61. Zeno's Paradox Non Archimedean version of Zeno s paradox. Recall the usual versionhas Achilles chasing a turtle, where the turtle is given a ten http://www.lix.polytechnique.fr/Labo/Ilan.Vardi/zeno.html

Non Archimedean version of Zeno's paradox. Recall the usual version has Achilles chasing a turtle, where the turtle is given a ten cubit head start and Achilles runs ten times faster than the turtle. When Achilles runs 10 cubits, the turtle goes 1 cubit. When Achilles runs 1 cubit, the turtle goes 1/10 of a cubit. It follows that Achilles catches the turtle in 10 + 1 + 1/10 + ... cubits. In modern notation, one can write these numbers as decimals yielding It follows that Achilles catches the turtle in 11 1/9 cubits. The non Archimedean case has the turtle trying to catch Achilles who is now given a ten cubit head start. When the turtle goes 10 cubits, Achilles will have run 100 cubits. When the turtle goes 100 cubits then Achilles will have run 1000 cubits, and so on. It follows that the turtle goes 10 + 100 + 1000 + .... Ordinarily, this would be considered as meaningless or ``infinity''. However, one can consider this purely formally, in other words, without caring too much about the actual meaning of the numbers and just using algebraic manipulations. So let x = 10 + 100 + 1000+ ..., then 9(x + 1) = 9 + 90 + 900 + 9000 + ... = ... 9999.

62. Zeno's Paradox NebulaSearch Home NebulaSearch Encyclopedia Top Zeno s paradox. Zeno s paradox, NebulaSearcharticle for Zeno s paradox. Please see Zeno s_paradoxes. NebulaSearch. http://www.nebulasearch.com/encyclopedia/article/Zeno's_paradox.html

NebulaSearch Home NebulaSearch Encyclopedia Top Zeno's paradox Main Index Tiphys..................Øvre_Eiker Yakima_River..................Østensjø Zendo..................Zenogantner Zeno's paradox NebulaSearch article for Zeno's paradox Please see Zeno's_paradoxes Related Links Zeno's Paradox of the Tortoise - An article in the Platonic Realms. http://www.mathacademy.com/pr/prime/articles/zeno_tort/ Zeno's Paradox of the Tortoise - An article in the Platonic Realms. http://www.mathacademy.com/pr/prime/articles/zeno_tort/ Paradoxes and Dilemmas - Common paradoxes and dilemmas, particularly of the social type: the Voting Paradox, Prisoner's Dilemma, Newcomb's Paradox, Unexpected Hanging, Execution Paradox, and the Self-Amendment Paradox. http://perspicuity.net/paradox/paradox.html Corel: Paradox forum at Tek-Tips - Corel: Paradox technical support forums and mutual help system for computer profes sionals. Selling and recruiting forbidden. http://www.tek-tips.com/gthreadminder.cfm/lev2/4/lev3/27/pid/177 The Epimenides Paradox - An analysis of several attempted resolutions of the Epimenides Paradox (also known as the Liar Paradox), showing how they all fail. http://www.ocf.berkeley.edu/~sjblatt/notes/nottrue.html

63. The Frontier - Zeno's Paradox Zeno s paradox. Over two thousand years ago, a philosopher namedZeno made a Zeno s paradox, copyright 1998 by George Beckingham. http://members.tripod.com/~geobeck/frontier/zeno.html

