Pre-Socratic Philosopher: Zeno - Essay Example

Many of the Pre-Socratic philosophers are known now just by their contributions to the thought and writings of Plato and Aristotle, whose works have survived through the ages to the present date. If their work did survive, it was likely in the form of fragments (as in the case of Heraclitus) or through second-hand reproductions of their quotes in the works of others. Zeno of Elea was a widely-regarded Pre-Socratic philosopher, as we can tell from how many ancient writings make reference to his work, including those from Plato and from Aristotle. Just as in modern academia, frequent references to a work means that members of the field know of and respect the opinions of its author. We know of Zeno’s work primarily through Aristotle’s arguments against the paradoxes that Zeno proposed in defense of his teacher, Parmenides, and their school of thought. Although very little about Zeno’s life and his work is known, and what is known coming from only second-hand sources, ultimately what defines the legacy of a philosopher is not the volume of work that survives or whether his or her name is attached to it – what defines the legacy is whether his or her ideas pushed the field along to a state of higher enlightenment and more productive conversations, which Zeno definitely did. The primary source of biographical information about Zeno comes from a work by Plato dealing with Zeno’s teacher, Parmenides. In the text, Plato describes Zeno’s visit to Athens. The author writes, “Zeno was nearly 40 years of age, tall and fair to look upon; in the days of his youth he was reported to have been beloved by Parmenides” (127b-e). At the time of its writing at approximately 450 BCE, Zeno’s birth date thus would have been around 490 BCE, making him double the age of the young Socrates at the time. This glimpse at Zeno is realistic in the sense that Zeno’s relationship with Parmenides was likely one of an apprentice to a teacher, given the closeness of the beliefs of the two men. Zeno’s paradoxes, coming from the assumption that Zeno composed a single work including as many as forty antimonies, are defenses of Parmenides’ way of seeing the world (Proclus, Morrow and Dillon 29). Ultimately, the point of Zeno’s work and of his discussion with Socrates is to defend these arguments that attempt to destroy traditional understandings of motion and ontology (Palmer). Much attention to Zeno’s work has been paid to reconstructing his arguments from the texts of other writings, which is an activity that produces nine existing paradoxes from the 40 or more that were originally written according to biographers. One must take great care to reconstruct the paradoxes faithfully, for how one describes the paradoxes will determine an effective response by modern thinkers. Of the nine surviving paradoxes, some are merely rewordings of each other and are not significantly different to warrant individualized attention, and for that reason, philosophers thinking on this subject mainly focus on three of the strongest and most famous cases, which are the paradox of the arrow, the dichotomy paradox, and Achilles and the tortoise (Huggett). Some allege that Zeno’s paradoxes are simple mathematical problems that can be solved using modern mathematical techniques rooted in calculus, which looks at infinitesimals and very small measurements of motion. This applies particularly for the paradox of the arrow, which has a straightforward mathematical explanation (Boyer 295-297). However, some philosophers still believe that Zeno’s paradoxes offer very real metaphysical problems, beyond just the mathematical aspects of the problem, that still need to be solved (Brown). Zeno’s arrow paradox begins with the argument that for motion to occur, an object must move (that is, change the position it is occupying). As an example, we take an arrow in flight, which at any one duration-less instant in time, the arrow is neither moving to where it is nor to where it is not. The arrow cannot move to where it is not, for no time has gone by for it to move, and it cannot move to where it is, for it is already there. The conclusion of this paradox is that at any instant of time, there is no motion occurring and so, if everything is without motion at each instant, and time is entirely made up of instants, then motion itself is impossible. The antimony works by dividing time into points or instants within which there is no motion. Aristotle, who preserves and reconstructs this argument in his Physics, responds by claiming Zeno is false in assuming that time is composed of indivisible points. Modern critics, however, take a different approach, by denying that motion is a property of an instant with some “desire” to be elsewhere based on Aristotelian physics. Instead, motion is a property of an object at an instant. In calculus, the speed of an object under Zeno’s view