A new twist on Zeno’s Arrow Paradox
Zeno of Elea’s arrow paradox, like some of his other extant paradoxes, aims to show that our observations of change and becoming in the world are illusions. Our senses suggest that there are all kinds of change and motion of things around us, but our reason concludes otherwise. As good philosophers, we should listen to the voice of our reason, rather than the evidence of our senses, and reject the reality of motion and change in the world.
Here’s how the Arrow Paradox is supposed to help show the unreality of motion. (What follows is a common reconstruction of Zeno’s argument.) Consider an object like an arrow which our visual experience describes as moving in its trajectory in the air. Zeno claims that at every instant of its supposed flight, the arrow occupies a region of space exactly coinciding the size and shape of the arrow. But if an object occupies a region of space coinciding with the size and shape of the object, then the object must be at rest. The arrow at every instant during its supposed flight, therefore, is at rest; it is at no moment in that time interval in motion. So, contrary to the judgment of our senses, motion is impossible.
A popular solution to Zeno’s Arrow Paradox is Russell’s “at-at theory of motion.” According to Russell, an object cannot be in motion (nor can it be at rest) at an instant. To be in motion is to be at different locations at different times. (And to be at rest during an interval of time is to occupy the same location at every instant of that time interval.) Location of an object at a single instant does not tell us anything about its kinematic status.
Let us take, as Zeno seems to be doing, an instant or moment of time to be a time magnitude of zero duration. Call the space region occupied by the arrow at instant t the location L of the arrow at that instant. Notice that Zeno has no qualms about the arrow being at location L at time t ; he thinks the arrow is there, but is just not in motion there.
Let us take note of the following fact: The arrow is at location L for a zero duration of time. This is, by the way, the kind of claim that, not only Zeno, but all physicists today would be apt to make in Newtonian contexts. But what sense does it make to say that an object is at some place for an instant, that is, for zero seconds? Is there any difference between asserting, “The object was there for zero seconds” and “The object wasn’t there at all”? Imagine yourself saying to your friend, “I was in my office yesterday.” Your friend asks, “For how long were you there?” And you answer, “For zero seconds (or minutes or hours).” Aren’t you saying (in a weird way) that you were not in your office yesterday? It would seem that being present at some location for a zero duration of time is equivalent to being absent at that location. If that is so, then it follows that the arrow did not occupy any of the locations on its supposed trajectory! It’s not that the arrow was at rest at every point of the trajectory, as Zeno contends; the arrow was not at any point of the trajectory. Nevertheless, his and my conclusions are the same: The arrow could not possibly traverse that trajectory.
Now, the at-at theory of motion cannot solve the new paradox, for it presupposes that the arrow can occupy different locations on the trajectory at different times. But this is precisely what the “new twist” is rejecting.
If the arrow is not to be found at any point of its presumed path, does this imply that the arrow simply didn’t budge and that it stayed put the whole time at its initial rest position? Things get more depressing: Being at rest during a certain interval of time is supposed to consist in an object’s occupying the same location at every instant of that time interval—for zero seconds. By a similar argument to the one given above, we are forced to conclude that the arrow is not to be found at its rest location either, at any of the instants during the time period of its supposed rest. Hence rest does not seem to be any more possible than motion.
The new paradox goes even further and gets more confusing: At a temporal instant no object, event, state or quality can exist. Thus no presence nor absence (or existence nor nonexistence), no change nor changelessness of any sort is possible either. This means we are equally barred from saying that the arrow is absent at all of the points of its presumed path.
For those who think the original Arrow Paradox still awaits solution, the new twist even more pressingly does so.
 Arisotle also denies that an object can be at rest or in motion at an instant of time (Physics 234a32, 239b1).
 A much more detailed version of this post can be found in my paper at: https://www.academia.edu/28269523/A_NEW_TWIST_TO_ZENOS_ARROW_PARADOX
Written by Erdinç Sayan
February 25, 2012 at 12:36 am
Posted in Metaphysics
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