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Roger Antonsen: Math is the hidden secret to understanding the world
Roger Antonsen: Matematika je skrivena tajna za razumijevanje svijeta
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Otključajte misterije i unutarnji mehanizam svijeta kroz jednu od najmaštovitijih umjetničkih izričaja ikada -- matematiku -- s Rogerom Antonsenom, dok on objašnjava kako male promjene u perspektivi mogu otkriti uzorke, brojeve i formule kao puteve prema suosjećanju i razumijevanju.
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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Bok.
00:14
I want to talk about understanding,
and the nature of understanding,
and the nature of understanding,
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Želim razgovarati o razumijevanju
i prirodi razumijevanja,
i prirodi razumijevanja,
00:18
and what the essence of understanding is,
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te što je srž razumijevanja,
00:21
because understanding is something
we aim for, everyone.
we aim for, everyone.
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jer je razumijevanje nešto čemu
svi težimo.
svi težimo.
00:24
We want to understand things.
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Želimo razumjeti stvari.
00:27
My claim is that understanding has to do
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Ja tvrdim da je razumijevanje povezano
00:30
with the ability to change
your perspective.
your perspective.
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sa sposobnošću mijenjanja perspektive.
00:32
If you don't have that,
you don't have understanding.
you don't have understanding.
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Ako je nemaš, nemaš razumijevanja.
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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I želim se usredotočiti na matematiku.
00:40
Many of us think of mathematics
as addition, subtraction,
as addition, subtraction,
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Mnogi od nas misle da je matematika
zbrajanje, oduzimanje,
zbrajanje, oduzimanje,
00:43
multiplication, division,
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množenje, dijeljenje,
00:45
fractions, percent, geometry,
algebra -- all that stuff.
algebra -- all that stuff.
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razlomci, postotci, geometrija,
algebra -- takve stvari.
algebra -- takve stvari.
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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Zapravo, želim razgovarati
i o srži matematike.
i o srži matematike.
00:53
And my claim is that mathematics
has to do with patterns.
has to do with patterns.
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Tvrdim da matematika ima veze
s uzorcima.
s uzorcima.
00:57
Behind me, you see a beautiful pattern,
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Iza mene vidite prekrasan uzorak,
00:59
and this pattern actually emerges
just from drawing circles
just from drawing circles
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i ovaj uzorak zapravo proizlazi iz
crtanja krugova
crtanja krugova
01:03
in a very particular way.
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na vrlo neobičan 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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Moja svakodnevna definicija
matematike koju svaki dan koristim
matematike koju svaki dan koristim
01:10
is the following:
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je sljedeća:
01:12
First of all, it's about finding patterns.
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Prije svega, najvažnije je pronaći uzorke.
01:16
And by "pattern," I mean a connection,
a structure, some regularity,
a structure, some regularity,
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A pod "uzorcima" mislim na veze,
strukturu, neku pravilnost,
strukturu, neku pravilnost,
01:21
some rules that govern what we see.
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pravila koja upravljaju onim što vidimo.
01:24
Second of all,
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Drugo,
01:25
I think it is about representing
these patterns with a language.
these patterns with a language.
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mislim da se radi o prikazivanju
ovih uzoraka jezikom.
ovih uzoraka jezikom.
01:29
We make up language if we don't have it,
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Mi izmislimo jezik ako ga nemamo,
01:31
and in mathematics, this is essential.
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a u matematici to je najbitnije.
01:35
It's also about making assumptions
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Važno je pretpostavljati
01:36
and playing around with these assumptions
and just seeing what happens.
and just seeing what happens.
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poigravati se pretpostavkama i
samo vidjeti što će se dogoditi.
samo vidjeti što će se dogoditi.
01:40
We're going to do that very soon.
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Napravit ćemo to uskoro.
01:42
And finally, it's about doing cool stuff.
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I na kraju, radi se o super stvarima.
01:46
Mathematics enables us
to do so many things.
to do so many things.
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Matematika nam omogućuje da
činimo mnogo stvari.
činimo mnogo stvari.
01:50
So let's have a look at these patterns.
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Pa pogledajmo ove uzorke.
01:52
If you want to tie a tie knot,
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Ako želite zavezati kravatu,
01:55
there are patterns.
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postoje uzorci.
01:56
Tie knots have names.
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Čvorovi za kravate
imaju imena.
imaju imena.
01:58
And you can also do
the mathematics of tie knots.
the mathematics of tie knots.
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A možete i otkriti 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 lijevi-van, desni-unutra,
sredina-van i zaveži.
sredina-van i zaveži.
