View source for Zeno's paradoxes

{{Redirect|Arrow paradox}}
{{short description|Set of philosophical problems}}
'''Zeno's paradoxes''' are a series of [[philosophy|philosophical]] [[Argument|arguments]] presented by the [[Ancient Greece|ancient Greek]] philosopher [[Zeno of Elea]] (c. 490–430 BC),<ref name=":0" /><ref name=":1" /> primarily known through the works of [[Plato]], [[Aristotle]], and later commentators like [[Simplicius of Cilicia]].<ref name=":1" /> Zeno devised these paradoxes to support his teacher [[Parmenides]]'s philosophy of [[monism]], which posits that despite our sensory experiences, reality is singular and unchanging. The paradoxes famously challenge the notions of plurality (the [[existence]] of many things), motion, space, and time by suggesting they lead to [[Contradiction|logical contradictions]].

Zeno's work, primarily known from [[Secondary source|second-hand accounts]] since his [[Primary source|original texts]] are lost, comprises forty "paradoxes of plurality," which argue against the [[Consistency|coherence]] of believing in multiple existences, and several arguments against motion and change.<ref name=":1" /> Of these, only a few are definitively known today, including the renowned "Achilles Paradox", which illustrates the problematic concept of infinite divisibility in [[space]] and [[time]].<ref name=":0" /><ref name=":1" /> In this paradox, Zeno argues that a swift runner like [[Achilles]] cannot overtake a slower moving [[tortoise]] with a head start, because the [[distance]] between them can be infinitely subdivided, implying Achilles would require an [[Infinity|infinite]] number of steps to catch the tortoise.<ref name=":0" /><ref name=":1" />

These paradoxes have stirred extensive philosophical and mathematical discussion throughout [[history]],<ref name=":0" /><ref name=":1" /> particularly regarding the nature of infinity and the continuity of space and time. Initially, [[Aristotle]]'s interpretation, suggesting a potential rather than actual infinity, was widely accepted.<ref name=":0" /> However, modern solutions leveraging the mathematical framework of [[calculus]] have provided a different perspective, highlighting Zeno's significant early insight into the complexities of infinity and continuous motion.<ref name=":0" /> Zeno's paradoxes remain a pivotal reference point in the philosophical and mathematical exploration of reality, motion, and the infinite, influencing both ancient thought and modern scientific understanding.<ref name=":0" /><ref name=":1" />

== History ==
The origins of the paradoxes are somewhat unclear, but they are generally thought to have been developed to support [[Parmenides]]' doctrine of [[monism]], that all of reality is one, and that ''all change is impossible'', that is, that nothing ever [[Motion|changes in location]] or in any other respect.<ref name=":0" /><ref name=":1" /> [[Diogenes Laërtius]], citing [[Favorinus]], says that Zeno's teacher Parmenides was the first to introduce the paradox of Achilles and the tortoise. But in a later passage, Laërtius attributes the origin of the paradox to Zeno, explaining that Favorinus disagrees.<ref>Diogenes Laërtius, ''Lives'', 9.23 and 9.29.</ref> [[University|Modern academics]] attribute the paradox to Zeno.<ref name=":0" /><ref name=":1" /> 

Many of these paradoxes argue that contrary to the evidence of one's senses, [[motion (physics)|motion]] is nothing but an [[illusion]].<ref name=":0" /><ref name=":1" /> In [[Plato|Plato's]] [[Parmenides (dialogue)|''Parmenides'']] (128a–d), Zeno is characterized as taking on the project of creating these [[paradoxes]] because other philosophers claimed paradoxes arise when considering Parmenides' view. Zeno's arguments may then be early examples of a method of proof called ''[[reductio ad absurdum]]'', also known as [[proof by contradiction]]. 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."<ref>''Parmenides'' 128d</ref> Plato has [[Socrates]] claim that Zeno and Parmenides were essentially arguing exactly the same point.<ref>''Parmenides'' 128a–b</ref> They are also credited as a source of the [[dialectic]] method used by Socrates.<ref>([fragment 65], Diogenes Laërtius. [http://classicpersuasion.org/pw/diogenes/dlzeno-eleatic.htm IX] {{Webarchive|url=https://web.archive.org/web/20101212095647/http://classicpersuasion.org/pw/diogenes/dlzeno-eleatic.htm |date=2010-12-12 }} 25ff and VIII 57).</ref>

