| Zeno's Paradoxes |
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Zeno's paradoxes are a set of Paradox es devised by Zeno Of Elea 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 . Several of Zeno's eight surviving paradoxes (preserved in Aristotle 's ''Physics'' and Simplicius 's commentary thereon) are essentially equivalent to one another; and most of them were regarded, even in ancient times, as very easy to refute. Three of the strongest and most famous—that of Achilles and the Tortoise , the Dichotomy argument, and that of an arrow in flight—are given here. Zeno's arguments are perhaps the first examples of a method of proof called '' Reductio Ad Absurdum '' also known as proof by Contradiction . They are also credited as a source of the Dialectic method used by Socrates . Zeno's paradoxes were a major problem for ancient and medieval Philosopher s, who found most proposed solutions somewhat unsatisfactory. More modern solutions using Calculus have generally satisfied mathematicians and engineers. Many philosophers still hesitate to say that all paradoxes are completely solved, while pointing out also that attempts to deal with the paradoxes have resulted in many intellectual discoveries. Variations on the paradoxes (see Thomson's Lamp ) continue to produce at least temporary puzzlement in elucidating what, if anything, is wrong with the argument. THE PARADOXES OF MOTION Achilles and the tortoise ''"You can never catch up."'' "In a race, the quickest runner can never overtake the slowest, since the pursuer must first reach the point whence the pursued started, so that the slower must always hold a lead." (Aristotle Physics VI:9, 239b15) In the paradox of Achilles and the Tortoise , we imagine the Greek hero Achilles in a footrace with the plodding reptile. Because he is so fast a runner, Achilles graciously allows the tortoise a head start of a hundred feet. If we suppose that each racer starts running at some constant speed (one very fast and one very slow), then after some finite Time , Achilles will have run a hundred feet, bringing him to the tortoise's starting point; during this time, the tortoise has "run" a (much shorter) distance, say one foot. It will then take Achilles some further period of time to run that distance, during which the tortoise will advance farther; and then another period of time to reach this third point, while the tortoise moves ahead. Thus, whenever Achilles reaches somewhere the tortoise has been, he still has farther to go. Therefore, Zeno says, swift Achilles can never overtake the tortoise. Thus, while common sense and common experience would hold that one runner can catch another, according to the above argument, he cannot; this is the paradox. The dichotomy paradox ''"You cannot even start."'' "That which is in locomotion must arrive at the half-way stage before it arrives at the goal." (Aristotle Physics VI:9, 239b10) Suppose Homer wants to catch a stationary bus. Before he can get there, he must get halfway there. Before he can get halfway there, he must get a quarter of the way there. Before traveling a fourth, he must travel one-eighth; before an eighth, one-sixteenth; and so on. The resulting sequence can be represented as: : This description requires one to travel an infinite number of finite distances, which Zeno argues would take an infinite time -- which is to say, it can never be completed. This sequence also presents a second problem in that it contains no first distance to run, for any possible first distance could be divided in half, and hence would not be first after all. Hence, the trip cannot even be begun. The paradoxical conclusion then would be that travel over any finite distance can neither be completed nor begun, and so all motion must be an Illusion . This argument is called the '' Dichotomy '' because it involves repeatedly splitting a distance into two parts. It contains some of the same elements as the ''Achilles and the Tortoise'' paradox, but with a more apparent conclusion of motionlessness. It is also known as the Race Course paradox. The arrow paradox ''"You cannot even move."'' "If everything when it occupies an equal space is at rest, and if that which is in locomotion is always occupying such a space at any moment, the flying arrow is therefore motionless." (Aristotle Physics VI:9, 239b5) Finally, in the arrow paradox, we imagine an arrow in flight. At every moment in time, the arrow is located at a specific position. If the moment is just a single instant, then the arrow does not have time to move and is at rest during that instant. Now, during the following instances, it then must also be at rest for the same reason. The arrow is always at rest and cannot move: motion is impossible. Whereas the first two paradoxes presented divide space, this paradox starts by dividing time - and not into segments, but into points. It is also known as the ''fletcher's paradox''. PROPOSED SOLUTIONS Proposed solutions to the arrow paradox Aristotle , who recorded Zeno's arguments in his work '' Physics '' disputes Zeno's reasoning. Aristotle denies that times is composed of "nows", as implied by the argument by Zeno. If there is just a collection of "nows" then there is no such thing as temporal magnitude. Therefore, if Aristotle is correct in denying that time is composed of indivisible nows, then Zeno is wrong in saying that the arrow was stationary throughout its flight despite saying that in each now the moving arrow is at rest. Another objection to the arrow paradox is that the arrow paradox seems to be a play on words more than anything else. In particular, 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. A mathematical account would be 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. Another solution may be that the instantaneous physical state of the arrow cannot be fully specified by its position alone: one must specify both its position and its Momentum . Classical Mechanics asserts that things such as rate of change of acceleration of at an instant. See also Uncertainty Principle . Proposed solutions both to Achilles and the tortoise, and to the dichotomy Both the paradoxes of Achilles and the tortoise and that of the dichotomy depend on 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. 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 using mathematical series notation These solutions have at their core Geometric Series . A general geometric series can be written as :: |