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1. Zeno's Paradox: Unraveling the
2. Zeno's Paradoxes
3. Key Contemporary Concepts: From
4. Modern science and Zeno's paradoxes
5. The Paradoxes of Zeno (Avebury
6. Zeno's paradox and the problem
7. Zeno's Paradox
8. The Universal Book of Mathematics:
9. Why mathematical solutions of
10. Supertasks: Zeno's Paradoxes,
11. Paradoxes: Paradox, Russell's
12. Zeno of Elea: An entry from Gale's
13. ZENO OF ELEAc. 490430 BCE: An
14. Towards a definitive solution

1. Zeno's Paradox: Unraveling the Ancient Mystery Behind the Science of Space and Time
by Joseph Mazur
Paperback: 272 Pages (2008-03-25)
list price: US$15.00 -- used & new: US$5.59
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Asin: B001G8WKBM
Average Customer Review: 3.5 out of 5 stars
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The fascinating story of an ancient riddle—and what it reveals about the nature of time and space

Three millennia ago, the Greek philosopher Zeno constructed a series of logical paradoxes to prove that motion is impossible. Today, these paradoxes remain on the cutting edge of our investigations into the fabric of space and time. Zeno’s Paradox uses the motion paradox as a jumping-off point for an exploration of the twenty-five-hundred-year quest to uncover the true nature of the universe. From Galileo to Einstein to Stephen Hawking, some of the greatest minds in history have tackled the problem and made spectacular breakthroughs—but through it all, the paradox of motion remains. ... Read more

Customer Reviews (3)

5-0 out of 5 stars Fascinating read
This is another book written in the same engaging formula, seen before in Mazur's other books. Besides the scientific facts, there is always a human element, and bits of trivia about the people that made these amazing discoveries. A fascinating mix of facts and literature, math and philosophy that draws you into a joyfully thought-provoking read.

1-0 out of 5 stars Run faster than Achilles!,
and perhaps the book won't overtake you and you won't have to buy it.

I purchased this product some time ago, but didn't feel up to the task of reviewing it. What for? Who heeds bad revs?

It's a bad (or rather, unworthy of its theme), bad book all right. I'll be brief:

1) Its exposure of philosophy is superficial and biased (I don't have the space here to give examples, but trust me).

2) It's repetitive. For example, the stadium paradox is covered at least thrice: in page 4 of the Introduction (where it's stated that Aristotle exposed it as based on a fallacy); in pp. 29/31, where Mazur gives Zeno his due; and in page 41/42, where the book says Aristotle failed to understand the nature of the paradox. The other paradoxes (especially the arrow) are also analized several times.

3) It's incoherently written. For example, in page 132, Mazur writes "The arrow paradox also requires an understanding of limits as a mathematical model for instantaneous velocity, which calculus treats as a derivative, an instrument that creates limits of average changes in a dependent variety in small intervals on an independent variable. The model here is to view each point on the arrow's trajectory as though it were a limit of a sequence of rational numbers on the number line, so the arrow's path is assured a persistent even flow of space in the continuity of time. In effect it assumes, quite correctly, that all numbers on the number line are convergent sequences of rational numbers". Half a page (in the book's oversize font) completely wasted. And don't you think that a reader who understands what "convergent sequences of rational numbers" means would also know what is a real number?

4) Mazur manages to be at the same time irrelevant, at the limit of his knowledge, and a provider of meaningless detail. Now hear this (p. 196)!: "The quantum mechanics story began when a German physicist named Max Karl Ernst Ludwig Planck asked why subatomic particles radiate a blue light when they travel through a non vacuum-medium faster than the speed of light in that medium". Why, Mr. Mazur, methought Cerenkov radiation was discovered much later. What Planck (what would have happened had
he had only three names?) was looking for was a way to avoid the so called "ultraviolet catastrophe" in the black body radiation formula.

Bear in mind that each of these examples could be multiplied almost indefinitely. On the other hand, nothing on supertasks, or the legitimacy of conflating the geometrical and the real number continua, or the conceptual resemblance between Zeno's paradoxes and Kant's antinomies, and the question these place on the possibility of a represtation-based understanding of Nature; ... .

Do you think I'm unfair? But Mazur strikes me as intellectually dishonest in the same sense as Lacan when he equated, before an innocent crowd of bewildered and awed disciples, the phallus with i, the unity of imaginary numbers.

