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

Prevodilac: Vesna Radovic Lektor: Mile Živković

Hi.

Zdravo.

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

Želim da govorim o razumevanju, i prirodi razumevanja,

and what the essence of understanding is,

i o tome šta je bit razumevanja,

because understanding is something we aim for, everyone.

jer je razumevanje nešto čemu svi mi težimo.

We want to understand things.

Želimo da razumemo stvari.

My claim is that understanding has to do

Tvrdim da razumevanje mora biti u vezi

with the ability to change your perspective.

sa sposobnošću da promenimo našu perspektivu.

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

Ako nema toga, nema ni razumevanja.

So that is my claim.

To je moja tvrdnja.

And I want to focus on mathematics.

Želim da se usredsredim na matematiku.

Many of us think of mathematics as addition, subtraction,

Mnogi misle o matematici kao o sabiranju, oduzimanju,

multiplication, division,

množenju, deljenju,

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

razlomcima, procentima, geometriji, algebri - tim stvarima.

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

Ali u stvari, želim takođe da govorim o suštini matematike.

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

Moja tvrdnja je da matematika ima veze sa šablonima.

Behind me, you see a beautiful pattern,

Iza mene, vidite jedan prelep šablon,

and this pattern actually emerges just from drawing circles

i ovaj šablon je zapravo nastao prosto iz crtanja krugova

in a very particular way.

na jedan poseban način.

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

Tako da je moja ustaljena definicija matematike koju koristim svaki dan

is the following:

sledeća:

First of all, it's about finding patterns.

Kao prvo, to je pronalaženje šablona.

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

I pod "šablonima" mislim vezu, strukturu, neku pravilnost,

some rules that govern what we see.

neka pravila koja određuju šta vidimo.

Second of all,

Kao drugo,

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

smatram da je to predstavljanje ovih šema uz pomoć jezika.

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

Mi izmišljamo jezik ako ne postoji,

and in mathematics, this is essential.

a u matematici, ovo je suština.

It's also about making assumptions

To je takođe pravljenje pretpostavki

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

i igranje sa ovim pretpostavkama samo da bismo videli šta će se desiti.

We're going to do that very soon.

To ćemo ubrzo i raditi.

And finally, it's about doing cool stuff.

I konačno, ona je rađenje kul stvari.

Mathematics enables us to do so many things.

Matematika nam omogućava da radimo mnoge stvari.

So let's have a look at these patterns.

Pa hajde da pogledamo ove šablone.

If you want to tie a tie knot,

Ako želite da vežete kravatu,

there are patterns.

tu je šablon,

Tie knots have names.

Čvorovi na kravati imaju ime.

And you can also do the mathematics of tie knots.

A takođe možete i uraditi matematiku čvorova.

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

Ovo je levo-van, desno-u, sredina-van i veži.

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

Ovo je levo-u, desno-van, levo-u, sredina-van i veži.

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

Ovo je jezik koji smo osmislili za šablone vezivanja kravate,

and a half-Windsor is all that.

i polu-Vindzor je isto to.

This is a mathematics book about tying shoelaces

Ovo je knjiga za matematiku o vezivanju pertli

at the university level,

na univerzitetskom nivou,

because there are patterns in shoelaces.

jer postoje šabloni za pertle.

You can do it in so many different ways.

Možete ih vezati na razne načine.

We can analyze it.

Možemo to proanalizirati.

We can make up languages for it.

Možemo da stvorimo jezik za to.

And representations are all over mathematics.

Predstavljanja su svuda u matematici.

This is Leibniz's notation from 1675.

Ovo je Lajbnicova beleška iz 1675.

He invented a language for patterns in nature.

On je izumeo jezik za šablone u prirodi.

When we throw something up in the air,

Kada nešto bacimo u vis,

it falls down.

to pada.

Why?

Zašto?

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

Nismo sigurni, ali možemo to predstaviti sa matematikom šablona.

This is also a pattern.

Ovo je takođe šablon.

This is also an invented language.

Ovo je takođe izmišljen jezik.

Can you guess for what?

Možete li da pogodite za šta?

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

To je u stvari sistem nota za ples, za irski ples.

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

Ovo omogućava njemu kao koreografu da radi kul, nove stvari,

because he has represented it.

zato što ih je predstavio.

