Hi. I want to talk about understanding, and the nature of understanding, and what the essence of understanding is, because understanding is something we aim for, everyone. We want to understand things. My claim is that understanding has to do with the ability to change your perspective. If you don't have that, you don't have understanding. So that is my claim.
Zdravo. Želim da govorim o razumevanju, i prirodi razumevanja, i o tome šta je bit razumevanja, jer je razumevanje nešto čemu svi mi težimo. Želimo da razumemo stvari. Tvrdim da razumevanje mora biti u vezi sa sposobnošću da promenimo našu perspektivu. Ako nema toga, nema ni razumevanja. To je moja tvrdnja.
And I want to focus on mathematics. Many of us think of mathematics as addition, subtraction, multiplication, division, fractions, percent, geometry, algebra -- all that stuff. But actually, I want to talk about the essence of mathematics as well. And my claim is that mathematics has to do with patterns.
Želim da se usredsredim na matematiku. Mnogi misle o matematici kao o sabiranju, oduzimanju, množenju, deljenju, razlomcima, procentima, geometriji, algebri - tim stvarima. Ali u stvari, želim takođe da govorim o suštini matematike. Moja tvrdnja je da matematika ima veze sa šablonima.
Behind me, you see a beautiful pattern, and this pattern actually emerges just from drawing circles in a very particular way. So my day-to-day definition of mathematics that I use every day is the following: First of all, it's about finding patterns. And by "pattern," I mean a connection, a structure, some regularity, some rules that govern what we see. Second of all, I think it is about representing these patterns with a language. We make up language if we don't have it, and in mathematics, this is essential. It's also about making assumptions and playing around with these assumptions and just seeing what happens. We're going to do that very soon. And finally, it's about doing cool stuff. Mathematics enables us to do so many things.
Iza mene, vidite jedan prelep šablon, i ovaj šablon je zapravo nastao prosto iz crtanja krugova na jedan poseban način. Tako da je moja ustaljena definicija matematike koju koristim svaki dan sledeća: Kao prvo, to je pronalaženje šablona. I pod "šablonima" mislim vezu, strukturu, neku pravilnost, neka pravila koja određuju šta vidimo. Kao drugo, smatram da je to predstavljanje ovih šema uz pomoć jezika. Mi izmišljamo jezik ako ne postoji, a u matematici, ovo je suština. To je takođe pravljenje pretpostavki i igranje sa ovim pretpostavkama samo da bismo videli šta će se desiti. To ćemo ubrzo i raditi. I konačno, ona je rađenje kul stvari. Matematika nam omogućava da radimo mnoge stvari.
So let's have a look at these patterns. If you want to tie a tie knot, there are patterns. Tie knots have names. And you can also do the mathematics of tie knots. This is a left-out, right-in, center-out and tie. This is a left-in, right-out, left-in, center-out and tie. This is a language we made up for the patterns of tie knots, and a half-Windsor is all that. This is a mathematics book about tying shoelaces at the university level, because there are patterns in shoelaces. You can do it in so many different ways. We can analyze it. We can make up languages for it.
Pa hajde da pogledamo ove šablone. Ako želite da vežete kravatu, tu je šablon, Čvorovi na kravati imaju ime. A takođe možete i uraditi matematiku čvorova. Ovo je levo-van, desno-u, sredina-van i veži. Ovo je levo-u, desno-van, levo-u, sredina-van i veži. Ovo je jezik koji smo osmislili za šablone vezivanja kravate, i polu-Vindzor je isto to. Ovo je knjiga za matematiku o vezivanju pertli na univerzitetskom nivou, jer postoje šabloni za pertle. Možete ih vezati na razne načine. Možemo to proanalizirati. Možemo da stvorimo jezik za to.
And representations are all over mathematics. This is Leibniz's notation from 1675. He invented a language for patterns in nature. When we throw something up in the air, it falls down. Why? We're not sure, but we can represent this with mathematics in a pattern.
Predstavljanja su svuda u matematici. Ovo je Lajbnicova beleška iz 1675. On je izumeo jezik za šablone u prirodi. Kada nešto bacimo u vis, to pada. Zašto? Nismo sigurni, ali možemo to predstaviti sa matematikom šablona.
