Kind: captions
Language: en
the following is a conversation with
jordan ellenberg a mathematician
at university of wisconsin and an author
who
masterfully reveals the beauty and power
of mathematics
in his 2014 book how not to be wrong
in his new book just released recently
called
shape the hidden geometry of information
biology
strategy democracy and everything else
quick mention of our sponsors
secret sauce expressvpn blinkist
and indeed check them out in the
description to support this podcast
as a side note let me say that geometry
is what made me fall in love with
mathematics
when i was young it first showed me that
something definitive could be stated
about this
world through intuitive visual proofs
somehow
that convinced me that math is not just
abstract numbers
devoid of life but a part of life part
of this world
part of our search for meaning this is
the lex friedman podcast and here is my
conversation
with jordan ellenberg if the brain is a
cake
it is well let's just let's go with me
on this okay okay
we'll pause it so for noam chomsky
language the universal grammar
the framework from which language
springs is like
most of the cake the delicious chocolate
center and then the rest of
cognition that we think of is built on
top
extra layers maybe the icing on the cake
maybe just
maybe consciousness is just like a
cherry on top
where do you put in this cake
mathematical thinking
is it as fundamental as language in the
chomsky
view is it more fundamental in language
is it echoes of the same kind of
abstract framework that he's thinking
about in terms of language that they're
all
like really tightly interconnected
that's a really interesting
question you're getting me to reflect on
this question of whether the feeling of
producing mathematical output if you
want
is like the process of you know uttering
language or producing linguistic output
i think it feels something like that and
it's certainly the case
let me put it this way it's hard to
imagine doing mathematics in a
completely non-linguistic way
it's hard to imagine doing mathematics
without
talking about mathematics and sort of
thinking in propositions but you know
maybe it's just because that's the way i
do mathematics so maybe i can't imagine
it any other way right it's a
well what about visualizing shapes
visualizing concepts to which language
is not
obviously attachable ah that's a really
interesting question and
you know one thing it reminds me of is
one thing i talk about uh
in the book is dissection proofs these
very beautiful proofs of geometric
propositions
um there's a very famous one by bhaskara
of the the pythagorean theorem
proofs which are purely visual proofs
where you show that
two quantities are the same by taking
the same pieces
and putting them together one way uh and
making one shape
and putting them together another way
and making a different shape and then
observing those two shapes must have the
same area because they were built out of
the same
pieces um you know there's a there's a
famous
story and it's a little bit disputed
about how accurate this is but that in
bhaskara's manuscript he sort of gives
this proof just gives the diagram and
then the
the entire uh verbal content of the
proof is he just writes under it
behold like that's it it's like
um there's some dispute about exactly
how accurate that is but
so then that's an interesting question
um if your proof
is a diagram if your proof is a picture
or even if your proof is like a movie of
the same pieces like coming together in
two different formations to make two
different things is that
language i'm not sure i have a good
answer what do you think i think it is
i think the process of manipulating
the visual elements is the same as the
process of
manipulating the elements of language
and i think
probably the manipulating the
aggregation the stitching stuff together
is the important part it's not the
actual specific elements it's more
more like to me language is a process
and math is a process it's not a
it's not just specific symbols it's uh
it's in action it's it's ultimately
created through action through change
and uh so you're constantly evolving
ideas
of course we kind of attach there's a
certain destination you arrive to that
you attach to and you call that a proof
but that's not that doesn't need to end
there it's just at the end of the
chapter and then it goes on
and on and on in that kind of way but i
got to ask you about geometry and it's a
prominent
topic in your new book shape so for me
geometry is the thing just like as
you're saying made me fall in love with
mathematics when i was young
so being able to prove something
visually
just did something to my brain that
it had this it planted this hopeful seed
that
you can understand the world like
perfectly
maybe it's an ocd thing but from a
mathematics perspective
like humans are messy the world is messy
biology is messy
your parents are yelling or making you
do stuff but
you know you can cut through all that bs
and truly understand the world through
mathematics and
nothing like geometry did that for me
for you
you did not immediately fall in love
with geometry so
uh how do you how do you think about
geometry
why is it a special field in mathematics
and
how did you fall in love with it if you
have wow you've given me like a lot to
say and
certainly the experience that you
describe is so
typical but there's two versions of it
um you know one thing i say in the book
is that geometry is the cilantro of math
people are not neutral about it there's
people who are like who like you
are like the rest of it i could take or
leave but then at this one moment
it made sense this class made sense why
wasn't it all like that there's other
people i can tell you because they come
and talk to me all the time
who are like i understood all the stuff
we were trying to figure out what x was
there's some mystery or trying to solve
it x is a number i figured it out but
then there was this geometry like what
was that
what happened that year like i didn't
get it i was like lost the whole year
and i didn't understand like why we even
spent the time doing that
so um but what everybody agrees on is
that it's somehow different
right there's something special about it
um
we're gonna walk around in circles a
little bit but we'll get there you asked
me
um how i fell in love with math i have a
story about this
um when i was a
small child i don't know maybe like i
was six or seven i don't know um
i'm from the 70s i think you're from a
different decade than that but you know
in the 70s we
had them you had a cool wooden box
around your
stereo that was the look everything was
dark wood uh and the box had a bunch of
holes in it
to lift the sound out yeah um and the
holes
were in this rectangular array a six by
eight array
um of holes and i was just kind of like
you know zoning out in the living room
as kids do
looking at this six by eight rectangular
array of holes
and if you like just by kind of like
focusing in and out just by kind of
looking at this box looking at this
rectangle
i was like well there's six rows of
eight holes each
but there's also eight columns of six
holes each
whoa so eight sixes and six eighths it's
just like the
section bruce you were just talking
about but it's the same holes it's the
same 48 holes that's how many there are
no matter of whether you count them
as rows or count them as columns and
this was like
unbelievable to me am i allowed to cost
on your podcast i don't know if that's
uh we fcc regulated
okay it was fucking unbelievable okay
that's the last time get it in this
story merits it
so two different perspectives and the
same physical reality
exactly and it's just as you say um
you know i knew that 6 times 8 was the
same as 8 times 6 when i knew my times
table like i knew that that was a fact
but did i really know it until that
moment that's the question
right i knew that i sort of knew that
the times table was symmetric
but i didn't know why that was the case
until that moment and in that moment i
could see like oh i didn't have to have
somebody tell me that
that's information that you can just
directly access that's a really amazing
moment and as math teachers that's
something that we're really trying to
bring
to our students and i was one of those
who did not love the kind of euclidean
geometry
ninth grade class of like prove that an
isosceles triangle has
equal angles at the base like this kind
of thing it didn't vibe with me the way
that algebra and numbers did
um but if you go back to that moment
from my adult perspective looking back
at what happened with that rectangle i
think that is a very geometric moment in
fact that moment
exactly encapsulates the
the intertwining of algebra and geometry
this algebraic
fact that well in the instance 8 times 6
is equal to 6 times 8 but
in general that whatever two numbers you
have you multiply them one way and it's
the same as if you multiply them in the
other order
it attaches it to this geometric fact
about a rectangle which in some sense
makes it true so you know who knows
maybe i was always faded to be an
algebraic geometer which is what i am as
a
as a researcher so that's the kind of
transformation
and you talk about symmetry in your book
what the heck is symmetry
what the heck is these kinds of
transformation on objects that uh once
you transform them they seem to be
similar
uh what do you make of it what's its use
in mathematics or
maybe broadly in understanding our world
well it's an absolutely fundamental
concept and it starts with the word
symmetry in the way that we usually use
it when we're just like
talking english and not talking
mathematics right sort of something is
when we say something is symmetrical
we usually means it has what's called an
axis of symmetry maybe
like the left half of it looks the same
as the right half that would be like a
left-right axis of symmetry or maybe the
top half looks like the bottom half
or both right maybe there's sort of a
four-fold symmetry where the top looks
like the bottom and the left looks like
the right
or more and that can take you in a lot
of different directions the abstract
study of what the possible combinations
of symmetries there are a subject which
is called group theory was actually
um one of my first loves in mathematics
what i thought about a lot when i was
in college but the notion of symmetry is
actually
much more general than the things that
we would call symmetry if we were
looking at like a classical
building or a painting or or something
like that
um you know nowadays in
in math um
we could use a symmetry to to refer to
any kind of transformation of an image
or a space or an object
you know so what i talk about in in the
book is
take a figure and stretch it vertically
make it twice as
make it twice as big vertically and make
it
half as wide
that i would call a symmetry it's not a
symmetry in the classical sense
but it's a well-defined transformation
that
has an input and an output i give you
some shape um
and it gets kind of i call this in the
book of scronch i just made it had to
make up some sort of funny sounding
name for it because it doesn't really
have um
a name um and just as you can sort of
study which kinds of objects are
symmetrical
under the operations of switching left
and right or switching top and bottom
or rotating 40 degrees or what have you
you could study what kinds of things are
preserved by
this kind of scratch symmetry and this
kind of
more general idea of what a symmetry can
be um
let me put it this way um a fundamental
mathematical idea
in some sense i might even say the idea
that dominates contemporary mathematics
or by contemporary by the way i mean
like the last like 150 years we're on a
very long time scale
in math i don't mean like yesterday i
mean like a century or so
up till now is this idea that's a
fundamental question of
when do we consider two things to be the
same
that might seem like a complete
triviality it's not for instance
if i have a triangle and i have a
triangle of the exact same dimensions
but it's over here
um are those the same or different well
you might say like well look there's two
different things this one's over here
this one's over there
on the other hand if you prove a theorem
about this one
it's probably still true about this one
if it has like all the same side lanes
and angles and like looks exactly the
same
the term of art if you want it you would
say they're congruent
but one way of saying it is there's a
symmetry called translation which just
means
move everything three inches to the left
and we want
all of our theories to be translation
invariant
what that means is that if you prove a
theorem about a thing if it's over here
and then you move it three inches to the
left it would be kind of weird if all of
your theorems like didn't still
work so this question of like what are
the symmetries
and which things that you want to study
or invariant under those symmetries is
absolutely fundamental but this is
getting a little abstract right
it's not at all abstract i think this
this this is completely central
to everything i think about in terms of
artificial intelligence i don't know if
you know about
the mnist dataset with handwritten
digits yeah
and uh you know i don't smoke much weed
or any really but it certainly feels
like it when i look at eminence and
think about this stuff which is like
what's the difference between one and
two and why are all the twos
similar to each other what kind of
transformations
are within the category of what makes a
thing the same
and what kind of transformations are
those that make it different
and symmetries core to that in fact our
whatever the hell our brain is doing
it's really good at constructing these
arbitrary and sometimes novel which is
really important
when you look at like the iq test or
they feel novel
uh ideas of symmetry of like what like
playing with objects we're able to see
things that are the same and not
and uh construct almost like little
geometric
theories of what makes things the same
and not and how to make
uh programs do that in ai is a total
open question
and so i kind of stared and wonder
how what kind of symmetries are enough
to solve
the mnist handwritten digit recognition
problem and write that down
and exactly and what's so fascinating
about the work in that direction
from the point of view of a
mathematician like me and a geometer
um is that the kind of groups and of
symmetries the types of symmetries that
we know of
um are not sufficient right so in other
words like
we're just going to keep on going with
the weeds on this
the deeper the better you know a kind of
symmetry that we understand very well is
rotation
yeah right so here's what would be easy
if if if humans
if we recognize the digit as a one if it
was like literally a rotation by some
number of degrees of some
fixed one in some typeface like palatino
or something
that would be very easy to understand
right it would be very easy to like
write a program that
could detect whether something was a
rotation
of a fixed net digit one um whatever
we're doing when you recognize the digit
one and distinguish it from the digit
two
it's not that it's not just
incorporating
uh one of the types of symmetries that
we
understand now i would say that
i would be shocked if there was some
kind of
classical symmetry type formulation that
captured what we're doing when we tell
the difference between a two and a three
to be honest i think i think what we're
doing is actually more
complicated than that i feel like it
must be they're so simple these numbers
i mean they're
really geometric objects like we can
draw out one two three
it does seem like it's it should be
formalizable that's why it's so
strange you think it's formalizable when
something stops being a two and starts
being a three right you can imagine
something continuously deforming from
being a two to a three
yeah but that's there is a moment
i have uh myself have written programs
that literally morph
twos and threes and so on and you watch
and there is moments that you notice
depending on the trajectory of that
transformation
that morphing that
it uh it is a three and a two there's a
hard line
wait so if you ask people if you show
them this morph if you ask a bunch of
people do they all agree about where the
transformation
i'm questioning because i would be
surprised i think so oh my god okay we
have an empirical but here's the problem
dude
here's the problem that if i just showed
that moment that i agreed on
well that's not fair no but say i said
so
i want to move away from the agreement
because that's a fascinating uh actually
question that i want to
backtrack from because i just
dogmatically said uh because
i could be very very wrong but the
morphing really helps that like the
change
because i mean partially because our
perception systems see this it's all
probably tied in there
somehow the change from one to the other
like seeing the video of it
allows you to pinpoint the place where
two becomes a three much better
if i just showed you one picture i think
uh
you you might you might really
really struggle you might call a seven
like
i i think there's something uh also that
we don't often think about which is
it's not just about the static image
it's the transformation
of the image or it's not a static shape
it's the transformation of the shape
there's something in the movement that's
seems to be
not just about our perception system but
fundamental to our cognition like how we
think stuff
about stuff yeah and it's so and and you
know that's part of geometry
too and in fact again another insight of
modern geometry is this idea that
you know maybe we would naively think
we're going to study
i don't know let's you know like
poincare we're going to study the
three-body problem we're going to study
sort of like three
objects in space moving around subject
only to the force of each other's
gravity which sounds very simple right
and if you don't know about this problem
you're probably like okay so you just
like put it in your computer and see
what they do well guess what that's like
a problem that poincare won a huge prize
for like making the first real progress
on in the 1880s
and we still don't know that much about
it
um 150 years later i mean it's a
humongous mystery you just open the door
and we're going to
walk right in before we return to uh
symmetry
what's the uh who's ponca and what's uh
what's this conjecture that he came up
with
oh why is this such a hard problem okay
so poincare
he ends up being a major figure in the
book and i don't i didn't even really
intend for him to be such a big figure
but he's so
he's um he's first and foremost a
geometer right so he's a mathematician
who kind of comes up
in late 19th century france
um at a time when french math was really
starting to flower actually i learned a
lot
i mean you know in math we're not really
trained on our own history when we get a
phd in math one about math so i learned
a lot
there's this whole kind of moment where
france has just been
beaten in the franco-prussian war and
they're like oh my god what did we do
wrong and they were like
we got to get strong in math like the
germans we have to be like more like the
germans so this never happens to us
again so it's very much
it's like the sputnik moment you know
like what happens in america in the 50s
and 60s
uh with the soviet union this is
happening to france and they're trying
to kind of like
instantly like modernize that that's
fascinating
the humans and mathematics are
intricately connected to the history
of humans the cold war is
uh i think fundamental to the way people
saw
science and math in the soviet union i
don't know if that was true in the
united states but certainly was in the
