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ANALYSE AND ARGUE FOR THE VALIDITY OR FALSENESS OF ZENO’S PARADOXES ON MOVEMENT.

According to The Penguin Dictionary of Ancient History Parmenides denied all motion and change, following his deductive logic, and concluded that what exists must be single, indivisible, and changeless. Zeno of Elea defended these surprising conclusions by revealing paradoxes in generally accepted ideas of plurality, divisibility, and change. They were designed to reduce the hypotheses of Parmenides’ opponents to absurdity by realizing incongruous consequences from them. They were not however conceived to attack a particular physical or metaphysical theory such as Pythagoreanism, as many scholars have claimed, but to generally invalidate popular beliefs and so circuitously support the deductions of Parmenides.

A paradox can be defined as a statement that seems self-contradictory but may be true. Zeno has inferred four such statements with regards movement and its impossibility. The obvious problem with any idea that denies movement is that, rather noticeably, movement does occur. It is Ayer that says, “If Zeno had been right in asserting that the concept of motion was self-contradictory, he could not have asserted it; for whether he spoke or wrote some part of his body would have to move”. But let us look at each paradox in turn.

“Achilles and the Tortoise” sees Zeno demonstrate that the faster runner can never overtake the slower one if the latter has a head start. Achilles allows the tortoise to start ten meters ahead of him, it takes Achilles a very short time to make up this distance but while he has been thus engaged the tortoise has advanced ten centimeters, as Achilles covers this distance the tortoise has advance another centimeter, then while Achilles covers that he moves another millimeter, and so on with diminishing distance ad infinitum. As long as the tortoise continues to move Achilles must first make it to the last position of the creature before he can catch it, and so he never will.

The Dichotomy paradox works in an opposite way from this, in so much as before you can traverse a distance you must first cover half of it, but before that you must cover half of that half, that is to say a quarter, and before this a half of that quarter, or an eighth, so on ad infinitum. When one considers these problems together it becomes clear that, conceptually, one can neither start nor finish a journey because space is infinitely divisible. At least that is the conclusion one might first arrive at, however, one must remember that Zeno wrote these paradoxes as a way to support Parmenides’ theories, one of which being that the world is indivisible.

This presents a problem if one takes the journey from start to finish being the whole. However, if one considers the smallest part, that can no longer be divided, to be the whole, and say that the journey is made up of an infinite number of wholes it presents no contradiction to the idea it is set to uphold. And yet, if the latter is the case then is it such a fundamental discovery? There must of course come a point when things can no longer be divided, but it does not necessarily follow that this affects the existence of motion.

Hobbes defines motion as “the continual relinquishing of one place and acquiring of another”. His use of continual implies the necessity of time in the conception of motion. If we keep this in mind we find Zeno to be correct in his paradox of the arrow in which he claims that at each instant of its journey it is at rest. That is to say that an instant in its shortest form is independent of time, and so without time motion does not occur. Yet Bergson refutes the idea with his own that the arrow is not ever anywhere.

“After stating Zeno’s argument, he replies: ‘Yes, if we suppose that the arrow can ever be in a point of its course. Yes, again, if the arrow, which is moving, ever coincides with a position, which is motionless. But the arrow never is in any point of its course.” It is difficult to realize if Bergson is comfortable with concept of infinity, one may counter argue that he does not divide the distance the arrow travels into its smallest denomination. Despite this he makes a valid point that “the mathematical view of change ‘implies the absurd proposition that movement is made of immobility’s’”.

It is comments such as this that validate Zeno’s point, however, Russell highlights that “motion implies relation”, just as a pie is made up of ingredients rather than pies, so motion is made of what is moving, not motions. It remains that an object can be, at different times, in different places no matter how close these times are to each other. To return to Achilles and the race he can never begin or complete, and as I have already pointed out, there is one glaring problem, he can physically both overtake the tortoise and complete the course.

One may say the mathematical Achilles’ heel of the Dichotomy is that ½ added to ¼ and ½ diminishing successively ad infinitum will equal 1, and thus prove no problem for Thetis’ son to traverse. Although the distance can be divided, in one stride he will cover an infinite number of points, and the thought of this does not hinder his progression, as he will cover infinity in his next stride.

