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{{Redirect|Achilles and the Tortoise}}
== the vertical arm shrug ==
{{Redirect|Arrow paradox}}


'''Zeno's paradoxes''' are a set of [[philosophy|philosophical]] problems generally thought to have been devised by [[Ancient Greece|Greek]] philosopher [[Zeno of Elea]] (ca. 490–430 BC) to support [[Parmenides|Parmenides's]] doctrine that contrary to the evidence of one's senses, the belief in [[Ontological pluralism|plurality]] and change is mistaken, and in particular that [[motion (physics)|motion]] is nothing but an [[illusion]]. It is usually assumed, based on [[Plato|Plato's]] [[Parmenides (dialogue)|''Parmenides'']] (128a-d), that Zeno took on the project of creating these [[paradox]]es because other philosophers had created paradoxes against Parmenides's view. Thus Plato has Zeno say the purpose of the paradoxes "is to show that their hypothesis that existences are many, if properly followed up, leads to still more absurd results than the hypothesis that they are one." (''Parmenides'' 128d). Plato has [[Socrates]] claim that Zeno and Parmenides were essentially arguing exactly the same point (''Parmenides'' 128a-b).
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Some of Zeno's nine surviving paradoxes (preserved in [[Aristotle|Aristotle's]] [[Physics (Aristotle)|''Physics'']]<ref name=aristotle>[http://classics.mit.edu/Aristotle/physics.html Aristotle's ''Physics''] "Physics" by Aristotle translated by R. P. Hardie and R. K. Gaye
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</ref><ref>
{{cite web|title=Greek text of "Physics" by Aristotle (refer to §4 at the top of the visible screen area)|url=http://web.archive.org/web/20080516213308/http://remacle.org/bloodwolf/philosophes/Aristote/physique6gr.htm#144}}
</ref>
and [[Simplicius of Cilicia|Simplicius's]] commentary thereon) are essentially equivalent to one another. Aristotle offered a refutation of some of them.<ref name=aristotle/>  Three of the strongest and most famous—that of [[Achilles]] and the [[tortoise]], the [[Dichotomy]] argument, and that of an arrow in flight—are presented in detail below.


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]].<ref>([fragment 65], Diogenes Laertius. [http://classicpersuasion.org/pw/diogenes/dlzeno-eleatic.htm IX] 25ff and VIII 57).</ref>
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Some mathematicians and historians, such as [[Carl Boyer]], hold that Zeno's paradoxes are simply mathematical problems, for which modern [[calculus]] provides a mathematical solution.<ref name=boyer>{{cite book |last=Boyer |first=Carl |title=The History of the Calculus and Its Conceptual Development |url=http://books.google.com/?id=w3xKLt_da2UC&dq=zeno+calculus&q=zeno#v=snippet&q=zeno |year=1959 |publisher=Dover Publications  |accessdate=2010-02-26 |page=295 | quote=If the paradoxes are thus stated in the precise mathematical terminology of continuous variables (...) the seeming contradictions resolve themselves. |isbn=978-0-486-60509-8 }}
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Some [[philosopher]]s, however, say that Zeno's paradoxes and their variations (see [[Thomson's lamp]]) remain relevant [[Metaphysics|metaphysical]] problems.<ref name=KBrown/><ref name=FMoorcroft>{{cite web |first=Francis |last=Moorcroft |title=Zeno's Paradox |url=http://www.philosophers.co.uk/cafe/paradox5.htm |archiveurl=http://web.archive.org/web/20100418141459id_/http://www.philosophers.co.uk/cafe/paradox5.htm |archivedate=2010-04-18 }}
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{{ cite journal |url=http://philsci-archive.pitt.edu/2304/1/zeno_maths_review_metaphysics_alba_papa_grimaldi.pdf |first=Alba |last=Papa-Grimaldi |title=Why Mathematical Solutions of Zeno's Paradoxes Miss the Point: Zeno's One and Many Relation and Parmenides' Prohibition |format=PDF  |work=The Review of Metaphysics |volume= 50 |year=1996|pages=299–314 }}</ref>
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The origins of the paradoxes are somewhat unclear. [[Diogenes Laertius]], a fourth source for information about Zeno and his teachings, citing [[Favorinus]], says that Zeno's teacher [[Parmenides]] was the first to introduce the Achilles and the tortoise paradox. But in a later passage, Laertius attributes the origin of the paradox to Zeno, explaining that Favorinus disagrees.<ref>Diogenes Laertius, ''Lives'', 9.23 and 9.29.</ref>
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== Paradoxes of motion ==
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=== Achilles and the tortoise ===
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[[File:Race between Achilles and the tortoise.gif|thumb|Distance vs. time, assuming the tortoise to run at Achilles' half speed]]
 
