Aristotle in his Physics explains Zenoâ€™s Paradox as, â€œIn a race, the quickest runner can never overtake the slowest, since the pursuer must first reach the point whence the pursued started, so that the slower must always hold a lead.â€
The inherent problem in the paradox is the ability to infinitely divide space and time. The question is whether this is a problem in measurement i.e. the mathematics or the conception i.e. our perception of space and time. The mathematician M. Black in the 1950â€™s goes so far as to claim that the paradox shows mathematics to be inapplicable to space, time and motion. He claims the application of mathematics incorrectly represents a finite race to appear as an infinite series of half runs.
The best mathematical solution to the paradox using calculus makes it a simple calculation. However, critics of the calculus approach have claimed that it only answers the geometry of the paradox and not its dynamics.
They claim calculus shows that the sum of an infinite number of terms can be finite, but this does not explain how one is able to finish going through an infinite number of points, if one has to go through each point one by one. Thus, according to calculus an infinite number of tasks can be done within a given time intervals, reaffirming the idea that space and time are infinitely divisible. Thus, it still suffers from the basic question as to how one can possibly reach the end of an endless series.