ZENO'S PARADOXES (450 B.C.) Megan Hennessy firstname.lastname@example.org Written and Presented on 5/8/95 Zeno of Elea, born about 490BC, is famous for his four paradoxes of motion and his paradox of plurality. He is a student of Parmenides and his paradoxes were designed to show the absurdity of views of those who made fun of Parmenides. Zeno tries to cut off all possible avenues of escape from the conclusion that space, time, and motion aren't real but illusory. Paradox of Plurality Zeno argues that if extended things exist, they must be composed of parts; thus there is a plurality of parts. Furthermore, these parts must themselves have parts. Since the process of subdivision is indefinitely repeatable, there must be an infinity of parts. Allen explains two difficulties which arise: 1) the ultimate parts must not have a magnitude, because if they did, they can be further subdivided. But the whole object can't be made up of parts which have no magnitude, for no matter how many of them are put together, the result will have no magnitude, and 2) yet the parts must have magnitude. But the addition of an infinite number of magnitudes, all greater than zero, will yield an infinite magnitude. Hence, whole objects, if they exist, are "so small as to have no magnitude and so large as to be infinite." Achilles and the Tortoise Achilles, the fastest of Greek warriors, is to run a footrace against a tortoise. The tortoise gets a headstart. Achilles can never catch up with the tortoise, no matter how fast he runs. In order to overtake the tortoise, Achilles must run from his starting point A to the tortoise's original starting point T. While he is doing that, the tortiose will have moved ahead to T1. Now Achilles must reach the point T1. While Achilles is traveling the new distance, the tortoise moves still farther to T2, and again Achilles must reach this new point. And so it continues; whenever Achilles arrives at a point where the tortoise was, the tortoise has already moved a bit ahead. Achilles can narrow the gap between him and the tortoise, but he can never actually catch up with him. The Dichotomy (1st Form) Achilles can never reach the end of the race course, much less the original starting point of the tortoise. Zeno argues that before Achilles can cover the whole distance he must cover the first half. Then he must cover the first half of the remaining distance, and so on. Therefore, he is always somewhat short of his goal. Hence, he can never reach it. (2nd Form) Achilles can never actually get started. Before he can complete the full distance, he must run half of it. But before he can complete the first half, he must run the first quarter, and so on. In order to cover any distance, no matter how short, the runner must already have completed an infinite number of runs. Since the sequence of runs completed form a regression, it has no first member, and hence, Achilles can never get started. The Arrow An arrow in flight is always at rest. At any given moment, the arrow is where it is, occupying a portion of space equal to itself. During the instant it cannot move, for that would require the instant to have parts, and by definition, and instant cannot have parts. For the arrow to move during the instant would require that during the instant it must occupy a space larger than itself, for otherwise it has no room to move. The Stadium Consider these two positions of rows: First Position Second Position A1 A2 A3 A1 A2 A3 B1 B2 B3 B1 B2 B3 C1 C2 C3 C1 C2 C3 While row A remains at rest, rows B and C move in opposite directions until all three rows are lined up. In the process, C1 passes twice as many Bs as As; it lines up with the first A to its left, but with the second B to its left. Zeno concluded that "double the time is equal to half." Cauchy's work on the summing of an infinite series converging, shed some light on the Dichotomy and Achilles paradoxes. Some philosophers argue that the running of a race as "completing an infinite sequence of tasks" is not a praper way to describe it. The use of infinity machines, like "The Desk Lamp," show that its impossible to say consistently that the machine has completed an infinite sequence of operations, and likewise, that the runner has completed an infinite number of runs. Exercise B. Russell disposed of the Arrow paradox definitively on the assumption of the mathematical continuity of space, time, and motion. Zeno has been accused of the inability to distinguish between instantaneous rest and instantaneous motion. The Stadium paradox depends on disregarding the relative motions of the bodies. What is meant by these statements and how do they help explain the paradoxes? Resolution of the Paradox (A Philosophical Puppet Play) by Abner Shimony Characters: Zeno, Pupil, Lion Setting: The school of Zeno at Elea Pupil: Master! There is a lion in the streets! Zeno: Very good. You have learned your lesson in geography well. The fifteenth meridian, as measured from Greenwich, coincides with the high road from the Temple of Poseidon to the Agora-- but you must not forget that it is an imaginary line. Pupil: Oh no, Master! I must humbly disagree. It is a real lion, a menagerie lion, and it is coming toward the school! Zeno: My boy, in spite of your proficiency at geography, which is commendable in its way-- albeit essentially the art of the surveyor and hence separated by the hair of the theodolite from the craft of a slave-- you are deficient in philosophy. That which is real cannot be imaginary, and that which is imaginary cannot be real. Being is, and non-being is not, as my revered teacher Parmenides demonstrated first, last, and continually, and as I have attempted to convey to you. Pupil: Forgive me, Master. In my haste and excitement, themselves expressions of passion unworthy of you and of our school, I have spoken obscurely. Into the gulf between the thought and the word, which, as you have taught us, is the trap set by non-being, I have again fallen. What I meant to say is that a lion has escaped from the zoo, and with deliberate speed it is rushing in the direction of the school and soon will be here! The lion appears in the distance. Zeno: O my boy, my boy! It pains me to contemplate the impenetrability of the human intellect and its incommensurability with the truth. Furthermore, I now recognize that a thirty-year novitiate is too brief-- sub specie aeternitatis-- and must be extended to forty years, before the apprenticeship proper can begin. A real lion, perhaps; but really running, impossible; and really arriving here, absurd! Pupil: Master... Zeno: In order to run from the zoological garden to the Eleatic school, the lion would first have to traverse half the distance. The lion traverses half the distance. Zeno: But there is a first half of that half, and a first half of that half, and yet again a first half of that half to be traversed. And so the halves would of necessity regress to the first syllable of recorded time-- nay, they would recede yet earlier than the first syllable. To have traveled but a minute part of the interval from the zoological garden to the school, the lion would have been obliged to embark upon his travels infinitely long ago. The lion bursts into the schoolyard. Pupil: O Master, run, run! He is upon us! Zeno: And thus, by reductio ad absurdum, we have proved that the lion could never have begun the course, the mere fantasy of which has so unworthily filled you with panic. The pupil climbs an Ionic column, while the lion devours Zeno. Pupil: My mind is in a daze. Could there be a flaw in the Master's argument? Bibliography Allen, R.E. Plato's Parmenides: Translation and Analysis. Minneapolis: Univ. of Minnesota Press, 1983. Grunbaum, Adolf. Modern Science and Zeno'x Paradoxes. Middletown: Wesleyan University Press, 1967. Meinwald, Constance C. Plato's Parmenides. New York: Oxford University Press, 1991. Rossvaer, Viggo. The Laborious Game: A Study of Plato's Parmenides. Norway: Universitetsforlaget, 1983. Salmon, Wesley C.(Editor) Zeno's Paradoxes. New York: Bobbs-Merrill Co. Inc., 1970. Sternfeld, Robert and Harold Zyskind. Meaning, Relation, and Existence in Plato's Parmenides. New York: Peter Lang Publishing Inc., 1987.