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Roger Antonsen: Math is the hidden secret to understanding the world
Rodžer Antonsen (Roger Antonsen): Matematika je skrivena tajna za razumevanje sveta
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Otključajte misterije i unutrašnje delanje sveta kroz jedan od najmaštovitijih oblika umetnosti ikada - matematiku - sa Rodžerom Antonsenom, dok on objašnjava kako mala promena u perspektivi može otkriti obrasce, brojeve i formule kao osnov za empatiju i razumevanje.
Roger Antonsen - Logician, mathematician, computer scientist
Roger Antonsen combines science, mathematics and computer science with entertainment, philosophy and visualizations. Full bio
Roger Antonsen combines science, mathematics and computer science with entertainment, philosophy and visualizations. Full bio
Double-click the English transcript below to play the video.
00:13
Hi.
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Zdravo.
00:14
I want to talk about understanding,
and the nature of understanding,
and the nature of understanding,
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Želim da govorim o razumevanju,
i prirodi razumevanja,
i prirodi razumevanja,
00:18
and what the essence of understanding is,
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i o tome šta je bit razumevanja,
00:21
because understanding is something
we aim for, everyone.
we aim for, everyone.
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jer je razumevanje nešto
čemu svi mi težimo.
čemu svi mi težimo.
00:24
We want to understand things.
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Želimo da razumemo stvari.
00:27
My claim is that understanding has to do
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Tvrdim da razumevanje mora biti u vezi
00:30
with the ability to change
your perspective.
your perspective.
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sa sposobnošću da promenimo
našu perspektivu.
našu perspektivu.
00:32
If you don't have that,
you don't have understanding.
you don't have understanding.
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Ako nema toga, nema ni razumevanja.
00:36
So that is my claim.
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To je moja tvrdnja.
00:37
And I want to focus on mathematics.
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Želim da se usredsredim na matematiku.
00:40
Many of us think of mathematics
as addition, subtraction,
as addition, subtraction,
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Mnogi misle o matematici kao o
sabiranju, oduzimanju,
sabiranju, oduzimanju,
00:43
multiplication, division,
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množenju, deljenju,
00:45
fractions, percent, geometry,
algebra -- all that stuff.
algebra -- all that stuff.
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razlomcima, procentima, geometriji,
algebri - tim stvarima.
algebri - tim stvarima.
00:50
But actually, I want to talk
about the essence of mathematics as well.
about the essence of mathematics as well.
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Ali u stvari, želim takođe da
govorim o suštini matematike.
govorim o suštini matematike.
00:53
And my claim is that mathematics
has to do with patterns.
has to do with patterns.
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Moja tvrdnja je da matematika
ima veze sa šablonima.
ima veze sa šablonima.
00:57
Behind me, you see a beautiful pattern,
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Iza mene, vidite jedan prelep šablon,
00:59
and this pattern actually emerges
just from drawing circles
just from drawing circles
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i ovaj šablon je zapravo nastao
prosto iz crtanja krugova
prosto iz crtanja krugova
01:03
in a very particular way.
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na jedan poseban način.
01:05
So my day-to-day definition
of mathematics that I use every day
of mathematics that I use every day
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Tako da je moja ustaljena definicija
matematike koju koristim svaki dan
matematike koju koristim svaki dan
01:10
is the following:
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sledeća:
01:12
First of all, it's about finding patterns.
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Kao prvo, to je pronalaženje šablona.
01:16
And by "pattern," I mean a connection,
a structure, some regularity,
a structure, some regularity,
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I pod "šablonima" mislim vezu,
strukturu, neku pravilnost,
strukturu, neku pravilnost,
01:21
some rules that govern what we see.
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neka pravila koja određuju šta vidimo.
01:24
Second of all,
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Kao drugo,
01:25
I think it is about representing
these patterns with a language.
these patterns with a language.
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smatram da je to predstavljanje
ovih šema uz pomoć jezika.
ovih šema uz pomoć jezika.
01:29
We make up language if we don't have it,
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Mi izmišljamo jezik ako ne postoji,
01:31
and in mathematics, this is essential.
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a u matematici, ovo je suština.
01:35
It's also about making assumptions
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To je takođe pravljenje pretpostavki
01:36
and playing around with these assumptions
and just seeing what happens.
and just seeing what happens.
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i igranje sa ovim pretpostavkama
samo da bismo videli šta će se desiti.
samo da bismo videli šta će se desiti.
01:40
We're going to do that very soon.
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To ćemo ubrzo i raditi.
01:42
And finally, it's about doing cool stuff.
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I konačno, ona je rađenje kul stvari.
01:46
Mathematics enables us
to do so many things.
to do so many things.
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Matematika nam omogućava
da radimo mnoge stvari.
da radimo mnoge stvari.
01:50
So let's have a look at these patterns.
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Pa hajde da pogledamo ove šablone.
01:52
If you want to tie a tie knot,
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Ako želite da vežete kravatu,
01:55
there are patterns.
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tu je šablon,
01:56
Tie knots have names.
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Čvorovi na kravati imaju ime.
01:58
And you can also do
the mathematics of tie knots.
the mathematics of tie knots.
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A takođe možete i uraditi
matematiku čvorova.
matematiku čvorova.
02:00
This is a left-out, right-in,
center-out and tie.
center-out and tie.
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Ovo je levo-van, desno-u,
sredina-van i veži.
sredina-van i veži.
02:04
This is a left-in, right-out,
left-in, center-out and tie.
left-in, center-out and tie.
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Ovo je levo-u, desno-van,
levo-u, sredina-van i veži.
levo-u, sredina-van i veži.
02:08
This is a language we made up
for the patterns of tie knots,
for the patterns of tie knots,
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Ovo je jezik koji smo osmislili
za šablone vezivanja kravate,
za šablone vezivanja kravate,
02:12
and a half-Windsor is all that.
