Paste your essay in here… A Case Study: Zeno’s Dichotomy
When discussing Zeno’s notable contributions to the philosophy of motion, it seems appropriate to quote Bertrand Russell’s assertion: “Zeno's arguments, in some form, have afforded grounds for almost all theories of space and time and infinity which have been constructed from his time to our own”(Russell 178). During his career in the 5th century B.C, Zeno of Elea served as a fervent advocate for Parmenidean notions of space and time. As such, several of his most distinguished paradoxes serve to bolster and protect Parmenides’ theories from critics of the Eleatic school. For instance, in the famed “Dichotomy” paradox, Zeno brilliantly extended Parmenides’ thesis on change by expanding its application to motion. A testament to the impossibility of movement, “The Dichotomy” argues that any travel requires the passage of infinite mid-points and that, on account of this unending division, motion becomes an absurdity. This paper will begin with a short clarification of Zeno’s “Dichotomy” before examining its merits and deficiencies. Ultimately, I demonstrate that the “Dichotomy” is a flawed theory of motion because it misrepresents the divisibility of space and neglects to recognize the finite sum of infinite mid-ways points.
“The Dichotomy” can be reduced to a rather simple image. In order to get to Point B from Point A, a traveller must first reach the mid-way point A1. Likewise, in order to get to A1 , the traveller needs to hit mid-way point A2. According to Zeno, because this process can be repeated ad infinitum, the traveller can never reach Point B and movement is rendered impossible. However, a new question now arises: what does he mean by “this process?” Traditionally, Zeno’s “Dichotomy” has been interpreted in two ways. Primarily, in attempting to understand Zeno’s assertion that “it is always necessary to traverse half the distance, but these are infinite, and it is impossible to get through things that are infinite”(McKirahan 184), some philosophers understand Zeno to be arguing that infinitely divisible spaces can not have a finite sum. In other words, we can never get to point b from point a because this journey requires an infinite amount of mid-way points which do not amount to b. Alternatively, following Aristotle's path, some consider Zeno’s doctrine to be dealing with time. In this view, motion is impossible because the passage of infinite mid-way points takes an infinite amount of time. Since the amount of time it takes to cross a given interval is not quantifiable, we can not conclude that motion is possible. With these readings in mind, the subsequent section will concern itself with the merits of Zeno’s “Dichotomy.”
Though the extreme implications of Zeno’s theory might prompt us to disregard his ideas, “The Dichotomy” presents several profound difficulties. In fact, the logical deductions proposed by Zeno appear to withstand intense criticism. Afterall, if one grants the idea that time and space are infinitely divisible, it does seem that motion is impossible. Therefore, in order to refute Zeno’s conclusions, one must return to his premise. As Bertrand Russell points out in Our Knowledge of the External World, if we wish to escape Zeno’s paradox we can either establish the continuity of space and time or argue that finite spaces contain an infinite amount of traversable points. In this view, in order to disprove Zeno, it seems as though one must make additional assumptions about time and space. Indeed, though many contemporary philosophers disagree with Zeno’s instantaneous view of these entities, it is far more difficult to criticize his methodical arguments. Ultimately, while Zeno’s theory of motion relies upon various critiqueable presumptions, “The Dichotomy” remains notable for its dialectical achievement.
Though Zeno’s paradox is worthy of praise for its reasonable deductions, depending on one’s interpretation, there are some serious logical flaws. This paragraph will address the problems with the first of the aforementioned readings of “The Dichotomy”. That is, I will examine the issues associated with the claim that infinite divisions cannot render a finite sum. As was discussed before, some philosophers understand Zeno’s paradox to be proposing that a traveller cannot move between Point A and Point B because the mid-way portions of space between these points are infinite and therefore can not render a finite Point B. However, if one examines an analogous mathematical problem, this form of Zeno’s argument is successfully discredited. Assume, for the sake of this metaphor, that Point A is equal to 0 and Point B is equal to 1. In such a scenario, the mid-way point A1 would be represented by the fraction ½ and A2 becomes ¼. Now, as long as the divisions continue ad infinitum, the sum of infinite series eventually renders 1, or in this case, Point B. For example: (Point A)=0…+ ½+¼+⅛+…=1(Point B). Thus, if one understands Zeno to mean that motion does not exist because passing through an infinitely divisible space seems impossible, this mathematical equation demonstrates that infinite series can, in fact, have a finite sum.
On the other hand, the Aristotelian tradition argues that Zeno considers space to be divisible into portions that take infinitely long to traverse. The objection here is that Zeno conflates infinitely divisible areas with spaces that contain infinite extremities. In this reading, Aristotle himself provides the most convincing counter-argument. Particularly, he contends that an area’s divisibility has no bearing on its extremities. For example, though the distance between Point A and Point B may be infinitely divided such that A1 and A2 can exist, Point A and Point B are definitive boundaries, and thus they measure a finite area. The same theory applies to the amount of time it takes to pass this space. Say, for instance, that it takes 60 minutes to travel from Point A to Point B. Aristotle would argue that this 60 minutes is a finite period of time that happens to be infinitely divisible. In Philosophy Before Socrates, Richard McKirahan provides a clear synopsis of this rebuttal: “There is no need to suppose that the infinite number of distances needing to be passed in getting from A to B will take an amount of time that is infinite in extent, only one that is (harmlessly) infinitely divisible (McKirahan 183). While this critique recognizes that Zeno is correct in assuming the infinite divisibility of time and space, it does not consider this to be a sufficient objection to the possibility of motion.
Overall, as the two preceding sections showed, Zeno’s “Dichotomy” is a untenable theory of motion because it conflates the infinitely divisible with infinite extremities, and it overlooks the fact that infinite series may render a finite sum. Despite its shortcomings, “The Dichotomy” is a serious paradox that, due to its compelling use of reason and deduction, forces us to re-evaluate the fundamental notions that govern our lives. Referred to by Aristotle as the “father of the dialectic,” Zeno’s robust charges against change, plurality and motion often managed to cast doubt upon popular opinions via logical inquiry. Thus, even when we disagree with his premises, Zeno’s dialectical achievements are worth of our praise.