soviet union
it definitely was and i would love to
hear more about how it was in the soviet
union
i mean there's uh and we'll talk about
the the olympia
i just remember that there was this
feeling
like the world hung in a balance
and you could save the world
with the tools of science and
mathematics
was like the super power
that fuels science and so like
people were seen as you know people in
america often idolize athletes
but ultimately the best athletes in the
world
they just throw a ball into a basket
so like there's not what people really
enjoy about sports
and i love sports is like excellence at
the highest level
but when you take that with mathematics
and science people also enjoyed
excellence in science and mathematics
and the soviet union
but there's an extra sense that that
excellence would
lead to a better world so
that created uh all the usual things you
you think about with the olympics which
is like
extreme competitiveness right but it
also created this sense that
in the modern era in america somebody
like elon musk
whatever you think of them like jeff
bezos those folks
they inspire the possibility that one
person
or a group of smart people can change
the world like
not just be good at what they do but
actually change the world
mathematics is at the core of that uh
and i don't know
there's a romanticism around it too like
when you read uh books about
in america people romanticize certain
things like baseball for example
there's like these beautiful poetic uh
writing about
the game of baseball the same was the
feeling with mathematics
and science in the soviet union and it
was it was in the air
everybody was forced to take high-level
mathematics courses like they you took a
lot of math
you took a lot of science and a lot of
like really rigorous literature
like they the the level of education in
russia
this could be true in china i'm not sure
uh in a lot of countries
is uh in um whatever that's called
it's k-12 in america but like young
people education the level they were
challenged
to to learn at is incredible it's like
america falls far behind i would say
america then quickly catches up and then
exceeds everybody else
at the like the as you start approaching
the end of high school to college
like the university system in the united
states arguably is the best in the world
but like what what we uh challenge
everybody it's not just like the good
the ace students but
everybody to learn in in the soviet
union was fascinating
i think i'm gonna pick up on something
you said i think you would love a book
called
duel at dawn by amir alexander which
i think some of the things you're
responding to what i wrote i think i
first got turned on to by amir's work
he's a historian of math
and he writes about the story of every
east galwa which is a story that's well
known to all mathematicians this kind of
like
very very romantic figure who
he really sort of like begins the
development of this
well this theory of groups that i
mentioned earlier this general
theory of symmetries um and then dies in
a duel in his early 20s like all this
stuff
mostly unpublished it's a very very
romantic story that we all
learn um and much of it is true but
alexander really lays out just how much
the way people
thought about math in those times in the
early 19th century
was wound up with as you say romanticism
i mean that's when the romantic
movement takes place and he really
outlines how
people were were predisposed to think
about mathematics in that way because
they thought about poetry that way and
they thought about music that way
it was the mood of the era to think
about we're reaching for the
transcendent we're sort of reaching for
sort of direct
contact with the divine and so part of
the reason that we think of gawa that
way
was because gawa himself was a creature
of that era and he romanticized himself
yeah i mean now now you know he like
wrote lots of letters and like he was
kind of like
i mean in modern terms we would say he
was extremely emo like that's
like just we wrote all these letters
about his like floored feelings and like
the fire within him about the
mathematics and you know so he
so it's just as you say that
the math history touches human history
they're never separate because
math is made of people yeah i mean
that's what it's
it's it's people who do it and we're
human beings doing it and we do it
within whatever
community we're in and we do it affected
by uh
the morals of the society around us so
the french the germans and the
pancreatic yes okay so back to ponca ray
so
um he's you know it's funny this book is
filled with kind of you know
mathematical characters who
often are kind of peevish or get into
feuds or sort of have like
weird enthusiasms um because those
people are fun to write about and they
sort of like say
very salty things poincare is actually
none of this as far as i can tell
he was an extremely normal dude he
didn't get into fights with people
and everybody liked him and he was like
pretty personally modest and he had
very regular habits you know what i mean
he did math for like
four hours in the morning and four hours
in the evening and that was it like he
had his
schedule um i actually i was like i
still am feeling like
somebody's going to tell me now the book
is out like oh didn't you know about
this like incredibly sordid episode
as far as i could tell a completely
normal guy
but um he just kind of
in many ways creates uh the geometric
world
in which we live and and you know his
first really big success
uh is this prize paper he writes for
this prize offered by the king of sweden
for the study of the three-body problem
um
the study of what we can say about yeah
three
astronomical objects moving and what you
might think would be this very simple
way nothing's going on except
gravity uh releasing the three-body
problem why is it a problem
so the problem is to understand um when
this motion is stable and when it's not
so stable meaning they would sort of
like end up in some kind of periodic
orbital or i guess it would mean sorry
stable would mean they never sort of fly
off far apart from each other and
unstable would mean like eventually they
fly apart
so understanding two bodies is much
easier yeah
third uh two bodies this is what
newton knew two bodies they sort of
orbit each other and some kind of uh
uh either in an ellipse which is the
stable case you know that's what
the planets do that we know um
or uh one travels on a hyperbola around
the other that's the unstable case it
sort of like zooms in from far away sort
of like whips around the
heavier thing and like zooms out um
those are basically the two options so
it's a very simple and easy to classify
story
with three bodies just the small switch
from two to three uh
it's a complete zoo it's the first
example what we would say now is it's
the first example of what's called
chaotic dynamics
where the stable solutions and the
unstable solutions
they're kind of like wound in among each
other and a very very very tiny change
in the initial conditions can
make the long-term behavior of the
system completely different so poincare
was the first to recognize that that
phenomenon even
uh even existed what about the uh
conjecture that carries his name right
so
he also um
was one of the pioneers of taking
geometry um which until that point
had been largely the study of two and
three-dimensional objects because that's
like
what we see right that's those are the
objects we interact with
um he developed that subject we now
called topology he called it analysis
situs he was a very
well-spoken guy with a lot of slogans
but that name did not
you can see why that name did not catch
on so now it's called topology now
um sorry what was it called before
analysis situs
which i guess sort of roughly means like
the analysis of location or something
like that like um
it's a latin phrase
partly because he understood that even
to understand stuff that's going on
in our physical world you have to study
higher dimensional spaces
how does this how does this work and
this is kind of like where my brain went
to it because you were talking about
not just where things are but what their
path is how they're moving when we were
talking about the path from two to three
um he understood that if you want to
study three the three bodies
moving in space well each
uh each body it has a location where it
is so it has an x coordinate a y
coordinate and a z coordinate right i
can specify a point in space by giving
you three numbers
but it also at each moment has a
velocity
so it turns out that really to
understand what's going on
you can't think of it as a point or you
could but it's
better not to think of it as a point in
three-dimensional space that's moving
it's better to think of it as a point in
six dimensional space where the
coordinates are where is it
and what's its velocity right now that's
a higher dimensional space called
phase space and if you haven't thought
about this before i admit that it's a
little bit
mind-bending but
what he needed then was a geometry that
was flexible enough
not just to talk about two-dimensional
spaces or three-dimensional spaces but
any dimensional space so the sort of
famous first line of this paper where he
introduces analysis
is is no one doubts nowadays that the
geometry of
n-dimensional space is an actually
existing thing right i think that
maybe that had been controversial and
he's saying like look let's face it just
because it's not physical
doesn't mean it's not there it doesn't
mean we shouldn't stop
interesting he wasn't jumping to the
physical the physical interpretation
like it does
it can be real even if it's not
perceivable to human
cognition i think i think that's right i
think
don't get me wrong poincare never strays
far from physics he's always motivated
by physics
but the physics drove him to need to
think about
spaces of higher dimension and so he
needed a formalism that was rich enough
to enable him to do that
and once you do that that formalism is
also going to include things that are
not physical
and then you have two choices you can be
like oh well that stuff's trash
or but and this is more the
mathematicians frame of mind
if you have a formalistic framework that
like seems really good and sort of seems
to be like very elegant and work well
and it includes all the physical stuff
maybe we should think about all of it
like maybe we should think about it
thinking maybe there's some gold to be
mined there
um and indeed like you know guess what
like before long there's relativity and
there's space time and like all of a
sudden it's like oh yeah maybe it's a
good idea we already have this geometric
apparatus like set up for like how to
think about
four-dimensional spaces like turns out
they're real after all
you know this is a a story much told
right in mathematics not just in this
context but in many i'd love to dig in a
little deeper on that actually because
i have some uh intuitions to work out
okay my brain well i'm not a
mathematical physicist so we can work
them out together
good we'll uh we'll we'll together walk
along the path of curiosity
but pancreatic uh conjecture
what is it the point conjecture is about
curved
three-dimensional spaces so i was on my
way there i promise
um the idea is that we perceive
ourselves as living in
we don't say a three-dimensional space
we just say three-dimensional space you
know you can go up and down you can go
left and right you can go forward and
back there's three dimensions in which
you can move
in poincare's theory there are many
possible three-dimensional spaces
in the same way that going down one
dimension to sort of
capture our intuition a little bit more
we know there are lots of different
two-dimensional surfaces right there's
a balloon and that looks one way and a
doughnut looks another way and a mobius
strip
looks a third way those are all like
two-dimensional surfaces that we can
kind of really
uh get a global view of because we live
in three-dimensional space so we can see
a two-dimensional surface sort of
sitting in our three-dimensional space
well to see a three-dimensional space
whole we'd have to kind of have
four-dimensional eyes right which we
don't so we have to use our mathematical
lines we have to envision
um the poincare conjecture
uh says that there's a very simple way
to determine whether a three-dimensional
space
um is the standard one the one that
we're used to
um and essentially it's that it's what's
called fundamental group
has nothing interesting in it and not
that i can actually say without saying
what the fundamental group is i can tell
you what the criterion is
this would be oh look i can even use a
visual aid so for the people watching
this on youtube you'll just see this for
the people
uh on the podcast you'll have to
visualize it so lex has been nice enough
to like
give me a surface with some interesting
topology it's a mug right here in front
of me
a mug yes i might say it's a genus one
surface but we could also say it's a mug
same thing
so if i were to draw a little circle
on this mug oh which way should i draw
it so it's visible like here okay
yeah if i draw a little circle on this
mug imagine this to be a loop of string
i could pull that loop of string closed
on the surface of
the mug right that's definitely
something i could do i could shrink it
shrink and shrink it until it's a point
on the other hand if i draw a loop that
goes around the handle
i can kind of judge it up here and i can
judge it down there and i can sort of
slide it up and down the handle but i
can't pull it closed can't i it's
trapped
not without breaking the surface of the
mug right now without like going inside
so um the condition of being what's
called simply connected this is
one of punk ray's inventions says that
any loop of string can be pulled shut so
it's a feature that the mug
simply does not have this is a
non-simply connected
mug and a simply connected mug would be
a cup right you would burn your hand
when you drank coffee out of it
so you're saying the universe is not a
mug
well i can't speak to the universe but
what i can say is that
um regular old space
is not a mug regular old space if you
like sort of actually physically have
like a loop of string
you can always close your shot you're
going to pull a shit
but you know what if your piece of
string was the size of the universe like
what if your poi
your piece of string was like billions
of light years long like like how do you
actually know
i mean that's still an open question of
the shape of the universe exactly
whether it's uh i think there's a lot
there is
ideas of it being a tourist i mean
there's there's some trippy ideas and
they're not
like weird out there controversial
there's a legitimate
at the center of uh cosmology debate
i mean i think i think somebody who
thinks that there's like some kind of
dodecahedral symmetry or i mean i
remember reading something crazy about
somebody saying that they saw the
signature of that and the
cosmic noise or what have you i mean to
make the flat earthers happy
i do believe that the current main
belief is
it's fl it's flat it's flat-ish
or something like that the shape of the
universe is flat-ish i don't know what
the heck that means i think that
i think that has like a very i mean how
are you even supposed to think about
the shape of a thing that doesn't have
anything outside of it
i mean ah but that's exactly what
topology does topology is what's called
an intrinsic theory
that's what's so great about it this
question about the mug
you could answer it without ever leaving
the mug
right because it's a question about a
loop drawn on the surface of the mug and
what happens if it never leaves that
surface so it's like
always there see but that's the the
difference between
then topology and say if you're like uh
trying to visualize a mug that you can't
visualize a mug while living inside the
mug
well that's true that visualization is
harder but in some sense no you're right
but if the tools of mathematics are
there
i i i don't want to fight but i think
the tools and mathematics are exactly
there to enable you to think about what
you cannot visualize
in this way let me give let's go always
to make things easier go downward
dimension
um let's think about we live on a circle
okay
you can tell whether you live on a
circle
or a line segment because if you live in
a circle if you walk a long way in one
direction you find yourself back where
you started and if you live in a line
segment
you walk for a long enough one direction
you come to the end of the world or if
you live on a line
like a whole line an infinite line then
you walk in
one direction for a long time and like
well then there's not a sort of
terminating algorithm to figure out
whether you live on a line or a circle
but at least you sort of
um at least you don't discover that you
live on a circle
so all of those are intrinsic things
right all of those are things that you
can figure out about your
world without leaving your world on the
other hand ready now we're going to go
from intrinsic to extrinsic why did i
not know we were going to talk about
this but why not
why not if you can't tell whether you
live in a circle
or a not like imagine like a knot
floating in three-dimensional space the
person who lives on that knot to them
it's a circle
yeah they walk a long way they come back
to where they started now we with our
three-dimensional eyes can be like
oh this one's just a plain circle and
this one's knotted up but that's an
that's a
that has to do with how they sit in
three-dimensional space it doesn't have
to do with intrinsic features of those
people's world
we can ask you one ape to another does
it make you
how does it make you feel that you don't
know if you live in a circle
or on a knot in a knot
in inside the string that forms the knot
i'm going to even know how to say i'm
going to be honest with you i don't know
if like
i i fear you won't like this answer but
it
does not bother me at all it does i
don't lose one minute of sleep over it
so like does it bother you that if we
look at like a mobius strip
that you don't have an obvious way of
knowing
whether you are inside of cylinder if
you live on a surface of a cylinder
or you live on the surface of a mobius
strip
no i think you can tell if you live if
which one because
if what you do is you like tell your
friend hey stay right here i'm just
gonna go for a walk
and then you like walk for a long time
in one direction and then you come back
and you see your friend again and if
your friend is reversed then you know
you live on a mobius strip well
no because you won't see your friend
right okay
fair fair point fair point on that and
you have to believe the story is about
no i don't even know
i i i would would you even know would
you really oh no you're i know your
point is right let me try to think of it
better
let's see if i can do this may not be
correct to talk about
cognitive beings living on a mobius
strip because
there's a lot of things taken for
granted there and we're constantly
imagining actual like
three-dimensional creatures like how it
actually
feels like to uh to live on a mobius
strip is tricky to
internalize i think that on what's
called the real projective plane which
is kind of even more sort of like messed
up version of the
mobius strip but with very similar
features this feature of kind of like
only having one side that has the
feature that there's a loop of string
which can't be pulled closed but if you
loop
it around twice along the same path that
you can pull closed
that's extremely weird yeah
um but that would be a way you could
know without leaving your world that
something very funny is going on you
know what's extremely weird
maybe we can comment on hopefully it's