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i polu-Vindzor je isto to.
02:15
This is a mathematics book
about tying shoelaces
about tying shoelaces
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Ovo je knjiga za matematiku
o vezivanju pertli
o vezivanju pertli
02:18
at the university level,
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na univerzitetskom nivou,
02:19
because there are patterns in shoelaces.
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jer postoje šabloni za pertle.
02:21
You can do it in so many different ways.
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Možete ih vezati na razne načine.
02:23
We can analyze it.
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Možemo to proanalizirati.
02:25
We can make up languages for it.
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Možemo da stvorimo jezik za to.
02:28
And representations
are all over mathematics.
are all over mathematics.
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Predstavljanja su svuda u matematici.
02:31
This is Leibniz's notation from 1675.
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Ovo je Lajbnicova beleška iz 1675.
02:35
He invented a language
for patterns in nature.
for patterns in nature.
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On je izumeo jezik za šablone u prirodi.
02:39
When we throw something up in the air,
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Kada nešto bacimo u vis,
02:41
it falls down.
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to pada.
02:42
Why?
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Zašto?
02:43
We're not sure, but we can represent
this with mathematics in a pattern.
this with mathematics in a pattern.
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Nismo sigurni, ali možemo to predstaviti
sa matematikom šablona.
sa matematikom šablona.
02:48
This is also a pattern.
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Ovo je takođe šablon.
02:49
This is also an invented language.
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Ovo je takođe izmišljen jezik.
02:52
Can you guess for what?
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Možete li da pogodite za šta?
02:55
It is actually a notation system
for dancing, for tap dancing.
for dancing, for tap dancing.
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To je u stvari sistem nota za
ples, za irski ples.
ples, za irski ples.
02:59
That enables him as a choreographer
to do cool stuff, to do new things,
to do cool stuff, to do new things,
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Ovo omogućava njemu kao koreografu
da radi kul, nove stvari,
da radi kul, nove stvari,
03:04
because he has represented it.
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zato što ih je predstavio.
03:07
I want you to think about how amazing
representing something actually is.
representing something actually is.
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Žeim da razmislite koliko sjajno
predstavljanje može stvarno biti.
predstavljanje može stvarno biti.
03:12
Here it says the word "mathematics."
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Ovde piše reč "matematika".
03:15
But actually, they're just dots, right?
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Ali u stvari, to su samo tačke, zar ne?
03:18
So how in the world can these dots
represent the word?
represent the word?
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Pa kako onda, zaboga, ove tačke
mogu da predstavljaju reč?
mogu da predstavljaju reč?
03:21
Well, they do.
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Pa, mogu.
03:23
They represent the word "mathematics,"
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One predstavljaju reč "matematika",
03:25
and these symbols also represent that word
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a ovi simboli takođe predstavljaju reč
03:27
and this we can listen to.
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i nju možemo da slušamo.
03:29
It sounds like this.
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Zvuči ovako.
03:30
(Beeps)
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(Zvučni signal)
03:32
Somehow these sounds represent
the word and the concept.
the word and the concept.
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Nekako ovi zvukovi predstavljaju
i reč i koncept.
i reč i koncept.
03:36
How does this happen?
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Kako se to dešava?
03:37
There's something amazing
going on about representing stuff.
going on about representing stuff.
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Postoji nešto sjajno što se
dešava pri predstavljanju stvari.
dešava pri predstavljanju stvari.
03:41
So I want to talk about
that magic that happens
that magic that happens
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Želim da govorim o magiji
koja se dešava
koja se dešava
03:47
when we actually represent something.
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kada zapravo predstavljamo nešto.
03:49
Here you see just lines
with different widths.
with different widths.
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Ovde vidite samo linije
različitih širina.
različitih širina.
03:52
They stand for numbers
for a particular book.
for a particular book.
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One predstavljaju brojeve
za određenu knjigu.
za određenu knjigu.
03:55
And I can actually recommend
this book, it's a very nice book.
this book, it's a very nice book.
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Mogu zapravo da vam preporučim
ovu knjigu, veoma je dobra.
ovu knjigu, veoma je dobra.
03:58
(Laughter)
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(Smeh)
03:59
Just trust me.
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Verujte mi na reč.
04:01
OK, so let's just do an experiment,
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OK, hajde da uradimo jedan eksperiment,
04:03
just to play around
with some straight lines.
with some straight lines.
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samo da se igramo sa
nekim pravim liniijama.
nekim pravim liniijama.
04:06
This is a straight line.
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Ovo je prava linija.
04:07
Let's make another one.
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Nacrtajmo još jednu.
04:08
So every time we move,
we move one down and one across,
we move one down and one across,
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Pri svakom pokretu crtamo,
jednu malo niže i jednu popreko,
jednu malo niže i jednu popreko,
04:11
and we draw a new straight line, right?
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i crtamo novu pravu liniju, tačno?
04:13
We do this over and over and over,
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Ponavljamo ovo iznova i iznova,
04:16
and we look for patterns.
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i tražimo šablone.
04:17
So this pattern emerges,
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Izdvaja se ovaj šablon,
04:20
and it's a rather nice pattern.
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i prilično je lep.
04:22
It looks like a curve, right?
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Izgleda kao jedna kriva, zar ne?
04:24
Just from drawing simple, straight lines.
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Iz prostog crtanja jednostavnih,
pravih linija.
pravih linija.
04:27
Now I can change my perspective
a little bit. I can rotate it.
a little bit. I can rotate it.
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Sad mogu malo da promenim perspektivu.
Mogu da rotiram ovo.
Mogu da rotiram ovo.
04:30
Have a look at the curve.
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Pogledajte krivu.
04:32
What does it look like?