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 lijevi-unutra, desni-van,
lijevi-unutra, sredina-vani zaveži.
lijevi-unutra, sredina-vani zaveži.
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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To je jezik koji smo izmislili za
uzorke čvorova za kravate,
uzorke čvorova za kravate,
02:12
and a half-Windsor is all that.
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i polu-Windsor je sve to.
02:15
This is a mathematics book
about tying shoelaces
about tying shoelaces
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Ovo je matematička knjiga
o vezanju vezica na cipelama
o vezanju vezica na cipelama
02:18
at the university level,
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na sveučilišnoj razini,
02:19
because there are patterns in shoelaces.
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jer u vezanju vezica
ima uzoraka.
ima uzoraka.
02:21
You can do it in so many different ways.
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Mogu se zavezati
na puno različitih načina.
na puno različitih načina.
02:23
We can analyze it.
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Možemo ih analizirati.
02:25
We can make up languages for it.
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Možemo izmisliti jezike za njih.
02:28
And representations
are all over mathematics.
are all over mathematics.
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A primjeri su u cijeloj matematici.
02:31
This is Leibniz's notation from 1675.
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Ovo je Leibnizova bilješka iz 1675.
02:35
He invented a language
for patterns in nature.
for patterns in nature.
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On je izmislio jezik za uzorke u prirodi.
02:39
When we throw something up in the air,
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Kad nešto bacimo u zrak,
02:41
it falls down.
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to padne.
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
prikazati matematikom u uzroku.
prikazati matematikom u uzroku.
02:48
This is also a pattern.
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To je također uzorak.
02:49
This is also an invented language.
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To je također izmišljeni jezik.
02:52
Can you guess for what?
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Možete li pogoditi za što služi?
02:55
It is actually a notation system
for dancing, for tap dancing.
for dancing, for tap dancing.
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To je zapravo sustav
bilježenja za ples, za step.
bilježenja za ples, za step.
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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To omogućuje njemu kao koreografu
da radi super stvari, nove stvari,
da radi super stvari, nove stvari,
03:04
because he has represented it.
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jer ih je on prikazao.
03:07
I want you to think about how amazing
representing something actually is.
representing something actually is.
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Želim da pomislite kako je divno
prikazati nešto.
prikazati nešto.
03:12
Here it says the word "mathematics."
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Ovdje piše riječ "matematika".
03:15
But actually, they're just dots, right?
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Ali zapravo, to su samo točkice, zar ne?
03:18
So how in the world can these dots
represent the word?
represent the word?
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Pa kako je moguće da te točkice
prikazuju riječ?
prikazuju riječ?
03:21
Well, they do.
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Pa, moguće je.
03:23
They represent the word "mathematics,"
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Prikazuju riječ "matematika",
03:25
and these symbols also represent that word
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i ovi simboli pokazuju istu riječ,
03:27
and this we can listen to.
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i ovo što možemo poslušati.
03:29
It sounds like this.
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Zvuči ovako.
03:30
(Beeps)
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(Pištanje)
03:32
Somehow these sounds represent
the word and the concept.
the word and the concept.
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Nekako ovi zvukovi prikazuju
riječ i koncept.
riječ i koncept.
03:36
How does this happen?
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Kako se ovo događa?
03:37
There's something amazing
going on about representing stuff.
going on about representing stuff.
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Ima nešto nevjerojatno u
prikazivanju stvari.
prikazivanju stvari.
03:41
So I want to talk about
that magic that happens
that magic that happens
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Zato želim razgovarati o magiji
koja se događa
koja se događa
03:47
when we actually represent something.
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kad zapravo prikazujemo nešto.
03:49
Here you see just lines
with different widths.
with different widths.
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Ovdje vidite samo crte
različite širine.
različite širine.
03:52
They stand for numbers
for a particular book.
for a particular book.
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Svaki označava broj određene knjige.
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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I stvarno preporučam ovu knjigu,
to je vrlo dobra knjiga.
to je vrlo dobra knjiga.
03:58
(Laughter)
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(Smijeh)
03:59
Just trust me.
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Vjerujte mi.
04:01
OK, so let's just do an experiment,
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U redu, napravimo eksperiment,
04:03
just to play around
with some straight lines.
with some straight lines.
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kako bismo se igrali s ravnim crtama.
04:06
This is a straight line.
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Ovo je ravna crta.
04:07
Let's make another one.
