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.
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. 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. 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 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 paraphrases in his Physica: "it turns out that half the time is equal to its double."
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. 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.
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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