Tom McCurdy
Sep7-04, 08:55 PM
Achilles and the Tortoise
Suppose the swift Greek warrior Achilles is to run a race with a tortoise. Because the tortoise is the
slower of the two, he is allowed to begin at a point some distance ahead. Once the race has started
however, Achilles can never overtake his opponent. For to do so, he must first reach the point from
where the tortoise began. But by the time Achilles reaches that point, the tortoise will have advanced
further yet. It is obvious, Zeno maintains, that the series is never ending: there will always be some
distance, however small, between the two contestants. More specifically, it is impossible for Achilles to
preform an infinite number of acts in a finite time.
Distance behind the Tortoise:
------5, 2.5, 1.25, 0.625, 0.3125, 0.015625, ….
Time: 1, 0.5, 0.25, 0.125, 0.625, 0.03125,
Solution:
extended version
http://philsci-archive.pitt.edu/archive/00001197/02/Zeno_s_Paradoxes_-_A_Timely_Solution.pdf
short
In the modern analysis, the paradox is resolved with the fundamental insight of calculus that a sum of infinitely many terms can yield a finite result. Adding the (infinitely many) times together that Achilles needs to reach the previous positions of the tortoise results in a finite total time, and that is indeed the time when Achilles overtakes the tortoise.
Suppose the swift Greek warrior Achilles is to run a race with a tortoise. Because the tortoise is the
slower of the two, he is allowed to begin at a point some distance ahead. Once the race has started
however, Achilles can never overtake his opponent. For to do so, he must first reach the point from
where the tortoise began. But by the time Achilles reaches that point, the tortoise will have advanced
further yet. It is obvious, Zeno maintains, that the series is never ending: there will always be some
distance, however small, between the two contestants. More specifically, it is impossible for Achilles to
preform an infinite number of acts in a finite time.
Distance behind the Tortoise:
------5, 2.5, 1.25, 0.625, 0.3125, 0.015625, ….
Time: 1, 0.5, 0.25, 0.125, 0.625, 0.03125,
Solution:
extended version
http://philsci-archive.pitt.edu/archive/00001197/02/Zeno_s_Paradoxes_-_A_Timely_Solution.pdf
short
In the modern analysis, the paradox is resolved with the fundamental insight of calculus that a sum of infinitely many terms can yield a finite result. Adding the (infinitely many) times together that Achilles needs to reach the previous positions of the tortoise results in a finite total time, and that is indeed the time when Achilles overtakes the tortoise.