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.
Bok. Želim razgovarati o razumijevanju i prirodi razumijevanja, te što je srž razumijevanja, jer je razumijevanje nešto čemu svi težimo. Želimo razumjeti stvari. Ja tvrdim da je razumijevanje povezano sa sposobnošću mijenjanja perspektive. Ako je nemaš, nemaš razumijevanja. 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.
I želim se usredotočiti na matematiku. Mnogi od nas misle da je matematika zbrajanje, oduzimanje, množenje, dijeljenje, razlomci, postotci, geometrija, algebra -- takve stvari. Zapravo, želim razgovarati i o srži matematike. Tvrdim da matematika ima veze s uzorcima.
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 prekrasan uzorak, i ovaj uzorak zapravo proizlazi iz crtanja krugova na vrlo neobičan način. Moja svakodnevna definicija matematike koju svaki dan koristim je sljedeća: Prije svega, najvažnije je pronaći uzorke. A pod "uzorcima" mislim na veze, strukturu, neku pravilnost, pravila koja upravljaju onim što vidimo. Drugo, mislim da se radi o prikazivanju ovih uzoraka jezikom. Mi izmislimo jezik ako ga nemamo, a u matematici to je najbitnije. Važno je pretpostavljati poigravati se pretpostavkama i samo vidjeti što će se dogoditi. Napravit ćemo to uskoro. I na kraju, radi se o super stvarima. Matematika nam omogućuje da činimo mnogo 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 pogledajmo ove uzorke. Ako želite zavezati kravatu, postoje uzorci. Čvorovi za kravate imaju imena. A možete i otkriti matematiku čvorova. Ovo je lijevi-van, desni-unutra, sredina-van i zaveži. Ovo je lijevi-unutra, desni-van, lijevi-unutra, sredina-vani zaveži. To je jezik koji smo izmislili za uzorke čvorova za kravate, i polu-Windsor je sve to. Ovo je matematička knjiga o vezanju vezica na cipelama na sveučilišnoj razini, jer u vezanju vezica ima uzoraka. Mogu se zavezati na puno različitih načina. Možemo ih analizirati. Možemo izmisliti jezike za njih.
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.
A primjeri su u cijeloj matematici. Ovo je Leibnizova bilješka iz 1675. On je izmislio jezik za uzorke u prirodi. Kad nešto bacimo u zrak, to padne. Zašto? Nismo sigurni, ali možemo to prikazati matematikom u uzroku.
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.
To je također uzorak. To je također izmišljeni jezik. Možete li pogoditi za što služi? To je zapravo sustav bilježenja za ples, za step. To omogućuje njemu kao koreografu da radi super stvari, nove stvari, jer ih je on prikazao.
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.
Želim da pomislite kako je divno prikazati nešto. Ovdje piše riječ "matematika". Ali zapravo, to su samo točkice, zar ne? Pa kako je moguće da te točkice prikazuju riječ? Pa, moguće je. Prikazuju riječ "matematika", i ovi simboli pokazuju istu riječ, i ovo što možemo poslušati. Zvuči ovako.
(Beeps)
(Pištanje)
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 prikazuju riječ i koncept. Kako se ovo događa? Ima nešto nevjerojatno u prikazivanju 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.
Zato želim razgovarati o magiji koja se događa kad zapravo prikazujemo nešto. Ovdje vidite samo crte različite širine. Svaki označava broj određene knjige. I stvarno preporučam ovu knjigu, to je vrlo dobra knjiga.
(Laughter)
(Smijeh)
Just trust me.
Vjerujte mi.
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.
U redu, napravimo eksperiment, kako bismo se igrali s ravnim crtama. Ovo je ravna crta. Napravimo još jednu. Svakim novim potezom stavimo jednu dolje i jednu preko, i nacrtamo novu ravnu crtu, u redu? Činimo ovo iznova i iznova, i tražimo uzorke. Javlja se ovaj uzorak, i baš je lijep uzorak. Izgleda kao krivulja, zar ne? Samo od crtanja jednostavnih, ravnih crta.
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 promijeniti malo perspektivu. Mogu okrenuti crtu. Pogledajte krivulju. Na što sliči? Je li dio kruga? Zapravo nije dio kruga. Moram nastaviti istraživanje i potražiti pravi uzorak. Možda ako je kopiram i pretvorim u umjetnost? Pa i ne. Možda bih trebao produljiti crte ovako, i potražiti uzorak u tome. Napravimo još crta. Učinimo ovo. Odmaknimo se i ponovno promijenimo perspektivu. Tada zapravo vidimo da ono što je počelo kao ravna crta zapravo je krivulja zvana parabola. Prikazana je jednostavnom jednadžbom, i to je prekrasan uzorak.