== Paradoxes ==
Some of Zeno's nine surviving paradoxes (preserved in [[Physics (Aristotle)|Aristotle's ''Physics'']]<ref name=aristotle>[http://classics.mit.edu/Aristotle/physics.html Aristotle's ''Physics''] {{Webarchive|url=https://web.archive.org/web/20110106095547/http://classics.mit.edu/Aristotle/physics.html |date=2011-01-06 }} "Physics" by Aristotle translated by R. P. Hardie and R. K. Gaye</ref><ref>{{cite web|title=Greek text of "Physics" by Aristotle (refer to §4 at the top of the visible screen area)|url=http://remacle.org/bloodwolf/philosophes/Aristote/physique6gr.htm#144|archive-url=https://web.archive.org/web/20080516213308/http://remacle.org/bloodwolf/philosophes/Aristote/physique6gr.htm#144|archive-date=2008-05-16}}</ref> and [[Simplicius of Cilicia|Simplicius's]] commentary thereon) are essentially equivalent to one another. Aristotle offered a response to some of them.<ref name=aristotle/>   Popular literature often misrepresents Zeno's arguments. For example, 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= [https://archive.org/details/momentofproofmat00bens/page/14 14]|url= https://archive.org/details/momentofproofmat00bens|url-access= registration}}</ref>  However, none of the original ancient sources has Zeno discussing the sum of any infinite series. [[Simplicius of Cilicia|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 encyclopedia |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 |encyclopedia=[[Stanford Encyclopedia of Philosophy]] |access-date=2011-03-07 |archive-date=2022-03-01 |archive-url=https://web.archive.org/web/20220301174333/https://plato.stanford.edu/entries/paradox-zeno/#ZenInf |url-status=live }}</ref>

=== Paradoxes of motion ===
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.
==== Dichotomy paradox ====
[[File:Zeno Dichotomy Paradox alt.png|thumb|The dichotomy]]
{{ quote
| 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
}}
Suppose [[Atalanta]] wishes to walk to the end of a path. Before she can get there, she must get halfway there. Before she can get halfway there, she must get a quarter of the way there. Before traveling a quarter, she must travel one-eighth; before an eighth, one-sixteenth; and so on.
<timeline>
ImageSize= width:800 height:100
PlotArea= width:720 height:55 left:65 bottom:20
AlignBars= justify
Period= from:0 till:100
TimeAxis= orientation:horizontal
ScaleMajor= unit:year increment:10 start:0
ScaleMinor= unit:year increment:1 start:0
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 id:homer    value:rgb(0.4,0.8,1)   # light purple
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  bar:homer fontsize:L color:homer
  from:0 till:100
  at:50                        mark:(line,red)
  at:25                        mark:(line,black)
  at:12.5                      mark:(line,black)
  at:6.25                      mark:(line,black)
  at:3.125                     mark:(line,black)
  at:1.5625                    mark:(line,black)
  at:0.78125                   mark:(line,black)
  at:0.390625                  mark:(line,black)
  at:0.1953125                 mark:(line,black)
  at:0.09765625                mark:(line,black)
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The resulting sequence can be represented as:
:<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.<ref>{{cite book|last1=Lindberg|first1=David|title=The Beginnings of Western Science|date=2007|publisher=University of Chicago Press|isbn=978-0-226-48205-7|page=33|edition=2nd}}</ref>

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 be neither completed nor begun, and so all motion must be an [[illusion]].<ref>{{cite encyclopedia |last=Huggett |first=Nick |url=http://plato.stanford.edu/entries/paradox-zeno/#Dic |title=Zeno's Paradoxes: 3.1 The Dichotomy |year=2010 |encyclopedia=[[Stanford Encyclopedia of Philosophy]] |access-date=2011-03-07 |archive-date=2022-03-01 |archive-url=https://web.archive.org/web/20220301174333/https://plato.stanford.edu/entries/paradox-zeno/#Dic |url-status=live }}</ref>

This argument is called the "[[Dichotomy]]" because it involves repeatedly splitting a distance into two parts. An example with the original sense can be found in an [[asymptote]]. It is also known as the '''Race Course''' paradox.