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Kako vam izgleda?
04:33
Is it a part of a circle?
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Je li to deo kruga?
04:35
It's actually not a part of a circle.
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U stvari, nije deo kruga.
04:37
So I have to continue my investigation
and look for the true pattern.
and look for the true pattern.
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Moram da nastavim sa istragom
i tražim pravi šablon.
i tražim pravi šablon.
04:41
Perhaps if I copy it and make some art?
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Možda ako ovo prekopiram i dodam
malo umetnoti?
malo umetnoti?
04:45
Well, no.
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Ipak ne.
04:46
Perhaps I should extend
the lines like this,
the lines like this,
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Možda da produžim linije, ovako,
04:49
and look for the pattern there.
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i pogledajte ovaj šablon ovde.
04:50
Let's make more lines.
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Nacrtajmo još linija.
04:52
We do this.
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Uradimo ovo.
04:53
And then let's zoom out
and change our perspective again.
and change our perspective again.
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I hajde da smanjimo zum
i ponovo promenimo perspektivu.
i ponovo promenimo perspektivu.
04:57
Then we can actually see that
what started out as just straight lines
what started out as just straight lines
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Onda zapravo vidimo da ono
što su u početku bile samo prave linije,
što su u početku bile samo prave linije,
05:01
is actually a curve called a parabola.
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su sada u stvari krive, parabole.
05:03
This is represented by a simple equation,
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Ovo se može predstaviti
jedostavnom jednačinom,
jedostavnom jednačinom,
05:07
and it's a beautiful pattern.
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a i to je lep šablon.
05:09
So this is the stuff that we do.
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To je ono što mi radimo.
05:11
We find patterns, and we represent them.
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Pronalazimo šablone i predstavljamo ih.
05:13
And I think this is a nice
day-to-day definition.
day-to-day definition.
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Mislim da je to fina,
jednostavna definicija.
jednostavna definicija.
05:16
But today I want to go
a little bit deeper,
a little bit deeper,
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Ali danas želim da malo
produbim temu,
produbim temu,
05:18
and think about
what the nature of this is.
what the nature of this is.
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i razmislim o prirodi svega ovoga.
05:22
What makes it possible?
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Zbog čega je ovo moguće?
05:24
There's one thing
that's a little bit deeper,
that's a little bit deeper,
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Postoji jedna stvar
koja je malo dublja,
koja je malo dublja,
05:26
and that has to do with the ability
to change your perspective.
to change your perspective.
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i ima veze sa sposobnošću
da promenite ugao gledanja.
da promenite ugao gledanja.
05:30
And I claim that when
you change your perspective,
you change your perspective,
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I tvrdim da kada
promenite ugao gledanja,
promenite ugao gledanja,
05:32
and if you take another point of view,
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i počnete da gledate stvari
iz drugog ugla,
iz drugog ugla,
05:35
you learn something new
about what you are watching
about what you are watching
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naučite nešto novo o
onome što posmatrate
onome što posmatrate
05:39
or looking at or hearing.
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ili gledate ili čujete.
05:41
And I think this is a really important
thing that we do all the time.
thing that we do all the time.
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Smatram da je stvarno važno
da to radimo sve vreme.
da to radimo sve vreme.
05:45
So let's just look at
this simple equation,
this simple equation,
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Pa hajde da pogledamo
ovu jednostavnu jednačinu,
ovu jednostavnu jednačinu,
05:49
x + x = 2 • x.
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x + x = 2 • x.
05:52
This is a very nice pattern,
and it's true,
and it's true,
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Ovo je veoma lep šablon,
i tačan je,
i tačan je,
05:54
because 5 + 5 = 2 • 5, etc.
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zato što je 5 + 5 = 2 • 5, itd.
05:57
We've seen this over and over,
and we represent it like this.
and we represent it like this.
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Videli smo ovo mnogo puta,
i predstavljamo to ovako.
i predstavljamo to ovako.
06:00
But think about it: this is an equation.
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Ali razmislite: ovo je jednačina.
06:03
It says that something
is equal to something else,
is equal to something else,
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Kaže da je nešto
jednako nečemu drugom,
jednako nečemu drugom,
06:05
and that's two different perspectives.
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a to su dve različite perspektive.
06:07
One perspective is, it's a sum.
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Jedna perspektiva je zbir.
06:09
It's something you plus together.
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Dodajete nešto nečemu.
06:11
On the other hand, it's a multiplication,
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Sa druge strane, imamo množenje,
06:14
and those are two different perspectives.
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i to su dve različite perspektive.
06:17
And I would go as far as to say
that every equation is like this,
that every equation is like this,
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I išao bih toliko daleko da kažem
da je svaka jednačina ovakva,
da je svaka jednačina ovakva,
06:20
every mathematical equation
where you use that equality sign
where you use that equality sign
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svaka matematička jednačina
gde koristite znak plus
gde koristite znak plus
06:25
is actually a metaphor.
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je zapravo metafora.
06:26
It's an analogy between two things.
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To je analogija između dve stvari.
06:28
You're just viewing something
and taking two different points of view,
and taking two different points of view,
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Prosto posmatrate nešto
i imate dva ugla gledanja,
i imate dva ugla gledanja,
06:32
and you're expressing that in a language.
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a to izražavate jezikom.
06:34
Have a look at this equation.
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Pogledajte ovu jednačinu.
06:36
This is one of the most
beautiful equations.
beautiful equations.
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Ovo je jedna od
najlepših jednačina.
najlepših jednačina.
06:38
It simply says that, well,
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Ona jednostavno kaže
06:41
two things, they're both -1.
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ove dve stvari obe imaju rešenje -1.
06:44
This thing on the left-hand side is -1,
and the other one is.
and the other one is.
