Editing Zeno's paradoxes (section)

=== 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>