Geometry.Net - Theorems_And_Conjectures: Zeno's Paradox

Election Snared In Zeno's Paradox: Absurdity Knows No Bounds
zeno's paradox and the 2000 Election The Limits of Absurdity. As the marginnarrows we are trapped in ever finer distinctions zeno's paradox.
http://www.nexial.org/ION/zeno.htm

Extractions: November 13, 2000 Xenophanes was a Greek philosopher who lived in the 6 th Century BC. He is famous for "Zeno's Paradoxes," which were logical, or mathematical puzzlers. One of them is told as a race between a tortoise and a hare (or Achilles, in the original version). The Hare, wanting very badly to race, agrees to give the tortoise a half-way head start. The tortoise agrees, but on the condition that this is done repeatedly for each segment. It then follows that dividing the distance in half indefinitely this way will never end, and thus the hare cannot win. Zeno's paradox is a logical trap that appears in certain problems; and, as it turns out, in popular elections. Take another example: the distance from Los Angeles to San Francisco. What paradox can exist in that, you ask? We can certainly calculate the flying distance. We all know that the road distance will be longer because of all the curves. What about the distance along the coastline? Well, if you follow all the coves and inlets, it is quite long. And if you follow smaller features the variations within the coves, the individual rocks, the sand grains, and so on then how long is it? If the wiggliness of the water's edge (the "fractal dimension" of the coastline) remains about the same at any scale, then the distance is infinite Zeno's Paradox! There is a practical solution, of course. The paradox is a matter of
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

Extractions: Go to my home page 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.
Zeno's Paradox
zeno's paradox. In order for a person to cross a room, that personmust first cross the halfway point of the room. In order to reach
http://www.geocities.com/CapitolHill/Lobby/3022/zeno.html

Extractions: 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.
Zeno's Paradox
zeno's paradox of Tortoise and Achilles Zeno of Elea (circa 450 bc) is creditedwith creating several famous paradoxes, but by far the best known is the
http://www.geocities.com/paradox_berrin/eng-5.html

Extractions: Zeno's Paradox of Tortoise and Achilles: Zeno 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.
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

Extractions: 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.
Example 4.1.1: Zeno Paradox (Achilles And The Tortoise)
The problem with zeno's paradox is that Zeno was uncomfortable withadding infinitely many numbers. In fact, his basic argument
http://www.shu.edu/projects/reals/numser/answers/zeno.html

Extractions: Example 4.1.1: Zeno Paradox (Achilles and the Tortoise) 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 ? Back 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
Zeno's Paradox
zeno's paradox and Other Musings. Hit with insomnia the other night, I began thinkingabout zeno's paradox and what made it a paradox. zeno's paradox, Resolved.
http://www.ugcs.caltech.edu/~thumper/info/zeno/

Extractions: Hit with insomnia the other night, I began thinking about Zeno's Paradox and what made it a paradox. Zeno, in case you don't know, was a Greek philosopher who believed that existence was static and that motion and change were rather impossible. (I got this out of an encyclopaedeia, okay? This is the "official" line on what he was talking about.) The classic argument that Zeno hit upon to demonstrate this (based on a technique called reductio ad absurdum ) involves a runner on a race course. Zeno argued that the runner could never finish the course because first he must run the first half of the course, and then he must run half of what is left, and so on an infinite number of times... The end result of this infinite halving is that the racer must never reach the finish because he always has some distance left, of which he must always first travel half. Applying this to any distance shows that it is impossible for an object to move at all! A couple of things occured to me while thinking about this. The most obvious thing was that this is a good argument for space (or time) being quantized. If either time or space is quantized, then that cuts off the infinite recursion and allows the runner to bridge the last quanta. Still, the fact that the paradox doesn't happen doesn't actually imply that space is quantized. (Note: the latest issue of Science News has a cover article on the latest physics theory "space-time foam" which implies that space is quantized; the smallest length you can have is 10^-35 meters.)
Snapping And Zeno's Paradox , By Rev. Chuan Zhi Shakya
Home Literature Essays by Chuan Zhi Shakya » Snapping and Zeno'sParadox. More Features back. Snapping and zeno's paradox. by Chuan
http://www.hsuyun.org/Dharma/zbohy/Literature/essays/czs/snapping.html