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Napravimo 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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Svakim novim potezom
stavimo jednu dolje i jednu preko,
stavimo jednu dolje i jednu preko,
04:11
and we draw a new straight line, right?
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i nacrtamo novu ravnu crtu,
u redu?
u redu?
04:13
We do this over and over and over,
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Činimo ovo iznova i iznova,
04:16
and we look for patterns.
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i tražimo uzorke.
04:17
So this pattern emerges,
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Javlja se ovaj uzorak,
04:20
and it's a rather nice pattern.
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i baš je lijep uzorak.
04:22
It looks like a curve, right?
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Izgleda kao krivulja, zar ne?
04:24
Just from drawing simple, straight lines.
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Samo od crtanja jednostavnih,
ravnih crta.
ravnih crta.
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 promijeniti malo perspektivu.
Mogu okrenuti crtu.
Mogu okrenuti crtu.
04:30
Have a look at the curve.
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Pogledajte krivulju.
04:32
What does it look like?
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Na što sliči?
04:33
Is it a part of a circle?
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Je li dio kruga?
04:35
It's actually not a part of a circle.
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Zapravo nije dio 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 nastaviti istraživanje
i potražiti pravi uzorak.
i potražiti pravi uzorak.
04:41
Perhaps if I copy it and make some art?
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Možda ako je kopiram i
pretvorim u umjetnost?
pretvorim u umjetnost?
04:45
Well, no.
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Pa i ne.
04:46
Perhaps I should extend
the lines like this,
the lines like this,
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Možda bih trebao produljiti crte ovako,
04:49
and look for the pattern there.
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i potražiti uzorak u tome.
04:50
Let's make more lines.
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Napravimo još crta.
04:52
We do this.
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Učinimo ovo.
04:53
And then let's zoom out
and change our perspective again.
and change our perspective again.
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Odmaknimo se i ponovno
promijenimo perspektivu.
promijenimo 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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Tada zapravo vidimo da ono
što je počelo kao ravna crta
što je počelo kao ravna crta
05:01
is actually a curve called a parabola.
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zapravo je krivulja zvana parabola.
05:03
This is represented by a simple equation,
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Prikazana je jednostavnom jednadžbom,
05:07
and it's a beautiful pattern.
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i to je prekrasan uzorak.
05:09
So this is the stuff that we do.
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Dakle ovo radimo.
05:11
We find patterns, and we represent them.
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Pronalazimo uzorke 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 lijepa
svakodnevna definicija.
svakodnevna definicija.
05:16
But today I want to go
a little bit deeper,
a little bit deeper,
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Ali danas želim ići malo dublje,
05:18
and think about
what the nature of this is.
what the nature of this is.
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i razmisliti o prirodi ovoga.
05:22
What makes it possible?
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Što omogućuje ovo?
05:24
There's one thing
that's a little bit deeper,
that's a little bit deeper,
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Jedna stvar 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 promijenite perspektivu.
da promijenite perspektivu.
05:30
And I claim that when
you change your perspective,
you change your perspective,
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I tvrdim da kad promijenite perspektivu,
05:32
and if you take another point of view,
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i ako pogledate s druge točke gledišta
05:35
you learn something new
about what you are watching
about what you are watching
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naučit ćete nešto novo o tome
što promatrate,
što promatrate,
05:39
or looking at or hearing.
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gledate ili slušate.
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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Mislim da je ovo što stalno
radimo zaista važno.
radimo zaista važno.
05:45
So let's just look at
this simple equation,
this simple equation,
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Zato pogledajmo ovu
jednostavnu jednadžbu.
jednostavnu jednadžbu.
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 vrlo lijep uzorak,
i točan je,
i točan je,
05:54
because 5 + 5 = 2 • 5, etc.
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jer 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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Vidjeli smo ovo već,
i prikazujemo to ovako.
i prikazujemo to ovako.
06:00
But think about it: this is an equation.
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Ali razmislite o tome: ovo je jednadžba.
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 nečem drugom,
06:05
and that's two different perspectives.
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i to su dvije različite perspektive.
06:07
One perspective is, it's a sum.
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Jedna perspektiva je, to je zbroj.
06:09
It's something you plus together.
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To je nešto što zbrojite zajedno.
06:11
On the other hand, it's a multiplication,
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U drugu ruku, to je množenje,
06:14
and those are two different perspectives.
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i to su dvije 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 usuđujem se reći da je
svaka jednadžba ovakva,
svaka jednadžba ovakva,
06:20
every mathematical equation
where you use that equality sign
where you use that equality sign
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svaka matematička jednadžba
u kojoj možete koristiti znak jednakosti
u kojoj možete koristiti znak jednakosti
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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Ona je analogija između dvije 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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Gledate nešto, zauzimate
dva stajališta,
dva stajališta,
06:32
and you're expressing that in a language.
