Math is the hidden secret to understanding the world | Roger Antonsen

Prevoditelj: Viktorija Đuđik Recezent: Ivan Stamenković

Hi.

Bok.

I want to talk about understanding, and the nature of understanding,

Želim razgovarati o razumijevanju i prirodi razumijevanja,

and what the essence of understanding is,

te što je srž razumijevanja,

because understanding is something we aim for, everyone.

jer je razumijevanje nešto čemu svi težimo.

We want to understand things.

Želimo razumjeti stvari.

My claim is that understanding has to do

Ja tvrdim da je razumijevanje povezano

with the ability to change your perspective.

sa sposobnošću mijenjanja perspektive.

If you don't have that, you don't have understanding.

Ako je nemaš, nemaš razumijevanja.

So that is my claim.

To je moja tvrdnja.

And I want to focus on mathematics.

I želim se usredotočiti na matematiku.

Many of us think of mathematics as addition, subtraction,

Mnogi od nas misle da je matematika zbrajanje, oduzimanje,

multiplication, division,

množenje, dijeljenje,

fractions, percent, geometry, algebra -- all that stuff.

razlomci, postotci, geometrija, algebra -- takve stvari.

But actually, I want to talk about the essence of mathematics as well.

Zapravo, želim razgovarati i o srži matematike.

And my claim is that mathematics has to do with patterns.

Tvrdim da matematika ima veze s uzorcima.

Behind me, you see a beautiful pattern,

Iza mene vidite prekrasan uzorak,

and this pattern actually emerges just from drawing circles

i ovaj uzorak zapravo proizlazi iz crtanja krugova

in a very particular way.

na vrlo neobičan način.

So my day-to-day definition of mathematics that I use every day

Moja svakodnevna definicija matematike koju svaki dan koristim

is the following:

je sljedeća:

First of all, it's about finding patterns.

Prije svega, najvažnije je pronaći uzorke.

And by "pattern," I mean a connection, a structure, some regularity,

A pod "uzorcima" mislim na veze, strukturu, neku pravilnost,

some rules that govern what we see.

pravila koja upravljaju onim što vidimo.

Second of all,

Drugo,

I think it is about representing these patterns with a language.

mislim da se radi o prikazivanju ovih uzoraka jezikom.

We make up language if we don't have it,

Mi izmislimo jezik ako ga nemamo,

and in mathematics, this is essential.

a u matematici to je najbitnije.

It's also about making assumptions

Važno je pretpostavljati

and playing around with these assumptions and just seeing what happens.

poigravati se pretpostavkama i samo vidjeti što će se dogoditi.

We're going to do that very soon.

Napravit ćemo to uskoro.

And finally, it's about doing cool stuff.

I na kraju, radi se o super stvarima.

Mathematics enables us to do so many things.

Matematika nam omogućuje da činimo mnogo stvari.

So let's have a look at these patterns.

Pa pogledajmo ove uzorke.

If you want to tie a tie knot,

Ako želite zavezati kravatu,

there are patterns.

postoje uzorci.

Tie knots have names.

Čvorovi za kravate imaju imena.

And you can also do the mathematics of tie knots.

A možete i otkriti matematiku čvorova.

This is a left-out, right-in, center-out and tie.

Ovo je lijevi-van, desni-unutra, sredina-van i zaveži.

This is a left-in, right-out, left-in, center-out and tie.

Ovo je lijevi-unutra, desni-van, lijevi-unutra, sredina-vani zaveži.

This is a language we made up for the patterns of tie knots,

To je jezik koji smo izmislili za uzorke čvorova za kravate,

and a half-Windsor is all that.

i polu-Windsor je sve to.

This is a mathematics book about tying shoelaces

Ovo je matematička knjiga o vezanju vezica na cipelama

at the university level,

na sveučilišnoj razini,

because there are patterns in shoelaces.

jer u vezanju vezica ima uzoraka.

You can do it in so many different ways.

Mogu se zavezati na puno različitih načina.

We can analyze it.

Možemo ih analizirati.

We can make up languages for it.

Možemo izmisliti jezike za njih.

And representations are all over mathematics.

A primjeri su u cijeloj matematici.

This is Leibniz's notation from 1675.

Ovo je Leibnizova bilješka iz 1675.

