# Metaphors, mathematics & the imagination | Roger Antonsen | TEDxOslo

Hi. I want to talk about understanding,

and the nature of understanding, and what the essence of understanding is, because understanding is something

we aim for, everyone. We want to understand things. My claim is that understanding has to do with the ability to change

your perspective. If you don’t have that,

you don’t have understanding. So that is my claim. And I want to focus on mathematics. Many of us think of mathematics

as addition, subtraction, multiplication, division, fractions, percent, geometry,

algebra — all that stuff. But actually, I want to talk

about the essence of mathematics as well. And my claim is that mathematics

has to do with patterns. Behind me, you see a beautiful pattern, and this pattern actually emerges

just from drawing circles in a very particular way. So my day-to-day definition

of mathematics that I use every day is the following: First of all, it’s about finding patterns. And by “pattern,” I mean a connection,

a structure, some regularity, some rules that govern what we see. Second of all, I think it is about representing

these patterns with a language. We make up language if we don’t have it, and in mathematics, this is essential. It’s also about making assumptions and playing around with these assumptions

and just seeing what happens. We’re going to do that very soon. And finally, it’s about doing cool stuff. Mathematics enables us

to do so many things. So let’s have a look at these patterns. If you want to tie a tie knot, there are patterns. Tie knots have names. And you can also do

the mathematics of tie knots. This is a left-out, right-in,

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

left-in, center-out and tie. This is a language we made up

for the patterns of tie knots, and a half-Windsor is all that. This is a mathematics book

about tying shoelaces at the university level, because there are patterns in shoelaces. You can do it in so many different ways. We can analyze it. We can make up languages for it. And representations

are all over mathematics. This is Leibniz’s notation from 1675. He invented a language

for patterns in nature. When we throw something up in the air, it falls down. Why? We’re not sure, but we can represent

this with mathematics in a pattern. This is also a pattern. This is also an invented language. Can you guess for what? It is actually a notation system

for dancing, for tap dancing. That enables him as a choreographer

to do cool stuff, to do new things, because he has represented it. I want you to think about how amazing

representing something actually is. Here it says the word “mathematics.” But actually, they’re just dots, right? So how in the world can these dots

represent the word? Well, they do. They represent the word “mathematics,” and these symbols also represent that word and this we can listen to. It sounds like this. (Beeps) Somehow these sounds represent

the word and the concept. How does this happen? There’s something amazing

going on about representing stuff. So I want to talk about

that magic that happens when we actually represent something. Here you see just lines

with different widths. They stand for numbers

for a particular book. And I can actually recommend

this book, it’s a very nice book. (Laughter) Just trust me. OK, so let’s just do an experiment, just to play around

with some straight lines. This is a straight line. Let’s make another one. So every time we move,

we move one down and one across, and we draw a new straight line, right? We do this over and over and over, and we look for patterns. So this pattern emerges, and it’s a rather nice pattern. It looks like a curve, right? Just from drawing simple, straight lines. Now I can change my perspective

a little bit. I can rotate it. Have a look at the curve. What does it look like? Is it a part of a circle? It’s actually not a part of a circle. So I have to continue my investigation

and look for the true pattern. Perhaps if I copy it and make some art? Well, no. Perhaps I should extend

the lines like this, and look for the pattern there. Let’s make more lines. We do this. And then let’s zoom out

and change our perspective again. Then we can actually see that

what started out as just straight lines is actually a curve called a parabola. This is represented by a simple equation, and it’s a beautiful pattern. So this is the stuff that we do. We find patterns, and we represent them. And I think this is a nice

day-to-day definition. But today I want to go

a little bit deeper, and think about

what the nature of this is. What makes it possible? There’s one thing

that’s a little bit deeper, and that has to do with the ability

to change your perspective. And I claim that when

you change your perspective, and if you take another point of view, you learn something new

about what you are watching or looking at or hearing. And I think this is a really important

thing that we do all the time. So let’s just look at

this simple equation, x + x=2 • x. This is a very nice pattern,

and it’s true, because 5 + 5=2 • 5, etc. We’ve seen this over and over,

and we represent it like this. But think about it: this is an equation. It says that something

is equal to something else, and that’s two different perspectives. One perspective is, it’s a sum. It’s something you plus together. On the other hand, it’s a multiplication, and those are two different perspectives. And I would go as far as to say

that every equation is like this, every mathematical equation

where you use that equality sign is actually a metaphor. It’s an analogy between two things. You’re just viewing something

and taking two different points of view, and you’re expressing that in a language. Have a look at this equation. This is one of the most

beautiful equations. It simply says that, well, two things, they’re both -1. This thing on the left-hand side is -1,

and the other one is. And that, I think, is one

of the essential parts of mathematics — you take

different points of view. So let’s just play around. Let’s take a number. We know four-thirds.

