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)

I wanted to listen to something with pulp…I searched on MIT lectures, and voila! I am so excited to have found this channel!!!

love you bro

So for the example at 4:00, wouldn't that repeated pattern create a hyperbola rather than a parabola? Drawing lines that intersect the axes and gradually become "more parallel" to them would imply that the axes are asymptotes, yes?

Emergence or correlation of pattern does not occure if there is no outside influence. The more complicated patterns of emergence are the more it needs an intelligent designer. BUT NICE TRY.

8:55 to 9:03 its just amazing

4/3rd 10:47

Amazing๐ฅ๐ฅโค

This is why we don't understand the world.

i hate ted talks with a passions. If you say ted i will hate you all my life !!!!

The same goosebumps and chills that when I listen to Grumiaux Brahms violin sonatas: sublime

Awesome

YouTube are YTb = 3 consonants ouue = 4 vowels. Nice perspective!

Math is all about empathy, so true! Thanks for the Great Talk!

time and maths are infinite?

After noticing the crowd's attention and interest to such magnificent explanation of the understanding of the essence of perceptions, I've come to conclude that most of us human's are selfish, fearful, and self-centered creatures. The speaker was trying to make the crowd understand the power of empathy, yet, most never grasped his point of view, nor tried to sense what he was feeling or seeing while explaining the beauty of different points of views/paradigms. This proves my point that no one really wants to step on someone else's shoes to try understand them. Why? Could it be fear?… Or could it simply be selfishness?

The Title of the video wants something more than what HE is telling.

Certain mathematics subjects need to be fixed. C/D equals 3.17157, not 3.14159

You know what? I am thinking on what is the probability that a random viewer here will reply in this comment.

Perhaps the same thing, phenomenon could be applicable & relative to the system of beliefs, believes, spirituality, divine, god, ideologies, etc…i guess??

Amazing how the 4/3 draw look so similar to some viking art like the node's . This is an Interesting connection

CHANGE THE PERSPECTIVE โค๏ธ

Exact.y. Itยดs math…NOT A CNN REPORT!

That mathematics trains our ability to empathize by pushing us to see things from different perspectives is exactly right. That's what it did to me. I never knew it until Antonsen said it. It's like my life makes more sense now. It's like he solved the puzzle of my life. Wow. Thanks.

Who knew chris pontius was also a math genius

And suddenly, the quote from Bruce Lee: "Be like Water", makes huge sense.

Never thought the miracles behind math…Mathematics could be the next language of the universe…

OMG. This guy goes far beyond beyond.

Wish you were my teacher. great video.

idiotic to the highest degree, might as well teach creationism to little 2 year old and brainwash them early on. get a life, whoever this guy is has a non-job

Empathy = changed perspective ๐ค

No, first understand the world first then use maths for the rest.

Evidence: You cannot calculate the height of the sun if you don't observe related possible parametric values.

"Be Water my friend, water can Flow or it can Crash!" ~ Bruce Lee

I asked my mom why I needed to memorize my time table in primary school and she told me I would need it when I grow up and drive a car, build anything, and I understand now.

The mistake we all make as humans is trying to change our own and otherโs perspectives while breaking the laws of reason and logic. We see a pattern that appears beautiful to us or that fits our narrative and limited personal experience, but it unfortunately doesnโt actually fit the pattern of reality.

bruce lee said: be water my friend

Amazing lecture and amazing person. He has reached a level of humanity that we should all strive for. Bravo!

Well said …I'll make sure that I change my perspective

Empathy is similar to an infinite summation. Just as one can not arrive at the limit of an infinite summation,ย one can notย trulyย reach empathy. ย However, byย putting on the other's shoesย one begins toย see the other's perspective. One must do this over and overย to approach Fairness. ย ..Greed is the opposite of Fairness and Greed accounts for the waste in a society. One of the most significant art works which logically explains this is Magritte's artistic master piece "This is not anย Apple" https://www.collette.co.nz/blog/385272

Don't always act on empathy. Empathy with a predator can get you killed.

Temper empathy with a healthy dose of self preservation.

Sadly migrants from the third world are trying to squeeze this Man out of his country.

Music is math

Just me that scanned the barcode to see the book was:

Book of Numbers: A Novelย by Joshua Cohen ?

Understanding And Changing Perspective

Beautiful Patterns, Structure, Regularity, Langauge

1:52 Tie Knots

Inventing A Language For Patterns In Nature

3:08 Symbolism, Sound, Representing Something Else

4:05 Drawing Lines

5:20 Changing Perspective

Addition, Mulitplication,

Metaphor

8:00 4/3

11:07 Understand if you can view from multiple perspectives

to understanding? is that grammarly correct?

I love maths. Thanks

Btw I would also like to know how you've maintained such a wonderful mane. I am flabbergasted ๐

Thank you.

