Lex Fridman Podcast - #64 - Grant Sanderson: 3Blue1Brown and the Beauty of Mathematics

The following is a conversation with Grant Sanderson.

He’s a math educator and creator of 3Blue1Brown,

a popular YouTube channel

that uses programmatically animated visualizations

to explain concepts in linear algebra, calculus,

and other fields of mathematics.

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And now, here’s my conversation with Grant Sanderson.

If there’s intelligent life out there in the universe,

do you think their mathematics is different than ours?

Jumping right in.

I think it’s probably very different.

There’s an obvious sense the notation is different, right?

I think notation can guide what the math itself is.

I think it has everything to do with the form

of their existence, right?

Do you think they have basic arithmetic?

Sorry, I interrupted.

Yeah, so I think they count, right?

I think notions like one, two, three,

the natural numbers, that’s extremely, well, natural.

That’s almost why we put that name to it.

As soon as you can count,

you have a notion of repetition, right?

Because you can count by two, two times or three times.

And so you have this notion of repeating

the idea of counting, which brings you addition

and multiplication.

I think the way that we extend it to the real numbers,

there’s a little bit of choice in that.

So there’s this funny number system

called the servial numbers

that it captures the idea of continuity.

It’s a distinct mathematical object.

You could very well model the universe

and motion of planets with that

as the back end of your math, right?

And you still have kind of the same interface

with the front end of what physical laws you’re trying to,

or what physical phenomena you’re trying

to describe with math.

And I wonder if the little glimpses that we have

of what choices you can make along the way

based on what different mathematicians

I’ve brought to the table

is just scratching the surface

of what the different possibilities are

if you have a completely different mode of thought, right?

Or a mode of interacting with the universe.

And you think notation is a key part of the journey

that we’ve taken through math.

I think that’s the most salient part

that you’d notice at first.

I think the mode of thought is gonna influence things

more than like the notation itself.

But notation actually carries a lot of weight

when it comes to how we think about things,

more so than we usually give it credit for.

I would be comfortable saying.

Do you have a favorite or least favorite piece of notation

in terms of its effectiveness?

Yeah, yeah, well, so least favorite,

one that I’ve been thinking a lot about

that will be a video I don’t know when, but we’ll see.

The number e, we write the function e to the x,

this general exponential function

with a notation e to the x

that implies you should think about a particular number,

this constant of nature,

and you repeatedly multiply it by itself.

And then you say, oh, what’s e to the square root of two?

And you’re like, oh, well, we’ve extended the idea

of repeated multiplication.

That’s all nice, that’s all nice and well.

But very famously, you have like e to the pi i,

and you’re like, well, we’re extending the idea

of repeated multiplication into the complex numbers.

Yeah, you can think about it that way.

In reality, I think that it’s just the wrong way

of notationally representing this function,

the exponential function,

which itself could be represented

a number of different ways.

You can think about it in terms of the problem it solves,

a certain very simple differential equation,

which often yields way more insight

than trying to twist the idea of repeated multiplication,

like take its arm and put it behind its back

and throw it on the desk and be like,

you will apply to complex numbers, right?

That’s not, I don’t think that’s pedagogically helpful.

So the repeated multiplication is actually missing

the main point, the power of e to the x.

I mean, what it addresses is things where the rate

at which something changes depends on its own value,

but more specifically, it depends on it linearly.

So for example, if you have like a population

that’s growing and the rate at which it grows

depends on how many members of the population

are already there,

it looks like this nice exponential curve.

It makes sense to talk about repeated multiplication

because you say, how much is there after one year,

two years, three years, you’re multiplying by something.

The relationship can be a little bit different sometimes

where let’s say you’ve got a ball on a string,

like a game of tetherball going around a rope, right?

And you say, its velocity is always perpendicular

to its position.

That’s another way of describing its rate of change

is being related to where it is,

but it’s a different operation.

You’re not scaling it, it’s a rotation.

It’s this 90 degree rotation.

That’s what the whole idea of like complex exponentiation

is trying to capture,

but it’s obfuscated in the notation

when what it’s actually saying,

like if you really parse something like e to the pi i,

what it’s saying is choose an origin,

always move perpendicular to the vector

from that origin to you, okay?

Then when you walk pi times that radius,

you’ll be halfway around.

Like that’s what it’s saying.

It’s kind of the, you turn 90 degrees and you walk,

you’ll be going in a circle.

That’s the phenomenon that it’s describing,

but trying to twist the idea

of repeatedly multiplying a constant into that.

Like I can’t even think of the number of human hours

of like intelligent human hours that have been wasted

trying to parse that to their own liking and desire

among like scientists or electrical engineers

or students everywhere,

which if the notation were a little different

or the way that this whole function was introduced

from the get go were framed differently,

I think could have been avoided, right?

And you’re talking about

the most beautiful equation in mathematics,

but it’s still pretty mysterious, isn’t it?

Like you’re making it seem like it’s a notational.

It’s not mysterious.

I think the notation makes it mysterious.

I don’t think it’s, I think the fact that it represents,

it’s pretty, it’s not like the most beautiful thing

in the world, but it’s quite pretty.

The idea that if you take the linear operation

of a 90 degree rotation,

and then you do this general exponentiation thing to it,

that what you get are all the other kinds of rotation,

which is basically to say,

if your velocity vector is perpendicular

to your position vector, you walk in a circle,

that’s pretty.

It’s not the most beautiful thing in the world,

but it’s quite pretty.

The beauty of it, I think comes from perhaps

the awkwardness of the notation

somehow still nevertheless coming together nicely,

because you have like several disciplines coming together

in a single equation.

Well, I think.

In a sense, like historically speaking.

That’s true.

You’ve got, so like the number E is significant.

Like it shows up in probability all the time.

It like shows up in calculus all the time.

It is significant.

You’re seeing it sort of mated with pi,

this geometric constant and I,

like the imaginary number and such.

I think what’s really happening there

is the way that E shows up is when you have things

like exponential growth and decay, right?

It’s when this relation that something’s rate of change

has to itself is a simple scaling, right?

A similar law also describes circular motion.

Because we have bad notation,

we use the residue of how it shows up

in the context of self reinforcing growth,

like a population growing or compound interest.

The constant associated with that

is awkwardly placed into the context

of how rotation comes about,

because they both come from pretty similar equations.