var cm_role = "live" var cm_host = "tripod.lycos.com" var cm_taxid = "/memberembedded" Zeno's Paradox Over two thousand years ago, a philosopher named Zeno made a startling discovery, one that had profound ramifications for life, the universe, and everything. In order to prove his idea, he needed two accomplices: Achilles, the mighty warrior, and a tortoise. Zeno called Achilles and the tortoise together one day, and proposed a race between them. Achilles agreed immediately. Being the strongest warrior and fastest runner in all of Greece, he knew he could easily win a race against an animal that was essentially a paperweight with legs. He even suggested that he have both arms tied behind his back, that he be blindfolded, and that he hop through the race on one foot, so great was his confidence. Zeno declined Achilles’ suggestions, but gave the tortoise one concession: a head start of one stadium (a unit of length whose approximate size is obvious). Achilles laughed at the futility of the gesture, but Zeno smugly stated that he had given the tortoise the only concession he would need. When Zeno saw the result of the race, he was very troubled. It turned out that the tortoise had fallen asleep at the start of the race, and hadn’t moved at all. Yet Achilles still wasn’t able to catch him. Zeno followed his reasoning to its logical conclusion and realized that no matter how small the distance between Achilles and the tortoise was (as long as it was a finite distance), Achilles could never cross it because he would always have to travel an infinite number of half distances, each one being finite (half of any finite number is a finite number), thus taking a finite amount of time to travel. Zeno had proven that all motion is impossible.

64. Ephilosopher :: Puzzles And Paradoxes :: Zeno's Paradox Solved? Philosophy Puzzles and paradoxes Zeno s paradox Solved? Moderated by adimantis,talking_dog. Author, Zeno s paradox Solved? Wisewoman skia Posts 4 http://www.ephilosopher.com/phpBB_14-action-viewtopic-topic-329&27.html

@import url("themes/PostNukeBlue2/style/style.css"); E P H I L O S O P H E R Jun 06, 2004 - 02:28 AM Hey You! Want to join? or Login Search advanced web Chatter Box Forum Stats 5 recent Topics: more topics Why our brain oriented like 'this'? JibberJabber Jun 06, 2004 - 02:20 AM Violence is good ,and is the only way forward. mnelson Jun 06, 2004 - 02:14 AM Can the problem of evil be a good argument against the existence of the Judeo-Christian God? myriadshalak Jun 06, 2004 - 02:10 AM Highly annoying habits of posters YadaYada Jun 05, 2004 - 10:55 PM Definition of selfconsciousness YadaYada Jun 05, 2004 - 05:08 PM Main Menu Site Map Home QuickLinks Student Help Grad School Department Rankings Academic Jobs ... Philosophy Blogs Community Submissions Philosophy Forums Members List Suggested Books ... Top 10 Lists Archives 2000-4 Archives Special Sections Interviews Articles Columns EP Newsletter You are currently not logged in , but you can still subscribe to our newsletter. Blog Headlines Scepticism with a 'c' More Philosophy Department Blogs "Science isn't a democracy" Has Bush Lost His Mind? ... On Applying Twice to Grad Schools in Philosophy Quotables Certainly it is wrong to be cruel to animals and the destruction of a whole species can be a great evil. The capacity for feelings of pleasure and pain and for the form of life of which animals are capable clearly impose duties of compassion and humanity in their case. - A Theory of Justice

65. Ephilosopher :: Puzzles And Paradoxes :: Zeno's Paradox Solved? Philosophy Puzzles and paradoxes Zeno s paradox Solved? Moderated by adimantis,talking_dog. Author, Zeno s paradox Solved? bollinger noesis Posts 180 http://www.ephilosopher.com/phpBB_14-action-viewtopic-topic-329-start-15&27.html

@import url("themes/PostNukeBlue2/style/style.css"); E P H I L O S O P H E R Jun 06, 2004 - 02:28 AM Hey You! Want to join? or Login Search advanced web Chatter Box Forum Stats 5 recent Topics: more topics Why our brain oriented like 'this'? JibberJabber Jun 06, 2004 - 02:20 AM Violence is good ,and is the only way forward. mnelson Jun 06, 2004 - 02:14 AM Can the problem of evil be a good argument against the existence of the Judeo-Christian God? myriadshalak Jun 06, 2004 - 02:10 AM Highly annoying habits of posters YadaYada Jun 05, 2004 - 10:55 PM Definition of selfconsciousness YadaYada Jun 05, 2004 - 05:08 PM Main Menu Site Map Home QuickLinks Student Help Grad School Department Rankings Academic Jobs ... Philosophy Blogs Community Submissions Philosophy Forums Members List Suggested Books ... Top 10 Lists Archives 2000-4 Archives Special Sections Interviews Articles Columns EP Newsletter You are currently not logged in , but you can still subscribe to our newsletter. Blog Headlines Scepticism with a 'c' More Philosophy Department Blogs "Science isn't a democracy" Has Bush Lost His Mind? ... On Applying Twice to Grad Schools in Philosophy Quotables All the choir of heaven and furniture of earth - in a word, all those bodies which compose the mighty frame of the world - have not any subsistence without a mind. - Principles of Human Knowledge