In short, if you think that the story of math and physics consists of knowing that in 1586 Stevin, Maurice & alia often met at a tavern where "water, dripping from cracks in its massive stone walls, kept [it] cool and damp. Candles and torch sconces provided moderate light in the windowless room. An intoxicating smell of fermenting spirits seeped from a whiskey and brandy distillery next door. Beer was cheap", and that they "would often sit together at a long sticky oak table coated with layers of sugars dried from decades of beer spills" (pp. 68/69), or that Dirichlet's names were Johann Peter Gustav Lejeune (p. 116), then you'll learn a lot from this book.

In any other case, avoid this travesty.

5-0 out of 5 stars dialectic
Fun to see this book. The subject is still alive. For someone not acquainted with Zeno's paradoxes, here is a book by a contemporary author supported by a contemporary publisher for a contemporary audience. But the subject is ancient, having been discussed by authors since Aristotle. An older literature is certainly available for those who would like to learn more of the details.

Dialectic is the flow of peace from micro to macro in Plato's Republic: Book I

I do have a critique. I would've preferred a healthier skepticism about Plato, especially where Plato uses second-hand sources about Parmenides, Zeno, and Socrates. Courts of law disregard hearsay and here I would apply the same rationale. For example, Aristotle said the "forms" were from Plato, not from Socrates. So when reading Socrates' story of the cave, because of Aristotle's warning I try to strip away what might be suspect as Plato's and instead look underneath for a basic story that Socrates might actually have told. The basic story of the cave seems to be that only through "dialectic" can you get out of the cave of darkness to see what you really are. Plato's writings also seem inaccurate about "dialectic." Whatever "dialectic" is, I know you are not going to get out of the cave of darkness to see what you really are by participating in dialectic with me. Based on what Aristotle said, I don't think you could've gotten out of the cave of darkness to see what you really are by dialectic with Plato, either. Only dialectic with Socrates could have guided you out of the cave of darkness to see what you really are.

Given these problems of partial information and hearsay, taking full account of context may help an interested reader. Socrates and Parmenides were of different times. Socrates followed Parmenides. And although our other main source about Socrates-- Xenophon-- was neither poet nor author like Plato, neither Xenophon nor Plato was in that closest circle of friends (those most intent in their practice of Socrates' message) who called Socrates "Master." But today for our information about Socrates we mostly rely on the writings of Plato and Xenophon, Plato is preferred to Xenophon, and neither was as attuned to Socrates' message as those who called Socrates Master. By comparison, Zeno seems to have heeded the message of Parmenides more closely than Plato heeded the message of Socrates. Rather than after he died, writing down the history of this amazing person who had tried to help them (as did Plato and Xenophon), Zeno constructed his paradoxes in order to help Parmenides in Parmenides' own lifetime. So it is a possibility that Zeno heard some feedback from Parmenides about his paradoxes and applied it, while it was impossible for Plato and Xenophon to get that kind of feedback from Socrates, who had died. From this point of view, Parmenides' message is the basic context for reading about Zeno.

Translating the Greek word "auto" into "self," Parmenides' most famous quote is "the self is for thinking and being." Thinking is a pragmatic tool for getting along in the world but does have limits. In Zeno's imagined race, by relying on thinking the clever tortoise could beforehand try to get Achilles to imagine that Achilles could never catch up from the head start being granted to the tortoise and so give up the race without even trying. Because before catching up, Achilles would always have to, first, get to an imagined midpoint between himself and the tortoise. But after reaching a midpoint, how could he ever get past another midpoint? The paradox imagined in the story comes from thinking that can't get beyond the imagined next midpoint. But recall that Parmenides said "the self is for thinking AND being." To win the actual race, all Achilles would have to do is Be what he is-- a human being with extreme physical capabilities. Simply by being the human being that he is, Achilles could win any race against a tortoise. Thinking does rule the imagination and without doubt works for many purposes. But as Zeno's story of the race illustrates, being what you are-- in the paradox, being a highly fit human being racing against a tortoise-- determines the real (not imagined) outcome.

The way it looks to me is that in their own time Parmenides and Socrates each had a gift for dialectic that could guide a person beyond thinking to be "the self" that truly-- beyond imagination and thought-- exists. In my own experience, that turns out to be the most interesting context for reading about Zeno. Maybe you will find it to be that way, too.

... Read more

2. Zeno's Paradoxes
Paperback: 320 Pages (2001-03)
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Asin: 0872205606
Average Customer Review: 4.5 out of 5 stars
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A reprint of the Bobbs-Merrill edition of 1970.