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.
Dakle ovo radimo. Pronalazimo uzorke i predstavljamo ih. Mislim da je to lijepa svakodnevna definicija. Ali danas želim ići malo dublje, i razmisliti o prirodi ovoga. Što omogućuje ovo? Jedna stvar je malo dublja, i ima veze sa sposobnošću da promijenite perspektivu. I tvrdim da kad promijenite perspektivu, i ako pogledate s druge točke gledišta naučit ćete nešto novo o tome što promatrate, gledate ili slušate. Mislim da je ovo što stalno radimo zaista važno.
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.
Zato pogledajmo ovu jednostavnu jednadžbu. x + x = 2 • x Ovo je vrlo lijep uzorak, i točan je, jer je 5 + 5 = 2 • 5, itd. Vidjeli smo ovo već, i prikazujemo to ovako. Ali razmislite o tome: ovo je jednadžba. Kaže da je nešto jednako nečem drugom, i to su dvije različite perspektive. Jedna perspektiva je, to je zbroj. To je nešto što zbrojite zajedno. U drugu ruku, to je množenje, i to su dvije različite perspektive. I usuđujem se reći da je svaka jednadžba ovakva, svaka matematička jednadžba u kojoj možete koristiti znak jednakosti je zapravo metafora. Ona je analogija između dvije stvari. Gledate nešto, zauzimate dva stajališta, i izražavate to 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 jednadžbu. Ovo je jedna od najljepših jednadžbi. Jednostavno kaže da, pa, dvije stvari, one su obje -1. Ovo na lijevoj strani je -1, a i ovo na drugoj. I to je, mislim, jedan od najvažnijih dijelova matematike -- zauzimaš različite točke gledišta.
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.
Ajmo se malo igrati. Odaberimo neki broj. Znamo za četiri trećine. Znamo što je četiri trećine. To je 1.333, ali moramo imati te tri točke, inače nije točno četiri trećine. Ali to je samo na bazi 10. Mi koristimo brojevni sustav od 10 znamenki. Ako za promjenu koristimo samo dvije znamenke to nazivamo binarni sustav. Piše se ovako. Dakle sad govorimo o broju. Broj je četiri trećine. Možemo ga napisati ovako, i možemo promijeniti bazu, promijeniti broj znamenki, 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.
Ovo su sve prikazi istog broja. Možemo ga napisati jednostavno, kao 1.3 ili 1.6. Sve ovisi o tome koliko znamenki imate. Ili da pojednostavimo i napišemo ovako. Sviđa mi se ovako, jer kaže da je četiri podijeljeno na tri. A ovaj broj izražava vezu između dva broja. Imate četiri na jednoj strani i tri na drugoj. Možete to vizualizirati na puno načina. Sada upravo gledam taj broj iz različitih perspektiva. Igram se. Igram se načinom na koji vidimo nešto, i radim to vrlo svojevoljno. Možemo uzeti mrežu. Ako su dimenzije 4 x 3, ova crta je 5, uvijek. Mora biti ovako. Ovo je prekrasan uzorak. Četiri, tri i pet. I ovaj pravokutnik koji je 4 x 3, vidjeli ste puno puta. Ovo je vaš prosječni ekran računala. 800 x 600, ili 1600 x 1200 je televizija ili ekran računala.
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 lijepi prikazi, ali želim ići dalje i još se igrati ovim brojem. Ovdje vidite dva kruga. Okretat ću ih ovako. Promatrajte ovaj gore-lijevo. Ide malo brže, zar ne? Možete to vidjeti. Zapravo ide točno četiri trećine brzine drugog kruga. To znači da kad se on okrene četiri puta, drugi se okrene tri puta. Napravimo sada dvije crte, i stavimo točku gdje se crte spajaju. Dobijemo da ova točka pleše okolo.
(Laughter)
(Smijeh)
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 točka proizlazi iz tog broja. Zar ne? Sad bismo je trebali pratiti. Pratimo je i pogledajmo što će se dogoditi. Ovo je najvažnije u matematici. Važno je vidjeti što će se dogoditi. A ovo proizlazi iz četiri trećine. Volim reći da je ovo slika četiri trećine. Puno je ljepša -- (Klicanje)
Thank you!
Hvala.