And so what we see is the E and the pi juxtaposed

a little bit closer than they would be

with a purely natural representation, I would think.

Here’s how I would describe the relation between the two.

You’ve got a very important function we might call exp.

That’s like the exponential function.

When you plug in one,

you get this nice constant called E

that shows up in like probability and calculus.

If you try to move in the imaginary direction,

it’s periodic and the period is tau.

So those are these two constants

associated with the same central function,

but for kind of unrelated reasons, right?

And not unrelated, but like orthogonal reasons.

One of them is what happens

when you’re moving in the real direction.

One’s what happens when you move in the imaginary direction.

And like, yeah, those are related.

They’re not as related as the famous equation

seems to think it is.

It’s sort of putting all of the children in one bed

and they’d kind of like to sleep in separate beds

if they had the choice, but you see them all there

and there is a family resemblance, but it’s not that close.

So actually thinking of it as a function

is the better idea.

And that’s a notational idea.

And yeah, and like, here’s the thing.

The constant E sort of stands

as this numerical representative of calculus, right?

Calculus is the like study of change.

So at the very least there’s a little cognitive dissonance

using a constant to represent the science of change.

I never thought of it that way.




It makes sense why the notation came about that way.

Because this is the first way that we saw it

in the context of things like population growth

or compound interest.

It is nicer to think about as repeated multiplication.

That’s definitely nicer.

But it’s more that that’s the first application

of what turned out to be a much more general function

that maybe the intelligent life

your initial question asked about

would have come to recognize as being much more significant

than the single use case,

which lends itself to repeated multiplication notation.

But let me jump back for a second to aliens

and the nature of our universe.


Do you think math is discovered or invented?

So we’re talking about the different kind of mathematics

that could be developed by the alien species.

The implied question is,

yeah, is math discovered or invented?

Is fundamentally everybody going to discover

the same principles of mathematics?

So the way I think about it,

and everyone thinks about it differently,

but here’s my take.

I think there’s a cycle at play

where you discover things about the universe

that tell you what math will be useful.

And that math itself is invented in a sense,

but of all the possible maths that you could have invented,

it’s discoveries about the world

that tell you which ones are.

So like a good example here is the Pythagorean theorem.

When you look at this,

do you think of that as a definition

or do you think of that as a discovery?

From the historical perspective, right, it’s a discovery

because they were,

but that’s probably because they were using physical object

to build their intuition.

And from that intuition came the mathematics.

So the mathematics wasn’t in some abstract world

detached from physics,

but I think more and more math has become detached from,

you know, when you even look at modern physics

from string theory to even general relativity,

I mean, all math behind the 20th and 21st century physics

kind of takes a brisk walk outside of what our mind

can actually even comprehend

in multiple dimensions, for example,

anything beyond three dimensions, maybe four dimensions.

No, no, no, no, higher dimensions

can be highly, highly applicable.

I think this is a common misinterpretation

that if you’re asking questions

about like a five dimensional manifold,

that the only way that that’s connected

to the physical world is if the physical world is itself

a five dimensional manifold or includes them.

Well, wait, wait, wait a minute, wait a minute.

You’re telling me you can imagine

a five dimensional manifold?

No, no, that’s not what I said.

I would make the claim that it is useful

to a three dimensional physical universe,

despite itself not being three dimensional.

So it’s useful meaning to even understand

a three dimensional world,

it’d be useful to have five dimensional manifolds.

Yes, absolutely, because of state spaces.

But you’re saying there in some deep way for us humans,

it does always come back to that three dimensional world

for the usefulness that the dimensional world

and therefore it starts with a discovery,

but then we invent the mathematics

that helps us make sense of the discovery in a sense.

Yes, I mean, just to jump off

of the Pythagorean theorem example,

it feels like a discovery.

You’ve got these beautiful geometric proofs

where you’ve got squares and you’re modifying the areas,

it feels like a discovery.

If you look at how we formalize the idea of 2D space

as being R2, right, all pairs of real numbers,

and how we define a metric on it and define distance,

you’re like, hang on a second,

we’ve defined a distance

so that the Pythagorean theorem is true,

so that suddenly it doesn’t feel that great.

But I think what’s going on is the thing that informed us

what metric to put on R2,

to put on our abstract representation of 2D space,

came from physical observations.

And the thing is, there’s other metrics

you could have put on it.

We could have consistent math

with other notions of distance,

it’s just that those pieces of math

wouldn’t be applicable to the physical world that we study

because they’re not the ones

where the Pythagorean theorem holds.

So we have a discovery, a genuine bonafide discovery

that informed the invention,

the invention of an abstract representation of 2D space

that we call R2 and things like that.

And then from there,

you just study R2 as an abstract thing

that brings about more ideas and inventions and mysteries

which themselves might yield discoveries.

Those discoveries might give you insight

as to what else would be useful to invent

and it kind of feeds on itself that way.

That’s how I think about it.

So it’s not an either or.

It’s not that math is one of these

or it’s one of the others.

At different times, it’s playing a different role.

So then let me ask the Richard Feynman question then,

along that thread,

is what do you think is the difference

between physics and math?

There’s a giant overlap.

There’s a kind of intuition that physicists have

about the world that’s perhaps outside of mathematics.

It’s this mysterious art that they seem to possess,

we humans generally possess.

And then there’s the beautiful rigor of mathematics

that allows you to, I mean, just like as we were saying,

invent frameworks of understanding our physical world.

So what do you think is the difference there

and how big is it?

Well, I think of math as being the study

of abstractions over patterns and pure patterns in logic.

And then physics is obviously grounded in a desire

to understand the world that we live in.

I think you’re gonna get very different answers

when you talk to different mathematicians

because there’s a wide diversity in types of mathematicians.

There are some who are motivated very much by pure puzzles.

They might be turned on by things like combinatorics.

And they just love the idea of building up

a set of problem solving tools applying to pure patterns.

There are some who are very physically motivated,

who try to invent new math or discover math in veins

that they know will have applications to physics

or sometimes computer science.

And that’s what drives them.

Like chaos theory is a good example of something

that’s pure math, that’s purely mathematical.

A lot of the statements being made,

but it’s heavily motivated by specific applications

to largely physics.

And then you have a type of mathematician

who just loves abstraction.

They just love pulling it to the more and more abstract

things, the things that feel powerful.

