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
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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 stra