Des jets aux infiniment petits: l'intuition - rigeur

0:00 I'm sorry to interrupt you, but I have to say that I'm very happy to be here with you today, and I'm very happy to be here with you today, and I'm very happy to be here with you today, and I'm very happy to be here today, Let's say it worked, I think, a little for you as an additional difficulty in my time, so how to explain things quite... And in addition, I had the misfortune of transmitting to him the first exchanges that had taken place with a psychologist who was an exchange... Thank you very much to all of you for your hard work, and I thank you very much to all of you for your hard work, and I thank you very much for your hard work, and I thank you very much for your hard work, I would also like to point out that he made the effort to write the text and to send it to me before the text of his speech, which is iconic. I preferred not to send it to him before, obviously, so that he could read it to us now, but know that it will be accessible and I will send it to everyone. I would like to thank you very much for your time and for intervening on the subject of G, G or G-morphism. I would also like to mention that this afternoon, as usual, the day is extended to a quarter. This time, it is not a mixed seminar, as usual. It is in the school of mathematics for mathematicians and other mathematicians. This is the second season, the second year.

2:30 There is always André who does this work, who has also been a member of the didactic department. It is supported from the point of view of Satchelman, who is a professor of mathematics and mathematics at the University of Paris, so it is also well known in France. Thank you. Well, to explain the problem of this theory of mathematical intuitions, we can start from a mathematical problem, which is the most simple one. If we consider the surface, One of the things we are interested in is, given a point on the surface, to build a tangent plane on this surface. And of course, the tangent plane, all those who have encountered it know that it is calculated at the cost of derivatives. We can look at the tangent, we have here a delta y and a delta x, and the derivative of f compared to x at the point we consider, at this point x0, is therefore, in this case, delta y over delta x. And it is with derivatives, with such operations, that we can calculate the tangent plane to a surface. What do we derive? The equation of the surface. We have a surface. So we write the equation. When we are in space, we have to give x, y, z. And we have an equation of any kind. x squared, y, z, plus 3, x, z, plus x squared, squared, x, squared, y, z. It doesn't matter. You have an equation of any kind. It's the equation of your surface. And it's this equation that you will derive to calculate the tangent plane.

5:00 And so, over the years, mathematicians have managed to do so. And they have managed to do so for the following reasons. If I tell you about a sphere and its tangent plane, I have not given you the equation of the sphere. I have not told you the size of the sphere. I have not told you where it is located in space. If I take a sphere of a different size from another, it is another equation that will have to be derived from the tangent plane. However, when I say the tangent plane has a sphere, you all know what I mean. So mathematicians have long sought to describe a surface and the tangent plane a surface afterwards, without necessarily referring to a dive, whether it is three, four or five dimensions, because when I speak from the right hand side, Is it a plane law? Is it a three-dimensional space law? How can we do this? The idea of the study of the differential variety is based on the following principle. If you are interested in the problem of geography, You can find a path where you don't climb too high because you don't want to suffer too much, etc. You want to know where the city is, you want to know if there is a road that goes from one place to another. Of course, if you don't think too much, the most brutal way to do it is to board a spaceship,

7:30 soak in the sweat, look at the Earth and say, ah, but I don't see the other way. What we do when we are interested in problems like this is to take a geography atlas and to look in this atlas and to review in this atlas all the information. So there has already been an important approach. Simply, we do not look at things, at visions, in great detail. In other words, what is the problem with geography? I simply look at it and I decide to follow this path. If it's a very short scale, at the very least, we can do it. At the very least, if it's not too far from Paris, we can go to the south of the Eiffel Tower to look at it. But in the end, we realize that it's not practical. So we're going to use a map. And a map, what is it? Not a map, it's a set of maps. A part of the surface that we are interested in, in this case the surface of the Earth. And of course, the maps move a little bit, the first map moves a little bit, the second, etc. So I'm not going to worry about the problem of mapping today, of maps moving a little bit, etc. and so on, and to know when we are on the first map if it is the same point as on the second map, and so on. Let's just look at one map of geography and ask ourselves, but what is a map of geography? So I have my surface and I would like to draw a map of geography of the surface or a part of that surface. What is a map? For example, think of the National Geographic Institute's well-designed helmet. What is it? Well, it's a piece of paper, quite flat, about the size of this table, on which there are a lot of indications, a lot of things drawn, including metric indications. You have level curves, you have attitudes. If you observe that the levels are very close, the slope is very narrow at this point. If they are not close at all, it means that the slope is quite soft.

10:00 So by looking at this map, which is quite flat, you can deduce a lot of information about the shape of the head. On a piece of the Earth that you would identify as a piece of a sphere, and by saying, well, there is a piece of a sphere, we have represented it there, well, we represent the radius as well, we really give you the entire surface of the Earth with its radius, that you can deduce, that you can calculate. It is not by taking the angle and measuring that you will know. Because if you measure, you will be able to do something proportional to the scale and you will find that the collater is quite large. So you have to keep in mind that when suddenly the level lines are very close, it means that it goes up and so if you see 1 cm on the map, in fact it was 1 cm, but horizontally, the level lines are close, it goes up. So you have a full metric indication. They allow us to reconstruct exactly what is on the surface, and of course, each point on the map corresponds to a point on the surface. A point on the map, let's say, is a point of the plan. These are the two coordinates, x and y. When we learned about dv, it is the tradition that sells in geometry. We have a certain function that tells us that we have here a point f of uv that belongs to the surface. So we have a function f which has two coordinates. Think of the length and the latitude. Knowing the length and the latitude, well, it describes a point of the surface. And then, in addition, here, there is a set of metric indications. Think of the levels, think of anything you want. A set of metric indications is called the Riemann metric. And so it allows us to reconstruct everything we want on the surface. This example, of course, this approach of what could be a representation of a sphere of space is obviously a very particular one. It is the idea that we will be able to reconstruct things exactly with metric indications. There are obviously much more general situations where we do things only in topological matters, without worrying about metrics.