These are the ones that initially invented like topology

and then later on get really into category theory

and go on about like infinite categories and whatnot.

These are the ones that love to have a system

that can describe truths about as many things as possible.

People from those three different veins of motivation

into math are gonna give you very different answers

about what the relation at play here is.

Cause someone like Vladimir Arnold,

who has written a lot of great books,

many about like differential equations and such,

he would say, math is a branch of physics.

That’s how he would think about it.

And of course he was studying

like differential equations related things

because that is the motivator behind the study

of PDEs and things like that.

But you’ll have others who,

like especially the category theorists

who aren’t really thinking about physics necessarily.

It’s all about abstraction and the power of generality.

And it’s more of a happy coincidence

that that ends up being useful

for understanding the world we live in.

And then you can get into like, why is that the case?

It’s sort of surprising

that that which is about pure puzzles and abstraction

also happens to describe the very fundamentals

of quarks and everything else.

So why do you think the fundamentals of quarks

and the nature of reality is so compressible

into clean, beautiful equations

that are for the most part simple, relatively speaking,

a lot simpler than they could be?

So you have, we mentioned somebody like Stephen Wolfram

who thinks that sort of there’s incredibly simple rules

underlying our reality,

but it can create arbitrary complexity.

But there is simple equations.

What, I’m asking a million questions

that nobody knows the answer to, but.

I have no idea, why is it simple?

It could be the case that

there’s like a filter iteration at play.

The only things that physicists find interesting

are the ones that are simple enough

they could describe it mathematically.

But as soon as it’s a sufficiently complex system,

like, oh, that’s outside the realm of physics,

that’s biology or whatever have you.

And of course, that’s true.

Maybe there’s something where it’s like,

of course there will always be something that is simple

when you wash away the like non important parts

of whatever it is that you’re studying.

Just from like an information theory standpoint,

there might be some like,

you get to the lowest information component of it.

But I don’t know, maybe I’m just having

a really hard time conceiving of what it would even mean

for the fundamental laws to be like intrinsically

complicated, like some set of equations

that you can’t decouple from each other.

Well, no, it could be that sort of we take for granted

that the laws of physics, for example,

are for the most part the same everywhere

or something like that, right?

As opposed to the sort of an alternative could be

that the rules under which the world operates

is different everywhere.

It’s like a deeply distributed system

where just everything is just chaos,

not in a strict definition of chaos,

but meaning like just it’s impossible for equations

to capture, for to explicitly model the world

as cleanly as the physical does.

I mean, we almost take it for granted that we can describe,

we can have an equation for gravity,

for action at a distance.

We can have equations for some of these basic ways

the planet’s moving.

Just the low level at the atomic scale,

how the materials operate,

at the high scale, how black holes operate.

But it doesn’t, it seems like it could be,

there’s infinite other possibilities

where none of it could be compressible into such equations.

So it just seems beautiful.

It’s also weird, probably to the point you’re making,

that it’s very pleasant that this is true for our minds.

So it might be that our minds are biased

to just be looking at the parts of the universe

that are compressible.

And then we can publish papers on

and have nice E equals empty squared equations.

Right, well, I wonder would such a world

with uncompressible laws allow for the kind of beings

that can think about the kind of questions

that you’re asking?

That’s true.

Right, like an anthropic principle coming into play

in some weird way here?

I don’t know, like I don’t know what I’m talking about at all.

Maybe the universe is actually not so compressible,

but the way our brain, the way our brain evolved

we’re only able to perceive the compressible parts.

I mean, we are, so this is the sort of Chomsky argument.

We are just descendants of apes

over like really limited biological systems.

So it totally makes sense

that we’re really limited little computers, calculators,

that are able to perceive certain kinds of things

and the actual world is much more complicated.

Well, but we can do pretty awesome things, right?

Like we can fly spaceships

and we have to have some connection of reality

to be able to take our potentially oversimplified models

of the world, but then actually twist the world

to our will based on it.

So we have certain reality checks

that like physics isn’t too far a field

simply based on what we can do.

Yeah, the fact that we can fly is pretty good.

It’s great, yeah, like it’s a proof of concept

that the laws we’re working with are working well.

So I mentioned to the internet that I’m talking to you

and so the internet gave some questions.

So I apologize for these,

but do you think we’re living in a simulation

that the universe is a computer

or the universe is a computation running on a computer?

It’s conceivable.

What I don’t buy is, you know, you’ll have the argument

that, well, let’s say that it was the case

that you can have simulations.

Then the simulated world would itself

eventually get to a point where it’s running simulations.

And then the second layer down

would create a third layer down and on and on and on.

So probabilistically, you just throw a dart

at one of those layers,

we’re probably in one of the simulated layers.

I think if there’s some sort of limitations

on like the information processing

of whatever the physical world is,

like it quickly becomes the case

that you have a limit to the layers that could exist there

because like the resources necessary

to simulate a universe like ours clearly is a lot

just in terms of the number of bits at play.

And so then you can ask, well, what’s more plausible?

That there’s an unbounded capacity

of information processing

in whatever the like highest up level universe is,

or that there’s some bound to that capacity,

which then limits like the number of levels available.

How do you play some kind of probability distribution

on like what the information capacity is?

I have no idea.

But I don’t, like people almost assume

a certain uniform probability

over all of those meta layers that could conceivably exist

when it’s a little bit like a Pascal’s wager

on like you’re not giving a low enough prior

to the mere existence of that infinite set of layers.

Yeah, that’s true.

But it’s also very difficult to contextualize the amount.

So the amount of information processing power

required to simulate like our universe

seems like amazingly huge.

But you can always raise two to the power of that.

Yeah, like numbers get big.

And we’re easily humbled

by basically everything around us.

So it’s very difficult to kind of make sense of anything

actually when you look up at the sky

and look at the stars and the immensity of it all,

to make sense of the smallness of us,

the unlikeliness of everything

that’s on this earth coming to be,

then you could basically anything could be,

all laws of probability go out the window to me

because I guess because the amount of information

under which we’re operating is very low.

We basically know nothing about the world around us,

relatively speaking.

And so when I think about the simulation hypothesis,

I think it’s just fun to think about it.

But it’s also, I think there is a thought experiment

kind of interesting to think of the power of computation,

whether the limits of a Turing machine,

sort of the limits of our current computers,

when you start to think about artificial intelligence,

how far can we get with computers?