66. Eclecticism > Zeno's Paradox Zeno s paradox may be rephrased as follows. Suppose I wish to cross the room.First, of course, I must cover half the distance. here. Zeno s paradox. http://www.michaelhanscom.com/eclecticism/2003/03/zenos_paradox.html

hostName = '.michaelhanscom.com'; eclecticism I'm too sexy for my blog. home about past comments ... photos here Zeno's Paradox March 11:04 AM Links Ever since I read Douglas Hofstadter's , I've had the paradox postulated by Zeno of Elea (c. 450 B.C.) bouncing around in my head. To summarize the paradox: 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. What this actually does is to make all motion impossible, for before I can cover half the distance I must cover half of half the distance, and before I can do that I must cover half of half of half of the distance, and so on, so that in reality I can never move any distance at all, because doing so involves moving an infinite number of small intermediate distances first. I knew there must be a solution, as we all do manage to move around quite handily, I just never knew what it was. Luckily enough, I managed to stumble across an explantion of the paradox

67. Theorized Solution To Zeno's Paradox (Philosophistry) Theorized Solution to Zeno s paradox. Lynd emailed me the otherday in response to my complaint that Why hasn t anybody solved http://www.philosophistry.com/archives/2003/08/000504.html

Philosophistry Theorized Solution to Zeno's Paradox Lynd e-mailed me the other day in response to my complaint that "Why hasn't anybody solved Zeno's Paradox." This is an exerpt of what he had to say... Lynds' solution to the Achilles and the tortoise paradox, submitted to Philosophy of Science, helped explain the work. A tortoise challenges Achilles, the swift Greek warrior, to a race, gets a 10m head start, and says Achilles can never pass him. When Achilles has run 10m, the tortoise has moved a further metre. When Achilles has covered that metre, the tortoise has moved 10cm...and so on. It is impossible for Achilles to pass him. The paradox is that in reality, Achilles would easily do so. A similar paradox, called the Dichotomy, stipulates that you can never reach your goal, as in order to get there, you must firstly travel half of the distance. But once you've done that, you must still traverse half the remaining distance, and half again, and so on. What's more, you can't even get started, as to travel a certain distance, you must firstly travel half of that distance, and so on. According to both ancient and present day physics, objects in motion have determined relative positions. Indeed, the physics of motion from Zeno to Newton and through to today take this assumption as given. Lynds says that the paradoxes arose because people assumed wrongly that objects in motion had determined positions at any instant in time, thus freezing the bodies motion static at that instant and enabling the impossible situation of the paradoxes to be derived. "There's no such thing as an instant in time or present moment in nature. It's something entirely subjective that we project onto the world around us. That is, it's the outcome of brain function and consciousness."

68. Rishabh Ðara | Paradox | Infinity | Zeno's Paradox's Rishabh Ðara The paradox Archive Maths Infinity Zeno s paradox s.Zeno s paradox s. Zeno of Elea was an ancient Greek (born http://www.rishabhdara.com/paradox/maths/infinity/zeno.html