He invented a language for patterns in nature.

On je izmislio jezik za uzorke u prirodi.

When we throw something up in the air,

Kad nešto bacimo u zrak,

it falls down.

to padne.

Why?

Zašto?

We're not sure, but we can represent this with mathematics in a pattern.

Nismo sigurni, ali možemo to prikazati matematikom u uzroku.

This is also a pattern.

To je također uzorak.

This is also an invented language.

To je također izmišljeni jezik.

Can you guess for what?

Možete li pogoditi za što služi?

It is actually a notation system for dancing, for tap dancing.

To je zapravo sustav bilježenja za ples, za step.

That enables him as a choreographer to do cool stuff, to do new things,

To omogućuje njemu kao koreografu da radi super stvari, nove stvari,

because he has represented it.

jer ih je on prikazao.

I want you to think about how amazing representing something actually is.

Želim da pomislite kako je divno prikazati nešto.

Here it says the word "mathematics."

Ovdje piše riječ "matematika".

But actually, they're just dots, right?

Ali zapravo, to su samo točkice, zar ne?

So how in the world can these dots represent the word?

Pa kako je moguće da te točkice prikazuju riječ?

Well, they do.

Pa, moguće je.

They represent the word "mathematics,"

Prikazuju riječ "matematika",

and these symbols also represent that word

i ovi simboli pokazuju istu riječ,

and this we can listen to.

i ovo što možemo poslušati.

It sounds like this.

Zvuči ovako.

(Beeps)

(Pištanje)

Somehow these sounds represent the word and the concept.

Nekako ovi zvukovi prikazuju riječ i koncept.

How does this happen?

Kako se ovo događa?

There's something amazing going on about representing stuff.

Ima nešto nevjerojatno u prikazivanju stvari.

So I want to talk about that magic that happens

Zato želim razgovarati o magiji koja se događa

when we actually represent something.

kad zapravo prikazujemo nešto.

Here you see just lines with different widths.

Ovdje vidite samo crte različite širine.

They stand for numbers for a particular book.

Svaki označava broj određene knjige.

And I can actually recommend this book, it's a very nice book.

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,

U redu, napravimo eksperiment,

just to play around with some straight lines.

kako bismo se igrali s ravnim crtama.

This is a straight line.

Ovo je ravna crta.

Let's make another one.

Napravimo još jednu.

So every time we move, we move one down and one across,

Svakim novim potezom stavimo jednu dolje i jednu preko,

and we draw a new straight line, right?

i nacrtamo novu ravnu crtu, u redu?

We do this over and over and over,

Činimo ovo iznova i iznova,

and we look for patterns.

i tražimo uzorke.

So this pattern emerges,

Javlja se ovaj uzorak,

and it's a rather nice pattern.

i baš je lijep uzorak.

It looks like a curve, right?

Izgleda kao krivulja, zar ne?

Just from drawing simple, straight lines.

Samo od crtanja jednostavnih, ravnih crta.

Now I can change my perspective a little bit. I can rotate it.

Sad mogu promijeniti malo perspektivu. Mogu okrenuti crtu.

Have a look at the curve.

Pogledajte krivulju.

What does it look like?

Na što sliči?

Is it a part of a circle?

Je li dio kruga?

It's actually not a part of a circle.

Zapravo nije dio kruga.

So I have to continue my investigation and look for the true pattern.

Moram nastaviti istraživanje i potražiti pravi uzorak.

Perhaps if I copy it and make some art?

Možda ako je kopiram i pretvorim u umjetnost?

Well, no.

Pa i ne.

Perhaps I should extend the lines like this,

Možda bih trebao produljiti crte ovako,

and look for the pattern there.

i potražiti uzorak u tome.

Let's make more lines.

Napravimo još crta.

We do this.

Učinimo ovo.

And then let's zoom out and change our perspective again.

Odmaknimo se i ponovno promijenimo perspektivu.

Then we can actually see that what started out as just straight lines

Tada zapravo vidimo da ono što je počelo kao ravna crta

is actually a curve called a parabola.

zapravo je krivulja zvana parabola.

This is represented by a simple equation,

Prikazana je jednostavnom jednadžbom,

and it's a beautiful pattern.

i to je prekrasan uzorak.

