Grant Sanderson: 3Blue1Brown and the Beauty of Mathematics
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§01Transcript · 00:00–07:29
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.
This is the Artificial Intelligence Podcast.
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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?
(Grant laughs)
- 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 arithmetics?
Sorry to interrupt.
- 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?
'Cause 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 to the real numbers,
there's a little bit of choice in that.
So there's this funny number system
called the surreal 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
have 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 the 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
'cause 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
as 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 notation.
- 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
§02Transcript · 07:33–15:07
somehow still nevertheless coming together nicely.
'Cause you have like several disciplines coming together
in a single equation.
- Well, I think-- - In a sense,
historically speaking.
- That's true.
So like the number E is significant.
It shows up in probability all the time.
It 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.
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, you know,
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?
- Yeah.
- 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.
- Never thought of it that way.
Yeah.
(laughs)
- Right?
- Yeah.
- It makes sense why the notation came about that way.
- Yes.
- It's 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.
- Let me jump back for a second to aliens
and the nature of our universe.
Okay.
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, you know, 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 was in some abstract world
detached from the 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, 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 in some deep way for us humans,
it does always come back to that three-dimensional world
for the usefulness of the three-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, 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 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
'cause they're not the ones
where the Pythagorean theorem holds.
So we have a discovery, a genuine bona fide 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
§03Transcript · 15:10–23:31
and how big is it?
- Well, I think of math as being the study
of like 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
'cause 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 into the more and more abstract
things, the things that feel powerful.
These are the ones that initially invented topology
and then later on get really into category theory
and go on about 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.
I'm asking a million questions
that nobody knows the answer to.
- Yeah, I have no idea.
(laughing)
Why is it simple?
- It could be the case that
there's like a filtration 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 some thing that is simple
when you wash away the 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 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.
Like not in a strict definition of chaos,
but meaning like just it's impossible
for equations to capture,
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 planets move and 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.
It just seems beautiful.
It's also weird, probably to the point you were 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 mt squared equations.
- Right.
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, I don't know what I'm talking about at all.
- Or maybe the universe is actually not so compressible,
but the way our brain evolved,
we're only able to perceive the compressible parts.
I mean, we are, so this is a sort of Chomsky argument.
We are just descendants of apes.
We're 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 physics isn't too far afield,
simply based on what we can do.
- Yeah, the fact that we can fly is pretty good.
- It's great, yeah.
It's a pretty good 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 the information processing
of whatever the physical world is,
it quickly becomes the case that you have a limit
to the layers that could exist there,
because 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 highest up level universe is,
or that there's some bound to that capacity,
which then limits the number of levels available.
How do you place some kind of probability distribution
on 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.
§04Transcript · 23:34–31:40
- But you can always raise two to the power of that.
- Yeah, exactly.
- 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,
where there are 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
is useful to 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?
You know, 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.
Like the idea that there even exists such a limit
is at the very least thought provoking,
when naively you might assume,
oh, well, you know,
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.
- It's that 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
that's 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 saying them,
or writing them, or listing them.
- That's an instantiation.
Okay, how many abstract words exist?
- Well, the idea of 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 change 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 non-physical, 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 this 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?
(both laughing)
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-- - Generalization, yeah.
- 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 a little just for the fun of it.
- Yeah, I know. - I'm the 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 would venture to say it's very difficult.
And so there's some concepts in mathematics,
like maybe multiple dimensions, we could sort of talk about,
that are impossible for us to truly intuit.
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.
- No, heck no, heck no.
I think to that point, for any particular idea like this,
there's multiple angles of attack.
I think when we do visualize infinity,
what we're actually doing, you write dot, dot, dot.
One, two, three, four, dot, dot, dot.
Those are symbols on the page that are insinuating
a certain infinity.
§05Transcript · 31:41–39:28
What you're capturing with a little bit of design there
is the I can always add one more property.
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, 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, 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.
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.
It's always tempting to start an explanation
from the top to the bottom.
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
'cause 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.
- 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.
- Yeah, 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.
- Yes.
- 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?
The, yeah, so that's, you're going,
the visualization itself is actually going to the bottom.
- 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 going to 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 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
'cause of some of the things it 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.
You know, 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 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 had properly 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
that are 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, in this case, I think if anyone wants
to start to understand it,
if they have like a math background of some,
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 anything has more references
and it gives you this like tree to start pawing through.
And like, 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,
§06Transcript · 39:30–46:56
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 1/100 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,
not, 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, 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,
it 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?"
- Do 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 described 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 was gonna look like?
- I had no idea.
I had no idea.
And it was wonderful.
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.
So you have a list of ideas.
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.
- Yeah, I love the head.
Absolutely.
- But there's some sorting procedure,
depending on mood and interest and so on.
But, okay, so you pick an idea,
then you have to try to write a narrative arc
that's 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.
'Cause 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's Chris, 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
§07Transcript · 46:57–54:34
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.
'Cause 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
in 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 want to 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
'cause you have incredible videos on neural networks,
for example.
And from my sort of perspective,
'cause I've probably, I mean,
I looked at the,
it's 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 are 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
is, 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're 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 theory, - Statistics.
- 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, at the, at the Euler equation, right?
The, 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.
- 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,
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 ideas 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.
It's like it's the real-- - So you're past
the dating stage, and you're now married.
- I'm 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.
- If you could live forever, would you?
- I think so, yeah.
- So you don't think mortality's 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,
like it's a very small percentage of my time
that mortality is salient,
that I'm like 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 super ego?
- 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,
like 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 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 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.
§08Transcript · 54:37–1:02:28
- 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.
- 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 they might be actually deep in their career,
some type, they've taken college,
they've taken calculus and so on,
but still wanna sort of explore math.
What would be your advice in sort of education at all ages?
- Your temptation will be to spend more time
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 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 text,
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
or calculus or something, Khan Academy.
- Programming is actually, I think, a great,
like learn to program and 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 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 sometimes in like playing music,
there was one, my brother and a friend of mine,
so this slightly violates the family and friends,
but there was a music that made me happy.
They were just accompanying.
We had like played a gig at a ski resort,
such that you like take a gondola to the top
and like did a thing.
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 over, 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 though.
- 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.
- I'll 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 "The 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.
Download it, use code LEXPODCAST,
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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
§09Transcript · 1:02:30–1:02:39
"if you go into it deeply enough."
Thank you for listening and hope to see you next time.
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