Did anyone notice at 2:20 the shoelace book was written by Burkard Polster! That's the Mathologer! Just search Mathologer on YouTube if you unfamiliar with the channel he uploads some beautiful mathematically related videos.

Ahhhh TedX, the pedo friendly channel.

Uma das palestras mais inspiradoras do TED! Parabรฉns Roger Antonsen, excelente aula!

I tried looking for the book referred to, IBSN 778215022741, but different searches all return an error! It seems it is a number short or two too long . Would anyone care to help, please?

Amazing

Maths is an exact language using symbols which has been invented by the human mind to communicate relationships precisely. It is used to construct models of the world we experience. It is important not to confuse the model with the real thing it represents. The model is always simpler, and emphasises certain features of immediate interest while ignoring others. Maths does not exist out there, it exists in our minds, like verbal languages.

Explanation in a mathematical sense means showing how a new model relates to other older models we already know. This can be thought of as changing our perspective in a broad sense.

THANK YOU

So what is the word truth, earth, in bible? Garden ? Eden ? I totally get that..the flower of life, Jesus knew. The father, the whole dna/blood, Jacobs ladder. Grandfathered in..nice show !! Totally enjoyed it.

The correct answer is: The Authorized King James Bible is the key to understanding everything.

Wonderful ! Yes we have to look at life from INSIDE out !

Great talk, except for the liberal political bs at the end.

Try looking at the world through the perspective of someone who BELIEVES in an ideology written by a 7th century warlord who married a 6 year old girl.

I love u

Ok, we must make morse code based on this video.

โYou do not really understand something unless you can explain it to your grandmother.โ

โ Albert Einstein

the new math makes a person use the brain but ALSO CONFUSE == the old adage remains true — if it is NOT BROKEN ,DO NOT FIX IT !!! this another way to render calculators obsolete

amazing perspective

Wish we had a better school system that would interest me the way you did.

17 minutes doesn't do justice to his explanation

Jes. Another fn tweed.

Math is a tool to explain what is, not what is. Or a way to explain what is, not what is,

I've been teaching maths for over 5yrs now but never had such perspective. I'm glad I know now how to make maths more enjoyable

Math requires one to think critically, without bias, which can be applied to any field.

Mathmatically… Jesus Christ is the only Way to Heaven. He is the only way %100 can be sins can be washed clean.

๐๐๐๐๐๐๐๐๐๐น๐ฅplease rtrn

Yes this guy knws abt maths bt don't knw the spelling of it, case is same as per prev. cmnts

best explained wow.

I like it, but since its a language and have to be precise; the 4/3 example there was a mistake of young comma which is a separator instead of point which is a radix marker. So, 4/3 is 1.3333 – – – rather than 1,333 and so on

Can someone please ask Mr Antonsen to show us the "base 3, 6 & 9" representation of 4/3, and to let us know if they're used in any science(s). Takk for belysningen, Sir

Thats why ted grows

Love u ted

Super. I always say my students – Math is music

If only colleges did not demand a particular form of mathematics, moreover, if the mathematicians advising colleges, and anyone determining college entrance exams, opened mathematics up to everyone beyond the boring textbook structure of equation, theorem, maybe a proof, examples and dozens of problems, to the beautiful patterns, to play with these, and attempt oneโs own curious proofs, the we may have creative problem solving individuals!

leftists first have to discover that 1 + 1 != 3

Logic is the hidden secret to understanding the Universe

Simply speaking, mathematics is about patterns, finding it and stuffs

ูุงุฒู ู ุฏุฑุณูู ุนุฏูุง ูุชุนูู ูู ู ูู

change the respective ๐๐

Change your perspectives and be like water! Water is so much more than just a liquid!

Loved it!

Very good! mimagobill.com

90% of speech:

"patterns"

Mathematics is like religion it teaches you other people's bullshit in patterned way, of course in their prospective and not yours this way most people do not understand mathematics and does not make sense to them.

Random disjointed mutterings.

Many people only go 3/4 the way falling short of 4/3. They only have 3/4 of their own perspective where they need an additional 1/4 to complete their own plus 1/3 more to consider another's. So they are 7/12 short from his perspective. But if you put it all together you'll end up with 48/12 which in turn equals 4. Now repeat this 3 times and you will see the pattern…

Math is a language with no basis in reality. It's laughable. Many actual mathematicians have been quoted saying exactly that. It's all theoretical BS. The theory of relativity is philosophy.

Bravo! Inspired!

nikoli tesla

Without empathy there is no depth.

tropaaaa

The world is dominated by rules/patterns. And Mathematics is the rule. Thus, the world is dominated by Mathematics.

This is the best Ted Talk I have ever heard

The book referenced in the barcode at 3:54 is "Book of Numbers: A Novel" by Joshua Cohen

This was so boring

๐๐๐amazing genius, to see all those other perspectives of simple occurrences. Thank you. Makes you humble.