In the 5th century B.C.E., Zeno of Elea offered arguments that led to conclusions contradicting what we all know from our physical experience–that arrows fly, that runners run, and that there are many different things in the world. The arguments were paradoxes for the ancient Greek philosophers. Because the arguments turn crucially on the notion that space and time are infinitely divisible, Zeno can be credited with being the first person in history to show that the concept of infinity is problematical.
In his Achilles Paradox, the fastest runner of antiquity, Achilles, races to catch a slower runner–for example, a tortoise that is crawling slowly away from him. The tortoise has a head start, so if Achilles hopes to overtake it, he must run at least to the place where the tortoise is when he sees it, but by the time he arrives there, it will have crawled to a new place, so then Achilles must run to this new place, but the tortoise meanwhile will have crawled on, and so forth. Achilles will never catch the tortoise, says Zeno, because the series of goals has no final member. Therefore, good reasoning shows that fast runners never can catch slow ones. So much the worse for the claim that motion really occurs, Zeno says in defense of his mentor Parmenides who had argued that motion is an illusion.
Although very few scholars today would agree with Zeno’s conclusions, we can not escape them by jumping up from our seat and chasing down a tortoise. What is required is a principled way out that does not get us embroiled in new paradoxes nor impoverish our mathematics and science.
This article explains his nine known paradoxes and considers the treatments that have been offered. Aristotle accused Zeno of not paying sufficient attention to the fact that there are no actual infinities but only potential infinities, and that durations divide into intervals but never into indivisible instants, and his treatment became the generally accepted solution until the late 19th century. The Standard Solution that is favored by most philosophers today employs the apparatus of calculus which has proved its indispensability for the development of modern science. Its key ideas are that space, time and motion are continua; that positions, distances, times, durations and speeds all should be treated as continuous variables whose values are real numbers or intervals of real numbers; that an object can have a positive speed at a point-place; and that some infinite series of positive terms can have a finite sum. From this perspective, Zeno was working with false assumptions. This Standard Solution took hundreds of years to perfect and was due to the flexibility of intellectuals who were willing to replace old theories and their concepts with more fruitful ones, despite the damage done to common sense and our naive intuitions. The article ends by exploring newer treatments developed since the 1960s.
Source: http://www.iep.utm.edu
In his Achilles Paradox, the fastest runner of antiquity, Achilles, races to catch a slower runner–for example, a tortoise that is crawling slowly away from him. The tortoise has a head start, so if Achilles hopes to overtake it, he must run at least to the place where the tortoise is when he sees it, but by the time he arrives there, it will have crawled to a new place, so then Achilles must run to this new place, but the tortoise meanwhile will have crawled on, and so forth. Achilles will never catch the tortoise, says Zeno, because the series of goals has no final member. Therefore, good reasoning shows that fast runners never can catch slow ones. So much the worse for the claim that motion really occurs, Zeno says in defense of his mentor Parmenides who had argued that motion is an illusion.
Although very few scholars today would agree with Zeno’s conclusions, we can not escape them by jumping up from our seat and chasing down a tortoise. What is required is a principled way out that does not get us embroiled in new paradoxes nor impoverish our mathematics and science.This article explains his nine known paradoxes and considers the treatments that have been offered. Aristotle accused Zeno of not paying sufficient attention to the fact that there are no actual infinities but only potential infinities, and that durations divide into intervals but never into indivisible instants, and his treatment became the generally accepted solution until the late 19th century. The Standard Solution that is favored by most philosophers today employs the apparatus of calculus which has proved its indispensability for the development of modern science. Its key ideas are that space, time and motion are continua; that positions, distances, times, durations and speeds all should be treated as continuous variables whose values are real numbers or intervals of real numbers; that an object can have a positive speed at a point-place; and that some infinite series of positive terms can have a finite sum. From this perspective, Zeno was working with false assumptions. This Standard Solution took hundreds of years to perfect and was due to the flexibility of intellectuals who were willing to replace old theories and their concepts with more fruitful ones, despite the damage done to common sense and our naive intuitions. The article ends by exploring newer treatments developed since the 1960s.
Source: http://www.iep.utm.edu
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