So this is the stuff that we do.

Dakle ovo radimo.

We find patterns, and we represent them.

Pronalazimo uzorke i predstavljamo ih.

And I think this is a nice day-to-day definition.

Mislim da je to lijepa svakodnevna definicija.

But today I want to go a little bit deeper,

Ali danas želim ići malo dublje,

and think about what the nature of this is.

i razmisliti o prirodi ovoga.

What makes it possible?

Što omogućuje ovo?

There's one thing that's a little bit deeper,

Jedna stvar je malo dublja,

and that has to do with the ability to change your perspective.

i ima veze sa sposobnošću da promijenite perspektivu.

And I claim that when you change your perspective,

I tvrdim da kad promijenite perspektivu,

and if you take another point of view,

i ako pogledate s druge točke gledišta

you learn something new about what you are watching

naučit ćete nešto novo o tome što promatrate,

or looking at or hearing.

gledate ili slušate.

And I think this is a really important thing that we do all the time.

Mislim da je ovo što stalno radimo zaista važno.

So let's just look at this simple equation,

Zato pogledajmo ovu jednostavnu jednadžbu.

x + x = 2 • x.

x + x = 2 • x

This is a very nice pattern, and it's true,

Ovo je vrlo lijep uzorak, i točan je,

because 5 + 5 = 2 • 5, etc.

jer je 5 + 5 = 2 • 5, itd.

We've seen this over and over, and we represent it like this.

Vidjeli smo ovo već, i prikazujemo to ovako.

But think about it: this is an equation.

Ali razmislite o tome: ovo je jednadžba.

It says that something is equal to something else,

Kaže da je nešto jednako nečem drugom,

and that's two different perspectives.

i to su dvije različite perspektive.

One perspective is, it's a sum.

Jedna perspektiva je, to je zbroj.

It's something you plus together.

To je nešto što zbrojite zajedno.

On the other hand, it's a multiplication,

U drugu ruku, to je množenje,

and those are two different perspectives.

i to su dvije različite perspektive.

And I would go as far as to say that every equation is like this,

I usuđujem se reći da je svaka jednadžba ovakva,

every mathematical equation where you use that equality sign

svaka matematička jednadžba u kojoj možete koristiti znak jednakosti

is actually a metaphor.

je zapravo metafora.

It's an analogy between two things.

Ona je analogija između dvije stvari.

You're just viewing something and taking two different points of view,

Gledate nešto, zauzimate dva stajališta,

and you're expressing that in a language.

i izražavate to jezikom.

Have a look at this equation.

Pogledajte ovu jednadžbu.

This is one of the most beautiful equations.

Ovo je jedna od najljepših jednadžbi.

It simply says that, well,

Jednostavno kaže da, pa,

two things, they're both -1.

dvije stvari, one su obje -1.

This thing on the left-hand side is -1, and the other one is.

Ovo na lijevoj strani je -1, a i ovo na drugoj.

And that, I think, is one of the essential parts

I to je, mislim, jedan od najvažnijih dijelova

of mathematics -- you take different points of view.

matematike -- zauzimaš različite točke gledišta.

So let's just play around.

Ajmo se malo igrati.

Let's take a number.

Odaberimo neki broj.

We know four-thirds. We know what four-thirds is.

Znamo za četiri trećine. Znamo što je četiri trećine.

It's 1.333, but we have to have those three dots,

To je 1.333, ali moramo imati te tri točke,

otherwise it's not exactly four-thirds.

inače nije točno četiri trećine.

But this is only in base 10.

Ali to je samo na bazi 10.

You know, the number system, we use 10 digits.

Mi koristimo brojevni sustav od 10 znamenki.

If we change that around and only use two digits,

Ako za promjenu koristimo samo dvije znamenke

that's called the binary system.

to nazivamo binarni sustav.

It's written like this.

Piše se ovako.

So we're now talking about the number.

Dakle sad govorimo o broju.

The number is four-thirds.

Broj je četiri trećine.

We can write it like this,

Možemo ga napisati ovako,

and we can change the base, change the number of digits,

i možemo promijeniti bazu, promijeniti broj znamenki,

and we can write it differently.

i možemo ga napisati drugačije.

So these are all representations of the same number.

Ovo su sve prikazi istog broja.

