This follows a previous article I wrote on Zeno of Elea. The first article reviewed Zeno's paradoxes and philosophical intentions. Although many debate Zeno's motives in writing his paradoxes, most scholars hold the traditional interpretation, by which they read Zeno as supporting the philosophy of his teacher Parmenides. This article, not for scholarly reasons but for simplicity, will hold Zeno's paradoxes to the traditional interpretation. After laying out the rest of the paradoxes, I will briefly note Zeno's importance not only to Western Philosophy but also to the academic world at large.
Large and Small Paradox. If there exists a plurality, Zeno argues, then each part of any plurality will be so small as to have no size and also so large as to be infinite in size. Why is this the case?
First, Zeno's states that parts of a plurality will be so small that they will have no size. In this case, we must assume that these parts of a plurality must not be pluralities themselves because if they were pluralities themselves, they would be further divisible an no longer parts. That which is not a plurality necessarily has no size, because anything possessing size will be divisible into parts. We can thus conclude that parts of a plurality must have absolutely no size at all, lest the cease to be parts.
Secondly, he claims that parts of a plurality must likewise be infinite in size. A plurality itself must have size, so that it may be divided into parts. If the parts have no size, as we saw above, then the sum of all the parts' sizes, equal to the size of the plurality, will then have no size. If we assume, in light of this premise, that parts to a plurality must have a size greater than zero, the parts themselves will be divisible into parts. Parts of a plurality will have a size greater than zero, the sub-parts will then have a size greater than zero, the sub-sub-parts will have a size greater than zero, ad infinitum. If we can infinitely divide something into parts that all have size, then the sum of all those parts will be equal to infinity.
Here we see that Zeno wishes to show how problematic pluralities are to metaphysics. In doing so, he further proves via negativa the monistic metaphysics of Parmenides.
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, we may say that the elements are nothing, and that these elements collectively make up the original object. However, adding a series of nothing's can never make something. The sum of these parts would make the original object nothing as well. And we cannot concede to object being nothing, because that would be absurd. Secondly, we may decide that the elements are something but have no size. Again, adding up elements with no size would result in an object with no size. If an object has no size then it cannot be divisible. Thirdly, we may say the elements are something and have size. However, if something has size, then it can be divided. Since elements are intrinsically something that cannot be divided, then the third inference fails. But if we end up dividing the elements, then we are left with the original problem.
Hence, infinite divisibility must not exist because this would require plurality. The world is not a plurality, according to Zeno, but rather the world must One, as we see in Parmenides.
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 monistic metaphysics is more plausible than a metaphysics of plurality.
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.
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 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."
Large and Small Paradox. If there exists a plurality, Zeno argues, then each part of any plurality will be so small as to have no size and also so large as to be infinite in size. Why is this the case?
First, Zeno's states that parts of a plurality will be so small that they will have no size. In this case, we must assume that these parts of a plurality must not be pluralities themselves because if they were pluralities themselves, they would be further divisible an no longer parts. That which is not a plurality necessarily has no size, because anything possessing size will be divisible into parts. We can thus conclude that parts of a plurality must have absolutely no size at all, lest the cease to be parts.
Secondly, he claims that parts of a plurality must likewise be infinite in size. A plurality itself must have size, so that it may be divided into parts. If the parts have no size, as we saw above, then the sum of all the parts' sizes, equal to the size of the plurality, will then have no size. If we assume, in light of this premise, that parts to a plurality must have a size greater than zero, the parts themselves will be divisible into parts. Parts of a plurality will have a size greater than zero, the sub-parts will then have a size greater than zero, the sub-sub-parts will have a size greater than zero, ad infinitum. If we can infinitely divide something into parts that all have size, then the sum of all those parts will be equal to infinity.
Here we see that Zeno wishes to show how problematic pluralities are to metaphysics. In doing so, he further proves via negativa the monistic metaphysics of Parmenides.
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, we may say that the elements are nothing, and that these elements collectively make up the original object. However, adding a series of nothing's can never make something. The sum of these parts would make the original object nothing as well. And we cannot concede to object being nothing, because that would be absurd. Secondly, we may decide that the elements are something but have no size. Again, adding up elements with no size would result in an object with no size. If an object has no size then it cannot be divisible. Thirdly, we may say the elements are something and have size. However, if something has size, then it can be divided. Since elements are intrinsically something that cannot be divided, then the third inference fails. But if we end up dividing the elements, then we are left with the original problem.
Hence, infinite divisibility must not exist because this would require plurality. The world is not a plurality, according to Zeno, but rather the world must One, as we see in Parmenides.
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 monistic metaphysics is more plausible than a metaphysics of plurality.
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.
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 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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