{{Redirect|Achilles and the Tortoise|the 2008 Japanese film|Achilles and the Tortoise (film)}}
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{{ quote
 
| ''In a race, the quickest runner can never overtake the slowest, since the pursuer must first reach the point whence the pursued started, so that the slower must always hold a lead.'' – as recounted by [[Aristotle]], [[Physics (Aristotle)|''Physics'']] VI:9, 239b15
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In the paradox of [[Achilles]] and the [[Tortoise]], Achilles is in a footrace with the tortoise. Achilles allows the tortoise a [[Head start (positioning)|head start]] of 100 metres, for example. If we suppose that each racer starts running at some constant speed (one very fast and one very slow), then after some finite [[time]], Achilles will have run 100 metres, bringing him to the tortoise's starting point. During this time, the tortoise has run a much shorter distance, say, 10 metres. It will then take Achilles some further time to run that distance, by which time the tortoise will have advanced farther; and then more time still to reach this third point, while the tortoise moves ahead. Thus, whenever Achilles reaches somewhere the tortoise has been, he still has farther to go. Therefore, because there are an infinite number of points Achilles must reach where the tortoise has already been, he can never overtake the tortoise.<ref>{{cite web |url=http://mathforum.org/isaac/problems/zeno1.html |title=Math Forum}}, mathforum.org
 
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{{ cite web |last=Huggett |first=Nick |url=http://plato.stanford.edu/entries/paradox-zeno/#AchTor |title=Zeno's Paradoxes: 3.2 Achilles and the Tortoise |year=2010 |work=[[Stanford Encyclopedia of Philosophy]] |accessdate=2011-03-07 }}</ref>
 
 
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=== Dichotomy paradox ===
 
{{ quote
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| ''That which is in locomotion must arrive at the half-way stage before it arrives at the goal.''– as recounted by [[Aristotle]], [[Physics (Aristotle)|''Physics'']] VI:9, 239b10
 
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  <li>[http://www.66bbs.net/home.php?mod=space&uid=73338 http://www.66bbs.net/home.php?mod=space&uid=73338]</li>
 
 
<math>H-\frac{B}{8}-\frac{B}{4}---\frac{B}{2}-------B</math>
  <li>[http://rustika-bogdanovic.com/galerija/picture.php?/313/list/335,519,123,96,344,12,458,77,480,313,217,364,232,396,283&comments_order=desc#comments&comments_order=desc#comments&comments_order=DESC#comments http://rustika-bogdanovic.com/galerija/picture.php?/313/list/335,519,123,96,344,12,458,77,480,313,217,364,232,396,283&comments_order=desc#comments&comments_order=desc#comments&comments_order=DESC#comments]</li>
 
 
The resulting sequence can be represented as:
  <li>[http://yutong.dz4a.com/bbs/upload/forum.php?mod=viewthread&tid=904745&extra= http://yutong.dz4a.com/bbs/upload/forum.php?mod=viewthread&tid=904745&extra=]</li>
:'''<math> \left\{ \cdots,  \frac{1}{16},  \frac{1}{8},  \frac{1}{4},  \frac{1}{2},  1 \right\}</math>'''
 
This description requires one to complete an infinite number of tasks, which Zeno maintains is an impossibility.
  <li>[http://www13.plala.or.jp/gakuki3/cgi_bin/aska/aska.cgi http://www13.plala.or.jp/gakuki3/cgi_bin/aska/aska.cgi]</li>
 
 
This sequence also presents a second problem in that it contains no first distance to run, for any possible ([[wikt:Finite|finite]]) first distance could be divided in half, and hence would not be first after all. Hence, the trip cannot even begin. The paradoxical conclusion then would be that travel over any finite distance can neither be completed nor begun, and so all motion must be an [[illusion]].
  <li>[http://www.jdxzgw.com/E_GuestBook.asp http://www.jdxzgw.com/E_GuestBook.asp]</li>
 
 
This argument is called the ''[[Dichotomy]]'' because it involves repeatedly splitting a distance into two parts. It contains some of the same elements as the ''Achilles and the Tortoise'' paradox, but with a more apparent conclusion of motionlessness. It is also known as the '''Race Course''' paradox. Some, like Aristotle, regard the Dichotomy as really just another version of ''Achilles and the Tortoise''.<ref>{{Cite web |last=Huggett |first=Nick |url=http://plato.stanford.edu/entries/paradox-zeno/#Dic |title=Zeno's Paradoxes: 3.1 The Dichotomy |year=2010 |work=[[Stanford Encyclopedia of Philosophy]] |accessdate=2011-03-07}}</ref>
  <li>[http://www.xuan6.com/home.php?mod=spacecp&ac=blog&blogid= http://www.xuan6.com/home.php?mod=spacecp&ac=blog&blogid=]</li>
 
