41. Zeno's Paradox Example zeno's paradox. Achilles The problem with zeno's paradox is thatZeno was uncomfortable with adding infinitely many numbers. In http://pirate.shu.edu/projects/reals/numser/answers/zeno.html

Example: Zeno's Paradox Achilles is racing 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 ? Let us look at the difference between Achilles and the tortoise: Time Difference t = 10 meters t = 1 5 = 10 / 2 meters t = 1 + 1/2 2.5 = 10 / 4 meters t = 1 + 1/2 + 1/4 1.25 = 10 / 8 meters t = 1 + 1/2 + 1/4 + 1/8 0.625 = 10 / 16 meters and so on. In general we have: Time Difference t = 1 + 1 / 2 + 1 / 2 n n meters Now we want to take the limit as n goes to infinity to find out when the distance between Achilles and the tortoise is zero. But that involves adding infinitely many numbers in the above expression for the time, and we don't know how to do that. However, if we define S n n then, dividing by 2 and subtracting the two expressions: S n - 1/2 S n n+1 or equivalently, solving for S n S n n+1 But now S n is a simple sequence, for which we know how to take limits. In fact, from the last expression it is clear that lim S n as n approaches infinity Hence, we have - mathematically correct - computed that Achilles reaches the tortoise after exactly 2 seconds, and then, of course passes it and wins the race.

42. Series And Convergence the Tortoise. zeno's paradox (Achilles and the Tortoise). Achilles,a fast runner, was asked to race against a tortoise. Achilles http://pirate.shu.edu/projects/reals/numser/series.html

Series and Convergence 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. 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. This continuous for a while, but whenever Achilles manages to reach the spot where the tortoise has just been a split-second ago, the tortoise has again covered a little bit of distance, and is still ahead of Achilles. Hence, as hard as he tries, Achilles only manages to cut the remaining distance in half each time, implying, of course, that Achilles can actually never reach the tortoise. So, the tortoise wins the race, which does not make Achilles very happy at all.

43. Zeno's Paradox No. 5 zeno's paradox. Francis Moorcroft. Click here to comment on zeno's paradox.Click here to return to the Philosophy Café. Join Our Café mailing list. http://www.philosophers.co.uk/cafe/paradox5.htm

Home Articles Games Portals ... Contact Us Paradoxes The fifth in Francis Moorcroft's series looking at some the classic philosophical paradoxes. No. 5 Zeno's Paradox Francis Moorcroft The four Paradoxes of Zeno, which attempt to show that motion is impossible, are most conveniently treated as two pairs of paradoxes. The reasons for this will hopefully become clearer later. The first two paradoxes are as follows. The Racecourse or Stadium argues that an athlete in a race will never be able to start. The reason for this is that before the runner can complete the whole course they have to complete half the course; and before they can complete half the course they have to complete a quarter; and before they can complete a quarter they have to complete an eighth; and so on. Therefore the runner has to complete an infinite amount of events in a finite amount of time - assuming that the race is to be run in a finite amount of time. As it is impossible to do an infinite amount of things in a finite amount of time, the race can never be started and so motion is impossible! The second paradox is that of Achilles and the Tortoise, where in a race, Achilles gives the Tortoise a head start. The argument attempts to show that even though Achilles runs faster than the Tortoise, he will never catch her. The argument is as follows: when Achilles reaches the point at which the Tortoise started, the Tortoise is no longer there, having advanced some distance; when Achilles arrives at the point where the Tortoise was when Achilles arrived at the point where the Tortoise started, the Tortoise is no longer there, having advanced some distance; and so on. Hence Achilles can never catch the Tortoise, no matter how much faster he may run!