(Applause) This is not new. This has been known for a long time, but --
(Pljesak) Ovo nije novost. Ovo je poznato već dugo, ali --
(Laughter)
(Smijeh)
But this is four-thirds.
Ali ovo su četiri trećine.
Let's do another experiment. Let's now take a sound, this sound: (Beep)
Napravimo drugi eksperiment. Sada uzmimo zvuk, ovaj zvuk: (Pištanje)
This is a perfect A, 440Hz. Let's multiply it by two. We get this sound. (Beep)
Ovo je savršeni ton A, 440 Hz. Pomnožimo ga s dva. Dobijemo ovo. (Pištanje)
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.
Kad ih pustimo zajedno, zvuče ovako. Ovo je oktava, zar ne? Možemo igrati ovu igru. Pustimo zvuk, pustimo isti A. Možemo ga pomnožiti s tri polovine.
(Beep)
(Pištanje)
This is what we call a perfect fifth.
Ovo zovemo čistom kvintom.
(Beep)
(Pištanje)
They sound really nice together. Let's multiply this sound by four-thirds. (Beep)
Zvuče baš lijepo zajedno. Pomnožimo taj zvuk s četiri trećine. (Pištanje)
What happens? You get this sound. (Beep)
Što se dogodi? Dobijemo ovaj zvuk. (Pištanje)
This is the perfect fourth. If the first one is an A, this is a D. They sound like this together. (Beeps)
Ovo je čista kvarta. Ako je prvi ton A, ovo je D. Ovako zvuče zajedno. (Pištanje)
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. Ovo što sad radim je mijenjanje perspektive. Samo gledam broj iz druge perspektive.
I can even do this with rhythms, right? I can take a rhythm and play three beats at one time (Drumbeats)
Ovo čak mogu raditi s ritmovima, zar ne? Mogu odabrati ritam i pustiti tri takta istovremeno (Bubnjevi)
in a period of time, and I can play another sound four times in that same space.
u određenom vremenskom razdoblju i mogu pustiti neki zvuk četiri puta u istom razdoblju.
(Clanking sounds)
(Zveckanje)
Sounds kind of boring, but listen to them together.
Zvuči malo dosadno, ali poslušajte ih zajedno.
(Drumbeats and clanking sounds)
(Bubnjevi i zveckanje)
(Laughter)
(Smijeh)
Hey! So.
Hej! Tako.
(Laughter)
(Smijeh)
I can even make a little hi-hat.
Čak mogu napraviti mali fuš.
(Drumbeats and cymbals)
(Bubnjevi i činele)
Can you hear this? So, this is the sound of four-thirds. Again, this is as a rhythm.
Možete li ovo čuti? Dakle, ovo je zvuk četiri trećine. Opet, ovo je njezin ritam.
(Drumbeats and cowbell)
(Bubnjevi i kravlje zvono)
And I can keep doing this and play games with this number. Four-thirds is a really great number. I love four-thirds!
Mogu nastaviti ovako i igrati se ovim brojem. Četiri trećine je odličan broj. Volim četiri trećine!
(Laughter)
(Smijeh)
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 podcijenjen broj. Ako uzmete kuglu i pogledate njezin volumen, on je zapravo četiri trećine određenog valjka. Dakle, četiri trećine je u kugli. To je volumen kugle.
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 radim sve ovo? Pa, želim govoriti o tome što znači razumjeti nešto i što mislimo pod razumijevanjem nečega. To je moj cilj ovdje. I tvrdim da razumijete nešto ako to možete pogledati iz različitih perspektiva. Pogledajmo ovo slovo. To je prekrasno R, zar ne? Kako to znate? Zapravo, vidjeli ste puno slova R, i generalizirali ste i usvojili sve njih te ste pronašli uzorak. Zato znate da je ovo 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.
Ovime pokušavam reći nešto o tome kako su razumijevanje i promjena perspektive povezani. Ja sam učitelj i predavač, i mogu zaista iskoristiti ovo za učenje jer kad pričam nekome drugu priču, metaforu, analogiju, ako pričam priču s drugog gledišta, omogućujem razumijevanje. Ja stvaram uvjete za razumijevanje, jer morate generalizirati sve što vidite i čujete, a ako vam dam drugu perspektivu, bit će 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.
