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Roger Antonsen: Math is the hidden secret to understanding the world
Roger Antonsen: Matematik er den gemte hemmelighed til forståelse af verden
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Oplev verdens mystik og kernen af livet, igennem en af de mest kreative kunstformer der findes -- matematikken-- sammen med Roger Antonsen. I dette oplysende foredrag fortæller han hvordan en lille forskydning i perspektiv kan afsløre mønstre, numre og formler som bliver genveje til empati og forståelse.
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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Hej.
00:14
I want to talk about understanding,
and the nature of understanding,
and the nature of understanding,
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Jeg vil tale om forståelse, og om begrebet forståelse,
00:18
and what the essence of understanding is,
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og hvad essensen af forståelse er,
00:21
because understanding is something
we aim for, everyone.
we aim for, everyone.
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for det at forstå, er noget vi alle efterstræber.
00:24
We want to understand things.
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Vi ønsker at forstå ting.
00:27
My claim is that understanding has to do
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Min påstand er, at forståelse har at gøre med
00:30
with the ability to change
your perspective.
your perspective.
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evnen til at ændre sit perspektiv.
00:32
If you don't have that,
you don't have understanding.
you don't have understanding.
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Evner du ikke det, har du ikke forståelse.
00:36
So that is my claim.
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Så det er min påstand.
00:37
And I want to focus on mathematics.
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Og jeg vil fokusere på matematik.
00:40
Many of us think of mathematics
as addition, subtraction,
as addition, subtraction,
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Mange af os tænker på matematik som addition, subtraktion,
00:43
multiplication, division,
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multiplikation, divison, fraktioner,
00:45
fractions, percent, geometry,
algebra -- all that stuff.
algebra -- all that stuff.
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procenter, geometri, algebra -- alt det der.
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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Men faktisk, vil jeg også tale om essensen af matematik.
00:53
And my claim is that mathematics
has to do with patterns.
has to do with patterns.
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Min påstand er, at matematik har noget at gøre med mønstre.
00:57
Behind me, you see a beautiful pattern,
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Bag mig ser I et smukt mønster,
00:59
and this pattern actually emerges
just from drawing circles
just from drawing circles
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dette mønster opstår, blot ved at anvende cirkler
01:03
in a very particular way.
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på en ganske særlig måde.
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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Så min hverdags definition af matematik, som jeg bruger hver dag,
01:10
is the following:
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er følgende:
01:12
First of all, it's about finding patterns.
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Først og fremmest handler det om, at finde mønstre.
01:16
And by "pattern," I mean a connection,
a structure, some regularity,
a structure, some regularity,
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Og med "mønstre", mener jeg en sammenhæng, en struktur, noget regelmæssighed,
01:21
some rules that govern what we see.
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nogle regler, der bestemmer hvad vi ser.
01:24
Second of all,
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Dernæst,
tænker jeg på hvordan jeg omsætter disse mønstre til et sprog.
01:25
I think it is about representing
these patterns with a language.
these patterns with a language.
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01:29
We make up language if we don't have it,
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Vi må opfinde et sprog, hvis vi endnu ikke har det,
01:31
and in mathematics, this is essential.
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og i matematik, er dette essentielt.
01:35
It's also about making assumptions
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Matematik handler også om, at antage,
01:36
and playing around with these assumptions
and just seeing what happens.
and just seeing what happens.
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og at lege med disse antagelser og se hvad der sker.
01:40
We're going to do that very soon.
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Det kommer vi snart til.
01:42
And finally, it's about doing cool stuff.
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Og endelig, drejer det sig om at gøre seje ting.
01:46
Mathematics enables us
to do so many things.
to do so many things.
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Matematik gør os i stand til at gøre så mange ting.
01:50
So let's have a look at these patterns.
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Så lad os se på disse mønstre.
01:52
If you want to tie a tie knot,
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Hvis du ønsker at binde en slipseknude,
01:55
there are patterns.
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er der mønstre.
01:56
Tie knots have names.
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Slipseknuder har navne.
01:58
And you can also do
the mathematics of tie knots.
the mathematics of tie knots.
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Og vi kan også bruge matematik på slipseknuder.
02:00
This is a left-out, right-in,
center-out and tie.
center-out and tie.
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Dette er en venstre-ud, højre-ind, midt-ud og stram til.
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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Dette er en venstre-ind, højre-ud, venstre-ind, midt-ud og stram til.
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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Dette er et sprog vi har opfundet, for at beskrive slipseknude-mønstre,
02:12
and a half-Windsor is all that.
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og en halv-Windsor er alt dette.
02:15
This is a mathematics book
about tying shoelaces
about tying shoelaces
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Dette er en matematikbog om at binde snørebånd,
02:18
at the university level,
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på universitets niveau,
02:19
because there are patterns in shoelaces.
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fordi, der er mønstre i at binde snørebånd.
02:21
You can do it in so many different ways.
