This article concludes a two-part series on Zeno of Elea. In the previous article, I discussed Zeno’s paradoxes and his philosophical agenda, which most scholars claim supported the metaphysics of his teacher, Parmenides. I should note that many scholars do debate his agenda; however, for the sake of this article, we will assume the traditional interpretation to easily lay out the remainder of his nine paradoxes. The last of his paradoxes will be followed by the significance of his thought in the history of philosophy and other areas of academia.

Large and Small Paradox. In this paradox, Zeno considers the nature of a plurality. He states that parts of plurality will not only be so small as to have size but also so large that they will be infinite in size. How might Zeno support such a contradictory position?

First, parts of plurality must be so small that they have no size. These parts of a plurality must not be pluralities themselves. However, that which is not a plurality cannot have a size because anything having a size is also divisible into parts. So, parts of plurality must have no size.

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. Imagine an object that you divide in half, then divide each half in half, and divide the resulting halves in half, ad infinitum. Assuming you can reach the end of this process, you will reach the metaphysical “elements,” from which we could infer three things.

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 of Wheat Paradox. Suppose you drop a bushel of wheat on the ground, and it makes a sound. However, the bushel is comprised of many individual grains, and each individual grains is comprised of hundreds, thousands, or even millions of parts. However, we hear no sound when one-millionth of a grain hits the ground. How is it that the millions of parts of grains do not makes sounds when dropped while the whole grains and entire bushel makes a sound? Again, this leans toward a monistic metaphysics of Parmenides.

The Place(s) Paradox. It is a sensible proposition when we say that every single thing has a corresponding place. However, we may also say that a place is also a thing and must have its own place, and that place has its own place, so on and so forth, ad infinitum. Therefore, every single thing has an infinite number of places which is a contradiction to the original statement. This paradox does not directly support Parmenides, but many scholars believe he is criticizing a popular belief in his day that all places must have corresponding 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.

Zeno also stands apart for his writing, as he chose to write in prose as to the traditional poetry of the Pre-Socratics. Aristotle also praised Zeno for his innovation, as Aristotle credited Zeno with inventing the “dialectic.”

The dialectic still remains an important topic today, but was most extensively examined by Hegel. In fact, Hegel justified his intrinsically paradoxical metaphysics by citing the paradoxes of Zeno. Not only did Hegel see Zeno’s brilliance and innovation, but Bertrand Russell sums up Zeno’s philosophy most appropriately, when he said, “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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