We can even write it simply, like 1.3 or 1.6.

Možemo ga napisati jednostavno, kao 1.3 ili 1.6.

It all depends on how many digits you have.

Sve ovisi o tome koliko znamenki imate.

Or perhaps we just simplify and write it like this.

Ili da pojednostavimo i napišemo ovako.

I like this one, because this says four divided by three.

Sviđa mi se ovako, jer kaže da je četiri podijeljeno na tri.

And this number expresses a relation between two numbers.

A ovaj broj izražava vezu između dva broja.

You have four on the one hand and three on the other.

Imate četiri na jednoj strani i tri na drugoj.

And you can visualize this in many ways.

Možete to vizualizirati na puno načina.

What I'm doing now is viewing that number from different perspectives.

Sada upravo gledam taj broj iz različitih perspektiva.

I'm playing around.

Igram se.

I'm playing around with how we view something,

Igram se načinom na koji vidimo nešto,

and I'm doing it very deliberately.

i radim to vrlo svojevoljno.

We can take a grid.

Možemo uzeti mrežu.

If it's four across and three up, this line equals five, always.

Ako su dimenzije 4 x 3, ova crta je 5, uvijek.

It has to be like this. This is a beautiful pattern.

Mora biti ovako. Ovo je prekrasan uzorak.

Four and three and five.

Četiri, tri i pet.

And this rectangle, which is 4 x 3,

I ovaj pravokutnik koji je 4 x 3,

you've seen a lot of times.

vidjeli ste puno puta.

This is your average computer screen.

Ovo je vaš prosječni ekran računala.

800 x 600 or 1,600 x 1,200

800 x 600, ili 1600 x 1200

is a television or a computer screen.

je televizija ili ekran računala.

So these are all nice representations,

Ovo su sve lijepi prikazi,

but I want to go a little bit further and just play more with this number.

ali želim ići dalje i još se igrati ovim brojem.

Here you see two circles. I'm going to rotate them like this.

Ovdje vidite dva kruga. Okretat ću ih ovako.

Observe the upper-left one.

Promatrajte ovaj gore-lijevo.

It goes a little bit faster, right?

Ide malo brže, zar ne?

You can see this.

Možete to vidjeti.

It actually goes exactly four-thirds as fast.

Zapravo ide točno četiri trećine brzine drugog kruga.

That means that when it goes around four times,

To znači da kad se on okrene četiri puta,

the other one goes around three times.

drugi se okrene tri puta.

Now let's make two lines, and draw this dot where the lines meet.

Napravimo sada dvije crte, i stavimo točku gdje se crte spajaju.

We get this dot dancing around.

Dobijemo da ova točka pleše okolo.

(Laughter)

(Smijeh)

And this dot comes from that number.

I ova točka proizlazi iz tog broja.

Right? Now we should trace it.

Zar ne? Sad bismo je trebali pratiti.

Let's trace it and see what happens.

Pratimo je i pogledajmo što će se dogoditi.

This is what mathematics is all about.

Ovo je najvažnije u matematici.

It's about seeing what happens.

Važno je vidjeti što će se dogoditi.

And this emerges from four-thirds.

A ovo proizlazi iz četiri trećine.

I like to say that this is the image of four-thirds.

Volim reći da je ovo slika četiri trećine.

It's much nicer -- (Cheers)

Puno je ljepša -- (Klicanje)

Thank you!

Hvala.

(Applause)

(Pljesak)

This is not new.

Ovo nije novost.

This has been known for a long time, but --

Ovo je poznato već dugo, ali --

(Laughter)

(Smijeh)

But this is four-thirds.

Ali ovo su četiri trećine.

Let's do another experiment.

Napravimo drugi eksperiment.

Let's now take a sound, this sound: (Beep)

Sada uzmimo zvuk, ovaj zvuk: (Pištanje)

This is a perfect A, 440Hz.

Ovo je savršeni ton A, 440 Hz.

Let's multiply it by two.

Pomnožimo ga s dva.

We get this sound. (Beep)

Dobijemo ovo. (Pištanje)

When we play them together, it sounds like this.

Kad ih pustimo zajedno, zvuče ovako.

This is an octave, right?

Ovo je oktava, zar ne?

