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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