[Music]
mathematics is in some ways
kind of scary in how useful it is at
really describing how the universe works
around us
now i mean to give you an idea the
origin of mathematics seems very
straightforward you know we can count on
our fingers up to 10
and maybe it was useful to understand
how many sheep you had so you could
start counting sheep
and then you know you either added or
subtracted sheep as you got more or as
you lost some
it was a simple thing we learned how to
count we learned how to add and subtract
the idea of multiplying and dividing is
a little more abstract but that also
makes
sense know that's something that we can
kind of visualize but then what amazes
me
is that this led us on a tremendously
complicated journey that's still going
on to this day and we had
no idea where this would lead us and
it's a very unfortunate situation when
you can't even begin a conversation
about mathematics without people saying
oh my gosh i don't want to talk about it
and it's kind of strange because
no one would ever say i hate literature
or i hate art or i hate music at least
intelligent people would never say that
it's kind of shameful to say that but
it's perfectly okay in our society to
say i
i hate mathematics and so what i dream
of the society in which
not it's not that everyone has a phd in
mathematics but rather
i would like to live in a society in
which if a mathematics is brought up
someone would say oh mathematics
interesting uh how do i find out more
can you give me a gist of the idea i'm
not scared of it i'm curious about it
the way i'm curious about solar system
about the atoms and the dna
all these things which are in our
collective consciousness in our
public discourse which are no no simpler
than mathematical concepts mathematical
concepts are no more complicated than
these concepts in physics and biology
that are in the air that are that we are
aware of so i would like people to be
aware of this
of this mathematical ideas i'd like to
be more and more aware of how
mathematics invades our lives how it's
controlling our lives
in the 1600s isaac newton asked a simple
question
if an apple falls then does the moon
also fall that is perhaps one of the
greatest questions
ever asked by a member of homo sapiens
since the six million years since we
parted ways
with the apes if an apple falls
does the moon also fall isaac newton
said yes
the moon falls because of the inverse
square law so does an apple
he had a unified theory of the heavens
but he didn't have the mathematics to
solve the falling moon problem
so what did he do he invented calculus
so calculus is a direct consequence of
solving the falling moon
problem i do think that everyone in
america
could benefit from having that
mathematical background in reasoning
just to help everyone make very good
decisions and here i'm distinguishing
already
between math as people usually conceive
of it and
decision making and analysis which is
actually what i think math is
so for example i don't think that being
a math person
means that you can recite the formulas
between sines cosines tangents
and to use logarithms and exponentials
interchangeably
that's not necessarily what i think
everyone should
try to concentrate to understand the
main things to concentrate to understand
are the mathematical principles of
reasoning and i understand why people
are scared and frightened
it's not their fault it's because of how
mathematics is taught
in our schools i think that actually one
reason mathematics is difficult to
understand
is actually because of that network of
prerequisites you see math is one of
these strange subjects
for which the concepts are chained in
sequences of dependencies
when you have long chains there are very
few starting points in mathematics you
have very few that you memorize and the
rest you deduce
as you go through and this chain of
deductions is actually what's critical
now let me contrast that with other
subjects like say history
history doesn't have this long chain in
fact if you
fully understand the war of 1812 that's
great and
it is true that that will influence
perhaps your understanding later of the
women's movement
but it won't be as absolutely
prerequisite
in the sense that if you think about the
concepts i actually think that history
has more concepts than mathematics it's
just that they're spread out broader
and they don't depend on each other as
strongly math
has fewer concepts but they're chained
deeper and because of the way that we
usually learn
when you have deep chains it's very
fragile because you can lose any one
link
meaning if you miss a few concepts along
the chain you can actually be
completely lost now i think that the way
to help to address this
is to provide a way for everyone to
learn at their own pace
and in fact to fill in the holes
whenever they are sensed
and i actually feel like if everyone was
able to pick up every one of those
prerequisites as necessary filling in
any gap they have
mathematics would change from being the
hardest subject
to the easiest subject so after thinking
for some time
i actually came to an idea which was
based on
using these core mathematical areas that
i'd been working with
to actually build a solution for
education that could be delivered for
free
on every smartphone this is actually the
project i'm working on
right now called xp our principle is
that
actually you could turn that smartphone
into a virtual
tutor which automates what a person
would get
if they hired a tutor it's not that the
class comes first
and then the homework and then the exam
the first thing that comes
is the exam essentially followed by
these practice problems which adapt to
you followed by the class for anything
that you don't know
the idea is that this should cure
boredom at the high end
and also cure confusion at the
struggling end some people ask the
question of what
good is math what is the relationship
between
math and physics well sometimes math
leads sometimes physics leads sometimes
they come
together because of course there's a use
for the mathematics
as a theoretical physicist i rely very
much on
calculating to understand
a a result so let's say i want to know
what happens when a big black hole
swallows a little tiny black hole
my starting point will be to think
what's my first mathematical sentence
that i know if i crack it open we'll
answer this question
not even that step can be very hard and
then once i do that i'll be
moving very in a very structured way
through the steps to unlock that when i
work with much younger students
i like to kind of sit back in the room
when we're calculating now and
and ask them to try to find the solution
and in the way that's you know we do
know how to go one step after another
but i'll be trying at the same time to
try to find that
cute way of unpacking it that's highly
non-obvious and sometimes i'm sitting
there for two days
while they're you know generating 12
pages of very