We can do this game. We can play a sound, play the same A.

Možemo igrati ovu igru. Pustimo zvuk, pustimo isti A.

We can multiply it by three-halves.

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.

Zvuče baš lijepo zajedno.

Let's multiply this sound by four-thirds. (Beep)

Pomnožimo taj zvuk s četiri trećine. (Pištanje)

What happens?

Što se dogodi?

You get this sound. (Beep)

Dobijemo ovaj zvuk. (Pištanje)

This is the perfect fourth.

Ovo je čista kvarta.

If the first one is an A, this is a D.

Ako je prvi ton A, ovo je D.

They sound like this together. (Beeps)

Ovako zvuče zajedno. (Pištanje)

This is the sound of four-thirds.

Ovo je zvuk četiri trećine.

What I'm doing now, I'm changing my perspective.

Ovo što sad radim je mijenjanje perspektive.

I'm just viewing a number from another perspective.

Samo gledam broj iz druge perspektive.

I can even do this with rhythms, right?

Ovo čak mogu raditi s ritmovima, zar ne?

I can take a rhythm and play three beats at one time (Drumbeats)

Mogu odabrati ritam i pustiti tri takta istovremeno (Bubnjevi)

in a period of time,

u određenom vremenskom razdoblju

and I can play another sound four times in that same space.

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?

Možete li ovo čuti?

So, this is the sound of four-thirds.

Dakle, ovo je zvuk četiri trećine.

Again, this is as a rhythm.

Opet, ovo je njezin ritam.

(Drumbeats and cowbell)

(Bubnjevi i kravlje zvono)

And I can keep doing this and play games with this number.

Mogu nastaviti ovako i igrati se ovim brojem.

Four-thirds is a really great number. I love four-thirds!

Četiri trećine je odličan broj. Volim četiri trećine!

(Laughter)

(Smijeh)

Truly -- it's an undervalued number.

Zaista -- to je podcijenjen broj.

So if you take a sphere and look at the volume of the sphere,

Ako uzmete kuglu i pogledate njezin volumen,

it's actually four-thirds of some particular cylinder.

on je zapravo četiri trećine određenog valjka.

So four-thirds is in the sphere. It's the volume of the sphere.

Dakle, četiri trećine je u kugli. To je volumen kugle.

OK, so why am I doing all this?

OK, zašto radim sve ovo?

Well, I want to talk about what it means to understand something

Pa, želim govoriti o tome što znači razumjeti nešto

and what we mean by understanding something.

i što mislimo pod razumijevanjem nečega.

That's my aim here.

To je moj cilj ovdje.

And my claim is that you understand something

I tvrdim da razumijete nešto

if you have the ability to view it from different perspectives.

ako to možete pogledati iz različitih perspektiva.

Let's look at this letter. It's a beautiful R, right?

Pogledajmo ovo slovo. To je prekrasno R, zar ne?

How do you know that?

Kako to znate?

Well, as a matter of fact, you've seen a bunch of R's,

Zapravo, vidjeli ste puno slova R,

and you've generalized

i generalizirali ste

and abstracted all of these and found a pattern.

i usvojili sve njih te ste pronašli uzorak.

So you know that this is an R.

Zato znate da je ovo R.

So what I'm aiming for here is saying something

Ovime pokušavam reći nešto

about how understanding and changing your perspective

o tome kako su razumijevanje i promjena perspektive

are linked.

povezani.

And I'm a teacher and a lecturer,

Ja sam učitelj i predavač,

and I can actually use this to teach something,

i mogu zaista iskoristiti ovo za učenje

because when I give someone else another story, a metaphor, an analogy,

jer kad pričam nekome drugu priču, metaforu, analogiju,

if I tell a story from a different point of view,

ako pričam priču s drugog gledišta,

I enable understanding.

omogućujem razumijevanje.

I make understanding possible,

Ja stvaram uvjete za razumijevanje,

because you have to generalize over everything you see and hear,

jer morate generalizirati sve što vidite i čujete,

and if I give you another perspective, that will become easier for you.

a ako vam dam drugu perspektivu, bit će vam lakše.

Let's do a simple example again.

Napravimo opet jednostavan primjer.

This is four and three. This is four triangles.

Ovo su četiri i tri. Ovo su četiri trokuta.

