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