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i izražavate to jezikom.
06:34
Have a look at this equation.
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Pogledajte ovu jednadžbu.
06:36
This is one of the most
beautiful equations.
beautiful equations.
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Ovo je jedna od najljepših jednadžbi.
06:38
It simply says that, well,
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Jednostavno kaže da, pa,
06:41
two things, they're both -1.
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dvije stvari, one su obje -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 na lijevoj strani je -1,
a i ovo na drugoj.
a i ovo na drugoj.
06:47
And that, I think, is one
of the essential parts
of the essential parts
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I to je, mislim, jedan od
najvažnijih dijelova
najvažnijih dijelova
06:50
of mathematics -- you take
different points of view.
different points of view.
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matematike -- zauzimaš
različite točke gledišta.
različite točke gledišta.
06:52
So let's just play around.
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Ajmo se malo igrati.
06:53
Let's take a number.
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Odaberimo neki broj.
06:55
We know four-thirds.
We know what four-thirds is.
We know what four-thirds is.
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Znamo za četiri trećine.
Znamo što je četiri trećine.
Znamo što je četiri trećine.
06:58
It's 1.333, but we have to have
those three dots,
those three dots,
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To je 1.333, ali moramo
imati te tri točke,
imati te tri točke,
07:01
otherwise it's not exactly four-thirds.
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inače nije točno četiri trećine.
07:03
But this is only in base 10.
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Ali to je samo na bazi 10.
07:05
You know, the number system,
we use 10 digits.
we use 10 digits.
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Mi koristimo
brojevni sustav od 10 znamenki.
brojevni sustav od 10 znamenki.
07:08
If we change that around
and only use two digits,
and only use two digits,
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Ako za promjenu koristimo samo
dvije znamenke
dvije znamenke
07:10
that's called the binary system.
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to nazivamo binarni sustav.
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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Dakle sad govorimo o broju.
07:15
The number is four-thirds.
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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 promijeniti bazu,
promijeniti broj znamenki,
promijeniti broj znamenki,
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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Ovo su sve prikazi 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 ovisi o tome koliko znamenki imate.
07:34
Or perhaps we just simplify
and write it like this.
and write it like this.
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Ili da pojednostavimo i napišemo 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, jer kaže
da je četiri podijeljeno na tri.
da je četiri podijeljeno na tri.
07:41
And this number expresses
a relation between two numbers.
a relation between two numbers.
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A ovaj broj izražava vezu
između dva broja.
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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Imate četiri na jednoj strani
i tri na drugoj.
i tri na drugoj.
07:47
And you can visualize this in many ways.
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Možete to vizualizirati na puno načina.
07:49
What I'm doing now is viewing that number
from different perspectives.
from different perspectives.
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Sada upravo gledam taj broj
iz različitih perspektiva.
iz različitih perspektiva.
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 načinom na koji vidimo nešto,
07:57
and I'm doing it very deliberately.
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i radim to vrlo svojevoljno.
07:58
We can take a grid.
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Možemo uzeti mrežu.
08:00
If it's four across and three up,
this line equals five, always.
this line equals five, always.
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Ako su dimenzije 4 x 3,
ova crta je 5, uvijek.
ova crta je 5, uvijek.
08:04
It has to be like this.
This is a beautiful pattern.
This is a beautiful pattern.
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Mora biti ovako. Ovo je prekrasan uzorak.
08:07
Four and three and five.
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Četiri, tri i pet.
08:09
And this rectangle, which is 4 x 3,
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I ovaj pravokutnik koji je 4 x 3,
08:11
you've seen a lot of times.
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vidjeli ste puno puta.
08:13
This is your average computer screen.
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Ovo je vaš prosječni ekran računala.
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 televizija ili ekran računala.
08:21
So these are all nice representations,
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Ovo su sve lijepi prikazi,
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 ići dalje
i još se igrati ovim brojem.
i još se igrati ovim brojem.
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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Ovdje vidite dva kruga.
Okretat ću ih ovako.
Okretat ću ih ovako.
08:31
Observe the upper-left one.
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Promatrajte ovaj gore-lijevo.
08:32
It goes a little bit faster, right?
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Ide malo brže, zar ne?
08:35
You can see this.
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Možete to vidjeti.