So this is also four-thirds, in a way.

Dakle i to su četiri trećine na neki način.

Let's just join them together.

Spojimo ih.

Now we're going to play a game; we're going to fold it up

Sad ćemo igrati igru, savit ćemo ih

into a three-dimensional structure.

u trodimenzionalnu strukturu.

I love this.

Volim ovo.

This is a square pyramid.

Ovo je četverostrana piramida.

And let's just take two of them and put them together.

Uzmimo dvije i spojimo ih.

So this is what is called an octahedron.

Ovo se zove oktaedar.

It's one of the five platonic solids.

On je jedan od pet pravilnih poliedara.

Now we can quite literally change our perspective,

Sad možemo posve doslovno promijeniti perspektivu,

because we can rotate it around all of the axes

jer ga možemo rotirati oko svih osi

and view it from different perspectives.

i pogledati iz različitih perspektiva.

And I can change the axis,

Mogu i promijeniti os,

and then I can view it from another point of view,

i tada ga pogledati s različitog gledišta,

but it's the same thing, but it looks a little different.

ali to je isto, iako izgleda malo drugačije.

I can do it even one more time.

Mogu to učiniti čak još jednom.

Every time I do this, something else appears,

Svaki put kad to učinim, nešto drugo se pojavi,

so I'm actually learning more about the object

pa zapravo učim više o predmetu

when I change my perspective.

kad promijenim perspektivu.

I can use this as a tool for creating understanding.

Mogu to koristiti kao alat za stvaranje razumijevanja.

I can take two of these and put them together like this

Mogu uzeti dva ovakva i spojiti ih zajedno, ovako

and see what happens.

i vidjeti što će se dogoditi.

And it looks a little bit like the octahedron.

I pomalo izgleda kao oktaedar.

Have a look at it if I spin it around like this.

Pogledajte ga kad ga zavrtim.

What happens?

Što se dogodi?

Well, if you take two of these, join them together and spin it around,

Ako uzmete dva, spojite ih i zavrtite,

there's your octahedron again,

opet dobijete oktaedar,

a beautiful structure.

prekrasnu strukturu.

If you lay it out flat on the floor,

Ako ga rasklopite,

this is the octahedron.

ovo je oktaedar.

This is the graph structure of an octahedron.

Ovo je grafički prikaz strukture oktaedra.

And I can continue doing this.

I mogu to nastaviti raditi.

You can draw three great circles around the octahedron,

Možete nacrtati tri velika kruga oko oktaedra

and you rotate around,

i kad ga okrećete

so actually three great circles is related to the octahedron.

zapravo su tri velika kruga s njim povezana.

And if I take a bicycle pump and just pump it up,

Ako uzmem pumpu za bicik i napumpam ga,

you can see that this is also a little bit like the octahedron.

vidjet ćete da je i ovo pomalo nalik oktaedru.

Do you see what I'm doing here?

Vidite li što radim?

I am changing the perspective every time.

Svaki put mijenjam perspektivu.

So let's now take a step back --

Učinite odmak,

and that's actually a metaphor, stepping back --

i to je zapravo metafora, učiniti odmak --

and have a look at what we're doing.

i pogledajte što radimo.

I'm playing around with metaphors.

Igram se metaforama.

I'm playing around with perspectives and analogies.

Igram se s perspektivama i analogijama.

I'm telling one story in different ways.

Pričam jednu priču na različite načine.

I'm telling stories.

Pričam priče.

I'm making a narrative; I'm making several narratives.

Pišem pripovijest. Pišem nekoliko pripovijesti.

And I think all of these things make understanding possible.

Mislim da sve ovo omogućuje razumijevanje.

I think this actually is the essence of understanding something.

Mislim da je ovo srž razumijevanja nečega.

I truly believe this.

Zaista to vjerujem.

So this thing about changing your perspective --

Tako da je ovo o mijenjanju perspektive --

it's absolutely fundamental for humans.

apsolutni temelj za ljude.

Let's play around with the Earth.

Igrajmo se sa Zemljom.

Let's zoom into the ocean, have a look at the ocean.

Pogledajmo izbliza ocean, pogledajmo ocean.

We can do this with anything.

Možemo ovo učiniti sa svime.

We can take the ocean and view it up close.

