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, 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.
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.
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. Yet again, Zeno attacks any metaphysical account of plurality. Consider an object with size, and we cut this object in half, then we divide the halves in half, then those halves in half, so on and so forth, ad infinitum. If it were ever possible to complete this process, we would be left with the most basic stuff of the world: the elements. If we consider these elements, we may make three inferences.
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.
Therefore, there is no such thing as infinite indivisibility, because we must suppose a metaphysics of plurality. Zeno, in the footsteps of Parmenides, disproves plurality in order to justify a monistic metaphysics.
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. Assume that every thing has a single corresponding place. Now, everything that exists must have a place, and since a place itself exists as well, it too must have a place, ad infinitum. Therefore, there are an infinite number of places for every single thing, which contradicts the original statement. This paradox highlights the difficulty of the assumption that every place must also have place itself, a commonly held view in his Greek world. There is not enough evidence to suggest that this paradox was directly related to Parmenides' philosophy.
Zeno made very profound statements on infinity through his paradoxes. His thought was so advanced that mathematicians could not appropriately resolve some of his paradoxes until the introduction of calculus. Even chemists and physicists today buy into the paradox of infinite divisibility, as they continue to search for the most basic particles, or the "God-particle."
Furthermore, Zeno set himself apart by writing in prose as opposed to poetry, the most common genre for the Pre-Socratics before him. Aristotle also highlighted Zeno's innovation, as Aristotle attributed the invention of the philosophical "dialectic" to Zeno.
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. 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, 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.
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.
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. Yet again, Zeno attacks any metaphysical account of plurality. Consider an object with size, and we cut this object in half, then we divide the halves in half, then those halves in half, so on and so forth, ad infinitum. If it were ever possible to complete this process, we would be left with the most basic stuff of the world: the elements. If we consider these elements, we may make three inferences.
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.
Therefore, there is no such thing as infinite indivisibility, because we must suppose a metaphysics of plurality. Zeno, in the footsteps of Parmenides, disproves plurality in order to justify a monistic metaphysics.
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. Assume that every thing has a single corresponding place. Now, everything that exists must have a place, and since a place itself exists as well, it too must have a place, ad infinitum. Therefore, there are an infinite number of places for every single thing, which contradicts the original statement. This paradox highlights the difficulty of the assumption that every place must also have place itself, a commonly held view in his Greek world. There is not enough evidence to suggest that this paradox was directly related to Parmenides' philosophy.
Zeno made very profound statements on infinity through his paradoxes. His thought was so advanced that mathematicians could not appropriately resolve some of his paradoxes until the introduction of calculus. Even chemists and physicists today buy into the paradox of infinite divisibility, as they continue to search for the most basic particles, or the "God-particle."
Furthermore, Zeno set himself apart by writing in prose as opposed to poetry, the most common genre for the Pre-Socratics before him. Aristotle also highlighted Zeno's innovation, as Aristotle attributed the invention of the philosophical "dialectic" to Zeno.
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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