tough beautifully done calculations um
where i'm trying to find a way to try to
do it in a half a page
and sometimes i succeed and sometimes i
don't you know sometimes i can't do that
it's just not possible or i haven't
figured out how to do it
and sometimes i do and when you do it's
not just that it's shorter
it's that you're working with such
bigger structures that you can see so
much more at a time
then here comes einstein asking a
different question
and that is what is the nature and
origin of gravity
einstein said that gravity is nothing
but the byproduct of curved space
so why am i sitting in this chair
a normal person would say i'm sitting in
this chair because gravity
pulls me to the ground but einstein said
no no no no
there's no such thing as gravitational
pull the earth has curved the space
over my head and around my body so space
is pushing me into my chair
so to summarize einstein's theory
gravity does not pull
space pushes but you see the pushing of
the fabric of space and time
requires differential calculus that is
the language of curved surfaces
differential calculus
which you learned in fourth year
calculus so
again here's a situation where math and
physics were very closely combined but
this time math came first the theory
of curved surfaces came first einstein
took that theory of curved surfaces
and then imported it into into
physics if you can do multiplication and
subtraction
it's not too long before you begin to
develop the basic building blocks of
calculus
and calculus describes how moving
objects can
change how things can accelerate if you
want to describe an
apple falling from a tree to the ground
or a ball rolling down a hill
that's calculus it's the the mathematics
of how things can change over time
that's really interesting and the
amazing thing is it works so well
if you use these equations to predict
how a ball will roll down a hill
reality matches that it really does tell
you
how something is going to behave so now
we've gone from counting on our fingers
how many sheep we have
to be able to predict what the universe
around us is going to do
that's incredibly powerful just taking
mammals that
the largest mammal the whale is
in terms of measurable quantities that
is of its physiology and its life
history
is actually a scaled-up version of
the smallest mammal which is actually
the shrew but a mouse
is you know very close to that and
everything in between that they are
scaled versions of one another
the most well known of these is the
scaling of metabolic rate
and metabolic rate is maybe
the most fundamental quantity of life
because metabolic rate simply means
how much energy or maybe just how much
food
does an animal need to eat each day
in order to stay alive um
and uh you know everybody's used to that
but it's familiar with that is sort of
roughly 2
000 food calories a day for a human
being so you can ask how what is that
for different mammals
and what you find is that they're
related to one another
in a very simple way despite the fact
that metabolism
is maybe the most complex physical
chemical process in the universe
it's phenomenal because metabolism is
you know taking essentially almost
inorganic something that's inorganic
and making it into life so that's
and um so here's this extraordinary
complex
process and uh yet it scales in a very
simple way and
and you can express it in english it can
express quite precisely in a math
very simple mathematical equation but in
english
it's roughly speaking that every time
you double
the size of an organism
from say 2 grams to 4 grams or from 20
grams
to 40 grams or 20 kilograms to 40
kilograms
or whatever every just doubling anywhere
instead of what you might naively expect
double the size you double the number of
cells
roughly speaking therefore you would
expect to double
the amount of energy the amount of
metabolic energy you need to keep that
organism alive because you're twice as
many cells
quite contrary you don't need twice as
much
systematically you only need roughly
speaking 75 percent as much
so there's this kind of systematic 25
percent
one-quarter savings
and uh it turns out that
um any anything else you measure
as i mentioned a moment ago scales in a
similar way with this sort of 25 percent
rule
occurring in some interesting ways the
same mathematica and
this is extremely important it's the
same mathematical and physical
principles
applied to a mammal which has
a beating heart as applies to a tree
and a mammal you know our circuitry
system is a bunch of tubes like in your
house the plumbing and this building
that we're sitting in
that's our circuitry system but a tree
and a plant
they're not like that they're a bunch of
fiber bundles
kind of joined together like electrical
cables that spray out and that's what
you see when you see a tree
but even though the and they don't have
beating hearts as we well
know and yet they satisfy the same
mathematical principles and
those mathematical principles give rise
to this quarter power scaling
in mammals but also in plants and trees
but also
in fish and birds and crustacea in
principle and insects and so on and
that's the idea
now we look around us and we see things
like planets orbiting the stars
or the galaxy turning around and we
realize those equations of motion apply
to
everything else in the universe it's not
just here
it's not just on the surface of the
earth but we can look at things
literally billions of light years out in
space
and they're following those same rules
of mathematics
so we keep getting led farther and
farther down this rabbit hole where does
math lead us
now we realize that you can describe
physics incredibly well
if you allow the universe to exist in
many different dimensions more than
three dimensions that we're familiar
with
in fact specifically if you want to do
particle physics it requires
11 dimensions that's not something our
minds comprehend but we can do the math
we can do the math of how things would
behave if they could move in 11
different directions
and it turns out to predict exactly the
results we get from particle physics
that's kind of scary does that mean
that's real are there really 11
dimensions
the math works so well and the
predictions are so strong
that it can't just be nonsense but now
we've gone to the limit of what i can
tell you is is it real or not our math
has given us something
incredibly useful but it's taken us
completely out of our realm of common
sense
of human scale of how our minds work and
even our sense of space and time
i don't think that journey is over yet
where is math going to lead us
it may lead us to understand things like
the universe is a type of a hologram
that was a mathematical solution to how
things work around a black hole
and it works really really well so i
think it's
wonderful and a little bit scary that
you start counting on your fingers
you get to 11 dimensions of space and
time and where else
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