Napravimo opet jednostavan primjer. Ovo su četiri i tri. Ovo su četiri trokuta. Dakle i to su četiri trećine na neki način. Spojimo ih. Sad ćemo igrati igru, savit ćemo ih u trodimenzionalnu strukturu. Volim ovo. Ovo je četverostrana piramida. Uzmimo dvije i spojimo ih. Ovo se zove oktaedar. On je jedan od pet pravilnih poliedara. Sad možemo posve doslovno promijeniti perspektivu, jer ga možemo rotirati oko svih osi i pogledati iz različitih perspektiva. Mogu i promijeniti os, i tada ga pogledati s različitog gledišta, ali to je isto, iako izgleda malo drugačije. Mogu to učiniti čak još jednom.
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 kad to učinim, nešto drugo se pojavi, pa zapravo učim više o predmetu kad promijenim perspektivu. Mogu to koristiti kao alat za stvaranje razumijevanja. Mogu uzeti dva ovakva i spojiti ih zajedno, ovako i vidjeti što će se dogoditi. I pomalo izgleda kao oktaedar. Pogledajte ga kad ga zavrtim. Što se dogodi? Ako uzmete dva, spojite ih i zavrtite, opet dobijete oktaedar, prekrasnu strukturu. Ako ga rasklopite, ovo je oktaedar. Ovo je grafički prikaz strukture oktaedra. I mogu to nastaviti raditi. Možete nacrtati tri velika kruga oko oktaedra i kad ga okrećete zapravo su tri velika kruga s njim povezana. Ako uzmem pumpu za bicik i napumpam ga, vidjet ćete da je i ovo pomalo nalik oktaedru. Vidite li što radim? Svaki put mijenjam perspektivu.
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.
Učinite odmak, i to je zapravo metafora, učiniti odmak -- i pogledajte što radimo. Igram se metaforama. Igram se s perspektivama i analogijama. Pričam jednu priču na različite načine. Pričam priče. Pišem pripovijest. Pišem nekoliko pripovijesti. Mislim da sve ovo omogućuje razumijevanje. Mislim da je ovo srž razumijevanja nečega. Zaista to vjerujem.
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 o mijenjanju perspektive -- apsolutni temelj za ljude. Igrajmo se sa Zemljom. Pogledajmo izbliza ocean, pogledajmo ocean. Možemo ovo učiniti sa svime. Možemo ocean pogledati izbliza. Možemo pogledati valove. Možemo ići na plažu. Možemo pogledati ocean iz druge perspektive. Svaki put kad to učinimo, naučimo još nešto o oceanu. Ako odemo na obalu možemo je namirisati, zar ne? Možemo čuti zvuk valova. Možemo osjetiti sol na jeziku. Sve to su različite perspektive. A ovo je najbolja. Možemo ući u vodu. Možemo vidjeti vodu iznutra. I znate što? Ovo je posve neophodno u matematici i informatici. Ako možete vidjeti strukturu iznutra, tada zaista naučite nešto o njoj. To je nekako jezgra nečega.
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.
Kad radimo ovo, i putujemo u ocean, koristimo našu maštu. Mislim da je ovo jedna razina dublje, i zapravo je uvijet za mijenjanje perspektive. Možemo se igrati. Možete zamisliti da sjedite ondje. Možete zamisliti da ste ovdje gore, i da sjedite ovdje. Možete se pogledati izvana. To je zaista čudno. Možete promijeniti svoju perspektivu. Koristite maštu, i vidite se izvana. To zahtijeva maštu.
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 računarstvo su najmaštovitiji umjetnički izrazi ikada. I ovo o mijenjanju perspektive bi vam trebalo zvučati poznato, jer to radimo svaki dan. A tada to zovemo suosjećanjem Kad ja gledam svijet iz tvoje perspektive, ja suosjećam s vama. Ako ja stvarno, zaista razumijem kako svijet izgleda iz vaše perspektive, ja sam suosjećajan. To zahtijeva maštu. Tako mi dobivamo razumijevanje. To je diljem matematike i to je diljem računarstva, i postoji zaista duboka veza između suosjećanja i ovih znanosti.
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 je moj zaključak sljedeći: razumjeti nešto na zaista dubokoj razini ima veze sa sposobnošću da promijenite perspektivu. Zato je moj savjet vama: pokušajte promijeniti perspektivu. Možete proučavati matematiku. To je prekrasan način da vježbate mozak. Mijenjanje perspektive čini vaš um fleksibilnijim. To vas otvara novim stvarima, i omogućuje vam da razumijete stvari. Da iskoristim drugu metaforu: neka vam um bude kao voda. To je lijepo.
Thank you.
Hvala.
(Applause)
(Pljesak)