would be the derivative of position x with respect to time t (dx/dt), which would be zero divided by zero, or undefined. On that basis, the assumption that the arrow is at rest at each instant is unfounded (Dowden). Zeno’s dichotomy paradox deals with motion in space. According to Aristotle’s reconstruction, “That which is in locomotion must arrive at the half-way stage before it arrives at the goal” (VI:9, 239b10). In other words, in order for an object to get to its destination, it must first get half-way there; before it gets half-way there, it must get a quarter of the way there; before it gets a quarter of the way there, it must get one-eighth of the way there, and so on ad infinitum. Zeno believes it is impossible for an object to complete an infinite number of steps toward a goal. Paradoxically, motion must be an illusion because if one plans to run a distance, and any possible distance can be divided in half, there would be no first motion as it would require an infinite number of first motions to complete. The dichotomy paradox is based on repeatedly splitting distances into two sections, the second of which is always impossible to reach because the first part is always being split in half (Huggett). Aristotle responded to this quandary by asserting that it is possible for a runner to reach potential infinites in a finite time provided the time intervals become shorter and shorter. While Aristotle was ultimately right, he lacked the mathematical tools to demonstrate why he was right. Today, using calculus to show differentials, modern critics of the dichotomy paradox use calculus to compute the finite amount of time it takes to reach these potentially infinite distances. That account rejects the idea that there must be a final step or a last member of an infinite sequence of steps (Dowden). The final paradox of note has to deal with a footrace between Achilles and a tortoise. In the story, Achilles allows the tortoise a head start (in this example, we will say 100 feet). If each participant in the race starts running at a different, constant speed, then Achilles will eventually reach the Tortoise’s head start position. During the time it took Achilles to get there, the tortoise would have gone a further distance, which requires Achilles to go some further distance, by which time the tortoise has already gone farther (Huggett). The paradox is that whenever Achilles reaches a place where the tortoise had been, he will still have more distance to cover. Accordingly, there is an infinite number of points that Achilles must go where the tortoise has already been, so he must not be able to overtake the tortoise in this race. It is not clear whether Zeno is assuming it is too far to run, there is not enough time, there are too many places to go, there is no final step, or there are too many tasks to complete (Dowden). Modern critics of the paradox argue that all are false. As we saw in the response to the dichotomy paradox, modern calculus can explain why it is not too far to run, why there are not too many places to go, why there is no final step, and where there are not too many tasks. One can see the applications of calculus’ answer to the arrow paradox apply also here to the assumption that there is not enough time, for motion is not in an instant but rather at an instant as described in calculus (Palmer). Zeno’s legacy as a philosopher is not judged by how much of his work survived or how much credit he received for his ideas; rather, his legacy is a product of the impact that his contributions made to the overall philosophical conversation on topics of motion, time, and space. Citations of his work are common throughout the ages, which shows the overall effect that he had on pushing philosophy and related sciences of mathematics and physics toward more knowledge of the world. While some believe philosophy began with the writings of Plato, a better understanding of history gives credit to Pre-Socratic philosophers for laying the conceptual foundation for Plato and Aristotle to put forward their theories. Works Cited Aristotle. Physics. Trans. R. P. Hardie and R. K. Gaye. n.d. http://classics.mit.edu/Aristotle/physics.html. Boyer, Carl. The History of the Calculus and its Conceptual Development. Dover Publications, 1959. Print. Brown, Kevin. Zeno and the Paradox of Motion. n.d. http://www.mathpages.com/rr/s3-07/3-07.htm. 10 March 2014. Dowden, Bradley. Zenos Paradoxes. n.d. http://www.iep.utm.edu/zeno-par/. Huggett, Nick. Zenos Paradoxes. 30 April 2002. http://plato.stanford.edu/entries/paradox-zeno/. Palmer, John. Zeno of Elea. 9 January 2008. http://plato.stanford.edu/entries/zeno-elea/. Plato. Parmenides. Trans. Benjamin Jowett. n.d. http://classics.mit.edu/Plato/parmenides.html. Proclus, Glenn R. Morrow and John M. Dillon. Proclus Commentary on Platos Parmenides. Princeton, NJ: Princeton University Press, 1992. Print. Read More