Paradox of division

The paradox of division or dichotomy paradox is an apparent contradiction in which it is to be shown that movement is in reality not possible. It is closely related to the paradox of Achilles and the turtle and, like the latter, goes back to the philosopher Zeno of Elea (approx. 490-430 BC).

Exact formulation

The example used to refute the existence of motion is as follows: A runner wants to travel a distance of positive length. To do this, he must first cover half of this distance. And in order to achieve this, it must first cover half of the half, i.e. a quarter of the total length. This procedure divides the route into an infinite number of sections, each of which takes a positive, finite time to be overcome. As a result, it must take the runner an infinitely long time to cover the entire distance.

Resolution of the paradox

Of course, this does not prove that no movement exists; On closer reflection it turns out that the paradox is based on a mistake in reasoning: By dividing a finite segment into an infinite number of finite segments, Zeno inadvertently shows himself that, conversely, a sum of infinitely many finite summands can have a finite value. The quantitative solution to the paradox lies in the geometric series , which Archimedes (287–212 BC) was already familiar with; in the context of his squaring the parabola he showed that

${\displaystyle \sum _{k=0}^{\infty }{\frac {1}{4^{k}}}={\frac {4}{3}}}$

is.
For the sake of illustration without loss of generality , let us assume a distance of 100 meters, and let us further assume that the runner needs exactly one second to overcome one meter. It is calculated that the runner needs 100 seconds to cover 100 meters.

If you now dismantle the route according to the above procedure, you get the following:

The runner must

• at the end 50 m,
• before 25 m,
• in front of it 12.5 m,
• in front of it 6.25 m
• Etc.

return. Now consider the time it takes for it. Using the summation sign follows: . ${\displaystyle 50\mathrm {\ s} +25\mathrm {\ s} +12{,}5\mathrm {\ s} +6{,}25\mathrm {\ s} +\dotsb =50\cdot (1+{\frac {1}{2}}+{\frac {1}{4}}+\dotsb )\mathrm {s} =50\cdot \sum _{k=0}^{\infty }\left({\frac {1}{2}}\right)^{k}\mathrm {s} =(50\cdot 2)\mathrm {s} =100\mathrm {\ s} }$

I.e. Here, too, the runner covers the distance of 100 meters in 100 seconds, which is why the contradiction does not arise.

The wrong idea behind it

The reason why it was assumed that there was a contradiction is that at the time this paradox arose, one had not yet dealt with infinite consequences and, in particular, with their convergence . The paradox is based on the ignorance of the fact that an infinite series converges if , like the one under consideration, it fulfills a convergence criterion. In this case, however, the wrong assumption is obvious, because by dividing the by definition finite total distance into an infinite number of finite total distances, Zeno actually proved that there are convergent infinite series.

Web links

• Entry in Edward N. Zalta (Ed.): Stanford Encyclopedia of Philosophy .

References and comments

1. ↑ The following transformations arithmetically illustrate how the infinite sum gets closer and closer to the number : ${\displaystyle 2}$

   ${\displaystyle 1+{\frac {1}{2}}={\frac {2}{2}}+{\frac {1}{2}}={\frac {3}{2}}=2-{\frac {1}{2}}}$

   ${\displaystyle {\frac {3}{2}}+{\frac {1}{4}}={\frac {6}{4}}+{\frac {1}{4}}={\frac {7}{4}}=2-{\frac {1}{4}}}$

   ${\displaystyle {\frac {7}{4}}+{\frac {1}{8}}={\frac {14}{8}}+{\frac {1}{8}}={\frac {15}{8}}=2-{\frac {1}{8}}}$ etc.,

   more general:

   ${\displaystyle {\frac {2^{k+1}-1}{2^{k}}}+{\frac {1}{2^{k+1}}}=}$ ${\displaystyle {\frac {2^{k+2}-2}{2^{k+1}}}+{\frac {1}{2^{k+1}}}=}$ ${\displaystyle {\frac {2^{k+2}-1}{2^{k+1}}}=}$ ${\displaystyle 2-{\frac {1}{2^{k+1}}};}$,

   from this for the notation used in the text:

   ${\displaystyle \sum _{k=0}^{\infty }({\frac {1}{2}})^{k}=}$ ${\displaystyle \lim _{k\to \infty }\sum _{k=0}^{n}({\frac {1}{2}})^{k}=}$ ${\displaystyle 2-\lim _{n\to \infty }{\frac {1}{2^{n+1}}}=}$ ${\displaystyle 2-0=}$ ${\displaystyle 2}$;

   here denotes the limit value of a sequence of numbers . - The "core" of the consideration is to be understood that the fraction of sufficiently high falls below any (however small) positive number, and thus at infinity (the limit) reached . ${\displaystyle lim}$ ${\displaystyle {\frac {1}{2^{k+1}}}}$ ${\displaystyle k}$ ${\displaystyle 0}$