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Ovo sa leve strane jednako je -1,
kao i ovo sa druge.
kao i ovo sa druge.
06:47
And that, I think, is one
of the essential parts
of the essential parts
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I smatram da je to
jedan od suštinskih delova
jedan od suštinskih delova
06:50
of mathematics -- you take
different points of view.
different points of view.
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matematike - imate
različite uglove gledanja.
različite uglove gledanja.
06:52
So let's just play around.
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Hajde malo da se igramo.
06:53
Let's take a number.
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Uzmimo jedan broj.
06:55
We know four-thirds.
We know what four-thirds is.
We know what four-thirds is.
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Znamo četiri trećine.
Znamo koliko je četiri trećine.
Znamo koliko je četiri trećine.
06:58
It's 1.333, but we have to have
those three dots,
those three dots,
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1.333, ali moramo da
imamo ove tri tačke,
imamo ove tri tačke,
07:01
otherwise it's not exactly four-thirds.
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inače to nisu tačno četiri trećine.
07:03
But this is only in base 10.
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Ali to je samo za osnovu 10.
07:05
You know, the number system,
we use 10 digits.
we use 10 digits.
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Znate, numerički sistem,
uzimamo 10 cifri.
uzimamo 10 cifri.
07:08
If we change that around
and only use two digits,
and only use two digits,
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Ako to preokrenemo i
uzmemo samo dve cifre,
uzmemo samo dve cifre,
07:10
that's called the binary system.
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to se zove binarni sistem.
07:12
It's written like this.
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Piše se ovako.
07:13
So we're now talking about the number.
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Sada pričamo o ovom broju.
07:15
The number is four-thirds.
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Taj broj je četiri trećine.
07:17
We can write it like this,
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Možemo ga napisati ovako,
07:19
and we can change the base,
change the number of digits,
change the number of digits,
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i možemo promeniti osnovu,
promeniti broj cifara,
promeniti broj cifara,
07:22
and we can write it differently.
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i možemo ga napisati drugačije.
07:24
So these are all representations
of the same number.
of the same number.
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Sve ovo su reprezentacije
istog broja.
istog broja.
07:28
We can even write it simply,
like 1.3 or 1.6.
like 1.3 or 1.6.
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Možemo ga napisati jednostavno,
kao 1.3 ili 1.6.
kao 1.3 ili 1.6.
07:31
It all depends on
how many digits you have.
how many digits you have.
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Sve zavisi od toga
koliko cifara imate.
koliko cifara imate.
07:34
Or perhaps we just simplify
and write it like this.
and write it like this.
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Ili možda možemo da pojednostavimo
i napišemo ga ovako.
i napišemo ga ovako.
07:37
I like this one, because this says
four divided by three.
four divided by three.
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Sviđa mi se ovako, zato što kaže
četiri podeljeno sa tri.
četiri podeljeno sa tri.
07:41
And this number expresses
a relation between two numbers.
a relation between two numbers.
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I ovaj broj izražava
odnos između dva broja.
odnos između dva broja.
07:44
You have four on the one hand
and three on the other.
and three on the other.
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Sa jedne strane imate četiri
i tri sa druge.
i tri sa druge.
07:47
And you can visualize this in many ways.
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I možete zamisliti ovo na više načina.
07:49
What I'm doing now is viewing that number
from different perspectives.
from different perspectives.
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Sada posmatram taj broj
iz drugačije perspektive.
iz drugačije perspektive.
07:53
I'm playing around.
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Igram se.
07:54
I'm playing around with
how we view something,
how we view something,
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Igram se sa tim
kako na nešto gledamo,
kako na nešto gledamo,
07:57
and I'm doing it very deliberately.
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i radim to namerno.
07:58
We can take a grid.
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Možemo uzeti rešetku.
08:00
If it's four across and three up,
this line equals five, always.
this line equals five, always.
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To je četiri udesno i tri gore,
ova linija je uvek pet.
ova linija je uvek pet.
08:04
It has to be like this.
This is a beautiful pattern.
This is a beautiful pattern.
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Mora da bude ovako.
Ovo je prelepi šablon.
Ovo je prelepi šablon.
08:07
Four and three and five.
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Četiri i tri i pet.
08:09
And this rectangle, which is 4 x 3,
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A ovaj pravougaonik, koji je 4 x 3,
08:11
you've seen a lot of times.
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videli ste mnogo puta.
08:13
This is your average computer screen.
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Ovo je prosečan ekran računara,
08:15
800 x 600 or 1,600 x 1,200
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800 x 600 ili 1600 x 1200
08:18
is a television or a computer screen.
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je ekran TV-a ili računara.
08:21
So these are all nice representations,
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Ovo su sve lepe reprezentacije,
08:23
but I want to go a little bit further
and just play more with this number.
and just play more with this number.
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ali želim da idem malo dalje
i malo više da se igram sa brojevima.
i malo više da se igram sa brojevima.
08:27
Here you see two circles.
I'm going to rotate them like this.
I'm going to rotate them like this.
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Ovde vidite dva kruga.
Rotiraću ih ovako.
Rotiraću ih ovako.
08:31
Observe the upper-left one.
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Gledajte ovaj
u gornjem levom uglu.
u gornjem levom uglu.
08:32
It goes a little bit faster, right?
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Kreće se malo brže, zar ne?
08:35
You can see this.
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To može da se vidi.
08:36
It actually goes exactly
four-thirds as fast.
four-thirds as fast.
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Zapravo kreće se tačno
brzinom od četiri trećine.
brzinom od četiri trećine.
08:39
That means that when it goes
around four times,
around four times,
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To znači da kada se
kreće okolo četiri puta,
kreće okolo četiri puta,
08:42
the other one goes around three times.
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drugi to radi tri puta.
