21. Zeno's Paradox Of The Arrow For an application of the Arrow paradox to atomism, click here. Go to previouslecture 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

22. Math Lair - Zeno's Paradox Zeno s Racecourse paradox involves the story of a race between Achilles and a This is quite similar to Zeno s bisection paradox, which is examined in http://www.stormloader.com/ajy/zeno.html

Zeno's Paradox Zeno's Racecourse Paradox involves the story of a race between Achilles and a tortoise. In this race, Achilles, being much faster, gives the tortoise a head start. Zeno's assertion is that Achilles can never overtake the tortoise, since when Achilles reaches the point where the tortoise started, the tortoise has moved ahead somewhat, say to point A. When Achilles reaches point A, the tortoise has moved ahead to point B. When Achilles reaches point B, the tortoise has moved further. Therefore, the tortoise must always hold a lead. This is quite similar to Zeno's bisection paradox, which is examined in detail below . This conclusion is very counter-intuitive. For example, everyone can remember overtaking someone while walking, driving or biking. If Zeno's assertions were true, motion would be impossible. Zeno's Bisection Paradox: Zeno's Assertion: A runner can never reach the end of a racecourse in a finite time. Statement: Reason: The remaining interval is divided in half. There are an infinite number of such halfway points which the runner must reach. Each of these points will take a finite time.

23. Zeno's Paradox A link to an unusual and strange discussion of Zeno s paradox in Reality Inspector,a novel about chess and computerhacking. 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?"

24. Mathematical Mysteries: Zeno's Paradoxes Zeno and the paradox of Motion Zeno s race course, part 2 Lecture notes fromthe University of Washington Zeno at St Andrews site http://plus.maths.org/issue17/xfile/

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 ... posters! 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.

25. Deconstructing Infinity: An Analysis Of Zeno's Paradox || Kuro5hin.org Deconstructing Infinity An Analysis of Zeno s paradox 262 comments (207 topical,55 editorial, 0 hidden). but (none / 0) ( 262) by keleyu on Thu May 5th, http://www.kuro5hin.org/story/2005/1/5/111446/4154

create account help/FAQ contact links ... MLP We need your support: buy an ad premium membership k5 store Deconstructing Infinity: An Analysis of Zeno's Paradox ... Science By gdanjo Fri Jan 7th, 2005 at 08:21:40 AM EST Zeno of Elea famously postulated his many paradoxes in defence of Parmenides' worldview: that all is "oneness," and pluralism is merely an illusion. Of his forty paradoxes, the four most enduring have fascinated philosophers, mathematicians, and regular pundits for millennia. As one of the many pundits ensnared by Zeno's challenges, I will present my own analysis of arguably his most famous paradox of all: The race between Achilles and the tortoise Introduction Zeno, son of Teleutagoras, was born in Elea, Lucania (now southern Italy) around 490 B.C. Zeno was a member of the Eleatic school, and most of what we know personally about him is from Plato's dialogue Parmenides - "tall and fair to look upon", he is thought to have been adopted by Parmenides as his own son (it is even suggested that they may have been lovers). Parmenides, founder of the Eleatic school, saw the physical world as an elaborate illusion devoid of Truth, for it is constantly in flux and seems to defy what logic might conclude: that

26. Zeno's_paradox This version of Zeno s paradox has even made it to Hollywood, Another versionof Zeno s paradox involves a race between Achilles and a Tortoise. http://faculty.ssu.edu/~kmshanno/zeno.htm

Zeno's Paradox(es) As everyone knows, it is impossible to ever get anywhere. If you are currently at point A and wish to move to a different point, B you must first traverse half the distance from A to B then half the remaining distance, then half the still remaining distance, ad infinitum. No matter what you do, you will always have half the remaining distance left, right? This version of Zeno's paradox has even made it to Hollywood, featured in the 1994 film, IQ, where Meg Ryan's character uses the paradox in an attempt to fend off the charismatic mechanic played by Tim Robbins. Of course you can debunk this one as easily as he did. Simply walk across the room and out the door. You know you get there. So what was wrong with Zeno? Another version of Zeno's paradox involves a race between Achilles and a Tortoise. Achilles can run 10 times as fast as the tortoise and therefore gives the tortoise a ten meter head start. However, if the tortoise has a ten meter head start how can Achilles ever catch him? By the time Achilles reaches the 10 meter mark, the tortoise will be at 11 meters. By the time Achilles gets there the tortoise will be at 11.1 meters and so on. This process of looking at where the tortoise will be when Achilles catches up to where he WAS can be repeated indefinitely creating an infinite sequence of snapshots all showing the tortoise still ahead. Therefore, Achilles, even though he runs ten times as fast as the tortoise, will never catch up. Next Page Outline Home K.M.Shannon

