1. 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 ... Z This document uses XHTML/Unicode to format the display. If you think special symbols are not displaying correctly, see our guide Displaying Special Characters last substantive content change MAR The Encyclopedia Now Needs Your Support Please Read How You Can Help Keep the Encyclopedia Free 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 2. The Paradoxes of Plurality 2.1 The Argument from Denseness 2.2 The Argument from Finite Size ... Related Entries 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 of 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

2. Zeno's Paradox Of The Tortoise And Achilles (PRIME) An article in the Platonic Realms. http://www.mathacademy.com/pr/prime/articles/zeno_tort/

BROWSE ALPHABETICALLY LEVEL: Elementary Advanced Both INCLUDE TOPICS: Basic Math Algebra Analysis 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.

3. Zeno And The Paradox Of Motion 3.7 Zeno and the Paradox of Motion http://tmsyn.wc.ask.com/r?t=an&s=hb&uid=24312681243126812&sid=343126

4. Math Forum Zeno's Paradox zeno's paradox. A Math Forum Project. Table of Contents Famous Problems Home http://tmsyn.wc.ask.com/r?t=an&s=hb&uid=24312681243126812&sid=343126

5. Zeno's Paradox Of The Tortoise And Achilles (PRIME) Zeno's classic paradox, from the Platonic Realms Interactive Math Encyclopedia. http://tmsyn.wc.ask.com/r?t=an&s=hb&uid=24312681243126812&sid=343126

6. 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

7. Zeno's Paradoxes 4. Two more paradoxes 4.1 The Paradox of Place 4.2 The Grain of Millet 5. Zeno's Influence on Philosophy. Further Reading. Bibliography http://tmsyn.wc.ask.com/r?t=an&s=hb&uid=24312681243126812&sid=343126

8. Zeno Of Elea [Internet Encyclopedia Of Philosophy] Zeno was an Eleatic philosopher, a native of Elea (Velia) in Italy, son of Teleutagoras, and the favorite disciple of Parmenides. http://tmsyn.wc.ask.com/r?t=an&s=hb&uid=24312681243126812&sid=343126

9. Zeno's Paradox Of The Arrow zeno's paradox of the Arrow http://tmsyn.wc.ask.com/r?t=an&s=hb&uid=24312681243126812&sid=343126

10. Zeno's Paradox Of The Race Course zeno's paradox of the Race Course The Paradox. Zeno argues that it is impossible for a runner to traverse a race course. His reason is that http://tmsyn.wc.ask.com/r?t=an&s=hb&uid=24312681243126812&sid=343126

11. AlterNet Rights And Liberties The Making Of A Movement For his drawings recall Borges' library of Babel, his Garden of Forking Paths, the Zohar, zeno's paradox or the aphorism by Pascal Borges http://tmsyn.wc.ask.com/r?t=an&s=hb&uid=24312681243126812&sid=343126

12. Zeno's Paradoxes founded Stoicism in the 4th Century BC. And Zeno of Elea (5th Century BC) was the Zeno of the paradoxes. To me, Zeno's arrow paradox seems much http://tmsyn.wc.ask.com/r?t=an&s=hb&uid=24312681243126812&sid=343126

13. Zeno's Paradoxes IV Zeno's Stadium Paradox. From Aristotle http://tmsyn.wc.ask.com/r?t=an&s=hb&uid=24312681243126812&sid=343126

14. Zeno's Paradox Of The Tortoise And Achilles (PRIME) Zeno s classic paradox, from the Platonic Realms Interactive Math Encyclopedia. http://www.mathacademy.com/pr/prime/articles/zeno_tort/index.asp

BROWSE ALPHABETICALLY LEVEL: Elementary Advanced Both INCLUDE TOPICS: Basic Math Algebra Analysis 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.

15. Math Forum: Zeno's Paradox Zeno s first paradox attacks the notion held by many philosophers of his day thatspace was infinitely divisible, and that motion was therefore continuous. 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 ... 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.