And that’s kind of where the simulation hypothesis

used with me as a thought experiment

is the universe just a computer?

Is it just a computation?

Is all of this just a computation?

And sort of the same kind of tools we apply

to analyzing algorithms, can that be applied?

If we scale further and further and further,

will the arbitrary power of those systems

start to create some interesting aspects

that we see in our universe?

Or is something fundamentally different

needs to be created?

Well, it’s interesting that in our universe,

it’s not arbitrarily large, the power,

that you can place limits on, for example,

how many bits of information can be stored per unit area.

Right, like all of the physical laws,

you’ve got general relativity and quantum coming together

to give you a certain limit on how many bits you can store

within a given range before it collapses into a black hole.

The idea that there even exists such a limit

is at the very least thought provoking,

when naively you might assume,

oh, well, technology could always get better and better,

we could get cleverer and cleverer,

and you could just cram as much information as you want

into like a small unit of space, that makes me think,

it’s at least plausible that whatever the highest level

of existence is doesn’t admit too many simulations

or ones that are at the scale of complexity

that we’re looking at.

Obviously, it’s just as conceivable that they do

and that there are many, but I guess what I’m channeling

is the surprise that I felt upon learning that fact,

that there are, that information is physical in this way.

There’s a finiteness to it.

Okay, let me just even go off on that.

From a mathematics perspective

and a psychology perspective, how do you mix,

are you psychologically comfortable

with the concept of infinity?

I think so.

Are you okay with it?

I’m pretty okay, yeah.

Are you okay?

No, not really, it doesn’t make any sense to me.

I don’t know, like how many words,

how many possible words do you think could exist

that are just like strings of letters?

So that’s a sort of mathematical statement as beautiful

and we use infinity in basically everything we do,

everything we do in science, math, and engineering, yes.

But you said exist, the question is,

you said letters or words?

I said words. Words.

To bring words into existence to me,

you have to start like saying them or like writing them

or like listing them.

That’s an instantiation.

Okay, how many abstract words exist?

Well, the idea of an abstract.

The idea of abstract notions and ideas.

I think we should be clear on terminology.

I mean, you think about intelligence a lot,

like artificial intelligence.

Would you not say that what it’s doing

is a kind of abstraction?

That like abstraction is key

to conceptualizing the universe?

You get this raw sensory data.

I need something that every time you move your face

a little bit and they’re not pixels,

but like analog of pixels on my retina changed entirely,

that I can still have some coherent notion of this is Lex,

I’m talking to Lex, right?

What that requires is you have a disparate set

of possible images hitting me

that are unified in a notion of Lex, right?

That’s a kind of abstraction.

It’s a thing that could apply

to a lot of different images that I see

and it represents it in a much more compressed way

and one that’s like much more resilient to that.

I think in the same way,

if I’m talking about infinity as an abstraction,

I don’t mean nonphysical woo woo,

like ineffable or something.

What I mean is it’s something that can apply

to a multiplicity of situations

that share a certain common attribute

in the same way that the images of like your face

on my retina share enough common attributes

that I can put the single notion to it.

Like in that way, infinity is an abstraction

and it’s very powerful and it’s only through

such abstractions that we can actually understand

like the world and logic and things.

And in the case of infinity,

the way I think about it,

the key entity is the property

of always being able to add one more.

Like no matter how many words you can list,

you just throw an A at the end of one

and you have another conceivable word.

You don’t have to think of all the words at once.

It’s that property, the oh, I could always add one more

that gives it this nature of infiniteness

in the same way that there’s certain like properties

of your face that give it the Lexness, right?

So like infinity should be no more worrying

than the I can always add one more sentiment.

That’s a really elegant,

much more elegant way than I could put it.

So thank you for doing that as yet another abstraction.

And yes, indeed, that’s what our brain does.

That’s what intelligent systems do.

That’s what programming does.

That’s what science does is build abstraction

on top of each other.

And yet there is at a certain point abstractions

that go into the quote woo, right?

Sort of, and because we’re now,

it’s like we built this stack of, you know,

the only thing that’s true is the stuff that’s on the ground.

Everything else is useful for interpreting this.

And at a certain point you might start floating

into ideas that are surreal and difficult

and take us into areas that are disconnected

from reality in a way that we could never get back.

What if instead of calling these abstract,

how different would it be in your mind

if we called them general?

And the phenomenon that you’re describing

is overgeneralization.

When you try to have a concept or an idea

that’s so general as to apply to nothing in particular

in a useful way, does that map to what you’re thinking

of when you think of?

First of all, I’m playing little just for the fun of it.

Devil’s advocate.

And I think our cognition, our mind is unable

to visualize.

So you do some incredible work with visualization and video.

I think infinity is very difficult to visualize

for our mind.

We can delude ourselves into thinking we can visualize it,

but we can’t.

I don’t, I mean, I don’t,

I would venture to say it’s very difficult.

And so there’s some concepts of mathematics,

like maybe multiple dimensions,

we could sort of talk about that are impossible

for us to truly intuit, like,

and it just feels dangerous to me to use these

as part of our toolbox of abstractions.

On behalf of your listeners,

I almost fear we’re getting too philosophical.


Heck no.

I think to that point for any particular idea like this,

there’s multiple angles of attack.

I think the, when we do visualize infinity,

what we’re actually doing, you know,

you write dot, dot, dot, right?

One, two, three, four, dot, dot, dot, right?

Those are symbols on the page

that are insinuating a certain infinity.

What you’re capturing with a little bit of design there

is the I can always add one more property, right?

I think I’m just as uncomfortable with you are

if you try to concretize it so much

that you have a bag of infinitely many things

that I actually think of, no, not one, two, three, four,

dot, dot, dot, one, two, three, four, five, six, seven, eight.

I try to get them all in my head and you realize,

oh, you know, your brain would literally collapse

into a black hole, all of that.

And I honestly feel this with a lot of math

that I try to read where I don’t think of myself

as like particularly good at math in some ways.

Like I get very confused often

when I am going through some of these texts.

And often what I’m feeling in my head is like,

this is just so damn abstract.

I just can’t wrap my head around it.

I just want to put something concrete to it

that makes me understand.