Extractions: More Features (click here) Essays by Ming Zhen Shakya Inspirational Writings Chants Prayers Dharma Talks Martial Arts The Diamond Sutra Poetry by Master Hsu Yun The Seventh World of Chan Ruminations on Zen's Cows Empty Cloud: Zen Teachings Maxims of Master Han Shan Site Index Search back by Chuan Zhi Shakya, OHY czs@hsuyun.org "Around puberty, the young male of the Ananda tribe of central Australia is taken from his mother, isolated in the wilderness and deprived of food for a prolonged period of time. He is kept awake at night in a state of constant fear by the eerie, whirling sound of the bullroarer, a native hunting device, until the combined physical and emotional stresses reach their maximum effect. At that moment, the elders of the tribe converge on the terrified youth wearing grotesque masks and covered with vivid body paints and proceed to subject him to a painful ritual of initiation into manhood. If he survives the ordeal, he young man emerges from the ritual in a drastically reduced state of mind, his awareness continuing at a level only sufficient to allow absolute adherence to the strict laws and taboos of the tribe. The adult Ananda tribesman may spend his entire life in this altered state the natives call Dream Time . He will stand on one leg for hours, completely motionless, in a waking trance so deep that flies may crawl across his eyeballs without causing him to blink."
Zeno's Paradox
back to list, zeno's paradox. Q Hi, Two Questions First one is aboutmatter,light and energy. If there is such thing as antimatter
http://van.hep.uiuc.edu/van/qa/section/New_and_Exciting_Physics/Quantum_Mechanic

Extractions: to list Hi, Two Questions... First one is about matter,light and energy. If there is such thing as anti-matter, could that mean the energy would be negative. And since light is a form of energy, could there be anti-light? For example, could there be such a thing as a "Flashdark" or Anti-FlashLight that would cast a shadow, or beam of anti-light? Next question is about math in respect to time. Say we were to throw something at a wall. And the time it took to reach the wall (from my hand) was 1 second. It would then make sence to say that at about .5 seconds, the ball was half way there. And at .25 seconds it was approximately a quarter of the way there, if you kept dividing in half, time will have never reached zero, therefore the ball will have never been in my hand. This seems weird, but mathematically it doesn't seem to make sence.. Thanks :)
Zeno's Paradox
zeno's paradox Time 4'54 Lyrics Andy Wagner First performed November 22,1997 show 4 Appears as track 2 on Pastor of Muppets Tim electric
http://www.losingblueprint.com/hmmobzenos.html
Zeno's Paradox
Non Archimedean version of zeno's paradox. Recall the usual versionhas Achilles chasing a turtle, where the turtle is given a ten
http://www.lix.polytechnique.fr/~ilan/zeno.html

Extractions: Recall the usual version has Achilles chasing a turtle, where the turtle is given a ten cubit head start and Achilles runs ten times faster than the turtle. When Achilles runs 10 cubits, the turtle goes 1 cubit. When Achilles runs 1 cubit, the turtle goes 1/10 of a cubit. It follows that Achilles catches the turtle in 10 + 1 + 1/10 + ... cubits. In modern notation, one can write these numbers as decimals yielding It follows that the turtle goes 10 + 100 + 1000 + .... Ordinarily, this would be considered as meaningless or ``infinity''. However, one can consider this purely formally, in other words, without caring too much about the actual meaning of the numbers and just using algebraic manipulations. So let x = 10 + 100 + 1000+ ..., then
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

Extractions: 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. A runner can never reach the end of a racecourse in a finite time.
Zeno's Paradox
zeno's paradox. Our bodies pass, brushed like a short whisk of a paintbrush, streaking sunspotted land scapes, carved and stroked
http://www.bc.edu/bc_org/svp/st_org/stylus/s96/poems/zeno.html

Extractions: Our bodies pass, brushed like a short whisk of a paint brush, streaking sun-spotted land scapes, carved and stroked tu- lips which wet and taut create space and distance between dancing atoms, melting molecules into corporal, colliding bodies. Zeno never balanced my x with your y; might our halves never cover half of the distance between us. Skies may fall, north calls to south, I call to you, electrons force boundaries tearing stars from their infinite heavens and my lips from losing you. Julie A. Crowley
ArtCar Fest: John Wilson's "Zeno's Paradox"
The West Coast's Largest Gathering of Art Cars! Thursday,September 26 Sunday, September 29, 2002.
http://www.artcarfest.com/vehicles/zenos.paradox.html
Three Interpretations Of Zeno's Paradox
Quorum of One is intended for adult readers. This issue Three modernday interpretationsof. zeno's paradox. With a double zeugma. - I -. Figs Zeno, p.
http://mapage.noos.fr/qoo/Quorum35.html

Extractions: - I - Figs Zeno, p Sometimes I wonder why I ever got out of the philosophy racket. I had a good thing going there for a while. Scroll contracts with hefty advances, personal appearances at archery contests and inter-species footraces, after-bacchanal speeches... I was clearing a million a year easy. Then one New Year's Eve I checked my bank balance and I only had half a mill. A year later I was down to 250 grand, and the next year... Well, let's not dwell on the past. I'm in a different line of work now. Private investigation is my game. Hunting down missing persons, tailing unfaithful spouses, checking out the backgrounds of quiz-show winners who marry recently bereaved royalty, that kind of thing. A lot of people think there's something glamorous about being a private eye. Well, I can tell you one thing about those people: none of them were in my office at about 2:00 pm last Thursday. Let me tell you how it happened... I was sitting at my desk doing nothing after lunch. To tell you the truth, I had been sitting at my desk doing nothing before lunch. And during lunch too, unless you call chomping down a corned lamb on pita and slugging back a bolt of Retsina "doing something".
Zeno's Paradox
If the classic paradox does not suit you, listed next are two more examples of impossiblestatements. OneSentence Paradox Example. Two-Sentence Paradox Example.
http://members.aol.com/trwstrong/zeno.html

