1. Zeno's Race Course, Part 1 Thoughtful lecture notes for discussing this paradox, presented by S. Marc Cohen. http://faculty.washington.edu/smcohen/320/zeno1.htm

The Paradox Zeno argues that it is impossible for a runner to traverse a race course. His reason is that Physics Why is this a problem? Because the same argument can be made about half of the race course: it can be divided in half in the same way that the entire race course can be divided in half. And so can the half of the half of the half, and so on, ad infinitum So a crucial assumption that Zeno makes is that of infinite divisibility : the distance from the starting point ( S ) to the goal ( G ) can be divided into an infinite number of parts. Progressive vs. Regressive versions How did Zeno mean to divide the race course? That is, which half of the race course Zeno mean to be dividing in half? Was he saying (a) that before you reach G , you must reach the point halfway from the halfway point to G ? This is the progressive version of the argument: the subdivisions are made on the right-hand side, the goal side, of the race-course. Or was he saying (b) that before you reach the halfway point, you must reach the point halfway from S to the halfway point? This is the

2. Zeno's Paradoxes Discusses the paradoxes of Zeno of Elea, e.g., Achilles and the Tortoise; by Nick Huggett. http://plato.stanford.edu/entries/paradox-zeno/

version history HOW TO CITE THIS ENTRY Stanford Encyclopedia of Philosophy A B C D ... Z This document uses XHTML-1/Unicode to format the display. Older browsers and/or operating systems may not display the formatting correctly. last substantive content change MAR Zeno's Paradoxes Almost everything that we know about Zeno of Elea is to be found in the opening pages of Plato's Parmenides 1. Background 2. The Paradoxes of Plurality 2.1 The Argument from Denseness 2.2 The Argument from Finite Size 2.3 The Argument from Complete Divisibility 3. The Paradoxes of Motion 3.1 The Dichotomy 3.2 Achilles and the Tortoise 3.3 The Arrow ... Related Entries 1. Background Before we look at the paradoxes themselves it will be useful to sketch some of their historical and logical significance. First, Zeno sought to defend Parmenides by attacking his critics. Parmenides rejected pluralism and the reality of any kind of change: for him all was one indivisible, unchanging reality, and any appearances to the contrary were illusions, to be dispelled by reason and revelation. Not surprisingly, this philosophy found many critics, who ridiculed the suggestion; after all it flies in the face of some of our most basic beliefs about the world. (Interestingly, general relativity particularly quantum general relativity arguably provides a novel if novelty is As we read the arguments it is crucial to keep this method in mind. They are always directed towards a more-or-less specific target: the views of some person or school. We must bear in mind that the arguments are

3. Zeno's Paradox Of The Tortoise And Achilles (PRIME) An article in the Platonic Realms. http://www.mathacademy.com/pr/prime/articles/zeno_tort/

BROWSE ALPHABETICALLY LEVEL: Elementary Advanced Both INCLUDE TOPICS: Basic Math Algebra Analysis Biography Calculus Comp Sci Discrete Economics Foundations Geometry Graph Thry History Number Thry Physics Statistics Topology Trigonometry eno of Elea ( circa 450 b.c.) is credited with creating several famous paradoxes , but by far the best known is the paradox of the Tortoise and Achilles. (Achilles was the great Greek hero of Homer's The Illiad .) It has inspired many writers and thinkers through the ages, notably Lewis Carroll and Douglas Hofstadter, who also wrote dialogues involving the Tortoise and Achilles. The original goes something like this: The Tortoise challenged Achilles to a race, claiming that he would win as long as Achilles gave him a small head start. Achilles laughed at this, for of course he was a mighty warrior and swift of foot, whereas the Tortoise was heavy and slow. Achilles said nothing. Zeno's Paradox may be rephrased as follows. Suppose I wish to cross the room. First, of course, I must cover half the distance. Then, I must cover half the remaining distance. Then, I must cover half the remaining distance. Then I must cover half the remaining distance . . . and so on forever. The consequence is that I can never get to the other side of the room.

