Editing Zeno's paradoxes (section)

==== 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.