These essays lead the reader through the land of the wonderful shrinking genie to the warehouse where the "infinity machines" are kept. By careful examination of a lamp that is switched on and off infinitely many times, or the workings of a machine that prints out an infinite decimal expansion of pi, we begin to understand how it is possible for Achilles to overtake the tortoise. The concepts that form the basis of modern science-space, time, motion, change, infinity-are examined and explored in this edition. Includes an updated bibliography. ... Read more

Customer Reviews (2)

4-0 out of 5 stars Great Investigation of the Notion of Infinity
I really enjoyed this book.It's a great survey of the notion of infinity, and the quality of the prose is, for the most part, very good.I would definitely recommend this book to anyone interested in philosophy, mathematics, or the history of thought.

5-0 out of 5 stars Good Survey of Modern Reaction
This is a great introduction to the modern reaction to Zeno's paradoxes. The most important articles from Russell to the debates on infinity machines are included. The bibliography, at over 200 works, is the best I've seen. There is a mathematical bent to most of the articles, usually in the form of questions of infinity, or measure theory. Nonetheless, there are articles by philosophers who reject the idea of a completed infinity. I did a semester-long college project on Zeno's paradoxes, focusing on mathematical implications, and this was the most useful sourcebook by far. ... Read more

3. Key Contemporary Concepts: From Abjection to Zeno's Paradox (Sage Key Concepts)
by Dr John Lechte
Paperback: 222 Pages (2003-02-24)
list price: US$43.95 -- used & new: US$34.19
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Asin: 0761965351
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Companion volume to Lechte's best-selling 50 Key Contemporary Thinkers. Guides readers in understanding society and culture in the twenty-first century. Encyclopedic format covers such topics as cybernetics, quantum theory, ideology, and aesthetics. For anyone interested in the human sciences. Softcover, hardcover available. ... Read more

4. Modern science and Zeno's paradoxes
by Adolf Grunbaum
 Hardcover: 153 Pages (1968)

Asin: B0006E038E
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5. The Paradoxes of Zeno (Avebury Series in Philosophy)
by J. A. Faris
 Hardcover: 136 Pages (1996-10)
list price: US$120.00 -- used & new: US$120.00
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Asin: 1859723683
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In the 5th century BC Zeno of Elea propounded four arguments about motion leading to paradoxical propositions: that a finite distance cannot be traversed by a moving object, that a faster runner cannot overtake a slower one. that an arrow flight is at rest, that half a given time is equal to the whole. These paradoxes were intended as support for the doctrine of Parmenides that all apparent motion is illusory, and of the attempts that have been made to show wherein they are fallacious none has met with universal acceptance. In this work solutions are suggested for the first three paradoxes. There is an exposition of the atomic theory of space and time which has been thought to underlie the fourth, and attempts which some commentators have made to explain the fourth on that basis are examined. ... Read more

6. Zeno's paradox and the problem of free will.: An article from: Skeptic (Altadena, CA)
by Phil Mole
 Digital: 19 Pages (2004-01-01)
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This digital document is an article from Skeptic (Altadena, CA), published by Skeptics Society & Skeptic Magazine on January 1, 2004. The length of the article is 5630 words. The page length shown above is based on a typical 300-word page. The article is delivered in HTML format and is available in your Amazon.com Digital Locker immediately after purchase. You can view it with any web browser.

Citation Details
Title: Zeno's paradox and the problem of free will.
Author: Phil Mole
Publication: Skeptic (Altadena, CA) (Refereed)
Date: January 1, 2004
Publisher: Skeptics Society & Skeptic Magazine
Volume: 10Issue: 4Page: 58(8)

Distributed by Thomson Gale ... Read more

7. Zeno's Paradox
by F. Gordon Robinson
Paperback: 272 Pages (2007-10-25)
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Asin: 1601453396
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Zeno's Paradox, was inspired by the philosophy of the novelist Ayn Rand.

Howard Rhodes' father was a renowned physicist. He and his wife were killed in a suspicious car accident when Howard was fourteen years old. Howard is now sixteen and living on his own as an emancipated minor. He is also an intellectually gifted and well-adjusted teenager whose mother stimulated his love for learning, explained Christianity's fatal flaw to him, and taught him that his most valuable asset was his brain and his ability to reason.