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Vi kan gøre det på så mange forskellige måder.
02:23
We can analyze it.
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Vi kan analysere det.
02:25
We can make up languages for it.
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Vi kan opfinde sprog kun til det.
02:28
And representations
are all over mathematics.
are all over mathematics.
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Og der findes den slags eksempler over det hele i matematik.
02:31
This is Leibniz's notation from 1675.
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Dette er Leibnitz's notation fra 1675.
02:35
He invented a language
for patterns in nature.
for patterns in nature.
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Han opfandt et sprog for mønstre i naturen.
02:39
When we throw something up in the air,
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Når vi kaster noget op i luften,
02:41
it falls down.
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falder det ned.
02:42
Why?
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Hvorfor?
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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Vi er ikke sikre, men vi kan beskrive dette med matematik i mønstre.
02:48
This is also a pattern.
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Dette er også et mønster.
02:49
This is also an invented language.
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Dette er også et opfundet sprog.
02:52
Can you guess for what?
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Kan I gætte for hvad?
02:55
It is actually a notation system
for dancing, for tap dancing.
for dancing, for tap dancing.
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Det er faktisk et notations-system for dans, for stepdans.
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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Dette gør det muligt for koreografen at gøre nogle seje ting, at gøre nye ting,
03:04
because he has represented it.
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fordi han har opfundet en beskrivelse for det.
03:07
I want you to think about how amazing
representing something actually is.
representing something actually is.
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Prøv at tænke over, hvor fantastisk det at kunne beskrive noget egentlig er.
03:12
Here it says the word "mathematics."
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Her står der "matematik".
03:15
But actually, they're just dots, right?
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Men faktisk, er det bare prikker. Ikke?
03:18
So how in the world can these dots
represent the word?
represent the word?
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Så hvordan i alverden kan disse prikker rent faktisk repræsentere dette ord?
03:21
Well, they do.
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Jamen, det gør de.
03:23
They represent the word "mathematics,"
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De repræsenterer ordet: "matematik",
03:25
and these symbols also represent that word
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og disse symboler repræsenterer også det ord,
03:27
and this we can listen to.
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og dette kan vi høre.
03:29
It sounds like this.
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Det lyder sådan her:
03:30
(Beeps)
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(bippelyde)
03:32
Somehow these sounds represent
the word and the concept.
the word and the concept.
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På forunderlig vis, repræsenterer disse lyde ordet og konceptet bag ordet.
03:36
How does this happen?
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Hvordan sker dette?
03:37
There's something amazing
going on about representing stuff.
going on about representing stuff.
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Der er noget forbløffende ved at kunne beskrive ting.
03:41
So I want to talk about
that magic that happens
that magic that happens
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Jeg vil gerne snakke om, den magiske ting der opstår
03:47
when we actually represent something.
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når vi rent faktisk beskriver noget.
03:49
Here you see just lines
with different widths.
with different widths.
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Her ser du bare linjer med forskellig afstand.
03:52
They stand for numbers
for a particular book.
for a particular book.
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De står for numre i en bestemt bog.
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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Og jeg kan faktisk anbefale denne bog, det er en god bog.
03:58
(Laughter)
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(latter)
03:59
Just trust me.
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Bare stol på mig.
04:01
OK, so let's just do an experiment,
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Ok, så lad os lave et eksperiment,
04:03
just to play around
with some straight lines.
with some straight lines.
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bare for at lege med nogle lige linjer.
04:06
This is a straight line.
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Det er en lige linje.
04:07
Let's make another one.
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Lad os lave en ny.
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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Så hver gang vi bevæger den, bevæger vi den en ned og en på tværs,
04:11
and we draw a new straight line, right?
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så vi tegner en ny, lige linje, ikke sandt?
04:13
We do this over and over and over,
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Dette gør vi igen og igen,
04:16
and we look for patterns.
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og leder efter mønstre.
04:17
So this pattern emerges,
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Dette mønster opstår,
04:20
and it's a rather nice pattern.
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og det er et ret pænt mønster.
04:22
It looks like a curve, right?
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Det ligner en kurve, ikke sandt?
04:24
Just from drawing simple, straight lines.
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Det opstår blot fra at tegne simple, lige linjer.
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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Nu kan jeg ændre mit perspektiv en smule. Jeg kan rotere det.
04:30
Have a look at the curve.
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Se lige den kurve.
04:32
What does it look like?
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Hvad ligner det?
04:33
Is it a part of a circle?
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Er det en del af en cirkel?
04:35
It's actually not a part of a circle.
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Det er faktisk ikke en del af en cirkel.
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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Så jeg må fortsætte mit detektivarbejde, og lede efter et korrekt mønster.
04:41
Perhaps if I copy it and make some art?
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Måske hvis jeg kopierer det, og laver noget kunst?
04:45
Well, no.
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Nå. Ikke alligevel.