4. Zeno's Paradox Of The Tortoise And Achilles (PRIME) Zeno’s paradox of the Tortoise and Achilles. eno race. Zeno s paradoxmay be rephrased as follows. Suppose I wish to cross the room. http://www.mathacademy.com/pr/prime/articles/zeno_tort/index.asp

BROWSE ALPHABETICALLY LEVEL: Elementary Advanced Both INCLUDE TOPICS: Basic Math Algebra Analysis Biography Calculus Comp Sci Discrete Economics Foundations Geometry Graph Thry History Number Thry Physics Statistics Topology Trigonometry eno of Elea ( circa 450 b.c.) is credited with creating several famous paradoxes , but by far the best known is the paradox of the Tortoise and Achilles. (Achilles was the great Greek hero of Homer's The Illiad .) It has inspired many writers and thinkers through the ages, notably Lewis Carroll and Douglas Hofstadter, who also wrote dialogues involving the Tortoise and Achilles. The original goes something like this: The Tortoise challenged Achilles to a race, claiming that he would win as long as Achilles gave him a small head start. Achilles laughed at this, for of course he was a mighty warrior and swift of foot, whereas the Tortoise was heavy and slow. Achilles said nothing. Zeno's Paradox may be rephrased as follows. Suppose I wish to cross the room. First, of course, I must cover half the distance. Then, I must cover half the remaining distance. Then, I must cover half the remaining distance. Then I must cover half the remaining distance . . . and so on forever. The consequence is that I can never get to the other side of the room.

5. Math Forum: Zeno's Paradox Zeno s paradox. A Math Forum Project http://mathforum.org/isaac/problems/zeno1.html

Zeno's Paradox A Math Forum Project Table of Contents: Famous Problems Home The Bridges of Konigsberg The Value of Pi Prime Numbers ... Links The great Greek philosopher Zeno of Elea (born sometime between 495 and 480 B.C.) proposed four paradoxes in an effort to challenge the accepted notions of space and time that he encountered in various philosophical circles. His paradoxes confounded mathematicians for centuries, and it wasn't until Cantor's development (in the 1860's and 1870's) of the theory of infinite sets that the paradoxes could be fully resolved. Zeno's paradoxes focus on the relation of the discrete to the continuous, an issue that is at the very heart of mathematics. Here we will present the first of his famous four paradoxes. Zeno's first paradox attacks the notion held by many philosophers of his day that space was infinitely divisible, and that motion was therefore continuous. Paradox 1: The Motionless Runner A runner wants to run a certain distance - let us say 100 meters - in a finite time. But to reach the 100-meter mark, the runner must first reach the 50-meter mark, and to reach that, the runner must first run 25 meters. But to do that, he or she must first run 12.5 meters. Since space is infinitely divisible, we can repeat these 'requirements' forever. Thus the runner has to reach an infinite number of 'midpoints' in a finite time. This is impossible, so the runner can never reach his goal. In general, anyone who wants to move from one point to another must meet these requirements, and so motion is impossible, and what we perceive as motion is merely an illusion.

6. Zeno's Paradox Again By Jesse Yoder Zeno s paradox again by Jesse Yoder. reply to this area. To my knowledge,Zeno s paradox has never been completely solved. Zeno s paradox http://mathforum.org/epigone/geometry-research/spolchordwom

Zeno's Paradox again by Jesse Yoder reply to this message post a message on a new topic Back to geometry-research Subject: Zeno's Paradox again Author: jesse@flowresearch.com Date: The Math Forum

