What is Zeno's Dichotomy Paradox? - Colm Kelleher

Translator: Andrea McDonough Reviewer: Bedirhan Cinar

Prevodilac: Marija Kojić Lektor: Jelena Kovačević

This is Zeno of Elea,

Ovo je Zenon od Eleje,

an ancient Greek philosopher

antički Grčki filozof

famous for inventing a number of paradoxes,

poznat po tome što je izumeo veliki broj paradoksa,

argumenata koji se čine logičnim,

arguments that seem logical,

but whose conclusion is absurd or contradictory.

ali čiji zaključak je apsurdan

ili kontradiktoran.

For more than 2,000 years,

Više od 2000 godina,

Zeno's mind-bending riddles have inspired

Zenonove zbunjujuće zagonetke

mathematicians and philosophers

inspirišu matematičare i filozofe

to better understand the nature of infinity.

da bolje razumeju prirodu beskonačnosti.

One of the best known of Zeno's problems

Jedan od najpoznatijih Zenonovih problema

is called the dichotomy paradox,

zove se paradoks dihotomije,

which means, "the paradox of cutting in two" in ancient Greek.

što znači „paradoks deljenja na dva" na antičkom Grčkom.

It goes something like this:

Paradoks se može objasniti ovako:

After a long day of sitting around, thinking,

Posle dugog dana sedenja

i razmišljenja

Zeno decides to walk from his house to the park.

Zenon odluči da prošeta od svoje kuće do parka.

The fresh air clears his mind

Svež vazduh ga osvežava

and help him think better.

i pomaže mu da bolje misli.

In order to get to the park,

Da bi došao do parka,

he first has to get half way to the park.

on prvo mora da prođe polovinu puta do parka.

This portion of his journey

Ovaj deo putovanja

takes some finite amount of time.

traje konačan period vremena.

Once he gets to the halfway point,

Kada pređe pola puta,

he needs to walk half the remaining distance.

on mora da pređe drugu polovinu puta.

Again, this takes a finite amount of time.

Ponovo, ovo traje konačan period vremena.

Once he gets there, he still needs to walk

Kada stigne tamo, još uvek mora da pređe

half the distance that's left,

polovinu puta koji mu preostaje,

which takes another finite amount of time.

što traje još jedan konačan period vremena.

This happens again and again and again.

Ovo se ponavlja iznova i iznova i iznova.

You can see that we can keep going like this forever,

Možete da vidite da ovako možemo zauvek,

dividing whatever distance is left

deleći preostali deo puta

into smaller and smaller pieces,

na manje i manje delove,

each of which takes some finite time to traverse.

od kojih svaki traje ograničen period vremena da se pređe.

So, how long does it take Zeno to get to the park?

Dakle, koliko Zenonu treba vremena da dođe do parka?

Well, to find out, you need to add the times

Pa, da biste saznali, morate da saberete vremena

of each of the pieces of the journey.

koja su potrebna za svaki deo putovanja.

The problem is, there are infinitely many of these finite-sized pieces.

Problem je u tome da postoji

beskonačno mnogo ovih vremenski ograničenih delova.

So, shouldn't the total time be infinity?

Dakle, zar ukupno vreme ne bi bilo beskonačno?

This argument, by the way, is completely general.

Ovaj argument je, inače, sasvim opšti.

It says that traveling from any location to any other location

On tvrdi da putovanje sa jednog mesta na drugo

should take an infinite amount of time.

treba da traje beskonačno mnogo vremena.

In other words, it says that all motion is impossible.

Drugim rečima, on tvrdi da je svako kretanje nemoguće.

This conclusion is clearly absurd,

Ovaj zaključak je očigledno apsurdan,

but where is the flaw in the logic?

ali gde je greška u logici?

To resolve the paradox,

Da rešimo paradoks,

it helps to turn the story into a math problem.

pomaže da pretvorimo priču u matematički problem.