And I think a lot of the motivation for the channel

is channeling that sentiment of, yeah,

a lot of the things that you’re trying to read out there,

it’s just so hard to connect to anything

that you spend an hour banging your head

against a couple of pages and you come out

not really knowing anything more

other than some definitions maybe

and a certain sense of self defeat, right?

One of the reasons I focus so much on visualizations

is that I’m a big believer in,

I’m sorry, I’m just really hampering on

this idea of abstraction,

being clear about your layers of abstraction, right?

It’s always tempting to start an explanation

from the top to the bottom, okay?

You give the definition of a new theorem.

You’re like, this is the definition of a vector space.

For example, that’s how we’ll start a course.

These are the properties of a vector space.

First from these properties, we will derive what we need

in order to do the math of linear algebra

or whatever it might be.

I don’t think that’s how understanding works at all.

I think how understanding works

is you start at the lowest level you can get at

where rather than thinking about a vector space,

you might think of concrete vectors

that are just lists of numbers

or picturing it as like an arrow that you draw,

which is itself like even less abstract than numbers

because you’re looking at quantities,

like the distance of the x coordinate,

the distance of the y coordinate.

It’s as concrete as you could possibly get

and it has to be if you’re putting it in a visual, right?

It’s an actual arrow. It’s an actual vector.

You’re not talking about like a quote unquote vector

that could apply to any possible thing.

You have to choose one if you’re illustrating it.

And I think this is the power of being in a medium

like video or if you’re writing a textbook

and you force yourself to put a lot of images

is with every image, you’re making a choice.

With each choice, you’re showing a concrete example.

With each concrete example,

you’re aiding someone’s path to understanding.

I’m sorry to interrupt you,

but you just made me realize that that’s exactly right.

So the visualizations you’re creating

while you’re sometimes talking about abstractions,

the actual visualization is an explicit low level example.


So there’s an actual, like in the code,

you have to say what the vector is,

what’s the direction of the arrow,

what’s the magnitude of the, yeah.

So that’s, you’re going, the visualization itself

is actually going to the bottom of that.

And I think that’s very important.

I also think about this a lot in writing scripts

where even before you get to the visuals,

the first instinct is to, I don’t know why,

I just always do, I say the abstract thing,

I say the general definition, the powerful thing,

and then I fill it in with examples later.

Always, it will be more compelling

and easier to understand when you flip that.

And instead, you let someone’s brain

do the pattern recognition.

You just show them a bunch of examples.

The brain is gonna feel a certain similarity between them.

Then by the time you bring in the definition,

or by the time you bring in the formula,

it’s articulating a thing that’s already in the brain

that was built off of looking at a bunch of examples

with a certain kind of similarity.

And what the formula does is articulate

what that kind of similarity is,

rather than being a high cognitive load set of symbols

that needs to be populated with examples later on,

assuming someone’s still with you.

What is the most beautiful or awe inspiring idea

you’ve come across in mathematics?

I don’t know, man.

Maybe it’s an idea you’ve explored in your videos,

maybe not.

What just gave you pause?

What’s the most beautiful idea?

Small or big.

So I think often, the things that are most beautiful

are the ones that you have a little bit of understanding of,

but certainly not an entire understanding.

It’s a little bit of that mystery

that is what makes it beautiful.

What was the moment of the discovery for you personally,

almost just that leap of aha moment?

So something that really caught my eye,

I remember when I was little, there were these,

I think the series was called like wooden books

or something, these tiny little books

that would have just a very short description

of something on the left and then a picture on the right.

I don’t know who they’re meant for,

but maybe it’s like loosely children

or something like that.

But it can’t just be children,

because of some of the things I was describing.

On the last page of one of them,

somewhere tiny in there was this little formula

that on the left hand had a sum

over all of the natural numbers.

It’s like one over one to the S plus one over two to the S

plus one over three to the S on and on to the infinity.

Then on the other side had a product over all of the primes

and it was a certain thing had to do with all the primes.

And like any good young math enthusiast,

I’d probably been indoctrinated with how chaotic

and confusing the primes are, which they are.

And seeing this equation where on one side

you have something that’s as understandable

as you could possibly get, the counting numbers.

And on the other side is all the prime numbers.

It was like this, whoa, they’re related like this?

There’s a simple description that includes

all the primes getting wrapped together like this.

This is like the Euler product for the Zeta function,

as I like later found out.

The equation itself essentially encodes

the fundamental theorem of arithmetic

that every number can be expressed

as a unique set of primes.

To me still there’s, I mean, I certainly don’t understand

this equation or this function all that well.

The more I learn about it, the prettier it is.

The idea that you can, this is sort of what gets you

representations of primes, not in terms of primes themselves,

but in terms of another set of numbers.

They’re like the non trivial zeros of the Zeta function.

And again, I’m very kind of in over my head

in a lot of ways as I like try to get to understand it.

But the more I do, it always leaves enough mystery

that it remains very beautiful to me.

So whenever there’s a little bit of mystery

just outside of the understanding that,

and by the way, the process of learning more about it,

how does that come about?

Just your own thought or are you reading?

Reading, yeah.

Or is the process of visualization itself

revealing more to you?

Visuals help.

I mean, in one time when I was just trying to understand

like analytic continuation and playing around

with visualizing complex functions,

this is what led to a video about this function.

It’s titled something like

Visualizing the Riemann Zeta Function.

It’s one that came about because I was programming

and tried to see what a certain thing looked like.

And then I looked at it and I’m like,

whoa, that’s elucidating.

And then I decided to make a video about it.

But I mean, you try to get your hands on

as much reading as you can.

You know, in this case, I think if anyone wants to start

to understand it, if they have like a math background

like they studied some in college or something like that,

like the Princeton Companion to Math

has a really good article on analytic number theory.

And that itself has a whole bunch of references

and you know, anything has more references

and it gives you this like tree to start piling through.

And like, you know, you try to understand,

I try to understand things visually as I go.

That’s not always possible,

but it’s very helpful when it does.

You recognize when there’s common themes,

like in this case, Cousins of the Fourier Transform

that come into play and you realize,

oh, it’s probably pretty important

to have deep intuitions of the Fourier Transform,

even if it’s not explicitly mentioned in like these texts.

And you try to get a sense of what the common players are.

But I’ll emphasize again, like,

I feel very in over my head when I try to understand

the exact relation between like the zeros

of the Riemann Zeta function

and how they relate to the distribution of primes.

