ZENO OF ELEA'S PARADOXES


Zeno of Elea is famous for his paradoxes. Zeno's paradoxes are a set of paradoxes devised to support Parmenides' doctrine that "all is one" and that contrary to the evidence of our senses, the belief in plurality and change is mistaken, and in particular that motion is nothing but an illusion. His most well known are the three paradoxes named "Dichotomy", "Achilles and the Tortoise", and "The Arrow".
Zeno's Paradoxes

Dichotomy: It is impossible to cover any distance, because half the distance must be traversed first, then half the remaining distance, then again half of what remains, and so on. Some portion of the distance to be covered is ALWAYS left to cover. Therefore, motion is impossible.
Achilles and the Tortoise: Achilles, the worlds fastest runner, is racing a tortoise. Since Achilles is so fast, he gracefully gives the tortoise a head start of 200 yards. After the race begins, he starts to overtake the tortoise. Since he must first reach the point where the tortoise started, from which it has already departed. Repeating indefinitely, Achilles gets to each new point in the race, the tortoise having been there, has already left. Therefore, even though Achilles is much faster than the tortoise, he can NEVER even tie.
The arrow: An arrow shot from a bow must be moving at every instant in its flight. But at every instant it must be somewhere in space. However, if it is always in some one place, it can't be in transit at every instant, for to be in transit is to be NOWHERE and thus cannot be moving.
Possible Solutions to Zeno's Paradoxes

Solution to Dichotomy: Both the paradoxes of Achilles and the Dichotomy depend on an dividing distances into a sequence of distances that become progressively smaller, and so are subject to the same counter-arguments. Aristotle pointed out that as the distance decreases, the time needed to cover those distances also decreases, so that the time needed also becomes increasingly small. Such an approach to solving the paradoxes would amount to a denial that it must take an infinite amount of time to traverse an infinite sequence of distances. Before 212 BC, Archimedes had developed a method to derive a finite answer for the sum of infinitely many terms that get progressively smaller that was equal to about 6/(π2). Theorems have been developed in more modern calculus to achieve the same result, but with a more rigorous proof of the method. These methods allow construction of solutions stating that (under suitable conditions) if the distances are always decreasing, the time is finite.
Solution to Achilles and the Tortoise: See Solution to Dichotomy.
Solution to The Arrow: The arrow paradox raises questions about the nature of motion that are not answered by the mathematical attempts to solve the Achilles and Dichotomy paradoxes. This paradox may be resolved mathematically as follows: in the limit, as the length of a moment approaches zero, the instantaneous rate of change or velocity (which is the quotient of distance over length of the moment) does not have to approach zero. This nonzero limit is the velocity of the arrow at the instant. The problem with the calculus solution is that calculus can only describe motion as the limit is approached, based on the external observation that the arrow plainly moves forward. Zeno's paradox however implies that if Zeno's method is followed to its logical extent, concepts such as velocity lose all meaning and there is no causal agent that is not similarly affected by the paradox that could enable the arrow to progress. Another point of view is that the premises state that at any instant, the arrow is at rest. However, being at rest is a relative term. One cannot judge, from observing any one instant, that the arrow is at rest. Rather, one requires other, adjacent instants to assert whether, compared to other instants, the arrow at one instant is at rest. Thus, compared to other instants, the arrow would be at a different place than it was and will be at the times before and after. Therefore, the arrow moves.
Solution to All 3 Paradoxes in Understandable English: Basically, what Zeno forgot, and Archamedies pointed out, motion is usually thought of as a distance travelled over time. Here's a group of useful formula relating distance and time to speed: Zeno's Paradox ignores the changes in time and therefore the changes in speed of the moving object. The arrow in the first example would be pretty fast, so even in a small amount of time it would go pretty far. In the example, however, the time is reduced to almost nothing and since anything multiplied by almost nothing equals almost nothing, the distance, of course, becomes very small. In real-life, however, time keeps increasing - especially when you're having fun - and because of this, time multiplied by a speed gives a number, and motion occurs.
Also See: Zeno of Elea Biography

Back to Table of Contents