 
There are two versions of the dichotomy paradox. In the other version, before Homer could reach the stationary bus, he must reach half of the distance to it. Before reaching the last half, he must complete the next quarter of the distance. Reaching the next quarter, he must then cover the next eighth of the distance, then the next sixteenth, and so on. There are thus an infinite number of steps that must first be accomplished before he could reach the bus, with no way to establish the size of any "last" step. Expressed this way, the dichotomy paradox is very much analogous to that of ''Achilles and the tortoise''.
  <li>[http://365hk.net/guestbook.asp?guestid= http://365hk.net/guestbook.asp?guestid=]</li>
 
 
=== Arrow paradox ===
  <li>[http://gcpq.org/dy_GuestBook.asp http://gcpq.org/dy_GuestBook.asp]</li>
{{quote|''If everything when it occupies an equal space is at rest, and if that which is in locomotion is always occupying such a space at any moment, the flying arrow is therefore motionless.''<ref>{{cite web |url=http://classics.mit.edu/Aristotle/physics.6.vi.html#752 |work=The Internet Classics Archive |title=Physics |author=Aristotle |quote=Zeno's reasoning, however, is fallacious, when he says that if everything when it occupies an equal space is at rest, and if that which is in locomotion is always occupying such a space at any moment, the flying arrow is therefore motionless. This is false, for time is not composed of indivisible moments any more than any other magnitude is composed of indivisibles.}}</ref>
 
– as recounted by [[Aristotle]], [[Physics (Aristotle)|''Physics'']] VI:9, 239b5}}
  <li>[http://maituw.gotoip3.com/home.php?mod=space&uid=114 http://maituw.gotoip3.com/home.php?mod=space&uid=114]</li>
 
 
In the arrow paradox (also known as the '''[[fletching|fletcher's]] paradox'''), Zeno states that for motion to occur, an object must change the position which it occupies. He gives an example of an arrow in flight. He states that in any one (durationless) instant of time, the arrow is neither moving to where it is, nor to where it is not.<ref>{{ cite book | url=http://en.wikisource.org/wiki/Lives_of_the_Eminent_Philosophers/Book_IX#Pyrrho | first=Diogenes |last=Laertius |authorlink=Diogenes Laërtius | title=[[Lives and Opinions of Eminent Philosophers]] | volume=IX |chapter=Pyrrho  |at= passage 72 | year=about 230 CE | isbn=1-116-71900-2  }}</ref>
  <li>[http://n.91234m.com/guestbook.asp?guestid= http://n.91234m.com/guestbook.asp?guestid=]</li>
It cannot move to where it is not, because no time elapses for it to move there; it cannot move to where it is, because it is already there. In other words, at every instant of time there is no motion occurring. If everything is motionless at every instant, and time is entirely composed of instants, then motion is impossible.
 
 
  <li>[http://bbs.wdlaks.com/home.php?mod=space&uid=158565 http://bbs.wdlaks.com/home.php?mod=space&uid=158565]</li>
Whereas the first two paradoxes divide space, this paradox starts by dividing time—and not into segments, but into points.<ref name=HuggettArrow>{{Cite web |last=Huggett |first=Nick |url=http://plato.stanford.edu/entries/paradox-zeno/#Arr |title=Zeno's Paradoxes: 3.3 The Arrow |year=2010 |work=[[Stanford Encyclopedia of Philosophy]] |accessdate=2011-03-07}}</ref>
 
 
  <li>[http://bbs.0552jie.com/home.php?mod=space&uid=20701 http://bbs.0552jie.com/home.php?mod=space&uid=20701]</li>
== Three other paradoxes as given by Aristotle ==
 
===Paradox of Place===
  <li>[http://www.rookiesgate.com/cgi-bin/dobbs/dobbsr.cgi http://www.rookiesgate.com/cgi-bin/dobbs/dobbsr.cgi]</li>
From Aristotle:
 
{{quote
  <li>[http://bbs.zbhouse.cn/home.php?mod=space&uid=2596 http://bbs.zbhouse.cn/home.php?mod=space&uid=2596]</li>
|''if everything that exists has a place, place too will have a place, and so on ''[[ad infinitum]]''.<ref>Aristotle [http://classics.mit.edu/Aristotle/physics.4.iv.html ''Physics'' IV:1, 209a25]</ref>}}
 
 
  <li>[http://www.infodoctornet.com.br/novo/index.php?option=com_blog&view=comments&pid=972497&Itemid=0 http://www.infodoctornet.com.br/novo/index.php?option=com_blog&view=comments&pid=972497&Itemid=0]</li>
===Paradox of the Grain of Millet===
 