12:30 But hey, it doesn't matter for the math. A local map of a surface, something that describes a piece of the surface, is a piece of white paper with a function that, at each point of this piece, associates a point of the surface and then a bunch of metric indications that will allow us to recalculate what is happening on the surface. We have these metric indications. So, what we wanted to do was talk about the white paper and, in fact, when we study surfaces, It is also important to see how the tangent plane will vary when we walk along it, and certainly you have already seen this example. Here I turn around the tangent plane, then I turn around the ribbon, and I come back and I fall on the tangent plane at the starting point. On the other hand, you certainly already have there too, well, there are some things that I have already described. Look at the ribbon of the disc. I don't know how to describe it, I don't know how to describe it very well. It's the one where we take this and instead of looking at it, we contour it, we see the door. And there, we have the tangent plane that we did once all around. Well, the tangent plane was found turned, I would say, far away. So we are interested in educating tangent spaces like that on a surface to derive information about the nature of the surface. And we would like to educate that. From our map, because if our idea is that we can define a sphere by saying that it is a piece of a sphere, it is quite simply like that, a piece of a plant, and I give you metric indications that will tell you that, for example, when you are here and you take two things that are one centimeter apart on your map, and when you take two things that are one centimeter apart, the distance in reality is not the same. For this sphere, I had to pull on it as if it were elastic to flatten it, and so on the edges, I had to extend this edge.

15:00 So these are my metric indications that allow me to say that when we measure a centimeter, I have indications that tell me that this is not a centimeter in reality, and that's why I can only know that it is a sphere. So what about the tangent plane? What is the tangent plane? Well, we would like to say that the tangent plane is when we are in a plane... So, the tangent of all tangent vectors in the image. Tangent of all tangent vectors on the surface in this picture. Not so much, because the tangent vectors are not in the surface, they are outside the surface, as the tangent plane is called. That's my problem. I want to define the tangent plane from my map. My map represents the surface, but the tangent plane is not in the surface, it is outside. The tangent vectors are outside, so they are not represented in the map, they are outside. Yes, but a tangent vector is a tangent vector. Each vector has a curve, I can always find a curve, and this vector is a tangent vector to a curve. And this curve here, this curve here, I can represent it in my chart. And of course, if I take this vector here, I take the tangent vector. So this given tangent vector, this given curve, is about the same. Oh yes, oh yes, but there are not a lot of curves on the surface that have the same vector. There are three curves, let's take another curve that has the same tangent vector.

17:30 These two curves are big, they just touch the surface. They will be tangent on the surface. They will give rise to the tangent vector on the surface. These representations here are tangent, because it goes without saying that all the functions that I consider today are good functions. So in particular, all the functions are neutral, etc. I will not specify each time. I am constantly using the truth, so it goes without saying that all my functions are neutral as much as I need them. Admit that without this, the map would really be of a mediocre quality, two curves are tangent on the surface, if and only if their representations are tangent in my map of the state. So what is a curve? If I have here a curve, to give a curve, we give each other an integral. On the other hand, the union gives itself a certain curve. Here I had my function f, the integral gives itself a certain function g, I had an integral. Here, I have my map, my function g, it will deform this interval into a certain curve in the plane, and then I deform all this by f, it gives me a surface, and it gives me others on the surface. And so, I will say that the two curves, in two curves, two curves, lg, are equivalent, if and only if they have the same tangent vector. If they all have the same tangent vector, if I put them in the same package of courses that have the same tangent vector, it will give me the same information as the tangent vector.

20:00 If I want to give all the tangent vectors, I have to give them all the packages of courses that have the tangent vector, the tangent plan. For example, I have a set of equivalent curves, and for each set of equivalent curves, I have a tangent vector, so we have a good bijection between the two, I no longer have the number of elements from one part to the other, and so I could define the tangent plane as the set of equivalent curves, which we call equivalence classes, for the equivalence relation that I have found. And so it would be a way of doing it, but not really. What do I use to define the equivalence? I use the tangent vector. And what was I trying to define? Ah, I was trying to define what a tangent vector is. So I use the tangent vector to define what a tangent vector is. It's not very good. But how do we do it? The two curves over there are tangent, precisely because their representations are tangent. So I can take the whole package. They are equal. This is the plan. Finally, if I want to say that the two curves on the surface, the two curves on the surface, the two curves drawn in the plane, and then we define them, so I have as many curves in the plane as I have on the surface, it becomes the same, and say that they are tangent over there and that they are tangent here. So, I have as many pairs of curves at this level as I have at the same notion of dimensions.

22:30 Come on, let's try to rewrite all this. The g-curve and the h-curve that I have drawn on my map of geography in this very equivalent way, if the g-derivative, ah, in relation to the parameters, so let's see how this parameter, let's put the parameter in the same room, if the g-derivative in relation to d equals the h-derivative in relation to d, the tangent vector, as we saw earlier, the tangent is given by the derivative. Where do I derive this? Well, I derive it from the point I'm interested in. It's at this point that I'm interested in what's going on elsewhere, but others are different. It's at this point. So, let's simplify the list a bit. Without further ado, a general term. Our interval does not prevent us from saying that we are going to take an interval that, for example, is less a real number than a real number. And that the point we are interested in is the point obtained for the parameter zero, so that we do not always have to say, ah, the point obtained for the value at the time of the parameter. This is simply to translate a little the domain of definition. So we will suppose that we have an integral that contains zero and that the point we are interested in is obtained for the parameter zero. It is simply a small change in the probability. This is what always happens. These are two curves that I have drawn in my notebook and I would say that these two curves are equivalent in this common point G0 to G0 in which I want to calculate the length of the leg. It is only if they have the same derivative in this point. I have a relationship of equivalence. There are also equivalence classes, i.e. a set of equivalence codes.

25:00 This is what we call the white-and-white sequence. An equivalence class of these equivalence classes is what we call a check. It is a class of equivalence of codes, like this, in one. This is what we call a check. It is a notion that is given to Chapelsman, who introduced this notion. At the very beginning of the 1950s, I was in the rush to study singularities, to study surface advantages and to study in particular the singularities that can be presented on surfaces. Of course, here I am in a very, very, very particular case of G. I am in the case that Erosma calls a 1-G, because I used the first derivatives. And Roslam is also interested in more demanding equivalence relations, where he will say that the two curves G and H are equivalent, not only because they have the same first derivative, but the first derivative, the second derivative, the third and the fourth also coincide. We call this a four-object, the four first derivatives are the same. So at this point, the curves are more than just tangents, if you will, at the second degree, at the third, at the fourth, etc. We can't distinguish them yet. And so he uses D, R, G. And of course, he does it not only for surfaces, but he does it for varieties of any dimension. A variety of dimension 1 is a curve, a variety of dimension 2 is a surface, and then you could have a volume. For example, I would define in a space a higher dimension, a variety of three dimensions, etc. So he does this in any dimension and with the consideration of any number of fields. But today, I will limit myself to studying the Gets in the sense that I just described.

