Geometry.Net - Theorems_And_Conjectures: Zeno's Paradox

81. Philosophy Teaching Sequence 4 Zeno's Paradox
Philosophy Teaching Sequence 4 zeno's paradox. zeno's paradox of the stadium providesan apparent counterexample to any theory that claims that motion occurs.
http://sitemason.vanderbilt.edu/site/liumxq/pts4
Home Page
Philosophy Teaching Sequence 4: Zeno's Paradox Teaching Sequence: Second week of classes
By this time, we have decided as a group which topics within the frame of this course are of general interest, constructed a detailed day-by-day syllabus that incorporates the students' interests, and discussed at length what is philosophy. Aim: Philosophical theories are tested by looking for counterexamples, that is, situations where the theory should hold but doesn't. In this unit I introduce Zeno's paradox of bisection, which is used to demonstrate how thought experiments generate counterexamples. Concerning writing skills I will focus on the nature of a critical philosophy essay, formulating a thesis statement, and constructing an introductory paragraph. This unit will take 3 classes to complete (we meet only two times a week for longer class periods). Day 1: In-class Collaborative Work
The first day of this unit employs in-class collaborative work that focuses on Zeno's argument for the claim that motion (or change) is impossible. Zeno's paradox of the stadium provides an apparent counterexample to any theory that claims that motion occurs. As a result, it helps explain why change is impossible. Have the students identify the issues orally, and list them on the board.
Homework: Write a page stating your personal beliefs about Zeno's paradox; (2) read pages 17-18 and 21-31 from

82. Not From Here To Eternity
Dr. David Harbater of the University of Pennsylvania explains zeno's paradox zeno's paradox http//www.seanet.com/~ksbrown/kmath440.htm.
http://www.msri.org/activities/jir/bwachtel/NotFromHeretoEternity.html
Not From Here to Eternity Say you started in the middle of a room, and you walked to the wall. First you walked half way, and then half of that, and then half again, and so on. So how did you ever reach the wall? An ancient paradox of motion - after this on Earth and Sky. Date 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 0:12:43-0:13:10) 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:

83. Talk:Zeno's Paradoxes - Wikipedia
zeno's paradox is based on the given that to get from point a topoint b, you have to pass through all points in between. Since
http://www.wikipedia.org/wiki/Talk:Zeno's_paradoxes
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Talk:Zeno's paradoxes
From Wikipedia, the free encyclopedia. Removed from the main page:
These paradoxes can also be explained by quantum physics, since certain particles can only occupy discrete positions and can "move instantaneously" from one point to another without crossing the space between. (this is known as "tunneling" and is related to the "probability wave" nature of particles at the subatomic level.)
Sorry, I don't think this does anything to resolve the paradox. In quantum mechanics, position is a continuous variable. If the person who added this thinks it is correct, please explain further. CYD You are the quantum physics guy, not me, and I didn't even write the stuff removed but:

84. Cafe Irreal: Fiction: Borges
KAKFA'S PRECURSORS (according to Borges). zeno's paradox against movement . Zeno'sparadox against movement . Zeno of Elea (as related by Philoponus).
http://home.sprynet.com/~awhit/fborges.htm
KAKFA'S PRECURSORS (according to Borges)
"Zeno's paradox against movement"
"Apologue of Han Yu" (Han Yu)
A Christian auditing (Soren Kierkegaard)
Yet he endeavours (Soren Kierkegaard)
Fears and scruples (Robert Browning)
Les Captifs de Lonjumeau
Carcassonne (Lord Dunsany)
[Editor's note: In his essay "Kafka and his precursors," the great Argentinian writer Jorge Luis Borges stated, "At first I had considered him [Kafka] to be as singular as the phoenix of rhetorical praise; after frequenting his pages a bit, I came to think I could recognize his voice, or his practices, in texts from diverse literatures and periods." Presented here are the full texts of the examples mentioned by Borges]
"Zeno's paradox against movement"
Zeno of Elea (as related by Philoponus) To show that this one is also unmoved, he made use of the following argument. If anything moves along a given finite straight line, it must, before moving along the whole of it, move along the half of it, and, before moving along the half of the whole, it must first move along a quarter of it, and before a quarter an eighth, and so on ad infinitum ; for the continuum is infinitely divisible. So if anything moves along a finite straight line, it must, before completing its movement, have moved through an infinite number of magnitudes: but if this is so, and if every movement occupies a definite finite time (for there is no motion that occupies an infinite time), then we find that in a finite time a motion through an infinite number of magnitudes has taken place, which is an impossibility; for the infinite is interminable absolutely. (Zeno of Elea, H.D.P. Lee, Amsterdam, 1967, reprint of 1936 edition)