==== Achilles and the tortoise<!--'Achilles and the Tortoise' and 'Achilles and the tortoise' redirects here--> ====
{{Redirect|Achilles and the Tortoise}}
{{See also|Infinity#Zeno: Achilles and the tortoise|selfref=yes}}
[[File:Zeno Achilles Paradox.png|thumb|Achilles and the tortoise]]

{{ quote
| In a race, the quickest runner can never over&shy;take 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
}}
In the paradox of '''Achilles and the tortoise'''<!--boldface per WP:R#PLA-->, [[Achilles]] is in a footrace with a tortoise. Achilles allows the tortoise a head start of 100 meters, for example. Suppose that each racer starts running at some constant speed, one faster than the other. After some finite time, Achilles will have run 100 meters, bringing him to the tortoise's starting point. During this time, the tortoise has run a much shorter distance, say 2 meters. 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 arrives somewhere the tortoise has been, he still has some distance to go before he can even reach the tortoise. As Aristotle noted, this argument is similar to the Dichotomy.<ref>{{cite encyclopedia |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 |encyclopedia=[[Stanford Encyclopedia of Philosophy]] |access-date=2011-03-07 |archive-date=2022-03-01 |archive-url=https://web.archive.org/web/20220301174333/https://plato.stanford.edu/entries/paradox-zeno/#AchTor |url-status=live }}</ref> It lacks, however, the apparent conclusion of motionlessness.

==== Arrow paradox ====
[[File:Zeno Arrow Paradox.png|thumb|The arrow]]
{{distinct|text = [[Arrow paradox (disambiguation)|''other paradoxes of the same name'']]}}

{{quote|If everything when it occupies an equal space is at rest at that instant of time, and if that which is in locomotion is always occupying such a space at any moment, the flying arrow is therefore motionless at that instant of time and at the next instant of time but if both instants of time are taken as the same instant or continuous instant of time then it is in motion.<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. |access-date=2012-08-21 |archive-date=2008-05-15 |archive-url=https://web.archive.org/web/20080515224131/http://classics.mit.edu//Aristotle/physics.6.vi.html#752 |url-status=live }}</ref>|as recounted by [[Aristotle]], [[Physics (Aristotle)|''Physics'']] VI:9, 239b5|title=|source=}}

In the arrow 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 at any one (durationless) instant of time, the arrow is neither moving to where it is, nor to where it is not.<ref>{{cite book | chapter-url=http://en.wikisource.org/wiki/Lives_of_the_Eminent_Philosophers/Book_IX#Pyrrho | first=Diogenes | last=Laërtius | author-link=Diogenes Laërtius | title=Lives and Opinions of Eminent Philosophers | volume=IX | chapter=Pyrrho | at=passage 72 | year=c. 230 | isbn=1-116-71900-2 | title-link=Lives and Opinions of Eminent Philosophers | access-date=2011-03-05 | archive-date=2011-08-22 | archive-url=https://web.archive.org/web/20110822084058/http://en.wikisource.org/wiki/Lives_of_the_Eminent_Philosophers/Book_IX#Pyrrho | url-status=live }}</ref>
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.