Zeno's Paradox's Zeno of Elea was an ancient Greek (born around 490 B.C.) who lived in what is now southern Italy. He became a disciple of the philosopher Parmenides, a philosopher who went around telling people that reality was an absolute, unchanging whole, and that therefore many things we take for granted, such as motion and plurality, were simply illusions. This kind of thing must no doubt have brought on ridicule from the more common-sensical Eleatics, and so Zeno set out to defend his master’s position by inventing ingenious problems for the common-sense view. Ever since then, Zeno’s paradoxes have been debated by philosophers and mathematicians. Zeno's writings have not survived, so his paradoxes are known to us chiefly through Aristotle's criticisms of them. Aristotle analyzed four paradoxes of motion: the Racetrack (or Dichotomy), Achilles and the Tortoise, the Arrow, and the Stadium (or Moving Rows). However, based on Aristotle's description of it, it is much less clear what Zeno intended by the Stadium paradox than by the other three. I have therefore left out this fourth paradox. The Racetrack (or Dichotomy) One can never reach the end of a racecourse, for in order to do so one would first have to reach the halfway mark, then the halfway mark of the remaining half, then the halfway mark of the final fourth, then of the final eighth, and so on

69. BBC - H2g2 - Zeno's Paradox - A541937 Assuming everyone is familiar with the story of the Tortoise and the Hare (if you renot, ask your parents what they were thinking), Zeno s paradox shows how http://www.bbc.co.uk/dna/h2g2/alabaster/A541937

@import url('/includes/tbenh.css') ; Home TV Radio Talk ... Advanced Search New visitors: Returning BBCi members: Everything Philosophy Everything Mathematics Zeno's Paradox Zeno, an ancient Greek possibly too smart for his own good, developed a paradox. It postulated that motion is impossible because the moving object always has to cover half the distance. Since the number of halves is infinite and they become infinitely small, the moving object never really gets itself going. The Basic Arrow Example Picture an arrow being fired at a target 16 yards away. Before it reaches the target, the arrow has to get to a point eight yards away, but before it gets to that point, the arrow has to get four yards away. Get it yet? If you don't, keep dividing the distance travelled by two. See how small the numbers get? Those numbers represent shorter distances travelled. After 12 of these permutations, the arrow is moving a little more than an eighth of an inch. And these divisions go on forever, so the arrow eventually moves so little that the change is virtually undetectable. The Advanced Fable Example Assuming everyone is familiar with the story of the Tortoise and the Hare (if you're not, ask your parents what they were thinking), Zeno's paradox shows how the Hare never would have stood a chance had the tortoise been given a head start.

70. Three Interpretations Of Zeno's Paradox Quorum of One is intended for adult readers. This issue Three modernday interpretationsof. Zeno s paradox. With a double zeugma. - I -. Figs Zeno, p. http://mapage.noos.fr/qoo/Quorum35.html

David Jaggard's Quorum of One Issue number 35 April Wet humor on the Web since Quorum of One is intended for adult readers This issue: Three modern-day interpretations of Zeno's Paradox With a double zeugma - I - Figs Zeno, p Sometimes I wonder why I ever got out of the philosophy racket. I had a good thing going there for a while. Scroll contracts with hefty advances, personal appearances at archery contests and inter-species footraces, after-bacchanal speeches... I was clearing a million a year easy. Then one New Year's Eve I checked my bank balance and I only had half a mill. A year later I was down to 250 grand, and the next year... Well, let's not dwell on the past. I'm in a different line of work now. Private investigation is my game. Hunting down missing persons, tailing unfaithful spouses, checking out the backgrounds of quiz-show winners who marry recently bereaved royalty, that kind of thing. A lot of people think there's something glamorous about being a private eye. Well, I can tell you one thing about those people: none of them were in my office at about 2:00 pm last Thursday. Let me tell you how it happened... I was sitting at my desk doing nothing after lunch. To tell you the truth, I had been sitting at my desk doing nothing before lunch. And during lunch too, unless you call chomping down a corned lamb on pita and slugging back a bolt of Retsina "doing something".