04:46
Perhaps I should extend
the lines like this,
the lines like this,
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Måske skulle jeg udvide linjerne på denne måde,
04:49
and look for the pattern there.
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og lede efter mønstret dér.
04:50
Let's make more lines.
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Lad os tegne flere linjer.
04:52
We do this.
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Vi gør sådan.
04:53
And then let's zoom out
and change our perspective again.
and change our perspective again.
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Lad os således zoome ud, og ændre vort perspektiv igen.
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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Så kan vi faktisk se, at hvad der startede som lige linjer
05:01
is actually a curve called a parabola.
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faktisk er en kurve der hedder en parabel.
05:03
This is represented by a simple equation,
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Dette er repræsenteret ved en simpel ligning,
05:07
and it's a beautiful pattern.
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og det er et smukt mønster.
05:09
So this is the stuff that we do.
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Så, dette er det arbejde vi gør.
05:11
We find patterns, and we represent them.
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Vi finder mønstre, og så beskriver vi dem.
05:13
And I think this is a nice
day-to-day definition.
day-to-day definition.
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Jeg synes at dette er en god almen definition.
05:16
But today I want to go
a little bit deeper,
a little bit deeper,
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Men i dag vil jeg grave lidt dybere ned,
05:18
and think about
what the nature of this is.
what the nature of this is.
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og tænke på hvad dette arbejde egentlig er.
05:22
What makes it possible?
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Hvad gør det muligt?
05:24
There's one thing
that's a little bit deeper,
that's a little bit deeper,
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Der er en ting som graver et spadestik dybere ned,
05:26
and that has to do with the ability
to change your perspective.
to change your perspective.
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og det har at gøre med evnen til at ændre sit perspektiv.
05:30
And I claim that when
you change your perspective,
you change your perspective,
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Og min påstand er, at når du ændrer perspektiv,
05:32
and if you take another point of view,
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og hvis du indtager et andet synspunkt,
05:35
you learn something new
about what you are watching
about what you are watching
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så lærer du noget nyt, om det du ser på,
05:39
or looking at or hearing.
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betragter eller lytter til.
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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Jeg tror det er en umådeligt vigtig ting, som vi gør hele tiden.
05:45
So let's just look at
this simple equation,
this simple equation,
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Så lad os se på denne simple ligning,
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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Det er et meget fint mønster, og det er sandt,
05:54
because 5 + 5 = 2 • 5, etc.
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fordi 5 + 5 = 2 • 5, og så videre.
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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Vi har set dette igen og igen - og vi beskriver det sådan her.
06:00
But think about it: this is an equation.
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Men tænk på dette: Dette er en ligning.
06:03
It says that something
is equal to something else,
is equal to something else,
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Den siger at noget er lig med noget andet,
06:05
and that's two different perspectives.
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og det er to forskellige perspektiver.
06:07
One perspective is, it's a sum.
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Ét perspektiv er, at det er en sum.
06:09
It's something you plus together.
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Det er noget du adderer.
06:11
On the other hand, it's a multiplication,
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På den anden side er det en multiplikation,
06:14
and those are two different perspectives.
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og det er to forskellige perspektiver.
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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Jeg vil gå så langt som at sige, at hver eneste ligning er sådan her;
06:20
every mathematical equation
where you use that equality sign
where you use that equality sign
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hver eneste matematiske ligning hvor du bruger dette lighedstegn,
06:25
is actually a metaphor.
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faktisk er en metafor.
06:26
It's an analogy between two things.
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Det er en analogi imellem to ting.
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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Du betragter blot noget, og indtager to forskellige syn på det,
06:32
and you're expressing that in a language.
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og dette beskriver du så i et sprog.
06:34
Have a look at this equation.
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Se denne ligning.
06:36
This is one of the most
beautiful equations.
beautiful equations.
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Det er én af de smukkeste ligninger.
06:38
It simply says that, well,
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Ligningen siger blot, at
06:41
two things, they're both -1.
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to ting er lig med -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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På venstre side er den -1, og det er den anden også.
06:47
And that, I think, is one
of the essential parts
of the essential parts
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Dette er, synes jeg, én af de essentielle dele af matematik--
06:50
of mathematics -- you take
different points of view.
different points of view.
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du indtager to forskellige synspunkter.
06:52
So let's just play around.
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Så, lad os lege lidt.
06:53
Let's take a number.
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Vi tager et nummer.
06:55
We know four-thirds.
We know what four-thirds is.
We know what four-thirds is.
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Vi kender fire tredjedele.
Vi ved hvad fire tredjedele er.
Vi ved hvad fire tredjedele er.
06:58
It's 1.333, but we have to have
those three dots,
those three dots,
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Det er 1.333..., men vi skal have de sidste tre prikker med,
07:01
otherwise it's not exactly four-thirds.
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ellers er det faktisk ikke præcist fire tredjedele.
07:03
But this is only in base 10.
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Men, dette er kun i 10-talssystemet.