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

Žeim da razmislite koliko sjajno predstavljanje može stvarno biti.

Here it says the word "mathematics."

Ovde piše reč "matematika".

But actually, they're just dots, right?

Ali u stvari, to su samo tačke, zar ne?

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

Pa kako onda, zaboga, ove tačke mogu da predstavljaju reč?

Well, they do.

Pa, mogu.

They represent the word "mathematics,"

One predstavljaju reč "matematika",

and these symbols also represent that word

a ovi simboli takođe predstavljaju reč

and this we can listen to.

i nju možemo da slušamo.

It sounds like this.

Zvuči ovako.

(Beeps)

(Zvučni signal)

Somehow these sounds represent the word and the concept.

Nekako ovi zvukovi predstavljaju i reč i koncept.

How does this happen?

Kako se to dešava?

There's something amazing going on about representing stuff.

Postoji nešto sjajno što se dešava pri predstavljanju stvari.

So I want to talk about that magic that happens

Želim da govorim o magiji koja se dešava

when we actually represent something.

kada zapravo predstavljamo nešto.

Here you see just lines with different widths.

Ovde vidite samo linije različitih širina.

They stand for numbers for a particular book.

One predstavljaju brojeve za određenu knjigu.

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

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,

OK, hajde da uradimo jedan eksperiment,

just to play around with some straight lines.

samo da se igramo sa nekim pravim liniijama.

This is a straight line.

Ovo je prava linija.

Let's make another one.

Nacrtajmo još jednu.

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

Pri svakom pokretu crtamo, jednu malo niže i jednu popreko,

and we draw a new straight line, right?

i crtamo novu pravu liniju, tačno?

We do this over and over and over,

Ponavljamo ovo iznova i iznova,

and we look for patterns.

i tražimo šablone.

So this pattern emerges,

Izdvaja se ovaj šablon,

and it's a rather nice pattern.

i prilično je lep.

It looks like a curve, right?

Izgleda kao jedna kriva, zar ne?

Just from drawing simple, straight lines.

Iz prostog crtanja jednostavnih, pravih linija.

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

Sad mogu malo da promenim perspektivu. Mogu da rotiram ovo.

Have a look at the curve.

Pogledajte krivu.

What does it look like?

Kako vam izgleda?

Is it a part of a circle?

Je li to deo kruga?

It's actually not a part of a circle.

U stvari, nije deo kruga.

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

Moram da nastavim sa istragom i tražim pravi šablon.

Perhaps if I copy it and make some art?

Možda ako ovo prekopiram i dodam malo umetnoti?

Well, no.

Ipak ne.

Perhaps I should extend the lines like this,

Možda da produžim linije, ovako,

and look for the pattern there.

i pogledajte ovaj šablon ovde.

Let's make more lines.

Nacrtajmo još linija.

We do this.

Uradimo ovo.

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

I hajde da smanjimo zum i ponovo promenimo perspektivu.

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

Onda zapravo vidimo da ono što su u početku bile samo prave linije,

is actually a curve called a parabola.

su sada u stvari krive, parabole.

This is represented by a simple equation,

Ovo se može predstaviti jedostavnom jednačinom,

and it's a beautiful pattern.

a i to je lep šablon.

So this is the stuff that we do.

To je ono što mi radimo.

We find patterns, and we represent them.

Pronalazimo šablone i predstavljamo ih.

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

Mislim da je to fina, jednostavna definicija.

But today I want to go a little bit deeper,

Ali danas želim da malo produbim temu,

and think about what the nature of this is.

i razmislim o prirodi svega ovoga.

What makes it possible?

Zbog čega je ovo moguće?

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

Postoji jedna stvar koja je malo dublja,

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

i ima veze sa sposobnošću da promenite ugao gledanja.

And I claim that when you change your perspective,

I tvrdim da kada promenite ugao gledanja,

and if you take another point of view,

i počnete da gledate stvari iz drugog ugla,

you learn something new about what you are watching

naučite nešto novo o onome što posmatrate

or looking at or hearing.

ili gledate ili čujete.

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

Smatram da je stvarno važno da to radimo sve vreme.

So let's just look at this simple equation,

Pa hajde da pogledamo ovu jednostavnu jednačinu,

x + x = 2 • x.

x + x = 2 • x.