Let's supposed that Zeno's house is one mile from the park

Pretpostavimo da je Zenonova kuća od parka udaljena 1,6 kilometara

and that Zeno walks at one mile per hour.

i da Zenon prelazi 1,6 kilometara na sat.

Common sense tells us that the time for the journey

Zdrav razum nam govori da vreme putovanja

should be one hour.

treba da bude 1 sat.

But, let's look at things from Zeno's point of view

Ipak, hajde da sagledamo stvari iz Zenonovog ugla

and divide up the journey into pieces.

i podelimo putovanje u delove.

The first half of the journey takes half an hour,

Prva polovina putovanja traje pola sata,

the next part takes quarter of an hour,

sledeća traje četvrt sata,

the third part takes an eighth of an hour,

sledeća osminu sata,

and so on.

Summing up all these times,

i tako dalje.

Kada saberemo sva ova vremena,

we get a series that looks like this.

dobijamo niz koji izgleda ovako.

"Now", Zeno might say,

„Sada,” Zenon bi mogao da kaže,

"since there are infinitely many of terms

budući da postoji beskonačan broj izraza

on the right side of the equation,

sa desne strane jednačine,

and each individual term is finite,

i da je svaki pojedinačan izraz konačan,

the sum should equal infinity, right?"

zbir bi trebao da bude jednak beskonačnom, tako?"

This is the problem with Zeno's argument.

U tome je problem sa Zenonovim argumentom.

As mathematicians have since realized,

Kao što su matematičari od tada shvatili,

it is possible to add up infinitely many finite-sized terms

moguće je zbrajati beskonačan niz mnogih konačnih izraza

and still get a finite answer.

i opet dobiti rezultat koji je konačan.

"How?" you ask.

„Kako?”, pitate se vi.

Well, let's think of it this way.

Razmišljajmo o tome ovako.

Let's start with a square that has area of one meter.

Počnimo sa kvadratom koji zauzima površinu od jednog metra.

Now let's chop the square in half,

Sada, hajde da presečemo kvadrat na pola,

and then chop the remaining half in half,

i onda da presečemo preostalu polovinu na pola,

and so on.

i tako dalje.

While we're doing this,

Dok ovo radimo,

let's keep track of the areas of the pieces.

hajde da beležimo površinu ovih delova.

The first slice makes two parts,

Prvi rez pravi dva dela,

each with an area of one-half

svaki sa površinom jedne polovine.

The next slice divides one of those halves in half,

Sledeći rez deli ove polovine na pola,

and so on.

i tako dalje.

But, no matter how many times we slice up the boxes,

Ipak, koliko god puta mi da presečemo kvadrat,

the total area is still the sum of the areas of all the pieces.

celokupna površina je još uvek zbir površina svih delova.

Now you can see why we choose this particular way

Sada uviđate zašto smo izabrali baš ovaj način

of cutting up the square.

da presecamo kvadrat.

We've obtained the same infinite series

Dobili smo isti beskonačan niz

as we had for the time of Zeno's journey.

koji smo imali za vreme Zenonovog putovanja.

As we construct more and more blue pieces,

Kako mi kombinujemo sve više i više plavih delova,

to use the math jargon,

da kažemo to matematički,

as we take the limit as n tends to infinity,

kako mi za limit uzimamo „n” koji teži beskonačnom,

the entire square becomes covered with blue.

ceo kvadrat postaje prekriven plavim.

But the area of the square is just one unit,

Ipak, površina kvadrata je samo jedna jedinica,

and so the infinite sum must equal one.

stoga beskonačan zbir mora biti jednak jedinici.

Going back to Zeno's journey,

U Zenonovom putovanju,

we can now see how how the paradox is resolved.

vidimo kako se paradoks može rešiti.

Not only does the infinite series sum to a finite answer,

Ne samo da je zbir beskonačnog niza

konačan odgovor,

but that finite answer is the same one

nego je i konačan odgovor isti onaj

that common sense tells us is true.

za koji nam zdrav razum govori da je tačan.

Zeno's journey takes one hour.

Zenonovo putovanje traje jedan sat.