08:36
It actually goes exactly
four-thirds as fast.
four-thirds as fast.
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Zapravo ide točno četiri trećine
brzine drugog kruga.
brzine drugog kruga.
08:39
That means that when it goes
around four times,
around four times,
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To znači da kad se on okrene
četiri puta,
četiri puta,
08:42
the other one goes around three times.
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drugi se okrene 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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Napravimo sada dvije crte,
i stavimo točku gdje se crte spajaju.
i stavimo točku gdje se crte spajaju.
08:47
We get this dot dancing around.
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Dobijemo da ova točka pleše okolo.
08:49
(Laughter)
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(Smijeh)
08:50
And this dot comes from that number.
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I ova točka proizlazi iz tog broja.
08:52
Right? Now we should trace it.
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Zar ne? Sad bismo je trebali pratiti.
08:55
Let's trace it and see what happens.
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Pratimo je i pogledajmo
što će se dogoditi.
što će se dogoditi.
08:57
This is what mathematics is all about.
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Ovo je najvažnije u matematici.
08:59
It's about seeing what happens.
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Važno je vidjeti
što će se dogoditi.
što će se dogoditi.
09:01
And this emerges from four-thirds.
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A ovo proizlazi iz 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 reći da je ovo slika četiri trećine.
09:07
It's much nicer -- (Cheers)
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Puno je ljepša -- (Klicanje)
09:08
Thank you!
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Hvala.
09:09
(Applause)
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(Pljesak)
09:16
This is not new.
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Ovo nije novost.
09:17
This has been known
for a long time, but --
for a long time, but --
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Ovo je poznato već dugo, ali --
09:19
(Laughter)
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(Smijeh)
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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Napravimo drugi eksperiment.
09:24
Let's now take a sound, this sound: (Beep)
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Sada uzmimo zvuk, ovaj zvuk: (Pištanje)
09:28
This is a perfect A, 440Hz.
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Ovo je savršeni ton A, 440 Hz.
09:31
Let's multiply it by two.
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Pomnožimo ga s dva.
09:33
We get this sound. (Beep)
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Dobijemo ovo. (Pištanje)
09:34
When we play them together,
it sounds like this.
it sounds like this.
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Kad ih pustimo zajedno, 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 igrati ovu igru.
Pustimo zvuk, pustimo isti A.
Pustimo zvuk, pustimo isti A.
09:41
We can multiply it by three-halves.
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Možemo ga pomnožiti s tri polovine.
09:42
(Beep)
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(Pištanje)
09:44
This is what we call a perfect fifth.
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Ovo zovemo čistom kvintom.
09:46
(Beep)
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(Pištanje)
09:47
They sound really nice together.
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Zvuče baš lijepo zajedno.
09:49
Let's multiply this sound
by four-thirds. (Beep)
by four-thirds. (Beep)
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Pomnožimo taj zvuk s četiri trećine.
(Pištanje)
(Pištanje)
09:53
What happens?
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Što se dogodi?
09:55
You get this sound. (Beep)
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Dobijemo ovaj zvuk. (Pištanje)
09:57
This is the perfect fourth.
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Ovo je čista kvarta.
09:58
If the first one is an A, this is a D.
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Ako je prvi ton A, ovo je D.
10:00
They sound like this together. (Beeps)
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Ovako zvuče zajedno. (Pištanje)
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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Ovo što sad radim je
mijenjanje perspektive.
mijenjanje perspektive.
10:07
I'm just viewing a number
from another perspective.
from another perspective.
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Samo gledam broj iz druge perspektive.
10:10
I can even do this with rhythms, right?
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Ovo čak mogu raditi s 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 odabrati ritam i pustiti
tri takta istovremeno (Bubnjevi)
tri takta istovremeno (Bubnjevi)
10:16
in a period of time,
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u određenom vremenskom razdoblju
10:18
and I can play another sound
four times in that same space.
four times in that same space.
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i mogu pustiti neki zvuk
četiri puta u istom razdoblju.
četiri puta u istom razdoblju.
10:22
(Clanking sounds)
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(Zveckanje)
10:23
Sounds kind of boring,
but listen to them together.
but listen to them together.
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Zvuči malo dosadno,
ali poslušajte ih zajedno.
ali poslušajte ih zajedno.
10:25
(Drumbeats and clanking sounds)
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(Bubnjevi i zveckanje)
10:28
(Laughter)
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(Smijeh)
10:30
Hey! So.
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Hej! Tako.
10:31
(Laughter)
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(Smijeh)
10:33
I can even make a little hi-hat.