27:30 This notion of the Gets of Habermas, is what is underlying to a notion of infinitely small. There is a theory of the infinitely small that was introduced by the American mathematician, who, I think, had a lecture at the Ecole Normale not very long ago. About a week ago, I think. About a week ago, yes. About a week ago, yes. I came to the class of Jean-Michel Aurelien. Well, we have two curves, G and H, which define the same G. We know that we can look at this in the map that is about to go on the surface. They define the same object at the same time. At this point, there are two different courses. You see them. Yes, well, let's take this. Different courses, you can see it. There are two different courses. Let's work on this interval. Sorry, you are sitting, you can no longer see the difference between the two. I am very close, I can no longer see the difference between the two. You can see it. Let's take another interval, even smaller. Well, the memory, as you can see on the table, I don't see the difference between them. The idea that mathematicians have is that if these two curves are tangent, the closer we get to the point of tangency, the more they are indistinguishable. If I look at my interval , and I take a very small r,

30:00 they become almost indistinguishable, almost indistinguishable. If R is infinitely small, if R is infinitely small, when I take the picture of infinitely small, then they are there. So, what does infinitely small mean? What does infinitely small mean? Real numbers, well, you know real numbers. There are some that are really infinitely small. It's zero. Apart from that, the zeros are small. So, the idea is to give a sentence. And when we think about the theory of scales, well, there are many ways, many theories of infinitesimal entities. Our non-standard analysis based on a theory of infinitesimal entities, well, it's made up of big theoretical theories, model theories, to build the extensions of the scales of infinitesimal entities. But here we are going to focus on the local thing, which we have already discussed. And here, I have t squared, when x is equal to t, y is equal to t squared. So this is the curve, my curve g, this is the curve which, therefore, takes t over t, t squared when x is equal to t, y is equal to t squared.

32:30 And I can calculate the vector of this at the origin. t squared over t is the derivative of 1, and the derivative of t squared is 2t, which is z. So I can calculate on dt, I have to derive 0, my dh on dt, at the point 0, it's a constant vector, it's always the vector 1, 0. So my curve of g and my curve of h have the same vector. So if for a moment, if for a few seconds, we played this absurd game of believing that there are infinite number of variables, If we pretend that there are infinite real numbers, if we pretend to believe it, if we pretend to believe it, what do we do? We must say that for t infinitely small, we have t because the two curves have the same tangent vector, so when I take the t smaller and smaller, they look more and more alike, and when t is infinitely small, they become the same.

35:00 For t infinitely small, the two are equal. The first component is equal to d squared, and the second component is equal to d squared. So when we have a number that is infinitely small, its square is null. But the idea is to avoid the following elements. And if we think about it, if you take a very small number, a thousandth, you raise it to the square, it's a millionth. It makes you 10 minus 12. Even your calculator will tell you that it makes you 0. In the past, mathematicians were told that a negative number is not a square root. It's positive, so a negative number is not a square root. We can't go against that. Yes, a negative number is a real square root.

37:30 But complex numbers are used every day. Mathematicians and physicists must use complex numbers every day, and also have square roots of negative numbers. Square roots of negative numbers are no longer real numbers, they are something else. Real numbers are not square roots, they are in an extension of reality. The question that is naturally asked in the book is to say, what is in the world of mathematics? One of the ways that we had real numbers and that we extended that to make them complex numbers, and now we have recovered square roots and negative numbers, could we not have real numbers to extend that and have a universe in which there are numbers, in which the square works with that? We ask ourselves the question that we have the real number of everyone, those of us who have all of us, and now we know that there is no real number in the square of zero. Something bigger, as we have done for the texts, something bigger, in which there are elements of the square of zero, which is precisely the role of the infinitely small, and which would make me say that two conditions are tangent if they are exact, if and only if they are equal on the infinitely small. To have real numbers that are infinitely small, even the extensions of infinitely small things, even if we think of complex numbers, if the square of a complex number is null, the complex number is null in the page. So, could it exist? You know, we are used to, in mathematics, to manipulate a lot of null square objects.

40:00 I suppose many of you have had the opportunity to manipulate matrices. If you take matrices, even already matrices, Two lines of Newton, if you know this matrix, and can be multiplied by n. So, elements whose square is null, when they are null, are often used in mathematics. For example, we have this mathematical calculation, and we know that this mathematical product is null, because if you multiply 0 by r, it becomes 0, and if you multiply r by 0, it becomes 0. I give them just to convince you that you should not be afraid to have square-null elements in certain contexts other than the number of them. The example I would like to give you of a situation where we have square-null elements is relatively different and much closer to what interests us. So you know what a polynomial is. A plus a 0x plus a 1x plus a 0x plus a 0x plus a 0x plus a 0x plus a 0x plus a 0x plus a 0x plus a 0x plus a 0x plus a 0x plus a 0x plus a 0x plus a 0x plus a 0x plus a 0x plus a 0x plus a 0x plus a 0x plus a 0x plus a 0x plus a 0x plus a 0x plus a 0x plus a 0x In addition to stress, it is easy to multiply without a lot of work, we always find ourselves somewhere in the calculation, we will start three times before finding the right answer, but in the end we can do all this. Well, I will now be interested in pretending that x² is equal to 0. But is it that absurd?

42:30 Let's take another example to convince us that there are really situations that answer extremely precise questions Squares, including the square in a real world. If you go to the supermarket and you buy a plastic box, you try very hard to find an electronic calculator in it, and you find anything with a plastic box. Well, in general, it's not a calculator. These are calculators and they are very useful. Because, well, mental calculators at some point are important. And so we like to take a calculator to make calculators. Find a calculator that works, let's say, with Wichita, so it can also help you to give a decimal. Give a millionth, that's a millionth. Find a calculator that works with a decimal, a millionth, that's it. A calculator, a millionth times a millionth. A millionth times a millionth, it doesn't matter, because it works. When it does an operation, it stops at the eighth decimal because it's done. Mathematics is very useful for multiplication and division. Mathematics is very useful for multiplication and division. It all depends on precision. A calculator like this, even in a powder box, does not work randomly. It has to work to calculate in such and such a way.