not too much of a tangent is
i remember thinking about this this
might be right
this might be wrong but if you're if we
now
talk about a sphere and you're living
inside a sphere
that you're going to see everywhere
around you the back of your own head
that i was because like i was
this is very counterintuitive to me to
think about maybe it's wrong
but because i was thinking like earth
you know your 3d
thing on sitting on a sphere but if
you're living inside the sphere
like you're going to see if you look
straight you're always going to see
yourself all the way around so
everywhere you look there's going to be
the back of your
head i think somehow this depends on
something of like how the physics of
light works in this scenario which i'm
sort of finding it hard to bend my
that's true the c is doing a lot of like
saying you see something's doing a lot
of work
people have thought about this i mean
this this metaphor of like what if we're
like little creatures in some sort of
smaller world like how could we
apprehend what's outside that metaphor
just comes back and back and actually i
didn't even realize like how frequent it
is it comes up in the book a lot
i know it from a book called flatland i
don't know if you ever read this when
you were a kid
an adult you know this this uh sort of
sort of comic novel from the 19th
century about
an entire two-dimensional world
uh it's narrated by a square that's the
main character
and um the kind of strangeness that
befalls him when
you know one day he's in his house and
suddenly there's like a little circle
there and there with him
and then the circle but then the circle
like starts getting bigger
and bigger and bigger and he's like what
the hell is going on it's like a horror
movie like for two-dimensional people
and of course what's happening is that a
sphere is entering his world and as the
sphere kind of like
moves farther and farther into the plane
it's cross-section the part of it that
he can see
to him it looks like there's like this
kind of bizarre being that's like
getting larger and larger and larger
um until it's exactly sort of halfway
through and then they have this kind of
like philosophical argument where the
sphere is like i'm a sphere i'm from the
third dimension the square is like what
are you talking about there's no such
thing
and they have this kind of like sterile
argument where the square is not
able to kind of like follow the
mathematical reasoning of the sphere
until the sphere just kind of grabs him
and like jerks him out of the plane
and pulls him up and it's like now like
now do you see like now do you see your
whole world that you didn't understand
before so do you think
that kind of process is possible for us
humans
so we live in the three-dimensional
world maybe with the time component
four-dimensional
and then math allows us to uh to go
high into high dimensions comfortably
and explore the world from those
perspectives
like is it possible
that the universe is uh many more
dimensions
than the ones we experience as human
beings so
if you look at uh the you know
especially in physics
theories of everything uh physics
theories that try to unify
general relativity and quantum field
theory
they seem to go to high dimensions
to work stuff out through the tools of
mathematics
is it possible so like the two options
are one
is just a nice way to analyze a universe
but the reality is is as exactly we
perceive it it is three-dimensional
or are we just seeing are we those
flatland creatures
they're just seeing a tiny slice of
reality
and the actual reality is many many
many more dimensions than the three
dimensions we perceive
oh i certainly think that's possible um
now how would you figure out whether it
was true or not is another question
um i suppose what you would do as with
anything else that you can't directly
perceive
is you would try to understand
what effect the presence of those extra
dimensions
out there would have on the things we
can
perceive like what else can you do right
and in some sense
if the answer is they would have no
effect
then maybe it becomes like a little bit
of a sterile question because what
question are you even
asking right you can kind of posit
however many entities that
is it possible to intuit how to mess
with the other dimensions
while living in a three-dimensional
world i mean that seems like a very
challenging thing to do
we the the reason flatland could be
written
is because it's coming from a
three-dimensional
writer yes but but what happens in the
book i didn't even tell you the whole
plot
what happens is the square is so excited
and so
filled with intellectual joy by the way
maybe to give the story some context
you ask like is it possible for us
humans to have this experience of being
transcendent transcendentally jerked out
of our world so we can sort of truly see
it from above
well edwin abbott who wrote the book
certainly thought so because
edward abbott was a minister so the
whole christian subtext to this book i
had completely not grasped
reading this as a kid that it means a
very different thing right if sort of a
theologian
is saying like oh what if a higher being
could like pull you out of
this earthly world you live in so that
you can sort of see the truth and like
really see it
uh from above as it were so that's one
of the things that's going on for him
and it's a testament to his skill as a
writer that his story just works whether
that's the framework you're coming to it
from
or not um but what happens in this book
and this part now looking at it through
a christian lens that becomes
a bit subversive is the square is so
excited about
what he's learned from the sphere and
the sphere explains them like what a
cube would be oh it's like you but
three-dimensional and the square is very
exciting and the square is like
okay i get it now so like now that you
explained to me how just by reason i can
figure out what a cube would be like
like a three-dimensional version of me
like let's figure out what a
four-dimensional version of me would be
like
and then this fear is like what the hell
are you talking about there's no fourth
dimension that's ridiculous like
there's three dimensions like that's how
many there are i can see like i mean so
it's the sort of comic moment where the
sphere is completely unable to
uh conceptualize that there could
actually be yet another
dimension so yeah that takes the
religious allegory to like a very weird
place that i don't really like
understand theologically but
that's a nice way to talk about religion
and myth
in general as perhaps us trying to
struggle with us meaning human
civilization trying to struggle
with ideas that are beyond our cognitive
capabilities but it's in fact not beyond
our capability it may be beyond our
cognitive capabilities
to visualize a four-dimensional cube a
tesseract as some like to call it or a
five-dimensional cube or a
six-dimensional cube
but it is not beyond our cognitive
capabilities
to figure out how many corners a
six-dimensional cube would have
that's what's so cool about us whether
we can visualize it or not we can still
talk about it we can still reason about
it
we can still figure things out about it
that's amazing
yeah if we go back to this first of all
to the mug
but to the example you give in the book
of the straw
uh how many holes does a straw have
and you listener may uh try to answer
that in your own head
yeah i'm gonna take a drink while
everybody thinks about it a slow
sip is it uh
zero one or two or more
more than that maybe maybe you get very
creative but
uh it's kind of interesting to uh each
uh dissecting each answer as you do in
the book is quite brilliant people
should definitely check it out
but if you could try to answer it now
like think about
all the options and why they may or may
not be right
yeah and it's one of it's one of these
questions where people on first hearing
it think it's a triviality and they're
like well the answer is obvious and then
what happens if you ever ask a group of
people this something
wonderfully comic happens which is that
everyone's like well it's completely
obvious
and then each person realizes that half
the person the other people in the room
have a different
obvious answer for the way that they
have and then people get really heated
people are like i can't believe that you
think it has two holes or like i can't
believe that you think it has one
and then you know you really like people
really learn something about each other
and people get heated
i mean can we go through the possible
options here
is it zero one two three ten
sure so i think you know most people
the zero holders are rare they would say
like well look
you can make a straw by taking a
rectangular piece of plastic and closing
it up the rectangular piece of plastic
doesn't have a hole in it uh i didn't
poke a hole in it
when i yeah so how can i have a hole
like it's just one thing okay
most people don't see it that way that's
like uh um
is there any truth to that kind of
conception yeah i think that would be
somebody whose account
i mean
what i would say is you could say the
same thing
um about a bagel you could say i can
make a bagel by taking like a long
cylinder of dough which doesn't have a
hole and then smooshing the ends
together
now it's a bagel so if you're really
committed you can be like okay bagel
doesn't have a hole either but like
who are you if you say a bagel doesn't
have a hole i mean i don't know yeah so
that's almost like an engineering
definition of it
okay fair enough so what's what about
the other options
um so you know one whole people
would say um i like how these are like
groups of people
like where we've planted our foot yes
one hole
there's books written about each belief
you know would say look there's like a
hole and it goes all the way through the
straw right there it's one region of
space that's the hole
yeah and there's one and two whole
people would say like well look there's
a hole in the top in the hole
at the bottom um i think a common thing
you see
when people um
argue about this they would take
something like this a bottle of water
i'm holding maybe i'll open it and they
say well how many holes are there in
this and you say like well there's one
there's one hole
at the top okay what if i like poke a
hole here so that all the water
spills out well now it's a straw
yeah so if you're a one hole or i say to
you like well how many holes are
in it now there was a there was one hole
in it before and i poked a new hole in
it
and then you think there's still one
hole even though there was
one hole and i made one more clearly not
this
is two holes yeah um and yet if you're a
two hole the one holder will say like
okay where does one hole begin in the
other whole end
yeah like what's it like and um
and in the in the book i sort of you
know in math there's two things we do
when we're faced with a problem that's
confusing us
um we can make the problem simpler
that's what we were doing a minute ago
and we were talking about high
dimensional space and i was like let's
talk about like
circles and line segments let's go down
a dimension to make it easier
uh the other big move we have is to make
the problem harder
and try to sort of really like face up
to what are the complications so
you know what i do in the book is say
like let's stop talking about straws for
a minute and talk about pants
how many holes are there in a pair of
pants
so i think most people who say there's
two holes in the straw would say there's
three holes in a pair of pants
i guess i mean i guess we're filming
only from here i could take up no i'm
not gonna do it
you'll just have to imagine the pants
sorry yeah um
lex if you want it no okay no
uh that's gonna be in the direction
that's the patreon-only footage
there you go so many people would say
there's three holes in the pair of pants
but you know for instance my daughter
when i asked
by the way talking to kids about this is
super fun i highly recommend it
um what did she say she said well
yeah i feel a pair of pants like just
has two holes because yes there's the
waist but that's just the two leg holes
stuck together
whoa okay two leg holes yeah okay right
i mean that's
she's a one caller for the straw so
she's a one-holer for the straw too
and um and that really does
capture something it captures this fact
which is central to the theory of what's
called homology which is like a central
part of modern topology that
um holes whatever we may mean by them
there are somehow things which have an
arithmetic to them they're things which
can be
added like the waste like waste equals
leg plus leg
is kind of an equation but it's not an
equation about numbers it's an equation
about some kind of
geometric some kind of topological thing
which is very strange and so
you know when i come down um
you know like a rabbi i like to kind of
like come up with these answers and
somehow like
dodge the original question and say like
you're both right my children okay so
yeah uh so for this for the
for the straw i think what a modern
mathematician would say is like
the first version would be to say like
well they're two holes but they're
really both the same hole
well that's not quite right a better way
to say it is there's two holes
but one is the negative of the other now
what can that mean
um one way of thinking about what it
means is that if you sip something like
a milkshake through the straw
no matter what the amount of milkshake
that's flowing in one end
that same amount is flowing out the
other end
so they're not independent from each
other there's some relationship
between them in the same way that if you
somehow could like
suck a milkshake through a pair of pants
the amount of milkshake just go with me
on this not experimenting
mom right there the amount of milkshake
that's coming in the left leg of the
pants
plus the amount of milkshake that's
coming in the right leg of the pants
is the same that's coming out the uh the
waist of the pants so just so you know i
fasted for 72 hours
yester uh the last three days so i just
broke the fast with a little bit of food
yesterday so
this is like this sounds uh food
analogies or metaphors for this podcast
work
wonderfully because i can intensely
picture it is that your weekly routine
or just in preparation for talking about
geometry for three hours
exactly this it's hardship
to purify the mind no it's for the first
time i just wanted to try the experience
oh wow
and just to uh to pause to do things
that are out of the ordinary to pause
and to uh reflect on how grateful i am
to be just
alive and be able to do all the cool
shit that i get to do so
did you drink water yes yes yes yes yes
water and salt so like electrolytes and
all those kinds of things
but anyway so the inflow on the top of
the pants
equals to the outflow on the bottom of
the pants
exactly so this idea that
i mean i think you know poincare really
have these i this idea this sort of
modern idea
i mean building on stuff other people
did uh betty is an important one
of this kind of modern notion of
relations between holes but the idea
that holes really had an arithmetic
the really modern view was really emmy
nerder's idea so she kind of comes in
and sort of truly puts the subject
uh on its modern footing that we have
that we have now so
you know it's always a challenge you
know in the book i'm not gonna say i
give like
a course so that you read this chapter
and then you're like oh it's just like i
took like
a semester of algebraic apology it's not
like this and it's always a you know
it's always a challenge
writing about math because there are
some things
that you can really do on the page and
the math is there and there's other
things which
it's too much in a book like this like
do them all the page you can only
say something about them if that makes
sense um
so you know in the book i try to do some
of both i try to do i try to
topics that are you can't really
compress
and really truly say exactly what they
are
in this amount of space um
i try to say something interesting about
them something meaningful about them so
that readers can get the flavor
um and then in other places i really try
to get up close and personal
and really do the math and have it take
place on the page
to some degree be able to give inklings
of the beauty of the subject
yeah i mean there's you know there's a
lot of books that are like i don't quite
know how to express this well i'm still
laboring to do it but um there's a lot
of books that
are about stuff
but i want my books to not only be about
stuff but to actually have some stuff
there on the page in the book for people
to interact with directly and not just
sort of hear me talk about distant
features about
just different distant features of it
right so
not be talking just about ideas but the
actually be
expressing the idea is there you know
somebody in the
maybe you can comment there's a guy his
youtube channel is
three blue one brown grant sanderson he
does that
masterfully well absolutely of uh
visualizing of expressing a particular
idea and then talking about it as well
back and forth uh what do you what do
you think about grant
it's fantastic i mean the flowering of
math youtube is like such a wonderful
thing because
you know math teaching there's so many
different venues
through which we can teach people math
there's the traditional one right
well where i'm in a classroom with
you know depending on the class it could
be 30 people it could be 100 people
it could god help maybe 500 people if
it's like the big calculus lecture or
whatever it may be
and there's sort of some but there's
some set of people of that order of
magnitude
and i'm with them for we have a long
time i'm with them for a whole semester
and i can ask them to do homework and we
talk together we have office hours that
they have one-on-one questions multiply
that's like a very high level of
engagement
but how many people am i actually
hitting at a time like not
that many right um and you can
and there's kind of an inverse
relationship where the
more and g the fewer people you're
talking to the more engagement you can
ask for the ultimate of course is like
the mentorship relation of like a phd
advisor and a graduate student where
you spend a lot of one-on-one time
together for like you know
three to five years and the ultimate
high level of engagement
to one person um you know books
i can this can get to a lot more people
that are ever gonna sit in my
classroom and you spend like uh
however many hours it takes to read a
book uh somebody like three blue one
brown or numberphile or um
people like vi heart i mean youtube
let's face it
has bigger reach than a book like
there's youtube videos that have many
many many more views than like
you know any hardback book like not
written by a kardashian or an obama is
gonna sell right so that's
i mean um any
and and then you know those are you know
some of them are like
longer 20 minutes long some of them are
five minutes long but they're you know
they're shorter and even so
look look like eugenia chang is a
wonderful category theorist in chicago i
mean
she was on i think the daily show or i
mean she was on you know
she has 30 seconds but then there's like
30 seconds to sort of say something
about math mathematics to like untold
millions of people so everywhere along
this curve isn't
is important one thing i feel like is
great right now is that people are just
broadcasting on all the channels because
we each have
our skills right somehow along the way
like i learned how to write books i had
this
kind of weird life as a writer where i
sort of spent a lot of time like
thinking about how to put
english words together into sentences
and sentences together into paragraphs
like
