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  1. Zeno's Paradox: Unraveling the Ancient Mystery Behind the Science of Space and Time by Joseph Mazur, 2008-03-25
  2. Zeno's Paradoxes
  3. Key Contemporary Concepts: From Abjection to Zeno's Paradox (Sage Key Concepts) by Dr John Lechte, 2003-02-24
  4. Modern science and Zeno's paradoxes by Adolf Grunbaum, 1968
  5. The Paradoxes of Zeno (Avebury Series in Philosophy) by J. A. Faris, 1996-10
  6. Zeno's paradox and the problem of free will.: An article from: Skeptic (Altadena, CA) by Phil Mole, 2004-01-01
  7. Zeno's Paradox by F. Gordon Robinson, 2007-10-25
  8. The Universal Book of Mathematics: From Abracadabra to Zeno's Paradoxes by David Darling, 2004-08-11
  9. Why mathematical solutions of Zeno's paradoxes miss the point: Zeno's one and many relation and Parmenides' prohibition.: An article from: The Review of Metaphysics by Alba Papa-Grimaldi, 1996-12-01
  10. Supertasks: Zeno's Paradoxes, Hilbert's Paradox of the Grand Hotel, Omega Point, Supertask, Thomson's Lamp
  11. Paradoxes: Paradox, Russell's Paradox, Problem of Evil, Impossible Object, Arrow's Impossibility Theorem, Zeno's Paradoxes, Epimenides Paradox
  12. Zeno of Elea: An entry from Gale's <i>Science and Its Times</i> by Judson Knight, 2001
  13. ZENO OF ELEAc. 490430 BCE: An entry from Gale's <i>Encyclopedia of Philosophy</i> by Richard McKirahan, 2006
  14. Towards a definitive solution of Zeno's paradoxes by Fazal Ahmad Shamsi, 1973

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

2. Zeno And The Paradox Of Motion
school calculus teachers to present them as "zeno's paradox", and then "resolve the paradox" by pointing out that an
http://www.mathpages.com/rr/s3-07/3-07.htm
3.7 Zeno and the Paradox of Motion The greatest of the Eleatic philosophers was Parmenides (born c. 539 BC). In addition to developing the theme of unchanging oneness, he is also credited with originating the use of logical argument in philosophy. His habit was to accompany each statement of belief with some kind of logical argument for why it must be so. It's possible that this was a conscious innovation, but it seems more likely that the habitual rationalization was simply a peculiar aspect of his intellect. In any case, on this basis he is regarded as the father of metaphysics, and, as such, a key contributor to the evolution of scientific thought. Parmenides's belief in the absolute unity and constancy of reality is quite radical and abstract, even by modern standards. He maintained that the universe is literally singular and unchangeable. However, his rationalism forced him to acknowledge that appearances are to the contrary, i.e., while he flatly denied the existence of plurality and change, he admitted the appearance of these things. Nevertheless, he insisted these were mere perceptions and opinions, not to be confused with "what is". Not surprisingly, Parmenides was ridiculed for his beliefs. One of Parmenides' students was Zeno, who is best remembered for a series of arguments in which he defends the intelligibility of the Eleatic philosophy by purporting to prove, by logical means, that change (motion) and plurality are impossible. We can't be sure how the historical Zeno intended his arguments to be taken, since none of his writings have survived. We know his ideas only indirectly through the writings of Plato, Aristotle, Simplicus, and Proclus, none of whom was exactly sympathetic to Zeno's philosophical outlook. Furthermore, we're told that Zeno's arguments were a "youthful effort", and that they were made public without his prior knowledge or consent. Also, even if we accept that his purpose was to defend the Eleatic philosophy against charges of logical inconsistency, it doesn't follow that Zeno necessarily regarded his counter-charges as convincing. It's conceivable that he intended them as