In school, Howard deals with a challenge from a classmate who is a class bully, and who is making everyone's life miserable. He also entertains his classmates and stuns his math and physics teachers when he unravels the error in the logic to Zeno's famous Paradox, which has had scientists baffled since 400 BC.

Howard's interest in physics leads him to investigate various possibilities regarding small particles, and he attracts everyone's attention, including Stefan Nacouski's who is a sleeper al Qaeda agent living in the United States, when he discovers a new source of energy.

Not only does Howard's new source of energy eliminate the need for burning fossil fuels, but also, he can create a particle beam shield that will protect the United States and its allies from a Ballistic Missile attack. Unfortunately, this same device can be refined to create a beam that will silently kill hundreds of innocent people. Stephan will resort to anything to get his hands on Howard's Killing Machine. ... Read more

8. The Universal Book of Mathematics: From Abracadabra to Zeno's Paradoxes
by David Darling
Hardcover: 400 Pages (2004-08-11)
list price: US$40.00 -- used & new: US$9.00
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Asin: 0471270474
Average Customer Review: 5.0 out of 5 stars
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Praise for David Darling

The Universal Book of Astronomy

"A first-rate resource for readers and students of popular astronomy and general science. . . . Highly recommended."
-Library Journal

"A comprehensive survey and . . . a rare treat."

The Complete Book of Spaceflight

"Darling's content and presentation will have any reader moving from entry to entry."
-The Observatory magazine

Life Everywhere

"This remarkable book exemplifies the best of today's popular science writing: it is lucid, informative, and thoroughly enjoyable."
-Science Books & Films

"An enthralling introduction to the new science of astrobiology."
-Lynn Margulis

Equations of Eternity

"One of the clearest and most eloquent expositions of the quantum conundrum and its philosophical and metaphysical implications that I have read recently."
-The New York Times

Deep Time

"A wonderful book. The perfect overview of the universe."
-Larry Niven ... Read more

Customer Reviews (5)

5-0 out of 5 stars It's in there!
It is a "dictionary" of mathematical terms.Seems to be pretty complete.Great resource.

4-0 out of 5 stars The Lighter Side of Mathematics
This encyclopedia of mathematics was quite worthwhile to read nevertheless.I expected an in-depth review of various famous math formulas and procedures for calculating numbers and short stories about the people who created them.Most of the items listed were interesting to read about.Yet,some were just trivial in nature.Based upon the glowing reviews,i felt compelled to investigate this book further.For the serious math student,this book is more crust than bread.For a book entitled,'Universal',it's fairly enclusive to the British and German mathematicians only. Now,puzzlers and ratzlers may be entertained by the charming entries within and even given the impetus to advance their research into weightier mathematic descriptions.However,this book only appears on the 'heavyside',not really enriching enough to nurture a sprouting engineer into fructation.A better title would simply be,"A Short History of Popular Mathematics and Puzzles".

5-0 out of 5 stars Recreational Mathematics ...exercise and enjoyment for the mind.
I have had an interest in Recreational Mathematics ever since my High School days; at least 55 years ago. I can still remember constructing a 15 Puzzle using a block of wood and my mother's wooden yardstick,with numbers pasted on from a calendar.Then there was the cardboard set of Tangrams I made after reading about them in a book.The next thing I can recall was Magic Squares and learning how to solve any odd-number.I was then nearly driven to distraction trying to find the "secret" to solving even- numbered Magic Squares. In High School,after encountering Logrithms for the first time,I couldn't get enough of them.I was talking to my Math teacher one day after school,telling him how fascinating I found them. He reached into his desk drawer and pulled out a Slide Rule.He asked me if I knew what it was. I had never seen or heard of one. He showed me how it was based on Logrithims and could be used to multiply,divide ,etc. Then he handed me a little booklet and sent me packing with it and the slide rule. It was the most fascinating thing I'd ever seen and almost as much fun as girls and shooting pool.I guess my love for puzzles and such was what led me into pursuing college and eventually becoming an Engineer.
All through the years,I've retained this interest in Puzzles and Recreational Mathematics. I started to acquire books on the subject,and once reading about Sam Loyd and the "Cyclopedia of Puzzles" ,I couldn't contain myself until I could at least see it and maybe even acquire a copy.It was published in 1914,and included over 5,000 puzzles,tricks,conundrums,riddles,etc.,of which about 1'000 are illustrated.Solutions are printed in the last pages.However,a number of puzzles were selected as "Prize Puzzles" and the solutions were withheld.A prize of $100 would be awarded to the person who sent in the best set of correct answers before January,1915.Because of errors,multiple solutions,impossible solutions,etc., it was impossible to determine a winner. It was quite a story!.After much effort,I finally got myself a copy and without doubt it is my prized puzzle book possession. I have never seen another copy,but it is sometimes shown in other puzzle books. I believe it has been reprinted ,and a condensed version was also published.
Over the years,I have added many Math and Puzzle books to my library and now have about 600.
I go through all this ,just to show you how much I think of this new book by David Darling.
I don't know whether it should be called a Dictionary,an Encyclopedia, (it's not a "Cyclopedia" as Sam Loyd called his),a Compendium, or what. It is all those things and much more.But not to worry,Darling had to call it something and I guess his title is as good as any.
It is a basic reference book dealing with all kinds of things,people ,definitions,theories,puzzles,terms, and Recreational mathematics. If you are reading about anything or anybody in these areas and want to know more;this will be the book to turn to to get started.
The book is organized alphabetically,like an encyclopedia,but also has an Index by category. It has an extensive and excellent Reference as well.
There is no doubt that I will turn to this book often in the future.It's only a shame I didn't have it ;or something similar to it many years ago.
Please don't take a few things I'm going to say as nitpicking. That is not my intent,So,here goes;

