Categories, Sites et Champs - lecture 3 of 3

0:00 Here we have defined the local epimorphisms. We can also define the local isomorphisms, and the local monomorphisms, but not by copying the epimorphism demonstration, but by marking the following. When you have a morphism of f in y, there is a diagonal arrow of x in the fibrous product of feta delta. Thank you for your attention. Thank you for watching this video.

2:30 So, when you have a morphine X in Y, you deduce it, you remember, when you have X going in X and going in Y, it's a common diagram. So, they send in the fibrous product the solution of this problem. We call it testa. F is a mono, if and only if, delta is an iso, and if and only if, in this case delta is always a mono, delta is an epis. We can give the definition, F is a local monomorphism, if and only if, delta, once we have given the axioms and we have this notion of local epimorphisms, we can define what is a local monomorphism. By saying, by definition, we will say that S is a monoplocal, that this part here, β, is a ploca-depi, it's a definition. So, we have the notion of local isomorphism and it verifies other axioms. This is where there are two out of three axioms. So, just to finish this paragraph, I will try to make you understand, because all this is still very abstract, so I will make you understand something. So we looked at what it means, if you will, that... So I always write U, V, etc. in CX, A, B, etc. in CX. So what does that mean?

5:00 Suppose we come back to the classic stuff, where we have CX and this by product, etc., and we have a topology. It's the same thing, I'll show you later, it's the same thing as saying that if I take, so A, it's not in his hat, it's in his hat, and I take all the open ones that go in A, so I take all the Vs, all the morphisms of Vs in A, think of the open ones, The family of V in U, which is what I said earlier, which is factored by 1, is a covering of U. So if you have a topological space, for example, a topological space, and then you take, for example, a family of 8, a covering of U, a covering of U. So what does it mean? It means the meeting of the 8. But we want to work in a more categorical way, with categories, so we will take the co-product of the UIs. The co-product of the UIs is not defined in the category of the UIs. This thing is perfectly defined in these hats. Even if the Ui were disjoints, by the way, it doesn't change anything, the disjoint region of the Ui, you have to see that as a thing in... And so, we'll say that this thing here is U, we'll say that this is an epimorphism of E4.

7:30 And that, now, is a morphism in these hats. You have to be careful. Why do you put quotes? Because you take men in these hats of U. In other words, I remind you of the problem of the inductive limit, take something a little more general, of Ai to calculate on U, by definition, that the inductive limit is Ai of U, it is calculated by the term. So this is a particular case, so this is worth the reunion of joints of O, of U. An application of U in the so-called joint meeting of the Ui is not at all an assembly application of U in the joint meeting of the Ui, even if the Ui are not together. Suppose that the Ui are joint. An application of U in the joint meeting of the Ui, it could overlap. Some points are in U1, others are in U2, etc. What we could do, we could look at the nature of the beams with local epimorphisms, we could see some things, but we only have the definition of them. So now you have, if X is a site, it means you have a family, you have given local epimorphisms. You would say that F, a pre-beam, we would say that F is separate, if whatever A in U, an epi, the K.

10:00 Well, F of U in F of A is a mono. On the other hand, of course, F, to say that F is a vessel, does not mean that for all epilocates, it is an iso. And we will say that F is a vessel if, whatever A in U, an epi-locate, then the following F of U, F of A, F of A, A. Exactly. That is to say, f of u is the core of this double arrow. This double arrow, everyone knows what it corresponds to. It's important because when we do the charts, we will have the same kind of exam. What is this double arrow that is there? Well, for that, a and b. Well, f is a counter-variant factor. And a times u times a, you have two arrows, b times a. b1 and b2. So you have two arrows P1 and B2. So I apply a counter-variant function, it gives me two arrows in the other direction. So obviously, if F is a vessel, this arrow is a nucleus. This arrow is exact, that is, F of U identifies itself in the nucleus of this double arrow. So a nucleus is a monomer. So if F is a vessel, F is separate.

12:30 So that was just a little incursion. To explain that we could very well do the technical logic without the hypothesis that there are fibrous products. Well, it's a lot of fibrous, it's going to change every day. But maybe, what does that mean in this case? Take, let's say, A equals reunion of joints of U. What does that mean in this case? Well, it's F of U, product, instead of taking coverings, reunions of joints, we take directly units. So, if you want, another way of saying A is an epi block, if there is a family U that has a recoupling of U and such U's are factorized in an epimorphism block, it goes from A to U. A is a pre-vessel. We are no longer in the open epi of U, we are out. It is important to understand that CX sends itself into CX hat, and that's the first jump.

15:00 But there are many other things out there. In particular, I give you the example of an open Dijon reunion, which is no longer an open, it no longer belongs to the theory of open X. For example, take this example, U1 and U2. You take U1, co-produced with U2. This is perfectly defined as pre-vessels, but it's not at all a new one. Well, it's normal to... it's quite... I don't know what to say about it. So now we're going to move on to the last chapter, on stacks. I prefer stacks than fields, because fields are still used in physics, in particular, whereas stacks... So, let's start with stacks. Hopefully, we won't have time to do a lot of introductions. There will be a lecture in January by Behrendt, I'm afraid it's too bad for French people, where he will talk about differential fields, people who take algebraic geometry, who are interested in algebraic fields, so he will say a word at the end, but there it's very, very ... fields in general, it's completely ... well, it contains differential fields, it's not the same spirit. So it's a function of Cx of in F. And I remind you, if you have U in V, so you go from F of V to F of U. And then if you have U of W, you go from F of W to S of V to S.