44. Transcript - Zeno's Paradox (September 17, 2000) Dr. David Harbater of the University of Pennsylvania explains zeno's paradox (Tape) Before you can get to the wall you have to walk halfway to the wall. http://www.earthsky.com/2000/es000917.html

Zeno's Paradox Say you start in the middle of a room and walk to the wall. First you have to walk halfway, and then half of that, and then half again, and so on. So how do you ever reach the wall? An ancient paradox of motion. Sunday, September 17, 2000 Photo courtesy of Greenwich 2000 DB: This is Earth and Sky, on a paradox concerning motion posed by the ancient philosopher Zeno. Say you walked from the middle of a room to the wall. You can imagine your journey as Zeno did as having been a series of steps. Dr. David Harbater of the University of Pennsylvania explains Zeno's paradox : (Tape) Before you can get to the wall you have to walk halfway to the wall. Once you do that you're still not at the wall, so you have to go halfway again of what remains. Then you're still not at the wall so you have to get to the wall, but before you do that you have to go halfway of what remains again, and so forth. Which means that before you get there, there's infinitely many things that have to happen more things that have to happen than you have time for. It sounds like it would go on forever and as a result you would never reach the wall. DB: Of course, you've already reached the wall, so you know it's possible. Here's Zeno's flaw:

45. More Info - Zeno's Paradox (September 17, 2000) More Information on zeno's paradox . zeno's paradox http//www.seanet.com/ksbrown/kmath440.htm.Mathematical Sciences Research Institute http//www.msri.org. http://www.earthsky.com/2000/esmi000917.html

More Information on "Zeno's Paradox" Thanks to the following individual who aided in the preparation of this script: Dr. David Harbater Department of Mathematics University of Pennsylvania Philadelphia, PA If you enjoyed this program, you may be interested in the following: Famous Paradoxes: http://forum.swarthmore.edu/isaac/problems/paradox.html Zeno's Paradox: http://www.seanet.com/ksbrown/kmath440.htm Mathematical Sciences Research Institute: http://www.msri.org Transcript for this show Let us know what you think! 2000 - Byrd and Block Communications, Inc.

46. NRICH Mathematics Enrichment Club (502.html) zeno's paradox By James Thimont (P369) on Monday, December 13, 1999 1243pm Does anyone know how to resolve (mathematically) zeno's paradox. http://www.nrich.maths.org.uk/askedNRICH/edited/502.html

Asked NRICH NRICH Prime NRICH Club ... Get Printable Page March 03 Magazine Site Update News Events Problems Solutions ... Games Archive Problems Solutions Articles Inspirations ... Interactivities Web board Ask NRICH Asked NRICH NRICH Club Register Tough Nuts About Help! ... Where is NRICH? Associated Projects Maths Thesaurus MOTIVATE EuroMaths Millennium Maths ... Project Display maths using fonts images Help Back Issues Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec Bernard's Bag(P) - solutions(P) Penta Probs(P) - solutions(P) Let Me Try(P) - solutions(P) Kid's Mag(P) Play Games(P) Staff Room(P) 6 Problems - solutions 15+Challenges - solutions Articles Games LOGOland Editorial News By James Thimont (P369) on Monday, December 13, 1999 - 12:43 pm Thanks James By Dan Goodman (Dfmg2) on Monday, December 13, 1999 - 04:04 pm n to P as it does to get from P n-1 to P n . So, the total time it takes to get from A to B is the S n=0 n which is equal to 2. It can be illuminating to draw the picture for this, have time on the x axis and distance on the y axis and plot the point corresponding to each of the P n n at a distance x n from A at time t n n ,x n By Michael Doré (P904) on Sunday, December 26, 1999 - 03:45 pm

47. NRICH Mathematics Enrichment Club (502.html) skip to content February 03 Asked NRICH zeno's paradox Does anyone know how to resolve(mathematically) zeno's paradox. It states that motion is impossible. http://www.nrich.maths.org.uk/askedNRICH/edited/502_printable.shtml