27. The Red Python Pages - Zeno's Achilles Paradox It is the infinite that lies at the heart of Zeno s paradox. Zeno had takencontinuous motion Ask Dr. Math Halving and Halving Again - Zeno s paradox http://www.redpython.co.uk/Paradoxes/zeno_achilles_paradox.htm

Home Page Chat Room Contact Me Family Only Guest Book Links Mathematics My Books Paul Bates's ... Click here for printable version Zeno's Paradoxes - The Achilles The Problem In his most famous paradox Zeno proves that anything that is moving can never catch up with anything that is moving slower than it. The slower when running will never be overtaken by the quicker; for that which is pursuing must first reach the point from which that which is fleeing started, so that the slower must necessarily always be some distance ahead.' Let's look at what he says. First assume that a hare is chasing a tortoise. The tortoise starts some distance ahead - lets say 10 metres. They both start running at the same time. The hare runs at a speed of 10 metres every second and the tortoise runs at a speed of 1 metre every second. After 1 second the hare has got to the tortoise's starting position, but the tortoise has moved 1 metre in the same time, so the hare has not caught up yet. The tortoise is now 1 metre ahead, but by the time the hare travels 1 metre the tortoise has traveled 10cm. The tortoise is now 10 cm ahead, but by the time the hare travels 10cm the tortoise has traveled 1cm. The tortoise is now 1 cm ahead. I could keep going on indefinitely but you should see that the tortoise is always a bit further on when the hare gets to where he was.

28. The Red Python Pages - Zeno's Achilles Paradox It is the infinite that lies at the heart of Zeno s paradox. Zeno had takencontinuous motion and divided it into infinite steps. The Greeks believed that http://www.redpython.co.uk/Printable/zeno_achilles_paradox.htm

The Red Python Pages Paradoxes Zeno's Paradox - The Achilles The Problem In his most famous paradox Zeno proves that anything that is moving can never catch up with anything that is moving slower than it. The slower when running will never be overtaken by the quicker; for that which is pursuing must first reach the point from which that which is fleeing started, so that the slower must necessarily always be some distance ahead.' Let's look at what he says. First assume that a hare is chasing a tortoise. The tortoise starts some distance ahead - lets say 10 metres. They both start running at the same time. The hare runs at a speed of 10 metres every second and the tortoise runs at a speed of 1 metre every second. After 1 second the hare has got to the tortoise's starting position, but the tortoise has moved 1 metre in the same time, so the hare has not caught up yet. The tortoise is now 1 metre ahead, but by the time the hare travels 1 metre the tortoise has traveled 10cm. The tortoise is now 10 cm ahead, but by the time the hare travels 10cm the tortoise has traveled 1cm. The tortoise is now 1 cm ahead. I could keep going on indefinitely but you should see that the tortoise is always a bit further on when the hare gets to where he was. Notes Everybody knows in the real world Zeno's Statement is wrong. After all, if it were true the world would be a

29. Zeno's Paradox Zeno s paradox. In order for a person to cross a room, that person must firstcross the halfway point of the room. In order to reach the halfway point, http://www.geocities.com/CapitolHill/Lobby/3022/zeno.html

Zeno's Paradox In order for a person to cross a room, that person must first cross the halfway point of the room. In order to reach the halfway point, the person must first reach the midpoint between the origin of the walk and the halfway point. And to reach halfway to the halfway point, the person must cross the halfway to the halfway to the halfway point. Zeno argued that the process could be continued forever. The gist of the argument is that in order to reach the other side of the room, an infinite number of points must be crossed. And logic tells us that an infinite number of points cannot be crossed in a finite period of time. Therefore, it is impossible to cross a room. QED. Back to the Paradox Page.

30. Mark's Paradox Page Zeno s paradox, where the sum of a set of infinite numbers can be derived to Zeno s paradox was not resolved until Newton and Liebnitz discovered the http://www.geocities.com/CapitolHill/Lobby/3022/

These are old fond paradoxes to make fools laugh i' the alehouse. - Othello, Act 1, Scene 1 Paradoxes are as old as humankind. The ancient Greeks studied them intensely which eventually helped lead to the discovery of irrational numbers, and paradoxes are mentioned in the Bible: "It was one of their own prophets who said 'Cretans were never anything but liars, dangerous animals, all greed and laziness;' and that is a true statement." (Titus 1:12-13) Even today, we are surrounded by paradoxes such as Blackwood's "the more terrible the prospect of thermonuclear war becomes the less likely it is to happen," or the Moebius Strip - a topological paradox. For this article, we define a paradox as a statement or sentiment that is seemingly contradictory or opposed to common sense and yet is perhaps true in fact. Another way of thinking of a paradox is a statement that is actually self-contradictory and hence false even though its true character is not immediately apparent. One of the oldest paradoxes is the one cited by the Apostle Paul in his letter to Titus (see above.) The Liar Paradox is interesting because it cannot be true because it would make the speaker a liar and therefore what he says is false. Neither can it be true because that would imply that Cretans are truth-tellers, and consequently what the speaker says would be true. (For classic

