Kind: captions
Language: en
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 and linear algebra
calculus and other fields of mathematics
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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 1 2 3 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
serial numbers that it captures the idea
of continuity it's a distinct
mathematical object you could very well
you know model the universe and motion
of planets with that as the backend 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 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 going to
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 I
would be comfortable saying give a favor
or least favorite piece of notation in
terms of its effectiveness yes yeah well
so at 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
you repeatedly multiply it by itself and
then you say well 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 and that's all
nice that's all nice and well but very
famously you have like e to the PI
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 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 and
so the repeater multiplication is
actually missing the main point the
power of e to the S I mean what it
addresses is things where the rate at
which something changes depends on its
own value but more specifically it
depends on it linearly so for example if
you have like a population that's
growing and the rate at which it grows
depends on how many members of the
population are already there it looks
like this nice exponential curve it
makes sense to talk about repeated
multiplication because you say how much
is there after one year two years three
years you're multiplying by something
the relationship can be a little bit
different sometimes where let's say
you've got a ball on a string like a
like a game of tetherball going around a
rope right and you say it's velocity is
always perpendicular to its position
that's another way of describing its
rate of change is being related to where
it is but it's a different operation
you're not scaling it it's a rotation
it's this 90 degree rotation that's what
the whole idea of like complex
exponentiation is trying to capture but
it's obfuscated in the notation when
what it's actually saying like if you
really parse something like e to the PI
I what it's saying is choose an origin
always move perpendicular to the vector
from that origin to you okay
then when you walk PI times that radius
you'll be halfway around like that's
what it's saying it's kind of the u-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
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 if 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 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 you if your
velocity vector is perpendicular to your
position vector you walk in a circle
that's pretty it's not the most
beautiful thing in the world but it's
quite pretty
the beauty of it I think comes from
perhaps the awkwardness of the notation
somehow still nevertheless coming
together nicely because you have like
several disciplines coming together in a
single equation well in a sense like
historically speaking but 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 is 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 the way that a shows up is when you
have things like exponential growth and
decay right it's when this relation that
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 X
that's like the exponential function
when you plug in one you get this nice
constant called EE 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
this 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 kind of like to
sleep in separate beds if they have the
choice but you see them all there and
you know there is a family resemblance
but it's not that close
so actually think of it as a function is
uh this is the better idea and that's a
notational idea and yeah and like here's
the thing the constant e sort of stands
is this numerical representative of
calculus right yeah
calculus is the like study of change
mm-hmm so it's very at least there's a
little cognitive dissonance using a
constant to represent the science of
change never thought of it that way yeah
yeah it makes sense why the notation
came about that way yes because this is
the first way that we saw it um in the
context of things like population growth
or compound interest it is nicer to
think about as repeated multiplication
that's definitely nicer but it's more
that that's the first application of
what turned out to be a much more
general function that maybe the
intelligent life your initial question
asked about would have come to recognize
as being much more significant than the
single use case which lends itself to
repeated multiplication notation but let
me jump back for a second to aliens and
the nature of our universe 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 is yeah it's 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 a discovery because
there 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 some abstract world detached from
physics but I think more and more math
has become detached from you know we
when you even look at modern physics
from string theory so 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 higher dimensions can
be highly highly applicable I think this
is a common misinterpretation the 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 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 I'm 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 meaningful
even understand a three dimensional
world it would be useful to have five
dimensional manifolds yes absolutely
because of state spaces but you're
saying there in some in some deep way
for us humans it does it does always
come back to that three dimensional
world for the useful usefulness that the
dimensional world and therefore it
starts with a discovery but then we
invent the mathematics that
helps us make sense of the discovery in
a sense yes I mean just to jump off of
the Pythagorean theorem it feels like a
discovery you've got these beautiful
geometric proofs where you've got
squares and you're modifying there is it
feels like a discovery if you look at
how we formalize the idea of 2d space as
being r2 right all pairs of real numbers
and how we define a metric on it and
define distance okay hang on a second
we've defined distance so that the
Pythagorean theorem is true so then
suddenly it doesn't feel that great but
I think what's going on is the thing
that informed us what metric to put on
r2 to put on our abstract representation
of 2d space came from physical
observations and the thing is there's
other metrics you could have put on it
we could have consistent math with other
notions of distance it's just that those
pieces of math wouldn't be applicable to
the physical world that we study because
they're not the ones where the
Pythagorean theorem holds so we have a
discovery a genuine 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 is 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 the Richard Fineman
question then along that thread is what
do you think is a 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 rigour of
mathematics that allows you to I mean
just like what 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 like abstractions over patterns and
pure
in logic and then physics is obviously
grounded in a desire to understand the
world that we live in yeah I think
you're going to get very different
answers when you talk to different
mathematicians because there's a wide
diversity and 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 right there
are some who are very physically
motivated who who tried 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 it'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 like topology and then later on
