Editing Zeno's paradoxes (section)

== 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
Colors=
 id:homer    value:rgb(0.4,0.8,1)   # light purple
PlotData=
  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)
</timeline>
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 ====
{{distinct|text = [[Arrow paradox (disambiguation)|''other paradoxes of the same name'']]}}
[[File:Zeno Arrow Paradox.png|thumb|The arrow]]

{{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=2009 |orig-date=c. 230 | isbn=1-116-71900-2 | title-link=Lives and Opinions of Eminent Philosophers | publisher=BiblioBazaar | 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>