From Aristotle:
  <li>[http://www.fxsoo.com/view/post:533266 http://www.fxsoo.com/view/post:533266]</li>
{{quote
 
|there is no part of the millet that does not make a sound: for there is no reason why any such part should not in any length of time fail to move the air that the whole bushel moves in falling. In fact it does not of itself move even such a quantity of the air as it would move if this part were by itself: for no part even exists otherwise than potentially.<ref>Aristotle [http://classics.mit.edu/Aristotle/physics.7.vii.html ''Physics'' VII:5, 250a20]</ref>}}
  <li>[http://n.91234m.com/guestbook.asp?guestid= http://n.91234m.com/guestbook.asp?guestid=]</li>
 
 
Description of the paradox from the ''Routledge Dictionary of Philosophy'':
  <li>[http://rustika-bogdanovic.com/galerija/picture.php?/15/list/170,251,354,247,191,318,15,263,463,78,428,62,308,387,333&comments_order=desc#comments&comments_order=desc#comments&comments_order=DESC#comments http://rustika-bogdanovic.com/galerija/picture.php?/15/list/170,251,354,247,191,318,15,263,463,78,428,62,308,387,333&comments_order=desc#comments&comments_order=desc#comments&comments_order=DESC#comments]</li>
{{quote
 
|The argument is that a single grain of [[millet]] makes no sound upon falling, but a thousand grains make a sound. Hence a thousand nothings become something, which is absurd.<ref>The Michael Proudfoot, A.R. Lace. Routledge Dictionary of Philosophy. Routledge 2009, p. 445</ref>}}
  <li>[http://andromeda.rutgers.edu/~jlynch/cgi-bin/18th.cgi http://andromeda.rutgers.edu/~jlynch/cgi-bin/18th.cgi]</li>
 
 
Description from Nick Huggett:
  <li>[http://exocetll.free.fr/e107_plugins/forum/forum_viewtopic.php?29906.last http://exocetll.free.fr/e107_plugins/forum/forum_viewtopic.php?29906.last]</li>
{{quote
 
|This a [[Parmenides|Parmenidean]] argument that one cannot trust one's sense of hearing. Aristotle's response seems to be that even inaudible sounds can add to an audible sound.<ref>Huggett, Nick, "Zeno's Paradoxes", The Stanford Encyclopedia of Philosophy (Winter 2010 Edition), Edward N. Zalta (ed.), http://plato.stanford.edu/entries/paradox-zeno/#GraMil</ref>}}
  <li>[http://www.pzty.net/wygkcn_GuestBook.asp http://www.pzty.net/wygkcn_GuestBook.asp]</li>
 
 
===The Moving Rows (or Stadium)===
  <li>[http://www.kangmeitw.com/thread-19597-1-1.html http://www.kangmeitw.com/thread-19597-1-1.html]</li>
From Aristotle:
 
{{quote
  <li>[http://oneway.tk/index.php?do=/blog/9315/the-layers-of-analysis/ http://oneway.tk/index.php?do=/blog/9315/the-layers-of-analysis/]</li>
|concerning the two rows of bodies, each row being composed of an equal number of bodies of equal size, passing each other on a race-course as they proceed with equal velocity in opposite directions, the one row originally occupying the space between the goal and the middle point of the course and the other that between the middle point and the starting-post. This...involves the conclusion that half a given time is equal to double that time.<ref>Aristotle [http://classics.mit.edu/Aristotle/physics.6.vi.html ''Physics'' VI:9, 239b33]</ref>}}
 
 
  <li>[http://ars.userfriendly.org/cartoons/read.cgi http://ars.userfriendly.org/cartoons/read.cgi]</li>
For an expanded account of Zeno's arguments as presented by Aristotle, see [[Simplicius of Cilicia|Simplicius']] commentary ''On Aristotle's Physics''.
 
 
</ul>
== Proposed solutions ==
According to [[Simplicius of Cilicia|Simplicius]], [[Diogenes the Cynic]] said nothing upon hearing Zeno's arguments, but stood up and walked, in order to demonstrate the falsity of Zeno's conclusions. To fully solve any of the paradoxes, however, one needs to show what is wrong with the argument, not just the conclusions. Through history, several solutions have been proposed, among the earliest recorded being those of Aristotle and Archimedes.
 
