Math is the hidden secret to understanding the world | Roger Antonsen

Summary

This inspiring talk explores the concept of understanding, positing that true comprehension is rooted in the ability to change one's perspective. The speaker uses mathematics as a lens to illustrate this idea, demonstrating how patterns, language, and assumptions in math enable us to view problems from various angles. From the simple beauty of equations to the complex patterns in nature, the talk encourages embracing different viewpoints to deepen our understanding and foster empathy, ultimately advocating for a flexible mind like water.

Takeaways

• 🧠 Understanding is fundamentally about the ability to change perspectives and view things from different angles.
• 🔢 Mathematics is not just about basic operations; it's about identifying and working with patterns, structures, and rules that govern what we see.
• 📚 The essence of mathematics involves finding patterns, representing them with language, making assumptions, and exploring the outcomes of those assumptions.
• 🎨 Representation in mathematics is powerful, allowing us to express abstract concepts through invented languages and notations, such as Leibniz's notation or dance notation.
• 👕 The concept of patterns extends to everyday objects like tie knots and shoelaces, which can be mathematically analyzed and described.
• 📐 Geometry is demonstrated through the drawing of circles to create patterns, emphasizing the visual aspect of mathematical patterns.
• 🎶 Mathematics is also present in music, with patterns evident in rhythm, melody, and harmony, and can be represented numerically.
• 🌐 The idea of perspective is crucial in understanding complex structures, such as the octahedron, by rotating and viewing them from different angles.
• 🌊 The ocean serves as a metaphor for understanding, where immersing oneself in the subject (like going underwater) provides deeper insights.
• 🤔 Imagination is key to changing perspectives, allowing us to view situations from outside our own experiences, akin to empathy.
• 💧 The speaker suggests that having a mind like water, flexible and open to change, is beneficial for understanding and learning.

Q & A

What is the central claim made about understanding in the script?

The central claim is that understanding is fundamentally about the ability to change one's perspective.

How is mathematics defined in the speaker's day-to-day definition?

Mathematics is defined as finding patterns, representing these patterns with a language, making assumptions, and then playing around with these assumptions to see what happens.

What is an example of a pattern in everyday life mentioned in the script?

Tie knots are given as an example of everyday patterns, with specific methods and a language created to describe them.

What is the significance of representing patterns in mathematics?

Representing patterns is significant because it allows us to communicate and manipulate mathematical ideas effectively, enabling us to do 'cool stuff' or solve problems.

How does the script relate the concept of patterns to the visual arts?

The script shows that by drawing straight lines in a particular pattern, one can create the visual appearance of a curve, demonstrating the connection between mathematical patterns and visual art.

What is the mathematical pattern that emerges from drawing straight lines in a specific manner as described in the script?

The pattern that emerges is a parabola, which is represented by a simple equation and is a beautiful example of a mathematical pattern.

How does the script use the concept of changing perspective to explore mathematical equations?

The script suggests that every mathematical equation is a metaphor, representing two different perspectives or viewpoints on the same concept.

What is the purpose of the experiment with the rotating circles in the script?

The experiment with rotating circles is used to visually represent the fraction four-thirds, demonstrating how changing perspective can lead to a deeper understanding of mathematical concepts.

How does the script connect the idea of changing perspective to the concept of empathy?

The script suggests that changing perspective is fundamental to empathy, as understanding the world from another person's viewpoint requires imagination and is akin to changing one's perspective in mathematics.

What advice does the speaker give for enhancing understanding?

The speaker advises to change one's perspective, study mathematics to train the brain, and to have a mind like water, which is flexible and open to new things.

How does the script use the octahedron to illustrate the concept of changing perspective?

The script uses the octahedron to show that by physically rotating it or changing the axis of view, one can gain new insights and understanding of the same object from different angles.

Outlines

00:00

🔍 The Essence of Understanding and Mathematics

The speaker introduces the concept of understanding as the ability to change perspectives, using mathematics as a lens to explore this idea. They define mathematics as the discovery and representation of patterns, using the example of a visual pattern created by drawing circles. The speaker emphasizes the importance of language in representing these patterns and the process of making assumptions to explore their consequences. They also touch on the ubiquity of patterns in everyday life, such as in tying knots and shoelaces, and highlight historical examples of pattern representation, including Leibniz's notation and a tap dancing notation system. The speaker concludes by reflecting on the power of representation and how it allows us to understand and interpret the world around us.

05:01

📚 Exploring Patterns and Perspectives in Mathematics

The speaker delves deeper into the nature of patterns and perspectives, using the simple equation x + x = 2 • x to illustrate how different viewpoints can lead to new insights. They argue that every equation is a metaphor, comparing two different perspectives on a concept. The discussion extends to the representation of numbers in different bases, such as binary, and the visual representation of the number four-thirds through geometric shapes and motion. The speaker also explores the connection between numbers and music, demonstrating how mathematical ratios can be heard in harmonious musical intervals. The underlying message is that changing one's perspective is fundamental to gaining a deeper understanding of mathematical concepts.

10:03

🎶 The Interplay of Numbers, Sound, and Rhythm

Continuing the theme of perspective and representation, the speaker experiments with sound and rhythm to further illustrate the concept of four-thirds. They demonstrate how this fraction can be represented as a sound, a rhythm, and even a physical motion, emphasizing the multifaceted nature of understanding. The speaker uses these examples to show that by changing the way we view and interact with a concept, we can gain a richer understanding of it. They also highlight the importance of imagination in this process, suggesting that the ability to imagine different perspectives is key to empathy and understanding in both mathematics and human relations.