45:00 It depends on precision, and it works with precision, and it gives you the result. It is not the mathematics that you would like, but in principle, according to what you do. Here is the result. A calculator that makes x² equal to zero means that it can also make x³ equal to zero, because x³ is x² times x, so it's zero times x, so it will be zero. If I make x² equal to zero, x³ will be equal to zero. And I can continue like this, x² will be x² times x², so it will be zero times x², so it will be zero. x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12, x13, x14, x15, x16, x17, x18, x19, x20, x21, x22, x23, x24, x25, x26, x27, x28, x29, x30, x31, x32, x33, x34, x35, x36, x37, x38, x39, x40, x41, x42, x43, x44, x44, x45, x46, x47, x48, x49, x49, x49, x50, x51, x52, x52, x53, x53, x54, x55, x56, x56, x57, x57, x58, x58, x58, x59, x59, x59, x59, x60, x60, x60, x This is a polynome calculator that is used to work with polynomes of the first degree. I say to myself, if I were to do X squared equals zero, I could still be a little more reasonable. I say to myself, I work with polynomes of the first degree, 1. When I close my brain, I say to myself, I work with polynomes of X squared equals zero. In the end, I am interested in working in the sense that sometimes I do not understand. I explain to myself that I have to work with polynomes of the first degree. The polynomial calculator only knows how to work with these polynomials. It doesn't have any more terms in its cases. And so, we visualize and we subtract. If we do AXV plus CX plus D, we get A plus CX plus B plus D.

47:30 And if we do a minus, we subtract a minus. And so we add and subtract that very easily. And then I call it the multiplier matrix. times, that makes AC times X squared, plus AD plus BC times X, plus B. The polynomial matrix that I found in the ABCD paper, it refers only to the level. So what will she give me as a result, my calculator arm, when I do a millionth square, you give me zero? Well, here, my calculator, it's going to give me that. And she's going to tell me that the product of these two polygons is equal to 100. It's not so great that when you do a millionth square with your calculator, it gives you zero. So if we think about it, great or not great? If I am interested in all the components of the first law, what forbids me to define... Three operations on the first degree polynomials. An operation I call plus, which is the parallel here, an operation I call minus, and an operation I call times. It's not the multiplication that makes the polynomials, but I introduce a multiplication on the polynomials of the first degree. A multiplication. Write it down as you like, write it down as you like, but I introduce an operation. There are two polynomials of the first degree, not this one. If I want to call the product of these two polygons, even if it is not the product of these two polygons, I have a product of an unusual multiplication on these polygons of the first two. This is legitimate. This is a natural mathematical approach. We have a set of elements that define the operations of a very rigorous matrix on these elements.

50:00 The product of these two polygons is not the definition of this. This is obviously not the usual product we know. So one day, he said, well, it's an operation that has two polygons, the first one and the second one, but there are also three of them. You say, well, if that's how it is, it's like with the calculator, that's more algebraic than logic. I want to quote all the rules, all the equalities that you know between the three algebraic operations, plus, minus, and times, all these equalities that we can write, like a plus b. times c is equal to a times c plus b times c times a times b times c is equal to a times b times c times a. All the rules that can bind the addition and multiplication and the subtraction, all these rules, contrary to what happens with the calculator that has only decimals, well, all these rules are still all satisfied for the operations that are here. That's a remarkable property. All the usual properties written by rigidity between addition, multiplication and subtraction, all those of the set of laws that are valid for the non-francés, well, all these are all valid for this bizarre operation that I have built. Of course, mathematicians know what I did. I took the ring of the volumens and I made the quotient of the ring of the volumens by what is called the principal ideal injected by the non-francés. I did an algebra quotient, and when you do an algebra quotient of a mathematical manoeuvre, it turns out to be something more analytical. But no, that's what I did. I built a mathematical structure with always my three operations, and which always responds to all the visual actions that we are used to for these three operations, which is not the case for the calculator of the case of all of this. That thing was more satisfying. We had one on one side, which was not the other. But here, everything remains satisfied. Everything remains satisfied, and of course, in A, x squared, x squared, and so on.

52:30 So I write my multiplication, not x squared, I simply have x squared that stops. In this set A, where I made special rules and equations, now, indeed, my polynomial x, which is a polynomial that is not double at all, When I write them in the square, for this multiplication, I write them in the square. So this kind of thing can exist. I will choose, I take the real numbers in the sensual sense, and I would like to find an extension. Of the greatest, there is one which contains the most of them all, and which has such a structure, that it is made of what we call a model, that is to say, it must have operations plus, minus, and times, it must have an addition, a subtraction, a multiplication, which will apply to all the actions, I can say, by means of equality between these three operations, all the actions will apply to the equalities that these three operations will apply. We are going to look for an extension of the numbers where we always have these three operations, plus, minus, times. You will tell me, why not division? If I really want to have something where there are infinitely small, if I want to divide by an infinitely small number, I should also introduce infinitely large numbers. If we do not want to use non-null real numbers by a non-null real number, we will try to use them by an infinitely large number.

55:00 So I have an anode, well, this anode interests me, it's the notation of the field, it's the set of powers, not because of numbers, it's an anode R, such as R square, equals 0, that's legitimate, I have an anode and I take the set of square elements, there are some or there are none, well, there are 0, in any case, there may be others, that I can do on my side. In the example I mentioned earlier, the first degree polygons with a strong multiplication, don't forget that the first degree polygons were contained, or if there was a real number, R, here is the first degree polynomial. So my first degree polynomial that I gave before is already a polynomial that contains reals, and in which there are elements of the square of the bubble as we have seen. The polynomial of the bubble is the square of the bubble. It contains all real numbers that must be identified at the first polynomial. The operations that I define on the polynomials are exactly, in the case of real numbers, the original operations. If you add, subtract or multiply, including in an unusual way, zero and x plus r, these are original operations. So, indeed, there are allusions. We ask for the real ones. I gave you one just before. In which there are elements of caricature. And I want to call them infinitely small. So far, this book has been called like that. If you want to call them differently, call them differently. This is simply a question of terminology. Well, I choose infinitely small names. Obviously, with that, we do not go very far. And to open it in collaboration with Scott, an academician of mine, we can work on it.