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Čak mogu napraviti mali fuš.
10:35
(Drumbeats and cymbals)
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(Bubnjevi i činele)
10:37
Can you hear this?
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Možete li ovo čuti?
10:38
So, this is the sound of four-thirds.
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Dakle, ovo je zvuk četiri trećine.
10:40
Again, this is as a rhythm.
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Opet, ovo je njezin ritam.
10:42
(Drumbeats and cowbell)
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(Bubnjevi i kravlje zvono)
10:44
And I can keep doing this
and play games with this number.
and play games with this number.
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Mogu nastaviti ovako
i igrati se ovim brojem.
i igrati se 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 odličan broj.
Volim četiri trećine!
Volim četiri trećine!
10:50
(Laughter)
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(Smijeh)
10:51
Truly -- it's an undervalued number.
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Zaista -- to je podcijenjen 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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Ako uzmete kuglu
i pogledate njezin volumen,
i pogledate njezin volumen,
10:56
it's actually four-thirds
of some particular cylinder.
of some particular cylinder.
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on je zapravo četiri trećine
određenog valjka.
određenog valjka.
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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Dakle, četiri trećine je u kugli.
To je volumen kugle.
To je volumen kugle.
11:03
OK, so why am I doing all this?
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OK, zašto radim sve ovo?
11:05
Well, I want to talk about what it means
to understand something
to understand something
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Pa, želim govoriti o tome što znači
razumjeti nešto
razumjeti nešto
11:09
and what we mean
by understanding something.
by understanding something.
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i što mislimo pod
razumijevanjem nečega.
razumijevanjem nečega.
11:11
That's my aim here.
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To je moj cilj ovdje.
11:13
And my claim is that
you understand something
you understand something
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I tvrdim da razumijete nešto
11:15
if you have the ability to view it
from different perspectives.
from different perspectives.
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ako to možete pogledati
iz različitih perspektiva.
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.
To je prekrasno R, zar ne?
To je prekrasno R, zar ne?
11:20
How do you know that?
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Kako to znate?
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, vidjeli ste puno slova R,
11:25
and you've generalized
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i generalizirali ste
11:27
and abstracted all of these
and found a pattern.
and found a pattern.
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i usvojili sve njih te ste
pronašli uzorak.
pronašli uzorak.
11:30
So you know that this is an R.
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Zato znate da je ovo R.
11:35
So what I'm aiming for here
is saying something
is saying something
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Ovime pokušavam reći nešto
11:38
about how understanding
and changing your perspective
and changing your perspective
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o tome kako su razumijevanje
i promjena perspektive
i promjena 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 učitelj i predavač,
11:45
and I can actually use this
to teach something,
to teach something,
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i mogu zaista iskoristiti ovo za učenje
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 pričam nekome
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 pričam priču s drugog gledišta,
11:55
I enable understanding.
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omogućujem razumijevanje.
11:56
I make understanding possible,
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Ja stvaram uvjete za razumijevanje,
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 generalizirati
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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a ako vam dam drugu perspektivu,
bit će vam lakše.
bit će vam lakše.
12:06
Let's do a simple example again.
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Napravimo opet jednostavan primjer.
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 trokuta.
Ovo su četiri trokuta.
12:10
So this is also four-thirds, in a way.
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Dakle i to su četiri trećine
na neki način.
na neki način.
12:13
Let's just join them together.
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Spojimo ih.
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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Sad ćemo igrati igru, savit ćemo ih
12:17
into a three-dimensional structure.
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u trodimenzionalnu strukturu.
12:19
I love this.
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Volim ovo.
12:20
This is a square pyramid.
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1416
Ovo je četverostrana piramida.
12:22
And let's just take two of them
and put them together.
and put them together.
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Uzmimo dvije i spojimo ih.
12:25
So this is what is called an octahedron.
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Ovo se zove oktaedar.
12:28
It's one of the five platonic solids.
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2707
On je jedan od pet pravilnih poliedara.
12:31
Now we can quite literally
change our perspective,
change our perspective,
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Sad možemo posve doslovno
promijeniti perspektivu,
promijeniti 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 osi
12:36
and view it from different perspectives.
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2012
i pogledati iz različitih perspektiva.
12:38
And I can change the axis,
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Mogu i promijeniti os,
12:40
and then I can view it
from another point of view,
from another point of view,
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2338
i tada ga pogledati s različitog gledišta,
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 to je isto, iako izgleda
malo drugačije.
malo drugačije.
12:45
I can do it even one more time.