07:05
You know, the number system,
we use 10 digits.
we use 10 digits.
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I ved, at vort nummersystem kun bruger 10 cifre.
07:08
If we change that around
and only use two digits,
and only use two digits,
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Hvis vi ændrer på dét, og kun anvender 2 cifre,
07:10
that's called the binary system.
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kaldes det det binære talsystem,
07:12
It's written like this.
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som skrives sådan:
07:13
So we're now talking about the number.
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Så nu taler vi om dette nummer.
07:15
The number is four-thirds.
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Tallet er tre fire tredjedele.
07:17
We can write it like this,
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Vi kan skrive det sådan her:
07:19
and we can change the base,
change the number of digits,
change the number of digits,
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Og vi kan ændre basen, altså antallet af cifre,
07:22
and we can write it differently.
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så kan vi skrive det anderledes.
07:24
So these are all representations
of the same number.
of the same number.
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Så, disse tal er alle beskrivelser af det samme nummer.
07:28
We can even write it simply,
like 1.3 or 1.6.
like 1.3 or 1.6.
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Vi kan endda skrive det helt simpelt, som 1.3 eller 1.6.
07:31
It all depends on
how many digits you have.
how many digits you have.
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Det afhænger alt sammen af, hvor mange cifre du har til rådighed.
07:34
Or perhaps we just simplify
and write it like this.
and write it like this.
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Eller vi kan simplificere det, og skrive det således:
07:37
I like this one, because this says
four divided by three.
four divided by three.
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Jeg kan godt lide denne, fordi den siger fire delt med tre.
07:41
And this number expresses
a relation between two numbers.
a relation between two numbers.
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Dette nummer udtrykker en relation mellem to numre.
07:44
You have four on the one hand
and three on the other.
and three on the other.
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På den ene side har du fire, og på den anden har du tre.
07:47
And you can visualize this in many ways.
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Du kan visualisere dette på mange måder.
07:49
What I'm doing now is viewing that number
from different perspectives.
from different perspectives.
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Det jeg gør nu, er at anskue dette nummer fra forskellige perspektiver.
07:53
I'm playing around.
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Jeg leger med det.
07:54
I'm playing around with
how we view something,
how we view something,
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Jeg leger med vores anskuelse af noget,
07:57
and I'm doing it very deliberately.
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og det gør jeg med vilje.
07:58
We can take a grid.
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Lad os tage dette gitter.
08:00
If it's four across and three up,
this line equals five, always.
this line equals five, always.
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Hvis det er fire på tværs og tre op, så er denne linje altid lig fem.
08:04
It has to be like this.
This is a beautiful pattern.
This is a beautiful pattern.
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Det er nødt til at være sådan.
Dette er et smukt mønster.
Dette er et smukt mønster.
08:07
Four and three and five.
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Fire og tre og fem.
08:09
And this rectangle, which is 4 x 3,
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Denne rektangel, som er 4 x 3,
08:11
you've seen a lot of times.
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har I set masser af gange.
08:13
This is your average computer screen.
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Dette er en gennemsnitlig computer skærm.
08:15
800 x 600 or 1,600 x 1,200
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800 x 600 eller 1,600 x 1,200
08:18
is a television or a computer screen.
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er et TV eller en computer skærm.
08:21
So these are all nice representations,
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Dette er alle gode beskrivelser.
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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Men jeg vil godt gå lidt længere,
og lege mere med dette nummer.
og lege mere med dette nummer.
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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Her ser du to cirkler.
Jeg roterer dem lige sådan her:
Jeg roterer dem lige sådan her:
08:31
Observe the upper-left one.
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Se på den øverste til venstre.
08:32
It goes a little bit faster, right?
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Den bevæger sig lidt hurtigere, ikke?
08:35
You can see this.
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Det kan I se.
08:36
It actually goes exactly
four-thirds as fast.
four-thirds as fast.
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Den går faktisk præcist fire tredjedele så hurtigt som den anden.
08:39
That means that when it goes
around four times,
around four times,
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Det betyder at når dén når rundt
fire gange,
fire gange,
08:42
the other one goes around three times.
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når den anden rundt tre gange.
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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Nu laver vi to linjer, og tegner en prik der hvor linjerne mødes.
08:47
We get this dot dancing around.
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Så danser denne prik rundt.
08:49
(Laughter)
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[Latter]
08:50
And this dot comes from that number.
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Denne prik kommer fra det nummer.
08:52
Right? Now we should trace it.
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Ikke? Nu bør vi spore den.
08:55
Let's trace it and see what happens.
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Lad os spore den og se hvad der sker.
08:57
This is what mathematics is all about.
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Dette er, hvad matematik i virkeligheden drejer sig om.
08:59
It's about seeing what happens.
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Det drejer sig om, at se hvad der sker.
09:01
And this emerges from four-thirds.
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Dette opstår ud fra fire tredjedele.