get really into category theory and go
on about like infinite categories and
whatnot these are the ones that love to
have a system that can describe truths
about as many things as possible right
people from those three different veins
of motivation into math are going to
give you very different answers about
what the relation at play here is
because someone like flightmare Arnold
who is this he's 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 well 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 that'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 and the nature of reality is so
compressible and too clean beautiful
equations that are for the most part
simple relatively speaking a lot simpler
than they could be so you have we
mentioned somebody like Stephen Wolfram
who thinks that sort of there's
incredibly simple rules underlying our
reality but it can create arbitrary
complexity but there is simple equations
what I'm asking a million questions that
nobody knows the answer to but no idea
why is it simple I it could be the case
that there's like a filter iteration I
played the only things that physicists
find interesting other ones little
simple enough they could describe it
mathematically but as soon as it's a
sufficiently complex system like now
that's outside the realm of physics
that's biology or whatever have you and
of course that's true all right you know
maybe there's something what'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 some like
an information theory standpoint there
might be some like you 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 some set of equations that you
can't decouple from each other well no
it could be it could be that sort of we
take for granted that they're the 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 are the world
operates is different everywhere it's
like a like a deeply distributed system
or just everything is just chaos like
not not in a strict definition of cast
but meaning like just it's impossible
for equations to capture for to
explicitly model the world as cleanly as
the physical does any we're almost take
it for granted that we can describe we
can have an equation for gravity
for action in a distance we can have
equations for some of these basic ways
the planets moving just the the
low-level at the atomic scale all the
materials operate at the high scale how
black holes operate but it doesn't it it
seems like it could be there's infinite
other possibilities where none of it
could be compressible into such equation
so it just seems beautiful it's also
weird probably to the point you're
making that it's very pleasant that this
is just for our minds right 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
mc-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
it or maybe the universe is actually not
so compressible but the way our brain
the the way our brain evolved were only
able to perceive the compressible parts
I mean we are so this is a sort of
Chomsky argument we are just the
sentence of apes over like really
limited biological systems so totally
make sense there were really limited
little computers calculators that are
able to perceive certain kinds of things
in the actual world is much more
complicated well but we can we can do
pretty awesome things right like we can
fly spaceships and that 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 and the fact
that we can fly is pretty good it's
great the laws were working with our 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 the computation running 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 yes and then the 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
unlike 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 they
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 I don't mean like people
almost assume a certain uniform
probability over all of those metal
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 what you can always raise
two to the power of that exactly yeah
like numbers bit big and we're easily
humbled but basically everything around
us so it's very difficult to to kind of
make sense of anything actually when you
look up at the sky and look at the stars
in 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 so when I
think about the simulation hypothesis I
think is 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 useless me as a
thought experiment is is the universe
just the computer is it just the
computation is all of this just the
computation and so the same kind of
tools we apply to analyzing algorithms
can that be applied you know if we scale
further and further and further
well the arbitrary power of those
systems start to create some interesting
aspects that we see in our universe or
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 like
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 that 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 I guess what
I'm channeling is the surprise that I
felt
learning that fact that there are the
information is physical in this way is
that there's a finance to it okay let me
just even go off on that from
mathematics perspective and the
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 okay no not really it doesn't make
any sense to me
I don't know like how many how many
words how many possible words do you
think could exist that are just like
strings of letters so that that's a sort
of mathematical statement as beautiful
and we use infinity basically everything
we do everything we do inside in science
math and engineering yes but you said
exist my the question is well you said
letters of words
I said words words the to bring words
into existence to me you have to start
like saying them or like writing them or
like listing them that's an
instantiation okay combination how many
abstract words exist it was the idea of
abstract yeah the the idea of abstract
notions and ideas I think we should be
clear around 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
they're like abstraction is key to
conceptualizing the universe you get
this raw sensory data you need I need
something that every time you move your
face a little bit and the they're not
pixels but like analog of pixels on my
retina change entirely yeah that I can
still have some coherent notion of this
is Lex and planet Lex yes right what
that requires is you have a disparate
set of possible images hitting me that
are unified in a notion of Lex yeah
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
woowoo
it like ineffable or something what I
mean is it something that can apply to a
multiplicity of situations that share
certain common attribute in the same way
that the images of like your face on my
retina Sharon
common attributes that I can put the
single notion to it like in that way
infinity is an abstraction and it's very
powerful and and it's a it's only
through such abstraction 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 infinite enos in the same way that
they're certain like properties of your
face that give it the Lexx miss 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 intelligence
systems do this 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 whoo right sort
of and because we're now it's like it's
like we built this stack of you know the
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 and take us into areas
that are disconnected from reality in a
way that we could never give back what
if instead of calling these abstract how
different would it be in your mind if we
call them general and the phenomenon
that you're describing is over
generalization when you try them
channelization 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 and that devil's advocate and uh I
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
that means I don't I would venture to
say it's very difficult and so there's
some concepts of mathematics like maybe
multiple dimensions we could sort of
