Name: Numbers and Free Will - Numberphile
Uploaded: 2020-04-13T16:39:03.000Z
Duration: 15 min 13 s
Description: Thousands of YouTube videos with English-Chinese subtitles! Now you can learn to understand native speakers, expand your vocabulary, and improve your pronunciation...

Well first of all I want to say that em-- it's such a pleasure to visit again with

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

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

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

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

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

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 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

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

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

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

- that's what we do in  linear algebra. but other people sometimes

it's you it's your choice. rather than just floating on the paper now, it kind of

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

so this is all great and this is very important to do that to kind of actualize