On page 278 it talks about the Rubik's Cube solution record being around 20 seconds. In 2003,the World Championship was held in Toronto.The winner was able to set a new record at around 16 seconds. This year,2007 a Canadian Championship was held. The winner did it with an average of 14.21 seconds.including one solving under 10 seconds.One solver managed to solve it in 4 minutes and 54 seconds...blindfolded.
On page 117,the 15 Puzzle is said to have been invented by Sam Loyd who could not obtain a patent. In 2006,Jerry Slocum published a magnificient book on this puzzle.In it you will see who the real inventer was,and how Sam Loyd fooled everyone about it for 115 years. It was actually invented by Matthias J. Rice in 1879,and was originally called the Gem Puzzle.This book does a marvelous job of researching the history of this puzzle.(See my review,June 6,2006.) Mr Slocum has also recently published another excellent book on Tangrams which I also reviewed on Jan 6,2004.He has written several other books on Puzzles,has the world's largest puzzle collection,and heads up the Slocum Puzzle Foundation and Museum in Beverly Hills,California.
The author has done a very good job of discussing Polyhedra,and has shown several in his book. I would like to point out the work done by Magnus Wenninger,who is the world's expert on constructing Polyhedron Models. He has written an excellent book on the subject as well. He has a web site showing many dozens of his models. Just search the netunder Mangus Wenninger .If you enjoy these models,it will blow you away. You can even purchase them at extremely reasonable prices. I first became aware of him through someone who constructs similar models.He takes a different approach,and builds his using balls and sticks. They are simply delightful and remind me of the illustration of the moleculeat the top of the cover of this book.
Just to keep things interesting,there are many puzzles included throughout the book. If you have been interested in this sort of thing ,you'll have encountered many of them over the years. If Mathematical Recreations is new to you;this book is a wonderful introduction to it all.
What more can I say? If you're into Mathemtical Recreations,Mathematics in general and Puzzles of all types ;you are going to love this book and want a copy.
I have been searching for the name to describe someone who is interested in solving puzzles. The best I have been able to find so far are;METAGROBOLOGIST andOMNIHEURIST. if you know of any others ,I would be pleased to hear from you.

Thank you David,it's a great book!

5-0 out of 5 stars It is on my essential reference shelf
I enjoy reading mathematical dictionaries. Whether I read it from cover to cover or scattershot style, I always learn something new. In this dictionary, I learned about the "Seventeen or Bust" distributed computing project, where the goal of the project is to check the remaining seventeen possibilities to be the smallest Sierpinski number. I also was reminded of many other mathematical facts that I have encountered sometime in the past.
The manuscripts that I receive as co-editor of Journal of Recreational Mathematics contain a wide variety of mathematical ideas. To handle them all, it is necessary to keep a mathematical dictionary handy. Since this book is well written and has over 1,800 entries, I have placed it on my essential reference shelf.

Published in Journal of Recreational Mathematics, reprinted with permission.

