|Zeno of Elea|
Zeno of Elea, a follower of Parmenides, is mainly remembered for his paradoxes which try to prove that movement does not exist, especially the paradox of Achilles and the tortoise, which asserted that it would be impossible for Achilles to catch the tortoise in a race, if he had accepted a starting handicap.
We know that Achilles runs faster than the tortoise (otherwise he could not catch it and the paradox would make no sense). As he has taken a handicap, when Achilles starts to run the tortoise will already be at a certain distance, at point A. When Achilles reaches point A, the tortoise will have advanced to point B. When Achilles reaches B, the tortoise is already in C, and so on, ad infinitum.
The time Achilles needs to catch the tortoise will be the sum of the times it takes him to reach points A, B, C... The total time is, therefore, the sum of an infinite series of numbers. The problem is that Zeno thinks that the sum of an infinite series of numbers must be infinite, so Achilles will never catch the tortoise (this is the conclusion of his reasoning). This, however, is not true: there are many infinite series whose sum is finite. One of them is precisely the series that computes the time needed by Achilles to catch the tortoise, according to Zeno’s reasoning.