We know what four-thirds is. It’s 1.333, but we have to have

those three dots, otherwise it’s not exactly four-thirds. But this is only in base 10. You know, the number system,

we use 10 digits. If we change that around

and only use two digits, that’s called the binary system. It’s written like this. So we’re now talking about the number. The number is four-thirds. We can write it like this, and we can change the base,

change the number of digits, and we can write it differently. So these are all representations

of the same number. We can even write it simply,

like 1.3 or 1.6. It all depends on

how many digits you have. Or perhaps we just simplify

and write it like this. I like this one, because this says

four divided by three. And this number expresses

a relation between two numbers. You have four on the one hand

and three on the other. And you can visualize this in many ways. What I’m doing now is viewing that number

from different perspectives. I’m playing around. I’m playing around with

how we view something, and I’m doing it very deliberately. We can take a grid. If it’s four across and three up,

this line equals five, always. It has to be like this.

This is a beautiful pattern. Four and three and five. And this rectangle, which is 4 x 3, you’ve seen a lot of times. This is your average computer screen. 800 x 600 or 1,600 x 1,200 is a television or a computer screen. So these are all nice representations, but I want to go a little bit further

and just play more with this number. Here you see two circles.

I’m going to rotate them like this. Observe the upper-left one. It goes a little bit faster, right? You can see this. It actually goes exactly

four-thirds as fast. That means that when it goes

around four times, the other one goes around three times. Now let’s make two lines, and draw

this dot where the lines meet. We get this dot dancing around. (Laughter) And this dot comes from that number. Right? Now we should trace it. Let’s trace it and see what happens. This is what mathematics is all about. It’s about seeing what happens. And this emerges from four-thirds. I like to say that this

is the image of four-thirds. It’s much nicer — (Cheers) Thank you! (Applause) This is not new. This has been known

for a long time, but — (Laughter) But this is four-thirds. Let’s do another experiment. Let’s now take a sound, this sound: (Beep) This is a perfect A, 440Hz. Let’s multiply it by two. We get this sound. (Beep) When we play them together,

it sounds like this. This is an octave, right? We can do this game. We can play

a sound, play the same A. We can multiply it by three-halves. (Beep) This is what we call a perfect fifth. (Beep) They sound really nice together. Let’s multiply this sound

by four-thirds. (Beep) What happens? You get this sound. (Beep) This is the perfect fourth. If the first one is an A, this is a D. They sound like this together. (Beeps) This is the sound of four-thirds. What I’m doing now,

I’m changing my perspective. I’m just viewing a number

from another perspective. I can even do this with rhythms, right? I can take a rhythm and play

three beats at one time (Drumbeats) in a period of time, and I can play another sound

four times in that same space. (Clanking sounds) Sounds kind of boring,

but listen to them together. (Drumbeats and clanking sounds) (Laughter) Hey! So. (Laughter) I can even make a little hi-hat. (Drumbeats and cymbals) Can you hear this? So, this is the sound of four-thirds. Again, this is as a rhythm. (Drumbeats and cowbell) And I can keep doing this

and play games with this number. Four-thirds is a really great number.

I love four-thirds! (Laughter) Truly — it’s an undervalued number. So if you take a sphere and look

at the volume of the sphere, it’s actually four-thirds

of some particular cylinder. So four-thirds is in the sphere.

It’s the volume of the sphere. OK, so why am I doing all this? Well, I want to talk about what it means

to understand something and what we mean

by understanding something. That’s my aim here. And my claim is that

you understand something if you have the ability to view it

from different perspectives. Let’s look at this letter.

It’s a beautiful R, right? How do you know that? Well, as a matter of fact,

you’ve seen a bunch of R’s, and you’ve generalized and abstracted all of these

and found a pattern. So you know that this is an R. So what I’m aiming for here

is saying something about how understanding

and changing your perspective are linked. And I’m a teacher and a lecturer, and I can actually use this

to teach something, because when I give someone else

another story, a metaphor, an analogy, if I tell a story

from a different point of view, I enable understanding. I make understanding possible, because you have to generalize

over everything you see and hear, and if I give you another perspective,

that will become easier for you. Let’s do a simple example again. This is four and three.