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Mogu to učiniti čak još jednom.
12:47
Every time I do this,
something else appears,
something else appears,
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Svaki put kad to učinim,
nešto drugo se pojavi,
nešto drugo se pojavi,
12:50
so I'm actually learning
more about the object
more about the object
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pa zapravo učim više o predmetu
12:52
when I change my perspective.
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kad promijenim perspektivu.
12:54
I can use this as a tool
for creating understanding.
for creating understanding.
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Mogu to koristiti kao alat
za stvaranje razumijevanja.
za stvaranje razumijevanja.
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 uzeti dva ovakva
i spojiti ih zajedno, ovako
i spojiti ih zajedno, ovako
13:02
and see what happens.
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i vidjeti što će se dogoditi.
13:03
And it looks a little bit
like the octahedron.
like the octahedron.
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I pomalo izgleda kao oktaedar.
13:07
Have a look at it if I spin
it around like this.
it around like this.
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Pogledajte ga kad ga zavrtim.
13:09
What happens?
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Što se dogodi?
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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Ako uzmete dva, spojite ih i zavrtite,
13:14
there's your octahedron again,
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opet dobijete oktaedar,
13:16
a beautiful structure.
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prekrasnu strukturu.
13:18
If you lay it out flat on the floor,
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Ako ga rasklopite,
13:20
this is the octahedron.
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ovo je oktaedar.
13:21
This is the graph structure
of an octahedron.
of an octahedron.
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Ovo je grafički prikaz strukture oktaedra.
13:25
And I can continue doing this.
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I mogu to nastaviti raditi.
13:27
You can draw three great circles
around the octahedron,
around the octahedron,
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Možete nacrtati tri velika kruga
oko oktaedra
oko oktaedra
13:31
and you rotate around,
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i kad ga okrećete
13:33
so actually three great circles
is related to the octahedron.
is related to the octahedron.
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zapravo su tri velika kruga
s njim povezana.
s njim povezana.
13:37
And if I take a bicycle pump
and just pump it up,
and just pump it up,
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Ako uzmem pumpu za bicik i napumpam ga,
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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809245
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vidjet ćete da je i ovo
pomalo nalik oktaedru.
pomalo nalik oktaedru.
13:44
Do you see what I'm doing here?
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Vidite li što radim?
13:47
I am changing the perspective every time.
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Svaki put mijenjam perspektivu.
13:50
So let's now take a step back --
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Učinite odmak,
13:53
and that's actually
a metaphor, stepping back --
a metaphor, stepping back --
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i to je zapravo metafora,
učiniti odmak --
učiniti odmak --
13:56
and have a look at what we're doing.
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i pogledajte što radimo.
13:58
I'm playing around with metaphors.
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Igram se metaforama.
14:00
I'm playing around
with perspectives and analogies.
with perspectives and analogies.
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Igram se s perspektivama i analogijama.
14:03
I'm telling one story in different ways.
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Pričam jednu priču na različite 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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Pišem pripovijest.
Pišem nekoliko pripovijesti.
Pišem nekoliko pripovijesti.
14:09
And I think all of these things
make understanding possible.
make understanding possible.
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Mislim da sve ovo omogućuje razumijevanje.
14:13
I think this actually is the essence
of understanding something.
of understanding something.
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Mislim da je ovo srž razumijevanja nečega.
14:16
I truly believe this.
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Zaista to vjerujem.
14:18
So this thing about changing
your perspective --
your perspective --
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Tako da je ovo o mijenjanju perspektive --
14:20
it's absolutely fundamental for humans.
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apsolutni temelj za ljude.
14:23
Let's play around with the Earth.
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Igrajmo se 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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Pogledajmo izbliza ocean,
pogledajmo ocean.
pogledajmo ocean.
14:28
We can do this with anything.
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Možemo ovo učiniti sa svime.
14:29
We can take the ocean
and view it up close.
and view it up close.
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Možemo ocean pogledati izbliza.
14:32
We can look at the waves.
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Možemo pogledati valove.
14:34
We can go to the beach.
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Možemo ići na plažu.
14:35
We can view the ocean
from another perspective.
from another perspective.
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Možemo pogledati ocean
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 učinimo,
naučimo još nešto o oceanu.
naučimo još nešto o oceanu.
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 na obalu
možemo je namirisati, zar ne?
možemo je namirisati, zar ne?
14:43
We can hear the sound of the waves.
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Možemo čuti zvuk valova.
14:45
We can feel salt on our tongues.
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Možemo osjetiti sol na jeziku.