I definitely understand it better than I did a year ago.

I definitely understand it on 100th as well as the experts

on the matter do, I assume.

But the slow path towards getting there is,

it’s fun, it’s charming,

and like to your question, very beautiful.

And the beauty is in the, what,

in the journey versus the destination?

Well, it’s that each thing doesn’t feel arbitrary.

I think that’s a big part,

is that you have these unpredictable,

no, yeah, these very unpredictable patterns

or these intricate properties of like a certain function.

But at the same time,

it doesn’t feel like humans ever made an arbitrary choice

in studying this particular thing.

So, you know, it feels like you’re speaking

to patterns themselves or nature itself.

That’s a big part of it.

I think things that are too arbitrary,

it’s just hard for those to feel beautiful

because this is sort of what the word contrived

is meant to apply to, right?

And when they’re not arbitrary means it could be,

you can have a clean abstraction and intuition

that allows you to comprehend it.

Well, to one of your first questions,

it makes you feel like if you came across

another intelligent civilization,

that they’d be studying the same thing.

Maybe with different notation.

Certainly, yeah, but yeah.

Like that’s what,

I think you talked to that other civilization,

they’re probably also studying the zeros

of the Riemann Zeta function

or like some variant thereof

that is like a clearly equivalent cousin

or something like that.

But that’s probably on their docket.

Whenever somebody does a lot of something amazing,

I’m gonna ask the question

that you’ve already been asked a lot

and that you’ll get more and more asked in your life.

But what was your favorite video to create?

Oh, favorite to create.

One of my favorites is,

the title is Who Cares About Topology?

You want me to pull it up or no?

If you want, sure, yeah.

It is about, well, it starts by describing

an unsolved problem that’s still unsolved in math

called the inscribed square problem.

You draw any loop and then you ask,

are there four points on that loop that make a square?

Totally useless, right?

This is not answering any physical questions.

It’s mostly interesting that we can’t answer that question.

And it seems like such a natural thing to ask.

Now, if you weaken it a little bit and you ask,

can you always find a rectangle?

You choose four points on this curve,

can you find a rectangle?

That’s hard, but it’s doable.

And the path to it involves things like looking at a torus,

this surface with a single hole in it, like a donut,

or looking at a mobius strip.

In ways that feel so much less contrived

to when I first, as like a little kid,

learned about these surfaces and shapes,

like a mobius strip and a torus.

Like what you learn is, oh, this mobius strip,

you take a piece of paper, put a twist, glue it together,

and now you have a shape with one edge and just one side.

And as a student, you should think, who cares, right?

Like, how does that help me solve any problems?

I thought math was about problem solving.

So what I liked about the piece of math

that this was describing that was in this paper

by a mathematician named Vaughn

was that it arises very naturally.

It’s clear what it represents.

It’s doing something.

It’s not just playing with construction paper.

And the way that it solves the problem is really beautiful.

So kind of putting all of that down

and concretizing it, right?

Like I was talking about how

when you have to put visuals to it,

it demands that what’s on screen

is a very specific example of what you’re describing.

The construction here is very abstract in nature.

You describe this very abstract kind of surface in 3D space.

So then when I was finding myself,

in this case, I wasn’t programming,

I was using a grapher that’s like built into OSX

for the 3D stuff to draw that surface,

you realize, oh man, the topology argument

is very non constructive.

I have to make a lot of,

you have to do a lot of extra work

in order to make the surface show up.

But then once you see it, it’s quite pretty

and it’s very satisfying to see a specific instance of it.

And you also feel like, ah,

I’ve actually added something

on top of what the original paper was doing

that it shows something that’s completely correct.

That’s a very beautiful argument,

but you don’t see what it looks like.

And I found something satisfying

in seeing what it looked like

that could only ever have come about

from the forcing function

of getting some kind of image on the screen

to describe the thing I was talking about.

So you almost weren’t able to anticipate

what it’s gonna look like.

I had no idea.

And it was wonderful, right?

It was totally, it looks like a Sydney Opera House

or some sort of Frank Gehry design.

And it was, you knew it was gonna be something

and you can say various things about it.

Like, oh, it touches the curve itself.

It has a boundary that’s this curve on the 2D plane.

It all sits above the plane.

But before you actually draw it,

it’s very unclear what the thing will look like.

And to see it, it’s very, it’s just pleasing, right?

So that was fun to make, very fun to share.

I hope that it has elucidated for some people out there

where these constructs of topology come from,

that it’s not arbitrary play with construction paper.

So let’s, I think this is a good sort of example

to talk a little bit about your process.

You have a list of ideas.

So that’s sort of the curse of having an active

and brilliant mind is I’m sure you have a list

that’s growing faster than you can utilize.

Now I’m ahead, absolutely.

But there’s some sorting procedure

depending on mood and interest and so on.

But okay, so you pick an idea

and then you have to try to write a narrative arc

that sort of, how do I elucidate?

How do I make this idea beautiful and clear

and explain it?

And then there’s a set of visualizations

that will be attached to it.

Sort of, you’ve talked about some of this before,

but sort of writing the story, attaching the visualizations.

Can you talk through interesting, painful,

beautiful parts of that process?

Well, the most painful is if you’ve chosen a topic

that you do want to do, but then it’s hard to think of,

I guess how to structure the script.

This is sort of where I have been on one

for like the last two or three months.

And I think that ultimately the right resolution

is just like set it aside and instead do some other things

where the script comes more naturally.

Because you sort of don’t want to overwork a narrative.

The more you’ve thought about it,

the less you can empathize with the student

who doesn’t yet understand the thing you’re trying to teach.

Who is the judger in your head?

Sort of the person, the creature,

the essence that’s saying this sucks or this is good.

And you mentioned kind of the student you’re thinking about.

Can you, who is that?

What is that thing?

That says, the perfectionist that says this thing sucks.

You need to work on that for another two, three months.

I don’t know.

I think it’s my past self.

I think that’s the entity that I’m most trying

to empathize with is like you take who I was,

because that’s kind of the only person I know.

Like you don’t really know anyone

other than versions of yourself.

So I start with the version of myself that I know

who doesn’t yet understand the thing, right?

And then I just try to view it with fresh eyes,

a particular visual or a particular script.