08:44
Now let's make two lines, and draw
this dot where the lines meet.
this dot where the lines meet.
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Hajde sada da nacrtamo dve linije
i tačku tamo gde se linije spajaju.
i tačku tamo gde se linije spajaju.
08:47
We get this dot dancing around.
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Dobijamo ovaj ples.
08:49
(Laughter)
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(Smeh)
08:50
And this dot comes from that number.
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I ova tačka dolazi od tog broja.
08:52
Right? Now we should trace it.
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Tačno? Sada treba da joj
uđemo u trag.
uđemo u trag.
Hajde da ga nađemo i vidimo
šta će se dogoditi.
šta će se dogoditi.
08:55
Let's trace it and see what happens.
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08:57
This is what mathematics is all about.
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O tome se radi u matematici.
08:59
It's about seeing what happens.
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O gledanju onog što se dešava.
09:01
And this emerges from four-thirds.
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I ovo smo dobill od četiri trećine.
09:04
I like to say that this
is the image of four-thirds.
is the image of four-thirds.
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Volim da kažem da je ovo slika
četiri trećine.
četiri trećine.
09:07
It's much nicer -- (Cheers)
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Lepše je - (Oduševljenje)
09:08
Thank you!
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Hvala vam!
09:09
(Applause)
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(Aplauz)
09:16
This is not new.
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Ovo nije novo.
09:17
This has been known
for a long time, but --
for a long time, but --
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Za ovo se zna odavno, ali -
09:19
(Laughter)
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(Smeh)
09:21
But this is four-thirds.
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Ali ovo su četiri trećine.
09:23
Let's do another experiment.
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Da uradimo još jedan
eksperiment.
eksperiment.
09:24
Let's now take a sound, this sound: (Beep)
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Da uzmemo neki zvuk, ovaj zvuk:
(Zvučni signal)
(Zvučni signal)
09:28
This is a perfect A, 440Hz.
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Ovo je savršeno A, 440Hz.
09:31
Let's multiply it by two.
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Da pomnožimo to sa dva.
09:33
We get this sound. (Beep)
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Dobijamo ovaj zvuk.
(Zvučni signal)
(Zvučni signal)
09:34
When we play them together,
it sounds like this.
it sounds like this.
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Kada ih pustimo zajedno,
zvuče ovako.
zvuče ovako.
09:37
This is an octave, right?
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Ovo je oktava, zar ne?
09:38
We can do this game. We can play
a sound, play the same A.
a sound, play the same A.
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Možemo ovako da se igramo. Pustimo
zvuk, čujemo isto A.
zvuk, čujemo isto A.
09:41
We can multiply it by three-halves.
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Pomnožićemo to sa tri polovine.
09:42
(Beep)
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(Zvučni signal)
09:44
This is what we call a perfect fifth.
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Ovo zovemo savršenom petinom.
09:46
(Beep)
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(Zvučni signal)
09:47
They sound really nice together.
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Zvuče stvarno lepo zajedno.
09:49
Let's multiply this sound
by four-thirds. (Beep)
by four-thirds. (Beep)
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Pomnožimo ovaj zvuk
sa četiri trećine. (Zvučni signal)
sa četiri trećine. (Zvučni signal)
09:53
What happens?
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Šta se dešava?
09:55
You get this sound. (Beep)
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Dobijamo ovaj zvuk.
(Signal)
(Signal)
09:57
This is the perfect fourth.
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Ovo je savršena četvrtina.
09:58
If the first one is an A, this is a D.
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Ako je prvi A, onda je ovo D.
10:00
They sound like this together. (Beeps)
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Ovako zvuče zajedno.
(Zvučni signal)
(Zvučni signal)
10:02
This is the sound of four-thirds.
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Ovo je zvuk četiri trećine.
10:05
What I'm doing now,
I'm changing my perspective.
I'm changing my perspective.
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Ono što radim sada je da
menjam ugao geldanja.
menjam ugao geldanja.
10:07
I'm just viewing a number
from another perspective.
from another perspective.
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Samo gledam broj iz
drugačije perspektive.
drugačije perspektive.
10:10
I can even do this with rhythms, right?
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Ovo mogu da radim i sa
ritmovima, zar ne?
ritmovima, zar ne?
10:12
I can take a rhythm and play
three beats at one time (Drumbeats)
three beats at one time (Drumbeats)
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Mogu uzeti jedan ritam i
svirati tri taktaistovremeno (Bubnjevi)
svirati tri taktaistovremeno (Bubnjevi)
10:16
in a period of time,
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neko vreme,
10:18
and I can play another sound
four times in that same space.
four times in that same space.
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mogu svirati drugi zvuk
četiri puta u istom prostoru.
četiri puta u istom prostoru.
10:22
(Clanking sounds)
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(Zveket)
10:23
Sounds kind of boring,
but listen to them together.
but listen to them together.
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Zvuče nekako dosadno
ali poslušajte ih zajedno.
ali poslušajte ih zajedno.
10:25
(Drumbeats and clanking sounds)
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(Bubnjevii zveket)
10:28
(Laughter)
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(Smeh)
10:30
Hey! So.
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Hej! Tako.
10:31
(Laughter)
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(Smeh)
10:33
I can even make a little hi-hat.
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Čak mogu malo da koristim i pedale.
10:35
(Drumbeats and cymbals)
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(Bubnjevi i cimbal)
10:37
Can you hear this?
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Čujete li ovo?
10:38
So, this is the sound of four-thirds.
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Pa, ovo je zvuk četiri trećine.
10:40
Again, this is as a rhythm.
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Opet, ovo je neki ritam.
10:42
(Drumbeats and cowbell)
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(Bubnjevi i medenica)
10:44
And I can keep doing this
and play games with this number.
and play games with this number.
