Zeno's Paradox - Achilles and the Tortoise - Solution

Zeno of Elea (c. 490 BC - 430 BC) was an ancient Greek philosopher. He devised certain paradoxes of logic. The paradox dealt with here is called the Paradox of Achilles and the Tortoise.

"In his Achilles Paradox, Achilles races to catch a slower runnerfor example, a tortoise that is crawling away from him. The tortoise has a head start, so if Achilles hopes to overtake it, he must run at least to the place where the tortoise presently is, but by the time he arrives there, it will have crawled to a new place, so then Achilles must run to this new place, but the tortoise meanwhile will have crawled on, and so forth. Achilles will never catch the tortoise, says Zeno. Therefore, good reasoning shows that fast runners never can catch slow ones," says the summary at Internet Encyclopedia of Philosophy (originally published 7 June 2009, last updated 1 April 2013, accessed 16 October 2013).

A distance issue, a ten meter head start for Achilles, is cited by "Platonic Realms PRIME Zenos Paradox of the Tortoise and Achilles" (1997-2013, accessed 16 October 2013).

The distance intervals involved are assumed to continue infinitely on and on, i.e., become permanently on and on without end, ever smaller and smaller, from the ten meters or about 33 yards, down to feet, down to inches, then centimeters, then millimeters etc., supposedly precluding Achilles ever catching the tortoise.

This paradox has been studied and analyzed for some 2,500 years. It is included in typical Ancient Philosophy classes. It is supposedly unanswerable. Supposedly no fault in logic has been found. See, e.g., Bryan Magee, The Story of Thought (DK Publishing, Inc., 1998), Chapter 1, "Before Socrates," page 19, "An Impeccably Logical Argument That Leads to A False Conclusion," on the ancient Greek philosopher Zeno (490-430 BC). There he discusses Zeno's 'Achilles and the tortoise' paradox, and says "there must be a fault in the logic," which is true, "but no one has yet been wholly successful in demonstrating what it is."

However, Leroy Pletten in September 1966 proposed a fault in the logic, that the size, i.e., the length, of Achilles' foot had been overlooked, and must needs be taken into account. Adult human feet are commonly known as a matter of public domain knowledge to be several inches in length.

This proposed solution questions, indeed, refutes, the assumption, that the distances involved become indefinitely on and on without end, ever smaller and smaller, from yards or meters, down to feet, down to inches, then centimeters, then millimeters etc.

On the contrary, in actual objective reality, adult human foot size is well established, well verified, with foot length in inches or centimeters substantially beyond the assumed infinite number of later-in-series steps, down to millimeters, and ever even smaller units. This refutes the assumed concept of them becoming infinitely smaller and smaller in size, i.e., smaller than foot length. This logic is contrary to reality, actual foot size.

This proposed solution shows that mere "logic" apart from actual reality can lead to false conclusions. This distance solution to the paradox works just as well whether citing foot size, paper airplanes, or bullets, or BBs, or ions in a vacuum attracted to a charge at one end, or any other entity the length of which is in excess of the postulated ever smaller latter steps in the sequence.

If you prefer a time based solution, rather than a distance based solution, the above-cited "Platonic Realms PRIME Zenos Paradox of the Tortoise and Achilles" is most useful. Although it cites the distance issue, its focus is the time for Achilles to do the first half the journey, two seconds, thus all the remaining postulated infinite time for the second half of the journey, when totalled, nonetheless involve only another two seconds, notwithstanding all the allegedly complex intervals and mathematics calculations purportedly involved. This Platonic Realms approach, a time-based approach, arrives at the same result as the Pletten approach, a distance-based approach. As noted in the above paragraph, this is so regardless of the physical entity involved.

For further reading:

Zeno's Paradox - Numberphile Youtube Video

Stanford Encyclopedia of Philosophy

Wolfram's Math World

Jm Loy's Physics

Bryan Magees In-Depth, Uncut TV Conversations
With Famous Philosophers (1978-87)

Wikipedia Zeno's Paradoxes

Copyright 1966
Current as of 17 October 2013

Email: lpletten at tir.com