This is also a pattern. This is also an invented language. Can you guess for what? It is actually a notation system for dancing, for tap dancing. That enables him as a choreographer to do cool stuff, to do new things, because he has represented it.
Ovo je takođe šablon. Ovo je takođe izmišljen jezik. Možete li da pogodite za šta? To je u stvari sistem nota za ples, za irski ples. Ovo omogućava njemu kao koreografu da radi kul, nove stvari, zato što ih je predstavio.
I want you to think about how amazing representing something actually is. Here it says the word "mathematics." But actually, they're just dots, right? So how in the world can these dots represent the word? Well, they do. They represent the word "mathematics," and these symbols also represent that word and this we can listen to. It sounds like this.
Žeim da razmislite koliko sjajno predstavljanje može stvarno biti. Ovde piše reč "matematika". Ali u stvari, to su samo tačke, zar ne? Pa kako onda, zaboga, ove tačke mogu da predstavljaju reč? Pa, mogu. One predstavljaju reč "matematika", a ovi simboli takođe predstavljaju reč i nju možemo da slušamo. Zvuči ovako.
(Beeps)
(Zvučni signal)
Somehow these sounds represent the word and the concept. How does this happen? There's something amazing going on about representing stuff.
Nekako ovi zvukovi predstavljaju i reč i koncept. Kako se to dešava? Postoji nešto sjajno što se dešava pri predstavljanju stvari.
So I want to talk about that magic that happens when we actually represent something. Here you see just lines with different widths. They stand for numbers for a particular book. And I can actually recommend this book, it's a very nice book.
Želim da govorim o magiji koja se dešava kada zapravo predstavljamo nešto. Ovde vidite samo linije različitih širina. One predstavljaju brojeve za određenu knjigu. Mogu zapravo da vam preporučim ovu knjigu, veoma je dobra.
(Laughter)
(Smeh)
Just trust me.
Verujte mi na reč.
OK, so let's just do an experiment, just to play around with some straight lines. This is a straight line. Let's make another one. So every time we move, we move one down and one across, and we draw a new straight line, right? We do this over and over and over, and we look for patterns. So this pattern emerges, and it's a rather nice pattern. It looks like a curve, right? Just from drawing simple, straight lines.
OK, hajde da uradimo jedan eksperiment, samo da se igramo sa nekim pravim liniijama. Ovo je prava linija. Nacrtajmo još jednu. Pri svakom pokretu crtamo, jednu malo niže i jednu popreko, i crtamo novu pravu liniju, tačno? Ponavljamo ovo iznova i iznova, i tražimo šablone. Izdvaja se ovaj šablon, i prilično je lep. Izgleda kao jedna kriva, zar ne? Iz prostog crtanja jednostavnih, pravih linija.
Now I can change my perspective a little bit. I can rotate it. Have a look at the curve. What does it look like? Is it a part of a circle? It's actually not a part of a circle. So I have to continue my investigation and look for the true pattern. Perhaps if I copy it and make some art? Well, no. Perhaps I should extend the lines like this, and look for the pattern there. Let's make more lines. We do this. And then let's zoom out and change our perspective again. Then we can actually see that what started out as just straight lines is actually a curve called a parabola. This is represented by a simple equation, and it's a beautiful pattern.
Sad mogu malo da promenim perspektivu. Mogu da rotiram ovo. Pogledajte krivu. Kako vam izgleda? Je li to deo kruga? U stvari, nije deo kruga. Moram da nastavim sa istragom i tražim pravi šablon. Možda ako ovo prekopiram i dodam malo umetnoti? Ipak ne. Možda da produžim linije, ovako, i pogledajte ovaj šablon ovde. Nacrtajmo još linija. Uradimo ovo. I hajde da smanjimo zum i ponovo promenimo perspektivu. Onda zapravo vidimo da ono što su u početku bile samo prave linije, su sada u stvari krive, parabole. Ovo se može predstaviti jedostavnom jednačinom, a i to je lep šablon.