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

Ovo je veoma lep šablon, i tačan je,

because 5 + 5 = 2 • 5, etc.

zato što je 5 + 5 = 2 • 5, itd.

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

Videli smo ovo mnogo puta, i predstavljamo to ovako.

But think about it: this is an equation.

Ali razmislite: ovo je jednačina.

It says that something is equal to something else,

Kaže da je nešto jednako nečemu drugom,

and that's two different perspectives.

a to su dve različite perspektive.

One perspective is, it's a sum.

Jedna perspektiva je zbir.

It's something you plus together.

Dodajete nešto nečemu.

On the other hand, it's a multiplication,

Sa druge strane, imamo množenje,

and those are two different perspectives.

i to su dve različite perspektive.

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

I išao bih toliko daleko da kažem da je svaka jednačina ovakva,

every mathematical equation where you use that equality sign

svaka matematička jednačina gde koristite znak plus

is actually a metaphor.

je zapravo metafora.

It's an analogy between two things.

To je analogija između dve stvari.

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

Prosto posmatrate nešto i imate dva ugla gledanja,

and you're expressing that in a language.

a to izražavate jezikom.

Have a look at this equation.

Pogledajte ovu jednačinu.

This is one of the most beautiful equations.

Ovo je jedna od najlepših jednačina.

It simply says that, well,

Ona jednostavno kaže

two things, they're both -1.

ove dve stvari obe imaju rešenje -1.

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

Ovo sa leve strane jednako je -1, kao i ovo sa druge.

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

I smatram da je to jedan od suštinskih delova

of mathematics -- you take different points of view.

matematike - imate različite uglove gledanja.

So let's just play around.

Hajde malo da se igramo.

Let's take a number.

Uzmimo jedan broj.

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

Znamo četiri trećine. Znamo koliko je četiri trećine.

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

1.333, ali moramo da imamo ove tri tačke,

otherwise it's not exactly four-thirds.

inače to nisu tačno četiri trećine.

But this is only in base 10.

Ali to je samo za osnovu 10.

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

Znate, numerički sistem, uzimamo 10 cifri.

If we change that around and only use two digits,

Ako to preokrenemo i uzmemo samo dve cifre,

that's called the binary system.

to se zove binarni sistem.

It's written like this.

Piše se ovako.

So we're now talking about the number.

Sada pričamo o ovom broju.

The number is four-thirds.

Taj 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 promeniti osnovu, promeniti broj cifara,

and we can write it differently.

i možemo ga napisati drugačije.

So these are all representations of the same number.

Sve ovo su reprezentacije 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 zavisi od toga koliko cifara imate.

Or perhaps we just simplify and write it like this.

Ili možda možemo da pojednostavimo i napišemo ga ovako.

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

Sviđa mi se ovako, zato što kaže četiri podeljeno sa tri.

And this number expresses a relation between two numbers.

I ovaj broj izražava odnos između dva broja.

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

Sa jedne strane imate četiri i tri sa druge.

And you can visualize this in many ways.

I možete zamisliti ovo na više načina.

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

Sada posmatram taj broj iz drugačije perspektive.

I'm playing around.

Igram se.

I'm playing around with how we view something,

Igram se sa tim kako na nešto gledamo,

and I'm doing it very deliberately.

i radim to namerno.

We can take a grid.

Možemo uzeti rešetku.

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

To je četiri udesno i tri gore, ova linija je uvek pet.

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

Mora da bude ovako. Ovo je prelepi šablon.

Four and three and five.

Četiri i tri i pet.

And this rectangle, which is 4 x 3,

A ovaj pravougaonik, koji je 4 x 3,

you've seen a lot of times.

videli ste mnogo puta.

This is your average computer screen.

Ovo je prosečan ekran računara,

800 x 600 or 1,600 x 1,200

800 x 600 ili 1600 x 1200

is a television or a computer screen.

je ekran TV-a ili računara.

So these are all nice representations,

Ovo su sve lepe reprezentacije,

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

ali želim da idem malo dalje i malo više da se igram sa brojevima.

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

Ovde vidite dva kruga. Rotiraću ih ovako.

Observe the upper-left one.

Gledajte ovaj u gornjem levom uglu.

It goes a little bit faster, right?

Kreće se malo brže, zar ne?

You can see this.

To može da se vidi.

It actually goes exactly four-thirds as fast.