at length which is this kind of like
weird specialized scale
and that's one thing but like sort of
being able to make like you know winning
good-looking eye-catching videos is like
a totally
different skill and you know probably
you know somewhere out there there's
probably sort of some
like heavy metal band that's like
teaching math through
heavy metal and like using their skills
to do that i hope there is
at any rate through music and so on yeah
but there is something to the process
i mean grant does this especially well
which is
in order to be able to visualize
something now he writes
programs so it's programmatic
visualization so like the
the things he is basically mostly
through his uh
madam library in python everything is
drawn
through python you have to um
you have to truly understand the topic
to be able to
to visualize it in that way and not just
understand it but really kind of
think in a very novel way it's funny
because i i've spoken with him a couple
times
i've spoken to him a lot offline as well
he really doesn't think
he's doing anything new meaning like
he sees himself as very different from
maybe like a researcher
but it feels to me
like he's creating something totally new
like that act of understanding
visualizing
is as powerful or has the same kind of
inkling of power as does
the process of proving something you
know
it just it doesn't have that clear
destination but it's
it's pulling out an insight and creating
multiple sets of perspective
that arrive at that insight and to be
honest it's something that i think we
haven't
quite figured out how to value
inside academic mathematics in the same
way and this is a bit older that i think
we haven't quite
figured out how to value the development
of computational infrastructure you know
we all have computers as our partners
now and people
build computers that sort of assist and
participate in our mathematics they
build those systems and that's
a kind of mathematics too but not in the
traditional form of
proving theorems and writing papers but
i think it's coming look i mean i think
you know for example the institute for
computational experimental mathematics
at brown which is like a
you know it's a nsf-funded math
institute very much part of sort of
traditional math academia they did an
entire theme semester about visualizing
mathematics looking at the same kind of
thing that they would do for like
an up-and-coming research topic like
that's pretty cool so i think there
really is
buy-in from uh the mathematics community
to recognize that this kind of stuff is
important and counts as part of
mathematics like part of what we're
actually here to do
yeah i'm hoping to see more and more of
that from like mit faculty from faculty
from all the
the top universities in the world let me
ask you this weird question about the
fields medal which is the nobel prize in
mathematics
do you think since we're talking about
computers there will one
day come a time
when a computer an ai system will win a
field medal
no of course that's what a human would
say why not
it's is that like you're that that's
like my captcha that's like the proof
that i'm a human
how does he want me to answer is there
something interesting to be said about
that
yeah i mean i am tremendously interested
in
what ai can do in pure mathematics i
mean it's of course
it's a parochial interest right you're
like why am i interested in like how it
can like help feed the world or help
solve like there's problems i'm like can
i do
more math like what can i do we all have
our interests right um
but i think it is a really interesting
conceptual
question and here too i think
um it's important to be kind of
historical because it's certainly true
that there's lots of things that we used
to call research and mathematics
that we would now call computation yeah
tasks that we've now offloaded to
machines like
you know in 1890 somebody could be like
here's my phd thesis i
computed all the invariants of this
polynomial ring under the action of some
finite group doesn't matter what those
words mean just it's like something that
in 1890
would take a person a year to do and
would be a valuable thing that you might
want to know and it's still a valuable
thing that you might want to know but
now you type a few lines of code in
macaulay or sage
or magma and you just have it so we
don't think of that as
math anymore even though it's the same
thing what's macaulay sage and magma
oh those are computer algebra programs
so those are like sort of bespoke
systems that lots of mathematicians use
that's similar to maple and
yeah oh yeah so similar to maple and
mathematica yeah okay but a little more
specialized but yeah
it's programs that work with symbols and
allow you to do can you do proofs can
you do
kind of kind of little little leaps and
proofs they're not really built for that
that's a whole other
story but these tools are part of the
process of mathematics now
right they are now for most
mathematicians i would say part of the
process of mathematics and so
um you know there's a story i tell in
the book which i'm fascinated by which
is
you know so far attempts
to get ais to prove interesting theorems
have not
done so well doesn't mean they can it's
actually a paper i just saw which
has a very nice use of a neuron that
defined counter examples to conjecture
somebody said like well maybe this is
always that
yeah and you can be like well let me
sort of train an ai to sort of try
to find things where that's not true and
it actually succeeded now in this case
if you look at the things that it found
you say like okay i mean
these are not famous conjectures yes
okay so like somebody wrote this down
maybe this is so
um looking at what the ai came up with
you're like
you know i bet if like five grad
students had thought about that problem
they wouldn't
i mean when you see it you're like okay
that is one of the things you might try
if you sort of like
put some work into it still it's pretty
awesome right but the story i tell
in the book which i'm fascinated by is
um
there is there's okay we're gonna go
back to knots
it's cool there's a knot called the
conway knot after john conway maybe
we'll talk about a very interesting
character also he has a small tangent
somebody i was supposed to talk to and
unfortunately he passed away and
he's he's somebody i find is an
incredible mathematician incredible
human being
oh and i am sorry that you didn't get a
chance because having had the chance to
talk to him a lot when i was you know
when i was a postdoc
um yeah you missed out there's no way to
sugarcoat it i'm sorry that you didn't
get that chance
yeah it is what it is so knots
yeah so there was a question and again
it doesn't matter the technicalities of
the question but it's a question of
whether the knot is slice it has to do
with um
something about what kinds of
three-dimensional surfaces in four
dimensions can be bounded by this knot
but never mind what it means it's some
question
uh and it's actually very hard to
compute whether or not
is slice or not um
and in particular the question of the
conway knot
whether it was slice or not was
particularly
vexed um until it was solved just a few
years ago by lisa picarillo who actually
now that i think of it was here in
austin i believe she was a grad student
at ut austin at the time i didn't even
realize there was an austin connection
to this story until i started
telling it she is in fact i think she's
now at mit so she's basically following
you around
if i remember correctly the reverse
there's a lot of really interesting
richness to this
story one thing about it is her paper
was rather
was very short it was very short and
simple nine pages of which two were
pictures
uh very short for like a paper solving a
major conjecture
and it really makes you think about what
we mean by difficulty in mathematics
like do you say oh actually the problem
wasn't difficult because you could solve
it so simply or do you say like well no
evidently it was difficult because like
the world's top
topologist many you know worked on it
for 20 years and nobody could solve it
so therefore it is difficult
or is it that we need sort of some new
category of things that about which
it's difficult to figure out that
they're not difficult
i mean this is the computer science
formulation but
the sort of the the journey to arrive
at the simple answer may be difficult
but once you have the answer
it will then appear simple and i mean
there might be a large category
i hope there's a large uh set
of such solutions
because um you know
once we stand at the end of the
scientific process that we're at the
very beginning
of or at least it feels like i hope
there's just simple answers to
everything
that we'll look and it'll be simple laws
that govern the universe
simple explanation of what is
consciousness of what is
love is mortality fundamental to life
what's the meaning of life uh
are are humans special or we're just
another sort of reflection of uh
and all that is beautiful uh in the
universe in terms of like
life forms all of it is life and just
has different
when taken from a different perspective
is all life can seem more valuable or
not but really it's all part of the same
thing
all those will have a nice like two
equations maybe one equation
why do you think you want those
questions to have simple answers
i think just like symmetry and the
breaking of symmetry is beautiful
somehow there's something beautiful
about simplicity
i think it so it's static it's aesthetic
yeah i
or but it's aesthetic in the way that uh
happiness is an aesthetic
like uh why is that so joyful that a
simple explanation
that governs a large number of cases is
really appealing
even when it's not like obviously we
get a huge amount of trouble with that
because oftentimes
it doesn't have to be connected with
reality or even that explanation could
be exceptionally harmful
most of like the world's history that
has
that was governed by hate and violence
had a very simple explanation at the
court
that was used to cause the violence and
the hatred so like we get into trouble
with that
but why is that so appealing and in this
nice forms
in mathematics like you look at the
einstein papers
why are those so beautiful and why is
the andrew wiles proof of the farm
ozilized theorem
not quite so beautiful like what's
beautiful about that story
is the human struggle of like the human
story of perseverance
of the drama of not knowing if the proof
is correct and
ups and downs and all those kinds of
things that's the interesting part but
the fact that the proof is huge and
nobody understands well
from my outsider's perspective nobody
understands what the heck it is
uh is is not as beautiful as it could
have been
i wish it was what fermat originally
said which is
you know it's it's not
it's not small enough to fit in the
margins of this page
but maybe if he had like a full page or
maybe a couple post-it notes he would
have enough to do the proof
what do you make of if we could take
another of a multitude of tangents
what do you make of fermat's last
theorem because the statement there's a
few
theorems there's a few problems that are
deemed by the world throughout its
history to be exceptionally difficult
and
that one in particular is uh really
simple to formulate
and really hard to come up with a proof
for
and it was like taunted as simple
uh by from himself there's something
interesting to be said about
that x to the n plus y to the n equals z
to the n for n
of three or greater is there a solution
to this
and then how do you go about proving
that like how would you
uh try to prove that and what do you
learn from
the proof that eventually emerged by
andrew wiles yeah so right to sort of
give
let me just say the background because i
don't know if everybody listening knows
the story so you know fermat
uh was an early number theorist not
really sort of an early mathematician
those
special adjacent didn't really exist
back then
he comes up in the book actually in the
context of um a different theorem of his
that has to do with testing whether a
number is prime
or not so i write about he was one of
the ones who was salty and like he would
exchange these letters where
he and his correspondence would like try
to top each other and
vex each other with questions and stuff
like this but this particular thing
um it's called fermazl's theorem because
it's a note he wrote uh in
in his uh in his copy of the description
arithmetic eye like he wrote
here's an equation it has no solutions i
can prove it but
the proof's like a little too long to
fit in this in the margin of this
book he was just like writing a note to
himself now let me just say historically
we know that vermont did not have a
proof of this theorem for a long time
people like
you know people were like this
mysterious proof that was lost a very
romantic story right
but fairmont later
he did prove special cases of this
theorem and
wrote about it to talk to people about
the problem uh it's very clear from the
way that he wrote where he can solve
certain examples of this type of
equation
that he did not know how to do the whole
thing
he may have had a deep
simple intuition about this how to solve
the whole thing that he had at that
moment
without ever being able to come up with
a complete proof and that intuition may
be lost to time
maybe but i think we so but you're right
that is unknowable but i think what we
can know is that later he certainly did
not think that he had a proof that he
was concealing from people he
yes uh he thought he didn't know how to
prove it and i also think he didn't know
how to prove it
now i understand the appeal of saying
like
wouldn't it be cool if this very simple
equation there was like a very simple
clever wonderful proof that you could do
in a page or two and
that would be great but you know what
there's lots of equations like that that
are solved by very clever methods like
that including
the special cases that female wrote
about the method of descent which is
like very wonderful and important
but in the end those are
nice things that like you know you teach
in an undergraduate class
um and it is what it is but they're not
big
um on the other hand work on the fermat
problem that's what we like to call it
because
it's not really his theorem because we
don't think he proved it so i mean
work on the vermont problem developed
this like incredible
richness of number theory that we now
live in today like and not by the way
just wilds andrew wiles being
the first new together with richard
taylor finally proved this theorem
but you know how you have this whole
moment that people try to prove this
theorem and they fail
and there's a famous false proof by
lemay from the 19th century
where kumar in understanding what
mistake lemay had made
in this incorrect proof basically
understands something incredible which
is that
you know a thing we know about numbers
is that um
you can factor them and you can factor
them uniquely
there's only one way to break a number
up into primes
like if we think of a number like 12 12
is two times three
times two i had to think about it right
or it's two times two times three of
course you can reorder them right
but there's no other way to do it
there's no universe in which 12 is
something times five
or in which there's like four threes in
it nope 12 is like two twos and a three
like that
is what it is and that's such a
fundamental feature
of arithmetic that we almost think of it
like god's law you know what i mean it
has to be that way
that's a really powerful idea it's it's
so cool
that every number is uniquely made up of
other numbers
and like made up meaning like there's
these like basic atoms
that form molecules that for that
get built on top of each other i love it
i mean when i teach you know
undergraduate number theory it's like
it's the first really deep theorem that
you prove
what's amazing is you know the fact that
you can factor a number into primes
is much easier essentially euclid knew
it although he didn't quite put it in
that
in that way the fact that you can do it
at all what's deep is the fact that
there's only one way to do it that or
however you sort of chop the number up
you end up with the same set of
prime factors um and indeed what people
finally understood uh at the end of the
19th century is that if you work in
number systems
slightly more general than the ones
we're used to
which it turns out are relevant for ma
all of a sudden this stops being true
things get i mean things get more
complicated and now
because you were praising simplicity
before you were like it's so beautiful
unique factorization
uh it's so great like so when i tell you
that in more general number
systems there is no unique factorization
maybe you're like that's bad i'm like no
that's good because there's like a whole
new world of phenomena to study that you
just can't see
through the lens of the numbers that
we're used to so i'm
i'm for complication i'm highly in favor
of complication
and every complication is like an
opportunity for new things to study
and is that the big uh kind of uh one of
the big insights for you from uh
andrew wiles is proof is there
interesting insights about
the process they use to prove that sort
of
resonates with you as a mathematician is
there an interesting concept that
emerged from it
is there interesting human aspects to
the proof
whether there's interesting human
aspects to the proof itself is an
interesting question
certainly it has a huge amount of
richness sort of at its heart
is an argument of on of what's called
deformation theory
um which was in part
created by my my phd advisor barry mazar
can you speak to what deformation theory
is i can speak to what it's like
sure how about that what does it rhyme
with right
well the reason that barry called it
deformation theory
i think he's the one who gave it the
name um i hope i'm not wrong and saying
this one dave
in your book you have calling different
things by the same name
as one of the things in the beautiful
map that opens the book
yes and this is a perfect example so
this is another phrase of
poincare this like incredible generator
of slogans and aphorisms he said
mathematics is the art of calling
different things by the same name
that very thing that very thing we do
right when we're like this triangle and
this triangle come on they're the same
triangle they're just in a different
place right
so in the same way um it came to be
understood that
the kinds of objects that you study uh
when you study when you study for
maslow's theorem
and let's not even be too careful about
what these objects are i can tell you
there are gal
representations in modular forms but
saying those words
is not going to mean so much but
whatever they are they're things that
can be
deformed moved around a little bit and
um i think the inside of what andrew and
and then andrew and richard were able to
do
was to say something like this um
deformation means
moving something just a tiny bit like an
infinitesimal amount
um if you really are good at
understanding which ways a thing can
move
in a tiny tiny tiny infinitesimal amount
in certain directions
maybe you can piece that information
together to understand the whole global
space in which it can move
and essentially their argument comes
down to showing that two of those
big global spaces are actually the same
the fabled r equals t
part of uh part of their proof which is
at the heart of it
um and it involves this very
careful uh principle like that but that
being said
what i just said it's probably not what
you're thinking because what you're
thinking
when you think oh i have a point in
space and i move it around like a little
tiny bit
um you're using
um your notion of distance that's you