Like, is this motivating?

Does this make sense?

Which has its downsides,

because sometimes I find myself speaking to motivations

that only myself would be interested in.

I don’t know, like I did this project on quaternions

where what I really wanted was to understand

what are they doing in four dimensions?

Can we see what they’re doing in four dimensions, right?

And I came up with a way of thinking about it

that really answered the question in my head

that made me very satisfied

and being able to think about concretely with a 3D visual,

what are they doing to a 4D sphere?

And so I’m like, great,

this is exactly what my past self would have wanted, right?

And I make a thing on it.

And I’m sure it’s what some other people wanted too.

But in hindsight, I think most people who wanna learn

about quaternions are like robotics engineers

or graphics programmers who want to understand

how they’re used to describe 3D rotations.

And like their use case was actually a little bit different

than my past self.

And in that way, like,

I wouldn’t actually recommend that video

to people who are coming at it from that angle

of wanting to know, hey, I’m a robotics programmer.

Like, how do these quaternion things work

to describe position in 3D space?

I would say other great resources for that.

If you ever find yourself wanting to say like,

but hang on,

in what sense are they acting in four dimensions?

Then come back.

But until then, that’s a little different.

Yeah, it’s interesting

because you have incredible videos on neural networks,

for example.

And from my sort of perspective,

because I’ve probably, I mean,

I looked at the,

is sort of my field

and I’ve also looked at the basic introduction

of neural networks like a million times

from different perspectives.

And it made me realize

that there’s a lot of ways to present it.

So you were sort of, you did an incredible job.

I mean, sort of the,

but you could also do it differently

and also incredible.

Like to create a beautiful presentation of a basic concept

requires sort of creativity, requires genius and so on,

but you can take it from a bunch of different perspectives.

And that video on neural networks made me realize that.

And just as you’re saying,

you kind of have a certain mindset, a certain view,

but from a, if you take a different view

from a physics perspective,

from a neuroscience perspective,

talking about neural networks

or from a robotics perspective,

or from, let’s see,

from a pure learning, statistics perspective.

So you can create totally different videos.

And you’ve done that with a few actually concepts

where you’ve have taken different cuts,

like at the Euler equation, right?

You’ve taken different views of that.

I think I’ve made three videos on it

and I definitely will make at least one more.


Never enough.

So you don’t think it’s the most beautiful equation

in mathematics?

Like I said, as we represent it,

it’s one of the most hideous.

It involves a lot of the most hideous aspects

of our notation.

I talked about E, the fact that we use pi instead of tau,

the fact that we call imaginary numbers imaginary,

and then, hence, I actually wonder if we use the I

because of imaginary.

I don’t know if that’s historically accurate,

but at least a lot of people,

they read the I and they think imaginary.

Like all three of those facts,

it’s like those are things that have added more confusion

than they needed to,

and we’re wrapping them up in one equation.

Like boy, that’s just very hideous, right?

The idea is that it does tie together

when you wash away the notation.

Like it’s okay, it’s pretty, it’s nice,

but it’s not like mind blowing greatest thing

in the universe,

which is maybe what I was thinking of when I said,

like once you understand something,

it doesn’t have the same beauty.

Like I feel like I understand Euler’s formula,

and I feel like I understand it enough

to sort of see the version that just woke up

that hasn’t really gotten itself dressed in the morning

that’s a little bit groggy,

and there’s bags under its eyes.

So you’re past the dating stage,

you’re no longer dating, right?

I’m still dating the Zeta function,

and like she’s beautiful and right,

and like we have fun,

and it’s that high dopamine part,

but like maybe at some point

we’ll settle into the more mundane nature of the relationship

where I like see her for who she truly is,

and she’ll still be beautiful in her own way,

but it won’t have the same romantic pizzazz, right?

Well, that’s the nice thing about mathematics.

I think as long as you don’t live forever,

there’ll always be enough mystery and fun

with some of the equations.

Even if you do, the rate at which questions comes up

is much faster than the rate at which answers come up, so.

If you could live forever, would you?

I think so, yeah.

So you think, you don’t think mortality

is the thing that makes life meaningful?

Would your life be four times as meaningful

if you died at 25?

So this goes to infinity.

I think you and I, that’s really interesting.

So what I said is infinite, not four times longer.

I said infinite.

So the actual existence of the finiteness,

the existence of the end, no matter the length,

is the thing that may sort of,

from my comprehension of psychology,

it’s such a deeply human,

it’s such a fundamental part of the human condition,

the fact that there is, that we’re mortal,

that the fact that things end,

it seems to be a crucial part of what gives them meaning.

I don’t think, at least for me,

it’s a very small percentage of my time

that mortality is salient,

that I’m aware of the end of my life.

What do you mean by me?

I’m trolling.

Is it the ego, is it the id, or is it the superego?

The reflective self, the Wernicke’s area

that puts all this stuff into words.

Yeah, a small percentage of your mind

that is actually aware of the true motivations

that drive you.

But my point is that most of my life,

I’m not thinking about death,

but I still feel very motivated to make things

and to interact with people,

experience love or things like that.

I’m very motivated,

and it’s strange that that motivation comes

while death is not in my mind at all.

And this might just be because I’m young enough

that it’s not salient.

Or it’s in your subconscious,

or that you’ve constructed an illusion

that allows you to escape the fact of your mortality

by enjoying the moment,

sort of the existential approach to life.

Could be.

Gun to my head, I don’t think that’s it.

Yeah, another sort of way to say gun to the head

is sort of the deep psychological introspection

of what drives us.

I mean, that’s, in some ways to me,

I mean, when I look at math, when I look at science,

is a kind of an escape from reality

in a sense that it’s so beautiful.

It’s such a beautiful journey of discovery

that it allows you to actually,

it sort of allows you to achieve a kind of immortality

of explore ideas and sort of connect yourself

to the thing that is seemingly infinite,

like the universe, right?

That allows you to escape the limited nature

of our little, of our bodies, of our existence.

What else would give this podcast meaning?

That’s right.

If not the fact that it will end.

This place closes in 40 minutes.

And it’s so much more meaningful for it.

How much more I love this room

because we’ll be kicked out.

So I understand just because you’re trolling me

doesn’t mean I’m wrong.

But I take your point.

I take your point.

Boy, that would be a good Twitter bio.

