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| 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.