Možemo ocean pogledati izbliza.

We can look at the waves.

Možemo pogledati valove.

We can go to the beach.

Možemo ići na plažu.

We can view the ocean from another perspective.

Možemo pogledati ocean iz druge perspektive.

Every time we do this, we learn a little bit more about the ocean.

Svaki put kad to učinimo, naučimo još nešto o oceanu.

If we go to the shore, we can kind of smell it, right?

Ako odemo na obalu možemo je namirisati, zar ne?

We can hear the sound of the waves.

Možemo čuti zvuk valova.

We can feel salt on our tongues.

Možemo osjetiti sol na jeziku.

So all of these are different perspectives.

Sve to su različite perspektive.

And this is the best one.

A ovo je najbolja.

We can go into the water.

Možemo ući u vodu.

We can see the water from the inside.

Možemo vidjeti vodu iznutra.

And you know what?

I znate što?

This is absolutely essential in mathematics and computer science.

Ovo je posve neophodno u matematici i informatici.

If you're able to view a structure from the inside,

Ako možete vidjeti strukturu iznutra,

then you really learn something about it.

tada zaista naučite nešto o njoj.

That's somehow the essence of something.

To je nekako jezgra nečega.

So when we do this, and we've taken this journey

Kad radimo ovo, i putujemo

into the ocean,

u ocean,

we use our imagination.

koristimo našu maštu.

And I think this is one level deeper,

Mislim da je ovo jedna razina dublje,

and it's actually a requirement for changing your perspective.

i zapravo je uvijet za mijenjanje perspektive.

We can do a little game.

Možemo se igrati.

You can imagine that you're sitting there.

Možete zamisliti da sjedite ondje.

You can imagine that you're up here, and that you're sitting here.

Možete zamisliti da ste ovdje gore, i da sjedite ovdje.

You can view yourselves from the outside.

Možete se pogledati izvana.

That's really a strange thing.

To je zaista čudno.

You're changing your perspective.

Možete promijeniti svoju perspektivu.

You're using your imagination,

Koristite maštu,

and you're viewing yourself from the outside.

i vidite se izvana.

That requires imagination.

To zahtijeva maštu.

Mathematics and computer science are the most imaginative art forms ever.

Matematika i računarstvo su najmaštovitiji umjetnički izrazi ikada.

And this thing about changing perspectives

I ovo o mijenjanju perspektive

should sound a little bit familiar to you,

bi vam trebalo zvučati poznato,

because we do it every day.

jer to radimo svaki dan.

And then it's called empathy.

A tada to zovemo suosjećanjem

When I view the world from your perspective,

Kad ja gledam svijet iz tvoje perspektive,

I have empathy with you.

ja suosjećam s vama.

If I really, truly understand

Ako ja stvarno, zaista razumijem

what the world looks like from your perspective,

kako svijet izgleda iz vaše perspektive,

I am empathetic.

ja sam suosjećajan.

That requires imagination.

To zahtijeva maštu.

And that is how we obtain understanding.

Tako mi dobivamo razumijevanje.

And this is all over mathematics and this is all over computer science,

To je diljem matematike i to je diljem računarstva,

and there's a really deep connection between empathy and these sciences.

i postoji zaista duboka veza između suosjećanja i ovih znanosti.

So my conclusion is the following:

Stoga je moj zaključak sljedeći:

understanding something really deeply

razumjeti nešto na zaista dubokoj razini

has to do with the ability to change your perspective.

ima veze sa sposobnošću da promijenite perspektivu.

So my advice to you is: try to change your perspective.

Zato je moj savjet vama: pokušajte promijeniti perspektivu.

You can study mathematics.

Možete proučavati matematiku.

It's a wonderful way to train your brain.

To je prekrasan način da vježbate mozak.

Changing your perspective makes your mind more flexible.

Mijenjanje perspektive čini vaš um fleksibilnijim.

It makes you open to new things,

To vas otvara novim stvarima,

and it makes you able to understand things.

i omogućuje vam da razumijete stvari.

And to use yet another metaphor:

Da iskoristim drugu metaforu:

have a mind like water.

neka vam um bude kao voda.

That's nice.

To je lijepo.

Thank you.

Hvala.

(Applause)

(Pljesak)