know from calculus
we know what it means for like two
points on the real line to be close
together
so i get another thing that comes up in
the book a lot
is this fact that the notion of distance
is not given to us by god we could mean
a lot of different things by distance
and just in the english language we do
that all the time we talk about somebody
being a close relative
it doesn't mean they live next door to
you right it means something else
there's a different notion of distance
we have in mind and there are lots of
notions of distances
that you could use you know in the
natural language processing community
and ai there might be some notion of
semantic distance or lexical distance
between two words how much do they tend
to arise in the same context that's
incredibly important
for um you know doing autocomplete and
like
machine translation and stuff like that
and it doesn't have anything to do with
are they next to each other in the
dictionary right it's a different kind
of distance
okay ready in this kind of number theory
there was a
crazy distance called the periodic
distance i didn't write about this that
much in the book because even though i
love it it's a big part of my research
life it gets a little bit into the weeds
but
your listeners are going to hear about
it now please where
you know what a normal person says when
they say two numbers are close
they say like you know their difference
is like a small number like seven and
eight are close because their difference
is one and one's pretty small
um if we were to be what's called a two
attic number theorist
we'd say oh two numbers are close if
their difference
is a multiple of a large power of two
so like so like one and
49 are close because their difference is
48 and 48 is a multiple of 16 which is a
pretty large power of two
whereas whereas one and two are pretty
far away
because the difference between them is
one which is not even a multiple of a
power of 2 at all it's odd
you want to know what's really far from
1 like
1 and 1 64. because their difference is
a negative power of 2
2 to the minus 6. so those points are
quite quite fast
the power of a large n would be too
cool if that's the difference between
two numbers and they're close
yeah so two to a large power is this
multiplication
very small number and two to a negative
power
is a very big number that's two attic
okay uh
i can't even visualize that it takes
practice it takes practice if you've
ever heard of the cantor set it looks
kind of like that
so it is crazy that this is good for
anything right
i mean this just sounds like a
definition that someone would make up to
torment you
but what's amazing is there's a general
theory of distance where you say any
definition you make that satisfies
certain axioms deserves to be called a
distance and this
see i'm starting to interrupt uh my
brain you broke my brain now awesome
uh 10 seconds ago uh because i'm also
starting to map
for the two added case to binary numbers
and sure you know because
because we romanticized those sauce
trunks oh that's exactly the right way
to think of it i was trying to
mess with number you know i was trying
to see okay which ones are close
and then i'm starting to visualize
different binary numbers and how they
which ones are close to each other and
uh i'm not sure well i think there's
no it's very similar that's exactly the
way to think of it it's almost like
binary numbers written in reverse
right because in a in a binary expansion
two numbers are closed a number that's
small is like point zero zero zero zero
something
something that's the decimal and it
starts with a lot of zeros in the two
attic metric
a binary number is very small if it
ends with a lot of zeros and then the
decimal point
got you so it is kind of like binary
numbers written backwards is actually i
should have
that's what i should have said lex
that's a very good metaphor
okay but so why is that why is that
interesting except for the fact that uh
it's it's it's a beautiful kind of uh
framework different kind of framework
which you think about distances
and you're talking about not just the
two attic but the generalization of that
yeah the mep and so so that because
that's the kind of deformation that
comes up
in wiles is in wiles as proof that
defamation we're moving something a
little bit
means a little bit in this to addiction
okay no i mean it's such i mean i could
just get excited to talk about it and i
just taught this like in the fall
semester that um but it like
reformulating
why is uh
so you pick a different uh measure of
distance
over which you can talk about very tiny
changes
and then use that to then prove things
about
the entire thing yes although you know
honestly what i would say
i mean it's true that we use it to prove
things but i would say we use it to
understand things
and then because we understand things
better then we can prove things but you
know the goal is always the
understanding the goal is not
so much to prove things the goal is not
to know what's true or false i mean this
is something i write about in the book
near the end and it's something that
it's a wonderful wonderful essay by by
bill thurston
kind of one of the great geometers of
our time who unfortunately passed away a
few years ago
um called on proof and progress in
mathematics and he writes very
wonderfully about how
you know we're not it's not a theorem
factory where we have a
production quota i mean the point of
mathematics is to help humans understand
things
and the way we test that is that we're
proving new theorems along the way
that's the benchmark but that's not the
goal
yeah but just as a as a kind of
absolutely but as a tool
it's kind of interesting to approach a
problem by saying
how can i change the distance function
like what the the nature of distance
because that might start to lead to
insights
for deeper understanding like if i were
to try to describe
human society by a distance two people
are close
if they love each other right and then
and then start to uh
and do a full analysis on the everybody
that lives on earth currently the seven
billion people you know and
from that perspective as opposed to the
geographic perspective of distance
and then maybe there could be a bunch of
insights about the source of
uh violence the source of uh maybe
entrepreneurial success or invention or
economic success or different systems
of communism capitalism start to i mean
that's i guess what
economics tries to do but really saying
okay let's think outside the box about
totally new distance functions that
could unlock something
profound about the space yeah because
think about it okay here's
i mean now we're gonna talk about ai
which you know a lot more about than i
do so
just you know start laughing
uproariously if i say something that's
completely wrong we both know very
little
relative to what we will know centuries
from now
that is that is a really good humble way
to think about it i like it okay so
let's just go for it
um okay so i think you'll agree with
this that
in some sense what's good about ai is
that
we can't test any case in advance the
whole point of ai is to make our one
point of it i guess is to make
good predictions about cases we haven't
yet seen and in some sense that's always
going to involve some notion of distance
because it's always going to involve
somehow taking a case we haven't seen
and saying
what cases that we have seen is it close
to is it like is it somehow an
interpolation
between now when we do that in order to
talk about things being like other
things
implicitly or explicitly we're invoking
some notion of distance and boy we
better get it right
yeah right if you try to do natural
language processing and your idea about
of distance between words is
how close they are in the dictionary
when you write them in alphabetical
order you are going to get
pretty bad translations right no the
notion of distance has to come from
somewhere else yeah that that's
essentially what neural networks are
doing this what word and bettings are
doing is
yes coming up with uh in the case of
word embeddings literally like literally
what they are doing is learning a
distance but those are super
complicated distance functions and it's
almost nice to think
maybe there's a nice transformation
that's simple
uh sorry this there's a nice formulation
of the distance
again with the simple so you don't
let me ask you about this from an
understanding perspective there's the
richard feynman maybe attributed to him
but maybe many others
is this idea that if you can't explain
something simply that you don't
understand it
in how many cases how often
is that true do you find there's some
profound truth in that
oh okay so you were about to ask is it
true to which i would say flatly no but
then you said
you followed that up with is there some
profound truth in it and i'm like
okay sure so there's some truth in it
but it's not true
[Laughter]
this is your mathematician answer the
truth that is in it
yeah is that learning to explain
something helps you
understand it um but real things are not
simple
yeah a few things are most are not
um and i don't to be honest i don't i
mean i don't
we don't really know whether feynman
really said that right or something like
that is sort of
disputed but i don't think feynman could
have literally believed that
whether or not he said it and you know
he was the kind of guy i didn't know him
but i'm
reading his writing he liked to sort of
say stuff like stuff that sounded good
you know what i mean so it's it's
totally strikes me as the kind of thing
he could have said because he liked the
way
saying it made him feel but also knowing
that he didn't like literally mean it
well i definitely have
have a lot of friends and i've talked to
a lot of physicists and they do
derive joy from believing that they can
explain stuff simply
or believing it's possible to explain
style simply
even when the explanation is not
actually that simple like i've
heard people think that the explanation
is simple and they do the explanation
and i think
it is simple but it's not capturing the
phenomena that we're discussing
it's capturing it somehow maps in their
mind but it's it's taking as a
starting point as an assumption that
there's a deep knowledge and a deep
understanding that's
that's actually very complicated and the
simplicity is almost like a
almost like a poem about the more
complicated thing as opposed to
a distillation and i love poems but a
poem is not an explanation
well some people might disagree with
that but certainly from a mathematical
perspective no poet would disagree with
it
no poet would disagree you don't think
there's some things that can only be
described
imprecisely i said explanation i don't
think any poem
i don't think any poet would say their
poem is an explanation they might say
it's a description they might say it's
sort of capturing sort of
well some people might say the only
truth is like music
right that the the the not the only
truth but some truth can only be
expressed through art
and i mean that's the whole thing we're
talking about religion and myth and
there's some things that uh are limited
cognitive capabilities
and the tools of mathematics or the
tools of physics are just not going to
allow us to capture
like it's possible consciousness is one
of those things
yes that is definitely possible but i
would even say
look unconsciousness is a thing about
which we're still in the dark as to
where
whether there's an explanation we would
we would understand it as an explanation
at all by the way okay i got to give
yeah one more amazing poincare quote
because this guy just never stopped
coming out with great quotes that
um you know paul erdish another fellow
who appears in the book and by the way
he thinks about this notion of distance
of like personal affinity kind of like
what you're talking about that kind of
social network and that
notion of distance that comes from that
so that's something that erdos did
well he thought about distances and
networks i guess he didn't probably he
didn't think about the social media that
was fascinating and that's how it
started that story of virtus number yeah
okay
but you know eredish was sort of famous
for saying and this is sort of long
lines we're saying
he talked about the book capital t
capital b
the book and that's the book where god
keeps the right proof of every
theorem so when he saw a proof he really
liked it was like really elegant really
simple
like that's from the book that's like
you found one of the ones that's in
the book um he wasn't the religious guy
by the way he referred to god as the
supreme fascist he was like uh
but somehow he was like i don't really
believe in god but i believe in god's
book i mean it was uh
yeah um but poincare on the other hand
um and by the way there are other
members hilda hudson is one who comes up
in this book she also kind of
saw math um she's one of the people who
sort of develops
um the disease model that we now use
that we use to sort of track pandemics
this sir model that sort of originally
comes from
her work with ronald ross but she was
also super super super devout
and she also sort of from the other side
of the religious coin was like yeah math
is how we communicate with god
she has a great all these people are
incredibly quotable she says you know
math isn't
the truth the things about mathematics
is like they're not the most important
of god thoughts
but they're the only ones that we can
know precisely
so she's like this is the one place
where we get to sort of see what god's
thinking when we do
mathematics again not a fan of poetry or
music some people will say hendrix is
like
some some people say chapter one of that
book is mathematics and then chapter two
is like classic rock right so
like it's not clear that the i'm sorry
you just sent me off on a tangent just
imagining like eredish at a hendrick's
concert like trying to sort of figure
out if it was from
the book or not but i was what i was
coming to was justice it but one point
said about this is he's like you know
if like this has all worked out in the
language of the divine and if a divine
being like came down and
told it to us we wouldn't be able to
understand it so it doesn't matter
so poincare was of the view that there
were things that were sort of like
inhumanly complex
and that was how they really were our
job is to figure out the things that are
not like that
that are not like that all this talk of
primes got me hungry for primes
you uh your blog post the beauty of
bounding gaps a huge
discovery about prime numbers and what
it means for the future of math
can you tell me about prior numbers what
the heck are those what are twin primes
what are prime gaps what are bounding
gaps and primes what are all these
things
and what if anything or what exactly is
beautiful about them
yeah so you know prime numbers
are one of the things that number
theorists study the most and have
for millennia um they are numbers
which can't be factored and then you say
like like five and then you're like wait
i can't factor five five is
five times one okay not like that that
is a factorization it absolutely is a
way of expressing five as
a product of two things but don't you
agree there's like something trivial
about it it's something you can do to
any number
it doesn't have content the way that if
i say that 12 is 6
times 2 or 35 is 7 times 5 i've really
done something to it i've broken up so
those are the kind of factorizations
that count
and a number that doesn't have a
factorization like that is called prime
except historical side note one
which at some times in mathematical
history has been deemed to be a prime
but
currently is not and i think that's for
the best but i bring it up only because
sometimes people think that
you know these definitions are kind of
if we think about them hard enough we
can figure out which definition is true
no there's just an artifact in
mathematics so yeah one
so which definition is best for us
for our purposes well those edge cases
are weird right so uh
so so it can't you can't be it doesn't
count when you use yourself
as a number or one as part of the
factorization
or as the entirety of the factorization
so the so you somehow get to the meat of
the number
by factorizing it and that's seems to
get to the core of all of mathematics
yeah you take any number and you
factorize it until you can factorize no
more and what you have left is some big
pile of primes i mean by definition when
you
can't factor anymore when you when
you're done you can't
break the numbers up anymore what's left
must be prime you know 12 breaks into
two
and two and three um so these numbers
are the atoms the building blocks
of all numbers and there's a lot we know
about them but there's much more we
don't know them i'll tell you the first
few there's 2
3 5 7 11.
by the way they're all going to be odd
from the non because if they were even i
could factor out 2 out of them
but it's not all the odd numbers 9 isn't
prime because it's 3 times 3
15 isn't prime because it's 3 times 5
but 13 is where were we 2 3 5
7 11 13 17 19
not 21 but 23 is etcetera etcetera okay
so you could go on
how high could you go if we were just
sitting here by the way your own brain
continuous without interruption would
you be able to go over a hundred
i think so there's always those ones
that trip people up there's
there's a famous one the groton deak
prime 57 like sort of
alexander groddendyk the great algebraic
geometer was sort of giving
some lecture involving a choice of a
prime in general and somebody said like
can't you just choose a problem he said
okay 57 which is in fact not prime it's
three times
19. oh damn but it was like
i promise you in some circles it's a
funny story okay um
but um there's a humor in it uh
yes i would say over a hundred i
definitely don't
remember like 107 i think i'm not sure
okay like so is there a category of uh
like fake primes that that
are easily mistaken to be prime like 57
i wonder yeah so i would say
57 and take a small 27 and 51
are definitely like prime offenders oh i
didn't do that on purpose
well done didn't do it on purpose anyway
they're definitely ones that people uh
or 91 is another classic seven times 13.
it really
feels kind of prime doesn't it but it is
not yeah
um but there's also by the way but
there's also an actual notion of
pseudoprime which is
which is the thing with the formal
definition which is not a psychological
thing
it is a prime which passes a primality
test
devised by fermat which is a very good
test which
if if a number fails this test it's
definitely not prime
and so there was some hope that oh maybe
if a number passes the test then it
definitely is prime that would give a
very simple criterion for formality
unfortunately it's only perfect in one
direction
so there are numbers i want to say 341
is the smallest
uh which passed the test but are not
prime 341 is this test easily
explainable or no
uh yes actually um ready let me give you
the simplest version of it you can dress
it up a little bit but here's the basic
idea
uh i take the number the mystery number
i raised two to that power
so let's say your mystery number is six
yeah are you sorry you asked me are you
ready no i might you're breaking my
brain again but yes
let's let's let's do it we're gonna do a
live demonstration um
let's say your number is six so i'm
going to raise 2 to the sixth power
okay so if i were working out i'd be
like that's 2 cubes squared so that's 8
times 8 so that's 64.
now we're going to divide by 6 but i
don't actually care what the quotient is
only the remainder
so let's see 64 divided by 6
is uh well it's it there's a quotient of
10 but the remainder is 4.
so you failed because the answer has to
be 2.
