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  • 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

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

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