Numbers and Free Will - Numberphile

Instead of just floating on paper now, it almost has value, that's right, it becomes something very concrete, it becomes a couple of numbers, and it's very, very efficient because once you have it, every vector can be written In this way 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 example what interests us many times is some type of transformation of this plane and we can feed these vectors, you know, this vector represents pairs of numbers in those transformations, so this is all cool and it's very important to do that to update the vectors by numbers so that they update.

You can also think of this by way it's some kind of coordinate grid, so I impose a coordinate grid so that each of them can get a direction, but actually it's not, it's the direction of this vector with respect to this particular coordinate grid is very important. Realize that the vector exists even before we introduce a coordinate grid and think about it like you said the ship exists even before we look at the relationship to an island or another ship or a person, etc., or you know that I existed even before I existed. an address before you find out what the address of my house is or before you choose my house, you know that I already exist and in the same way a vector exists before we even enter the coordinate grid and it is clear why, because look, I drew it before I had any coordinate system.

I already drew it, it was already there on the brown paper, there's no denying the fact that it existed before, but I didn't impose anything on it and the vector, if you think about it, the vector doesn't care what we're doing. whether it's just sitting there and enjoying your life or whatever whatever it entails, you know, but we came, I came, I imposed myself on this place, I imposed by putting this coordinate system on a coordinate grid and with regarding this coordinate grid I have Now I represented that vector using a couple of numbers, but something very important to keep in mind 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 that someone else could come along and build a different coordinate system, they'll see different courses, or they could change their mind and they could create a different coordinate system. This is our idea that these are the two basic basis vectors that now go along the x. and and like I drew them originally but now imagine the tip well it could be like this and they could be like this like this and in principle no, they don't even have to be perpendicular to each other while they are.

We are not parallel, it is my

free

will if you want, you know it is my

free

will to choose that, but once you realize that there are many coordinate systems, many coordinate systems and I have the option to create this system of coordinates or someone else. could come, you could do it Brady, you could create your own coordinate system and we can't, I can't convince you that my concern is better than yours, everyone is on equal footing, but now that we realize that this involves some choice , the name is a choice of In the coordinate system, it is very clear that this pair of numbers is not the same as the vector and in this that is fine, that is how it works, but it is very important to realize that many times we hear that we are trapped in this process. and we get so excited that G we can represent a vector by a couple of numbers and we forget about the difference and we start convincing ourselves we start believing we start believing that there's actually 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 your coordinates are We know the record and we're like, "What?" What are you talking about? what coordinates he or she does not know does not know what the coordinates are.

The fact that it's just there, I just come and try to put it in the box, if you like, I try to sign some numbers to it, Professor, you're talking as if a vector is a real thing that we apply an abstraction to. to this is the vector itself an abstraction to begin with the effect is not something real a vector is as imagined as the coordinate system that you imposed yes and no because well you see that they are abstractions now we are in the world of abstraction And my point is precisely that even in the abstract world of mathematics there are entities, there are things like vectors that are not the same as numbers.

How can you, if you appreciate this, how can you believe that the human being is the sequence of numbers, 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 that exist and that make a lot of sense? We can work with them, like taking the sum of two vectors without any reference to coordinates or anything. So how can you believe that that is the same as the pair of numbers? It is not. If you look closely at how we got that pair of numbers from a vector, you realize that that involved an additional choice, so every time we do this procedure we are projecting. that vector in our particular frame of reference, let's look at this cup now I can project it on the plane ok I can project it on the plane when I project it on the plane I see what I see I'll say disk okay with a little thing sticking out that you obviously know but more or less the disk and on the other hand I could project it on this board on this wall which I will see well if I put it in a particular way you will only see a rectangle ok So let's say you look at this projection.

Do you see? You see a disk and/or you look here and you see a rectangle. So you might say: What is this? Is this a disk? Isn't this a parallelogram? Not again. Then someone else might come along and say AHA, maybe it's both a disk and a parallelogram and I'd still be wrong because this is something completely different. Yes, I can project it and I can record the information and it can be. Some useful information, but it doesn't do all this justice and neither does any other projection, and likewise with the vector, think of a vector as a cup, it's an object of a completely different nature to a pair of numbers. , the same can apply. technique not only to 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 center point and then you know. I'm teaching my class and it's kind of fun stuff. I am teaching in my class. This is a textbook that we use and they can get into this. I get to this point, the matrix representation of a linear transformation, kind of hit me. You know the matrix, you know it, and of course you know it. I remember the famous movie in this one, as you know. Remember that Morpheus was saying to Neo: Do ​​you want to know what it is?

And that's exactly what I'm asking you now. you want to know what it is, well the matrix, on the one hand, is a very efficient way of packaging 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 at the restaurant, you can read the menu everything youyou want, you can even call the person who comes to your table and explain each ingredient, you can ask the chef to come and explain the cooking process to you, you can get all this information, but I'm sorry, it's not the same as eating that food, eat that plate correctly, so it's something like this and my point is that this matrix representation can be very useful just like our computers are very useful, algorithms are very useful or they can know. if we forget where they come from, where these programs come from, where these numbers come from and when we forget the difference between the real things that they represent and the representation that's when we create the kind of matrix that Morpheus was talking about, you know, and Morpheus He said that you create the prison for your mind, that's what we do when we forget that difference between objects themselves and representations, so my point is that we use that representation, let's use those numbers, let's use computers, you know, for our benefit and we are using them, but let us not forget the difference between the essence of life, so to speak, about the things that are simply what we. we are trying to represent and the sequences of numbers we get as a result of that representation process.

I get asked about this all the time. You know, a famous author asked me recently. You know what he said. So you are a mathematician. Would you say that? life was an algorithm, you know, people ask me or is it 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 can upload their mind and their brain or whatever. be. Whatever they put you in those machines