Well first of all I want to say that em-- it's such a pleasure to visit again with
the Numberphile family
I haven't seen you guys in a long time so hope you're doing well
I wanted to talk to you about something which I've been thinking about lately
and part of the reason is that um-- I'm teaching this class at UC Berkeley
linear algebra
It has to do with numbers because of course you know, i love numbers
I do numbers for a living I'm a mathematician hello
and I know you guys also like numbers
because you know we are watching Numberphile
But also being a mathematician I think um--
I have a certain vantage point which sort of enables me
to see perhaps better than people who are not professional mathematicians
not only the the uses of of numbers, but also limitations of numbers and this in
particular you know i was interested in this as I was following a debate
a recent debate which many of you may have seen in a-- play out in the media
about artificial intelligence
what do we mean by artificial intelligence
I mean of course we can mean many different things
but essentially we talk-- we're talking about computers right?
We're talking about computers, we're talking about computer programs, we're talking about algorithms
How do they work,? they work with numbers
To me um-- when people say that humans are just specialized computers, and eventually we'll just bid-- build more
and more powerful computer so that eventually they will surpass the power of a human
em-- That kind of line of reasoning to me kind of betrays this idea that
somehow the human is nothing but a machine; the human is nothing but a
sequence of numbers
It is really something which were-- lives and dies by numbers you see so that's
why I'm not suggesting at all that there is nothing [more] to math than numbers.
Of course there are many other things, right
So for example there is geometry and so on, and in fact i will demonstrate now or I will--
I will I hope I will demonstrate (that-- that's my purpose) that in mathematics
there are many things which we often confuse with numbers but which are not
actually numbers.
or they could be represented by numbers but numbers do not really do justice to them
And so would like to eh-- show you this one example which came up
in my linear algebra class, and this has to do with vectors
So look at this brown paper, so it is on this-- on this-- on this table, right?
Imagine that it extends to infinity in all directions
So you can think about as a two-dimensional vector space; let me do-- be a little bit more concrete
Let me take a point, there we go, so this point will be the origin
this will be sort of like the zero point of this vector space, ok,?
and now i want to talk about vectors
And so a vector to me would be something like this, you see, it's an interval which has a length and a direction
and it starts at this origin which I have fixed once and for all-- this is fixed once and for all
Here is another example of a vector and so on, so a vector is right here
So the totality of all vectors is essentially the
totality of all the points of this brown paper right,? because for every point i
can just connect that point to the origin and point to that point
Now, it's not just a static thing; it's not just a collection of vectors, which it is, but there is more
For example, we can add any two vectors to each other, and many of you may know how to do this
It's-- it's called the parallelogram rule sooo that's the intersection point
Okay, very nice but it's not very functional because they are-- yes the
vectors are here they're concrete-- they're kind of concrete
they live here, they have the separation but it's very difficult to work with
them so we try to make it more functional and we do it more functional
by introducing a coordinate system but the coordinate system or in linear algebra
we call it introducing a basis so now we come to the crucial point
ok the crucial point is the basis I want to try to coordinate grid here
that's what I want to do I don't know just gotta coordinate grid and so for
example I can use
well let me use another this color so i will have two coordinate axis so what
them let's say goes like will go like this
my drawing is not particularly perfect you know i have two classes today so
please excuse my my wiggly lines but I hope the point is clear
so usually what we do we say this is x axis and this is y axis with all this
from school and we know this even before we started vectors right with with with
we think about it rather usually as representing points but now i want to
think about more as vectors and then we will see why
another way to think about this to give this two axis is the same as to give
two unit vectors along this axis so one of them would be this one say and let's
go you know I don't want to overload it with notation but it could be or
something and then this would be another basis vector so this is a basis. they're
units so you're the kind of unit relative to something but we'll discuss
this was meant to be unit here's what i can do i can represent now this vector
by a pair of numbers by simply taking the projection onto the x-axis and the
y-axis you see
so this point would be some multiple of this vector so it will be the
corresponding distance here so it looks like three halves right so this looks more
like this looks sort of like 3 halves close to 3/2. it is one and you
have the increment which looks like a kind of close to one and this one looks
like let's make it let me make it longer so it'd be 2. so its kind of like
just kind of rounded up
ok so then what i say is that this v1 can be represented by we usually what we
write it as a column, as a column of numbers
so the first we write the the first coordinate three halves and then we
write this
- that's what we do in linear algebra. but other people sometimes
people write also like this
it's you it's your choice. rather than just floating on the paper now, it kind of
almost has value
that's right it becomes a very concrete thing it becomes a pair of numbers
ok and it's very very efficient because once you have once you have that
so every vector can be written in this form you see my V2 can also be
written as a pair of numbers my V1 plus V2 can be written as the pair of
numbers and then we can work with them because for instance what we are
interested often is some kind of transformation of this plane and we can
feed these vectors this you know this vector representations pairs of numbers
into about those transformations
so this is all great and this is very important to do that to kind of actualize
vectors by numbers so they become actualized you can think also of this by
the way is a kind of a coordinate grid so I impose a coordinate grid
so then each of them is can get an address but this is really isn't it is
the address of this vector with respect to this particular coordinate grid it's
very important to realize that the vector exist even before we introduce a
coordinate grid and think about it like you said the ship exists even before we
look at the relative to an island or another ship or a person and so on
or you know I existed even before I have an address it before you find out what
my home addresses or before I choose my home
you know I already exists and likewise a vector exist even before we introduce
the coordinate grid and it's clear why because look I drew it before i had any
coordinate system I drew it already it was already there on the brown paper
there is no denying the fact that it existed before right
but what I did not impose something on it and the vector if you think about it
vector couldn't care less what we're doing whether it's just sitting there