This is four triangles. So this is also four-thirds, in a way. Let’s just join them together. Now we’re going to play a game;

we’re going to fold it up into a three-dimensional structure. I love this. This is a square pyramid. And let’s just take two of them

and put them together. So this is what is called an octahedron. It’s one of the five platonic solids. Now we can quite literally

change our perspective, because we can rotate it

around all of the axes and view it from different perspectives. And I can change the axis, and then I can view it

from another point of view, but it’s the same thing,

but it looks a little different. I can do it even one more time. Every time I do this,

something else appears, so I’m actually learning

more about the object when I change my perspective. I can use this as a tool

for creating understanding. I can take two of these

and put them together like this and see what happens. And it looks a little bit

like the octahedron. Have a look at it if I spin

it around like this. What happens? Well, if you take two of these,

join them together and spin it around, there’s your octahedron again, a beautiful structure. If you lay it out flat on the floor, this is the octahedron. This is the graph structure

of an octahedron. And I can continue doing this. You can draw three great circles

around the octahedron, and you rotate around, so actually three great circles

is related to the octahedron. And if I take a bicycle pump

and just pump it up, you can see that this is also

a little bit like the octahedron. Do you see what I’m doing here? I am changing the perspective every time. So let’s now take a step back — and that’s actually

a metaphor, stepping back — and have a look at what we’re doing. I’m playing around with metaphors. I’m playing around

with perspectives and analogies. I’m telling one story in different ways. I’m telling stories. I’m making a narrative;

I’m making several narratives. And I think all of these things

make understanding possible. I think this actually is the essence

of understanding something. I truly believe this. So this thing about changing

your perspective — it’s absolutely fundamental for humans. Let’s play around with the Earth. Let’s zoom into the ocean,

have a look at the ocean. We can do this with anything. We can take the ocean

and view it up close. We can look at the waves. We can go to the beach. We can view the ocean

from another perspective. Every time we do this, we learn

a little bit more about the ocean. If we go to the shore,

we can kind of smell it, right? We can hear the sound of the waves. We can feel salt on our tongues. So all of these

are different perspectives. And this is the best one. We can go into the water. We can see the water from the inside. And you know what? This is absolutely essential

in mathematics and computer science. If you’re able to view

a structure from the inside, then you really learn something about it. That’s somehow the essence of something. So when we do this,

and we’ve taken this journey into the ocean, we use our imagination. And I think this is one level deeper, and it’s actually a requirement

for changing your perspective. We can do a little game. You can imagine that you’re sitting there. You can imagine that you’re up here,

and that you’re sitting here. You can view yourselves from the outside. That’s really a strange thing. You’re changing your perspective. You’re using your imagination, and you’re viewing yourself

from the outside. That requires imagination. Mathematics and computer science

are the most imaginative art forms ever. And this thing about changing perspectives should sound a little bit familiar to you, because we do it every day. And then it’s called empathy. When I view the world

from your perspective, I have empathy with you. If I really, truly understand what the world looks

like from your perspective, I am empathetic. That requires imagination. And that is how we obtain understanding. And this is all over mathematics

and this is all over computer science, and there’s a really deep connection

between empathy and these sciences. So my conclusion is the following: understanding something really deeply has to do with the ability

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

try to change your perspective. You can study mathematics. It’s a wonderful way to train your brain. Changing your perspective

makes your mind more flexible. It makes you open to new things, and it makes you

able to understand things. And to use yet another metaphor: have a mind like water. That’s nice. Thank you. (Applause)

Simply Amazing

What you see here is a bunch of R's hahaha

nice video and helps to chage our perspective 🙂

I dislike.

One person said, "Dis I like"

Change of perspective!!!

What is the book he is referring to when he references the bar code?

subtitulo en español desfasado!

POSSIBLE to meet the only person who DISLIKE this vidéo ?

Nice perspective you are taking – Gregory Bateson & Arthur Koestler would have liked it a lot, and Gilles Fauconnier & Mark Turner, too.

Thats my teacher 🙂

video was recorded nearly silent

Superb sir..

Muy buenas intenciones, pero no conectó con el público. Se puede ver en los rostros de los espectadores.

1000 Gold coins for TEDx Oslo!

Metaphor and analogy are two different things. I cannot make out which one the equal sign represents! I really need to find it out as I'm going to use this video in a school project of mine.