Asked NRICH By James Thimont (P369) on Monday, December 13, 1999 - 12:43 pm Thanks James By Dan Goodman (Dfmg2) on Monday, December 13, 1999 - 04:04 pm n to P as it does to get from P n-1 to P n . So, the total time it takes to get from A to B is the S n=0 n which is equal to 2. It can be illuminating to draw the picture for this, have time on the x axis and distance on the y axis and plot the point corresponding to each of the P n n at a distance x n from A at time t n n ,x n By Michael Doré (P904) on Sunday, December 26, 1999 - 03:45 pm Perhaps one of Zeno's even more confusing paradoxes is the following. Consider a race between a tortoise and a hare. The hare can run much faster than the tortoise so the tortoise has a head start. In order to overtake the tortoise, the hare must first reach the initial position of the tortoise. But as the tortoise is also moving, by this time the tortoise will have moved on in front of its initial position and will still be ahead. So the hare must reach this new position – but by the time it does the tortoise will have moved further forward. And so on. Therefore it is impossible for the hare to catch up. Thanks

48. The Frontier - Zeno's Paradox zeno's paradox. Over two thousand years ago, a philosopher namedZeno made a zeno's paradox, copyright 1998 by George Beckingham. http://members.tripod.com/~geobeck/frontier/zeno.html

49. Zeno's Paradox zeno's paradox. We introduced zeno's paradox in the very beginningof this class. The purpose was not to frustrate you. I wanted http://www.angelfire.com/space/omakridis/zeno.html

Zeno's Paradox We introduced Zeno's Paradox in the very beginning of this class. The purpose was not to frustrate you. I wanted to sensitize you to a point that will come up again and again in this class: Our mental constructs - our more precise 'languages' including logical thinking and mathematical descriptions of phenomena - are NOT necessarily mirrors of the reality that we access through our senses. Here is an example of a situation - Achilles trying to catch up with the frustratingly slow tortoise - and yet he can't! We know, in our sensory universe, that this is impossible - of course Achilles can and will catch up. And, yet, when Zeno expresses this in an apparently more precise language, we cannot point to what mistake he is making. He comes up with an unexpected conclusion - a conclusion which 'reality' as we know it does not support: Zeno 'shows' that Achilles will never manage to catch up with the tortoise. Here is how it works: Assume that Achilles runs with a speed that is 10 times the speed of the tortoise. We give the animal a 100 yard start - just to be fair. By the time Achilles has covered the 100-yard distance, the tortoise has moved by 10 yards [1/10th of 100]. By the time Achilles has covered the 10 yards, the tortoise has covered 1/10 of that = 1 yard. By the time Achilles has covered that distance [1 yard] the tortoise has moved by 1/10 of a yard. By the time Achilles has covered that, the tortoise has moved by 1/10 of that = 1/100; and so on - 1/1000, 1/1000000, .... The distance becomes infinitely small, but it never becomes zero.

50. Zeno's Paradox And Chaos. Archimedes and the tortoise are in a race. Archimedes gives the tortoisea head start. Archimedes catches up the tortoise. No no http://www.sbu.ac.uk/dirt/museum/paradox.html

Archimedes and the tortoise are in a race. Archimedes gives the tortoise a head start. Archimedes catches up the tortoise. No no, let's try that again, this time using discrete numbers. We'll give the tortoise 100 metres start. Both archimedes and the tortoise start runnning. After time T, Archimedes has run 100 metres.... BUT the tortoise has covered another 0.1 metres. Lets run that again. After time dT, Archimedes has run 0.1 metres.... BUT the tortoise has covered another 0.0001 metres. and so on, while you can extend the decimal places to describe the result. Using discrete numbers to model the situation, Archimedes does not catch the tortoise. What about chaos? If you take any situation that you might want to model, you have to define your starting position. It will be a set of numbers. Your accuracy will depend on how precise your number is, the number of decimal places. Complex situations mean that imprecision in the starting position will mean great differences in the final position. Real life situations like the weather or human relations require numbers of infinite length to define the starting position. Since I'm a medic, I'll give you a medical example.

