The following is a conversation with Grant Sanderson.
He's a math educator and creator of Three Blue One Brown,
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?
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, 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?
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 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
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 the 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 a 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 to the idea of repeated multiplication,
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 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 tether ball 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 explanation 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 have we were,
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.
Cause 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, 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.
I never thought of it that way.
Yeah.
Right?
Yeah.
So 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...
So we 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 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 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, as 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,
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 their 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 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.
You 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 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
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, right?
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
and 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
and math are gonna give you very different answers
about what the relation at play here is
because 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 what 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 to me like Steven 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-
Yeah, I have no idea.
I'm like, 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,
other ones that are simple enough,
they could describe it mathematically.
But as soon as it's a sufficiently complex system,
they're like, yeah, that's outside the realm of physics.
That's biology or whatever have you.
And of course that's true, right?
You know, maybe there's something where it's like,
of course there will always be some thing 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.
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 plan is moving, just the low level
of the atomic scale, how the materials operate
at the high scale, how black holes operate.
But 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 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
at some weird way here.
I don't know.
Like I don't know what I'm talking about at all.
Well, 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, this is a 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 afield,
simply based on what we can do.
Yeah, the fact that we can fly is pretty good.
It's great.
Yeah.
And land.
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, 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 us, 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
used 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?
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,
we'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, 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 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 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.
How many possible words do you think could exist
that are just like strings of letters?
That's a sort of mathematical statement as beautiful
and we use infinity in basically everything we do
in science, math and engineering, yes.
But you said exist.
The question is, you said letters or words?
I said 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.
You think about intelligence a lot,
like artificial intelligence.
Would you not say that what it's doing
is a kind of abstraction?
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 analog of pixels on my retina change entirely,
that I can still have some coherent notion
of this is Lex, I'm talking about Lex.
What that requires is you have a disparate set
of possible images hitting me
that are unified in a notion of Lex.
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 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, 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 your face on my retina
share enough common attributes
that I can put this single notion to it.
In that way, infinity is an abstraction,
and it's very powerful, and it's only through such abstractions
that we can actually understand
the world and logic and things.
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.
I'm 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.
Oh, I could always add one more
that gives it this nature of infinitiveness
in the same way that there are certain properties
of your face that give it the lexness.
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 intelligence 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,
ooh, 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 phenomena
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.
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 dilute 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 in mathematics,
like maybe multiple dimensions,
we could sort of talk about it,
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, right?
No, heck no.
Heck no.
But 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, right?
One, two, three, four, dot, dot, dot.
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.
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 get them all in my head and you realize,
oh, your brain would literally collapse
into a black hole, all of that.
I honestly feel this with a lot of math
that I try to read where I don't think of myself
as 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, 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.
It's always tempting
to start an explanation from the top
to the bottom. You give
the definition of a new theorem.
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
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 a 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?
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 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 like
just gave you pause.
It's the most beautiful idea. Small
or big. So I think often the things
that are most beautiful are the ones
that you have like
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
like
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 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.
It's like 1 over 1 to the s plus
1 over 2 to the s plus 1 over 3 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 that 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
like all the primes getting wrapped together like this.
This is like the Euler product
for the zeta function
as I 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.
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 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
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.
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
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 plying 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's
data 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 one one hundredth as well as the
experts on the matter do I assume
but
the slow path towards getting there
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
each thing doesn't feel arbitrary I think that's
a big part is that you have
these unpredictable
yeah these very unpredictable patterns
where 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
and this is sort of what the word contrived
is meant to apply to right
and
the 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
that's what I think you talk 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
on their docket
whenever somebody does
a lot of something amazing
I'm going to ask the question
that you've already been asked a lot
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
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 4 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 4 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 von
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
used 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
I've actually added something on top of what the
original paper was doing that
it shows something that's completely correct
it'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 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 going to 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 Gary
design and it was
you knew it was going to be something and you
can say various things about it like oh it
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'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
a good sort of example to
talk a little about your process
you have a list of ideas
that's sort of the
the curse of having
having an active and brilliant mind
is I'm sure you have a list that's growing
faster than you can utilize
now on 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 because you sort of don't want to overwork
a
a narrative that 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
judge 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 it 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
because that's kind of the only person I know
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
and then
I just try to
view it with fresh eyes
a particular visual or a particular script
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, 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
and I
had 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
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
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
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
but hang on in what sense are they acting
in four dimensions then come back
but until then it'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
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 if 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
is
requires sort of
creativity requires genius and so on
but you can take it from
a bunch of different perspectives and that video
networks mean you realize that
and just as you're saying you kind of have a certain
mindset a certain view
but
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
you can create totally different videos
and you've done that with a few actually concepts
where you have taken different
classes like at the
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
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
the dating stage and you're now married
we'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
it's an 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
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, is it the superego
is it
the reflective self the vernici
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 like
make things and to like interact with people
like experience love or things like that
I'm very motivated and I
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 construct an illusion
that allows you to escape
the fact of your mortality by
enjoying the moment sort of the existential
approach life
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
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 it 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 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
that it 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 immortality and things like that
makes for
a better novel I think
oh yeah you need some sort of
existential threat to
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
as I animate it I often change
the
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
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 there might be actually
deep in their career
some type of type in college
taking calculus and so on but still want to
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
kind of 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 because I 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 a very basic sort of
just picking up getting comfortable
with a very basic linear algebra or calculus
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
but 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 because 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
I'm willing to say I learned nine times better
from teaching something than reading
that might even be a low ball
so doing something to teach
or to 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
music used to be a much bigger part
of my life than it is now
when I was a teenager
and I can think of sometimes
playing music
there was one where my brother
and a friend of mine
slightly violates the family and friends
but there was music that made me happy
they were just accompanying
we played a gig at a ski resort
such that you take a gondola to the top
and did a thing
and then on the gondola right down
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 were 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
I'll try to promise you a music video
by 2020
by the end of 2020
by myself a longer window
alright maybe we can
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 the 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 not recognized 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
well 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
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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