THE QUESTION: a Contemporary Version of One of Zeno S Paradoxes

THE QUESTION: A Contemporary Version of one of Zeno’s Paradoxes

You can never leave any room you are in, because before you can get to the door, you first have to go halfway to the door. But, before you get halfway to the door you must first get halfway to that place (that is, one fourth of the way to the door). But, there are an infinite number of such halfway places and you do not have time to go to an infinite number of places, so you can never get out of any room — or move anywhere. Right?

THE ANSWER

The ancient Greek philosopher Zeno, a disciple of Parmenides, lived in the city of

Elea in what is now southern Italy during the Fifth Century B. C. Zeno’s paradoxes, which purport to demonstrate that motion is impossible, have titillated the imagination for 2,500 years. Using the contemporary version of one of the paradoxes given above, and a little careful thinking, this essay will identify and explicate the flaws in Zeno’s reasoning.

The key to the paradox is found in its very first sentence: “You can never leave any room you are in, because before you can get to the door, you first have to go halfway to the door.” In saying that I, or any other person, must first go halfway to the door, the author of the paradox shows that he conceives of the distance between the door and me as a series of discreet spaces. Before I can pass on to a subsequent space within the series, I must “first” traverse the one before it. The author of the paradox goes on to reason that, since the distance between the door and me is infinitely divisible, this distance must be composed of an infinite series of discreet spaces. Thus, says the paradox’s author, since I cannot pass through an infinite series of discreet spaces in a finite amount of time, I cannot move to the door. Since what applies to the distance between the door and me applies to every other distance, I cannot move at all.

The fundamental error of the paradox is to conceive of the distance between the door and me as a series of discreet spaces. It is not such a series; rather, it is a continuum. Within the distance between the door and me, there is not first space No.1, then space No. 2, then space No. 3, ad infinitum, with each space in the distance separate and distinct from all of the others. Each of the “spaces” in the distance “bleeds” into the next, much like each color on a color chart “bleeds” into the next. As a result, the distance between the door and me is not a series of discreet spaces, but one total space, and a finite one at that. As a result, I can traverse the distance between the door and me in a finite time.

What led the author of the present paradox, and Zeno before him, to believe that the distance between the door and me is composed of an infinite series of discreet spaces is the fact that the one total, finite space between the door and me can, conceptually, be infinitely divided. To infer from this fact, however, that the one total, finite space actually is so divided, rendering an infinite series of discreet spaces, is bad logic. An analogy will illustrate why this inference is bad. By using fractions of cents, conceptually, I can divide ten dollars into an infinite number of smaller amounts. This fact, however, does not mean that the ten dollars “contains” an infinite number of values.

No matter how many times I divide the ten dollars, its value remains the same and finite.

As the old saying goes, you can’t make the pie bigger just by cutting it into smaller pieces. This analogy demonstrates that, although a finite distance, e.g. the distance between the door and me, can, conceptually, be infinitely divided, it always remains a finite distance. Since the distance between the door and me is finite, I can traverse it in a finite amount of time. Thus, motion from me to the door, and between all other sorts of things is possible and, indeed, happens all the time.

As stated at the beginning of this essay, Zeno’s paradoxes have titillated the imagination for centuries. With a little careful thinking, and common sense, however, one can easily see the flaws in Zeno’s reasoning.