Whereas the first two paradoxes divide space, this paradox starts by dividing time—and not into segments, but into points.<ref name=HuggettArrow>{{cite encyclopedia |last=Huggett |first=Nick |url=http://plato.stanford.edu/entries/paradox-zeno/#Arr |title=Zeno's Paradoxes: 3.3 The Arrow |year=2010 |encyclopedia=[[Stanford Encyclopedia of Philosophy]] |access-date=2011-03-07 |archive-date=2022-03-01 |archive-url=https://web.archive.org/web/20220301174333/https://plato.stanford.edu/entries/paradox-zeno/#Arr |url-status=live }}</ref>

=== Other paradoxes ===
Aristotle gives three other paradoxes.
==== Paradox of place ====
From Aristotle:
{{quote
|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] {{Webarchive|url=https://web.archive.org/web/20080509083946/http://classics.mit.edu//Aristotle/physics.4.iv.html |date=2008-05-09 }}</ref>}}

==== Paradox of the grain of millet ====
{{see also|Sorites paradox}}
Description of the paradox from the ''Routledge Dictionary of Philosophy'':
{{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, an absurd conclusion.<ref>The Michael Proudfoot, A.R. Lace. Routledge Dictionary of Philosophy. Routledge 2009, p. 445</ref>}}

Aristotle's response:
{{quote
|Zeno's reasoning is false when he argues that 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] {{Webarchive|url=https://web.archive.org/web/20080511153804/http://classics.mit.edu//Aristotle/physics.7.vii.html |date=2008-05-11 }}</ref>}}

Description from Nick Huggett:
{{quote
|This is 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 {{Webarchive|url=https://web.archive.org/web/20220301174333/https://plato.stanford.edu/entries/paradox-zeno/#GraMil |date=2022-03-01 }}</ref>}}

==== The moving rows (or stadium) ====
[[File:Zeno Moving Rows Paradox.png|thumb|The moving rows]]

From Aristotle:
{{quote
|... 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] {{Webarchive|url=https://web.archive.org/web/20080515224131/http://classics.mit.edu//Aristotle/physics.6.vi.html |date=2008-05-15 }}</ref>}}

An expanded account of Zeno's arguments, as presented by Aristotle, is given in [[Simplicius of Cilicia|Simplicius's]] commentary ''On Aristotle's Physics''.<ref name=":2">{{Cite book |last1=Simplikios |title=Simplicius on Aristotle's Physics 6 |last2=Konstan |first2=David |last3=Simplikios |date=1989 |publisher=Cornell Univ. Pr |isbn=978-0-8014-2238-6 |series=Ancient commentators on Aristotle |location=Ithaca N.Y}}</ref><ref name=":1">{{Citation |last=Huggett |first=Nick |title=Zeno's Paradoxes |date=2024 |encyclopedia=The Stanford Encyclopedia of Philosophy |editor-last=Zalta |editor-first=Edward N. |url=https://plato.stanford.edu/archives/spr2024/entries/paradox-zeno/ |access-date=2024-03-25 |edition=Spring 2024 |publisher=Metaphysics Research Lab, Stanford University |editor2-last=Nodelman |editor2-first=Uri}}</ref><ref name=":0">{{Cite web |title=Zeno's Paradoxes {{!}} Internet Encyclopedia of Philosophy |url=https://iep.utm.edu/zenos-paradoxes/ |access-date=2024-03-25 |language=en-US}}</ref>

According to Angie Hobbs of The University of Sheffield, this paradox is intended to be considered together with the paradox of Achilles and the Tortoise, problematizing the concept of discrete space & time where the other problematizes the concept of infinitely divisible space & time.<ref>{{Cite web |title=Zeno's Paradoxes: The Moving Rows |url=https://digitalmedia.sheffield.ac.uk/media/Zeno%27s+ParadoxesA+The+Moving+Rows/1_e2yi73na |access-date=2024-06-28 |website=The University of Sheffield Kaltura Digital Media Hub |language=en}}</ref>