5-0 out of 5 stars Everything you wanted to know about mathematics and far more
From over 300 references, David Darling has compiled what I, a non-mathematician, consider to be an excellent encyclopedia of mathematics.There are over 1800 entries.There are simple definitions, more in-depth explanations, graphs and many photos.He illustrates well the application and appearance of many abstract mathematical concepts in the real world of art, architecture, etc.

In addition to the hundreds of 'pure' mathematical references, he also includes many entries that are fun for everyone. These entries include puzzles, games and tricks.

I enjoyed the background and historical information included in the biographies of the many mathematicians covered.Historical information about concepts and values, e.g. pi, is also included, such as the time Indiana almost voted to round pi off to 3.2!

This book would be an excellent library builder.It is hard to read straight through - I tried it and failed - but as reference and reading here and there it is great.It is good enough that I want to find his other references and check them out as well.
... Read more

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
 Digital: 24 Pages (1996-12-01)
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Asin: B00096PO4W
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This digital document is an article from The Review of Metaphysics, published by Philosophy Education Society, Inc. on December 1, 1996. The length of the article is 7090 words. The page length shown above is based on a typical 300-word page. The article is delivered in HTML format and is available in your Amazon.com Digital Locker immediately after purchase. You can view it with any web browser.

From the supplier: Zeno's paradoxes are concerned with the relationship between the one and the many, and thus mathematical solutions are beside the point. In the paradoxes of Achilles and the tortoise, the dichotomy, the arrow and the stadium, Zeno is illustrating the view of Parmenides that only the one is real and that it is impossible to conceive of a transition from the one to the many or from being to becoming. Mathematical solutions are based on manipulating the unit and thus fall outside the parameters of Zeno's paradoxes. Zeno may have been attacking the Pythagorean idea that multiplying or adding could produce the many out of the one.

Citation Details
Title: Why mathematical solutions of Zeno's paradoxes miss the point: Zeno's one and many relation and Parmenides' prohibition.
Author: Alba Papa-Grimaldi
Publication: The Review of Metaphysics (Refereed)
Date: December 1, 1996
Publisher: Philosophy Education Society, Inc.
Volume: v50Issue: n2Page: p299(16)

Distributed by Thomson Gale ... Read more

10. Supertasks: Zeno's Paradoxes, Hilbert's Paradox of the Grand Hotel, Omega Point, Supertask, Thomson's Lamp
Paperback: 56 Pages (2010-05-06)
list price: US$19.99 -- used & new: US$19.99
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Asin: 1155647289
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Purchase includes free access to book updates online and a free trial membership in the publisher's book club where you can select from more than a million books without charge. Chapters: Zeno's Paradoxes, Hilbert's Paradox of the Grand Hotel, Omega Point, Supertask, Thomson's Lamp, Balls and Vase Problem. Excerpt:The balls and vase problem (also known as the Ross-Littlewood paradox or the ping pong ball problem ) is a hypothetical problem in abstract mathematics and logic designed to illustrate the seemingly paradoxical , or at least non-intuitive , nature of infinity . The problem starts with an empty vase and an infinite supply of balls at a starting time before noon. At each step in the procedure, balls are added and removed from the vase. The question is then posed: How many balls are in the vase at noon? At each step, balls are inserted into and removed from the vase in a particular order: As part of the problem statement, it is assumed that an infinite number of steps is performed. This is allowed by the following conditions: This guarantees that a countably infinite number of steps is performed by noon. Solutions Answers to the puzzle fall into several categories. Vase is empty Since by noon every ball n that is inserted into the vase (at step n 10) is eventually removed in a subsequent step (step n ), the vase is empty at noon. Vase contains infinite balls Since at each step ten balls are inserted but only one is removed, a net nine balls are added at every step before noon. Clearly, the number of balls (as a function of the step) equals 9 times the step: B = 9 n for all n . So as n tends to infinity, B likewise diverges towards infinity. Therefore, the vase is filled with an infinite number of balls by noon. Alternatively, since one ball out of ten is removed at every step, the remaining number of balls will be 10 of infinity, which is infinite. Answer to problem depends The number of balls that one ends up with depends on the order in which the balls... ... Read more