7. Puzzle: Zeno's Paradox Zeno s paradox. http://www.deltalink.com/dodson/html/puzzle.html

Zeno's Paradox Solution Zeno was a famous mathematician from Elea, a Greek city on the Italian coast. Zeno was well known for posing puzzling paradoxes that seemed impossible to resolve. One of his his most well known paradoxes was that of Achilles and the tortoise. Suppose you have a race between Achilles and a tortoise. Now suppose that Achilles runs 10 times as fast as the tortoise and that the tortoise has a 10 meter head start at the beginning of the race. Zeno argued that in such a situation, it would take Achilles an infinite amount of time to catch the tortoise. His argument went as follows: By the time Achilles runs the 10 meters to the point where the tortoise began, the tortoise will have traveled one meter and will therefore still be one meter ahead of Achilles. Then, by the time Achilles covers a distance ofone meter, the tortoise will have traveled one tenth of a meter and is still ahead of Achilles. After Achilles travels one tenth of a meter, the tortoise will have traveled 1/100th of a meter. Each time Achilles reaches the previous position of the tortoise, the tortoise has reached another position ahead of Achilles. As long as it takes Achilles some amount of time to traverse the distance between the point where he is and the point where the tortoise is, the tortoise will have time to move slightly beyond that point. No matter how long the race goes on, Achilles will have to move through every point where the tortoise has been before he can pass him. Each time Achilles reaches such a point, the tortoise is at another point. Therefore, Achilles will have to pass through an infinite number of points in order to catch up with the tortoise. If it takes him some time to pass through each one of these points, it will take hiim forever to catch up. Can you find the faulty logic in the above argument?

8. Zeno's Paradox Resolved Zeno s paradox Resolved. Mathematical Treatment of Zeno s paradox. For a mathematicaltreatment of Zeno s paradox, download the PDF file. Back to Puzzle. http://www.deltalink.com/dodson/html/zeno.solution.html

Zeno's Paradox Resolved T Mathematical Treatment of Zeno's Paradox F or a mathematical treatment of Zeno's Paradox, download the PDF file

9. Zeno And The Paradox Of Motion Regarding these first two arguments, there s a tradition among some high schoolcalculus teachers to present them as Zeno s paradox , and then resolve the http://www.mathpages.com/rr/s3-07/3-07.htm

3.7  Zeno and the Paradox of Motion The Eleatic school of philosophers was founded by the religious thinker and poet Xenophanes (born c. 570 BC), whose main teaching was that the universe is singular, eternal, and unchanging.  "The all is one."  According to this view, as developed by later members of the Eleatic school, the appearances of multiplicity, change, and motion are mere illusions.  Interestingly, the colony of Elea was founded by a group of Ionian Greeks who, in 545 BC, had been besieged in their seaport city of Phocaea by an invading Persian army, and were ultimately forced to evacuate by sea.  They sailed to the island of Corsica , and occupied it after a terrible sea battle with the navies of  Carthage and the Etruscans.  Just ten years later, in 535 BC, the Carthagians and Etruscans regained the island, driving the Phocaean refugees once again into the sea.  This time they landed on the southwestern coast of Italy and founded the colony of Elea , seizing the site from the native Oenotrians.  All this happened within the lifetime of Xenophanes, himself a wandering exile from his native city of

10. Zeno's Paradox Of The Arrow Zeno’s paradox of the Arrow. A reconstruction of the argument. Go toprevious lecture on the Zeno’s paradox of the Race Course, part 2. http://faculty.washington.edu/smcohen/320/ZenoArrow.html

A reconstruction of the argument (following Aristotle, Physics 239b5-7 = RAGP 10): 2. At every moment of its flight, the arrow is in a place just its own size. 3. Therefore, at every moment of its flight, the arrow is at rest. The velocity of x at instant t can be defined as the limit of the sequence of x t x is in a place just the size of x at instant i x is resting at i nor that x is moving at i Perhaps instants and intervals are being confused War and Peace 1a. At every instant false 2a. At every instant during its flight, the arrow is in a place just its own size. ( true 1b. During every interval true 2b. During every interval of time within its flight, the arrow occupies a place just its own size. ( false A final reconstruction The order in which these quantifiers occur makes a difference! (To find out more about the order of quantifiers, click here .) Observe what happens when their order gets illegitimately switched: 1c. If there is a place just the size of the arrow at which it is located at every instant between t and t , the arrow is at rest throughout the interval between t and t 2c. At every instant between

11. Zeno's Paradox A link to an unusual and strange discussion of Zeno s paradox in Reality Inspector,a novel about chess and computerhacking. I have solved Zeno s paradox. http://www.westgatehouse.com/zeno.html