== Proposed solutions ==

=== In classical antiquity ===
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.<ref name=":2" /><ref name=":1" /> To fully solve any of the paradoxes, however, one needs to show what is wrong with the argument, not just the conclusions. Throughout 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>{{failed verification|reason=In the section cited, Aristotle says nothing about the distance decreasing |date=October 2019}}<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. {{Original research inline|date=October 2020}} 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>
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 |title=Physics |url=http://classics.mit.edu/Aristotle/physics.6.vi.html |volume=VI |at=Part 9 verse: 239b5 |isbn=0-585-09205-2 |access-date=2008-08-11 |archive-date=2008-05-15 |archive-url=https://web.archive.org/web/20080515224131/http://classics.mit.edu//Aristotle/physics.6.vi.html |url-status=live }}</ref> [[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><ref>{{Cite book |last=Kiritsis |first=Paul |title=A Critical Investigation into Precognitive Dreams |date=2020-04-01 |publisher=Cambridge Scholars Publishing |year=2020 |isbn=978-1527546332 |edition=1 |pages=19 |language=en}}</ref><ref>{{Cite web |last=Aquinas |first=Thomas |author-link=Thomas Aquinas |title=Commentary on Aristotle's Physics |url=https://aquinas.cc/la/en/~Phys.Bk6.L11 |access-date=2024-03-25 |website=aquinas.cc}}</ref>

=== In modern mathematics ===
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=https://archive.org/details/historyofcalculu0000boye |url-access=registration |year=1959 |publisher=Dover Publications  |access-date=2010-02-26 |page=[https://archive.org/details/historyofcalculu0000boye/page/295 295] | 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 }}</ref> Infinite processes remained theoretically troublesome in mathematics until the late 19th century. With the [[(ε, δ)-definition of limit|epsilon-delta]] definition of [[Limit (mathematics)|limit]], [[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><ref name=russell>[[Bertrand Russell|B Russell]] (1956) ''Mathematics and the metaphysicians'' in "The World of Mathematics" (ed. [[James R. Newman|J R Newman]]), pp 1576-1590.</ref>

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 |archive-url=https://web.archive.org/web/20100418141459/http://www.philosophers.co.uk/cafe/paradox5.htm |archive-date=2010-04-18 }}</ref><ref name=Papa-G>{{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 |journal=The Review of Metaphysics |volume=50 |year=1996 |pages=299–314 |access-date=2012-03-06 |archive-date=2012-06-09 |archive-url=https://web.archive.org/web/20120609113959/http://philsci-archive.pitt.edu/2304/1/zeno_maths_review_metaphysics_alba_papa_grimaldi.pdf |url-status=live }}</ref> While mathematics can calculate where and when the moving Achilles will overtake the Tortoise of Zeno's paradox, philosophers such as Kevin Brown<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 |access-date = 2010-06-06 |url-status = dead |archive-url  = https://archive.today/20121205030717/http://www.mathpages.com/rr/s3-07/3-07.htm |archive-date = 2012-12-05}}</ref> and Francis Moorcroft<ref name=FMoorcroft/> hold 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. 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/>

==== Henri Bergson ====
An alternative conclusion, proposed by [[Henri Bergson]] in his 1896 book ''[[Matter and Memory]]'', is that, while the path is divisible, the motion is not.<ref>{{cite book|last=Bergson|first=Henri|title=Matière et Mémoire|trans-title=Matter and Memory|url=https://antilogicalism.com/wp-content/uploads/2017/07/matter-and-memory.pdf|author-link=Henri Bergson|date=1896|pages=77–78 of the PDF|publisher=Translation 1911 by Nancy Margaret Paul & W. Scott Palmer. George Allen and Unwin|access-date=2019-10-15|archive-date=2019-10-15|archive-url=https://web.archive.org/web/20191015184719/https://antilogicalism.com/wp-content/uploads/2017/07/matter-and-memory.pdf|url-status=live}}</ref><ref>{{Cite book |last=Massumi |first=Brian |title=Parables for the Virtual: Movement, Affect, Sensation |publisher=Duke University Press Books |year=2002 |isbn=978-0822328971 |edition=1st |location=Durham, NC |pages=5–6 |language=English}}</ref> 