So this is the stuff that we do. We find patterns, and we represent them. And I think this is a nice day-to-day definition. But today I want to go a little bit deeper, and think about what the nature of this is. What makes it possible? There's one thing that's a little bit deeper, and that has to do with the ability to change your perspective. And I claim that when you change your perspective, and if you take another point of view, you learn something new about what you are watching or looking at or hearing. And I think this is a really important thing that we do all the time.
To je ono što mi radimo. Pronalazimo šablone i predstavljamo ih. Mislim da je to fina, jednostavna definicija. Ali danas želim da malo produbim temu, i razmislim o prirodi svega ovoga. Zbog čega je ovo moguće? Postoji jedna stvar koja je malo dublja, i ima veze sa sposobnošću da promenite ugao gledanja. I tvrdim da kada promenite ugao gledanja, i počnete da gledate stvari iz drugog ugla, naučite nešto novo o onome što posmatrate ili gledate ili čujete. Smatram da je stvarno važno da to radimo sve vreme.
So let's just look at this simple equation, x + x = 2 • x. This is a very nice pattern, and it's true, because 5 + 5 = 2 • 5, etc. We've seen this over and over, and we represent it like this. But think about it: this is an equation. It says that something is equal to something else, and that's two different perspectives. One perspective is, it's a sum. It's something you plus together. On the other hand, it's a multiplication, and those are two different perspectives. And I would go as far as to say that every equation is like this, every mathematical equation where you use that equality sign is actually a metaphor. It's an analogy between two things. You're just viewing something and taking two different points of view, and you're expressing that in a language.
Pa hajde da pogledamo ovu jednostavnu jednačinu, x + x = 2 • x. Ovo je veoma lep šablon, i tačan je, zato što je 5 + 5 = 2 • 5, itd. Videli smo ovo mnogo puta, i predstavljamo to ovako. Ali razmislite: ovo je jednačina. Kaže da je nešto jednako nečemu drugom, a to su dve različite perspektive. Jedna perspektiva je zbir. Dodajete nešto nečemu. Sa druge strane, imamo množenje, i to su dve različite perspektive. I išao bih toliko daleko da kažem da je svaka jednačina ovakva, svaka matematička jednačina gde koristite znak plus je zapravo metafora. To je analogija između dve stvari. Prosto posmatrate nešto i imate dva ugla gledanja, a to izražavate jezikom.
Have a look at this equation. This is one of the most beautiful equations. It simply says that, well, two things, they're both -1. This thing on the left-hand side is -1, and the other one is. And that, I think, is one of the essential parts of mathematics -- you take different points of view.
Pogledajte ovu jednačinu. Ovo je jedna od najlepših jednačina. Ona jednostavno kaže ove dve stvari obe imaju rešenje -1. Ovo sa leve strane jednako je -1, kao i ovo sa druge. I smatram da je to jedan od suštinskih delova matematike - imate različite uglove gledanja.
So let's just play around. Let's take a number. We know four-thirds. We know what four-thirds is. It's 1.333, but we have to have those three dots, otherwise it's not exactly four-thirds. But this is only in base 10. You know, the number system, we use 10 digits. If we change that around and only use two digits, that's called the binary system. It's written like this. So we're now talking about the number. The number is four-thirds. We can write it like this, and we can change the base, change the number of digits, and we can write it differently.
Hajde malo da se igramo. Uzmimo jedan broj. Znamo četiri trećine. Znamo koliko je četiri trećine. 1.333, ali moramo da imamo ove tri tačke, inače to nisu tačno četiri trećine. Ali to je samo za osnovu 10. Znate, numerički sistem, uzimamo 10 cifri. Ako to preokrenemo i uzmemo samo dve cifre, to se zove binarni sistem. Piše se ovako. Sada pričamo o ovom broju. Taj broj je četiri trećine. Možemo ga napisati ovako, i možemo promeniti osnovu, promeniti broj cifara, i možemo ga napisati drugačije.
So these are all representations of the same number. We can even write it simply, like 1.3 or 1.6. It all depends on how many digits you have. Or perhaps we just simplify and write it like this. I like this one, because this says four divided by three. And this number expresses a relation between two numbers. You have four on the one hand and three on the other. And you can visualize this in many ways. What I'm doing now is viewing that number from different perspectives. I'm playing around. I'm playing around with how we view something, and I'm doing it very deliberately. We can take a grid. If it's four across and three up, this line equals five, always. It has to be like this. This is a beautiful pattern. Four and three and five. And this rectangle, which is 4 x 3, you've seen a lot of times. This is your average computer screen. 800 x 600 or 1,600 x 1,200 is a television or a computer screen.