for any prime let's do it with five
which is prime
two to the fifth is 32 divide 32 by five
uh and you get six with a remainder of
two
well the remainder of two here for seven
two to the seventh is 128
divide that by seven and let's see i
think that's seven times
14 is that right no
seven times 18
is 126 with a remainder of two right 128
is
a multiple of seven plus two so if that
remainder is not two
then that's definitely not that it's
definitely not prime
and then if it is it's likely a prime
but not for sure
it's likely a prime but not for sure and
there's actually a beautiful geometric
proof which is in the book actually
that's like one of the most granular
parts of the book because it's such a
beautiful proof i couldn't not give it
so you you draw a lot of like
opal and pearl necklaces and spin them
that's kind of the geometric nature of
the
of this proof of fermat's little theorem
um
so yeah so with pseudo primes there are
primes that are kind of faking if they
pass that test but there
are numbers that are faking it that pass
that test but are not actually prime
um but the point is
um there are many many many
theorems about prime numbers um
are there like there's a bunch of
questions to ask is there an infinite
number of primes
can we say something about the gap
between primes as the
numbers grow larger and larger and
larger and so on
yeah it's a perfect example of your
desire for simplicity in all things you
know it would be really simple
if there was only finitely many primes
yes and then there would be this sim
finite set of atoms that all numbers
would be built up right
that would be very simple and good in
certain ways but it's completely false
and number theory would be totally
different if that were the case it's
just not true
um in fact this is something else that
euclid knew so this is a very very old
fact like much before long before we had
anything like modern numbers that primes
are infinite
the primes that there are that write the
there's an infinite number of primes
so what about the gaps between the
primes right so so one thing that people
recognized
and really thought about a lot is that
the primes on average
seem to get farther and farther apart as
they get bigger and bigger in other
words it's less and less common
like i already told you of the first 10
numbers two three five seven four of
them are prime that's a lot forty
percent
if i looked at you know ten digit
numbers
no way would forty percent of those be
prime being prime would be a lot rarer
in some sense because there's a lot more
things for them to be divisible by
that's one way of thinking of it it's
it's a lot more possible for there to be
a factorization because there's a lot of
things you can try to factor out of it
as the numbers get bigger and bigger
primarily gets rarer and rarer
and the extent to which that's the case
that's pretty well understood but then
you can ask more fine-grained questions
and here is one
um a
twin prime is a pair of primes that are
two apart
like three and five or like 11 and 13
or like 17 and 19. and one thing we
still don't know is
are there infinitely many of those we
know on average they get farther and
farther apart but that doesn't mean
there couldn't be like occasional
folks that come close together and
indeed
uh we think that there are and one
interesting question
i mean this is because i think you might
say like well
why how could one possibly have a right
to have an opinion about something like
that
like what you know we don't have any way
of describing a process that makes
primes like sure you can like
look at your computer and see a lot of
them but the fact that there's a lot
why is that evidence that there's
infinitely many right maybe i can go on
the computer and find 10 million well 10
million 10 million is pretty far from
infinity right so how is that how is
that evidence there's a lot of things
there's like a lot more than 10 million
atoms that doesn't mean there's
infinitely many atoms in the universe
right i mean on most people's physical
theories there's probably not as i
understand it
okay so why would we think this
the answer is that we've that it turns
out to be like
incredibly productive and enlightening
to think about primes
as if they were random numbers as if
they were randomly distributed
according to a certain law now they're
not they're not random there's no chance
involved
it's completely deterministic whether a
number is prime or not and yet
it just turns out to be phenomenally
useful useful in mathematics
to say even if something is
governed by a deterministic law let's
just pretend it wasn't
let's just pretend that they were
produced by some random process and see
if the behavior is roughly the same
and if it's not maybe change the random
process maybe make the randomness a
little bit different and tweak it and
see if you can find a random process
that matches the behavior we see and
then maybe you predict that
other behaviors um of the system are
like that of the random process
and so that's kind of like it's funny
because i think when you talk to people
about the twin prime conjecture
people think you're saying wow there's
like some deep
structure there that like makes those
primes be like close together again and
again
and no it's the opposite of deep
structure what we say when we say we
believe the twin prime conjecture is
that we believe the primes are like sort
of
strewn around pretty randomly and if
they were then by chance you would
expect there to be infinitely many twin
primes and we're saying yep we expect
them to behave just like they would if
they were random dirt
the you know the fascinating parallel
here is uh
i just got a chance to talk to sam
harris and he uses the prime numbers
as an example often i don't know if
you're familiar
with who sam is he uses that as an
example
of there being no free will
wait where did you get this well he just
uses
as an example of it might seem like this
is a random number generator
but it's all like formally defined so if
we keep
getting more and more primes then
like that might feel like a new
discovery and that might feel like a new
experience but it's not
it was always written in the cards but
it's funny that you say that because a
lot of people think of like
randomness uh the fundamental
randomness within the nature of reality
might be the source
of something that we experience as free
will and you're saying it's like useful
to look at prime numbers as
um as a random process
in order to prove stuff about them but
fundamentally of course it's not a
random process
well not in order to prove some stuff
about them so much as to figure out
what we expect to be true and then try
to prove that here's what you don't want
to do try really hard to prove something
that's false that makes it really hard
to prove the thing if it's false
so you certainly want to have some
heuristic ways of guessing making good
guesses about what's true so yeah
here's what i would say let's you're
going to be imaginary sam harris now yes
like
you are talking about prime numbers and
you are like but prime numbers are
completely deterministic
and i'm saying like well but let's treat
them like a random process
and then you say but you're just saying
something that's not true they're not a
random process or deterministic and i'm
like okay great you hold to your
insistence that is honoring the process
meanwhile i'm generating insight about
the primes that you're not because i'm
willing to sort of pretend that there's
something that they're not in order to
understand what's going on yeah so it
doesn't matter what the reality is what
matters is
what's uh what framework
of thought results in the maximum number
of insights
yeah because i feel look i'm sorry but i
feel like you have more insights about
people if you
think of them as like beings that have
wants and needs and desires and do stuff
on
purpose even if that's not true you
still understand better what's going on
by treating them in that way don't you
find look when you work on machine
learning
don't you find yourself sort of talking
about what the machine is what the
machine is trying to do
in a certain instance do you not find
yourself drawn to that language
well oh it knows this it's trying to do
that it's learning that
i'm certainly drawn to that language to
the point where i
received quite a bit of criticisms for
it because i you know like
oh i'm on your side man so especially in
robotics
i don't know why but robotics people
don't like to
name their robots or they they certainly
don't like to gender their robots
because the moment you gender a robot
you start to anthropomorphize
if you say he or she you start to you in
your mind construct like a
um like a life story in your mind you
can't help it
this like you create like a humorous
story to this person you start to
understand this person this robot you
start to project your own
but i think that's what we do to each
other and i think that's actually really
useful from the engineering
process especially for human robot
interaction
and yes for machine learning systems for
helping you build an intuition about a
particular problem
it's almost like asking this question
you know when a machine learning system
fails in a particular edge case
asking like what were you thinking about
like
like asking like almost like when you're
talking about to a child
who just does something bad you're
you want to understand like what was um
how did they see the world maybe there's
a totally new maybe you're the one
that's thinking about the world
incorrectly and uh yeah that
anthropomorphization process i think is
ultimately good for insight and the same
is
i i i agree with you i tend to believe
about free will
as well let me ask you a ridiculous
question if it's okay
of course i've just recently
most people go on like rabbit hole like
youtube
things and i went on a rabbit hole often
do of wikipedia
and i found a page on uh
finitism ultra finitism
and intuitionism or you need to i'm i
forget what it's called
yeah intuitionism intuitionism that
seemed pretty
pretty interesting i have my to-do list
to actually like look into
like is there people who like formally
attract like real mathematicians are
trying to argue for this
but the belief there i think let's say
find nitism that
infinity is is fake
meaning um infinity might be like a
useful hack
for certain like a useful tool in
mathematics but it really
gets us into trouble because there's no
infinity
in the real world maybe i'm sort of
not expressing that uh fully correctly
but basically saying like there's things
that are
in once you add into mathematics things
that are not
provably within the physical world
you're starting to
inject to corrupt your
framework of reason what do you think
about that
i mean i think okay so first of all i'm
not an expert and
i couldn't even tell you what the
difference is between those three terms
finetism ultrafinitism and intuitionism
although i know they're related i tend
to associate them with the netherlands
in the 1930s
okay i'll tell you can i just quickly
comment because i read the wikipedia
page
the difference in ultra like the
ultimate sentence of the modern age can
i just comment because i read the
wikipedia page that sums up our moment
bro i'm basically an expert ultra ultra
finitism
so financialism says that the only
infinity you're allowed to have is that
the natural numbers are
infinite so like those numbers are
infinite
so like one two three four five the
integers
are internet the ultra financialism says
nope even that infinity's fake that's
fine
i bet ultra fanatism came second i bet
it's like when there's like a hardcore
scene and then one guy is like
oh now there's a lot of people in the
scene i have to find a way to be more
hardcore than the hardcore people
go back to the emo doc yeah okay so is
there any uh
are you ever because i'm not often
uncomfortable with infinity
like psychologically i i you know i have
i have
trouble when that sneaks in there
it's because it works so damn well i get
a little suspicious
um because it could be almost like a
crutch
or an oversimplification that's missing
something profound about reality
well so first of all okay if you say
like
is there like a serious way of doing
mathematics that doesn't
really treat infinity as a real
thing or maybe it's kind of agnostic and
it's like i'm not really gonna make a
firm statement about whether it's a real
thing or not yeah that's called most of
the history of mathematics
right so it's only after cantor right
that we really are
sort of okay we're gonna like have a
notion of like the cardinality of an
infinite set
and like um do something that you might
call like the modern theory of infinity
um that said obviously everybody was
drawn to this notion and no not
everybody was comfortable with it look i
mean this is what happens with newton
right i mean so
newton understands that to talk about
tangents and to talk about instantaneous
velocity
um he has to do something that we would
now call taking a limit
right the fabled dy for dx if you sort
of go back to your calculus class for
those who have taken calculus remember
this mysterious thing
and you know what is it what is it well
he'd say like
well it's like you sort of um divide the
length of this line segment
by the length of this other line segment
and then you make them a little shorter
and you divide again and then you make
them a little shorter and you divide
again and then you just keep on doing
that until they're like infinitely short
and then you divide them again
these quantities that are like they're
not zero
but they're also smaller than any
actual number these infinite decimals
well people were queasy about it and
they weren't wrong to be queasy about it
right from a modern perspective it was
not really well formed there's this very
famous critique of newton
by bishop berkeley where he says like
what these things you define like
you know they're not zero but they're
smaller than any number are they the
ghosts of departed quantities
that was this like ultra line
and on the one hand he was right
it wasn't really rigorously modern
standards on the other hand like newton
was out there doing calculus
and other people were not right it
worked and it worked
i think i think a sort of intuitionist
few for instance
i would say would express serious doubt
and by
it's not by the way it's not just
infinity it's like saying i think we
would express serious doubt that like
the real numbers exist now most people
are
comfortable with the real numbers
well computer scientists with floating
point number i mean the
floating point of arithmetic that is
that's a great point actually
i think in some sense this flavor of
doing math saying
we shouldn't talk about things that we
cannot specify in a finite amount of
time there's something very
computational in flavor about that and
it's probably not a coincidence that it
becomes popular
in the 30s and 40s which is also like
kind of like the dawn of
ideas about formal computation right you
probably know the timeline better than i
do sorry what because popular the
these ideas that maybe we should be
doing math in this more restrictive way
where
even a thing that you know because look
the origin of all this is like
you know number represents a magnitude
like the length of a line like so i mean
the idea that
there's a continuum there's sort of like
there's like
um it's pretty old but that you know
just cause something is old doesn't mean
we can't
reject it if we want to well a lot of
the fundamental ideas in computer
science when you talk about the
complexity
of problems uh to touring himself they
rely on
an infinity as well the ideas that kind
of challenge that
the whole space of machine learning i
would say challenges that
it's almost like the engineering
approach to things like the floating
point of arithmetic the other one that
back to john conway that challenges this
idea
i mean maybe to tying the the ideas of
deformation theory
and and uh
limits to infinity is this idea of
cellular automata
with uh john conway looking at the game
of life stephen wolfram's work
that i've been a big fan of for a while
of cellular autonomy i was i was
wondering if you have
if you have ever encountered these kinds
of objects
you ever looked at them as a
mathematician where you have very simple
rules
of tiny little objects that when taken
as a whole create
incredible complexities but are very
difficult to analyze
very difficult to make sense of even
though the one individual object
one part it's like what you're saying
about andrew wiles like you
you can look at the deformation of a
small piece to tell you about the hole
it feels like with cellular automata
or any kind of complex systems it's
it's often very difficult to say
something about the whole thing
even when you can precisely describe the
operation
of uh the sm the local neighborhoods
yeah i mean i love that subject i
haven't really done research in it
myself i've played around with it i'll
send you a fun blog post i wrote where i
made some cool
texture patterns from cellular
autonomous um but
um and those are really always
compelling it's like
you create simple rules and they create
some beautiful textures it doesn't make
any
actually did you see there was a great
paper i don't know if you saw this like
a machine learning paper
yes i don't know if you want to talk
about where they were like learning the
texture is like let's try to like
reverse engineer and like learn a
cellular automaton that can reduce
texture that looks like this
from the images very cool and as you say
the thing you said is i feel the same
way when i read machine learning paper
is that what's especially interesting is
the cases where it doesn't work
like what does it do when it doesn't do
the thing that you tried to train it
yeah to do that's extremely interesting
yeah yeah that was a cool paper
so yeah so let's start with the game of
life let's start with
um or let's start with john conway so
conway
so yeah so let's start with john conway
again just i don't know
from my outsider's perspective there's
not many mathematicians
that stand out throughout the history of
the 20th century
he's one of them i feel like he's not
sufficiently recognized
i think he's pretty recognized okay well
i mean
he was a full professor of princeton for
most of his life he was sort of
certainly at the pinnacle of
yeah but i found myself every time i
talk about conway and how excited
i am about him i have to constantly
explain to
people who he is and that's that's
always a
sad sign to me but that's probably true
for a lot of mathematicians
i was about to say like i feel like you
have a very elevated idea of how famous
but
this is what happens when you grow up in
the soviet union you know you think the
mathematicians are like very very famous
yeah but i'm not actually so convinced
at a tiny tangent that
that shouldn't be so i mean there's uh
it's not obvious to me that that's one
of the like if if i were to analyze
american society that uh
perhaps elevating mathematical and
scientific thinking to a little bit
higher
level would benefit the society well
both in discovering the beauty of what
it is to be human
and for actually creating cool
technology better iphones
but anyway john conway yeah and conway
is such a perfect example of somebody
whose humanity was
and his personality was like wound up
with his mathematics right so it's not
sometimes i think people
who are outside the field think of
mathematics as this kind of like
cold thing that you do separate from
your existence as a human being no way
your personality is in there just as it
would be in like a novel you wrote or a
painting you painted or just like the
way you walk down the street like it's
in there it's you doing it
and conway was certainly a singular
personality um
i think anybody would say that
he was playful like everything was a
game to him
now what you might think i'm going to
say and it's true is that he sort of
was very playful in his way of doing
mathematics
but it's also true it went both ways he
also sort of made mathematics
out of games he like looked like he was
a constant inventor of games with like
crazy names and
then he would sort of analyze those
games mathematically um
to the point that he and then later
collaborating with knuth like
you know created this number system the
surreal numbers
in which actually each number is a game
there's a wonderful book about this
called i mean there are his own books
and then there's like a book that he