An unusual and strange discussion of Zeno's paradox can be found in chapters , and of Reality Inspector , a novel about chess and computer-hacking. The appropriate excerpts are presented below. If you wish to read the story context surrounding the excerpts, go to the three linked chapters. If you have comments or questions about the ideas, please contact John Caris from chapter Then he sees a human figure close to the edge of the woods. Walking over, he notices that the person is painting, no doubt a landscape scene. "Hi, there." The painter turns around, brush in one hand, palette in the other. "Oh, hi." He is not too enthusiastic, but a little disconcerted about the interruption. "I'm John Ocean. And I seem to be lost. Can you tell me what place this is?" "I've heard of you. You're a reality inspector, aren't you?" "Yes." John feels flustered and confused, not so much by the response of the painter but by the overall strangeness of the situation. "I'm Achilles." "The Achilles?" "How many are there?" "The Greek who fought in the Trojan war?"

12. Zeno's Paradox, 2 A link to an unusual and strange discussion of Zeno s paradox in RealityInspector, a novel about chess and computerhacking. from http://www.westgatehouse.com/zeno2.html

from chapter Suddenly, a terrible force grabs her. She gulps a deep breath and looks up. Sitting across from her is a strange man dressed in a Greek toga. He smiles and says, "I'm Achilles. Do you wish to hear how I beat the tortoise?" Mary glances about. She is no longer in the Cow Palace; she is in a sunlit room. The walls are plain; on one wall hang ancient weapons of war. When in Greece do as the Greeks, she thinks. "Of course, I want to hear." "Well, when I realized that the tortoise could never win, I knew I had time to discover a solution. You see, the tortoise can't win because it can never cross over the finish line. For the finish line is one dimensional; it has only length but no width. So the tortoise is stopped by the abyss of non-dimensional space. Let me show you." Achilles draws on a piece of paper. "Anything stepping into the abyss will get lost forever because spatial coordinates don't exist there. How can you tell where you are unless you have some reference system? It's like being in a boat on the ocean without having any means for navigating. You just drift about. And in the abyss you drift for eternity. "Now notice that the abyss separates three dimensional space. Since the abyss lacks a spatial dimension, the three dimensional space is actually contiguous. But the abyss does have the dimension of time. Here eternity exists. The present is." Achilles looks at Mary and smiles.

13. Zeno's Paradoxes And Zeno of Elea (5th Century BC) was the Zeno of the paradoxes. To me, Zeno sarrow paradox seems much more interesting than his other paradoxes. http://www.jimloy.com/physics/zeno.htm

Return to my Physics pages Go to my home page Zeno's Paradoxes Among the most famous of Zeno's "paradoxes" involves Achilles and the tortoise, who are going to run a race. Achilles, being confident of victory, gives the tortoise a head start. Zeno supposedly proves that Achilles can never overtake the tortoise. Here, I paraphrase Zeno's argument: Before Achilles can overtake the tortoise, he must first run to point A, where the tortoise started. But then the tortoise has crawled to point B. Now Achilles must run to point B. But the tortoise has gone to point C, etc. Achilles is stuck in a situation in which he gets closer and closer to the tortoise, but never catches him. What Zeno is doing here, and in one of his other paradoxes, is to divide Achilles' journey into an infinite number of pieces. This is certainly permissible, as any line segment can be divided into an infinite number of points or line segments. This, in effect, divides Achilles' run into an infinite number of tasks. He must pass point A, then B, then C, etc. And what Zeno is arguing is that you can't do an infinite number of tasks in a finite amount of time. Why not? Zeno says that you can divide a line into an infinite number of pieces. And then he says that you cannot divide a time interval into an infinite number of pieces. This is inconsistent.