51. GIC - Zeno's Paradox Has A Solution RE zeno's paradox There are two major apparents regards the simple solution takento be that 'paradoxal infinities' do not exist because the length being http://visitastronomy.com/paradox.htm

RE: ZENO'S PARADOX There are two major apparents regards the simple solution taken to be that 'paradoxal infinities' do not exist because the length being subdivided by contiguous 1/2 slices are contained in a 'frame' being the start and end of the 'unit'. The first 'major' is the FACT of the UNIT itself. It is NOT dimensionless. Dimensionless implicits comparative rates, for instance comparison of two sizes one over the other to produce a 'dimensionless' ratio having no systems of units. A unit itself on the other hand, is its own system of units, i.e this length being contiguously divided by Zeno is one 'unit' long The second major is where an action is involved, for instance a walker crossing the length being subdivided. By walking is a way the paradox is often stated, that the walker takes steps until 1/2 the length has been crossed, then more steps until 3/4 of the length has been crossed. In the games of paradox the walker is percieved never to be able to complete the crossing because an increasingly small portion of the length still remains. But woah, pause, time out. The paradox stated

52. Zeno's Paradox zeno's paradox. © Copyright 1997, Jim Loy. Among the Zeno's arrow paradoxis much more interesting than his other paradoxes. It essentially http://www.voidprojects.plus.com/regular/bzeno.htm

Return to my Science pages Go to my home page Zeno's Paradox 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.

53. Totse.com | A New Quantum- Physics Twist On Zeno's Paradox Of M www.totse.com A new quantum- physics twist on zeno's paradox ofm - A new quantum-physics twist on zeno's paradox of motion. http://www.totse.com/en/fringe/fringe_science/zeno.html

About Community Bad Ideas Drugs ... ABOUT A new quantum- physics twist on Zeno's paradox of m twist on Zeno's paradox. Can't Get There from Here Two thousand years ago the Greek philosopher Zeno noted that an object moving from one place to another must first reach a halfway point, and before that a point half of the way to the halfway point, and so on. Any movement involves an infinite number of intermediate points, and so any motion must require an infinite amount of time Motion, Zeno concluded, is logically impossible In fact, things do move Zeno did not consider that an endless series could have a finite sum. But in the counter-intuitive realm of quantum physics, something akin to Zeno's paradox can occur: atoms can be paralyzed if they are closely scrutinized. The act of observing prevents the atom from passing a halfway point between two energy levels. In 1977 E. C. George Sudarshan and Baidyanath Misra of the University of Texas at Austin realized that an unstable object, such as a radioactive atom, would never decay if it were observed continuously. They called this surprising phenomenon the quantum Zeno effect. Now

54. Zeno's Paradox Of The Arrow Go to next lecture on Empedocles. Go to previous lecture on the Zeno'sParadox of the Race Course, part 2. Return to the PHIL 320 Home Page http://www.dsfss.uniud.it/~andrea/corso2000-01/S. Marc Cohen/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 argument falsely assumes that time is composed of "nows" (i.e., indivisible instants). There is no such thing as motion (or rest) "in the now" (i.e., at an instant). 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 " entails neither that x is resting at i nor that x is moving at i Perhaps instants and intervals are being confused: "When?" can mean either "at what instant?" (as in "When did the concert begin?") or "during what interval?" (as in "When did you read 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. (

55. Zeno's Quantum Paradox Reversed: Watching A Flying Arrow Increase Its Speed of the 19th century, who resolved zeno's paradox by showing that nonzero velocitycan exist in the limit of infinitesimal divisions of a trajectory. http://www.globaltechnoscan.com/7june-13june/Zeno.htm

Please register here to Search or Submit B usiness O pportunities REGISTER HERE LOGIN Zeno's quantum paradox reversed: Watching a flying arrow increase its speed For Business Opportunities in Engineering Industry please click here Is motion an illusion? Can"glimpses" freeze radioactive decay? For over 2,500 years, scientists and philosophers have been grappling with Zeno of Elea's famous paradox. More recently, scientists believed that the counterpart of this paradox, known as the quantum Zeno paradox, is realizable in the microscopic world governed by quantum physics. Now, scientists from the Weizmann Institute of Science have shown that in most cases, the quantum Zeno paradox should not take place. An article describing the calculations that lead to this surprising conclusion appears in today's Nature. The article is also surveyed in the journal's News and Views section. The Greek philosopher Zeno, who lived in the 5th Century B.C., decades before Socrates, dedicated his life's work to showing the logical paradoxes inherent to the idea of indefinite divisibility in space and time (i.e., that

