Zeno actually came up with several paradoxes. For example, the arrow paradox: take the flight of an arrow. It follows its path from some point A to some other point B. However, at each point of its flight, it must go through a sequence of intervening instants (a_0, a_1, ..., a_n). At each of these instants, the arrow is stationary. But how could a path made up of a sequence of "instants" (stationary points) ever make up the motion of the arrow from point A to point B? In other words, how would the arrow even move, given that if you sum up all of the instants that make up the arrow's path, then logically you should have no motion at all? Today, in calculus, this paradox is resolved with the introduction of a rigorous notion of instantaneous velocity:
https://www.themathpage.com/aCalc/instantaneous-velocity.htm
As the name suggests, instantaneous velocity is the velocity of an object at a certain time t. It is the limit of the average velocity. You find the instantaneous velocity by taking the derivative of the position function at some time t. With instantaneous velocity, an object actually does have a definite velocity at each instant (time t). Thus, it no longer makes sense to say that the object is stationary at each instant of its trajectory from point A to point B. Hence, paradox resolved.
The paradox you are referring to specifically is the Achilles and the Tortoise paradox. You have Achilles and the tortoise. Let's say the tortoise starts out any arbitrary distance ahead of Achilles (e.g. 5 meters). Then Achilles can never reach the Tortoise because he must first travel half the distance, then half of that distance (a fourth), then half of that distance (an eighth), and so on. Like the arrow paradox, this paradox is resolved with calculus. What we have is the infinite geometric series: 1/2 + 1/4 + 1/8 + 1/16 + ... which converges absolutely to 1. Thus, it is not the case that Achilles will never catch up with the tortoise simply because catching up with the tortoise requires traversing an infinite number of points. He will in fact catch up with the tortoise eventually. Hence, paradox resolved. Likewise, all of the other paradoxes Zeno gives can in one way or another be addressed with modern methods in mathematics such as the calculus.
Whether there is some ultimate scale of physical length (e.g. the Planck scale) beyond which spacetime can no longer be subdivided is a debated question. While mathematically, spacetime is typically modeled as a continuum, our current understanding of physics does suggest there should be a definite length scale beyond which spacetime would breakdown and no longer be continuous. |