11. Paradoxes: Paradox, Russell's Paradox, Problem of Evil, Impossible Object, Arrow's Impossibility Theorem, Zeno's Paradoxes, Epimenides Paradox
Paperback: 432 Pages (2010-09-15)
list price: US$49.93 -- used & new: US$49.93
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Asin: 1157713513
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Chapters: Paradox, Russell's Paradox, Problem of Evil, Impossible Object, Arrow's Impossibility Theorem, Zeno's Paradoxes, Epimenides Paradox, Liar Paradox, Barber Paradox, Raven Paradox, Voting Paradox, Newcomb's Paradox, Unexpected Hanging Paradox, Omnipotence Paradox, Strange Loop, Chicken or the Egg, All Horses Are the Same Color, List of Paradoxes, D'alembert's Paradox, Ship of Theseus, Quantitative Structure-Activity Relationship, French Paradox, Sorites Paradox, Moore's Paradox, Perceptual Paradox, When a White Horse Is Not a Horse, Barbershop Paradox, Ellsberg Paradox, Problem of Future Contingents, Apportionment Paradox, Mere Addition Paradox, Paradox of the Pesticides, Liberal Paradox, Catch-22, Lottery Paradox, Elevator Paradox, Paradoxes of Material Implication, Fitch's Paradox of Knowability, Chainstore Paradox, Buridan's Ass, Paradox of Hedonism, Round Square Copula, C-Value Enigma, Drinker Paradox, Abilene Paradox, Balls and Vase Problem, Grelling-nelson Paradox, Kavka's Toxin Puzzle, Hardy's Paradox, Boltzmann Brain, Observer's Paradox, Hobson's Choice, Paradox of the Court, Buttered Cat Paradox, the Treachery of Images, Paradox of Analysis, Quine's Paradox, Coastline Paradox, Paradox of the Plankton, Ludlul Bēl Nēmeqi, Mexican Paradox, Bracketing Paradox, Discursive Dilemma, Bonini's Paradox, Crocodile Dilemma, Irresistible Force Paradox, Exception Paradox, Movement Paradox, Paradoxology, Applicability Domain, Hispanic Paradox, Preface Paradox, Paradox of Tolerance, Faraday Paradox, Algol Paradox, Lombard's Paradox, Is the Glass Half Empty or Half Full?, Icarus Paradox, Prevention Paradox, Code-Talker Paradox, Nihilist Paradox, Socratic Paradox, Nikodym Set, Mandeville's Paradox, Milner-rado Paradox, Yablo's Paradox, Excusable Negligence, Absence Paradox. Source: Wikipedia. Pages: 430. Not illustrated. Free updates online. Purchase includes a free trial membership in the publisher's book club where you can se...More: http://booksllc.net/?id=37391 ... Read more

12. Zeno of Elea: An entry from Gale's <i>Science and Its Times</i>
by Judson Knight
 Digital: 2 Pages (2001)
list price: US$2.90 -- used & new: US$2.90
(price subject to change: see help)
Asin: B0027UWJMS
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Editorial Review

Product Description
This digital document is an article from Science and Its Times, brought to you by Gale®, a part of Cengage Learning, a world leader in e-research and educational publishing for libraries, schools and businesses.The length of the article is 678 words.The article is delivered in HTML format and is available in your Amazon.com Digital Locker immediately after purchase.You can view it with any web browser.The histories of science, technology, and mathematics merge with the study of humanities and social science in this interdisciplinary reference work. Essays on people, theories, discoveries, and concepts are combined with overviews, bibliographies of primary documents, and chronological elements to offer students a fascinating way to understand the impact of science on the course of human history and how science affects everyday life. Entries represent people and developments throughout the world, from about 2000 B.C. through the end of the twentieth century. ... Read more

13. ZENO OF ELEAc. 490430 BCE: An entry from Gale's <i>Encyclopedia of Philosophy</i>
by Richard McKirahan
 Digital: 9 Pages (2006)
list price: US$8.90 -- used & new: US$8.90
(price subject to change: see help)
Asin: B001SCK0K4
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Editorial Review

Product Description
This digital document is an article from Encyclopedia of Philosophy, brought to you by Gale®, a part of Cengage Learning, a world leader in e-research and educational publishing for libraries, schools and businesses.The length of the article is 6654 words.The article is delivered in HTML format and is available in your Amazon.com Digital Locker immediately after purchase.You can view it with any web browser.Explores major marketing and advertising campaigns from 1999-2006. Entries profile recent print, radio, television, billboard and Internet campaigns. Each essay discusses the historical context of the campaign, the target market, the competition, marketing strategy, and the outcome. ... Read more

14. Towards a definitive solution of Zeno's paradoxes
by Fazal Ahmad Shamsi
 Unknown Binding: 84 Pages (1973)

Asin: B0006CXZPY
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