16. About "Zeno's Paradox" Zeno s paradox. _ Library Home Full Tableof Contents Suggest a Link Library Help 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 Home The Math Library ... Help http://mathforum.org/

17. Zeno And The Paradox Of Motion We arrive at Zeno s paradox only when these arguments against infinite divisibilityare combined with the complementary set of arguments (The Arrow and The http://www.mathpages.com/rr/s3-07/3-07.htm

3.7  Zeno and the Paradox of Motion The Eleatic school of philosophers was founded by the religious thinker and poet Xenophanes (born c. 570 BC), whose main teaching was that the universe is singular, eternal, and unchanging.  "The all is one."  According to this view, as developed by later members of the Eleatic school, the appearances of multiplicity, change, and motion are mere illusions.  Interestingly, the colony of Elea was founded by a group of Ionian Greeks who, in 545 BC, had been besieged in their seaport city of Phocaea by an invading Persian army, and were ultimately forced to evacuate by sea.  They sailed to the island of Corsica , and occupied it after a terrible sea battle with the navies of  Carthage and the Etruscans.  Just ten years later, in 535 BC, the Carthagians and Etruscans regained the island, driving the Phocaean refugees once again into the sea.  This time they landed on the southwestern coast of Italy and founded the colony of Elea , seizing the site from the native Oenotrians.  All this happened within the lifetime of Xenophanes, himself a wandering exile from his native city of

18. Zeno's Paradoxes To me, Zeno s arrow paradox seems much more interesting than his other paradoxes.It essentially says that if you examine an arrow in flight at one instant http://www.jimloy.com/physics/zeno.htm

Return to my Physics pages Go to my home page Zeno's Paradoxes 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.

19. Zeno's Paradoxes - Wikipedia, The Free Encyclopedia Zeno s paradox however implies that if Zeno s method is followed to its logicalextent, concepts such as velocity lose all meaning and there is no causal http://en.wikipedia.org/wiki/Zeno's_paradoxes

Zeno's paradoxes From Wikipedia, the free encyclopedia. Zeno's paradoxes are a set of paradoxes devised by Zeno of Elea to support Parmenides ' doctrine that "all is one" and that contrary to the evidence of our senses, the belief in plurality and change is mistaken, and in particular that motion is nothing but an illusion Several of Zeno's eight surviving paradoxes (preserved in Aristotle 's Physics and Simplicius 's commentary thereon) are essentially equivalent to one another; and most of them were regarded, even in ancient times, as very easy to refute. Three of the strongest and most famous—that of Achilles and the tortoise , the Dichotomy argument, and that of an arrow in flight—are given here. Zeno's arguments are perhaps the first examples of a method of proof called reductio ad absurdum also known as proof by contradiction . They are also credited as a source of the dialectic method used by Socrates Zeno's paradoxes were a major problem for ancient and medieval philosophers , who found most proposed solutions somewhat unsatisfactory. More modern solutions using calculus have generally satisfied mathematicians and engineers. Many philosophers still hesitate to say that all paradoxes are completely solved, while pointing out also that attempts to deal with the paradoxes have resulted in many intellectual discoveries. Variations on the paradoxes (see

20. Zeno's Paradoxes - Wikipedia, The Free Encyclopedia (Redirected from Zeno s paradox). Zeno s paradoxes are a set of Zeno s paradoxhowever implies that if Zeno s method is followed to its logical extent, http://en.wikipedia.org/wiki/Zeno's_paradox

Zeno's paradoxes From Wikipedia, the free encyclopedia. (Redirected from Zeno's paradox Zeno's paradoxes are a set of paradoxes devised by Zeno of Elea to support Parmenides ' doctrine that "all is one" and that contrary to the evidence of our senses, the belief in plurality and change is mistaken, and in particular that motion is nothing but an illusion Several of Zeno's eight surviving paradoxes (preserved in Aristotle 's Physics and Simplicius 's commentary thereon) are essentially equivalent to one another; and most of them were regarded, even in ancient times, as very easy to refute. Three of the strongest and most famous—that of Achilles and the tortoise , the Dichotomy argument, and that of an arrow in flight—are given here. Zeno's arguments are perhaps the first examples of a method of proof called reductio ad absurdum also known as proof by contradiction . They are also credited as a source of the dialectic method used by Socrates Zeno's paradoxes were a major problem for ancient and medieval philosophers , who found most proposed solutions somewhat unsatisfactory. More modern solutions using calculus have generally satisfied mathematicians and engineers. Many philosophers still hesitate to say that all paradoxes are completely solved, while pointing out also that attempts to deal with the paradoxes have resulted in many intellectual discoveries. Variations on the paradoxes (see

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