[[Aristotle]] (384 BC−322 BC) remarked that as the distance decreases, the time needed to cover those distances also decreases, so that the time needed also becomes increasingly small.<ref>Aristotle. Physics 6.9
</ref><ref>
Aristotle's observation that the fractional times also get shorter does not guarantee, in every case, that the task can be completed. One case in which it does not hold is that in which the fractional times decrease in a [[Harmonic series (mathematics)|harmonic series]], while the distances decrease geometrically, such as: 1/2 s for 1/2 m gain, 1/3 s for next 1/4 m gain, 1/4 s for next 1/8 m gain, 1/5 s for next 1/16 m gain, 1/6 s for next 1/32 m gain, etc. In this case, the distances form a convergent series, but the times form a [[divergent series]], the sum of which has no limit. Archimedes developed a more explicitly mathematical approach than Aristotle.</ref>
Aristotle also distinguished "things infinite in respect of divisibility" (such as a unit of space that can be mentally divided into ever smaller units while remaining spatially the same) from things (or distances) that are infinite in extension ("with respect to their extremities").<ref>Aristotle. Physics 6.9; 6.2, 233a21-31</ref>
 
Before 212 BC, [[Archimedes]] had developed a method to derive a finite answer for the sum of infinitely many terms that get progressively smaller. (See: [[Geometric series]], [[1/4 + 1/16 + 1/64 + 1/256 + · · ·]], [[The Quadrature of the Parabola]].) Modern calculus achieves the same result, using more rigorous methods (see [[convergent series]], where the "reciprocals of powers of 2" series, equivalent to the Dichotomy Paradox, is listed as convergent). These methods allow the construction of solutions based on the conditions stipulated by Zeno, i.e. the amount of time taken at each step is geometrically decreasing.<ref name=boyer /><ref>George B. Thomas, ''Calculus and Analytic Geometry'', Addison Wesley, 1951</ref>
 
Aristotle's objection to the arrow paradox was that "Time is not composed of indivisible nows any more than any other magnitude is composed of indivisibles."<ref>{{cite book |author=Aristotle |url=http://classics.mit.edu/Aristotle/physics.6.vi.html |title=Physics |volume=VI |at=Part 9 verse: 239b5 |isbn=0-585-09205-2 }}
</ref>
 
[[Saint Thomas Aquinas]], commenting on Aristotle's objection, wrote "Instants are not parts of time, for time is not made up of instants any more than a magnitude is made of points, as we have already proved. Hence it does not follow that a thing is not in motion in a given time, just because it is not in motion in any instant of that time."<ref>Aquinas. Commentary on Aristotle's Physics, Book 6.861
</ref>
 
[[Bertrand Russell]] offered what is known as the "at-at theory of motion". It agrees that there can be no motion "during" a durationless instant, and contends that all that is required for motion is that the arrow be at one point at one time, at another point another time, and at appropriate points between those two points for intervening times. In this view motion is a function of position with respect to time.<ref name=HuggettBook>{{ cite book |title=Space From Zeno to Einstein |first=Nick |last=Huggett |year=1999 |isbn=0-262-08271-3 }}
</ref><ref>
{{ cite book |url=http://books.google.com/?id=uPRbOOv1YxUC&pg=PA198&lpg=PA198&dq=at+at+theory+of+motion+russell#v=onepage&q=at%20at%20theory%20of%20motion%20russell&f=false | title=Causality and Explanation | first=Wesley C. |last=Salmon |authorlink=Wesley C. Salmon | page=198 |isbn=978-0-19-510864-4 |year=1998 }}</ref>
 
Nick Huggett argues that Zeno is [[begging the question]] when he says that objects that occupy the same space as they do at rest must be at rest.<ref name=HuggettArrow/>
 
[[Peter Lynds]] has argued that all of Zeno's motion paradoxes are resolved by the conclusion that instants in time and instantaneous magnitudes do not physically exist.<ref>[http://philsci-archive.pitt.edu/1197/ Lynds, Peter. Zeno's Paradoxes: a Timely Solution]
</ref><ref>
Lynds, Peter. Time and Classical and Quantum Mechanics: Indeterminacy vs. Discontinuity. Foundations of Physics Letter s (Vol. 16, Issue 4, 2003). doi:10.1023/A:1025361725408
</ref><ref name="Time’s Up Einstein">
[http://www.wired.com/wired/archive/13.06/physics.html Time’s Up Einstein], Josh McHugh, [[Wired Magazine]], June 2005
</ref>
Lynds argues that an object in relative motion cannot have an instantaneous or determined relative position (for if it did, it could not be in motion), and so cannot have its motion fractionally dissected as if it does, as is assumed by the paradoxes. For more about the inability to know both speed and location, see [[Heisenberg uncertainty principle]].
 
