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Zeno's paradoxes are a set of Problem s 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 . It is usually assumed that Zeno took on the project of creating these paradoxes because other philosophers had created paradoxes against Parmenides' view. Thus Zeno can be interpreted as saying that to assume there is plurality is even more absurd than assuming there is only one. As such, if we are convinced by Zeno's paradoxes, we should take Parmenides' view more seriously. Plato , ''Parmenides'', 128c ; Kirk, p. 277. 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 . According to some historians of philosophy, Zeno's paradoxes were a major problem for ancient and medieval Philosophers . In modern times, Calculus has been widely accepted by mathematicians and engineers as at least a practical solution for calculating infinitesimal distances. Other proposed solutions to Zeno's paradoxes from past and present philosophers have included the denial that space and time are themselves infinitely divisible, and the denial that the terms ''space'' and ''time'' refer to any entity with any innate properties at all. Many philosophers still hesitate to say that all paradoxes are completely solved. Some philosophers state that these paradoxes still have modern relevance: attempts to deal with the paradoxes have resulted in intellectual discoveries, and variations on the paradoxes (see Thomson's Lamp ) continue to produce at least temporary puzzlement in discovering what, if anything, is wrong with the argument. The origins of the paradoxes are somewhat unclear. Diogenes Laertius says that Zeno's teacher, Parmenides , was "the first to use the argument known as 'Achilles and the Tortoise' ", and attributes this assertion to Favorinus . In a later statement, Laertius attributed the paradoxes to Zeno. PARADOXES OF MOTION Achilles and the tortoise ''"You can never catch up."'' In the paradox of Achilles and the Tortoise , we imagine the Greek hero Achilles in a footrace with the plodding reptile. Because he is such a fast 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, in which said period 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."'' 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 quarter, he must travel one-eighth; before an eighth, one-sixteenth; and so on. ImageSize = width:800 height:100 PlotArea = width:720 height:55 left:65 bottom:20 AlignBars = justify Period = from:0 till:100 TimeAxis = orientation:horizontal ScaleMajor = unit:year increment:10 start:0 ScaleMinor = unit:year increment:1 start:0 Colors= id:homer value:rgb(0.4,0.8,1) # light purple PlotData= bar:homer fontsize:L color:homer from:0 till:100 at:50 mark:(line,black) at:25 mark:(line,black) at:12.5 mark:(line,black) at:6.25 mark:(line,black) at:3.125 mark:(line,black) at:1.5625 mark:(line,black) The resulting sequence can be represented as: : This description requires one to complete an infinite number of tasks, which Zeno maintains is an impossibility. This sequence also presents a second problem in that it contains no first distance to run, for any possible ( Finite ) first distance could be divided in half, and hence would not be first after all. Hence, the trip cannot even begin. 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. Some, like Aristotle, regard the Dichotomy as really just another version of ''Achilles and the Tortoise''. However, they emphasise different points. In the ''Achilles and the Tortoise'', the focus is that movement by multiple objects is just an illusion whereas in the ''Dichotomy'' the focus is that movement is actually impossible. The arrow paradox ''"You cannot even move."'' In the arrow paradox, Zeno asks us to imagine an arrow in flight. He then asks us to divide up time into a series of indivisible nows or moments. At any given moment if we look at the arrow it has an exact location so it is not moving. Yet movement has to happen in the present; it can't be that there's no movement in the present yet movement in the past or future. So throughout all time, the arrow is at rest. Thus motion cannot happen. This paradox is also known as the fletcher's paradox—a '' Fletcher '' being a maker of arrows. Whereas the first two paradoxes presented divide space into segments, this paradox divides time into points. 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 time is composed of "nows", as implied by Zeno's argument. 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. According to Zeno, at any instant, the arrow must be at rest. However, this has been disputed, since 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, if the arrow is found to be at a different place than it was and will be at the times before and after, then we have reason to claim the arrow has moved. 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. 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 BCE , 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. Proposed solution using mathematical series notation Several proposed solutions have at their core Geometric Series . A general geometric series can be written as :: |
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