This is Zeno of Elea, an ancient Greek philosopher famous for inventing a number of paradoxes, arguments that seem logical, but whose conclusion is absurd or contradictory. For more than 2,000 years, Zeno's mind-bending riddles have inspired mathematicians and philosophers to better understand the nature of infinity. One of the best known of Zeno's problems is called the dichotomy paradox, which means, "the paradox of cutting in two" in ancient Greek. It goes something like this: After a long day of sitting around, thinking, Zeno decides to walk from his house to the park. The fresh air clears his mind and help him think better. In order to get to the park, he first has to get half way to the park. This portion of his journey takes some finite amount of time. Once he gets to the halfway point, he needs to walk half the remaining distance. Again, this takes a finite amount of time. Once he gets there, he still needs to walk half the distance that's left, which takes another finite amount of time. This happens again and again and again. You can see that we can keep going like this forever, dividing whatever distance is left into smaller and smaller pieces, each of which takes some finite time to traverse. So, how long does it take Zeno to get to the park? Well, to find out, you need to add the times of each of the pieces of the journey. The problem is, there are infinitely many of these finite-sized pieces. So, shouldn't the total time be infinity? This argument, by the way, is completely general. It says that traveling from any location to any other location should take an infinite amount of time. In other words, it says that all motion is impossible. This conclusion is clearly absurd, but where is the flaw in the logic? To resolve the paradox, it helps to turn the story into a math problem. Let's supposed that Zeno's house is one mile from the park and that Zeno walks at one mile per hour. Common sense tells us that the time for the journey should be one hour. But, let's look at things from Zeno's point of view and divide up the journey into pieces. The first half of the journey takes half an hour, the next part takes quarter of an hour, the third part takes an eighth of an hour, and so on. Summing up all these times, we get a series that looks like this. "Now", Zeno might say, "since there are infinitely many of terms on the right side of the equation, and each individual term is finite, the sum should equal infinity, right?" This is the problem with Zeno's argument. As mathematicians have since realized, it is possible to add up infinitely many finite-sized terms and still get a finite answer. "How?" you ask. Well, let's think of it this way. Let's start with a square that has area of one meter. Now let's chop the square in half, and then chop the remaining half in half, and so on. While we're doing this, let's keep track of the areas of the pieces. The first slice makes two parts, each with an area of one-half The next slice divides one of those halves in half, and so on. But, no matter how many times we slice up the boxes, the total area is still the sum of the areas of all the pieces. Now you can see why we choose this particular way of cutting up the square. We've obtained the same infinite series as we had for the time of Zeno's journey. As we construct more and more blue pieces, to use the math jargon, as we take the limit as n tends to infinity, the entire square becomes covered with blue. But the area of the square is just one unit, and so the infinite sum must equal one. Going back to Zeno's journey, we can now see how how the paradox is resolved. Not only does the infinite series sum to a finite answer, but that finite answer is the same one that common sense tells us is true. Zeno's journey takes one hour.
Ovo je Zenon od Eleje, antički Grčki filozof poznat po tome što je izumeo veliki broj paradoksa, argumenata koji se čine logičnim, ali čiji zaključak je apsurdan ili kontradiktoran. Više od 2000 godina, Zenonove zbunjujuće zagonetke inspirišu matematičare i filozofe da bolje razumeju prirodu beskonačnosti. Jedan od najpoznatijih Zenonovih problema zove se paradoks dihotomije, što znači „paradoks deljenja na dva" na antičkom Grčkom. Paradoks se može objasniti ovako: Posle dugog dana sedenja i razmišljenja Zenon odluči da prošeta od svoje kuće do parka. Svež vazduh ga osvežava i pomaže mu da bolje misli. Da bi došao do parka, on prvo mora da prođe polovinu puta do parka. Ovaj deo putovanja traje konačan period vremena. Kada pređe pola puta, on mora da pređe drugu polovinu puta. Ponovo, ovo traje konačan period vremena. Kada stigne tamo, još uvek mora da pređe polovinu puta koji mu preostaje, što traje još jedan konačan period vremena. Ovo se ponavlja iznova i iznova i iznova. Možete da vidite da ovako možemo zauvek, deleći preostali deo puta na manje i manje delove, od kojih svaki traje ograničen period vremena da se pređe. Dakle, koliko Zenonu treba vremena da dođe do parka? Pa, da biste saznali, morate da saberete vremena koja su potrebna za svaki deo putovanja. Problem je u tome da postoji beskonačno mnogo ovih vremenski ograničenih delova. Dakle, zar ukupno vreme ne bi bilo beskonačno? Ovaj argument je, inače, sasvim opšti. On tvrdi da putovanje sa jednog mesta na drugo treba da traje beskonačno mnogo vremena. Drugim rečima, on tvrdi da je svako kretanje nemoguće. Ovaj zaključak je očigledno apsurdan, ali gde je greška u logici? Da rešimo paradoks, pomaže da pretvorimo priču u matematički problem. Pretpostavimo da je Zenonova kuća od parka udaljena 1,6 kilometara i da Zenon prelazi 1,6 kilometara na sat. Zdrav razum nam govori da vreme putovanja treba da bude 1 sat. Ipak, hajde da sagledamo stvari iz Zenonovog ugla i podelimo putovanje u delove. Prva polovina putovanja traje pola sata, sledeća traje četvrt sata, sledeća osminu sata, i tako dalje. Kada saberemo sva ova vremena, dobijamo niz koji izgleda ovako. „Sada,” Zenon bi mogao da kaže, budući da postoji beskonačan broj izraza sa desne strane jednačine, i da je svaki pojedinačan izraz konačan, zbir bi trebao da bude jednak beskonačnom, tako?" U tome je problem sa Zenonovim argumentom. Kao što su matematičari od tada shvatili, moguće je zbrajati beskonačan niz mnogih konačnih izraza i opet dobiti rezultat koji je konačan. „Kako?”, pitate se vi. Razmišljajmo o tome ovako. Počnimo sa kvadratom koji zauzima površinu od jednog metra. Sada, hajde da presečemo kvadrat na pola, i onda da presečemo preostalu polovinu na pola, i tako dalje. Dok ovo radimo, hajde da beležimo površinu ovih delova. Prvi rez pravi dva dela, svaki sa površinom jedne polovine. Sledeći rez deli ove polovine na pola, i tako dalje. Ipak, koliko god puta mi da presečemo kvadrat, celokupna površina je još uvek zbir površina svih delova. Sada uviđate zašto smo izabrali baš ovaj način da presecamo kvadrat. Dobili smo isti beskonačan niz koji smo imali za vreme Zenonovog putovanja. Kako mi kombinujemo sve više i više plavih delova, da kažemo to matematički, kako mi za limit uzimamo „n” koji teži beskonačnom, ceo kvadrat postaje prekriven plavim. Ipak, površina kvadrata je samo jedna jedinica, stoga beskonačan zbir mora biti jednak jedinici. U Zenonovom putovanju, vidimo kako se paradoks može rešiti. Ne samo da je zbir beskonačnog niza konačan odgovor, nego je i konačan odgovor isti onaj za koji nam zdrav razum govori da je tačan.