Another proposed solution is to question one of the assumptions Zeno used in his paradoxes (particularly the Dichotomy), which is that between any two different points in space (or time), there is always another point. Without this assumption there are only a finite number of distances between two points, hence there is no infinite sequence of movements, and the paradox is resolved. The ideas of [[Planck length]] and [[Planck time]] in modern physics place a limit on the measurement of time and space, if not on time and space themselves. According to [[Hermann Weyl]], the assumption that space is made of finite and discrete units is subject to a further problem, given by the "tile argument" or "distance function problem".<ref>{{cite web|last=Van Bendegem|first=Jean Paul|title=Finitism in Geometry|url=http://plato.stanford.edu/entries/geometry-finitism/#SomParSolProDea|work=Stanford Encyclopedia of Philosophy|accessdate=2012-01-03|date=17 March 2010 }}
</ref><ref name="atomism uni of washington">
{{ cite web|last=Cohen|first=Marc|title=ATOMISM|url=https://www.aarweb.org/syllabus/syllabi/c/cohen/phil320/atomism.htm|work=History of Ancient Philosophy, University of Washington|accessdate=2012-01-03|date=11 December 2000 }}
</ref>
According to this, the length of the hypotenuse of a right angled triangle in discretized space is always equal to the length of one of the two sides, in contradiction to geometry. [[Jean Paul Van Bendegem]] has argued that the Tile Argument can be resolved, and that discretization can therefore remove the paradox.<ref name=boyer/><ref>{{cite journal |jstor=187807 |title=Discussion:Zeno's Paradoxes and the Tile Argument |first=Jean Paul |last=van Bendegem |location= Belgium |year=1987 |journal=Philosophy of Science |volume=54 |issue=2 |pages=295–302|doi=10.1086/289379}}</ref>
 
[[Hans Reichenbach]] has proposed that the paradox may arise from considering space and time as separate entities. In a theory like general relativity, which presumes a single space-time continuum, the paradox may be blocked.<ref>Hans Reichenbach (1958) The Philosophy of Space and Time. Dover</ref>
 
== The paradoxes in modern times ==
Infinite processes remained theoretically troublesome in mathematics until the late 19th century. The [[Continuous_function#Weierstrass_definition_.28epsilon-delta.29_of_continuous_functions|epsilon-delta]] version of [[Karl Weierstrass|Weierstrass]] and [[Augustin Louis Cauchy|Cauchy]] developed a rigorous formulation of the logic and calculus involved. These works resolved the mathematics involving infinite processes.<ref name=Lee>{{cite journal |last=Lee |first=Harold | title=Are Zeno's Paradoxes Based on a Mistake? |jstor=2251675 |year=1965 |journal=[[Mind (journal)|Mind]] |volume=74 |issue=296 |publisher=Oxford University Press |pages=563–570 |doi=10.1093/mind/LXXIV.296.563}}</ref>
 
While mathematics can be used to calculate where and when the moving Achilles will overtake the Tortoise of Zeno's paradox, philosophers such as Brown and Moorcroft<ref name=KBrown>{{cite web |first=Kevin |last=Brown |title=Zeno and the Paradox of Motion |work=Reflections on Relativity |url=http://www.mathpages.com/rr/s3-07/3-07.htm |accessdate=2010-06-06 }}
</ref><ref name=FMoorcroft/>
claim that mathematics does not address the central point in Zeno's argument, and that solving the mathematical issues does not solve every issue the paradoxes raise.
 
Zeno's arguments are often misrepresented in the popular literature. That is, Zeno is often said to have argued that the sum of an infinite number of terms must itself be infinite–with the result that not only the time, but also the distance to be travelled, become infinite.<ref>{{cite book|last=Benson|first=Donald C.|title=The Moment of Proof : Mathematical Epiphanies|year=1999|publisher=Oxford University Press|location=New York|isbn=978-0195117219|page=14|url=http://books.google.com/books?id=8_vbuzxrpfIC&printsec=frontcover#v=onepage&q&f=false}}</ref>  However, none of the original ancient sources has Zeno discussing the sum of any infinite series. Simplicius has Zeno saying "it is impossible to traverse an infinite number of things in a finite time". This presents Zeno's problem not with finding the ''sum'', but rather with ''finishing'' a task with an infinite number of steps: how can one ever get from A to B, if an infinite number of (non-instantaneous) events can be identified that need to precede the arrival at B, and one cannot reach even the beginning of a "last event"?<ref name=KBrown/><ref name=FMoorcroft/><ref name=Papa-G /><ref>{{Cite web |last=Huggett |first=Nick |url=http://plato.stanford.edu/entries/paradox-zeno/#ZenInf |title=Zeno's Paradoxes: 5. Zeno's Influence on Philosophy |year=2010 |work=[[Stanford Encyclopedia of Philosophy]] |accessdate=2011-03-07 }}</ref>
 