71. Zeno's Paradox And Chaos. Archimedes and the tortoise are in a race. Archimedes gives the tortoisea head start. Archimedes catches up the tortoise. No no http://myweb.lsbu.ac.uk/~dirt/museum/paradox.html

Archimedes and the tortoise are in a race. Archimedes gives the tortoise a head start. Archimedes catches up the tortoise. No no, let's try that again, this time using discrete numbers. We'll give the tortoise 100 metres start. Both archimedes and the tortoise start runnning. After time T, Archimedes has run 100 metres.... BUT the tortoise has covered another 0.1 metres. Lets run that again. After time dT, Archimedes has run 0.1 metres.... BUT the tortoise has covered another 0.0001 metres. and so on, while you can extend the decimal places to describe the result. Using discrete numbers to model the situation, Archimedes does not catch the tortoise. What about chaos? If you take any situation that you might want to model, you have to define your starting position. It will be a set of numbers. Your accuracy will depend on how precise your number is, the number of decimal places. Complex situations mean that imprecision in the starting position will mean great differences in the final position. Real life situations like the weather or human relations require numbers of infinite length to define the starting position. Since I'm a medic, I'll give you a medical example.

72. AGNI | 57 | Poetry | Zeno's Paradox, Or My Mother's Forsythia By Joyce Peseroff AGNI home page, Zeno s paradox, or My Mother s Forsythia. by Joyce Peseroff.By half and half and half and half again, I can approach but never touch. http://www.bu.edu/agni/poetry/print/2003/57-peseroff.html

Zeno's Paradox, or My Mother's Forsythia by Joyce Peseroff By half and half and half and half again, I can approach but never touch her bristling-yellow-paintbrush-flowing-green forsythia in April as it drops harmonized, a soundtrack to my grief, weeks after she died. If I describe the crack from a box of tranquilized bees, smudgy embers in the smoker, can you hear it? half-life could be a millisecond or an era. apple, almond, peach-perfumed, whatever (AGNI three books of poems are Mortal Education (Carnegie-Mellon, 2000), The Hardness Scale A Dog in the Lifeboat (Cornell, 1991). She teaches in the Creative Writing program at the University of Massachusetts, Boston. (5/03) Send this page to a friend AGNI Magazine :: published at Boston University :: page last updated May 29, 2004

73. Untitled1.html SWIFT, ACHILLES AND THE TORTOISE THE SUBLIME INNOCENCE, OR THE UNCANNY RETURN OFTHE REFERENT IN POSTSTRUCTURALIST THEORY ALONG THE LINES OF ZENO S paradox. http://www.pd.org/topos/perforations/perf6/uncanny_linetski_p6.html

NABOKOV AND SWIFT, ACHILLES AND THE TORTOISE: THE SUBLIME INNOCENCE, OR THE UNCANNY RETURN OF THE REFERENT IN POSTSTRUCTURALIST THEORY ALONG THE LINES OF ZENO'S PARADOX Vadim Linetski The vogue which the Zeno's famous paradox enjoys in poststructuralist theory is not surprising. It is this paradox which provides a point of convergence between the main strands associated with the illustrious names of Deleuze, Lacan and Derrida as well as between the crucial issues of poststructuralist project - those of difference/alterity, identity construction and the "bar games" with the referent (Genosko 1994: 7, 41) - furnishing the framework for the allegedly non-logocentric redefinition of the status of the work of art and aesthetic activity in general. However, as the attempts to attain a critical distance from what has already become a poststructuralist orthodoxy gain in strength and scope, the reasons that of yore have pushed our paradox to the fore become obscured if not obfuscated. This paradoxical fate of Zeno's paradox deserves most minute attention. If the efforts to carve the path beyond deconstruction have thus far so obviously failed to produce anything worthwhile, then precisely because the paradoxicality just mentioned points to the very heart of the problem with which theorists unsuccessfully grapple for the last thirty or so. In most general terms the problem in question is the undoubdtfully honorouble task of surpassing logocentric tradition in all its forms and disguises, the task for which the name of Derrida has become totemic. And yet the very mood characteristic of the current theoretical scene - that of melancholic scepticism - seems to suggest that the deconstructive project, by and large, has fizzled out. Fortunately, as we shall see, the funeral is not fatal, for the deceased has been buried alive. Put otherwise, it is not so easy to kill the mocking-bird of deconstruction, this Phoenix of sorts.

74. 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?