09:04
I like to say that this
is the image of four-thirds.
is the image of four-thirds.
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Jeg kan lide at sige, at dette er billedet på fire tredjedele.
09:07
It's much nicer -- (Cheers)
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Det er meget federe -- [Begejstret hujen]
09:08
Thank you!
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Tak!
09:09
(Applause)
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[Publikum klapper]
09:16
This is not new.
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Dette er ikke noget nyt.
09:17
This has been known
for a long time, but --
for a long time, but --
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Dette har vi vidst i lang tid, men...
09:19
(Laughter)
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[Latter]
09:21
But this is four-thirds.
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Men dette er fire tredjedele.
09:23
Let's do another experiment.
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Lad os lave et andet eksperiment.
09:24
Let's now take a sound, this sound: (Beep)
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Lad os nu tage denne lyd. [Bip]
09:28
This is a perfect A, 440Hz.
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Dette er et perfekt A, 440 Hz.
09:31
Let's multiply it by two.
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Lad os multiplicere det med to.
09:33
We get this sound. (Beep)
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Så får vi denne lyd. [Nyt bip]
09:34
When we play them together,
it sounds like this.
it sounds like this.
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Når vi spiller dem samtidig, så lyder det sådan her: [Nyt bip]
09:37
This is an octave, right?
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Det er en oktav, ikke?
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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Vi kan sagtens spille dette spil. Vi kan afspille en lyd, spille det samme A.
09:41
We can multiply it by three-halves.
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Så kan vi multiplicere det med tre halve.
09:42
(Beep)
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[Nyt bip]
09:44
This is what we call a perfect fifth.
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Dette er hvad vi kalder et perfekt femte.
09:46
(Beep)
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[Nyt bip]
09:47
They sound really nice together.
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De lyder ret godt sammen.
09:49
Let's multiply this sound
by four-thirds. (Beep)
by four-thirds. (Beep)
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Lad os multiplicere denne lyd med fire tredjedele. [Nyt bip]
09:53
What happens?
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Hvad sker der?
09:55
You get this sound. (Beep)
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Du får denne lyd. [Nyt bip]
09:57
This is the perfect fourth.
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Dette, er en perfekt fjerde.
09:58
If the first one is an A, this is a D.
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Hvis den første er et A, så er denne et D.
10:00
They sound like this together. (Beeps)
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De lyder sådan her sammen. [Nyt bip]
10:02
This is the sound of four-thirds.
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Dette er lyden af fire trejdedele.
10:05
What I'm doing now,
I'm changing my perspective.
I'm changing my perspective.
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Det jeg gør lige nu, er at ændre mit perspektiv.
10:07
I'm just viewing a number
from another perspective.
from another perspective.
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Jeg ser blot et nummer fra et andet perspektiv.
10:10
I can even do this with rhythms, right?
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Jeg kan endda gøre dette med rytmer, ikke?
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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Jeg kan tage en rytme, og spille tre beats på samme tid [Tromme rytme]
10:16
in a period of time,
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i et afgrænset tidsrum,
10:18
and I can play another sound
four times in that same space.
four times in that same space.
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og så kan jeg spille en anden lyd, fire gange i det samme tidsrum.
10:22
(Clanking sounds)
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[Anden rytme]
10:23
Sounds kind of boring,
but listen to them together.
but listen to them together.
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Lyder lidt kedeligt, men prøv at lytte til dem sammen.
10:25
(Drumbeats and clanking sounds)
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[Tromme rytme og Anden rytme spiller sammen, arytmisk]
10:28
(Laughter)
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[Latter i salen]
10:30
Hey! So.
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Hey! Såeh.
10:31
(Laughter)
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[Latter i salen]
10:33
I can even make a little hi-hat.
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Jeg kan endda lave en lille bækken lyd.
10:35
(Drumbeats and cymbals)
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[Trommerytme og bækken]
10:37
Can you hear this?
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Kan I høre dette?
10:38
So, this is the sound of four-thirds.
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Dette er så lyden af fire tredjedele.
10:40
Again, this is as a rhythm.
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Og igen; dette som en rytme.
10:42
(Drumbeats and cowbell)
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[Trommerytme og kobjælde]
10:44
And I can keep doing this
and play games with this number.
and play games with this number.
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Dette kan jeg blive ved med at gøre, og lege med dette nummer.
10:47
Four-thirds is a really great number.
I love four-thirds!
I love four-thirds!
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Fire tredjedele er et virkeligt fedt nummer. Jeg elsker fire tredjedele!
10:50
(Laughter)
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[Latter]
10:51
Truly -- it's an undervalued number.
250
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Det er i sandhed, et undervurderet nummer.
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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Hvis vi tager en sfære, og ser på massefylden af denne sfære,
10:56
it's actually four-thirds
of some particular cylinder.
of some particular cylinder.
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så er det faktisk en fire tredjedel af en bestemt cylinder.