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

4. Zeno's Race Course, Part 1
Thoughtful lecture notes for discussing this paradox, presented by S. Marc Cohen.
http://faculty.washington.edu/smcohen/320/zeno1.htm
  • The Paradox Zeno argues that it is impossible for a runner to traverse a race course. His reason is that Physics Why is this a problem? Because the same argument can be made about half of the race course: it can be divided in half in the same way that the entire race course can be divided in half. And so can the half of the half of the half, and so on, ad infinitum So a crucial assumption that Zeno makes is that of infinite divisibility : the distance from the starting point ( S ) to the goal ( G ) can be divided into an infinite number of parts.
  • Progressive vs. Regressive versions
    How did Zeno mean to divide the race course? That is, which half of the race course Zeno mean to be dividing in half? Was he saying (a) that before you reach G , you must reach the point halfway from the halfway point to G ? This is the progressive version of the argument: the subdivisions are made on the right-hand side, the goal side, of the race-course. Or was he saying (b) that before you reach the halfway point, you must reach the point halfway from S to the halfway point? This is the
  • 5. Zeno's Paradox Resolved
    zeno's paradox Resolved The faulty logic in Zeno's argument is the assumption that the sum of an infinite number of numbers is always infinite. While this seems intuitively logical, it is in fact wrong.
    http://www.deltalink.com/dodson/html/zeno.solution.html
    Zeno's Paradox Resolved T Mathematical Treatment of Zeno's Paradox F or a mathematical treatment of Zeno's Paradox, download the PDF file

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

    7. Zeno's Paradoxes
    Discusses the paradoxes of Zeno of Elea, e.g., Achilles and the Tortoise; by Nick Huggett.
    http://plato.stanford.edu/entries/paradox-zeno/
    version
    history HOW TO CITE
    THIS ENTRY
    Stanford Encyclopedia of Philosophy
    A B C D ... Z content revised
    APR
    Zeno's Paradoxes
    Almost everything that we know about Zeno of Elea is to be found in the opening pages of Plato's Parmenides
    1. Background
    Before we look at the paradoxes themselves it will be useful to sketch some of their historical and logical significance. First, Zeno sought to defend Parmenides by attacking his critics. Parmenides rejected pluralism and the reality of any kind of change: for him all was one indivisible, unchanging reality, and any appearances to the contrary were illusions, to be dispelled by reason and revelation. Not surprisingly, this philosophy found many critics, who ridiculed the suggestion; after all it flies in the face of some our most basic beliefs about the world. (Interestingly, general relativity particularly quantum general relativity arguably provides a novel if novelty is As we read the arguments it is crucial to keep this method in mind. They are always directed towards a more-or-less specific target: the views of some person or school. We must bear in mind that the arguments are

    8. Math Forum: Zeno's Paradox
    zeno's paradox. A Math Forum Project
    http://mathforum.org/isaac/problems/zeno1.html
    Zeno's Paradox
    A Math Forum Project Table of Contents: Famous Problems Home The Bridges of Konigsberg
    The Value of Pi

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

    9. About "Zeno's Paradox"
    zeno's paradox. Library Home Full Table of Contents Suggest a Link LibraryHelp Visit this site http//www.jimloy.com/physics/zeno.htm. Author Jim Loy.
    http://mathforum.org/library/view/8069.html
    Zeno's Paradox
    Library Home
    Full Table of Contents Suggest a Link Library Help
    Visit this site: http://www.jimloy.com/physics/zeno.htm Author: Jim Loy Description: 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. A discussion of the inconsistency in Zeno's argument. Levels: Elementary Middle School (6-8) High School (9-12) Languages: English Resource Types: Articles Math Topics: Infinity Logic/Foundations
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    10. Math Forum: Zeno's Paradox
    zeno's paradox. A Math Forum Project
    http://forum.swarthmore.edu/~isaac/problems/zeno1.html
    Zeno's Paradox
    A Math Forum Project Table of Contents: Famous Problems Home The Bridges of Konigsberg
    The Value of Pi

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

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

    12. Transcript - Zeno's Paradox (September 17, 2000)
    zeno's paradox. Say you start in the middle of a room and walk to the wall.
    http://www.earthsky.org/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:

    13. Zeno's Paradox, 2
    A link to an unusual and strange discussion of zeno's paradox in RealityInspector, a novel about chess and computerhacking. from
    http://www.westgatehouse.com/zeno2.html
    from chapter
    Suddenly, a terrible force grabs her. She gulps a deep breath and looks up. Sitting across from her is a strange man dressed in a Greek toga. He smiles and says, "I'm Achilles. Do you wish to hear how I beat the tortoise?" Mary glances about. She is no longer in the Cow Palace; she is in a sunlit room. The walls are plain; on one wall hang ancient weapons of war. When in Greece do as the Greeks, she thinks. "Of course, I want to hear." "Well, when I realized that the tortoise could never win, I knew I had time to discover a solution. You see, the tortoise can't win because it can never cross over the finish line. For the finish line is one dimensional; it has only length but no width. So the tortoise is stopped by the abyss of non-dimensional space. Let me show you." Achilles draws on a piece of paper. "Anything stepping into the abyss will get lost forever because spatial coordinates don't exist there. How can you tell where you are unless you have some reference system? It's like being in a boat on the ocean without having any means for navigating. You just drift about. And in the abyss you drift for eternity. "Now notice that the abyss separates three dimensional space. Since the abyss lacks a spatial dimension, the three dimensional space is actually contiguous. But the abyss does have the dimension of time. Here eternity exists. The present is." Achilles looks at Mary and smiles.