Just because you’re trolling me doesn’t mean I’m wrong.

Yeah, and sort of difference in backgrounds.

I’m a bit Russian, so we’re a bit melancholic

and seem to maybe assign a little too much value

to suffering and mortality and things like that.

Makes for a better novel, I think.

Oh yeah, you need some sort of existential threat

to drive a plot.

So when do you know when the video is done

when you’re working on it?

That’s pretty easy actually,

because I’ll write the script.

I want there to be some kind of aha moment in there.

And then hopefully the script can revolve around

some kind of aha moment.

And then from there, you’re putting visuals

to each sentence that exists,

and then you narrate it, you edit it all together.

So given that there’s a script,

the end becomes quite clear.

And as I animate it, I often change

certainly the specific words,

but sometimes the structure itself.

But it’s a very deterministic process at that point.

It makes it much easier to predict

when something will be done.

How do you know when a script is done?

It’s like, for problem solving videos,

that’s quite simple.

It’s once you feel like someone

who didn’t understand the solution now could.

For things like neural networks,

that was a lot harder because like you said,

there’s so many angles at which you could attack it.

And there, it’s just at some point

you feel like this asks a meaningful question

and it answers that question, right?

What is the best way to learn math

for people who might be at the beginning of that journey?

I think that’s a question that a lot of folks

kind of ask and think about.

And it doesn’t, even for folks

who are not really at the beginning of their journey,

like there might be actually deep in their career,

some type they’ve taken college

or taken calculus and so on,

but still wanna sort of explore math.

What would be your advice instead of education at all ages?

Your temptation will be to spend more time

like watching lectures or reading.

Try to force yourself to do more problems

than you naturally would.

That’s a big one.

Like the focus time that you’re spending

should be on like solving specific problems

and seek entities that have well curated lists of problems.

So go into like a textbook almost

and the problems in the back of a textbook kind of thing,

back of a chapter.

So if you can take a little look through those questions

at the end of the chapter before you read the chapter,

a lot of them won’t make sense.

Some of them might,

and those are the best ones to think about.

A lot of them won’t, but just take a quick look

and then read a little bit of the chapter

and then maybe take a look again and things like that.

And don’t consider yourself done with the chapter

until you’ve actually worked through a couple exercises.

And this is so hypocritical, right?

Cause I like put out videos

that pretty much never have associated exercises.

I just view myself as a different part of the ecosystem,

which means I’m kind of admitting

that you’re not really learning,

or at least this is only a partial part

of the learning process if you’re watching these videos.

I think if someone’s at the very beginning,

like I do think Khan Academy does a good job.

They have a pretty large set of questions

you can work through.

Just the very basics,

sort of just picking up,

getting comfortable with the very basic linear algebra,

calculus or so on, Khan Academy.

Programming is actually I think a great,

like learn to program and like let the way

that math is motivated from that angle push you through.

I know a lot of people who didn’t like math

got into programming in some way

and that’s what turned them on to math.

Maybe I’m biased cause like I live in the Bay area,

so I’m more likely to run into someone

who has that phenotype.

But I am willing to speculate

that that is a more generalizable path.

So you yourself kind of in creating the videos

are using programming to illuminate a concept,

but for yourself as well.

So would you recommend somebody try to make a,

sort of almost like try to make videos?

Like you do as a way to learn?

So one thing I’ve heard before,

I don’t know if this is based on any actual study.

This might be like a total fictional anecdote of numbers,

but it rings in the mind as being true.

You remember about 10% of what you read,

you remember about 20% of what you listen to,

you remember about 70% of what you actively interact with

in some way, and then about 90% of what you teach.

This is a thing I heard again,

those numbers might be meaningless,

but they ring true, don’t they, right?

I’m willing to say I learned nine times better

if I’m teaching something than reading.

That might even be a low ball, right?

So doing something to teach

or to like actively try to explain things

is huge for consolidating the knowledge.

Outside of family and friends,

is there a moment you can remember

that you would like to relive

because it made you truly happy

or it was transformative in some fundamental way?

A moment that was transformative.

Or made you truly happy?

Yeah, I think there’s times,

like music used to be a much bigger part of my life

than it is now, like when I was a, let’s say a teenager,

and I can think of some times in like playing music.

There was one, like my brother and a friend of mine,

so this slightly violates the family and friends,

but it was the music that made me happy.

They were just accompanying.

We like played a gig at a ski resort

such that you like take a gondola to the top

and like did a thing.

And then on the gondola ride down,

we decided to just jam a little bit.

And it was just like, I don’t know,

the gondola sort of came over a mountain

and you saw the city lights

and we’re just like jamming, like playing some music.

I wouldn’t describe that as transformative.

I don’t know why, but that popped into my mind

as a moment of, in a way that wasn’t associated

with people I love, but more with like a thing I was doing,

something that was just, it was just happy

and it was just like a great moment.

I don’t think I can give you anything deeper than that.

Well, as a musician myself, I’d love to see,

as you mentioned before, music enter back into your work,

back into your creative work.

I’d love to see that.

I’m certainly allowing it to enter back into mine.

And it’s a beautiful thing for a mathematician,

for a scientist to allow music to enter their work.

I think only good things can happen.

All right, I’ll try to promise you a music video by 2020.

By 2020?

By the end of 2020.

Okay, all right, good.

Give myself a longer window.

All right, maybe we can like collaborate

on a band type situation.

What instruments do you play?

The main instrument I play is violin,

but I also love to dabble around on like guitar and piano.

Beautiful, me too, guitar and piano.

So in a mathematician’s lament, Paul Lockhart writes,

the first thing to understand

is that mathematics is an art.

The difference between math and the other arts,

such as music and painting,

is that our culture does not recognize it as such.

So I think I speak for millions of people, myself included,

in saying thank you for revealing to us

the art of mathematics.

So thank you for everything you do

and thanks for talking today.

Wow, thanks for saying that.

And thanks for having me on.

Thanks for listening to this conversation

with Grant Sanderson.

And thank you to our presenting sponsor, Cash App.

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And now, let me leave you with some words of wisdom

from one of Grant’s and my favorite people, Richard Feynman.

Nobody ever figures out what this life is all about,

and it doesn’t matter.

Explore the world.

Nearly everything is really interesting

if you go into it deeply enough.

Thank you for listening, and hope to see you next time.

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