31. Zeno's Paradox@Everything2.com Zeno s paradox is primarily illustrative of the dangers of thinking too So we can resolve Zeno s paradox by saying that Zeno is not giving us the http://www.everything2.com/index.pl?node_id=78782

32. Zeno's Paradox Zeno s paradox While critical thinking may not make up for a lack of knowledge,it is essential for gaining knowledge. http://ronz.blogspot.com/

@import url(http://www.blogger.com/css/navbar/main.css); @import url(http://www.blogger.com/css/navbar/1.css); BlogThis! Zeno's Paradox While critical thinking may not make up for a lack of knowledge, it is essential for gaining knowledge. Friday, July 29, 2005: Science: Humans learn without explicit thought - Michael Hopkin, Nature.com Humans can learn skills without remembering what they have done, according to a study of patients with severe amnesia... Most people gather information and abilities through a process of 'declarative' learning, in which they remember the act of learning as well as the new skill or knowledge itself. Hopefully, you'll recall reading this article, as well as remembering the nuggets of information it contains... Other animals, which lack our cognitive powers, use a slower, more primitive method called habit learning. If one item in a pair of objects is designated the 'correct' item, monkeys can learn to select this particular item by simple trial and error. Without realizing they are doing it, they gradually acquire the habit of picking the right option. Interesting, especially if you're not familiar with more current theories of memory. Some of our learning takes place without our being aware of what was learned or that we even learned anything. This type of learning occurs primarly by repetition. It's no coincidence that repetition is a very effective method to influence people's thoughts and behaviors.

33. PlanetMath: Zeno's Paradox Zeno s paradox is owned by mathwizard. This is version 3 of Zeno s paradox,born on 200401-31, modified 2004-02-04. Object id is 5538, canonical name 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 ... Classification talkback Polls Forums Feedback Bug Reports downloads Snapshots PM Book information News Docs Wiki ChangeLog ... About Zeno's paradox (Topic) Imagine the great greek hero Achilles starting a race with a turtle. Achilles is a fast runner, running metres per second, while the turtle is slow and runs at one metre per second. Therefore Achilles agrees to give the turtle some advantage and the turtle starts 10 metres in front of Achilles. The ancient greek philosopher Zeno found the following ``paradox''. 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. To solve this ``paradox'' we have to take a look at the times needed to run these distances. It takes Achilles one second to get to where the turtle started. In this time the turtle runs one metre. It takes only the tenth of a second for Achilles to get there as well. The turtle now runs 10 centimetres, which Achilles passes in one hundredth of a second and so on. So Achilles reaches the turtle after

34. 10.12. Zeno Of Elea (495?-435? B.C.) debate continues on the validity of both the paradoxes and the rationalizations.For related information of Zeno, see Georg Cantor , Zeno s paradox. 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

35. 4.1. Series And Convergence Example 4.1.1 Zeno s paradox (Achilles and the Tortoise). Achilles, a fastrunner, was asked to race against a tortoise. Achilles can run 10 meters per 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.

36. Re: Zeno's Paradox Subject Re Zeno s paradox; From Vesselin Gueorguiev vesselin@baton.phys.lsu.edu Re Zeno s paradox. From baez@galaxy.ucr.edu (john baez) http://www.lns.cornell.edu/spr/1999-04/msg0015719.html

Date Prev Date Next Thread Prev Thread Next ... Thread Index Re: Zeno's paradox Subject : Re: Zeno's paradox From Date : 05 Apr 1999 00:00:00 GMT Approved : mmcirvin@world.std.com (sci.physics.research) Newsgroups : sci.physics.research Organization : Louisiana State University References 7dutdp$frn$1@pravda.ucr.edu Reply-To : vesselin@baton.phys.lsu.edu Sender : mmcirvin@world.std.com (Matthew J McIrvin) Follow-Ups Re: Zeno's paradox From: baez@galaxy.ucr.edu (john baez) Re: Zeno's paradox From: "Mark W. Meckes" <mark@b63358.STUDENT.CWRU.Edu> Re: Zeno's paradox From: wschmied@mail.blinx.de (Wolfram Schmied) Re: Zeno's paradox From: wschmied@mail.blinx.de (Wolfram Schmied) Re: Zeno's paradox From: "D. Schauer" <dan@cromwell.physics.uiuc.edu> Re: Zeno's paradox From: "D. Schauer" <dan@cromwell.physics.uiuc.edu> Re: Zeno's paradox From: omega@ihug.co.nz (Harry Johnston) References Zeno's paradox From: baez@galaxy.ucr.edu (john baez) Prev by Date: Re: Quantization procedures Next by Date: Re: Quantum groups (was: Re: Poisson groups/actions) Prev by thread: Zeno's paradox Next by thread: Re: Zeno's paradox Index(es): Date Thread