85. Re: ATL: Re: Zeno's Paradox (was Argument For Strong AI)
Re ATL Re zeno's paradox (was Argument for strong AI). To atlantis@wetheliving.com;Subject Re ATL Re zeno's paradox (was Argument for strong AI);
http://objectivism.cx/~atlantis/mailing-list/msg22145.html
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Re: ATL: Re: Zeno's Paradox (was Argument for strong AI)
To atlantis@wetheliving.com Subject : Re: ATL: Re: Zeno's Paradox (was Argument for strong AI) From nglover@clemson.edu Date : Fri, 13 Jul 2001 17:55:12 -0400 In-Reply-To F294XMCSI33BjEEzxsM0000f947@hotmail.com Sender owner-atlantis@wetheliving.com
http://hubcap.clemson.edu/~nglover/ "It's good to be open-minded, but not so open that your brains fall out." - Jacob Needleman
References
ATL: Re: Zeno's Paradox (was Argument for strong AI)
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86. Analysis WebNotes: Chapter 01, Class 01
zeno's paradox Suppose an arrow is flying through the air. Beforeit can reach its target, it first has to cross at least half
http://www.math.unl.edu/~webnotes/classes/class01/Zeno.htm
Zeno's Paradox: Suppose an arrow is flying through the air. Before it can reach its target, it first has to cross at least half the distance between the archer and the target. But after it has reached the midway point, it still has to cross half the remaining distance. And after crossing that, it must cross half of the yet remaining distance. In fact, this process goes on for ever, because the distance between the arrow and its target can always be halved, and the first half always stands between the arrow and its destination. So, there are infinitely many states the arrow must pass through before it can hit the target, and only a finite amount of time to do it in. The arrow can't possibly do infinitely many things in finite time, and so it can never reach the target. (in fact the same argument, applied to the journey from the bow to any point on its path, shows that any motion at all is impossible for it!) What's wrong with the argument? If your answer involves adding up infinitely many things, ask yourself how you can do infinitely many things (additions in this case) in finite time-a computer couldn't! What about a bouncing ball? If at every bounce it only reached half as high up as at the last bounce, does it ever completely stop bouncing?

87. Forays Into Mathematics
zeno's paradox The Race Between Achilles and the Tortoise. Achilles wasa mythological Greek warrior who was also famous as a very fast runner.
http://www.tetrakatus.com/logic/jacob/math.html
The Webster's New World Dictionary of Mathematics defines mathematics as "The investigation of numbers, space, and the many generalizations of these concepts created by the intellectual genius of man." And thus, a mathematician would be someone who investigates numbers, space, and the many generalizations of these concepts. I once fancied myself a pure mathematician. I enjoy seeing the strange beauty and intrigue behind the models of mathematics. I had come to mathematics in high school when I first learned geometry from Mr. Lasley. It was enlightening, it seemed to be able to show everything, based upon only a few assumptions it could show you truth. It seemed an infallible system. But then I discovered the paradox. The subtle ideas of infinity, truth, sets, and numbers all collided in my mind. And they tore at me like the logic tears at itself. I struggled long and hard to find their solution. But in my quest I found new paradoxes; new contradictions. But it was Godel's proof that any mathematical system will either be incomplete or inconsistent, that shredded my last hope for perfection in mathematics. At this point I would not have given up all hope for mathematics, while math could not be perfect, it still had merit. I still was interested in proving what I could, in making mathematics to be as complete as possible, without contradiction. But I found that mathematicians, like people, are paradoxical. They had understanding and lack of understanding in what I tried to say. Many would not answer my questions, or would attempt to ignore the answers. I did not find encouragement in college, but instead stone to try to germinate into. By the time I reached Calculus II I was frustrated in a most severe way.