Sve ovo su reprezentacije istog broja. Možemo ga napisati jednostavno, kao 1.3 ili 1.6. Sve zavisi od toga koliko cifara imate. Ili možda možemo da pojednostavimo i napišemo ga ovako. Sviđa mi se ovako, zato što kaže četiri podeljeno sa tri. I ovaj broj izražava odnos između dva broja. Sa jedne strane imate četiri i tri sa druge. I možete zamisliti ovo na više načina. Sada posmatram taj broj iz drugačije perspektive. Igram se. Igram se sa tim kako na nešto gledamo, i radim to namerno. Možemo uzeti rešetku. To je četiri udesno i tri gore, ova linija je uvek pet. Mora da bude ovako. Ovo je prelepi šablon. Četiri i tri i pet. A ovaj pravougaonik, koji je 4 x 3, videli ste mnogo puta. Ovo je prosečan ekran računara, 800 x 600 ili 1600 x 1200 je ekran TV-a ili računara.
So these are all nice representations, but I want to go a little bit further and just play more with this number. Here you see two circles. I'm going to rotate them like this. Observe the upper-left one. It goes a little bit faster, right? You can see this. It actually goes exactly four-thirds as fast. That means that when it goes around four times, the other one goes around three times. Now let's make two lines, and draw this dot where the lines meet. We get this dot dancing around.
Ovo su sve lepe reprezentacije, ali želim da idem malo dalje i malo više da se igram sa brojevima. Ovde vidite dva kruga. Rotiraću ih ovako. Gledajte ovaj u gornjem levom uglu. Kreće se malo brže, zar ne? To može da se vidi. Zapravo kreće se tačno brzinom od četiri trećine. To znači da kada se kreće okolo četiri puta, drugi to radi tri puta. Hajde sada da nacrtamo dve linije i tačku tamo gde se linije spajaju. Dobijamo ovaj ples.
(Laughter)
(Smeh)
And this dot comes from that number. Right? Now we should trace it. Let's trace it and see what happens. This is what mathematics is all about. It's about seeing what happens. And this emerges from four-thirds. I like to say that this is the image of four-thirds. It's much nicer -- (Cheers)
I ova tačka dolazi od tog broja. Tačno? Sada treba da joj uđemo u trag. Hajde da ga nađemo i vidimo šta će se dogoditi. O tome se radi u matematici. O gledanju onog što se dešava. I ovo smo dobill od četiri trećine. Volim da kažem da je ovo slika četiri trećine. Lepše je - (Oduševljenje)
Thank you!
Hvala vam!
(Applause) This is not new. This has been known for a long time, but --
(Aplauz) Ovo nije novo. Za ovo se zna odavno, ali -
(Laughter)
(Smeh)
But this is four-thirds.
Ali ovo su četiri trećine.
Let's do another experiment. Let's now take a sound, this sound: (Beep)
Da uradimo još jedan eksperiment. Da uzmemo neki zvuk, ovaj zvuk: (Zvučni signal)
This is a perfect A, 440Hz. Let's multiply it by two. We get this sound. (Beep)
Ovo je savršeno A, 440Hz. Da pomnožimo to sa dva. Dobijamo ovaj zvuk. (Zvučni signal)
When we play them together, it sounds like this. This is an octave, right? We can do this game. We can play a sound, play the same A. We can multiply it by three-halves.
Kada ih pustimo zajedno, zvuče ovako. Ovo je oktava, zar ne? Možemo ovako da se igramo. Pustimo zvuk, čujemo isto A. Pomnožićemo to sa tri polovine.
(Beep)
(Zvučni signal)
This is what we call a perfect fifth.
Ovo zovemo savršenom petinom.
