Ezio Fornero, A Zeno′s Paradox. Achilles and the Tortoise.doc

A ZENO’S PARADOX: Achilles and the Tortoise Perhaps Zeno’s paradoxes are the most famous in history. In general, philosophers and historians of Mathematics have understood them as paradoxes against movement, in the sense that Zeno and the school of Elea would have used them to deny the reality of movement, demonstrating its impossibility from a logical point of view (motion in itself would imply logical contradictions). If this were so, one could argue that the world real could not be understood by rational thought, or that truth should be purely ideal and phenomena just misleading appearances, etc., but this is not the problem of Zeno’s paradoxes against movement. Some people took seriously Zeno’s arguments (and do so today, as well, so that from time to time someone would come up with a solution of the paradox of Achilles and the Tortoise, as if the problem was justify the movement - not the search for some error in reasoning ...). Before proceeding, let’s see Achilles and the Tortoise. Achilles (a champion in the races) is chasing the Tortoise, to which he has given a head start. He reaches the point where the Tortoise started the race, but in the meantime it has travelled a certain distance. Achilles covers this distance and again, in the meantime, the Tortoise runs a further distance. So, when Achilles arrives at a position already reached by the Tortoise, it reaches an even more advanced, and Achilles will never overtake the Tortoise. This reasoning is tantamount to construct an infinite number of points, such that if ABC are three consecutive points in the series, the Turtle covers the distance BC, while Achilles runs AB. Call Zeno’s succession this endless series of points, and ‘Zeno’s points’ and ‘Zenos instants’ the points and the corresponding moments belonging to the succession. To avoid difficulties, we assume that Achilles and the Tortoise run at constant speed. To discuss the topic is better to establish some assumptions. Indeed, a paradox is a