Extractions: There are actually two ancient philosophers named Zeno. They are different, so be careful when talking about Zeno. There was the one Zeno, Zeno of Cittium. He was a philosopher who taught in Athens back in the late 300's to early 200's B.C. He is noted for being the founder of Stoic philosophy. He is not the philospher who inspired this page. There was also the other (earlier) Zeno, Zeno of Elea. He was a student of Parmenides in the 6th century B.C. This Zeno and his paradoxes (the Dichotomy, the Achilles, the Arrow, and the Stadium) are mentioned by both Plato and Aristotle. He was the inspiration for this page, even though it does not use any of his paradoxes. It does use a classic paradox though: the Cretan's saying that all Cretans are liars (found in the Bible in Titus 1:12.) If the classic paradox does not suit you, listed next are two more examples of impossible statements. This sentence is false. The following sentence is true.
Zeno's Paradoxes
Zeno's writings have not survived, so his paradoxes are known to us chiefly through it,it is much less clear what Zeno intended by the Stadium paradox than by
http://members.aol.com/kiekeben/zeno.html

Extractions: 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 masters position by inventing ingenious problems for the common-sense view. Ever since then, Zenos 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. 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
Zeno
zeno's paradox and the Creationist Demand for Transitional Forms. This certainlywould evade zeno's paradox if distance were infinitely divisible.
http://www.glenn.morton.btinternet.co.uk/zeno.htm

Extractions: This may be freely distributed so long as no chances are made and no monetary charge is assessed. http://www.glenn.morton.btinternet.co.uk/zeno.htm Visitors to these pages since 12-29-97 Now it is not difficult to see that Zeno's paradox doesn't apply to real life. Why? Because the mathematical laws which are used in Zeno's paradoxinfinite divisibility of spacedoes not happen. It is clear from the fact that Zeno's demonstration that infinite divisibility requires no motion combined with the observation that athletes actually finish races that there comes a point in the division process in which the distance to the finish line is so small that it can no longer be divided. Thus, this paradox hints at the quantization of space, the famous del X of Heisenberg's uncertainty principle. While Zeno didn't come to that obvious conclusion, it is one mathematical way out of the paradox. Similarly, Zeno presented a paradox that said that our athlete could not beat a tortoise. Give the tortoise a head start on our swift athlete. In order to pass the tortoise, the runner must first reach the point where the tortoise started from. But by the time our muscle-bound but inept hero has gotten there, the tortoise isn't there anymore. He has moved a bit. So in order to pass the tortoise, the muscleman must now run to the place where the tortoise is now, but once again, the tortoise has already moved and the athlete can continue this forever and never catch up with the tortoise. Both of these paradoxes show that the continuum doesn't exist. Space is not equally divisible.
Zeno's Paradox
Farewell, welfare. (Literary note) Borges, when finishing his short story Deathand the Compass, makes reference to an infinite labyrinth a straight line.
http://www.sfu.ca/~dkeller/TOUCH_PC/TEXT/FAREWEL.HTM

Extractions: Farewell , welfare (Literary note) Borges, when finishing his short story Death and the Compass, makes reference to an infinite labyrinth: a straight line. This maze is a variation of one of the paradoxes proposed by Zeno. Zeno states that to travel a distance d, you need to cover d/2 first. To get to d/2, you need to reach d/4. To. . . well, the story goes on. In other words, why ever move if we won't ever get anywhere?
Zeno's Paradox
crystal wisdomof-jams, critic very, VERY seriously. Got that? Zeno'sParadox. The great Greek philosopher Zeno of Elea (born sometime
http://www.northernheads.com/specialarticles/zeno.htm

Extractions: Email Me Maybe I shouldn't have worn my critics cap to the Attic but I was mostly hoping I would be denied entrance by one of the bouncers. Maybe it was this failed attempt at glory that embodied itself as skepticism toward the effort given by the Burt Nielson Band I remember the day well, the year 2000 was still a novelty, it was my frosh year, i was being introduced to all the beauties of University life. I woke at 5:00 in my residence room, stumbled down to supper, grabbed a coast, asked a friend if they knew the BNB and soon we were in the Attic emptying our tuition money into a $6.00 cover and ryes galore. The Dead was new, I'd heard a lot about Phish and it there was something encompassing that held the title "jam." The BNB tickled my brain with Funky Shoes or funkin' shoes....something like that. A year later, school sucks, I'm drinking cheap wine talking about "The recent events." I heard

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