Zapravo kreće se tačno brzinom od četiri trećine.

That means that when it goes around four times,

To znači da kada se kreće okolo četiri puta,

the other one goes around three times.

drugi to radi tri puta.

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

Hajde sada da nacrtamo dve linije i tačku tamo gde se linije spajaju.

We get this dot dancing around.

Dobijamo ovaj ples.

(Laughter)

(Smeh)

And this dot comes from that number.

I ova tačka dolazi od tog broja.

Right? Now we should trace it.

Tačno? Sada treba da joj uđemo u trag.

Let's trace it and see what happens.

Hajde da ga nađemo i vidimo šta će se dogoditi.

This is what mathematics is all about.

O tome se radi u matematici.

It's about seeing what happens.

O gledanju onog što se dešava.

And this emerges from four-thirds.

I ovo smo dobill od četiri trećine.

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

Volim da kažem da je ovo slika četiri trećine.

It's much nicer -- (Cheers)

Lepše je - (Oduševljenje)

Thank you!

Hvala vam!

(Applause)

(Aplauz)

This is not new.

Ovo nije novo.

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

Za ovo se zna odavno, ali -

(Laughter)

(Smeh)

But this is four-thirds.

Ali ovo su četiri trećine.

Let's do another experiment.

Da uradimo još jedan eksperiment.

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

Da uzmemo neki zvuk, ovaj zvuk: (Zvučni signal)

This is a perfect A, 440Hz.

Ovo je savršeno A, 440Hz.

Let's multiply it by two.

Da pomnožimo to sa dva.

We get this sound. (Beep)

Dobijamo ovaj zvuk. (Zvučni signal)

When we play them together, it sounds like this.

Kada 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 ovako da se igramo. Pustimo zvuk, čujemo isto A.

We can multiply it by three-halves.

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.

Zvuče stvarno lepo zajedno.

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

Pomnožimo ovaj zvuk sa četiri trećine. (Zvučni signal)

What happens?

Šta se dešava?

You get this sound. (Beep)

Dobijamo ovaj zvuk. (Signal)

This is the perfect fourth.

Ovo je savršena četvrtina.

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

Ako je prvi A, onda je ovo D.

They sound like this together. (Beeps)

Ovako zvuče zajedno. (Zvučni signal)

This is the sound of four-thirds.

Ovo je zvuk četiri trećine.

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

Ono što radim sada je da menjam ugao geldanja.

I'm just viewing a number from another perspective.

Samo gledam broj iz drugačije perspektive.

I can even do this with rhythms, right?

Ovo mogu da radim i sa ritmovima, zar ne?

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

Mogu uzeti jedan ritam i svirati tri taktaistovremeno (Bubnjevi)

in a period of time,

neko vreme,

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

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?

Čujete li ovo?

So, this is the sound of four-thirds.

Pa, ovo je zvuk četiri trećine.

Again, this is as a rhythm.

Opet, ovo je neki ritam.

(Drumbeats and cowbell)

(Bubnjevi i medenica)

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

I mogu da nastavim ovako da se igram sa ovim brojem.

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

Četiri trećine je stvarno sjajan broj. Obožavam ga!

(Laughter)

(Smeh)

Truly -- it's an undervalued number.

Zaista - to je potcenjen broj.

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

Tako ako uzmete loptu i pogledate njenu zapreminu,

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

imate u stvari četiri trećine nekog određenog cilindra.

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

Tako da su četiri trećine u lopti. One su u zapremini lopte.

OK, so why am I doing all this?

OK, zašto ja uopšte ovo radim?

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

Pa, želim da govorim o tome šta znači razumeti nešto

and what we mean by understanding something.

i šta smatramo pod tim.

That's my aim here.

To je moj cilj ovde.

And my claim is that you understand something

A moja tvrdnja je da razumemo nešto

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

ako imamo sposobnost da to glesamo iz različitih perspektiva.

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

Pogledajmo ovo slovo. Jedno lepo slovo R, zar ne?

How do you know that?

Kako to znamo?

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

Zapravo, videli smo milion puta slovo R,

and you've generalized

i generalizovali smo

and abstracted all of these and found a pattern.

i rezimirali sve ovo i pronašli šablon.

So you know that this is an R.

Tako da znamo da je ovo slovo R.

