I'm curious to see if Objectivists on here can offer a practical explanation for Zeno's paradox of Achilles vs. the Tortoise. I realize that Aristotle attempted to tackle it, but his explanation doesn't seem complete to me since he resorts to "potential halves" instead of "actual halves," thereby not actually resolving (or refuting) Zeno's reasoning; or perhaps I'm not understanding him correctly and we may agree more than I think. I have posted my resolution to the "paradox" beneath the selection.
"t is impossible for [Achilles] to overtake the tortoise when pursuing it. For in fact it is necessary that what is to overtake [something], before overtaking [it], first reach the limit from which what is fleeing set forth. In [the time in] which what is pursuing arrives at this, what is fleeing will advance a certain interval, even if it is less than that which what is pursuing advanced … .And in the time again in which what is pursuing will traverse this [interval] which what is fleeing advanced, in this time again what is fleeing will traverse some amount … . And thus in every time in which what is pursuing will traverse the [interval] which what is fleeing, being slower, has already advanced, what is fleeing will also advance some amount." http://plato.stanfor...ox-zeno/#ParPla
Here is my take on it:
Zeno's critique of motion as illusion does make sense in the realm of mathematics, and this is where I see Zeno's flaw. Mathematics doesn't exist in reality (similar, if not identical, to mathematical formalism [which happens to be contra mathematical Platonism], which I indirectly learned I agree with while researching solutions to this paradox). Mathematics is inherently a mental construction. Laws of physics, biology, properties of matter . . . none of these actually exist in physical reality.
We can take a system of any kind, superimpose it onto reality, and while it might line up as a perfect stencil (for instance, I'm not going to walk off a cliff anytime soon even though a system tells me I will fall to my death and I'm saying the system is fake), the system itself is purely subjective. It is the same with infinity. There is no way to even conceptualize infinity much less see it in nature. Thus there is no way for us to manipulate it and use it or view it outside of a subjective construct that doesn't actually exist.
Zeno's example of Achilles does not work in reality. Achilles would blow by the turtle in all accounts, and we know this from experience. Can we mathematically explain it? No, and that's why his paradox frustrates so many people. But the truth of the matter lies in the distinction between the mental system of mathematics or physics and what is real. Zeno maintained that his paradox showed that movement and change is an illusion, but it really doesn't. Mathematics and all subjectively constructed systems are the illusion.
Between any two independent objects, there is a gap. However, a kilometer is not a yard, and a yard is not a foot. It depends on how you look at it. The fact, though, is that no matter how you look at it, measurements don't actually exist. Basic dimensions of length, width and depth certainly exist at different degrees, but the fact remains: the middle of anything is purely subjective, just as the length of a distance between two objects is purely subjective; not only that, the systems we're using to measure and find the middle of anything don't actually exist! That's it, that's the answer: mathematics and other systems of measurements are mental constructs, therefore there is no paradox of any kind. Sense-experience gives us all the information we need. In other words, I'd put my money on Achilles; I don't know many who wouldn't.