Benardete’s Challenge

Benardete [4] discusses novel variants of a paradox of Zeno. Here is a version that suits our purposes: Imagine that an impenetrable barrier is erected at each point x = 1/2n for n = 1, 2,...; we suppose the barriers to be of zero thickness (or of decreasing thickness as they close in on x = 0, so that they do not overlap or touch each other, and so that they do not overlap or touch x = 0). Moreover, imagine that each barrier is immovable once so placed. Finally, imagine that a projectile is aimed at the barriers from a point to the left, i.e., from some x < 0.

Let us first of all note that this appears to be a case of dense infinity. There is an infinitude of physical entities in a finite region. To be sure, this particular setup is highly implausible; we are bringing it into the discussion as an easy warmup case, before proceeding to more physically plausible cases.

Now, what will happen as the projectile moves rightward? Since there is nothing apparent to impede the projectile at negative positions (x < 0), it would seem that it should continue its rightward motion until it strikes a barrier. But before it can strike a barrier at x = 1 /2n it must first strike (and pass through) all those to its left (at x = 1/2m for all m > n). This is impossible by the conditions of the problem. So it cannot strike any barrier at all! Hence it must stop its rightward motion, never passing zero, yet without touching anything that would be a cause for its rightward motion to cease.

This has been debated in various philosophical papers; see [13, 19, 30]. In [18] standard physics is brought to bear on the puzzle in the forms of classical mechanics, quantum mechanics, and relativity, showing for instance in the classical case that a field effect in the form of a repulsive force is mandated by Newton’s Laws, so that the projectile is bounced back to the left before passing zero. But the lesson for us here is that even a dense infinity need not be paradoxical when seen from within standard physical theory. (Of course, one can resurrect a paradox by insisting the barriers produce no forces outside their own immediate locations; and the lesson then would be that this is inconsistent with standard physics.)

Another version of the puzzle involves a continuous barrier-wall extending from some point b > 0 all the way back to, but not including, x = 0. That is, this is a wall of width b but with its left face missing. While a seeming bit of physical nonsense (at least in terms of materials made of atoms) it is a familiar enough entity in mathematics, essentially a half-open half-closed interval. And the same form of argument applies as in the earlier Benardete example. It would seem that physical entities cannot be isolated quite as well as our imaginations might like: physical interactions will occur and cannot be dismissed by mere stipulation.

Thus the Benardete examples provide a kind of dense infinity, but not apparently one that “breaks” anything. Perhaps this is because it does not directly involve an infinite density of standard physical quantities like mass or charge or energy. (A closer analysis might turn up an infinite sort of potential energy, however.) In any event, when we turn to something “real” such as an electron, the situation presents itself more starkly.

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