(Beep)
(Zvučni signal)
They sound really nice together. Let's multiply this sound by four-thirds. (Beep)
Zvuče stvarno lepo zajedno. Pomnožimo ovaj zvuk sa četiri trećine. (Zvučni signal)
What happens? You get this sound. (Beep)
Šta se dešava? Dobijamo ovaj zvuk. (Signal)
This is the perfect fourth. If the first one is an A, this is a D. They sound like this together. (Beeps)
Ovo je savršena četvrtina. Ako je prvi A, onda je ovo D. Ovako zvuče zajedno. (Zvučni signal)
This is the sound of four-thirds. What I'm doing now, I'm changing my perspective. I'm just viewing a number from another perspective.
Ovo je zvuk četiri trećine. Ono što radim sada je da menjam ugao geldanja. Samo gledam broj iz drugačije perspektive.
I can even do this with rhythms, right? I can take a rhythm and play three beats at one time (Drumbeats)
Ovo mogu da radim i sa ritmovima, zar ne? Mogu uzeti jedan ritam i svirati tri taktaistovremeno (Bubnjevi)
in a period of time, and I can play another sound four times in that same space.
neko vreme, mogu svirati drugi zvuk četiri puta u istom prostoru.
(Clanking sounds)
(Zveket)
Sounds kind of boring, but listen to them together.
Zvuče nekako dosadno ali poslušajte ih zajedno.
(Drumbeats and clanking sounds)
(Bubnjevii zveket)
(Laughter)
(Smeh)
Hey! So.
Hej! Tako.
(Laughter)
(Smeh)
I can even make a little hi-hat.
Čak mogu malo da koristim i pedale.
(Drumbeats and cymbals)
(Bubnjevi i cimbal)
Can you hear this? So, this is the sound of four-thirds. Again, this is as a rhythm.
Čujete li ovo? Pa, ovo je zvuk četiri trećine. Opet, ovo je neki ritam.
(Drumbeats and cowbell)
(Bubnjevi i medenica)
And I can keep doing this and play games with this number. Four-thirds is a really great number. I love four-thirds!
I mogu da nastavim ovako da se igram sa ovim brojem. Četiri trećine je stvarno sjajan broj. Obožavam ga!
(Laughter)
(Smeh)
Truly -- it's an undervalued number. So if you take a sphere and look at the volume of the sphere, it's actually four-thirds of some particular cylinder. So four-thirds is in the sphere. It's the volume of the sphere.
Zaista - to je potcenjen broj. Tako ako uzmete loptu i pogledate njenu zapreminu, imate u stvari četiri trećine nekog određenog cilindra. Tako da su četiri trećine u lopti. One su u zapremini lopte.
OK, so why am I doing all this? Well, I want to talk about what it means to understand something and what we mean by understanding something. That's my aim here. And my claim is that you understand something if you have the ability to view it from different perspectives. Let's look at this letter. It's a beautiful R, right? How do you know that? Well, as a matter of fact, you've seen a bunch of R's, and you've generalized and abstracted all of these and found a pattern. So you know that this is an R.
OK, zašto ja uopšte ovo radim? Pa, želim da govorim o tome šta znači razumeti nešto i šta smatramo pod tim. To je moj cilj ovde. A moja tvrdnja je da razumemo nešto ako imamo sposobnost da to glesamo iz različitih perspektiva. Pogledajmo ovo slovo. Jedno lepo slovo R, zar ne? Kako to znamo? Zapravo, videli smo milion puta slovo R, i generalizovali smo i rezimirali sve ovo i pronašli šablon. Tako da znamo da je ovo slovo R.
So what I'm aiming for here is saying something about how understanding and changing your perspective are linked. And I'm a teacher and a lecturer, and I can actually use this to teach something, because when I give someone else another story, a metaphor, an analogy, if I tell a story from a different point of view, I enable understanding. I make understanding possible, because you have to generalize over everything you see and hear, and if I give you another perspective, that will become easier for you.
Ono na šta ciljam je da kažem nešto o tome kako su razumevanje i promena perspektive povezani. Ja sam nastavnik i predavač, i mogu u stvari da iskoristim ovo da bih predavao jer kad nekom drugom dam drugu priču, metaforu, analogiju, ako ispričam priču sa drugačijeg stanovišta, ja omogućavam razumevanje. Činim razumevanje mogućim jer morate generalizovati sve što vidite i čujete, i ako vam dam drugačiju perspektivu, biće vam lakše.
