience is particularly unsettling.
Good luck to all those who are brave enough and disenchanted enough to understand it, and so to let this new picture emerge from the ashes of the old image of reality and of real events as they are described by our scientific model of reality, which the paradox of phenomenal observation relentlessly destroys before our very eyes.