Others have argued that Achilles would never exhaust the whole distance of the course, as there would always “Be a remainder, a fragmentary part, however small, wanting to complete the structure”. Is it possible to exhaust the inexhaustible? This question is not to be taken to mean that the length of the course is infinite in quantity, simply that it is infinite in divisibility, two concepts that are not to be confused.

Mathematicians may say that to attempt to deal with the full concept of infinity is futile. For example, those who choose to calculate ∏ to millions of decimal places do not come any closer to understanding its purpose than those who convert it to a vulgar fraction. Some believe they betray the above idea better than dealing with the empirical concept of motion, no matter the intentions of the author. With this in mind Zeno’s work is an interesting set of ideas for mental discourse, and but has no place in the workings of reality.

The fourth paradox deals with the concept of velocity and wonders how it is possible for a moving object to complete the same distance in the whole time and half the time simultaneously. That is to say that there are three objects, equal in length, A is stationary while B and C move with equal speed in opposite directions. It takes them half the time to pass each other than it takes them to pass A; verbally this appears impossible and so our concept of motion must be wrong. But it also seems glaringly obvious that the outcome is the only one possible and so presents no contradiction to the existence of motion. Just as Russell reduced Bergson’s theory to a play on words, Zeno’s paradox can be likewise reduced.

It is Aristotle that remarks, “The fallacy lies in assuming that a body takes equal time to pass with equal velocity a body that is in motion and a body of equal size at rest” (240 a I). But since Tannery, it has not been deemed likely that Zeno could have made such a basic logical blunder. Zellar then suggests that Zeno would not have considered this important; he is after all attempting to contradict the idea of motion and so creates a situation rife with inconsistency. Booth reminds us of the limitations of Zeno’s time and therefore makes us realize that errors we consider elementary may not have been conceived of at the time the paradoxes were first considered.

One may also question the validity of the accounts we have of the paradoxes, considering we receive them from Aristotle who was attempting to refute them. With only about two hundred of Zeno’s own words surviving we cannot expect a biased free report. It is not to be inferred that Aristotle did not fully comprehend Zeno’s proposals, simply that he phrased them in such a way as they were more easily refutable. According to Booth, towards the end of the Nineteenth Century many French Scholars reconstructed the paradoxes because they believed Aristotle was attempting to make Zeno appear less intelligent than he actually was.

Lee accepts the French revision and Ross agrees with their reworking of the fourth paradox, but sticks implicitly to Aristotle with regards the others. It seems presumptuous of the scholars to attempt to reconstruct the writings of a man they have not had access to, we have only Aristotle and other Greek commentators of the time to rely on, and yes they carry a bias opinion, but can we be expected to trust men who are writing two thousand years later? I am forced to agree with Russell when he said the historical correctness of Zeno’s work was not important and that they should be regarded as “merely a text for discussion”.

The French Scholars assumed that Zeno’s use of the word onkoi to describe the As, Bs and Cs in the fourth paradox, was synonymous with the definition of the Atomists, that they were indivisible atoms, and therefore indivisible units. However, Booth points out that onkos was an everyday word for ‘body’ and ‘mass’ and may not have carried the implication of a minimal unit in Zeno’s original work, thus highlighting the problem of reconstructing ancient cases. They also said Zeno’s argument is directed specifically in opposition to the theory that time and distance are composed of indivisible minimal units, as pointed out by Bergson’s comment.

Aristotle says, “Zeno’s paradox depends on an arbitrary selection of the points of division” and so refutes his predecessor. Yet Russell found that after two thousand years of such denunciations Weirstrass made the paradoxes the foundation of a mathematical renaissance. He proved that the arrow, at each point of its flight, is genuinely at rest, by firmly banishing infinitesimals. I have also said that Zeno’s paradoxes can only be substantiated conceptually because physically, movement is evidently possibly. I am inclined to agree with Russell that Zeno provides us with an interesting mental exercise, rather than an inarguably persuasive theory for the non-existence of motion. 


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Melissa

 

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Uploaded Date: Sep 19,2014

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I am a Shakespeare fanatic with six years` classroom experience. I enjoy reading a wide range of fiction and non-fiction and became a teacher to share my passion for language. Now that I have a young family, I have decided to indulge my love of teaching outside the classroom to fit arou.... Read More

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