The Duality of Time Theory

The above four paradoxes not only challenge all methodical theories of motion, but also our everyday experience. For this reason, they have been often dismissed as logical nonsense. Many attempts, however, have also been made to dispose of them by means of mathematical theorems, such as the theory of convergent series or the theory of sets. Some of these philosophical and mathematical treatments have offered convincing arguments that may explain how motion occurs outwardly in space and time, but none was ever really able to even tackle Zeno’s original arguments on the complications that emerge from the mystical and metaphysical relation between the One and the Many, or also: space and time.

According to Simplicius, Diogenes the Cynic (  404-323 BC) said nothing upon hearing these arguments, but he simply stood up and walked, in order to demonstrate the falsity of Zeno’s conclusions. Aristotle also did not fully appreciate the significance of Zeno’s arguments, since he called them “fallacies”, without actually being able to refute them. His refutation, however, include important counter arguments, such as Potential Infinity, that will be discussed further below.

On the other hand, many modern scientists like to believe that axiomatic mathematics has habitually dispelled these paradoxes, where now it is possible to talk about limits and infinity without reaching any mathematical contradiction, since it can be proven that the sum of an infinite number of halving intervals is still finite. For the purpose of this discussion, we will concentrate here only on the main concepts behind Aristotle’s response, on the one hand, and the modern mathematical treatment known as the Standard Solution, on the other hand.

In essence, Zeno’s paradoxes of motion, the Achilles as the most evident example, arise from the two main assumptions:

1.that distances and durations can be divided infinitely, 2.and thus the infinite number of segments become too many for the runner to complete.

Aristotle accepted the second assumption, but rejected the first one by inventing “potential infinity”, while the Standard Solution maintained the first assumption and then used complicated mathematical tools to prove that it is possible to complete infinite number of steps in a finite time.