The Better Solution to Zeno’s Paradox of Motion

This is about Zeno’s “dichotomy” paradox of motion.

As described by Wikipedia:

“Suppose Homer wants to catch a stationary bus. Before he can get there, he must get halfway there. Before he can get halfway there, he must get a quarter of the way there. Before traveling a fourth, he must travel one-eighth; before an eighth, one-sixteenth; and so on. […] This description requires one to complete an infinite number of tasks, which Zeno maintains is an impossibility.”

Since I’m not necessarily a mathematical Platonist, I won’t maintain that there “is” an infinite number of points to a number line; they don’t exist until you construct them individually (and then they only exists as abstractions/ideas). The paradox presumes distance should be idealized as a number line in which an infinitude of infinitesimally small distance-intervals exists already. Even if mathematics were Platonic, it would still be questionable whether an ideal number line characterizes real, physical distances.

Now, the standard calculus answer does work—that as time intervals and distance intervals diminish, they both approach the same limit (being zero) simultaneously. But that answer deals with infinities, as well as relies on mathematical equations, which are not process-oriented, and I believe that infinities are unnecessary—since distance intervals don’t exist until you create them—and that, as such, the problem could best be understood as a process—particularly a thought process, given that the problem is a thought experiment. (Yes, motion does actually happen in real life, but there’s obviously no paradox in motion intrinsically, or it wouldn’t happen; thus, to contemplate the paradox, per se, is to contemplate the thought experiment as a thought process.)

So here is my deconstruction of the paradox as a thought process.

Here’s what we do: we take one step in this thought process—we imagine the thing to move half-way (say, 4m). Then we take another step in the thought process—we imagine the thing to move half-way again (2m). Then we imagine it moving again (1m), again (.5m), and so on and so on. The problem here is that, while the distance traversed diminishes for every successive iteration of the thought-loop, the time it takes to think it through, doesn’t. Thus, it appears that, while time (being easily conflated with thinking-time) carries on indefinitely, the distances traveled get shorter and shorter incommensurately with the time intervals associated with them. If one could reduce one’s thought-process time by half for each successive iteration as the distance intervals diminish by half, then perhaps one would see things differently.

The proof that this mental effect is what’s responsible for the appearance of a paradox would be that we wouldn’t formulate the argument as so: the object must travel for half the time it takes to get there, then half the remaining time, then half the remaining time again, etc., and therefore it can never get there since it takes an infinite number of steps. That formulation is symmetrical to the distance-based one and carries the same implication, but somehow it just wouldn’t work—­it doesn’t have the same poignancy as the distance-based one.

One other point that comes to mind is that the idea alone that infinitesimal distance intervals don’t exist until you imagine them resolves Zeno’s paradox because there is no infinitude of steps; there is only the circular thought process, which carries on for exactly as long as one carries it on. In the same vein, contrary to popular belief, the value pi is not “infinite”…it’s a prescription for constructing a procession of digits of indefinite length. (Actually, it’s just a value. What was described is the algorithm for expressing that value in decimal form.) The actual value of pi can actually be expressed more succinctly, and precisely, as follows:

(Maybe there’s a more concise way to express that in summation notation, IDK.) Admittedly, this is actually another prescription for iterating in a loop; it just isn’t about creating digits, per se. The more iterations you step through, the closer your approximation is to the limit л. To use this as a prescription for generating accurate digits of pi, you would have to (a) convert your value to decimal (or another base, if you preferred), and (b) know how many digits are accurate at a given iteration and discard the rest. In any case, the above

formula itself is not an approximation, and taken as a whole, it is not infinite; that is, it does not contain an infinite amount of information, or it couldn’t be shown on your screen.

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