Today there is still a debate on the question of whether or not Zeno's paradoxes have been resolved. In ''The History of Mathematics'', Burton writes, "Although Zeno's argument confounded his contemporaries, a satisfactory explanation incorporates a now-familiar idea, the notion of a 'convergent infinite series.'".<ref>Burton, David, ''A History of Mathematics: An Introduction'', McGraw Hill, 2010, ISBN 978-0-07-338315-6</ref>
 
[[Bertrand Russell]] offered a "solution" to the paradoxes based on modern physics,{{Citation needed|date=June 2010}} but Brown concludes "Given the history of 'final resolutions', from Aristotle onwards, it's probably foolhardy to think we've reached the end. It may be that Zeno's arguments on motion, because of their simplicity and universality, will always serve as a kind of [[Rorschach test|'Rorschach image']] onto which people can project their most fundamental phenomenological concerns (if they have any)."<ref name=KBrown/>
 
Pat Corvini offers a solution to the paradox of Achilles and the tortoise by first distinguishing the physical world from the abstract mathematics used to describe it.<ref>{{Cite web |url=https://estore.aynrand.org/p/194/achilles-the-tortoise-and-the-objectivity-of-mathematics-mp3-download |title=Achilles, the Tortoise, and the Objectivity of Mathematics }}</ref> She claims the paradox arises from a subtle but fatal switch between the physical and abstract. Zeno's syllogism is as follows: P1: Achilles must first traverse an infinite number of divisions in order to reach the tortoise; P2: it is impossible for Achilles to traverse an infinite number of divisions; C: therefore, Achilles can never surpass the tortoise. Corvini shows that P1 is a mathematical abstraction which cannot be applied directly to P2 which is a statement regarding the physical world. The physical world requires a resolution amount used to distinguish distance while mathematics can use any resolution.
 
== Quantum Zeno effect ==
{{Main|Quantum Zeno effect}}
 
In 1977,<ref>{{Cite journal  |bibcode=1977JMP....18..756M |last=Sudarshan |first=E. C. G. |authorlink=E. C. G. Sudarshan |last2=Misra |first2=B. |author2-link= |title=The Zeno's paradox in quantum theory  |journal=Journal of Mathematical Physics |volume=18 |issue=4 |pages=756–763 |date= |year=1977 |doi=10.1063/1.523304 }}
</ref>
physicists [[E. C. G. Sudarshan]] and B. Misra studying quantum mechanics discovered that the dynamical evolution (motion) of a quantum system can be hindered (or even inhibited) through observation of the system.<ref name="u0">{{cite journal | url=http://www.boulder.nist.gov/timefreq/general/pdf/858.pdf |format=[[PDF]] |author=W.M.Itano |coauthors=D.J.Heinsen, J.J.Bokkinger, D.J.Wineland |title=Quantum Zeno effect |journal=[[Physical Review A|PRA]] |volume=41 | issue=5 |pages=2295–2300 |year=1990 |doi=10.1103/PhysRevA.41.2295 |bibcode=1990PhRvA..41.2295I }}
</ref>
This effect is usually called the "quantum Zeno effect" as it is strongly reminiscent of Zeno's arrow paradox. This effect was first theorized in 1958.<ref>{{Cite journal  |last=Khalfin |first=L.A. |author-link=  |journal=Soviet Phys. JETP |volume=6 |pages=1053 |date= |year=1958 |url= |doi=  |postscript=<!--None--> |bibcode = 1958JETP....6.1053K  |title=Contribution to the Decay Theory of a Quasi-Stationary State }}</ref>
 
== Zeno behaviour ==
In the field of verification and design of [[timed event system|timed]]  and [[hybrid systems]], the system behaviour is called ''Zeno'' if it includes an infinite number of discrete steps in a finite amount of time.<ref name="Fishwick2007">{{cite book | editor=Paul A.  Fishwick | title=Handbook of dynamic system modeling | url=http://books.google.com/?id=cM-eFv1m3BoC&pg=SA15-PA22 | accessdate=2010-03-05 | edition=hardcover | series=Chapman & Hall/CRC Computer and Information Science | date=1 June 2007 | publisher=CRC Press | location=Boca Raton, Florida, USA | isbn=978-1-58488-565-8 | pages=15–22 to 15–23 | chapter=15.6 "Pathological Behavior Classes" in chapter 15  "Hybrid Dynamic Systems: Modeling and Execution" by Pieter J.  Mosterman, The Mathworks, Inc. }}
</ref>
Some [[formal verification]] techniques exclude these behaviours from analysis, if they are not equivalent to non-Zeno behaviour.<ref>{{cite book |last=Lamport |first=Leslie |authorlink=Leslie Lamport |year=2002 |title=Specifying Systems |format=[[PDF]] |publisher=Addison-Wesley |isbn=0-321-14306-X |url=http://research.microsoft.com/en-us/um/people/lamport/tla/book-02-08-08.pdf |page=128 |accessdate=2010-03-06 }}
</ref><ref>
{{ cite journal  |last1=Zhang |first1=Jun |last2=Johansson| first2=Karl | first3=John  |last3=Lygeros |first4=Shankar |last4=Sastry  |url=http://aphrodite.s3.kth.se/~kallej/papers/zeno_ijnrc01.pdf  |title=Zeno hybrid systems | journal=International Journal for Robust and Nonlinear control |year=2001 |accessdate=2010-02-28  |doi=10.1002/rnc.592  |volume=11  |issue=5  |pages=435 }}</ref>
 