75. COMMENTS AND I THOUGHT ZENO S paradox WAS HARD Marshall has more, which I hope you willread. Now, about Heisenberg PS You can read about Zeno s paradox here. http://coldfury.com/reason/comments.php?id=1043_0_1_0_C

76. Zeno's Paradox: Physics Zeno s paradox Physics Campfire If ye would like to moderate the Physics Campfire,please drop becket@jollyroger.com a line. Physics Zeno s paradox Campfire. http://killdevilhill.com/physicschat/messages2/4379.html

Zeno's paradox: Physics Campfire If ye would like to moderate the Physics Campfire, please drop becket@jollyroger.com a line. //Required //var site = '681666'; //var mnum = '139010'; //Not Required var max_words = 4; var max_links_per_word = 4; var link_color = '0107A1'; var boxbg_color = 'FFFAEA'; var boxtitle_color = 'black'; var boxdesc_color = 'black'; var boxurl_color = 'red'; Open Source CMS Renaissance Postnuke Hosting Gallery Hosting Blog Hosting ... The World's Largest Literary Cafe Posted by JJJ on August 30, 19100 at 22:05:24: In Reply to: Zeno's paradox posted by JJJ on August 30, 19100 at 22:03:09: : Taggart, this is probably of interest to you... http://wis-wander.weizmann.ac.il/weizmann/doa_iis.dll/Serve/item/English/1.200.6.13.html press release Follow Ups: Yes it is interesting, however William Taggart How would you know what QM predicts? Bruce I am not going to get into another flame war. but! William Taggart This has nothing to do with matter being infinitely divisible. Bruce You just dont get it, do you ? William Taggart Re: You just dont get it, do you ?

77. 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

78. Tetsuboushi.com: Comment On Zeno's Paradox Of Employment Iron Comments. Zeno s paradox of employment. Post a comment. NameEmail Address URL Remember personal info? Yes No Comments http://www.tetsuboushi.com/cgi-bin/mt-files/mt-comments.cgi?entry_id=48

79. Self-Service Science Forum Message The problem you refer to is called Zeno s paradox . Hence it is sometimescalled Zeno s arrow paradox , although it is not really a paradox. JR. http://www2.abc.net.au/science/k2/stn/december1999/posts/topic18265.shtm

80. SPEED SpeedAt-An-Instant Zeno s paradox And Balla s Flight Of The Swifts. The FuturistTechnical Manifesto (1910) illustrates in artistic terms Zeno s paradox. http://www.aug.edu/dvskel/Justice-Malloy1992.htm

Speed-At-An-Instant: Zeno's Paradox And Balla's Flight Of The Swifts Rhona Justice-Malloy Department Of Drama University Of Georgia In On Physics Aristotle made a seemingly harmless statement. He wrote, "A body will move through a given medium in a given time." As is the way with philosophers, Zeno of Elea turned Aristotle's apparently obvious statement into a paradox that would puzzle mathematicians well into the seventeenth-century. Actually, Zeno would not have needed a camera to demonstrate his point had he been acquainted with the work of the futurist painter Giacomo Balla. Balla's Flight of the Swifts (1913) and Swifts: Paths of Movement + Dynamic Sequences (1913) clearly illustrates the concept of speed-at-an-instant-of-time. Balla's freezeframe style conveys the optical phenomena of movement as it might be captured on film when he presents one event (flight) in several instants of time within that event. Balla piles scores of frozen instants, one on another, to be synthesized in one glance or, to reverse the analogy, he gives many visual moments in one temporal moment. "The Futurist Technical Manifesto" (1910) illustrates in artistic terms Zeno's paradox. The goal of the futurist painter is to reproduce a gesture in motion to show not the force of the motion but the dynamic sensation that force produces. This sensation is perceived by the eye in much the same way as by a camera's lens. The Manifesto states, "On account of the persistence of an image upon the retina, moving objects constantly multiply themselves" (Kazloff, 1973, p. 147). To enforce the dynamic sensation the painter must not simply concentrate on reproducing multiple images of the moving subject but must also articulate the surrounding atmosphere. Aristotle aptly described this condition when he defined space as the potential for motion. Balla's

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