wrote with burleigh camping guy called
winning ways
which is such a rich source of ideas
um and he
too kind of has his own crazy number
system in which by the way there are
these
infinitesimals the ghosts of departed
quantities they're in there
now not as ghosts but as like certain
kind of two-player games
um so
you know he was a guy so i knew him when
i was a postdoc
um and i knew him at princeton and our
research overlapped in some ways now it
was on stuff that he had worked on many
years before the stuff i was working on
kind of connected with stuff in group
theory which
somehow seems to keep coming up um
and so i often would like sort of ask
him a question i would have come upon
him in the common room and i would ask
him a question about something
and just any time you turned him on you
know what i mean you sort of asked the
question
it was just like turning a knob and
winding him up
and he would just go and you would get a
response that was like
so rich and went so many places and
taught you so much
and usually had nothing to do with your
question yeah usually
your question was just to prompt to him
you couldn't count on actually getting
the questions
brilliant curious minds even at that age
yeah it was
definitely a huge loss uh but
on his game of life which was i think he
developed in the 70s
as almost like a side thing a fun little
experience
of life is this um it's a very simple
algorithm it's not really a game per se
in the sense of the kinds of games that
he liked whereas people played against
each other and
um but
essentially it's a game that you play
with marking
little squares on the sheet of graph
paper and in the 70s i think he was like
literally doing it with like a pen
on graph paper you have some
configuration of squares some of the
squares in the graphic
are filled in some are not and then
there's a rule a single rule
that tells you um at the next stage
which squares are filled in and which
squares are not
sometimes an empty square gets filled in
that's called birth sometimes a
square that's filled in gets erased
that's called death and there's rules
for which squares are born which squares
die
um it's um
the rule is very simple you can write it
on one line and then the great miracle
is
that you can start from some very
innocent looking
little small set of boxes and
get these results of incredible richness
and of course nowadays you don't do it
on paper nowadays you're doing a
computer there's actually a great
ipad app called golly which i really
like that has like
conway's original rule and like gosh
like hundreds of other
variants and it's lightning fast so you
can just be like i want to see
10 000 generations of this rule play out
like faster than your eye can even
follow and it's like amazing so i highly
recommend it if this is at all
intriguing to you getting golly on your
uh ios device and you can do this kind
of process which i really enjoy doing
which is almost from like putting a
darwin hat on or
a biologist head-on and doing analysis
of
a higher level of abstraction like the
organisms that spring up
because there's different kinds of
organisms like you can think of them as
species and they interact with each
other
they can uh there's gliders they shoot
different there's like
things that can travel around there's
things that can
glider guns that can generate those
gliders they they're
you can use the same kind of language as
you would about describing a biological
system
so it's a wonderful laboratory and it's
kind of a rebuke to someone who
doesn't think that like very very rich
complex structure can
come from very simple underlying laws
like it definitely can now
here's what's interesting if you just
picked like some random rule
you wouldn't get interesting complexity
i think that's one of the most
interesting things of these uh
one of these most interesting features
of this whole subject that the rules
have to be tuned just
right like a sort of typical rule set
doesn't generate any kind of interesting
behavior
yeah but some do i don't
think we have a clear way of
understanding what's doing which don't i
don't know maybe stephen thinks he does
i don't know
but no no it's a giant mystery what
stephen what stephen wolfram did
is um now there's a whole interesting
aspect of the fact that he's
a little bit of an outcast in the
mathematics and physics community
because he's so focused on a particular
his particular work
i think if you put ego aside which i
think
unfairly some people are not able to
look beyond i think his work is actually
quite brilliant
but what he did is exactly this process
of darwin-like exploration
is taking these very simple ideas and
writing a thousand page book on them
meaning like let's play around with this
thing let's see
and can we figure anything out spoiler
alert
no we can't in fact he does uh
he does a challenge uh i think it's like
a rule 30 challenge which is quite
interesting just
simply for machine learning people for
mathematics
people is can you predict the middle
column
for his it's a it's a it's a 1d cellular
automata
can you pre generally speaking can you
predict
anything about how a particular rule
will evolve
just in the future uh very simple
just look at one particular part of the
world just zooming in on that part
you know 100 steps ahead can you predict
something
and uh the the the challenge is to do
that kind of prediction
so far as nobody's come up with an
answer but the point is like
we can't we don't have tools or maybe
it's impossible or
i mean he has these kind of laws of
irreducibility they hear firstly but
it's poetry it's like we can't prove
these things it seems like we can't
that's the basic
uh it almost sounds like ancient
mathematics or something like that where
you
like the gods will not allow us to
predict the cellular automata
but uh that's fascinating that we can't
i'm not sure what to make of it and
there is power to calling this
particular
set of rules game of life as conway did
because not actually exactly sure but i
think he had a sense that there's
some core ideas here that are
fundamental
to life to complex systems to the way
life emerged on earth
i'm not sure i think conway thought that
it's something that i mean conway always
had a rather ambivalent
relationship with the game of life
because i think he
saw it as it was certainly the thing he
was most famous for in the outside world
and i think that he his view which is
correct
is that he had done things that were
much deeper mathematically than that you
know
and i think it always like grieved him a
bit that he was like the game of life
guy
when you know he proved all these
wonderful theorems and like did i mean
created all these wonderful games like
created the serial numbers like i mean
he did i mean
he was a very tireless guy who like just
like did like an incredibly
variegated array of stuff so he was
exactly the kind of person who
you would never want to like reduce to
like one achievement you know what i
mean
let me ask about group theory
you mentioned a few times what is group
theory
what is an idea from group theory that
you find beautiful
well so i would say group theory sort of
starts
as the general theory of symmetry is
that
you know people looked at different
kinds of things and said like
as we said like oh it could have maybe
all there is is the symmetry from left
to right like a human being right or
that's roughly bilateral bilaterally
symmetric as we say
so um so there's two symmetries and then
you're like well wait didn't i say
there's just one there's just left
to right well we always count the
symmetry of doing nothing
we always count the symmetry that's like
there's flip and don't flip those are
the two configurations that you can be
in so there's two
um you know something like a rectangle
is bilaterally symmetric you can flip it
left to right but you can also flip it
top to bottom
so there's actually four symmetries
there's do nothing
flip it left to right and flip it top to
bottom or do both of those things
um a square
um there's even more because now you can
rotate it
you can rotate it by 90 degrees so you
can't do that that's not a symmetry of
the rectangle if you try to rotate it 90
degrees you get a rectangle oriented in
a different way
so um a person has two symmetries a
rectangle four
a square eight different kinds of shapes
have different numbers of symmetries
um and the real observation is that
that's just not like a set of things
they can be combined you do one symmetry
then you do another
the result of that is some third
symmetry
so a group really abstracts away this
notion of saying
um
it's just some collection of
transformations you can do to a thing
where you
combine any two of them to get a third
so you know a place where this comes up
in computer sciences and
is in sorting because the ways of
permuting a set
the ways of taking sort of some set of
things you have on the table and putting
them in a different order
shuffling a deck of cards for instance
those are the symmetries of the deck and
there's a lot of them there's not two
there's not four there's not eight think
about how many different orders the deck
of card can be in each one of those is
the result of applying a symmetry
uh to the original deck so a shuffle is
a symmetry right you're reordering the
cards
if if i shuffle and then you shuffle the
result
is some other kind of thing you might
call it duffel
a double shuffle which is a more
complicated symmetry so group theory is
kind of
the study of the general abstract world
that encompasses
all these kinds of things but then of
course like lots of things that are way
more complicated than that
like infinite groups of symmetries for
instance so thank you
oh yeah okay well okay ready think about
the symmetries of the line you're like
okay i can
reflect it left to right you know around
the origin
okay but i could also reflect it left to
right grabbing
somewhere else like at one or two or pi
or anywhere
or i could just slide it some distance
that's a symmetry slide it five units
over so there's clearly infinitely many
symmetries of the line
that's an example of an infinite group
of symmetries is it possible to say
something that kind of captivates keeps
being
brought up by physicists which is gage
theory gauge symmetry
as one of the more complicated type of
symmetries is there is that
is there a easy explanation what the
heck it is is that something that comes
up
on your mind at all well i'm not a
mathematical physicist but i can say
this
it is certainly true that has been a
very
useful notion in physics to try to say
like
what are the symmetry groups like of the
world like what are the symmetries under
which things don't change right so
we just i think we talked a little bit
earlier about it should be a basic
principle that a theorem that's true
here
is also true over there yes and same for
a physical law right i mean
if gravity is like this over here it
should also be like this over there okay
what that's saying is we think
translation in space
should be a symmetry all the laws of
physics should be unchanged
if the symmetry we have in mind is a
very simple one like translation
and so then um there becomes a question
like what are the symmetries
of the actual world with its physical
laws
and one way of thinking isn't
oversimplification but like one way of
thinking of this big
um shift from
uh before einstein to after is that we
just changed our idea about what the
fundamental group of
symmetries were so that things like the
lorenz contraction things like these
bizarre
relativistic phenomena or lorenz would
have said
oh to make this work we need a thing to
um
to change its shape if it's moving
yeah nearly the speed of light well
under the new frame of framework
it's much better you feel like oh no it
wasn't changing its jeep you were just
wrong about what counted as a symmetry
now that we have this new group the
so-called lorenz group now that we
understand what the symmetries really
are we see it was just an illusion that
the the thing was changing its shape
yeah so you can then describe the
sameness of things under
this weirdness that exactly that is
general relativity for example
yeah yeah still um
i wish there was a simpler explanation
of like exactly i mean
get you know gauge symmetry is a pretty
simple
general concept about rulers being
deformed
i it's just i i uh i've actually just
personally been
on a search not a very uh rigorous
or aggressive search but for uh
something i personally enjoy which is
taking complicated
concepts and finding the sort of minimal
example
that i can play around with especially
programmatically that's great i mean
that this is what we try to train our
students to do right i mean in class
this is exactly what
this is like best pedagogical practice i
do
hope there's simple explanation
especially like
i've uh in my sort of uh
drunk random walk drunk walk whatever
that's called
uh sometimes stumble into the world of
topology
and like quickly like you know when you
like go to a party and you realize this
is not the right
party for me so whenever i go into
topology it's like
so much math everywhere i don't even
know what it feels like
this is me like being a hater is i think
there's way too much math like
they're two the cool kids who just want
to have like everything is expressed
through math
because they're actually afraid to
express stuff simply through language
that's that's my hater formulation of
topology but at the same time i'm sure
that's very necessary to do sort of
rigorous
discussion but i feel like but don't you
think that's what gauge symmetry is like
i mean it's not a field i know well but
it certainly seems like yes it is like
that okay but
my problem with topology okay and even
like differential geom and differential
geometry
is like you're talking about beautiful
things
like if they could be visualized it's
open question if
everything could be visualized but
you're talking about things that could
be
visually stunning i think
but they are hidden underneath all of
that math
like if you look at the papers that are
written in topology if you look at all
the discussions on stack exchange
they're all math dense math heavy and
the only kind of
visual things that emerge every once in
a while
is like uh something like a mobius strip
every once in a while some kind of uh um
[Music]
simple visualizations well there's the
the vibration there's the hop vibration
or all those kinds of things
that somebody some grad student from
like 20 years ago
wrote a program in fortran to visualize
it and that's it
and it's just you know it makes me sad
because
those are visual disciplines just like
computer vision is a visual discipline
so you can provide a lot of visual
examples i wish topology
was more excited and in love with
visualizing some of the ideas
i mean you could say that but i would
say for me a picture of the hop
vibration does nothing for me
whereas like when you're like oh it's
like about the quaternions it's like a
subgroup of the quaternions i'm like oh
so now i see what's going on like why
didn't you just say that why were you
like showing me this stupid picture
instead of telling me what you were
talking about
oh yeah yeah i'm just saying no but it
goes back to what you're saying about
teaching that like people are different
in what they'll respond to so i think
there's no i mean
i'm very opposed to the idea that
there's one right way to explain things
i think there's a huge variation in like
you know our brains like have all these
like weird like hooks and loops and it's
like very hard to know like what's going
to latch on and it's not going to be the
same thing
for everybody so well i think
monoculture is bad right i think that's
and i think we're agreeing on that point
that like
it's good that there's like a lot of
different ways in and a lot of different
ways to describe these ideas because
different people are going to find
different things illuminating but
that said i think there's a lot to be
discovered
when you
force little like silos of brilliant
people
to kind of find a middle ground
or like uh aggregate or
come together in a way so there's like
people that do love visual things
i mean there's a lot of disciplines
especially in computer science
that they're obsessed with visualizing
visualizing data
visualizing neural networks i mean
neural networks themselves are
fundamentally visual there's a lot of
work in computer vision that's very
visual
and then coming together with some some
folks that were like deeply rigorous
and are like totally lost in
multi-dimensional space where it's hard
to even bring them back down to 3d
[Laughter]
they're very comfortable in this
multi-dimensional space so forcing them
to kind of work together to communicate
because it's not just about public
communication of ideas
it's also i feel like when you're forced
to do that public communication like you
did with your book
i think deep profound ideas can be
discovered that's like applicable for
research and for science
like there's something about that
simplification or not simplification but
this distillation or condensation or
whatever the hell you call it
compression
of ideas that somehow actually
stimulates creativity
and uh i'd be excited to see more of
that in the
in in the mathematics community can you
let me make a crazy metaphor
maybe it's a little bit like the
relation between prose and poetry right
i mean if you you might say like why do
we need anything more than prose you're
trying to convey some information so you
just like say it
um well poetry does something right it's
sort of
you might think of it as a kind of
compression of course not all poetry is
compressed like not awesome
some of it is quite baggy but like
um you are kind of
often it's compressed right a lyric poem
is often sort of like a compression of
what would take a long
time and be complicated to explain in
prose into sort of
a different mode that it's going to hit
in a different way we talked about
poncare conjecture
there's a guy he's russian grigori
pearlman he proved poincare's conjecture
if you can comment on the proof itself
if that stands out to you something
interesting or the
human story of it which is he turned
down the fields medal
for the proof is there something
you find inspiring or insightful about
the proof itself
or about the man yeah i mean one thing
i really like about the proof and partly
that's because it's sort of a
thing that happens again and again in
this book i mean i'm writing about
geometry and the way it sort of appears
in all these kind of real world problems
and
but it happens so often that the
geometry you think you're studying
is somehow not enough you have to go one
level higher in abstraction
and study a higher level of geometry and
the way that plays out
is that you know poincare asks a
question about a certain kind of
three-dimensional object
is it the usual three-dimensional space
that we know or is it some kind of
exotic thing and so of course this
sounds like it's a question about the
geometry of the three-dimensional space
but no apparelment understands and by
the way in a tradition that involves
richard hamilton and many other people
like most really important mathematical
advances this doesn't happen alone it
doesn't happen in a vacuum it happens as
the culmination of a program that
involves many people
same with wiles by the way i mean we
talked about wiles and i want to
emphasize that
starting all the way back with kumar who
i mentioned in the 19th century but
um gerhard frye and mazer and ken ribbit
and like many other people
are involved in building the other
pieces of the arch before you put the
keystone in we stand on the shoulders of
giants
yes um
so what is this idea the idea is that
well of course the geometry of the
three-dimensional
object itself is relevant but the real
geometry you have to understand is
the geometry of the space of all
three-dimensional geometries
whoa you're going up a higher level
because when you do that you can say
now let's trace out a path
in that space yes there's a mechanism
called reachy flow and again we're
outside my research area so for all the
geometric analysts and differential
geometers out there listening to this
if i please i'm doing my best and i'm
roughly saying
so this the ritchie flow allows you to
say like okay let's start from some
mystery three-dimensional space
which poincare would conjecture is
essentially the same thing as our
familiar three-dimensional space but we
don't know that
and now you let it flow you
sort of like let it move in its natural