15:05

🌐 The Power of Perspective and Imagination in Understanding

In the final paragraph, the speaker synthesizes the ideas presented throughout the talk, concluding that the ability to change one's perspective is essential for deep understanding. They use the metaphor of viewing the ocean from various angles to demonstrate how each perspective offers unique insights. The speaker encourages the audience to embrace change in perspective as a means to foster imagination, flexibility, and open-mindedness. They advocate for studying mathematics as a way to train the brain in this skill, likening a flexible mind to water that can adapt to any shape. The talk ends with a call to action to cultivate empathy and understanding through the imaginative exploration of different viewpoints.

Keywords

💡Understanding

Understanding in the context of the video is the ability to grasp the nature, cause, or significance of something. It is the main theme of the video, where the speaker argues that true understanding is achieved by being able to change one's perspective. For example, the speaker suggests that understanding in mathematics is not just about performing calculations but recognizing patterns and representing them in various ways.

💡Perspective

Perspective is a key concept in the video, referring to the different viewpoints or approaches one can take to analyze and interpret information. The speaker emphasizes that changing one's perspective is fundamental to gaining a deeper understanding, as it allows for the discovery of new insights and connections, such as viewing mathematical equations from different angles.

💡Mathematics

Mathematics is presented as the central subject of exploration in the video. It is defined by the speaker not just as a set of operations like addition or algebra, but as the study of patterns, structures, and the creation of languages to represent these patterns. The video uses mathematics as a lens to discuss the broader concept of understanding through the ability to perceive patterns and change perspectives.

💡Patterns

Patterns are recurring structures or regularities that the speaker identifies as the core of mathematics and understanding. In the video, patterns are found in various forms, from the visual patterns in drawings to the numerical patterns in mathematical equations. The recognition and representation of these patterns are what enable deeper understanding and the ability to change perspectives.

💡Representation

Representation is the act of expressing or depicting patterns or ideas in a language or symbolic form. The video discusses how mathematics involves creating and using representations to communicate and explore patterns. For instance, the speaker mentions Leibniz's notation and the notation system for tap dancing as examples of representation in mathematics and art.

💡Language

Language in the video is used to refer to the systems of symbols and words created to describe and analyze patterns. It is an essential tool in mathematics for representing patterns and enabling communication of complex ideas. The speaker uses the term to highlight how we invent and utilize languages to understand and manipulate patterns in various contexts.

💡Assumptions

Assumptions are the starting points or premises made in mathematical reasoning and problem-solving. The speaker suggests that mathematics involves making assumptions and exploring their implications, which is a way of changing perspective and gaining understanding. The process of testing and playing with assumptions is shown in the video as a method to discover new insights.

💡Metaphor

Metaphor is used in the video to describe the way mathematical equations can represent analogies between different concepts or viewpoints. The speaker argues that equations are metaphors that allow us to see the relationship between two different perspectives, such as the sum and product in the equation x + x = 2 • x.

💡Empathy

Empathy is introduced towards the end of the video as a form of perspective-changing that involves understanding the world from another person's point of view. The speaker draws a connection between empathy and the ability to change perspectives in mathematics and computer science, suggesting that the imaginative act of seeing from another's viewpoint is crucial for deep understanding.

💡Imagination

Imagination is portrayed as a necessary component for changing perspectives and gaining understanding. The speaker argues that the ability to imagine different viewpoints, whether in mathematics, computer science, or empathy, is what allows for the exploration of new ideas and the deepening of comprehension.

💡Pi and Octahedron

The pi and octahedron are used in the video to illustrate the concept of perspective and representation. The speaker demonstrates how the number pi can be visualized through the rotation of circles and how the octahedron, a three-dimensional shape, can be viewed from multiple angles, emphasizing the idea that changing perspective reveals new aspects of a subject.

Highlights

Understanding is linked to the ability to change perspective.

Mathematics is defined as finding patterns, representing them with language, making assumptions, and doing innovative work.

Patterns in everyday life, such as tie knots and shoelaces, can be mathematically analyzed and represented.

Leibniz's notation from 1675 was a language for patterns in nature, particularly in the motion of objects.

Tap dancing notation is an example of representing movement with mathematical patterns.

The representation of language through symbols and sounds demonstrates the power of abstract representation.

Experiments with lines reveal patterns such as parabolas, showcasing the emergence of complex shapes from simple elements.

Changing perspective in mathematical equations allows for deeper insights and understanding.

The concept of four-thirds is explored through various representations, including binary, visual, and auditory forms.

The relationship between numbers and geometry is exemplified by the properties of a sphere and a cylinder.

Empathy and understanding in mathematics and computer science are facilitated by the ability to view problems from different angles.

Imagination is crucial for changing perspective and gaining a deeper understanding in both mathematics and computer science.

The speaker advocates for studying mathematics as a way to train the brain in flexibility and openness to new ideas.

Metaphors and narratives are powerful tools for teaching and understanding complex concepts.

The octahedron, a platonic solid, is used to illustrate the concept of perspective and understanding through physical manipulation.

Great circles on the Earth serve as a metaphor for understanding the interconnectedness of perspectives.

The final advice is to 'have a mind like water,' emphasizing adaptability and openness as keys to understanding.

Transcripts

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)