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I mogu da nastavim ovako
da se igram sa ovim brojem.
da se igram sa ovim brojem.
10:47
Four-thirds is a really great number.
I love four-thirds!
I love four-thirds!
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Četiri trećine je stvarno sjajan broj.
Obožavam ga!
Obožavam ga!
10:50
(Laughter)
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(Smeh)
10:51
Truly -- it's an undervalued number.
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Zaista - to je potcenjen broj.
10:53
So if you take a sphere and look
at the volume of the sphere,
at the volume of the sphere,
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Tako ako uzmete loptu
i pogledate njenu zapreminu,
i pogledate njenu zapreminu,
10:56
it's actually four-thirds
of some particular cylinder.
of some particular cylinder.
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imate u stvari četiri trećine
nekog određenog cilindra.
nekog određenog cilindra.
10:59
So four-thirds is in the sphere.
It's the volume of the sphere.
It's the volume of the sphere.
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Tako da su četiri trećine u lopti.
One su u zapremini lopte.
One su u zapremini lopte.
11:03
OK, so why am I doing all this?
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OK, zašto ja uopšte ovo radim?
11:05
Well, I want to talk about what it means
to understand something
to understand something
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Pa, želim da govorim o tome šta znači
razumeti nešto
razumeti nešto
11:09
and what we mean
by understanding something.
by understanding something.
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i šta smatramo pod tim.
11:11
That's my aim here.
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To je moj cilj ovde.
11:13
And my claim is that
you understand something
you understand something
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A moja tvrdnja je da razumemo nešto
11:15
if you have the ability to view it
from different perspectives.
from different perspectives.
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ako imamo sposobnost da to
glesamo iz različitih perspektiva.
glesamo iz različitih perspektiva.
11:18
Let's look at this letter.
It's a beautiful R, right?
It's a beautiful R, right?
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Pogledajmo ovo slovo.
Jedno lepo slovo R, zar ne?
Jedno lepo slovo R, zar ne?
11:20
How do you know that?
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Kako to znamo?
11:22
Well, as a matter of fact,
you've seen a bunch of R's,
you've seen a bunch of R's,
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Zapravo, videli smo
milion puta slovo R,
milion puta slovo R,
11:25
and you've generalized
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i generalizovali smo
11:27
and abstracted all of these
and found a pattern.
and found a pattern.
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i rezimirali sve ovo i pronašli šablon.
11:30
So you know that this is an R.
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Tako da znamo da je ovo slovo R.
11:35
So what I'm aiming for here
is saying something
is saying something
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Ono na šta ciljam je da kažem nešto
11:38
about how understanding
and changing your perspective
and changing your perspective
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3381
o tome kako su razumevanje i
promena perspektive
promena perspektive
11:41
are linked.
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povezani.
11:43
And I'm a teacher and a lecturer,
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Ja sam nastavnik i predavač,
11:45
and I can actually use this
to teach something,
to teach something,
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i mogu u stvari da iskoristim ovo
da bih predavao
da bih predavao
11:47
because when I give someone else
another story, a metaphor, an analogy,
another story, a metaphor, an analogy,
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jer kad nekom drugom dam
drugu priču, metaforu, analogiju,
drugu priču, metaforu, analogiju,
11:52
if I tell a story
from a different point of view,
from a different point of view,
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ako ispričam priču
sa drugačijeg stanovišta,
sa drugačijeg stanovišta,
11:55
I enable understanding.
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ja omogućavam razumevanje.
11:56
I make understanding possible,
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Činim razumevanje mogućim
11:58
because you have to generalize
over everything you see and hear,
over everything you see and hear,
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jer morate generalizovati
sve što vidite i čujete,
sve što vidite i čujete,
12:01
and if I give you another perspective,
that will become easier for you.
that will become easier for you.
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i ako vam dam drugačiju perspektivu,
biće vam lakše.
biće vam lakše.
12:06
Let's do a simple example again.
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Da uradimo ponovo jedan primer.
12:08
This is four and three.
This is four triangles.
This is four triangles.
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Ovo su četiri i tri.
Ovo su četiri trougla.
Ovo su četiri trougla.
12:10
So this is also four-thirds, in a way.
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Tako da je ovo takođe na neki način
četiri trećine.
četiri trećine.
12:13
Let's just join them together.
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Hajde da ih jednostavno povežemo.
12:15
Now we're going to play a game;
we're going to fold it up
we're going to fold it up
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2709
Sada ćemo da igramo jednu igru:
savićemo ih
savićemo ih
12:17
into a three-dimensional structure.
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1682
u trodimenzionalnu strukturu.
12:19
I love this.
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Obožavam ovo.
12:20
This is a square pyramid.
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1416
Ovo je četvrtasta piramida.
12:22
And let's just take two of them
and put them together.
and put them together.
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3150
Hajde da uzmemo dve i sastavimo ih.
12:25
So this is what is called an octahedron.
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2689
I ovo je ono što se zove oktahedron.
12:28
It's one of the five platonic solids.
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2707
Jedan je od pet platonskih čvrstih tela.
12:31
Now we can quite literally
change our perspective,
change our perspective,
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Sada i bukvalno možemo
promeniti našu perspektivu,
promeniti našu perspektivu,
12:33
because we can rotate it
around all of the axes
around all of the axes
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2695
jer ga možemo rotirati oko svih osa
12:36
and view it from different perspectives.
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2012
i posmatrati iz različitih perspektiva.
12:38
And I can change the axis,
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2066
Mogu i da promenim ovu osu,
12:40
and then I can view it
from another point of view,
from another point of view,
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2338
i onda mogu da posmatram
iz drugačije perspekte,
iz drugačije perspekte,
12:42
but it's the same thing,
but it looks a little different.
but it looks a little different.
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ali je to ista stvar
samo što izgleda drugačije.
samo što izgleda drugačije.