14. Math Lair - Zeno S Paradox Click Here! Zeno s paradox. Zeno s Racecourse paradox involves the story of a racebetween Achilles and a tortoise. Zeno s Bisection paradox Zeno s Assertion http://www.stormloader.com/ajy/zeno.html

15. 4.1. Series And Convergence Tortoise. Example 4.1.1 Zeno s paradox (Achilles and the Tortoise).Achilles, a fast runner, was asked to race against a tortoise. http://www.shu.edu/projects/reals/numser/series.html

4.1. Series and Convergence IRA So far we have learned about sequences of numbers. Now we will investigate what may happen when we add all terms of a sequence together to form what will be called an infinite series. The old Greeks already wondered about this, and actually did not have the tools to quite understand it This is illustrated by the old tale of Achilles and the Tortoise. Example 4.1.1: Zeno's Paradox (Achilles and the Tortoise) Achilles, a fast runner, was asked to race against a tortoise. Achilles can run 10 meters per second, the tortoise only 5 meter per second. The track is 100 meters long. Achilles, being a fair sportsman, gives the tortoise 10 meter advantage. Who will win ? Both start running, with the tortoise being 10 meters ahead. After one second, Achilles has reached the spot where the tortoise started. The tortoise, in turn, has run 5 meters. Achilles runs again and reaches the spot the tortoise has just been. The tortoise, in turn, has run 2.5 meters. Achilles runs again to the spot where the tortoise has just been. The tortoise, in turn, has run another 1.25 meters ahead.

16. 10.12. Zeno Of Elea (495?-435? B.C.) For related information of Zeno, see Georg Cantor , Zeno s paradox. Sources.Bell, ET Men of Mathematics. New York Simon and Schuster, Inc., 1937. http://www.shu.edu/projects/reals/history/zeno.html

10.12. Zeno of Elea (495?-435? B.C.) IRA Zeno of Elea was the first great doubter in mathematics. His paradoxes stumped mathematicians for millennia and provided enough aggravation to lead to numerous discoveries in the attempt to solve them. Zeno was born in the Greek colony of Elea in southern Italy around 495 B.C. Very little is known about him. He was a student of the philosopher Parmenides and accompanied his teacher on a trip to Athens in 449 B.C. There he met a young Socrates and made enough of an impression to be included as a character in one of Plato's books Parmenides . On his return to Elea he became active in politics and eventually was arrested for taking part in a plot against the city's tyrant Nearchus. For his role in the conspiracy, he was tortured to death. Many stories have arisen about his interrogation. One anecdote claims that when his captors tried to force him to reveal the other conspirators, he named the tyrant's friends. Other stories state that he bit off his tongue and spit it at the tyrant or that he bit off the Nearchus' ear or nose. Zeno was a philosopher and logician, not a mathematician. He is credited by Aristotle with the invention of the dialectic, a form of debate in which one arguer supports a premise while another one attempts to reduce the idea to nonsense. This style relied heavily on the process of

17. Zeno's Paradoxes tasks. The bestknown example of a current-day Zeno type paradox isthe Thomson Lamp, named after James F. Thomson. The Thomson Lamp. http://members.aol.com/kiekeben/zeno.html

Zeno's Paradoxes Zeno of Elea was an ancient Greek (born around 490 B.C.) who lived in what is now southern Italy. He became a disciple of the philosopher Parmenides, a philosopher who went around telling people that reality was an absolute, unchanging whole, and that therefore many things we take for granted, such as motion and plurality, were simply illusions. This kind of thing must no doubt have brought on ridicule from the more common-sensical Eleatics, and so Zeno set out to defend his master’s position by inventing ingenious problems for the common-sense view. Ever since then, Zeno’s paradoxes have been debated by philosophers and mathematicians. Zeno's writings have not survived, so his paradoxes are known to us chiefly through Aristotle's criticisms of them. Aristotle analyzed four paradoxes of motion: the Racetrack (or Dichotomy), Achilles and the Tortoise, the Arrow, and the Stadium (or Moving Rows). However, based on Aristotle's description of it, it is much less clear what Zeno intended by the Stadium paradox than by the other three. I have therefore left out this fourth paradox. The Racetrack (or Dichotomy) One can never reach the end of a racecourse, for in order to do so one would first have to reach the halfway mark, then the halfway mark of the remaining half, then the halfway mark of the final fourth, then of the final eighth, and so on

18. Mathematical Mysteries: Zeno's Paradoxes finite, indivisible elements is apparently incorrect. So, here is wherethe real paradox of Zeno lies. In his arguments, he manages to http://plus.maths.org/issue17/xfile/