88. Zeno's Paradox
The summary for this Japanese page contains characters that cannot be correctly displayed in this language/character set.
http://www.geocities.co.jp/Playtown-Spade/3812/novels/paradox.html
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89. Ô_ubNXg
Continuity Requirement (Samuel Gorovitz) Leaving the Past Alone (William Lane Craig)Tachyons, Time Travel, and Divine Omniscience zeno's paradox (A.Ushenko
http://www.ne.jp/asahi/takuo/time/Book.htm
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90. Hosted By Tripod
However, in the verbal statement of zeno's paradox as portrayed above,a geometric shrinkage of distancetime intervals is specified.
http://mindlight.tripod.com/achilles.htm
Get Four DVDs for $.49 each. Join now. Tell me when this page is updated
The Achilles Paradox
The Greeks were the first to consider the significance of infinite series, also known as infinite sums, or, in some cases, infinite sub-divisions. Let's take a look at the paradox of Achilles, as described by the ancient philosopher, Zeno:
Imagine a racecourse that stretches one kilometer from point A to point B. Then imagine a runner call him Achilles who starts at point A and runs at a uniform rate of one meter per second toward his goal, point B.
Now, consider these facts: Achilles must first traverse half the distance between points A and B, arriving midway between the two points, at point C. Then Achilles must travel half the remaining distance between point C and his goal, B, arriving at point D. This halving process continues ad infinitum, because regardless of how little distance remains to be crossed, it can still be halved.
Furthermore, each finite segment of the racecourse requires a finite length of time to be traversed; and, since we are dealing with an infinite number of finite intervals, we must conclude that Achilles will never reach his goal. (This version is taken from

91. Lexicon 2K - Z's
Z. ZEPELLIN TUBE A source of immense power, possessed by the SumatranRATs in an adventure of HEMLOCK STONES. zeno's paradox
http://www.benway.com/firesign/lexicon/Z.html
[Z]
ZEPELLIN TUBE:
A source of immense power, possessed by the Sumatran RAT s in an adventure of HEMLOCK STONES
ZENO'S PARADOX:
A paradox devised by the Greek philosopher Zeno, which seems to prove that motion as such is impossible; Reason: Consider an arrow flying towards a target. Before it gets to the target it must first get halfway there, but before it gets to that point it must first get 1/4 the way there, but before that (etc..) Since an infinite number of things must be done first, the arrow could never get *anywhere*; ergo, motion is impossible. This paradox is referred to indirectly in the TWO PLACES album, where BABE falls asleep in his car, while the talking freeway signs read off:
"Antelope Freeway, one mile" "Antelope Freeway, one half mile" "Antelope Freeway, one quarter mile" "Antelope Freeway, one eighth mile" "Antelope Freeway, one sixteenth mile" "Antelope Freeway, one thirtysecondth mile" "Antelope Freeway, one sixty-fourth mile" "Antelope Freeway, one one-hundred-and-twenty-eighth mile..." ZION (oh 'frocious Lion...):

92. ASA - June 2001: Re: Zeno's Paradox And The Creationist Demand
Re zeno's paradox and the creationist demand for transitional fossils. Atthis reference you write, concerning zeno's paradox of motion
http://www.asa3.org/archive/asa/200106/0220.html
Re: Zeno's paradox and the creationist demand for transitional fossils
From: Gordon Simons ( gsimons@email.unc.edu
Date: Mon Jun 25 2001 - 09:08:47 EDT
Next message: george murphy: "Re: Ikedaian Cabalism"
Previous message: George Hammond: "[Fwd: CREATION vs FORMATION (God vs Gravity)]"
Maybe in reply to: Glenn Morton: "Zeno's paradox and the creationist demand for transitional fossils"
Messages sorted by: [ date ] [ thread ] [ subject ] [ author ]
Glenn,
You wrote:
http://www.glenn.morton.btinternet.co.uk/zeno.htm
At this reference you write, concerning Zeno's paradox of motion:
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