12:45
I can do it even one more time.
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Mogu to da uradim još jedanput.
12:47
Every time I do this,
something else appears,
something else appears,
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Svaki put kada ovo uradim,
nešto se drugo pojavi,
nešto se drugo pojavi,
12:50
so I'm actually learning
more about the object
more about the object
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tako da zapravo učim
sve više o ovom predmetu
sve više o ovom predmetu
12:52
when I change my perspective.
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kada promenim perspektivu.
12:54
I can use this as a tool
for creating understanding.
for creating understanding.
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Mogu da ga koristim kao alat
za stvaranje razumevanja.
za stvaranje razumevanja.
12:58
I can take two of these
and put them together like this
and put them together like this
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Mogu da uzmem dva komada
i sastavim ih ovako
i sastavim ih ovako
13:02
and see what happens.
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i vidite šta se dešava.
13:03
And it looks a little bit
like the octahedron.
like the octahedron.
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Malo i liči na oktahedron.
13:07
Have a look at it if I spin
it around like this.
it around like this.
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Pogledajte ako ga okrenem ovako.
13:09
What happens?
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Šta se dešava?
13:11
Well, if you take two of these,
join them together and spin it around,
join them together and spin it around,
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Pa, ako uzmete dva tela,
spojite ih i okrenete,
spojite ih i okrenete,
13:14
there's your octahedron again,
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evo ga oktahedron opet,
13:16
a beautiful structure.
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prelepa struktura.
13:18
If you lay it out flat on the floor,
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Ako stavite pravo na pod,
13:20
this is the octahedron.
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ovo je oktahedron.
13:21
This is the graph structure
of an octahedron.
of an octahedron.
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Ovo je grafička struktura oktahedrona.
13:25
And I can continue doing this.
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I mogu ovako do sutra.
13:27
You can draw three great circles
around the octahedron,
around the octahedron,
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Možete da nacrtate tri velika kruga
oko oktahedrona,
oko oktahedrona,
13:31
and you rotate around,
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i da ih zarotirate,
13:33
so actually three great circles
is related to the octahedron.
is related to the octahedron.
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i onda su u stvari tri velika
kruga povezana sa oktahedronom.
kruga povezana sa oktahedronom.
13:37
And if I take a bicycle pump
and just pump it up,
and just pump it up,
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I ako uzmem pumpu za bicikl
i prosto ih naduvam,
i prosto ih naduvam,
13:41
you can see that this is also
a little bit like the octahedron.
a little bit like the octahedron.
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možete opet videti da
malo podseća na oktahedron.
malo podseća na oktahedron.
13:44
Do you see what I'm doing here?
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Da li vidite šta radim?
13:47
I am changing the perspective every time.
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Menjam perspektivu svaki put.
13:50
So let's now take a step back --
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Sad ćemo malo da se vratimo unazad -
13:53
and that's actually
a metaphor, stepping back --
a metaphor, stepping back --
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i to je u stvari
metafora, vraćanje unazad -
metafora, vraćanje unazad -
13:56
and have a look at what we're doing.
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i da pogledamo šta smo uradili.
13:58
I'm playing around with metaphors.
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Poigravam se sa metaforama.
14:00
I'm playing around
with perspectives and analogies.
with perspectives and analogies.
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Poigravam se sa
perspektivama i analogijama.
perspektivama i analogijama.
14:03
I'm telling one story in different ways.
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Prenosim jednu priču na razne načine.
14:05
I'm telling stories.
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Pričam priče.
14:06
I'm making a narrative;
I'm making several narratives.
I'm making several narratives.
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Pravim narativ.
Pravm nekoliko narativa.
Pravm nekoliko narativa.
14:09
And I think all of these things
make understanding possible.
make understanding possible.
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Smatram da je zbog svega
ovog razumevanje moguće.
ovog razumevanje moguće.
14:13
I think this actually is the essence
of understanding something.
of understanding something.
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Smatram da je ovo zapravo suština
razumevanja nečega.
razumevanja nečega.
14:16
I truly believe this.
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Istinski verujem u to.
14:18
So this thing about changing
your perspective --
your perspective --
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Tako da je ovo sa menjanjem
perspektive
perspektive
14:20
it's absolutely fundamental for humans.
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apsolutno neophodno za čovečanstvo.
14:23
Let's play around with the Earth.
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Hajde da se igramo sa Zemljom.
14:25
Let's zoom into the ocean,
have a look at the ocean.
have a look at the ocean.
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Zumirajmo okean, pogledajte okean.
14:28
We can do this with anything.
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Ovo možemo da radimo sa svim stvarima.
14:29
We can take the ocean
and view it up close.
and view it up close.
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Možemo da uzmemo okean
i posmatramo ga izbliza.
i posmatramo ga izbliza.
14:32
We can look at the waves.
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Možemo da gledamo talase.
14:34
We can go to the beach.
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Možemo da odemo do plaže.
14:35
We can view the ocean
from another perspective.
from another perspective.
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Možemo da posmatramo okean
iz druge perspektive.
iz druge perspektive.
14:37
Every time we do this, we learn
a little bit more about the ocean.
a little bit more about the ocean.
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Svaki put kad to uradimo, naučimo
nešto više o okeanu.
nešto više o okeanu.
14:41
If we go to the shore,
we can kind of smell it, right?
we can kind of smell it, right?
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Ako odemo do obale,
možemo da osetimo miris, zar ne?
možemo da osetimo miris, zar ne?
14:43
We can hear the sound of the waves.
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Možemo da čujemo zvuk talasa.
14:45
We can feel salt on our tongues.
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Možemo da osetimo so na jeziku.
14:47
So all of these
are different perspectives.
are different perspectives.
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Sve su to drugačije perspektive.
