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

**, which asserted that it would be impossible for Achilles to catch the tortoise in a race, if he had accepted a starting handicap.***Achilles and the tortoise*
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

**, so Achilles will never catch the tortoise (this is the conclusion of his reasoning). This, however, is not true:***Zeno thinks that the sum of an infinite series of numbers must be infinite***. One of them is precisely the series that computes the time needed by Achilles to catch the tortoise, according to Zeno’s reasoning.***there are many infinite series whose sum is finite*