Zeno's Paradoxes, Part I - The Problematic Nature of Infinity and Motion

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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Zeno's Paradoxes, Part II - The Problematic Nature of Infinity and Motion

This article is the second and final installment of a two-part series on Zeno of Elea. In the first article, we noted the first five paradoxes on Zeno, in addition to some interpretive gray areas. I should note again that scholars debate the most appropriate interpretation of Zeno, particularly in light of our limited primary sources. For the sake of this article, I hold to the traditional interpretation of Zeno, put forth by Plato, in which Zeno's paradoxes build on and support the work of this teacher Parmenides. I will conclude by briefly considering the huge impact Zeno has made on Western Philosophy.

Large and Small Paradox. Zeno contends that if a plurality exists, then any part of a plurality will be simultaneously so small as to have no size and so large as to be infinite in size. How does Zeno come to this conclusion?

To begin, Zeno demonstrates how parts of plurality may be so small as to have no size. We must first assume parts of plurality are not pluralities themselves because we can divide pluralities into parts but parts cannot be divided any further. Parts likewise must have no size because anything with size can be divided into parts, and parts cannot be divided any further. Therefore, parts are so small that they have no size at all.

On the other hand, all parts of a plurality must be infinite in size. A plurality must have a size to be divided into parts. However, if the parts have no size then the plurality as a whole will have no size and cease to be a plurality. Therefore, each part of a plurality must have a size greater than zero. And each sub-part of every part must have a size greater than zero, and each sub-sub-part must have a size greater than zero as well, ad infinitum, making the sizes of the parts of a plurality all equal to infinity because they are infinitely divisible and can be infinitely summed.

We see in this paradox that Zeno wishes to demonstrate the problems of a metaphysics of plurality. As a result, he further adds credibility to Parmenides' monistic metaphysics.

Infinite Divisibility Paradox. Again, we will see Zeno attack metaphysics of plurality. Consider, if you will, an object that you divide in half, then further split those halves in half, and divide the remainders in half, so on and so forth, ad infinitum. If we were to ever complete this task, Zeno suggests that we would end up with the metaphysical "elements," and we could make three inferences from these elements.

First, the elements are nothing(s), and the elements "add" up to make the original object, and you cannot add a series of nothing's to make something. Therefore, the elements cannot be nothing. Secondly, the elements are something(s) yet have no size(s). Again, adding elements that have no size will equal an object of no size, and an object of no size cannot be divisible. Thirdly and finally, the elements are something(s) and have size(s) as well. If the elements have size(s), then the elements can be divided further and cease to be elements, and we are left with the original problem.

Therefore, Zeno concludes that infinite divisibility is not a possible operation because it presupposes a metaphysics of plurality. Rather, the world would appropriately be one, unified whole that cannot be divided, as Parmenides argued.

The Grain/Bushel of Wheat Paradox. Imagine a bushel of wheat falling from a table to the floor. We all agree that the bushel will make a noise when hitting the ground. However, hundreds, even thousands, of parts make up the individual grains that make up the bushel. But we do not hear a sound when one-thousandth of grain hits the floor. How is that these parts do make sounds when they are dropped, but the whole bushel makes a sound? Zeno points out here that a monastic metaphysics is more plausible than a metaphysics of plurality.

The Place(s) Paradox. We may safely assume that every single thing corresponds to its own place. So, everything that exists has a place, since a place is a thing that does exist, it will also have its own place, and that place its own place, ad infinitum. We therefore conclude that there are an infinite number of places for every single thing, which contradicts our original premise. Although this paradox does not directly support Parmenides' philosophy, it does discredit a the commonly held belief in his time, that suggested all places must have their own places.

Zeno, in his brilliance, highlighted very important concepts, namely infinity and plurality, to show their shortcomings and further reinforce his teacher's philosophy. His works on infinity long baffled mathematicians, and it was not until the introduction of calculus that mathematicians could appropriately solve some of Zeno's paradoxes. Even now, Physicists and Chemists continue to search for the most basic particles, or the "God-particle," with Zeno's presupposition that infinity is not a practical possibility.

Not only was he brilliant, but he was innovative. Instead of writing his philosophy in poetic forms as the Pre-Socratics before him, he wrote very extensively in prose, which is still the most common genre in Philosophy and Science. Aristotle sang Zeno's praises for his innovation as well, but not for his writing. In fact, Aristotle attributes to Zeno the invention of the "dialectic."

The dialectic became hugely important to later philosophers, most notable Hegel. By relying on Zeno, Hegel even justified his inherently contradictory metaphysics. Bertrand Russell too noted Zeno's significance to the academy in general, when he stated, "Zeno's arguments, in some form, have afforded ground for almost all theories of space and time and infinity which have been constructed from his time to our own."

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