14:50
And this is the best one.
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A ovo je najbolja.
14:51
We can go into the water.
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Možemo da uđemo u vodu.
14:53
We can see the water from the inside.
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2009
Možemo da vidimo vodu iznutra.
14:55
And you know what?
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I znate šta?
14:56
This is absolutely essential
in mathematics and computer science.
in mathematics and computer science.
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884673
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Ovo je apsolutno neophodno
u matematici i informatici.
u matematici i informatici.
14:59
If you're able to view
a structure from the inside,
a structure from the inside,
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Ako možete da vidite
strukturu iznutra,
strukturu iznutra,
15:02
then you really learn something about it.
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onda ste je stvarno savladali.
15:05
That's somehow the essence of something.
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To je nekako suština svačega.
15:07
So when we do this,
and we've taken this journey
and we've taken this journey
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3643
Tako kada uradimo ovo,
i kada otputujemo
i kada otputujemo
15:11
into the ocean,
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u okean,
15:12
we use our imagination.
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koristimo našu maštu.
15:14
And I think this is one level deeper,
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2653
A mislim da je to još jedan korak dalje,
15:17
and it's actually a requirement
for changing your perspective.
for changing your perspective.
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i zapravo je neophodan da bismo
promenili perspektivuu.
promenili perspektivuu.
15:21
We can do a little game.
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1167
Odigraćemo jednu igru.
15:23
You can imagine that you're sitting there.
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Zamislite da sedite tamo.
15:25
You can imagine that you're up here,
and that you're sitting here.
and that you're sitting here.
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3227
Zamislite da ste ovde i da sedite tu.
15:28
You can view yourselves from the outside.
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2326
Možete videti sebe spolja.
15:30
That's really a strange thing.
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To je tek čudno.
15:32
You're changing your perspective.
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1823
Menjate perspektivu.
15:34
You're using your imagination,
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1859
Koristite maštu,
15:36
and you're viewing yourself
from the outside.
from the outside.
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i posmatrate sebe spolja.
15:39
That requires imagination.
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Za to je neophodna mašta.
15:41
Mathematics and computer science
are the most imaginative art forms ever.
are the most imaginative art forms ever.
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929126
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Matematika i informatika
su najmaštovitije umetničke forme od svih.
su najmaštovitije umetničke forme od svih.
15:46
And this thing about changing perspectives
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I ovo sa menjanjem perspektiva
15:49
should sound a little bit familiar to you,
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2508
bi trebalo da vam je malo jasnije,
15:51
because we do it every day.
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2212
jer je to nešto što radimo svaki dan.
15:54
And then it's called empathy.
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1620
I onda se to zove empatija.
15:56
When I view the world
from your perspective,
from your perspective,
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Kada ja posmatram svet iz vašeg ugla,
16:00
I have empathy with you.
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osećam empatiju sa vama.
16:02
If I really, truly understand
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Ako stvarno, istinski razumem
16:04
what the world looks
like from your perspective,
like from your perspective,
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kako svet izgleda iz vaše perspektive,
16:07
I am empathetic.
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955603
1471
ja sam empatičan.
16:09
That requires imagination.
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Za to je neophodna mašta.
16:11
And that is how we obtain understanding.
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Na taj način usvajamo razumevanje.
16:15
And this is all over mathematics
and this is all over computer science,
and this is all over computer science,
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I to je svuda u matematici i informatici
16:18
and there's a really deep connection
between empathy and these sciences.
between empathy and these sciences.
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5535
i postoji zaista velika povezanost
između empatije i ovih nauka.
između empatije i ovih nauka.
16:25
So my conclusion is the following:
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Stoga, moj zaključak je sledeći:
16:29
understanding something really deeply
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razumevanje nečega istinski
16:32
has to do with the ability
to change your perspective.
to change your perspective.
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mora da ima veze sa sposobnošću
da promenite perspektivu.
da promenite perspektivu.
16:35
So my advice to you is:
try to change your perspective.
try to change your perspective.
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Moj savet vam je:
probajte da promenite perspektivu.
probajte da promenite perspektivu.
16:39
You can study mathematics.
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Možete izučavati matematiku.
16:41
It's a wonderful way to train your brain.
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To je divan način da uvežbate svoj mozak.
16:44
Changing your perspective
makes your mind more flexible.
makes your mind more flexible.
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Kada menjate perspektivu
vaš um postaje fleksibilniji.
vaš um postaje fleksibilniji.
16:48
It makes you open to new things,
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Otvoreniji ste za nove stvari,
16:50
and it makes you
able to understand things.
able to understand things.
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i sposobni ste da razumete stvari.
16:53
And to use yet another metaphor:
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2017
I da upotrebim još jednu metaforu:
16:55
have a mind like water.
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neka vam um bude poput vode.
16:56
That's nice.
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To je lepo.
16:57
Thank you.
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Hvala vam.
16:59
(Applause)
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(Aplauz)
ABOUT THE SPEAKER
Roger Antonsen - Logician, mathematician, computer scientistRoger Antonsen combines science, mathematics and computer science with entertainment, philosophy and visualizations.
Why you should listen
Roger Antonsen is a logician, mathematician, computer scientist, researcher, inventor, author, lecturer, science communicator and public speaker. He teaches logical methods as an associate professor at the Department of Informatics in the research group Logic and Intelligent Data (LogID) at the University of Oslo.
Antonsen is also engaged in various forms of science communication and outreach. His academic interests are logical calculi, proof theory, mathematical logic, complexity theory, automata, combinatorics, philosophy of mathematics, but he is interested in most topics related to mathematics, computer science and philosophy. His vision is to communicate science differently, to inspire creative thinking and to remove the common misconceptions about mathematics and computer science.
Roger Antonsen | Speaker | TED.com