37. Zeno's Paradox Subject Zeno s paradox; From baez@galaxy.ucr.edu (john baez); Date 02 Apr 1999000000 GMT; Approved p.helbig@jb.man.ac.uk (sci.physics.research) http://www.lns.cornell.edu/spr/1999-04/msg0015687.html

Date Prev Date Next Thread Prev Thread Next ... Thread Index Zeno's paradox Subject : Zeno's paradox From : baez@galaxy.ucr.edu (john baez) Date : 02 Apr 1999 00:00:00 GMT Approved : p.helbig@jb.man.ac.uk (sci.physics.research) Newsgroups : sci.physics.research Organization : University of California, Riverside References Follow-Ups Re: Zeno's paradox From: Prev by Date: WWW: Physics Animations. Next by Date: Re: Spin Foams and Einstein-Cartan Prev by thread: WWW: Physics Animations. Next by thread: Re: Zeno's paradox Index(es): Date Thread

38. Zeno's Paradoxes: Information From Answers.com Zeno s paradox ( zee nohz) A paradox is an apparent falsehood that is true, oran apparent truth that is false. http://www.answers.com/topic/zeno-s-paradoxes

showHide_TellMeAbout2('false'); Arts Business Entertainment Games ... More... On this page: Literature Wikipedia Best of Web Mentioned In Or search: - The Web - Images - News - Blogs - Shopping Zeno's paradoxes Literature Zeno's paradox zee -nohz) A paradox is an apparent falsehood that is true, or an apparent truth that is false. Zeno, an ancient Greek, argued that a number of apparent truths such as motion and plurality are really false. A well-known, simplified version of one of his paradoxes is that an arrow can never reach its target, because the distance it must travel can be divided into an infinite number of subdistances, and therefore the arrow must take an infinite amount of time to arrive at its destination. Wikipedia Zeno's paradoxes Zeno's paradoxes are a set of paradoxes devised by Zeno of Elea to support Parmenides ' doctrine that "all is one" and that contrary to the evidence of our senses, the belief in plurality and change is mistaken, and in particular that motion is nothing but an illusion Several of Zeno's eight surviving paradoxes (preserved in Aristotle 's Physics and Simplicius Achilles and the tortoise , the Dichotomy Zeno's arguments are perhaps the first examples of a method of proof called reductio ad absurdum also known as proof by contradiction . They are also credited as a source of the dialectic method used by Socrates Zeno's paradoxes were a major problem for ancient and medieval philosophers , who found most proposed solutions somewhat unsatisfactory. More modern solutions using

39. Stephenson:Neal:Quicksilver:165:Zeno's Paradox (Matt Zwolinski) - Metaweb Zeno s paradox is actually shorthand for a series of paradoxes put forward bythe Greek The specific reference here is to Zeno s paradox of motion, http://www.metaweb.com/wiki/wiki.phtml?title=Stephenson:Neal:Quicksilver:165:Zen

40. American-Scientist-Open-Access-Forum: Zeno's Paradox And The Road To The Optimal Zeno s paradox was the one about the philosopher who thought How can I don t know what the theoretical solution to Zeno s paradox is, but http://cogsci.soton.ac.uk/~harnad/Hypermail/Amsci/0819.html

Zeno's Paradox and the Road to the Optimal/Inevitable From: Stevan Harnad ( harnad@COGPRINTS.SOTON.AC.UK Date: Sat Sep 02 2000 - 22:40:56 BST Next message: David Goodman: "Re: problem of the Ginsparg Archive as self-archiving model" Previous message: Marvin: "Re: question about other forums" Next in thread: ransdell, joseph m.: "Re: Zeno's Paradox and the Road to the Optimal/Inevitable" Maybe reply: ransdell, joseph m.: "Re: Zeno's Paradox and the Road to the Optimal/Inevitable" Maybe reply: Stevan Harnad: "Re: Zeno's Paradox and the Road to the Optimal/Inevitable" Maybe reply: David Goodman: "Re: Zeno's Paradox and the Road to the Optimal/Inevitable" Maybe reply: Stevan Harnad: "Re: Zeno's Paradox and the Road to the Optimal/Inevitable" Maybe reply: Stevan Harnad: "Re: Zeno's Paradox and the Road to the Optimal/Inevitable" Messages sorted by: [ date ] [ thread ] [ subject ] [ author ] ... [ attachment ] ZENO'S PARADOX AND THE ROAD TO THE OPTIMAL/INEVITABLE Zeno's Paradox was the one about the philosopher who thought: "How can I possibly get across this room? For before I can do that, I have to

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