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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Så fire tredjedele er i sfæren.
Det er sfærens masse.
Det er sfærens masse.
11:03
OK, so why am I doing all this?
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Ok, så hvorfor gør jeg alt dette?
11:05
Well, I want to talk about what it means
to understand something
to understand something
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Jeg prøver jo at snakke om, hvad det betyder at forstå noget,
11:09
and what we mean
by understanding something.
by understanding something.
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og hvad vi mener, når vi siger vi forstår noget.
11:11
That's my aim here.
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Det er mit mål her.
11:13
And my claim is that
you understand something
you understand something
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Min påstand er, at du forstår noget
11:15
if you have the ability to view it
from different perspectives.
from different perspectives.
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2992
hvis du har evnen til at se det fra forskellige perspektiver.
11:18
Let's look at this letter.
It's a beautiful R, right?
It's a beautiful R, right?
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2541
Lad os kigge på dette bogstav.
Det er et smukt R, ikke?
Det er et smukt R, ikke?
11:20
How do you know that?
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Hvordan véd du det?
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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3188
Jamen, du har jo allerede set en bunke R'er,
11:25
and you've generalized
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1645
og har ud fra dem generaliseret
11:27
and abstracted all of these
and found a pattern.
and found a pattern.
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og har udtrukket alle disse forskellige former, og har fundet et mønster.
11:30
So you know that this is an R.
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Så du véd dette er et R.
11:35
So what I'm aiming for here
is saying something
is saying something
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Så det jeg sigter efter her, er at sige noget om,
11:38
about how understanding
and changing your perspective
and changing your perspective
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hvordan forståelse og dét at ændre sit perspektiv
11:41
are linked.
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1332
er knyttet sammen.
11:43
And I'm a teacher and a lecturer,
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Jeg er lærer, og foredragsholder,
11:45
and I can actually use this
to teach something,
to teach something,
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og kan faktisk bruge dette til at undervise i noget.
11:47
because when I give someone else
another story, a metaphor, an analogy,
another story, a metaphor, an analogy,
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4840
for når jeg giver andre en ny historie, en metafor, en analogi,
11:52
if I tell a story
from a different point of view,
from a different point of view,
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hvis jeg fortæller en historie, fra et andet perspektiv,
11:55
I enable understanding.
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1513
så faciliterer jeg forståelse.
11:56
I make understanding possible,
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1866
Jeg gør forståelse mulig,
11:58
because you have to generalize
over everything you see and hear,
over everything you see and hear,
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3066
fordi du er nødt til at generalisere alt hvad du ser og hører,
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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4599
og hvis jeg forærer dig et andet perspektiv, bliver det nemmere for dig.
12:06
Let's do a simple example again.
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Lad os tage endnu et simpelt eksempel.
12:08
This is four and three.
This is four triangles.
This is four triangles.
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2641
Dette er fire og tre.
Dette er fire triangler.
Dette er fire triangler.
12:10
So this is also four-thirds, in a way.
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2448
Så, dette er også fire tredjedele på en måde.
12:13
Let's just join them together.
280
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1722
Lad os sætte dem sammen.
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
Nu skal vi lege en leg; hvor vi folder denne sammen
12:17
into a three-dimensional structure.
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1682
til en tredimensionel struktur.
12:19
I love this.
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1164
Jeg elsker det her.
12:20
This is a square pyramid.
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1416
Dette er en firkantet pyramide.
12:22
And let's just take two of them
and put them together.
and put them together.
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3150
Lad os tage to af disse, og sætte dem sammen.
12:25
So this is what is called an octahedron.
286
733703
2689
Dette hedder en oktaeder.
12:28
It's one of the five platonic solids.
287
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2707
Den er en af de fem platoniske legemer.
12:31
Now we can quite literally
change our perspective,
change our perspective,
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2445
Nu kan vi bogstaveligt talt ændre vort perspektiv,
12:33
because we can rotate it
around all of the axes
around all of the axes
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2695
fordi vi kan rotere den rundt om alle sine akser
12:36
and view it from different perspectives.
290
744335
2012
og se den fra forskellige perspektiver.
12:38
And I can change the axis,
291
746371
2066
Jeg kan ændre aksen,
12:40
and then I can view it
from another point of view,
from another point of view,
292
748461
2338
og så kan jeg se den fra et andet synspunkt,
12:42
but it's the same thing,
but it looks a little different.
but it looks a little different.
293
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2703
det er den samme ting, men ser en smule anderledes ud.
12:45
I can do it even one more time.
294
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1668
Jeg kan endda gøre det en gang mere.
12:47
Every time I do this,
something else appears,
something else appears,
295
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3302
Hver gang jeg gør dette, opstår noget andet
12:50
so I'm actually learning
more about the object
more about the object
296
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2179
så jeg lærer rent faktisk noget mere om objektet
12:52
when I change my perspective.