57:30 And we say, here it is. In order for things to work well, we need an axiom that we can take from any other allotment. In the same way that when we want to make square roots of negative numbers, we take complex ones, we don't take anything, we take something with negative numbers, we put them in square roots. Here, we take an r, and we want, indeed, that two functions are tangent when they are equal to the infinitives, and we say, that's what we want. That is to say, we want a function to be discernible from its tangent on the infinitesimals. Two functions that are tangent and indiscernible, so in particular a function and its tangent become indiscernible on the infinitesimals. What does this mean? Well, it means, I forgot the words, if you take a function defined on the infinitesimals, this function is indiscernible from its tangent. All functions are equal to their tangent. If the function is equal to its tangent, the tangent is a right, it is a function of the first two. So, on infinitely many objects, any function is equal to its tangent, so any function is a first two. This is the theory of Bohr. And so the idea is that we will develop the mathematics. We will develop geometry, we will develop mathematics by working with the immersion of N and L in a very long period of time without taking action. And of course, in this case, it is clear that the derivative of F in relation to T can be obtained. The function of the number a, if it is equal to that, then the derivative of the left member should be the derivative of the right member, and the derivative of the right member is the derivative of the function of the integral, it is simple to understand. So we have the notion of derivative, we have the function, we have the notion of derivative, we have to be able to develop all the analysis and all the differential geometry.

1:00:00 We have done that with complex numbers to find the square root of the number a. If you don't want to give an example of such a ring that satisfies this action, well, to put you at my speed, I'm going to show you that rings like that don't exist. Well, they don't exist. Demonstration. Take the function f , which is equal to zero, where 1 is different from 0. So, what do we have here? We have A T plus 8, where T is infinitely high, as we are told. S of 0 equals A times B plus 5. So, I already deduce that B is necessarily equal to 0. So, if I take my T, which is infinitely high, 2 T squared equals 4 times T squared, which equals 4 times O, which equals 5. All these terms also belong to the minimum value of t.

1:02:30 If t is infinitely small, it is square root of t. 2t is still an infinitely small number, so 2t, f of 2t is equal to a. 2t is a times a squared plus b, and b is negative. So we do 2 times 1, that's it. So 2at, at, at is f of t since b is equal to 0, and f of t since t is equal to 0, that's it. So what do we find? So I have come to the conclusion that 1 is equal to 2. If there is ever an element that is infinitely small, non-null, you arrive at the conclusion that the real number 1 is equal to the real number 2, in the sense that the real number is non-existent. Now look at what we have here. We said that t is equal to, or that t is different, d, that the number is 2, yes, because in logic we call it a third excuse.

1:05:00 One thing is true or the other, or the third excuse, or the other. The number is 2, or there is no other, there is no other possible. This is the theoretical approach to all mathematicians, especially to the man of the street, the man of the street. Excuse me, if I say to the man of the street, in France, it's beautiful. Yes, some days, especially in the south, it's more beautiful than in the north. He will not say in France it's beautiful, or in France it's not beautiful. Yes or no, it's the other way around. It's true some days in some places, it's more true in some places than in others, but finally, the value of truth in France, if it's true, it's not true, it's false. It's something false. Well, it's a little bit this third-party law that we have to build a counter-example. We started from the idea that a number is a place where it is not. It seems so obvious to us, but when we think of Mr. Dumont's logic, it is clear that we do not have this kind of thing. It's a little more subtle than that. Mathematics and logic, in which we refuse to see a third party, a theorist, together with all the visual rules of logic, are the third party. The difference is essentially that we cannot do things like that, that we cannot do demonstrations by the absurd. A demonstration by the absurd is what? It is to use the third party. The proposition is true or it is false. And we will demonstrate which case is false.

1:07:30 We have a contradiction. Is it true or is it not possible for it to be false? It is true. I did not build an argument showing that it could not be false. They wanted to do constructive mathematics. If I want to show that something is true, I have to really give an argument proving that it is true. If I want to show that something exists, I have to build it. I cannot pretend that something exists simply by saying, well, you know, if it did not exist, we would have problems. No, if I want to prove that they exist, then I have to construct them. This is what we call intuitionist mathematics. And it is well known that within the most usual classical mathematics, the one that everyone uses, the one that uses excluded thirds, there are intuitionist models of the theory of ensembles. And one of the examples, since we are talking about Leibniz, These are the topologists of the common topos, that is, the topos of beams, or beams. May I ask a question on this part? Can we, instead of touching on logic, say, well, we have to change logic and just say that its function does not work, because its function is not differentiable, that's it, and that's what gives existence, I don't know why, we don't know. Yes, there is no way to define differentiable without turning it around. Yes, on the infinitely small, the idea is that on the infinitely small, it is a bit what we have just shown, a function will not be able to have a jump, all functions will have to be differentiable on the infinitely small. In this theory, all functions are differentiable.

1:10:00 We are going to understand why, in the logic of the beams, the stone is excluded, it is not true. We are going to take a very simple example of a beam. We are going to take the beams on the usual plane. Astrology, I want to do it in a very particular way, so I'm going to work on the usual plane. And I'm going to define a beam on the usual plane. On the usual plane, there are parts that take a part of the plane. This part will have a border. If the border is part of U, it is closed, and if there is no point of the border in U, U is open. Since we have open and closed intergrals, we can have open and closed parts in the French plan. An open part is a part that contains no point of the border. What is a beam? But a set that varies according to a planet. We could imagine a set that varies according to time. We prefer the mathematics of a rigid set, but of a set that can evolve. And so, where a phenomenon can be true at a given moment, it can be true at another moment. You remember it. It evolves. Well, our beams on the plane are sets that evolve according to the opening on which we fall. We must immediately understand why these things exist. What are surfaces? Well, I'm going to take surfaces. What surfaces? We're going to take relatively simple surfaces. Surfaces that are of the equation z equals f of x square. That's three-dimensional.

