Motion in A Singular and Plural Worlds in the Light of Zeno's Paradoxes

Motion in A Singular and Plural Worlds in the Light of Zeno’s Paradoxes


Zeno’s paradoxes confound a reader by exposing him to experience two types of reality. First, the readers are allured to envisage the conventional reality, the reality of a plural world with continuous divisions of things, place, and space-time matrix. But by challenging a reader to produce an infinite number of destiny points by dividing the continuous length between the tortoise and Achilles, Zeno introduces him with the reality of a singular world that is continuous within itself due to the lack of sufficient object of references to produce the sense of relativity in such a world. The race set in a traditionally plural world provokes a reader to draw the conclusion of the race according to the rule of the singular reality ultimately to be confounded with the result that is contradictory to his expected traditional result. Indeed Achilles can move both in a singular and a plural world, but the motion in a singular world is not perceivable since such world does not have any object of references (Whitehead 45).


Before defining the proposed singular and plural world thesis, it is necessary to have a clear idea of Zeno’s paradoxes of plurality. In Zeno’s word, “the universe is singular, eternal, and unchanging. The all is one.” (Brown 34) But this singular universe has a lot to do with his paradoxes of motion. In this singular universe, if Achilles takes a step toward any direction from any from where is, he will find himself where he was. This statement essentially seems to fabulous, since it is quite contradictory to real life experience. But a deeper understanding makes sense. Indeed Zeno’s singular universe is such that it consists of the only One, not of two. As a result, it is as it is. Since it consists of one, it does not provide a viewer with any chance to compare it with other. Therefore it lacks diversity. Because of its lack of diversity and presence of the others, it does not have any objects of references by which distance can be measured and any event cannot take place in it. Again because of the lacks of distance and event, space and time collapse in such a world. In it.00000000001 meter is equal to infinity; but more accurately, the previous statement is simply meaningless. In such universe whether Achilles moves one hundred miles or so back or forth, he will be where he is now. Wherever Achilles goes at what distance, he will remain at the center since such singular universe evolves out of his singular existence. Indeed, there are no “early” and “later”. Simply there exists the “now” since there is no other event in term of which the ‘early’ can assigned a meaning. In Zeno’s singular world, one is both existent and non-existent. One is existent is the sense that it perceives itself in a self-submerged merged way and again it is nonexistent in the sense that there is no other that can prove its existence. (Grünbaum 172-83)


Indeed this singular universe is one and at the same time it is many, since such one contains infinite number of ones upon its division for infinite times. Therefore one is both finite and infinite, as Zeno says, “If there are many, they must be as many as they are and neither more nor less than that. But if they are as many as they are, they would be limited. If there are many, things that are are unlimited.” (Simplicius On Aristotle’s Physics, 140.29) Indeed in Zeno’s universe, one is the one. Therefore it does not have the possibility to join with other to produce the bigger one. The only thing that the one can do is to divide itself and upon the division, the plurality begins. Since plurality begins, relativity can grow giving birth to the sense of distance and events. As a result time starts from here. But the simplest plural world is composed of three ones, since if the simplest plural world is composed of two, they will be mirror images for each others. For example, if the plural universe consists of only Achilles and the tortoise and if they are in motion, each one of them will think that he is motionless but the other is moving.


Now for the sake of convenience, we suppose that the universe is composed of three: Achilles, the tortoise and a viewer. Since the three together produce a plural universe, in it they are each other’s object of references and both relativity and events are born. Consequently distance and time are perceivable in this universe. In the race, both Achilles and the tortoise consider the viewer as their reference object in relation to which they locate their position, measure their distances among themselves and measure their time.


Again in another way it can be proved that Achilles is in motion. When they are informed that Achilles tries to reach the ever divided distance, they are locked in Achilles’ singular. In a singular world Achilles can move, but remain at the same place, the center of the world, where he initially was. Since he will not be able to reach his immediate next point among an infinite number of points in a continuous world, he cannot overtake the tortoise. Indeed the infinitesimally divisibility of distance is essentially a feature of the singular world. But according to the common experience, the readers of a plural world, (when they are informed that Achilles tries to reach the ever divided distance, they are locked in Achilles’ singular) will see that Achilles struggles hard and covers little distance that is infinitely smaller than the distance between him and the tortoise. In reverse, the distance between them stretches to an infinite length. The more Achilles takes a newer step the more distance grows to infinity. Therefore, from the very beginning of the race Achilles is distanced infinitely from the tortoise in Zeno’s infinitely stretchable singular world. Indeed he cannot see the tortoise at all. He is simply informed about the tortoise’s position and starts to run in order to cover the infinite distance. Indeed the paradox is that though Zeno claims that both Achilles and the viewers see the tortoise according the rules of a plural world, in Achilles’ infinitesimally divisible or stretchable singular world the tortoise does not exist at all. But if it exists, theoretically the singular world collapses and turns into a plural world where relativity of distance is possible, and distance between the two would have limited and fixed in relation to other unit distance. Therefore in a singular and infinite world, Achilles can be in motion and can move but his world expands and stretches in such a way that he remains at its center, since a singular world has only one center. In this singular, since there is no secondary existence in relation to which he can relocate his relative position, whether he moves forward or not, he remains at the same place where he is, namely, the center.