Let's do a simple example again. This is four and three. This is four triangles. So this is also four-thirds, in a way. Let's just join them together. Now we're going to play a game; we're going to fold it up into a three-dimensional structure. I love this. This is a square pyramid. And let's just take two of them and put them together. So this is what is called an octahedron. It's one of the five platonic solids. Now we can quite literally change our perspective, because we can rotate it around all of the axes and view it from different perspectives. And I can change the axis, and then I can view it from another point of view, but it's the same thing, but it looks a little different. I can do it even one more time.
Da uradimo ponovo jedan primer. Ovo su četiri i tri. Ovo su četiri trougla. Tako da je ovo takođe na neki način četiri trećine. Hajde da ih jednostavno povežemo. Sada ćemo da igramo jednu igru: savićemo ih u trodimenzionalnu strukturu. Obožavam ovo. Ovo je četvrtasta piramida. Hajde da uzmemo dve i sastavimo ih. I ovo je ono što se zove oktahedron. Jedan je od pet platonskih čvrstih tela. Sada i bukvalno možemo promeniti našu perspektivu, jer ga možemo rotirati oko svih osa i posmatrati iz različitih perspektiva. Mogu i da promenim ovu osu, i onda mogu da posmatram iz drugačije perspekte, ali je to ista stvar samo što izgleda drugačije. Mogu to da uradim još jedanput.
Every time I do this, something else appears, so I'm actually learning more about the object when I change my perspective. I can use this as a tool for creating understanding. I can take two of these and put them together like this and see what happens. And it looks a little bit like the octahedron. Have a look at it if I spin it around like this. What happens? Well, if you take two of these, join them together and spin it around, there's your octahedron again, a beautiful structure. If you lay it out flat on the floor, this is the octahedron. This is the graph structure of an octahedron. And I can continue doing this. You can draw three great circles around the octahedron, and you rotate around, so actually three great circles is related to the octahedron. And if I take a bicycle pump and just pump it up, you can see that this is also a little bit like the octahedron. Do you see what I'm doing here? I am changing the perspective every time.
Svaki put kada ovo uradim, nešto se drugo pojavi, tako da zapravo učim sve više o ovom predmetu kada promenim perspektivu. Mogu da ga koristim kao alat za stvaranje razumevanja. Mogu da uzmem dva komada i sastavim ih ovako i vidite šta se dešava. Malo i liči na oktahedron. Pogledajte ako ga okrenem ovako. Šta se dešava? Pa, ako uzmete dva tela, spojite ih i okrenete, evo ga oktahedron opet, prelepa struktura. Ako stavite pravo na pod, ovo je oktahedron. Ovo je grafička struktura oktahedrona. I mogu ovako do sutra. Možete da nacrtate tri velika kruga oko oktahedrona, i da ih zarotirate, i onda su u stvari tri velika kruga povezana sa oktahedronom. I ako uzmem pumpu za bicikl i prosto ih naduvam, možete opet videti da malo podseća na oktahedron. Da li vidite šta radim? Menjam perspektivu svaki put.
So let's now take a step back -- and that's actually a metaphor, stepping back -- and have a look at what we're doing. I'm playing around with metaphors. I'm playing around with perspectives and analogies. I'm telling one story in different ways. I'm telling stories. I'm making a narrative; I'm making several narratives. And I think all of these things make understanding possible. I think this actually is the essence of understanding something. I truly believe this.
Sad ćemo malo da se vratimo unazad - i to je u stvari metafora, vraćanje unazad - i da pogledamo šta smo uradili. Poigravam se sa metaforama. Poigravam se sa perspektivama i analogijama. Prenosim jednu priču na razne načine. Pričam priče. Pravim narativ. Pravm nekoliko narativa. Smatram da je zbog svega ovog razumevanje moguće. Smatram da je ovo zapravo suština razumevanja nečega. Istinski verujem u to.