1:12:30 Which are of the equation z equals f of x square. For example, take the surface z is equal to x squared plus f squared. That's parabolic. This surface... It is defined for all x, for all y. But take now the surface z is equal to 1 over x, which is 1 over y. Well, it is defined on this note, minus the x axis. When x equals 0, when y equals 0, it doesn't work this way. So this surface is not defined everywhere. It is defined only on a piece. And of course, you can imagine a lot of other surfaces that are not defined as, for example, when x is positive or things like that. You can imagine a lot of surfaces. And so when I'm going to talk about the surfaces of the equation z equals f of x, y, well, I also want to be able to talk about this one. I want to be able to talk about a lot of surfaces that may not be defined everywhere. It feels a little more curious to say that the whole of surfaces, which are defined, we don't know too well, it would still be much more pertinent to say that I'm going to take my fresco so that I have an open U. They define these surfaces, because I can look at the surfaces that are defined on this group. And as we have seen, there are surfaces that are defined on a group, they are not defined on something bigger, there are points, it does not work. And so this idea of a stream, of a grain, of an ensemble, is a family of ensembles.

1:15:00 And these families of ensembles have, of course, properties, they can be defined on a large group, or they can be defined on a very small group. If I am defined on a sphere, and if I am defined on a sphere on the side, and that on the intersection, the two are equal, it coincides, it's the same surface. We can look at the properties of these things, of course, but it is the idea of working with surface beams. And then, all these beams, it constitutes what we call a topos of beams. And we can demonstrate that a topos of beams always has an intrinsic logic. This is an intuitionist model of the theory of ensembles. It tells me that if I want to demonstrate something in a proposal, which one is it? This one or another? I can only demonstrate in ensembles by using only the rules of the intuitionist logic. I demonstrate in ensembles by forbidding the use of the third excuse. I demonstrate in ensembles by forbidding the use of the third excuse. The result is automatically true in any theory. The topos is an inductionist model of the theory of the universe, so it is a very strong result that I do not even want to give you the demonstration here, but I would like to show you the logic of my stream that is here. The logic, in the particular case that is here on the video, the intrinsic logic of this topos is something well known, it is what we call, it is reduced to what we call local logic. So, phi is real, cos phi is non-zero. In the logic of this cos, if and only if, phi is real. This is what we call local.

1:17:30 That is, phi is real classically, in the most classic sense of the term, on an adjacent plane, on the edge of the plane. We understand that it is possible. We look in our sphere of surfaces. And I am interested in saying that xy is equal to 0.0 xy on the surface, and 0.0 xy is not on the surface. This is a denunciation. Classically, this denunciation is true for all x and y. xy is good or not good. Classically, this denunciation is true. In the local logic, this denunciation is false. Thank you for watching this video.

1:20:00 But it is not null on an adjacent point. It is the only point where it is null and all around it is nothing different from zero. So it is not locally true. There is at least one point where on the adjacent of this point, I can not say that it is different from zero because the point is null and around it is not null. So here at the adjacent of the summit, neither of the two properties is satisfied. It is neither different from zero everywhere nor equal to zero everywhere. So the probability is false locally, while Euth, well, it's the third section, which is well known in Mexico. Look at other quantities that are not bound by a number called x squared, y squared. When x, y becomes large, you can take values ​​as large as you want. Each point, each point on the surface, a value of x, y at the base, at this point, well, I take the base. All of this is bordered by an entire number. So, the neighborhood of each point, for each point, we take a neighborhood that is not too large, and it is bordered by an entire number. So this property is true locally. Locally, the function is always bordered by an entire number. The number does not change. So you see this local logic of saying, well, I look at the formula in the non-physical sense, but I would say that it is totally true when it is true in the classical sense of the term, on all points of the equation. And this for each point. Well, we see that in some cases, things do not work. But there are other properties that are classically wrong and locally wrong. How can we build such a universe in which logic will be an idiosyncratic, will refuse the T.S.S.C. and therefore the demonstration that I gave you that there was no new sense of cohomology, the demonstration did not work because it was based fundamentally on the T.S.S.C.

1:22:30 We have to build a real model, an intuitionist model, of real numbers in which we have infinitely good answers, in the sense of coques and parabolas, and so on. It is a set in which intuition, multiplication and subtraction are generated with the visual actions of these arithmetic laws. I will automatically have an operation that I still want from our degree on the same level that I had an addition, a subtraction, a multiplication. Each polynomial has an operation of its own. Why? Because when I take A, I can make what I want to call D of A, which will simply be two times the distance. A will be A times A times A, which is A squared plus A times A times A. That's what everyone wants to call two times A squared, right? So I do A times A times A, multiplication, then A times A times A, and then I write it down.

1:25:00 I want to call it two times A squared. I want to call it minus two. So I have a 2a1-3a2 plus a-2 with additions, subtractions and multiplications. Each time I am given a full-fledged polynome, I can evaluate this polynome with any element in it. So we could say that one to the other, one to the other, is not something with an addition, multiplication or subtraction, A whole coefficient, a whole polynomial with a whole coefficient, well, we have an operation that will show us what kind of equation we have, because we have all the equalities that we have. So an halo is a set in which we can perform any polynomial with a whole coefficient. Among all these polynomials, there will be the square polynomial, and of course, more generally, for a function. There is a polynome with several variables. You can do anything. You can construct anything. There is a polynome with n variables. There is a polynome with n variables. There is a polynome with n variables. So, an anode is a set under which any polynome with n variables and coefficients can be used in the world. The definition of an anode is this. I give you this. This is to give the definition of what we call a set of algebras.

1:27:30 It is a set of algebras for all volumes, but more generally for all real functions. Each time we have a variable polynomial, I give myself the polynomial operation that exists on the model. Here, the idea would be to do the same thing, but not only with the polynomes, but with all the functions of the class C1. It would only be limited to the polynomes, because what we would find would be exactly the inputs. The inputs are the sets that we can put a polynomial in. Here, I am interested in situations where I can perform any infinitely derivative function, plus, of course, the axioms. What are the axioms? Well, it's all the elements unsatisfactory for infinitely derivative functions, or in fact unsatisfactory in the normal sense. That is to say, first of all, all the infinitely derivative functions, there, the axiom is a barrier that happens in three dimensions. So I can define the functions, since I am in a three-dimensional space, in a real space, I can define the derivatives, etc.