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1525
når jeg ændrer mit perspektiv.
12:54
I can use this as a tool
for creating understanding.
for creating understanding.
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3394
Jeg kan bruge dette som et redskab til at skabe forståelse.
12:58
I can take two of these
and put them together like this
and put them together like this
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3592
Jeg kan tage to af disse og sætte dem sammen, sådan her
13:02
and see what happens.
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770164
1247
og se hvad der sker.
13:03
And it looks a little bit
like the octahedron.
like the octahedron.
301
771865
3411
Den ligner en smule den oktaeder vi så.
13:07
Have a look at it if I spin
it around like this.
it around like this.
302
775300
2478
Se den, når jeg snurrer den rundt sådan her.
13:09
What happens?
303
777802
1182
Hvad sker der?
13:11
Well, if you take two of these,
join them together and spin it around,
join them together and spin it around,
304
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3344
Hvis du tager to af disse figurer, sætter dem sammen og snurrer dem rundt,
13:14
there's your octahedron again,
305
782376
2401
så har du din oktaeder igen,
en smuk struktur.
en smuk struktur.
13:16
a beautiful structure.
306
784801
1631
13:18
If you lay it out flat on the floor,
307
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2164
Hvis du lægger den fladt ud på gulvet,
13:20
this is the octahedron.
308
788644
1217
så er dette oktaederen.
13:21
This is the graph structure
of an octahedron.
of an octahedron.
309
789885
2703
Dette er den grafiske struktur af en oktaeder.
13:25
And I can continue doing this.
310
793255
2373
Og sådan kan jeg blive ved.
13:27
You can draw three great circles
around the octahedron,
around the octahedron,
311
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3527
Du kan tegne store cirkler omkring oktaederen,
13:31
and you rotate around,
312
799203
1850
og du kan snurre rundt,
13:33
so actually three great circles
is related to the octahedron.
is related to the octahedron.
313
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4461
så hele tre, store cirkler er nu i forbindelse med oktaederen.
13:37
And if I take a bicycle pump
and just pump it up,
and just pump it up,
314
805562
3659
Hvis jeg tager en cykelpumpe
og puster denne op,
og puster denne op,
13:41
you can see that this is also
a little bit like the octahedron.
a little bit like the octahedron.
315
809245
3153
så kan du se, at denne struktur også minder en smule om oktaederen.
13:44
Do you see what I'm doing here?
316
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2296
Kan I se hvad jeg gør her?
13:47
I am changing the perspective every time.
317
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2681
Jeg ændrer perspektivet hver gang.
13:50
So let's now take a step back --
318
818801
2650
Så lad os tage et skridt tilbage --
13:53
and that's actually
a metaphor, stepping back --
a metaphor, stepping back --
319
821475
3037
det er jo faktisk en metafor, at tage et skridt tilbage --
13:56
and have a look at what we're doing.
320
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2363
og se på hvad vi egentlig laver.
13:58
I'm playing around with metaphors.
321
826923
1664
Nu leger jeg med metaforer.
14:00
I'm playing around
with perspectives and analogies.
with perspectives and analogies.
322
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2472
Jeg leger med perspektiver og analogier.
14:03
I'm telling one story in different ways.
323
831107
2032
Jeg fortæller den samme historie på forskellige måder.
14:05
I'm telling stories.
324
833472
1210
Jeg fortæller historier.
14:06
I'm making a narrative;
I'm making several narratives.
I'm making several narratives.
325
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3184
Jeg kreerer et narrativ; jeg skaber faktisk flere narrativer.
14:09
And I think all of these things
make understanding possible.
make understanding possible.
326
837914
3522
Jeg tror at alle disse metoder, er med til at gøre forståelse mulig.
14:13
I think this actually is the essence
of understanding something.
of understanding something.
327
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3379
Jeg tror faktisk det er essensen i, at forstå noget.
14:16
I truly believe this.
328
844863
1294
Jeg tror virkelig på, det er sådan det fungerer.
14:18
So this thing about changing
your perspective --
your perspective --
329
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2427
Så det med at ændre sit perspektiv,
14:20
it's absolutely fundamental for humans.
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er absolut fundamentalt for mennesker.
14:23
Let's play around with the Earth.
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Lad os lege lidt med Jordkloden.
14:25
Let's zoom into the ocean,
have a look at the ocean.
have a look at the ocean.
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Lad os zoome ind på havet, se på havet.
14:28
We can do this with anything.
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Dette er muligt med alting.
14:29
We can take the ocean
and view it up close.
and view it up close.
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Vi kan se på havet, helt tæt på.
14:32
We can look at the waves.
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Vi kan se på bølgerne.
14:34
We can go to the beach.
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Vi kan gå på stranden.
14:35
We can view the ocean
from another perspective.
from another perspective.
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Vi kan betragte havet fra et alternativt perspektiv.
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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Hver gang vi gør dét, lærer vi en smule mere om havet.