==== Peter Lynds ====

In 2003, Peter Lynds 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>{{cite web|url=http://philsci-archive.pitt.edu/1197/|title=Zeno's Paradoxes: A Timely Solution|date=January 2003|access-date=2012-07-02|archive-date=2012-08-13|archive-url=https://web.archive.org/web/20120813040121/http://philsci-archive.pitt.edu/1197/|url-status=live}}</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">[https://www.wired.com/wired/archive/13.06/physics.html Time’s Up, Einstein] {{Webarchive|url=https://web.archive.org/web/20121230100640/http://www.wired.com/wired/archive/13.06/physics.html |date=2012-12-30 }}, 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.  Nick Huggett argues that Zeno is [[begging the question|assuming the conclusion]] when he says that objects that occupy the same space as they do at rest must be at rest.<ref name=HuggettArrow/>

==== Bertrand Russell ====
Based on the work of [[Georg Cantor]],<ref>{{cite book |last=Russell |first=Bertrand |date=2002 |title=Our Knowledge of the External World: As a Field for Scientific Method in Philosophy |chapter=Lecture 6. The Problem of Infinity Considered Historically |publisher=Routledge |page=169 |orig-year=First published in 1914 by The Open Court Publishing Company |isbn=0-415-09605-7}}</ref> [[Bertrand Russell]] offered a solution to the paradoxes, 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 just change in position over time.<ref name=HuggettBook>{{ cite book |title=Space From Zeno to Einstein |first=Nick |last=Huggett |year=1999 |publisher=MIT Press |isbn=0-262-08271-3}}</ref><ref>{{cite book |url=https://books.google.com/books?id=uPRbOOv1YxUC&q=at+at+theory+of+motion+russell&pg=PA198 |title=Causality and Explanation |first=Wesley C. |last=Salmon |author-link=Wesley C. Salmon |page=198 |isbn=978-0-19-510864-4 |year=1998 |publisher=Oxford University Press |access-date=2020-11-21 |archive-date=2023-12-29 |archive-url=https://web.archive.org/web/20231229215244/https://books.google.com/books?id=uPRbOOv1YxUC&q=at+at+theory+of+motion+russell&pg=PA198#v=snippet&q=at%20at%20theory%20of%20motion%20russell&f=false |url-status=live }}</ref>

==== Hermann Weyl ====
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. According to [[Hermann Weyl]], the assumption that space is made of finite and discrete units is subject to a further problem, given by the "[[Weyl's tile argument|tile argument]]" or "distance function problem".<ref>{{cite encyclopedia| last=Van Bendegem| first=Jean Paul| title=Finitism in Geometry| url=http://plato.stanford.edu/entries/geometry-finitism/#SomParSolProDea| encyclopedia=Stanford Encyclopedia of Philosophy| access-date=2012-01-03| date=17 March 2010| archive-date=2008-05-12| archive-url=https://web.archive.org/web/20080512012132/http://plato.stanford.edu/entries/geometry-finitism/#SomParSolProDea| url-status=live}}</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| access-date=2012-01-03|date=11 December 2000  |url-status=dead |archive-url=https://web.archive.org/web/20100712095732/https://www.aarweb.org/syllabus/syllabi/c/cohen/phil320/atomism.htm |archive-date=July 12, 2010}}</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|s2cid=224840314 }}</ref>