So this thing about changing your perspective -- it's absolutely fundamental for humans. Let's play around with the Earth. Let's zoom into the ocean, have a look at the ocean. We can do this with anything. We can take the ocean and view it up close. We can look at the waves. We can go to the beach. We can view the ocean from another perspective. Every time we do this, we learn a little bit more about the ocean. If we go to the shore, we can kind of smell it, right? We can hear the sound of the waves. We can feel salt on our tongues. So all of these are different perspectives. And this is the best one. We can go into the water. We can see the water from the inside. And you know what? This is absolutely essential in mathematics and computer science. If you're able to view a structure from the inside, then you really learn something about it. That's somehow the essence of something.
Tako da je ovo sa menjanjem perspektive apsolutno neophodno za čovečanstvo. Hajde da se igramo sa Zemljom. Zumirajmo okean, pogledajte okean. Ovo možemo da radimo sa svim stvarima. Možemo da uzmemo okean i posmatramo ga izbliza. Možemo da gledamo talase. Možemo da odemo do plaže. Možemo da posmatramo okean iz druge perspektive. Svaki put kad to uradimo, naučimo nešto više o okeanu. Ako odemo do obale, možemo da osetimo miris, zar ne? Možemo da čujemo zvuk talasa. Možemo da osetimo so na jeziku. Sve su to drugačije perspektive. A ovo je najbolja. Možemo da uđemo u vodu. Možemo da vidimo vodu iznutra. I znate šta? Ovo je apsolutno neophodno u matematici i informatici. Ako možete da vidite strukturu iznutra, onda ste je stvarno savladali. To je nekako suština svačega.
So when we do this, and we've taken this journey into the ocean, we use our imagination. And I think this is one level deeper, and it's actually a requirement for changing your perspective. We can do a little game. You can imagine that you're sitting there. You can imagine that you're up here, and that you're sitting here. You can view yourselves from the outside. That's really a strange thing. You're changing your perspective. You're using your imagination, and you're viewing yourself from the outside. That requires imagination.
Tako kada uradimo ovo, i kada otputujemo u okean, koristimo našu maštu. A mislim da je to još jedan korak dalje, i zapravo je neophodan da bismo promenili perspektivuu. Odigraćemo jednu igru. Zamislite da sedite tamo. Zamislite da ste ovde i da sedite tu. Možete videti sebe spolja. To je tek čudno. Menjate perspektivu. Koristite maštu, i posmatrate sebe spolja. Za to je neophodna mašta.
Mathematics and computer science are the most imaginative art forms ever. And this thing about changing perspectives should sound a little bit familiar to you, because we do it every day. And then it's called empathy. When I view the world from your perspective, I have empathy with you. If I really, truly understand what the world looks like from your perspective, I am empathetic. That requires imagination. And that is how we obtain understanding. And this is all over mathematics and this is all over computer science, and there's a really deep connection between empathy and these sciences.
Matematika i informatika su najmaštovitije umetničke forme od svih. I ovo sa menjanjem perspektiva bi trebalo da vam je malo jasnije, jer je to nešto što radimo svaki dan. I onda se to zove empatija. Kada ja posmatram svet iz vašeg ugla, osećam empatiju sa vama. Ako stvarno, istinski razumem kako svet izgleda iz vaše perspektive, ja sam empatičan. Za to je neophodna mašta. Na taj način usvajamo razumevanje. I to je svuda u matematici i informatici i postoji zaista velika povezanost između empatije i ovih nauka.
So my conclusion is the following: understanding something really deeply has to do with the ability to change your perspective. So my advice to you is: try to change your perspective. You can study mathematics. It's a wonderful way to train your brain. Changing your perspective makes your mind more flexible. It makes you open to new things, and it makes you able to understand things. And to use yet another metaphor: have a mind like water. That's nice.
Stoga, moj zaključak je sledeći: razumevanje nečega istinski mora da ima veze sa sposobnošću da promenite perspektivu. Moj savet vam je: probajte da promenite perspektivu. Možete izučavati matematiku. To je divan način da uvežbate svoj mozak. Kada menjate perspektivu vaš um postaje fleksibilniji. Otvoreniji ste za nove stvari, i sposobni ste da razumete stvari. I da upotrebim još jednu metaforu: neka vam um bude poput vode. To je lepo.
Thank you.
Hvala vam.
(Applause)
(Aplauz)