14:47
So all of these
are different perspectives.
are different perspectives.
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Sve to su različite 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 ući u vodu.
14:53
We can see the water from the inside.
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2009
Možemo vidjeti vodu iznutra.
14:55
And you know what?
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I znate što?
14:56
This is absolutely essential
in mathematics and computer science.
in mathematics and computer science.
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Ovo je posve 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 vidjeti strukturu iznutra,
15:02
then you really learn something about it.
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tada zaista naučite nešto o njoj.
15:05
That's somehow the essence of something.
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2021
To je nekako jezgra nečega.
15:07
So when we do this,
and we've taken this journey
and we've taken this journey
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3643
Kad radimo ovo, i putujemo
15:11
into the ocean,
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899550
1173
u ocean,
15:12
we use our imagination.
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1890
koristimo našu maštu.
15:14
And I think this is one level deeper,
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2653
Mislim da je ovo jedna razina dublje,
15:17
and it's actually a requirement
for changing your perspective.
for changing your perspective.
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i zapravo je uvijet
za mijenjanje perspektive.
za mijenjanje perspektive.
15:21
We can do a little game.
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Možemo se igrati.
15:23
You can imagine that you're sitting there.
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Možete zamisliti da sjedite ondje.
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
Možete zamisliti da ste ovdje gore,
i da sjedite ovdje.
i da sjedite ovdje.
15:28
You can view yourselves from the outside.
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2326
Možete se pogledati izvana.
15:30
That's really a strange thing.
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To je zaista čudno.
15:32
You're changing your perspective.
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1823
Možete promijeniti svoju 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 vidite se izvana.
15:39
That requires imagination.
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2029
To zahtijeva maštu.
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
4933
Matematika i računarstvo su
najmaštovitiji umjetnički izrazi ikada.
najmaštovitiji umjetnički izrazi ikada.
15:46
And this thing about changing perspectives
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2182
I ovo o mijenjanju perspektive
15:49
should sound a little bit familiar to you,
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2508
bi vam trebalo zvučati poznato,
15:51
because we do it every day.
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2212
jer to radimo svaki dan.
15:54
And then it's called empathy.
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1620
A tada to zovemo suosjećanjem
15:56
When I view the world
from your perspective,
from your perspective,
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2699
Kad ja gledam svijet iz tvoje perspektive,
16:00
I have empathy with you.
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ja suosjećam s vama.
16:02
If I really, truly understand
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1848
Ako ja stvarno, zaista razumijem
16:04
what the world looks
like from your perspective,
like from your perspective,
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3078
kako svijet izgleda iz vaše perspektive,
16:07
I am empathetic.
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1471
ja sam suosjećajan.
16:09
That requires imagination.
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2180
To zahtijeva maštu.
16:11
And that is how we obtain understanding.
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959827
2459
Tako mi dobivamo razumijevanje.
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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To je diljem matematike
i to je diljem računarstva,
i to je diljem računarstva,
16:18
and there's a really deep connection
between empathy and these sciences.
between empathy and these sciences.
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i postoji zaista duboka veza
između suosjećanja i ovih znanosti.
između suosjećanja i ovih znanosti.
16:25
So my conclusion is the following:
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Stoga je moj zaključak sljedeći:
16:29
understanding something really deeply
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2222
razumjeti nešto na zaista dubokoj razini
16:32
has to do with the ability
to change your perspective.
to change your perspective.
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ima veze sa sposobnošću
da promijenite perspektivu.
da promijenite perspektivu.
16:35
So my advice to you is:
try to change your perspective.
try to change your perspective.
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Zato je moj savjet vama:
pokušajte promijeniti perspektivu.
pokušajte promijeniti perspektivu.
16:39
You can study mathematics.
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Možete proučavati matematiku.
16:41
It's a wonderful way to train your brain.
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To je prekrasan način da vježbate mozak.
16:44
Changing your perspective
makes your mind more flexible.
makes your mind more flexible.
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Mijenjanje perspektive
čini vaš um fleksibilnijim.
čini vaš um fleksibilnijim.
16:48
It makes you open to new things,
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To vas otvara novim stvarima,
16:50
and it makes you
able to understand things.
able to understand things.
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i omogućuje vam da razumijete stvari.
16:53
And to use yet another metaphor:
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Da iskoristim drugu metaforu:
16:55
have a mind like water.
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neka vam um bude kao voda.
16:56
That's nice.
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To je lijepo.
16:57
Thank you.
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Hvala.
16:59
(Applause)
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(Pljesak)
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