=== Applications ===
==== Quantum Zeno effect ====
{{Main article|Quantum Zeno effect}}
In 1977,<ref>{{Cite journal |bibcode=1977JMP....18..756M |last1=Sudarshan |first1=E. C. G. |author-link=E. C. G. Sudarshan |last2=Misra |first2=B. |title=The Zeno's paradox in quantum theory |journal=Journal of Mathematical Physics |volume=18 |issue=4 |pages=756–763 |year=1977 |doi=10.1063/1.523304 |osti=7342282 |url=http://repository.ias.ac.in/51139/1/211-pub.pdf |access-date=2018-04-20 |archive-date=2013-05-14 |archive-url=https://web.archive.org/web/20130514062722/http://repository.ias.ac.in/51139/1/211-pub.pdf |url-status=live }}</ref> physicists [[E. C. George Sudarshan]] and B. Misra 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 |author1=W.M.Itano |author2=D.J. Heinsen |author3=J.J. Bokkinger |author4=D.J. Wineland |title=Quantum Zeno effect |journal=[[Physical Review A]] |volume=41 |issue=5 |pages=2295–2300 |year=1990 |doi=10.1103/PhysRevA.41.2295 |pmid=9903355 |bibcode=1990PhRvA..41.2295I |access-date=2004-07-23 |archive-url=https://web.archive.org/web/20040720153510/http://www.boulder.nist.gov/timefreq/general/pdf/858.pdf |archive-date=2004-07-20 |url-status=dead }}
</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. |journal=Soviet Phys. JETP |volume=6 |page=1053 |year=1958 |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 system]]s, 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 | chapter-url=https://books.google.com/books?id=cM-eFv1m3BoC&pg=SA15-PA22 | access-date=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. | archive-date=2023-12-29 | archive-url=https://web.archive.org/web/20231229215249/https://books.google.com/books?id=cM-eFv1m3BoC&pg=SA15-PA22#v=onepage&q&f=false | url-status=live }}</ref> Some [[formal verification]] techniques exclude these behaviours from analysis, if they are not equivalent to non-Zeno behaviour.<ref>{{cite journal |last=Lamport |first=Leslie |author-link=Leslie Lamport |year=2002 |title=Specifying Systems |journal=Microsoft Research |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 |access-date=2010-03-06 |archive-date=2010-11-16 |archive-url=https://web.archive.org/web/20101116164613/http://research.microsoft.com/en-us/um/people/lamport/tla/book-02-08-08.pdf |url-status=live }}</ref><ref>{{cite journal  |last1=Zhang |first1=Jun |last2=Johansson| first2=Karl | first3=John  |last3=Lygeros |first4=Shankar |last4=Sastry   |title=Zeno hybrid systems | journal=International Journal for Robust and Nonlinear Control |year=2001 |access-date=2010-02-28  |doi=10.1002/rnc.592  |volume=11  |issue=5  |page=435 |s2cid=2057416 |url=http://aphrodite.s3.kth.se/~kallej/papers/zeno_ijnrc01.pdf |url-status=dead |archive-url=https://web.archive.org/web/20110811144122/http://aphrodite.s3.kth.se/~kallej/papers/zeno_ijnrc01.pdf |archive-date=August 11, 2011}}</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 book|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 |access-date=2010-03-02 |url-status=dead |archive-url=https://web.archive.org/web/20080528193234/http://mtc.epfl.ch/~tah/Publications/a_comparison_of_control_problems_for_timed_and_hybrid_systems.html |archive-date=May 28, 2008 }}</ref>

== Similar paradoxes ==
=== School of Names ===
[[File:Paradox of the stick.png|thumb|Diagram of Hui Shi's stick paradox]]
Roughly contemporaneously during the [[Warring States period]] (475–221 BCE), [[History of Science and Technology in China|ancient Chinese]] philosophers from the [[School of Names]], a school of thought similarly concerned with logic and dialectics, developed paradoxes similar to those of Zeno. The works of the School of Names have largely been lost, with the exception of portions of the [[Gongsun Long|''Gongsun Longzi'']]. The second of the Ten Theses of [[Hui Shi]] suggests knowledge of infinitesimals:''That which has no thickness cannot be piled up; yet it is a thousand li in dimension.'' Among the many puzzles of his recorded in the [[Zhuangzi (book)|''Zhuangzi'']] is one very similar to Zeno's Dichotomy: {{Quote|quote=<poem>"If from a stick a foot long you every day take the half of it, in a myriad ages it will not be exhausted."</poem>|source=''Zhuangzi'', chapter 33 (Legge translation)<ref>{{Cite book |title=Sacred Books of the East |publisher=[[Oxford University Press]] |year=1891 |editor-last=Müller |editor-first=Max |volume=40 |translator-last=Legge |translator-first=James |chapter=The Writings of Kwang Tse}}</ref>}} [[Mozi (book)|The Mohist canon]] appears to propose a solution to this paradox by arguing that in moving across a measured length, the distance is not covered in successive fractions of the length, but in one stage. Due to the lack of surviving works from the School of Names, most of the other paradoxes listed are difficult to interpret.<ref>{{Cite web |title=School of Names > Miscellaneous Paradoxes (Stanford Encyclopedia of Philosophy) |url=https://plato.stanford.edu/entries/school-names/paradoxes.html |access-date=2020-01-30 |website=plato.stanford.edu |archive-date=2016-12-11 |archive-url=https://web.archive.org/web/20161211103807/https://plato.stanford.edu/entries/school-names/paradoxes.html |url-status=live }}</ref>