1:30:00 And I take all the functions of S towards the real numbers, which are infinitely derivable from the surface. So I take all the functions of the sphere towards the real, which are infinitely derivable from the surface. Well, this is my A. I claim that it is an algebraic series. It's very easy. If you have S of Rn in A, that is to say that if I take a function here, I should deduce something like this. Then what are the h i? Well, they are n functions of S in R. So I can make the function of S of Rn, which is x, over h1 of x, a function in Rn. And then, from Rn to R, I have my condition F, from S to R, which is R to H1 to X1. So I have, on these sets, I have a structure of C-infinite to G, every time I have an operation of Rn to R. Well, I get an operation corresponding to this set. So, the surfaces, I can see them, I can associate each surface with a C-infinite to G.

1:32:30 Good. What is your topos, your universe, your universe? We have it. So, surfaces allow me to build a C-infinite algebra every time. There is one thing left to say. What is a C-infinite algebra of finite presentation? A C-infinite algebra of finite presentation? Well, when I can reconstruct it from a finite number of elements and a finite number of equations. An example for the anodes. When would I say that the anode A is engendered by three elements A? Well, when I can reconstruct... Anode A can be reconstructed from A, B, C by means of addition, multiplication and subtraction. I can reconstruct everything from these three elements. For example, you take the anode of the polynome, all the polynomes together. It is clear that the anode A is engendered by three polynomes, namely the polynome... x, y, z is equal to x, q of x, y, z is equal to y, and t of x, y, z is equal to 7. These three polynoms x, y, z, if you make additions, subtractions, multiplications of x, y, z, you will be able to construct any polynom in x, y, z. A polynom in x, y, z is something that was constructed from x, y, z. So this table is engendered by three elements. It is of finite generation, it is finally engendered, it is engendered by three polynoms from which I can reconstruct all the others using only my operations. And then, obviously, once we have done something like that, we could make a quotient of this. We did that, we made a quotient of x squared equals zero.

1:35:00 To see where all this comes from when we identify x squared equals zero. This is an example of a nuclear revolution. I could do the quotient by x squared equals zero, and I could also do the quotient by x squared equals zero. I do the quotient, I have a certain number of generators, a finite number, and then I do the quotient by a certain number of equations, a finite number of equations. And all the... All the nodes that I can reconstruct from a finite number of generators and a finite number of equations are what I call finite presentation nodes. Well, I do the same thing with the C-infinite objects. I use the C-infinite object of finite presentation. I use the C-infinite object that I can reconstruct entirely from a finite number of generators, operations of the C-infinite object, and subsequently by a finite number of equations. This is what we call the algebra of presentation. Well, the C-infinite algebra of finished presentation, these are the ones that will play the role, the open ones will play the role. A beam is an F2A family, indexed by the C-infinity of Algeria. Two presentations. In the same sense that earlier, when I was studying the beam of surfaces, where my beam of surfaces was indexed by all the openings of the plane, at each opening I was looking at all the surfaces that were defined by Toussus, and so my beam of surfaces was a family of ensembles.

1:37:30 The ensembles of all the defined surfaces will give us a list of all the parameters here, at the same time, instead of taking all the open ones of the plan, I take all the C-infinite algebras of final presentation and I take a beam, it is a family of ensembles, for each C-infinite algebra of dimension, I give myself an ensemble. Of course, there are also operations of restriction and collocation, such as when a surface is defined on something large, it is defined on something smaller. All of these are indexed by these affiliations of presentation. They are, I would say, the ones who describe the whole situation to me. Instead of being open-handed, they are the ones who play the role of the speaker. In here, there is a table. It must therefore be an indexed family. Well, it's the most trivial of all. It's the family. Among all the families, there is this one. At the level of A, I take A, nothing else. I find it easier to imagine. At the level of A, I take the whole of A. A is the infinity of the algebra, the multiplication of x and y. Is this a manual? Yes, it is a manual. If you look at plus, times, minus, these are the functions of r squared in R. The addition, the multiplication, the subtraction are the functions of r squared in R. And these are infinitely derivative functions. And since these are infinitely derivative functions, they exist on all C-infinity algebras, so on the C-infinity algebras, I have operations corresponding to them, I have them.

1:40:00 So my C-infinity algebras is only a nano, because it has the same operations as the real one, all C-infinity operations on the real one exist on A, and with OTS, all the actions are satisfied, that's all that's left to be done. So all the actions on A2 are satisfied. So, this is in fact an amour. And, and, and, and, what is that? It's quite simple. You see? You understand the construction. It's that the elements of a C-infinity algebra A are in a bijection, I put it here, in a bijection, with the amorphisms towards A. This is a C-infinity algebra. This is a C-infinite in G, in the same sense that I said I would do earlier with a surface. I did it earlier with a surface. If there is an F of Rn in R, well, I will be able to construct an F corresponding to C-infinite of R, R, exposing N in C-infinite of R, Hn, n functions, here, with a simple difference, f, f by f, exactly as I did with the surfaces. So this is another example of this Hn-influenced argument that for a surface like this, instead of putting the surface here, I put f here.

1:42:30 It's a nice dimension, isn't it? It's clear that what I did with the surfaces earlier had nothing to do with the fact that the surface was dimension 2. I would have taken a variation of dimension 17, the S, as I said, the fact that the S was a dimension. Let's take an objection. How does the rejection happen? Well, the rejection happens quite often. If you have A, you have to build this. So, given this, you have to associate something. Sorry, but G is an indication of C-infinity of R in R. Since it is an application of C-infinity of R in R, it is an operation of A. All things of Rn in R are operations of A. So, in particular, the function of R in R is an operation of A. There is, if we have a G of R in R, there is in a corresponding way a G of A in A. The G is an operation of a C-infinity of G. So it exists on a C-infinity of G. The function that sends G on G applied to A. And in the other direction, of course, if I have a function H, I have to create a random element. Well, I take an element H, the identity of this element. The identity is visible and exists here. It is obviously an infinitely variable indication. The function identity, the function that sends X is this X. And so I transform an image by H. This is immediate. I have built a bijection between the two. If I go back, I find that there are no recipes. The algebraists have noticed that I have just demonstrated that this is the C-infinity algebraic book to a general. So, so, I have just demonstrated the same thing as all the homomorphisms of C-infinity algebraic books that had been written since the 6th century.