93. ASA - June 2001: Zeno's Paradox And The Creationist Demand For
zeno's paradox and the creationist demand for transitional fossils. From GlennMorton (glenn.morton@btinternet.com) Date Mon Jun 25 2001 004232 EDT.
http://www.asa3.org/archive/asa/200106/0213.html
Zeno's paradox and the creationist demand for transitional fossils
From: Glenn Morton ( glenn.morton@btinternet.com
Date: Mon Jun 25 2001 - 00:42:32 EDT
Next message: george murphy: "Re: Ikedaian Cabalism"
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Reply: Gordon Simons: "Re: Zeno's paradox and the creationist demand for transitional fossils"
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I have just placed a new page on my web page which is a comparison of Zeno
with anti-evolutionists. I would be interested in any critiques, complaints
or comments. Please e-mail me as I am not on this list.
This can be found at http://www.glenn.morton.btinternet.co.uk/zeno.htm
glenn
see http://www.glenn.morton.btinternet.co.uk/dmd.htm
for lots of creation/evolution information
personal stories of struggle
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94. Archive Of Astronomy Questions And Answers
How do you reconcile zeno's paradox with modern physics? You reconcile itby refuting Zeno's presumption that nature is infinitely divisible.
http://itss.raytheon.com/cafe/qadir/q520.html
How do you reconcile Zeno's paradox with modern physics?
You reconcile it by refuting Zeno's presumption that nature is infinitely divisible. We know that at the atomic domain, particles are defined in terms of wave functions which connect all points in spacetime by a 'probability' that a particle will be found there. We also suspect that space-time itself cannot be subdivided below a scale of 10**-33 centimeters at which point space-time becomes a quantum mechanical 'thing' lacking a definite shape. Both of these things differ from Zeno's assumption that space and motion were subdividable infinitely, and it is in the TRUE character of space, time, and motion that the mathematical paradox is resolved once and for all. We do not live in an abstract mathematical universe, but one with lots of 'dirt' and things that go bump in the night!!! Return to Ask the Astronomer

95. Zeno's Fifth Paradox
zeno's Fifth paradox. Do events of probability 0 ever occur? (See the note on zeno'sparadox of Motion.) For example, zeno's Stadium paradox was really about
http://www.mathpages.com/home/kmath158.htm
Zeno's Fifth Paradox
Zeno's Paradox of Motion .) For example, Zeno's "Stadium" paradox was really about special relativity (not bad for 300 B.C!), and our "Target" paradox could be regarded as an argument for the fundamental quantum character of nature, and the uncertainty principle. Maybe if Plank's constant was *zero* it really *would* be impossible to hit the target. Has anyone ever produced a consistent version of the laws of physics with h=0? Return to MathPages Main Menu

96. Zeno's Stadium Paradox
vacuum. zeno's intention was to discredit the senses, which he soughtto do through a series of paradoxes on time and space. Zeno
http://www.faragher.freeserve.co.uk/stadium2.htm
The Stadium Zeno supported Parmenides' contention that the universe is a plenum, a single substance, a oneness, without boundary or separation, although the world appears diversified to our fallible senses. A plenum is space totally filled with matter; the opposite to a vacuum. Zeno's intention was to discredit the senses, which he sought to do through a series of paradoxes on time and space. Zeno asserts that a runner cannot reach a goal because, in order to do so, he must traverse a distance; but he cannot traverse that distance without first traversing half of it, and so on, ad infinitum. Because an infinite number of bisections exist in a spatial distance, one cannot travel any distance in finite time - however short the distance or great the speed. This argument, like several others of Zeno's, is intended to demonstrate the logical impossibility of motion. The argument is, then, that the evidence of the senses should be rejected in favour of the evidence of rationality and logic. We discover reality not by examining it, but by thinking logically. Return to Nature of Time and Space or Return to Home Page

97. We've Moved!
The PRIME Encyclopedia Article you have linked to Zenos paradox of the Tortoise and Achilles has moved to com/ pr/ prime/ articles/ zeno_ tort/ index. Please update your link thank you!
http://www.mathacademy.com/platonic_realms/encyclop/articles/zeno_tort.html
The PRIME Encyclopedia Article you have linked to:
has moved to:
http://www.mathacademy.com/pr/prime/articles/zeno_tort/index.asp

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Talk:Zeno's paradoxes

KAKFA'S PRECURSORS (according to Borges)

"Zeno's paradox against movement" "Apologue of Han Yu" (Han Yu) A Christian auditing (Soren Kierkegaard) Yet he endeavours (Soren Kierkegaard) Fears and scruples (Robert Browning) Les Captifs de Lonjumeau Carcassonne (Lord Dunsany)

"Zeno's paradox against movement"

Re: ATL: Re: Zeno's Paradox (was Argument for strong AI)

[Z]

Re: Zeno's paradox and the creationist demand for transitional fossils

Zeno's paradox and the creationist demand for transitional fossils

How do you reconcile Zeno's paradox with modern physics?

Zeno's Fifth Paradox

"Zeno's paradox against movement"
"Apologue of Han Yu" (Han Yu)
A Christian auditing (Soren Kierkegaard)
Yet he endeavours (Soren Kierkegaard)
Fears and scruples (Robert Browning)
Les Captifs de Lonjumeau
Carcassonne (Lord Dunsany)