=== Lewis Carroll's "What the Tortoise Said to Achilles" ===
{{Main article|What the Tortoise Said to Achilles}}
"What the Tortoise Said to Achilles",<ref>{{Cite journal|last=Carroll|first=Lewis|title=What the Tortoise Said to Achilles|date=1895-04-01|url=https://academic.oup.com/mind/article/IV/14/278/1046872|journal=Mind|language=en|volume=IV|issue=14|pages=278–280|doi=10.1093/mind/IV.14.278|issn=0026-4423|access-date=2020-07-20|archive-date=2020-07-20|archive-url=https://web.archive.org/web/20200720045852/https://academic.oup.com/mind/article/IV/14/278/1046872|url-status=live}}</ref> written in 1895 by [[Lewis Carroll]], describes a paradoxical infinite regress argument in the realm of pure logic. It uses Achilles and the Tortoise as characters in a clear reference to Zeno's paradox of Achilles.<ref>{{Cite book |last=Tsilipakos |first=Leonidas |title=Clarity and confusion in social theory: taking concepts seriously |date=2021 |publisher=Routledge |isbn=978-1-032-09883-8 |series=Philosophy and method in the social sciences |location=Abingdon New York (N.Y.) |pages=48}}</ref>

== See also ==
* [[Incommensurable magnitudes]]
* [[Infinite regress]]
* [[Philosophy of space and time]]
* [[Renormalization]]
* [[Ross–Littlewood paradox]]
* [[Supertask]]
* [[Zeno machine]]
* [[List of paradoxes]]

== 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 encyclopedia |encyclopedia=[[Stanford Encyclopedia of Philosophy]] |title=Zeno's Paradoxes |url=http://plato.stanford.edu/entries/paradox-zeno/ |first=Nick |last=Huggett |year=2010 |access-date=2011-03-07 |archive-date=2022-03-01 |archive-url=https://web.archive.org/web/20220301174333/https://plato.stanford.edu/entries/paradox-zeno/ |url-status=live }}
* [[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}}.
* {{cite book |last=Skyrms |first=Brian |authorlink=Brian Skyrms |chapter=Zeno's Paradox of Measure |title=Physics, Philosophy, and Psychoanalysis |editor-first=R. S. |editor-last=Cohen |editor2-first=L. |editor2-last=Laudan |editor2-link=Larry Laudan |location=Dordrecht |publisher=Reidel |year=1983 |isbn=90-277-1533-5 |pages=223–254 }}
{{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. "[https://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]].
* [https://archive.today/20121205030717/http://www.mathpages.com/rr/s3-07/3-07.htm Kevin Brown on Zeno and the Paradox of Motion]
* {{cite encyclopedia |url=http://plato.stanford.edu/entries/zeno-elea/ |encyclopedia=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]]|access-date=2013-04-13|archive-date=2018-10-03|archive-url=https://web.archive.org/web/20181003050912/http://www.numberphile.com/videos/zeno_paradox.html|url-status=dead}}

{{Paradoxes}}
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[[Category:Eponymous paradoxes]]
[[Category:Philosophical paradoxes]]
[[Category:Supertasks]]
[[Category:Mathematical paradoxes]]
[[Category:Paradoxes of infinity]]
[[Category:Physical paradoxes]]