In [[systems design]] these behaviours will also often be excluded from system models, since they cannot be implemented with a digital controller.<ref>{{cite journal |last2=Henzinger |first2=Thomas |  last1=Franck | first1=Cassez| first3=Jean-Francois |last3=Raskin  |url=http://mtc.epfl.ch/~tah/Publications/a_comparison_of_control_problems_for_timed_and_hybrid_systems.html  |title=A Comparison of Control Problems for Timed and Hybrid Systems  |year=2002 |accessdate=2010-03-02 }}</ref>
 
A simple example of a system showing Zeno behaviour is a bouncing ball coming to rest. The physics of a bouncing ball, ignoring factors other than rebound, can be mathematically analyzed to predict an infinite number of bounces.
 
== See also ==
* [[Incommensurable magnitudes]]
* [[Philosophy of space and time]]
* [[Ross-Littlewood paradox]]
* [[Solvitur ambulando]]
* [[Supertask]]
* [[Zeno machine]]
* [[Lewis Carroll]], [[What the Tortoise Said to Achilles]]
 
== Notes ==
{{Reflist|30em}}
 
== References ==
{{refbegin}}
* [[Geoffrey Kirk|Kirk, G. S.]], [[John Raven|J. E. Raven]], M. Schofield (1984) ''The Presocratic Philosophers: A Critical History with a Selection of Texts, 2nd ed.'' [[Cambridge University Press]]. ISBN 0-521-27455-9.
* {{cite web |work=[[Stanford Encyclopedia of Philosophy]] |title=Zeno's Paradoxes |url=http://plato.stanford.edu/entries/paradox-zeno/ |first=Nick |last=Huggett |year=2010 |accessdate=2011-03-07}}
* [[Plato]] (1926) ''Plato: Cratylus. Parmenides. Greater Hippias. Lesser Hippias'', H. N. Fowler (Translator), [[Loeb Classical Library]]. ISBN 0-674-99185-0.
* Sainsbury, R.M. (2003) ''Paradoxes'', 2nd ed. Cambridge University Press. ISBN 0-521-48347-6.
{{refend}}
 
== External links ==
{{Wikisource|Catholic Encyclopedia (1913)/Zeno of Elea|Zeno of Elea}}
* Dowden, Bradley. "[http://www.iep.utm.edu/zeno-par/ Zeno’s Paradoxes]." Entry in the [[Internet Encyclopedia of Philosophy]].
* {{springer|title=Antinomy|id=p/a012710}}
* [https://www.coursera.org/course/mathphil Introduction to Mathematical Philosophy], Ludwig-Maximilians-Universität München
* Silagadze, Z. K. "[http://uk.arxiv.org/abs/physics/0505042 Zeno meets modern science,]"
* ''[http://demonstrations.wolfram.com/ZenosParadoxAchillesAndTheTortoise/ Zeno's Paradox: Achilles and the Tortoise]'' by Jon McLoone, [[Wolfram Demonstrations Project]].
* [http://www.mathpages.com/rr/s3-07/3-07.htm Kevin Brown on Zeno and the Paradox of Motion]
* {{cite web |url=http://plato.stanford.edu/entries/zeno-elea/ |publisher=Stanford Encyclopedia of Philosophy |title=Zeno of Elea |first=John |last=Palmer |year=2008}}
* {{PlanetMath attribution|id=5538|title=Zeno's paradox}}
* {{cite web|last=Grime|first=James|title=Zeno's Paradox|url=http://www.numberphile.com/videos/zeno_paradox.html|work=Numberphile|publisher=[[Brady Haran]]}}
 
[[Category:Paradoxes]]
[[Category:Supertasks]]
[[Category:Mathematics paradoxes]]
[[Category:Paradoxes of infinity]]
 
{{Link GA|ru}}

Latest revision as of 14:55, 15 December 2014

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