1:45:00 I have just written in a well-written way that it is the same thing as homomorphisms. If I have my element A belonging to A square, it means that I have a corresponding H. I don't know very well how to determine what the A's of square are, but thanks to my projection here, I will be able to determine them. I will be able to determine them more easily. Well, I fail. I have the identity function that goes to square. This is a partial function. If I put them in the square, I don't know if you can see it, but it's always a function, but here it's a square. Don't get me wrong. F of the square identity, sorry, H of the square identity, what is it? H is a non-morphism of C-infinity algebra. It's something that is common with all operations of C-infinity algebra. And we just saw that C-infinity algebra was an operation of C-infinity algebra. So F is common with the operation of C-infinity algebra. We say that h is a moment in time, that is, h of the identity v to the square.

1:47:30 We make a commutative with this object. Yes, but h of the identity, remember, was a square equals to zero if and only if h of the function of an identity to the square, I will write it like everyone else, if and only if, h of the function of the system of the universe, if and only if, h of this function, force the square equals to zero. This is an example of the quotient that we talked about earlier. Forcing the square to be equal to zero, the objects on the other side of the formula, it means that we take all the homomorphisms, we make the quotient with x squared and x squared. We take the homomorphisms in the form of a function here, which sends x squared over zero, which are ultimately defined on the quotient. Take the example of the polygons we had earlier when we took x squared equal to zero and left the functions of the first degree, a function that sends x squared to zero because in the end a function is defined on the polygons of the first degree, otherwise it is equal to zero. This is the algebraic propriety. The functions, the homomorphisms here that send x squared to zero because it is exactly the homomorphisms that are defined on those that remain when I force x squared equal to zero because here it is equal to zero and I can say that it is equal to the risk. So here are the infinitives of R, which are the things that exist in the universe. We have taken some examples from another context. For example, earlier, when we took the function of C-infinite, RR,

1:50:00 in R, for example, we took the function of derivation, which sends F under the derivative of F to zero. When you take the function, the application that has a function on its derivative in zero, well, this function there, we can see it well, x squared over zero, because the derivative of x squared is 2x, and the derivative of x squared over zero is zero. So here are the functions of Mr. Dumont. The derivative at the origin, well, we can see it well, x squared over zero. It's absolutely not a criminality, that's it. So things seem to exist very well. And so we have now built a manual LR in our space. We have the square-null elements, we have the square-null elements. Look at these elements that we call infinitives. What are they? They are exactly the elements obtained by making the algebraic quotient x squared equals zero. We have here, you see, infinity, algebraic, very simple. And we made the quotient by the equation x squared equals zero. And what we obtain, it gives us exactly the infinitives. We demonstrate, we demonstrate that in the logic corresponding to the local logic, in the case of the sphere of curves, of the sphere of surfaces, in the intrinsic logic of this sphere, satisfied by the action of the polar bear, any function defined on the infinitives is automatically defined. This is exactly the same as what I did earlier with the polynomials when we force x squared to zero and the rest of the things are the same. This is a model of Haussier's axiom, but in logic. It shows that there are intuitionist models of the ensemble theory, for example this one, in which Haussier's axiom exists.

1:52:30 So, indeed, we can assume, in the ensemble theory, that there are sufficient rings. And so on and so forth, and so forth, and so on and so forth, and so on and so forth, and so on and so forth, We make geometry by working with this and by saying that the two functions are tangent when they are equal on the infinitives, etc. We make differential geometry, you know. We specify, in a sense, the notion of the algebra of K, but in the end, we have a precise definition of what a surface is. It is something such that in any point, if I look at a point x of the surface, I can then look at points that are infinitely close to each other, points whose distance from point X is infinitely small. For example, the points infinitely close to X. So the points infinitely close to X are called A, B, or K. That's the definition of a surface. What is a surface? Well, a surface is something that, when you place it infinitely close to a point, The points are infinitely close to the origin in the world. It is something that is indistinguishable from the origin of an infinitesimal part. A surface is something that is totally indistinguishable from an infinitesimal part. And of course, a variety of dimensions also has something to do with it. And with that, you develop the differential geometry. I'm not going to go into too much detail, because you're making very good arguments in terms of infinitely small things. And by always starting from the idea that a surface is the same thing locally, on an infinitely small space, it's the same thing in your head. And you're going to be, you're going to be revisiting the surface in a very interesting way.

1:55:00 If you're having fun with all that. What interests me are not these surrealist discoveries. These are the pre-substracts of Mr. Gounod, the real substracts. That's what I'm talking about, that you have in this new context a more general notion of substracts about which you can easily demonstrate things. So much the better, I'm not interested in your substracts, I'm interested in Mr. Gounod's substracts. But don't think about that, don't think about that. I want to demonstrate to you that all substracts, in my place, are infinite objects. And this C-infinity algebra, I'm going to make a beam that I'll call S-infinity, it must be S-infinity at some point, associated with a beam. At each surface, I can associate a C-infinity algebra, and so I can associate the beam of the homomorphism of this C-infinity algebra to the C-infinity algebra A, and that's where all the C-infinity algebras are present. This is an expression. And we can see that this is a surface, that is to say that locally, the infinitesimal neighborhood of points is isomorphic, not diagonal. So at each surface of Mr. Tout-le-Monde, I will associate a surface with the sense of Coq-Rouge.

1:57:30 This realist of my universe is a physicist, extremely profound, preserves and reflects the visual image of surfaces. If a theorem about the surface here is true, the same theorem is true, but reciprocally, if the theorem about the surface phi of s is true, then it is true for the surface of the Earth. A surface that has been broken. Concerning a surface that has been broken. I have shown that this is valid in the theory of the ensembles, by using the action of cohomology. The particular is valid in the theory of the ensembles, which is a particular model of the theory of the ensembles, for this particular ring, which is a particular model of the theory of cohomology. It is valid there, it is not valid when it happens to S, but it is not valid when it happens to S. So the result is this. In intuitionist theory of science, we can demonstrate a very classic theorem by using the infinitely small but by using this theorem, it is valid classically.

2:00:00 This is a somewhat curious situation. We can use the infinitely small in situations where there are none. We can pretend to demonstrate by using the infinitely small, although classically we know that there are none. The price to pay is to use only the intuitionist logic. If we can demonstrate the logic of intuition by using infinitely small things, we can be sure that the theoretical theory is true, that the classical theory does not use infinitely small things, but it doesn't matter. We have a completely different theory. This is the idea of this theory of infinitely small things.