So what I'm aiming for here is saying something

Ono na šta ciljam je da kažem nešto

about how understanding and changing your perspective

o tome kako su razumevanje i promena perspektive

are linked.

povezani.

And I'm a teacher and a lecturer,

Ja sam nastavnik i predavač,

and I can actually use this to teach something,

i mogu u stvari da iskoristim ovo da bih predavao

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

jer kad nekom drugom dam drugu priču, metaforu, analogiju,

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

ako ispričam priču sa drugačijeg stanovišta,

I enable understanding.

ja omogućavam razumevanje.

I make understanding possible,

Činim razumevanje mogućim

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

jer morate generalizovati sve što vidite i čujete,

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

i ako vam dam drugačiju perspektivu, biće vam lakše.

Let's do a simple example again.

Da uradimo ponovo jedan primer.

This is four and three. This is four triangles.

Ovo su četiri i tri. Ovo su četiri trougla.

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

Tako da je ovo takođe na neki način četiri trećine.

Let's just join them together.

Hajde da ih jednostavno povežemo.

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

Sada ćemo da igramo jednu igru: savićemo ih

into a three-dimensional structure.

u trodimenzionalnu strukturu.

I love this.

Obožavam ovo.

This is a square pyramid.

Ovo je četvrtasta piramida.

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

Hajde da uzmemo dve i sastavimo ih.

So this is what is called an octahedron.

I ovo je ono što se zove oktahedron.

It's one of the five platonic solids.

Jedan je od pet platonskih čvrstih tela.

Now we can quite literally change our perspective,

Sada i bukvalno možemo promeniti našu perspektivu,

because we can rotate it around all of the axes

jer ga možemo rotirati oko svih osa

and view it from different perspectives.

i posmatrati iz različitih perspektiva.

And I can change the axis,

Mogu i da promenim ovu osu,

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

i onda mogu da posmatram iz drugačije perspekte,

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

ali je to ista stvar samo što izgleda drugačije.

I can do it even one more time.

Mogu to da uradim još jedanput.

Every time I do this, something else appears,

Svaki put kada ovo uradim, nešto se drugo pojavi,

so I'm actually learning more about the object

tako da zapravo učim sve više o ovom predmetu

when I change my perspective.

kada promenim perspektivu.

I can use this as a tool for creating understanding.

Mogu da ga koristim kao alat za stvaranje razumevanja.

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

Mogu da uzmem dva komada i sastavim ih ovako

and see what happens.

i vidite šta se dešava.

And it looks a little bit like the octahedron.

Malo i liči na oktahedron.

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

Pogledajte ako ga okrenem ovako.

What happens?

Šta se dešava?

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

Pa, ako uzmete dva tela, spojite ih i okrenete,

there's your octahedron again,

evo ga oktahedron opet,

a beautiful structure.

prelepa struktura.

If you lay it out flat on the floor,

Ako stavite pravo na pod,

this is the octahedron.

ovo je oktahedron.

This is the graph structure of an octahedron.

Ovo je grafička struktura oktahedrona.

And I can continue doing this.

I mogu ovako do sutra.

You can draw three great circles around the octahedron,

Možete da nacrtate tri velika kruga oko oktahedrona,

and you rotate around,

i da ih zarotirate,

so actually three great circles is related to the octahedron.

i onda su u stvari tri velika kruga povezana sa oktahedronom.

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

I ako uzmem pumpu za bicikl i prosto ih naduvam,

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

možete opet videti da malo podseća na oktahedron.

Do you see what I'm doing here?

Da li vidite šta radim?

I am changing the perspective every time.

Menjam perspektivu svaki put.

So let's now take a step back --

Sad ćemo malo da se vratimo unazad -

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

i to je u stvari metafora, vraćanje unazad -

and have a look at what we're doing.

i da pogledamo šta smo uradili.

I'm playing around with metaphors.

Poigravam se sa metaforama.

I'm playing around with perspectives and analogies.

Poigravam se sa perspektivama i analogijama.

I'm telling one story in different ways.

Prenosim jednu priču na razne načine.

I'm telling stories.

Pričam priče.

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

Pravim narativ. Pravm nekoliko narativa.

And I think all of these things make understanding possible.

Smatram da je zbog svega ovog razumevanje moguće.

I think this actually is the essence of understanding something.