and enjoying its life or whatever their whatever it is whatever it involves you
know but we came your i came i imposed on
this place I imposed by putting this coordinate system is coordinate grid and
with respect to this coordinate grid
I have now represented that vector by a pair of numbers but very important very
important thing to realize at this point is that I had a choice I had a choice I
could choose this coordinate grid in a different way and this is what i teach
my students in linear algebra i tell them someone else could come and
construct a different coordinate system
you see a different courses or I could change my mind and I could create a
different coordinate system
this is our imagine that these are the two basic basis vectors now going along
the x and y as I drew them originally but now imagine the tip
well it could be like this and they could be like this like that and in
principle you don't you don't even have to be perpendicular to each other as
long as they're not parallel
it is my free will if you wish you know it is my free will is in choosing that
but once you realize that there are many coordinate systems many coordinate
systems and I have a choice of making creating this coordinate system, or someone else
could come you could do it
Brady you could make your own coordinate system and we cannot i cannot convince
you that my concern is better than yours
they all are on equal footing but now because we now realize that this
involves certain choice name is a choice of the coordinate system it becomes very
very clear that this pair of numbers is not the same as the vector and in this
is fine
this is how it works but it's very important to realize that because often
times we hear we get so caught up in this process and we get so excited that
G we can represent a vector by a pair of numbers and we forget the difference and
we start we convince ourselves we start believing we start believing that
actually there is no difference between them but what I'm arguing
is that there is a big difference and that's what I teach my
students and this is very important because you see I mean
let me put it this way if I could ask this vector if this vector could talk
and I could ask this vector
what are your coordinates you know the record we be like, "What?"
what are you talking about? what coordinates he doesn't he or she
you know doesn't know what the coordinates are. The fact that its just there
it just is I came and I try to sort of put it on the box
if you will I try to sign some numbers to it
professor you are talking like a vector is a real thing that we applied an
abstraction on to this is the vector itself an abstraction to start with the
effect is not a real thing
a vector is just as imagined as the coordinate system that you imposed on
yes and no because well you see they are abstractions
we are now in the world of abstraction and my point is precisely that
even in the abstract world of mathematics you have entities you have
things like vectors which are not the same as numbers
how can -- if you appreciate this -- how can you believe that the human being is the
sequence of numbers you see what I mean?
how can you believe that life is an algorithm if you already see in
mathematics in the abstract world of mathematics you find things, which
exist which makes perfect sense
we can work with them like take the sum of two vectors without any reference to
coordinates or anything like this
how can you believe that that is the same as the pair of numbers it's not if you
look closely at how we got that pair of numbers out of a vector you realize that
involved additional choice
so every time we make this procedure we are projecting that vector sort of on to
our particular frame of reference
let's look at this Cup now I can project it onto the plane
ok I can project it onto the plane when I project it onto the plane
I see what do I see I will say disk well with some little thing protruding which
you know obviously but more or less the disc and on the other hand I could
project it onto this whiteboard onto this wall
what will i see well if i put it in a particular way you will just see a
rectangle
ok so let's say you look at this projection
what do you see you see a disk and and or you look here and you see a rectangle
so you might say what is this is this a disk
no is this a parallelogram? again, no. Then someone else could come and say AHA
maybe it is both a disk and a parallelogram
and he or she was still be wrong because this is an entirely different thing you
see
yes I can project it down and i can record the information and it may be
some useful information but it doesn't do justice to this whole thing and
neither does any other projection and likewise with the vector think of a
vector as a cup
it's an object of an entirely different nature than a pair of numbers you can
apply the same technique not only two vectors but to other things related to
this vector space for example what we call linear transformations
a typical example of a linear transformation would be a rotation of
this brown paper around this special point around the central point and then you
know i'm teaching my class and it is a kind of funny things i'm
teaching my class this is textbook which we use and so and they can I get into
this
I get to this point the matrix representation of a linear transformation they
kind of hit me
you know the matrix you know and so of course you know i remember that the
famous movie in this like you know remember the Morpheus was saying to Neo: Do you
want to know what it is and that's exactly what I'm asking right now
do you want to know what it is well the matrix on the one hand is a very
efficient way to package information to convert objects like vectors and linear
transformations into collections of numbers
the menu is not the same as a meal you know you can read the menu you can order
restaurant it can read the menu all you want you can even call the way that come
to your table and explain every ingredient
you can ask the chef to come and explain the process of cooking you can get all
this information but I'm sorry it's not the same as eating that meal eating that
dish right
so it is something like this and my point is that this matrix representation
on the one can be very useful
just like our computers are very useful algorithms are very useful or they can
you know if we forget where they come from where this programs come from
where these numbers come from and when we forget the difference between
the actual things they represent
and the representation that's when we create the kind of matrix that Morpheus
was talking about it you know and Morpheus said you create the you know
the prison for your mind
that's what we do when we forget that difference between the objects
themselves and representations
so my point is let's use that representation
let's use those numbers let's use computers you know to our advantage and
we are using them but let's not forget the difference between the essence of
life so to speak about things which just are which we are trying to represent
and the sequences of numbers which we get as the result of that process of
representation
I get asked about this all the time you know a famous author recently asked me
you know he said
so you're a mathematician would you say that life was an algorithm
you know people ask me or is human just the sequence of zeros and ones you know
you have people like Ray Kurzweil, who believe that they will be able to build
machines so that they could upload their mind and the brain or whatever whatever
they got you know onto those machines