Mathematics

Some of the most Important Paradoxes of Zeno Including Achilles and the Tortoise



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Zeno of Elea was a Greek philosopher born circa 490 B.C. and died circa 430 B.C.  Zeno presented many paradoxes to support Parmenides’ hypothesis that “all is one”.  The three paradoxes of motion are most famous.  They are “Achilles And The Tortoise Paradox”, “The Dichotomy Paradox”, and the “Arrow Paradox”. 


The “Achilles And The Tortoise Paradox” has Achilles and the tortoise race with the tortoise 150 yards or so ahead.  The paradox has the rules that the tortoise will be slower than Achilles.  Assuming they run at a constant speed with Achilles running 150 yards a second and the tortoise going 20 yards a second, the paradox claims that Achilles cannot overtake the tortoise.  The reasoning is since Achilles will reach the tortoise’s starting point in one second, the tortoise will have moved 20 yards because it is moving 20 yards per second.  As soon as Achilles reaches each point where the tortoise was, the tortoise will have advanced past that point. 


There are several flaws in the reasoning that are still in debate today, although many think it is solved. One flaw is Zeno is using points, which is a term in geometry that is defined as an object that has zero dimensions, although two points determine a line (because it gives it direction) . 


Mathematically, all that is necessary is doing the math.  After one second, the tortoise is 20 yards ahead of Achilles because Achilles has run 150 yards in one second, “catching up” with the tortoise’s starting point; and the tortoise has run 20 yards in one second, advancing 20 yards from its starting point. After one more second, Achilles has gone another 150 yards (150 yards

ahead of the tortoise’s starting point) and the tortoise has gone another 20 yards (40 yards ahead of its (tortoise’s) starting point).  This clearly overtakes the tortoise because Achilles has gone 150 yards-40yards=110 yards past the starting point of the tortoise.


The tortoise is moving from each point, but Achilles moves faster and overtakes the tortoise.  Anything that goes faster than another object ahead of it will overtake it in a time that can be calculated.  Of course there is also the flaw that a constant speed is impossible because living beings become tired after a short time when running.  Also, speeds can only be calculated to the accuracy available at the time. 


The “Dichotomy Paradox” is similar to the Achilles and the tortoise paradox.  It claims there is no way to catch a train because before one reaches the train, he must travel half the distance; before he travels half the distance, he must travel one-fourth the distance; before he travels one-fourth the distance, he must travel one-eighth the distance, etc. 


The solution is analogous to the solution to the Achilles and the tortoise paradox. Points on a route are the same as the term points in geometry.  And, as already mentioned, points have zero dimensions. This means if one walks one-fourth the distance to the train, he did not walk exactly one-fourth the distance unless he happened to walk exactly one-fourth the distance.  This is because with each step he moves a distance that might or might not be exactly one-half, one-fourth, one-eighth, or even one-tenth. 


But he still moves past exactly one-half, one-fourth, one-eighth, one-tenth, etc. whether or not he does hit on the exact one-half, one-fourth, one-eighth, and one-tenth points when he lands his foot on the ground. 


The “Arrow Paradox” claims that an arrow cannot move in any instant because if it does, it must change the position it is in.  The position it moves to must be either where it is (if it does not move) or where it is not.  This is impossible because it cannot move to exactly where it is. 


The flaw to this argument is an instant is relative to where any object is.  If two arrows are moving the same speed, they will appear not to be moving to each other.  But they are moving according to everything else in the area.  This is explained in the physics subject reference frames.  Any object is moving from one point to another, but points have zero dimensions. 



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ARTICLE SOURCES AND CITATIONS
  • InfoBoxCallToAction ActionArrowhttp://en.wikipedia.org/wiki/Zeno_of_Elea
  • InfoBoxCallToAction ActionArrowhttp://en.wikipedia.org/wiki/Zeno's_paradoxes#Proposed_solutions
  • InfoBoxCallToAction ActionArrowhttp://www.mathacademy.com/pr/prime/articles/zeno_tort/index.asp