Smatram da je ovo zapravo suština razumevanja nečega.

I truly believe this.

Istinski verujem u to.

So this thing about changing your perspective --

Tako da je ovo sa menjanjem perspektive

it's absolutely fundamental for humans.

apsolutno neophodno za čovečanstvo.

Let's play around with the Earth.

Hajde da se igramo sa Zemljom.

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

Zumirajmo okean, pogledajte okean.

We can do this with anything.

Ovo možemo da radimo sa svim stvarima.

We can take the ocean and view it up close.

Možemo da uzmemo okean i posmatramo ga izbliza.

We can look at the waves.

Možemo da gledamo talase.

We can go to the beach.

Možemo da odemo do plaže.

We can view the ocean from another perspective.

Možemo da posmatramo okean iz druge perspektive.

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

Svaki put kad to uradimo, naučimo nešto više o okeanu.

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

Ako odemo do obale, možemo da osetimo miris, zar ne?

We can hear the sound of the waves.

Možemo da čujemo zvuk talasa.

We can feel salt on our tongues.

Možemo da osetimo so na jeziku.

So all of these are different perspectives.

Sve su to drugačije perspektive.

And this is the best one.

A ovo je najbolja.

We can go into the water.

Možemo da uđemo u vodu.

We can see the water from the inside.

Možemo da vidimo vodu iznutra.

And you know what?

I znate šta?

This is absolutely essential in mathematics and computer science.

Ovo je apsolutno neophodno u matematici i informatici.

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

Ako možete da vidite strukturu iznutra,

then you really learn something about it.

onda ste je stvarno savladali.

That's somehow the essence of something.

To je nekako suština svačega.

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

Tako kada uradimo ovo, i kada otputujemo

into the ocean,

u okean,

we use our imagination.

koristimo našu maštu.

And I think this is one level deeper,

A mislim da je to još jedan korak dalje,

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

i zapravo je neophodan da bismo promenili perspektivuu.

We can do a little game.

Odigraćemo jednu igru.

You can imagine that you're sitting there.

Zamislite da sedite tamo.

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

Zamislite da ste ovde i da sedite tu.

You can view yourselves from the outside.

Možete videti sebe spolja.

That's really a strange thing.

To je tek čudno.

You're changing your perspective.

Menjate perspektivu.

You're using your imagination,

Koristite maštu,

and you're viewing yourself from the outside.

i posmatrate sebe spolja.

That requires imagination.

Za to je neophodna mašta.

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

Matematika i informatika su najmaštovitije umetničke forme od svih.

And this thing about changing perspectives

I ovo sa menjanjem perspektiva

should sound a little bit familiar to you,

bi trebalo da vam je malo jasnije,

because we do it every day.

jer je to nešto što radimo svaki dan.

And then it's called empathy.

I onda se to zove empatija.

When I view the world from your perspective,

Kada ja posmatram svet iz vašeg ugla,

I have empathy with you.

osećam empatiju sa vama.

If I really, truly understand

Ako stvarno, istinski razumem

what the world looks like from your perspective,

kako svet izgleda iz vaše perspektive,

I am empathetic.

ja sam empatičan.

That requires imagination.

Za to je neophodna mašta.

And that is how we obtain understanding.

Na taj način usvajamo razumevanje.

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

I to je svuda u matematici i informatici

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

i postoji zaista velika povezanost između empatije i ovih nauka.

So my conclusion is the following:

Stoga, moj zaključak je sledeći:

understanding something really deeply

razumevanje nečega istinski

has to do with the ability to change your perspective.

mora da ima veze sa sposobnošću da promenite perspektivu.

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

Moj savet vam je: probajte da promenite perspektivu.

You can study mathematics.

Možete izučavati matematiku.

It's a wonderful way to train your brain.

To je divan način da uvežbate svoj mozak.

Changing your perspective makes your mind more flexible.

Kada menjate perspektivu vaš um postaje fleksibilniji.

It makes you open to new things,

Otvoreniji ste za nove stvari,

and it makes you able to understand things.

i sposobni ste da razumete stvari.

And to use yet another metaphor:

I da upotrebim još jednu metaforu:

have a mind like water.

neka vam um bude poput vode.

That's nice.

To je lepo.

Thank you.

Hvala vam.

(Applause)

(Aplauz)