Abstract, tabbed is an excerpt from an early paper of mine on the topic of this very argument. I was briefly interested in Zeno’s paradoxes at the time this paper was written (in my first year of school); but, alas: it’s been so long. The writing in it is sloppy, but I believe the argumentation to hold up adequately. I invite you to pick it apart. It’s been at least four years since I’ve concerned myself with Zeno, so hopefully this proves productive.
[tab]Zeno argues against the notion that many things exist. He intended for this argument to prove that only one thing could exist, and thereby discredit the widely held belief in plurality. In the following paragraphs we will examine this argument by analyzing its individual premises. Then, I will explain why I disagree with the effectiveness of the argument against plurality based on an inherent flaw in one of the principles Zeno employs. I will be analyzing Zeno’s argument through the lens of Simplicius’ commentary on Aristotle’s Physics, as it is the clearest reproduction of the original paradox.
In his famous argument against plurality, Zeno argues, quite literally, against the claim that more than one thing exists. Though the argument seems subtle, its conclusion is incompatible with the way we look at the world (which, interestingly enough, seems to have been Zeno’s intention). Zeno begins by arguing that everything that exists, physically, must have size, or we would not say that it existed. This is an easy enough premise to accept, as one can imagine an object x without size; if x was added to y, it would not increase the size of y. Therefore, x does not exist; it is nothing. This concept is further clarified in Simplicius’ statement that “if when is subtracted, the other thing is no smaller, nor is it increased when is added, clearly is nothing.” Second, Zeno proposes that whatever has size must have parts, as it can be divided. He explains that any object must have a part that protrudes from another part of the object; some parts must be in front and beside other parts. Simplicius explains that “if it exists, each thing must have some size and thickness, and part of it must be apart from the rest.” So, anything that exists must have size and parts. If you were to take a part of an object x, this part will, obviously, exist. Thus, x will also have size and parts, and if you were to take a part of it, that too would have size and parts, and so on. Here, Zeno invokes the principle of infinite regression or divisibility with respect to his claim that any object that can be said to exist must have infinitely many parts with size. Any object with infinitely many parts with size, must be infinitely large. Therefore, Zeno concludes, anything that exists must be infinitely large. Simplicius states that “if there are many things, they must be both small and large…but so large as to be unlimited.” Obviously, there cannot be more than one infinitely large thing in existence, and thus, we are left with the argument against plurality. For Zeno, to accept that an object can be infinitely divided is to accept that the object is also infinitely large, and consequently, that plurality cannot exist.
Although upon first glance Zeno’s argument may appear sound, it has an inherent flaw. Zeno’s argument against plurality works on the basis of two principles: the infinite divisibility principle (that an object can be infinitely divided into parts) and the infinite sum principle (that the sum of an infinite amount of numbers is infinitely large). Though, for the sake of argument, we will assume that any object can be infinitely divided, the sum of its divisions will not be infinitely large. If this were the case, there would be a clear contradiction in concepts. If I were to take the mouse currently sitting to the right of my keyboard, for example, and divide it infinitely (assuming, of course, that I have this ability), I would be left with an infinite amount of parts. If I were to, theoretically, reconstruct all these parts so that my mouse is once again intact, it would not be infinitely large. Unfortunately for my mouse, it would still fit under my hand, and there lies the problem for Zeno. The infinite sum principle is an inherently contradictory concept, and the reason that it seems to work within the context of Zeno’s argument is because he takes it for granted. I am apt to assume that Zeno was aware of the absurdity of this principle because he spent the better part of his argument explaining infinite divisibility, only to imply the principle of infinite sum within a single premise. Zeno did, by the looks of it, try to pull a fast one. So, although the aim of Zeno’s argument was to reduce the common sense notion of plurality to absurdity, his argument is far from sound.[/tab]