    14. American-Scientist-E-PRINT-Forum: Zeno's Paradox And The Road To The Optimal/Ine
    zeno's paradox and the Road to the Optimal/Inevitable. From Stevan ZENO'SPARADOX AND THE ROAD TO THE OPTIMAL/INEVITABLE. zeno's paradox
    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 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
    get half-way across, and that takes time. And before I can get half-way
    across, I have to get half-way-half-way across, and that takes time
    too. And so on. So how can I possibly even begin?"
    I don't know what the theoretical solution to Zeno's Paradox is, but

    15. Zeno's Paradox Of The Arrow
    Zeno’s Paradox of the Arrow. A reconstruction of the argument. (following space.For an application of the Arrow Paradox to atomism, click here.
    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
  • 16. Re: Zeno's Paradox
    Re zeno's paradox. Subject Re zeno's paradox; From baez@galaxy.ucr.edu (johnbaez); Date 06 Apr 1999 000000 GMT; Next by thread Re zeno's paradox; Index(es
    http://www.lns.cornell.edu/spr/1999-04/msg0015738.html
    Date Prev Date Next Thread Prev Thread Next ... Thread Index
    Re: Zeno's paradox

    17. Re: Zeno's Paradox
    Re zeno's paradox. Subject Re zeno's paradox; From Vesselin Gueorguiev vesselin@baton.phys.lsu.edu ; Prevby thread zeno's paradox; Next by thread Re
    http://www.lns.cornell.edu/spr/1999-04/msg0015719.html
    Date Prev Date Next Thread Prev Thread Next ... Thread Index
    Re: Zeno's paradox

    18. American-Scientist-E-PRINT-Forum: Re: Zeno's Paradox And The Road To The Optimal
    Re zeno's paradox and the Road to the Optimal/Inevitable. Maybe in reply toStevan Harnad zeno's paradox and the Road to the Optimal/Inevitable ;
    http://www.ecs.soton.ac.uk/~harnad/Hypermail/Amsci/0823.html
    Re: Zeno's Paradox and the Road to the Optimal/Inevitable
    From: Stevan Harnad ( harnad@COGPRINTS.SOTON.AC.UK
    Date: Sun Sep 03 2000 - 12:08:40 BST I leave it to others to reply further to Joseph Ransdell if they wish;
    to me it seems he is debating irrelevant points of ideology and
    interpretation, rather than points of susbtance and practice. He thinks
    self-archiving papers in the Ginsparg Archives is primary publication.
    Fine, let him call it what he wishes, as long as authrs actually go
    ahead and do it, rather than taking such interpretations as a further
    basis for Zeno's Paralysis. But his misinterpretation of the CALL to
    self-archive-so-as-to-free-the-literature rather than the
    self-archiving itself as the actual freeing of the literature [1]
    seems so off-the-mark that one can only think it willful.

    19. American-Scientist-E-PRINT-Forum: Re: Problem Of The Ginsparg Archive As Self-ar
    Date Sun Sep 03 2000 014013 BST Next message ransdell, josephm. Re zeno's paradox and the Road to the Optimal/Inevitable ;
    http://www.ecs.soton.ac.uk/~harnad/Hypermail/Amsci/0820.html
    Re: problem of the Ginsparg Archive as self-archiving model
    From: David Goodman ( dgoodman@PHOENIX.PRINCETON.EDU
    Date: Sun Sep 03 2000 - 01:40:13 BST As I interpret the authors' argument it accepts the
    viability of the model in
    the high energy physics community, but argues that it is not
    generally applicable because other scientific communities are not
    equally disciplined or responsible.
    As Steve wisely remarks, how can we tell unless we try?
    As a librarian, I have been an observer of academics in a number of
    different fields. The basic patterns of academic publication and
    status seem to be
    not that dissimilar all of the physical and biological sciences.

    20. ChuckJerry.com: Deep Thoughts: Russians
    Russians. D has stumbled upon zeno's paradox. Now for a little philosophy. I'veread something very interesting lately, and it is called zeno's paradox .
    http://www.chuckjerry.com/thoughts/zeno.shtml
    ChuckJerry.com
    haven for the stupid home
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    Russians D has stumbled upon Zeno's Paradox . It appears to have blown his mind.

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