talk about that are impossible for us to
truly into it like and it just feels
dangerous to me to use these as part of
our toolbox of abstractions on behalf of
your listeners I almost fear we're
getting too philosophical oh no I I
think to that point for any particular
idea like this there's multiple angles
of attack
I think the when we do visualize
infinity what we're actually doing you
know you write dot dot dot one two three
four dot dot right that's those are
symbols on the page that are insinuating
a certain infinity what you're capturing
with a little bit of design there is the
I can always add one more property right
I think I'm just as uncomfortable with
you are
if you try to concretize it so much that
you have a bag of infinitely many things
that I actually think of no not one two
three four dot dot dot one two three
four five six seven eight I try to get
them all and I had and you realize oh I
you know your your brain would literally
collapse into a black hole all of that
and and I honestly feel this with a lot
of math that I tried to read where I I
don't think of myself as like
particularly good at math in some ways
like I get very confused often when I am
going through some of these texts and
often when I'm feeling my head is like
this is just so damn have strict I just
can't wrap my head around I just wanted
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 out
this idea of abstraction being clear
about your layers of abstraction yes
right it's always tempting to start an
explanation from the top to the bottom
yeh you you give the definition of a new
theorem you're like this is the
definition of a vector space for example
we're gonna that's how well start of
course these are the properties of a
vector space Yuma 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 it 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 the 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 like
that 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 eating someone's
path to understanding you know I'm sorry
to interrupt you but you just made me
realize that that's exactly right so the
visualization is you're creating while
you're sometimes talking about
abstractions the actual visualization is
a explicit low-level example yes so
there there's an actual like in the code
you have to say what the what the vector
is what's the direction of the arrow
what's the magnitude of the yeah so
that's you're going the visualization
itself is actually going to the bottom I
think and I think that's very important
I also think about this a lot in writing
scripts where even before you get to the
visuals the first instinct is to I don't
know why I just always do I say the
abstract thing I say the general
definition the powerful thing and then I
fill it in with examples later always it
will be more compelling and easier to
understand when you flip that and
instead you let someone's brain do the
pattern recognition you just show them a
bunch of examples the brain is gonna
feel a certain similarity between them
then by the time you bring in the
definition or by the time you bring in
the formula its 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 still with you what is
the most beautiful or on inspiring idea
you've come across in mathematics I
don't know maybe it's an ID you've
explored in your videos maybe not what
mike 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 almost a moment of
the discovery for you personally almost
just that leap of haha moment so
something that really caught my eye I
remember when I was little there were
these I come I think the series was
called like wooden books or something
these tiny little books that would have
just a very short description of
something on the left and then a picture
on the right I don't know who they're
meant for but maybe it's like loosely
children or something like that but it
can't just be children because of some
of the things I was describing on the
last page of one of them some were tiny
in there was this little formula the on
the left hand had a sum over all of the
natural numbers you know 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 the
infinity then on the other side had a
product over all of the primes and it
was a certain thing I had to do with all
the primes and like any good young math
enthusiast I'd probably been
indoctrinated with how chaotic and
confusing the primes are which they are
and seeing this equation where on one
site you have something that's as
understandable as you could possibly get
the counting numbers yes and on the
other side is all the prime numbers it
was like this whoa they're related like
this there's there's a simple
description that includes like all the
primes getting wrapped together
this this is like the Euler product for
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 Prime's
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 its 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 yes
or is the 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 like well 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 you know 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 you know anything has
more references and it gives you this
like tree to start pawing 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 I 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 theirs 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
none yeah these very unpredictable
patterns were 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 one
they're not arbitrary means it could be
you can have a clean abstraction an
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
it may be with different notation but
personally yeah but yeah like that's
what I think you talk to that other
civilization they're probably also
studying the zeros of the Riemann zeta
function or it's 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 gonna ask the
question that 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 the title is who cares
about topology you want me to pull it up
or not if you want sure yeah it is about
well it starts by describing an unsolved
problem that 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 we're
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
Vaughan 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 cut 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 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
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 and 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 abut your most weren't able
to anticipate what its gonna look like I
don't know idea I had no idea and it was
wonderful right it was totally it looks
like a Sydney Opera House or some sort
of Frank Gehry design and it was you
knew it was gonna be something and you
can say various things about it like oh
it 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 so 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 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 that's I think
this is good a good sort of example to
talk a little bit about your process so
you have you have a list of ideas so
that 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 and but
there's some sorting procedure depending
on mood and interest and so on but okay
so pick an idea then you have to try to
write a narrative arc it's sort of how
do I elucidate how how do I make this
idea beautiful and clear and explain it
and then there's a set of visualizations
that'll 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 the ultimately the right
resolution
just like set it aside and instead do
some other things where the script comes
more naturally because you sort of don't
want to overwork a narrative 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 judger in your head sort of
the person the creature the essence
that's saying this sucks er this is good
and you mentioned kind of the student