In a traditionally familiar world or a plural world, motion is defined in term of the covered distance at any instant of time. Indeed motion in a plural world is possible because of the relativity of positions of the existing objects in such world. These objects essentially serve as objects of references for each other. Zeno’s paradox springs out from the fact that the readers are provoked to imagine Achilles and the tortoise both in a singular world and a plural world: first in a plural world, then only Achilles in a singular world and then again the result of the race according to the rules of a plural world. Before starting the race, Achilles looks at the tortoise and then measures the distance between them. Then he calculates the possible result. In his calculation relativity dominates. Blessed to be born in a plural world, he has learnt to consider a certain distance between two objects as a unit distance and a certain interval between two ticks of the clock as a unit of time. Now he calculates the distance between him and the tortoise in relation to the unit distance and the unit time. But since the second object or the third or the fourth and so on, except the first, does not exist in a singular world, the first such as Achilles cannot have a sense of relative unit with which he can measure the distance between where he was and where he is. Indeed by acknowledging the existence of the tortoise, the singularity of Achilles’ ever divisible world has been violated. Therefore it is fairly possible for Achilles to cover the distance between him and the tortoise.


When Zeno says that the tortoise is positioned 100 steps ahead of Achilles, each of these steps is a set of infinite points and this set of infinite points constitutes one step. Indeed the set of infinite points has been existent or got its oneness because its relativity with other distances. In a plural world Achilles is not obsessed with the infinite number of points that constitute the length of 100 meter. Rather he is occupied with the finite sets of infinite points, since finite sets of infinite points are more useful in his relative plural world. Therefore when Achilles takes one step, his step covers an infinite number of points in the finite one. Indeed he appears to be a clever giant who know how to bind the countless in a limitation.


For example, if the distance between A (Achilles) and T (Tortoise) is considered as one and divisibility is applied, AT is composed of infinite points. But in a plural world, the AT distance is divided into a finite number of equal ones. They are ones, because the oneness of each of them is established by other ones. As it has been said in the definition section of this paper, these ones are existent ones since each of their ones by others. For Achilles or the viewers of a plural world, each of the ones is one step that binds the infinite points in a finite boundary and the finiteness of which is adaptable to the finite plural world.


Upon evaluating the solutions proposed by different scholars, the one and only objection that may crucially threaten the entire epitome of my position is the rigorous division of the singular and the plural worlds. Some may argue that it is almost irrational to separate Achilles’ world as a singular one and to deny the existence of the tortoise in it. Again it is also irrational to imagine Achilles both in a singular and a plural world. Such arguments primarily depends on the assumption that since the viewers of Achilles’ plural world can imagine the divisibility of the distance between the two, it is almost fabulous to mark Achilles’ as a singular one and to separate it from the Achilles-tortoise-viewers plural world. Also critics may raise the question whether it is appropriate to assign the features such as singularity, non-relativity, infinitely divisible or stretchable, singular existence, Unicenter, etc to Achilles’ singular world. For example, the simplest solution proposed by Simplicus asserts that one should look into the flaws of Zeno arguments, not into his conclusion. What Simplicus indicates as a flaw in Zeno’s argument is most likely Zeno’s provocation to the readers to think of the process of Achilles’ movement through an infinitely divisible distance. Therefore he has Diogenes the Cynic’s to rise up and walk in order to prove the falsity of Zeno’s proposition. Indeed by doing so, Simplicus not only defies Achilles’ motion through infinity but also my proposed singular world. Indeed most of the scholars’ solutions may seem to be in apparent contradiction with my singular-plural reality solution of Zeno’s paradoxes of motion.


Indeed the singular-plural reality solution itself is a response to the aforementioned objections since my proposition is more of a self-evident explanation of Zeno’s paradoxes than a solution. Indeed the critics so far have tried to escape from Zeno’s maze by avoiding any one of the two realities. Because of not having any clear concept or idea about the singular world or simply because of not daring to envisage a world from which Achilles’ contestant, the Tortoise, will be banished, the critics have simply tried to justify Achilles’ motionlessness with denying the experiences of a plural world. For the same reason, they have failed to answer the question why Achilles can be considered both motionless and in motion. Unlike them, my rigorous division between the two aspects of reality will enrich anyone a better understanding of why Achilles cannot reach the tortoise and how, at the same time, he can easily overtake it. Though I have separated the singularity of Achilles’ world from the plurality of the viewers’ world, such division is for convenience’s sake. In the singular-plural proposition, singularity remains embedded in the plurality of the viewers’ world. For example, Aristotle proposes that “things are infinite in respect of divisibility” and this divisibility is only possible in one’s mental process. (Aristotle’s Physics, 140.36) For him, since it is possible in one’s mentality, the viewers will see that Achilles has really overtaken the tortoise. But if the viewers remain continually engaged in the dividing process in mentality, he will never reach the tortoise. Whereas Aristotle and others consider things’ divisibility as purely mental, I tried to prove it as another aspect of reality.


Obviously Zeno’s paradoxes are the embodiments of two apparently contradictory aspects of reality. In other words they are the paradoxes of continuity and discontinuity; the reality of a singular world and that of a plural world. Finding a solution to these paradoxes is essentially to achieve a better understanding of the contradictions or, more accurately, to compose the line of compromise, between them, that seems to be more convincing than their apparent outlooks. Since a paradox provides shelter to two apparently opposing realities in its womb, Zeno’s paradoxes, though finally they purport that Achilles will not be able to move to the next point, can be explained in such a way that they themselves yield motion.


Works Cited

Aristotle, ‘Physics’, W. D. Ross(trans), in The Complete Works of Aristotle, J. Barnes (ed.), Princeton: Princeton University Press, 1984.

Brown, Kelvin. Reflection on Relativity. London: Mike Publishers, 2009.

Grünbaum, Arthur., Modern Science and Zeno’s Paradoxes, Middletown: Connecticut Wesleyan University Press. 1967

Whitehead, A. Nigel., Process and Reality, New York: The Macmillan Co. 1929


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