56. Untitled zeno's paradox of Plurality and Proof by Contradiction. Here I will be concernedwith what has come to be known as Zeno's second paradox of plurality http://www.aug.edu/dvskel/Zeno2.htm

Zeno's Paradox of Plurality and Proof by Contradiction Stephen Campbell Department of Education University of CaliforniaIrvine It is all one to me where I begin, for I will return there again in timeParmenides Zeno's paradoxes have been a source of inspiration and bewilderment for almost two and a half thousand years. Indeed, Aristotle considered Zeno of Elea (c. 450 B.C.E.) the father of dialectic, a form of reasoning that does not seem to be unrelated to the logical forms of reasoning underlying mathematical proof. Of all Zeno's paradoxes, the most renown, or at least the most familiar, are his paradoxes of motion. Much lesser known are Zeno's paradoxes of plurality. According to Proclus, Zeno composed as many as forty of these paradoxes, all but three of which have been lost. Here I will be concerned with what has come to be known as Zeno's second paradox of plurality: If there are many, it is necessary that they be as many as they are, neither more nor fewer. But if they are as many as they are, they must be finitely many. If there are many, the existents must be infinitely many. For there are always other existents between existents, and again others between these. And thus the existents are infinitely many. (adapted from Vlastos, p. 371)

57. Untitled SPEEDAT-AN-INSTANT zeno's paradox AND BALLA'S FLIGHT OF THE SWIFTS. The FuturistTechnical Manifesto (1910) illustrates in artistic terms zeno's paradox. http://www.aug.edu/dvskel/justiceFA92.htm

SPEED-AT-AN-INSTANT: ZENO'S PARADOX AND BALLA'S FLIGHT OF THE SWIFTS RHONA JUSTICE-MALLOY DEPARTMENT OF DRAMA UNIVERSITY OF GEORGIA In On Physics Aristotle made a seemingly harmless statement. He wrote, "A body will move through a given medium in a given time." As is the way with philosophers, Zeno of Elea turned Aristotle's apparently obvious statement into a paradox that would puzzle mathematicians well into the seventeenth-century. The question Zeno posed to his students in the fifth century B.C. involved the idea of speed-at-an-instant-of-time. The paradox goes as follows. Question: if an 'instant' may mean an infinitely small period of time, then in an instant how much distance would, say, a flying arrow cover? Answer: It would not cover any distance. If Zeno had had a camera he might have encouraged his students to imagine how it would be if they photographed the arrow in flight. As they made the exposure shorter and shorter, the image of the arrow would get less and less blurred and an entirely 'instantaneous' photograph would freeze the arrow's motion completely. So, they might have concluded, considered at an instant of time, the arrow could not be said to have a 'speed' at all. Further, if time is regarded as a succession of infinitesimal instants added together, Zeno's paradox implies that the moving arrow is at no instant of time genuinely 'moving' at all (Toulmin and Goodfield, 1961, p. 104). Actually, Zeno would not have needed a camera to demonstrate his point had he been acquainted with the work of the futurist painter Giacomo Balla. Balla's

58. Zeno's Paradox Frank and Ernest by Bob Thaves. Return to Calculus Comics. http://www.pen.k12.va.us/Div/Winchester/jhhs/math/humor/comics/calculus/zeno.htm

Frank and Ernest by Bob Thaves Return to Calculus Comics

59. Zeno's_paradox zeno's paradox(es). As everyone knows, it is impossible to ever get anywhere. Anotherversion of 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

60. FirstMatter Sidebar: Zeno's Paradox zeno's paradox. An excerpt from the Reality Inspector by John Cariswith links. Go there Return to What I've Learned About Visioning http://www.firstmatter.com/newsletter/sidebar.asp?key=162&art=39

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