path
according to some almost physical
process and ask
where it winds up and what you find is
that it always winds up
you've continuously deformed it there's
that word deformation again
and what you can prove is that the
process doesn't stop until you get to
the usual three-dimensional space and
since you can get from the mystery thing
to the standard space by this process of
continually
changing and never kind of having any
sharp transitions
then the original shape must have been
the same as the standard
shape that's the nature of the proof now
of course it's incredibly technical i
think
as i understand it i think the hard part
is proving that
the favorite word of ai people you don't
get any singularities along the way
um but of course in this context
singularity just means
acquiring a sharp kink it just means uh
becoming non-smooth at some point so
just saying something interesting about
uh
formal about the smooth trajectory
through this weird space
yeah but yeah so what i like about it is
that it's just one of many examples of
where
it's not about the geometry you think
it's about it's about the geometry of
all geometries so to speak and it's only
by
kind of like kind of like being jerked
out of flat land right same idea it's
only by sort of
seeing the whole thing globally at once
that you can really make progress on
understanding like the one thing you
thought you were looking at
it's a romantic question but what do
what what do you think about him turning
down the fields medal
is uh is that just our nobel prizes and
feelings medals just
just the cherry on top of the cake and
really math itself
the process of uh curiosity of pulling
at the string of
the mystery before us that's the cake
and then
the awards are just icing and
uh clearly i've been fasting and i'm
hungry but uh
but do you think it's um it's it's
tragic or just
just a little curiosity that he turned
on the metal
well it's interesting because on the one
hand i think it's absolutely true
that right in some kind of like vast
spiritual sense like awards are not
important like not important the way
that sort of like
understanding the universe is important
um
on the other hand most people who are
offered that prize accept it you know
it's it is so there's something unusual
about his uh
his choice there um i i wouldn't say i
see it as
tragic i mean maybe if i don't really
feel like i have a clear picture of
of why he chose not to take it i mean
it's not he's not alone
in doing things like this people
sometimes turn down prizes for
ideological reasons
um probably more often in mathematics i
mean
i think i'm right in saying that peter
schulze like turned down sort of some
big monetary
prize because he just you know i mean i
think he
[Music]
at some point you have plenty of money
and maybe you think it sends the wrong
message about what the point of doing
mathematics
is um i do find that there's most people
accept
you know most people are given a prize
most people take it i mean people like
to be appreciated but
like i said we're people yes not that
different from most other people
but the important reminder that that
turning down the prize
serves for me it's not that there's
anything wrong with the prize
and there's something wonderful about
the prize i think the no
the nobel prize is trickier because so
many nobel prizes are given
first of all the nobel prize often
forgets many many of the
important people throughout history
second of all there's like these weird
rules to it there's only three people
and some projects have a huge number of
people and it's like this
it um i don't know it it doesn't
kind of highlight the way
science is done on some of these
projects in the best possible way
but in general the prizes are great but
what this kind of teaches me and reminds
me is
sometimes in your life there'll be
moments
when the thing that you
you would really like to do society
would really like you to do
is the thing that goes against something
you believe in
whatever that is some kind of principle
and stand your ground
in the face of that it's something um
i believe most people will have a few
moments like that in their life
maybe one moment like that and you have
to do it that's what integrity is
so like it doesn't have to make sense to
the rest of the world but to stand on
that like
to say no it's interesting because i
think do you know that he turned down
the prize
in service of some principle because i
don't know that
well yes that seems to be the inkling
but he has never made it super clear but
the the inkling is that there he had
some problems with the whole process of
mathematics that
includes awards like this hierarchies
and
reputations and all those kinds of
things and individualism that's
fundamental to american culture
he probably because he visited the
united states quite a bit
that he probably you know
it's it's like all about experiences and
he may have had you know some parts of
academia
some pockets of academia can be less
than inspiring perhaps sometimes
because of the individual egos involved
not academia people in general smart
people with egos
and if they if you interact with a
certain kinds of people you can become
cynical too easily
i'm one of those people that i've been
really fortunate to interact with
incredible people at mit and academia in
general but
i've met some assholes and i tend to
just kind of when i when i
run into difficult folks i just kind of
smile and send them all my love and just
kind of go
go around but for others those
experiences can be sticky
like they can become cynical about the
world
when uh folks like that exist so it's he
he may have uh he may have become a
little bit cynical about the process of
science
well you know it's a good opportunity
let's posit that that's his reasoning
because i truly don't know
um it's an interesting opportunity to go
back to almost the very first thing we
talked about the idea of the
mathematical olympiad because of course
that is so the international
mathematical olympiad is like a
competition
for high school students solving math
problems and
in some sense it's absolutely false to
the reality of mathematics because just
as you say
it is a contest where you win prizes
um the aim is to sort of be faster than
other people
uh and you're working on sort of canned
problems that someone already knows the
answer to like not
problems that are unknown so you know in
my own life
i think when i was in high school i was
like very motivated by those
competitions and like i went to the math
olympiad and
you won it twice and got i mean well
there's something i have to explain to
people because it says i think it says
on wikipedia that i won
a gold medal and in the real olympics
they only give one gold medal
in each event i just have to emphasize
that the international math olympiad
is not like that the gold medal gold
medals are awarded to the top 112th of
all participants
okay so sorry to bust the legend or
anything like well you're an exceptional
performer in terms of
uh achieving high scores on the problems
and they're very difficult
so you've achieved a high level of
performance on the
in this very specialized skill and by
the way it was very it was a very cold
war activity you know when in 1987 the
first year i went it was in havana
americans couldn't go to havana back
then it was a very complicated process
to get there
and they took the whole american team on
a field trip to the museum of american
imperialism
in havana so we could see what america
was all about
how would you recommend a person
learn math so somebody who's young
or somebody my age or somebody older
who've taken a bunch of math but wants
to rediscover the beauty of math
and maybe integrate into their work more
so than the research space
so and and so on is there something you
could say about the
process of uh
incorporating mathematical thinking into
your life
i mean if the thing is it's in part a
journey of self-knowledge
you have to know what's going to work
for you and that's going to be
different for different people so there
are totally people who at any stage of
life
just start reading math textbooks that
is a thing that you can do and it works
for some people and not for others
for others a gateway is you know i
always recommend like
the books of martin gardner or another
sort of person we haven't talked about
but who also
like conway embodies that spirit of play
um he wrote a column in scientific
american for decades called mathematical
recreations and there's such
joy in it and such fun and these books
the columns are collected into books and
the books are old now but for each
generation of people who discover them
they're completely
fresh and they give a totally different
way into the subject than
reading a formal textbook which for some
people would be the right
thing to do and you know working contest
style problems too
those are bound to books like especially
like russian and bulgarian problems
right there's book after book
problems from those contexts that's
going to motivate some people um
for some people it's gonna be like
watching well-produced videos like a
totally different
format like i feel like i'm not
answering your question i'm sort of
saying there's no one answer and like
it's a journey where you figure out what
resonates with you
for some people if the self-discovery is
trying to figure out why is it that i
want to know
okay i'll tell you a story once when i
was in grad school i was very frustrated
with my like lack of knowledge of a lot
of things
as we all are because no matter how much
we know we don't know much more and
going to grad school means just coming
face to face with like the incredible
overflowing fault of your ignorance
right so i told joe harris who was an
algebraic geometer
a professor in my department i was like
i really feel like i don't know enough
and i should just like take a year of
leave and just like read
ega the holy textbook el amon de
geometry algebraic the elements of
algebraic geometry this like
i'm just gonna i i feel like i don't
know enough so i'm just gonna sit and
like read this like
1500 page many volume book
um and he was like and the professor
hair was like that's a really stupid
idea
and i was like why is that a stupid idea
then i would know more algebraic
geometries like because you're not
actually gonna do it like you learn
i mean he knew me well enough to say
like you're gonna learn because you're
gonna be working on a problem and then
there's going to be a fact from ega you
need in order to solve your problem that
you want to solve and that's how you're
going to learn it
you're not going to learn it without a
problem to bring you into it and so for
a lot of people
i think if you're like i'm trying to
understand machine learning and i'm like
i can see that there's sort of some
mathematical technology that i
don't have i think you like let
that problem that you actually care
about
drive your learning i mean one thing
i've learned from advising students you
know
math is really hard in fact anything
that you do
right is hard um
and because it's hard like you might
sort of have some idea that somebody
else gives you oh i should learn
x y and z well if you don't actually
care you're not going to do it you might
feel like you should maybe somebody told
you you should
but i think you have to hook it to
something that you actually care about
so for a lot of people that's the way in
you have an engineering problem you're
trying
to handle you have a physics problem
you're trying to handle
you have a machine learning problem
you're trying to handle let
that not a kind of abstract idea of what
the curriculum is
drive your mathematical learning and
also just as a brief comment
that math is hard there's a sense to
which heart is a feature not a bug
in the sense that again this maybe this
is my own learning
preference but i think it's a value to
fall in love with the process of doing
something hard
overcoming it and becoming a better
person because of it like i hate running
i hate exercise to bring it down to like
the simplest hard
and i enjoy the part once it's done
the person i feel like for the in the
rest of the day once i've accomplished
it the actual process especially the
process of getting started
in the initial like i really i don't
feel like doing it
and i really have the way i feel about
running is the way i feel about
really anything difficult in the
intellectual space
especially mathematics but also
just something that requires like
holding a bunch of concepts in your mind
with some uncertainty like where this
the terminology or the notation is not
very clear
and so you have to kind of hold all
those things together
and like keep pushing forward through
the frustration of really
like obviously not understanding certain
like
parts of the picture like your giant
missing parts of the picture
and still not giving up it's the same
way i feel about running
and and there's something about falling
in love
with the feeling of after you went to
the journey of
not having a complete picture at the end
having a complete picture and then you
get to appreciate the beauty
and just remembering that it sucked for
a long time
and how great it felt when you figured
it out at least at the basic
that's not sort of research thinking
because with research you probably
also have to enjoy the dead ends
with uh with learning math from a
textbook or from video there's a nice
you have to enjoy the dead ends but i
think you have to accept the dead ends
let me
let's put it that way well yeah
enjoy the suffering of it so i i
the way i think about it i do uh there's
an uh i don't enjoy the suffering it
pisses me off but
i accept that it's part of the process
it's interesting there's a lot of ways
to kind of deal with that dead end
um there's a guy who's the ultra
marathon runner navy seal david goggins
who kind of i mean there's a certain
philosophy of like
most people would quit here
and so if most people would quit here
and i don't i'll have an opportunity to
discover something beautiful that others
haven't yet
so like anything any feeling that really
sucks it's like okay most people
would would just like go do something
smarter if i stick with this
um i will discover a new garden of uh
fruit trees that i can pick okay you say
that but like what about the guy who
like wins the nathan's hot dog eating
contest every year like when he eats his
35th hot dog he like correctly says like
okay most people would stop here
like are you like lotting that he's like
no i'm gonna eat the thirty times i am i
am
i am in the in the long arc of history
that man is onto something which brings
up
this question what advice would you give
to young people today
thinking about their career about their
life whether it's in mathematics
uh poetry or hot dog eating cockneys
and you know i have kids so this is
actually a live issue for me right i
actually it's a it's not a photograph i
actually do have to give
advice to two young people all the time
they don't listen but i still give it
um you know one thing i often say to
students i don't think i've actually
said this to my kids yet but i say
to students a lot is you know you come
to these decision points
and everybody is beset by self-doubt
right it's like not sure like what
they're capable of
like not sure what they're what they
really want to do
i always i sort of tell people like
often when you have a decision to make
um one of the choices is the high
self-esteem
choice and i always thought make the
high self-esteem choice make the choice
sort of take yourself out of it and like
if you didn't have those
you can probably figure out what the
version of you feels completely
confident would do
and do that and see what happens and i
think that's often like pretty good
advice that's interesting sort of like
uh
you know like with sims you can create
characters like create
a character of yourself that lacks all
the self-doubt
right but it doesn't mean i would never
say to somebody you should just go have
high self-esteem yeah you shouldn't have
doubts no you probably should have
doubts it's okay to have them
but sometimes it's good to act in the
way that the person who didn't have them
would act um that's a really nice way to
put it
yeah that's a that's a like from a third
person perspective
take the part of your brain that wants
to do big things what would they do
that's not afraid to do those things
what would they do
yeah that's that's really nice that's
actually a really nice way to formulate
it that's very practical advice you
should give it to your kids
do you think there's meaning to any of
it from a mathematical perspective
this life if i were to ask you
we're talking about primes talking about
proving stuff
can we say and then the book that god
has
that mathematics allows us to arrive at
something about
in that book there's certainly a chapter
on the meaning of life in that book
do you think we humans can get to it and
maybe
if you were to write cliff notes what do
you suspect those cliff notes would say
i mean look the way i feel is that you
know mathematics
as we've discussed like it underlies the
way we think about
constructing learning machines and
underlies physics
um it can be used i mean it does all
this stuff
and also you want the meaning of life i
mean it's like we already did a lot for
you like ask a rabbi
[Laughter]
no i mean yeah you know i wrote a lot in
the la in the last book how not to be
wrong
yeah i wrote a lot about pascal a
fascinating guy
um who is a sort of very serious
religious mystic as well as being an
amazing
mathematician and he's well known for
pascal's wager i mean he's probably
among all mathematicians he's the ones
who's best known for this
can you actually like apply mathematics
to kind of
these transcendent questions
um but what's interesting when i really
read pascal
about what he wrote about this you know
i started to see that people often think
oh this is him saying i'm gonna use
mathematics
to sort of show you why you should
believe in god
you know to really that's this
mathematics has the answer to this
question
um but he really doesn't say that he
almost kind of says
the opposite if you ask blaise pascal
like why do you believe in god
it's he'd be like oh because i met god
you know he had this kind of like
psychedelic experience it's like a
mystical experience where
as he tells it he just like directly
encountered god and he's like okay i
guess there's a god i met him last night
so that's
that's it that's why he believed it
didn't have to do with any kind you know
the mathematical argument was like
um about certain reasons for behaving in
a certain way but he basically said like
look like math doesn't tell you that
god's there or not like if god's there
he'll tell you
you know you don't i love this so you ha
you have mathematics you have uh what do
you what do you have
like a ways to explore the mind let's
say psychedelics
you have like incredible technology you
also have
love and friendship and like what what
the hell do you want to know what the
meaning of it all is just enjoy it
i don't think there's a better way to
end it jordan this was a fascinating
conversation i really love
the way you explore math in your writing
the the willingness to be specific
and clear and actually explore difficult
ideas but at the same time
stepping outside and figuring out
beautiful stuff and i love the chart
at the opening of uh your new book
that shows the chaos the mess that is
your mind yes this is what i was trying
to keep in my head
all at once while i was writing and um
i probably should have drawn this
picture earlier in the process maybe it
would have made my organization easier i
actually drew it only at the end
and many of the things we talked about
are on this map
the connections are yet to be fully
dissected and investigated and yes
god is in the picture right on the edge
right on the edge not in the center
thank you so much for talking it is a
huge honor that you would waste your
valuable time with me
thank you like we went to some amazing
places today this is really fun
thanks for listening to this
conversation with jordan ellenberg and
thank you to
secret sauce expressvpn blinkist
and indeed check them out in the
description to support this podcast
and now let me leave you with some words
from jordan in his book how not to be
wrong
knowing mathematics is like wearing a
pair of x-ray specs
that reveal hidden structures underneath
the messy and chaotic surface of the
world
thank you for listening and hope to see
you next time
you