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 <E1> 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