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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Hvis vi går ud til kysten, kan vi faktisk dufte havet. Ikke?
14:43
We can hear the sound of the waves.
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Vi kan høre bølgernes brusen.
14:45
We can feel salt on our tongues.
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Vi kan smage salt på tungen.
14:47
So all of these
are different perspectives.
are different perspectives.
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Alle disse oplevelser, er forskellige perspektiver.
14:50
And this is the best one.
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Og dette er den bedste;
14:51
We can go into the water.
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vi kan gå i vandet.
14:53
We can see the water from the inside.
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Vi kan se vandet indefra.
14:55
And you know what?
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Ved I hvad?
14:56
This is absolutely essential
in mathematics and computer science.
in mathematics and computer science.
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Dette er absolut essentielt for matematisk forskning og IT-videnskab.
14:59
If you're able to view
a structure from the inside,
a structure from the inside,
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Hvis du kan se en struktur indefra,
15:02
then you really learn something about it.
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kan du virkelig lære dig noget om den.
15:05
That's somehow the essence of something.
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Det er, på en måde, essensen af noget.
15:07
So when we do this,
and we've taken this journey
and we've taken this journey
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Så vi gør dette, og vi har taget denne rejse,
15:11
into the ocean,
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ud i havet.
15:12
we use our imagination.
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vi bruger vores fantasi.
15:14
And I think this is one level deeper,
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Jeg tror dette er et niveau dybere,
15:17
and it's actually a requirement
for changing your perspective.
for changing your perspective.
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og er faktisk en nødvendighed for at kunne ændre på sit perspektiv.
15:21
We can do a little game.
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Lad os lave en lille leg.
15:23
You can imagine that you're sitting there.
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Forestil dig, at du sidder dér.
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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Du an forestille dig, at du er heroppe, og at du sidder her.
15:28
You can view yourselves from the outside.
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Du kan se dig selv udefra.
15:30
That's really a strange thing.
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Det er virkelig mærkeligt.
15:32
You're changing your perspective.
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Du ændrer dit perspektiv.
15:34
You're using your imagination,
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Du bruger din fantasi.
15:36
and you're viewing yourself
from the outside.
from the outside.
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Ser dig selv udefra.
15:39
That requires imagination.
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Det kræver fantasi.
15:41
Mathematics and computer science
are the most imaginative art forms ever.
are the most imaginative art forms ever.
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Matematik og IT-videnskab er de mest kreative kunstarter der nogensinde har eksisteret.
15:46
And this thing about changing perspectives
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Det her med at skifte perspektiver;
15:49
should sound a little bit familiar to you,
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burde lyde genkendeligt for jer,
15:51
because we do it every day.
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for vi gør det hver eneste dag.
15:54
And then it's called empathy.
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Det hedder empati.
15:56
When I view the world
from your perspective,
from your perspective,
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Når jeg ser verden, fra dit perspektiv,
16:00
I have empathy with you.
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så har jeg empati for dig.
16:02
If I really, truly understand
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1848
Hvis jeg virkelig, i sandhed forstår
16:04
what the world looks
like from your perspective,
like from your perspective,
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hvordan verden ser ud fra dit perspektiv.
16:07
I am empathetic.
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så er jeg empatisk.
16:09
That requires imagination.
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Det kræver fantasi.
16:11
And that is how we obtain understanding.
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Og det er sådan vi opnår forståelse.
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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Dette går igen i hele matematikkens verden, og i al IT og computer videnskab,
16:18
and there's a really deep connection
between empathy and these sciences.
between empathy and these sciences.
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der er en virkelig dyb forbindelse mellem empati og disse videnskaber.
16:25
So my conclusion is the following:
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Så min konklusion er følgende:
16:29
understanding something really deeply
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At forstå noget, særligt dybt
16:32
has to do with the ability
to change your perspective.
to change your perspective.
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har at gøre med evnen til at ændre sit perspektiv.
16:35
So my advice to you is:
try to change your perspective.
try to change your perspective.
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Så mit råd til her er følgende:
Prøv at ændre dit perspektiv.
Prøv at ændre dit perspektiv.
16:39
You can study mathematics.
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Du kan studere matematik.
16:41
It's a wonderful way to train your brain.
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Det er en vidunderlig måde at træne din hjerne på.
16:44
Changing your perspective
makes your mind more flexible.
makes your mind more flexible.
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At ændre perspektiv gør dit sind mere fleksibelt.
16:48
It makes you open to new things,
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Gør dig mere åben overfor nye ting,
16:50
and it makes you
able to understand things.
able to understand things.
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gør dig i stand til at forstå ting.
16:53
And to use yet another metaphor:
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For nu at bruge endnu en metafor:
16:55
have a mind like water.
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hav et sind som vandet.
16:56
That's nice.
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Det er lækkert.
16:57
Thank you.
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Tak.
16:59
(Applause)
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[Bifald]
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