you're you're thinking about um what can
you uh who is that what is that thing
that's Chris that is the perfections
that says this thing sucks you need to
work on it in front of the two three
months I don't know I think it's my past
self I think that's the entity that I'm
most trying to empathize with is like
you take who I was because it's kind of
the only person I know like you don't
really know anyone other than versions
of yourself so I start with the version
of myself that I know who doesn't yet
understand the thing right and then I
just try to view it with fresh eyes a
particular visual or a particular script
like is this motivating does this make
sense which has its downsides because
sometimes I find myself speaking to
motivations that only myself would be
interested in I don't like it 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 can way of thinking about it
that really answered the question in my
head that maybe very satisfied and being
able to think about concretely with a 3d
visual what are they doing to a 4d
sphere and some 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 to it 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 program or like how do these
quarter neon 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 so you
have incredible videos on your networks
for example and for my certain
perspectives have probably I mean I
looked at the is served my field and
I've also looked at the basic
introduction of neural networks like a
million times from different
perspectives and it made me realize that
there's a lot of ways to present it so
you were sort of you did an incredible
job I mean sort of the
but you could also do it differently and
also incredible like to create a
beautiful presentation of a basic
concept is requires sort of creativity
requires genius and so on but you can
take it from a bunch of different
perspectives in that video and you'll
know which mean you realize that and
just as you're saying you kind of have a
certain mindset a certain view but from
if you take a different view from a
physics perspective from a neuroscience
perspective talking about neural
networks or from 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've have taken
different costs 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 no 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 II the fact that we use PI instead
of tau the fact that we call imaginary
numbers imaginary and then and 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 eye 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 want
away the notation look 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
it hasn't really gotten itself dressed
in the morning that's a little bit
groggy and there's bags under its eyes
so years like it's their past yeah the
the the dating stage you know we're no
longer dating right instead of dating
Bizet des function this head like she's
beautiful and right and like we have fun
and it's that that high dopamine part
for like maybe at some point will settle
into the more mundane nature 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 will 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 do 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 not four times longer
mm-hmm I said infinite so the 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 the fact that things and they
see 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 that the aid
there's that the super-ego is a
so you're the reflective self the Verna
Keys 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
experienced 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 force in your subconscious or
that you were instructed an illusion
that allows you to escape the fact of
your mortality by enjoying the moment so
to the existential approach life would
be gun to my head I don't think that's
it yeah another another sort of would
say gun to the head it's the deep
psychological introspection of what
drives us I mean that's uh in some ways
to me I mean when I look at math when I
look at science is the 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 is 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 right that allows you to escape
the 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 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 and sort of
difference in backgrounds I'm a bit
Russian so we're a bit melancholic and
it seemed to maybe assign a little too
much value just suffering immortality
and things like that
make makes for a better novel I think oh
yeah you need you need some sort of
existential threat yeah 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'm
you know 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 know 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 the end becomes quite clear
and you know you're as I as I animated 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 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 it's just at some point you
feel like this this asks a meaningful
question and it answers that question
right what is the best way to learn math
for people who might be at the beginning
of that journey I think that's a it'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 is deep in their
career some type they've taken college a
taking calculus and so on but still
wanna sort of explore math well 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 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 in and the
problems in the back of a text and 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 those
are the best ones to think about a lot
of them won't but just you know take a
quick look and then read a little bit of
the chapter
and 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 right and and this is so
hypocritical right because 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 a very basic sort of just picking
picking out getting getting comfortable
is a very basically algebra calculus fun
Khan Academy programming is actually I
think a great like learned to program
and like let the way the 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 cuz 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 what's the 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 percent
of what you read you remember about 20%
of what you listen to 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 bring true don't they
right I'm willing to say I learned the
nine times better than reading that
might even be a lowball yeah right so 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
transformed
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 way at my
command my brother and a friend of mine
this slightly violates the family and
friends but there was a music that made
me happy they were just company um we
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 the gondola sort of over came
over a mountain and you saw the the city
lights and were just like jamming like
playing some music I wouldn't describe
that nice 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 it 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
and back into your creative work I'd
love to see that I'm certainly allowing
you to enter back into mine and it's
it's a it's a beautiful thing for
mathematician for scientists 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 2021
but by 20 by the end of 2020 I 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 devil around on like
guitar and piano for me to eat our own
piano so in in a mathematicians limit
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 well
thanks for saying that and thanks for
having me on
thanks for listening to this
conversation of grants Anderson and
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me leave you with some words of wisdom
from one of grants and my favorite
people Richard Fineman 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
you