@import url(../../newinclude/plus_copy.css); @import url(../../newinclude/print.css); @import url(../../newinclude/plus.css); about Plus support Plus ... Plus search plus with google home latest issue explore the archive ... news Permission is granted to print and copy this page on paper for non-commercial use. For other uses, including electronic redistribution, please contact us. Issue 17 Nov 2001 Contents Features Cars in the next lane really do go faster Model Trains Measure for measure Maths on the tube Career interview Career interview: Maths editor Regulars Plus puzzle Mystery mix Reviews 'The Tyranny of Numbers' 'The Golden Section' 'MathInsight 2002 Calendar' News from Nov 2001 ... poster! Nov 2001 Regulars Mathematical mysteries: Zeno's Paradoxes by Rachel Thomas The paradoxes of the philosopher Zeno , born approximately 490 BC in southern Italy, have puzzled mathematicians, scientists and philosophers for millennia. Although none of his work survives today, over 40 paradoxes are attributed to him which appeared in a book he wrote as a defense of the philosophies of his teacher Parmenides. Parmenides believed in monism , that reality was a single, constant, unchanging thing that he called 'Being' . In defending this radical belief, Zeno fashioned 40 arguments to show that change (motion) and plurality are impossible.

19. ZENO'S PARADOX: A RESPONSE TO MR. LYNDS (by Eric Engle) $6.95 HOSTING 1 GIG, 1000 EMAILS, 50 GIG TRANSFER, CGI CLICK HERE.Zeno s paradox A response to Mr. Lynds by Eric Engle. paradoxes http://www.lexnet.bravepages.com/ZENO.html

FREE WEB SITE HOSTING Web Hosting Service Web Hosting Domain Free Website ... WebSite Hosting Zeno's Paradox: A response to Mr. Lynds by Eric Engle Paradoxes exist to point out flaws in our reasoning. They are thus heuristic devices. A paradox occurs when our presumptions are inadequate to solve a problem. Thus for example, if we believe (erroneously) that all statements must be either true or false we will quickly run into paradoxes. For example, the statement "this statement is false" is a classic paradox with no truth value. The statement is neither true nor false. It is indeterminate. (The tougher paradox of this art is in fact whether statements about unicorns have truth value - clearly unicorns do not exist - but does that mean that a statement about a non-existing entity is false or merely with no truth value?). Paradoxes such as these exist because people think that all statements must have a truth value, that is that all statements are either true or false. In fact Aristotle in Posterior Analytics ( Zeno's paradoxes all concern motion. Zeno effectively asks "How can motion be possible?" This paradox is arguably of little heuristic value today because we have since Einstein at least recognized that time and matter-energy are convertible elements, the same thing in fact. Thus rather than seeing a solid object, an arrow, existing at definite points in its trajectory, the correct view is to see a wave of energy following the arrow's trajectory with much greater mass/energy presence at certain instances of space time.

20. PlanetMath: Zeno's Paradox Zeno s paradox, (Topic). Imagine the great greek hero Achilles starting a racewith a turtle. Zeno s paradox is owned by mathwizard. (view preamble). http://planetmath.org/encyclopedia/ZenosParadox.html

(more info) Math for the people, by the people. Encyclopedia Requests Forums Docs ... Random Login create new user name: pass: forget your password? Main Menu sections Encyclop¦dia Papers Books Expositions meta Requests Orphanage Unclass'd Unproven ... Corrections talkback Polls Forums Feedback Bug Reports downloads Snapshots PM Book information Docs Classification News Legalese ... TODO List Zeno's paradox (Topic) Imagine the great greek hero Achilles starting a race with a turtle. Achilles is a fast runner, running If Achilles wants to get in front of the turtle he first has to run to where the turtle started. But in that time the turtle has bridged some distance, which Achilles now has to run in order to take up. But in this time again the turtle has gone for some distance and Achilles is still in behind of the turtle. This process continues forever and apparently Achilles cannot pass the turtle. The paradox can be solved, if we take into consideration the fact that an infinite series (a sum of infinitely many numbers) may well converge. and runs at a speed . Achilles runs at a speed with . Then the time needed for Achilles to reach the turtle is given as: which converges if and only if , so in any possible race Achilles can catch up with the turtle, as was expected.

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