17:30 And then there you have U of V, let's say. And here you have V'U, you have W, if you want, equals, I think I'm going to change the notation, U1, U2, you have U3, U2, V, and you have V equals U1 times U2, because you are in the quadrangle of the ensembles, you have two, or in the quadrangle of the ensembles, and you have two morphisms that are equal. So if now I want to do the same thing... So this is a pre-vessel. Now I'm going to do the same thing, but I'm going to replace 7 by 4. Now, before, these were ensembles. If now I'm in 4, I'll take other quotations, I'll call sigma a pre-stack, so it will be a pre-vessel, if you like, with a value that's a bit linear, in 4, and so what's going to change? There are a lot of things that change. So, we are always on X. X is a principle. There is no topology for it. So, at all open, I associate sigma with U. So, instead of being an ensemble, that is to say an object of the category of ensembles, now it will be an object of category 4, that is to say it will be a category. Then, if I have something that comes from U1 into U2,

20:00 So, I'll go from sigma to U2, no, sigma to U1, but before it was a morphism, now it's an intercategory, it's a functor, so I'll call it R as a restriction, RU, R index U, so now it's a functor, and then now if I have three, U1, U2, so I call, I take my, there are a lot of notations, U and D, so, I have sigma of u3, sigma of u2, sigma of u1, here I have r, v, here I have r, u, and here I have r, v times u. Unfortunately, we could ask for two equal functions, but to say that two functions are equal in practice does not apply. They are isomorphic. So the difficulty is that we have to control the isomorphisms. So here I need my notes. So, we give an isomorphism C, U, V, which is an isomorphism of functions R, U, composed with R, V, isomorphism A, R, 0, 1. So the problem is that we have to continue, because it's not that precise. These isomorphisms of functions must verify a condition on how they are composed. The identity of U is the identity of the sigma of U, and the composition of the identity of U is the identity of the identity of sigma of U, it is a functor, it is a morphism of functors.

22:30 This is the identical functor, this is a morphism of the identical functor towards itself, and we ask whether it is the identity. And then, we ask that when we have four things, U1, U2, U3, U4, U, V, W, we have a natural diagram, we have Ru composed with Rp, composed with Rw. In this case, we use C, V, W. To be more precise, we use R, U. I can also use C, U, V, U. And then I use R, U, W, V, U. There's an error in my notes. Here it's C, V, R, U, W. And we're asking what it is that a pre-champ, that is to say, where a pre-vessel has value in two categories. Well, here we took the fourth category, to put it by analogy with the seventh category.

25:00 I would have to take two strict categories in which... I'll let you write down what a professor is worth in three categories. As you can see, the difficulty is very low. I'll give you some examples. First, some small variations. Sigma is a pre-stack on the pre-site X. You can define what a pre-stack is. Additive, abelian... For example, if all the categories sigma of u are additive, and if the restrictors are additive, we would say that you have an additive pretext. In a Cabellian pretext, we ask that the restrictors are exact restrictors. There are examples. There is an example that you know well, an omnix and a precit. There is a category. All open U of X. This is the category of prefixes on U. So, with a restriction morphism, the prefixes are restricted. If you have V in U, well, we lose it. There you will have a function, a small u. Ru is the function Ju and 3. That's a morphism. In CX, it is defined by U in D, a precise morphism, that morphisms, in theory, define precise morphisms in the other sense, and the impressions are the same.

27:30 A morphism sends Cv in Cu, generates a Cv in Cu, so a precise morphism of U in D. Well, all of this is... So here is an example. You can verify that it is an example of a pre-stack. Another example, you can also assume that x, another example, if x is a site, and you can take the beams, it will still be a pre-stack. It will still be a stack if it is not a pre-stack. So it is not a pre-stack, it doesn't mean anything. SHx, where U gives SH, U, A, and it's a preface. So, an obvious remark. If you have a preface, a real preface, then immediately you associate F , which is a set, or even a preface, yes, a set preface. So it's a set. A set, you can see that as a discrete category. It's not interesting, but it's an example. Pre-beefs define pre-stacks, but it's not interesting. Discretionary stacks. Just as children define a category. It's not interesting. Another example. Take a beam, a group beam, not necessarily commutative,

30:00 over x, on a site. We can define a stack, sigma, in the following way. At all opened U, I associate the category with a single object and a single morphism to identify this object. And then I have to define the following category at all opened U. I associate sigma of u with the following equation. The objects of sigma of u are a point, but man, in sigma of u, the point is a point, g of u. I did this in groups. It's a bit the same thing, it's an example that I should have given at the beginning of my course on categories, which I may have already given. When you have a group, you can assign a category to it. If G is a group, for example, you can assign a category G, such that the objects of G are just a point, but man in G, the point, the point, equals G. It's better with an algebra, often it's not with an algebra. So that's the same thing, but a plus. Instead of having a group, you have a beam in a group. And so you define a pre-stack that openly associates you with the category whose objects are a point, but the morphisms are physical.

32:30 If you want, I can call it G+. So in this example, sigma of U is equal to G of U plus the associated category. So, one or two more obvious things. We'll get down to the serious stuff next week. Now we're going to make some definitions. So we can make direct images. So F is a pre-sit morphism. And then sigma is a pre-stack on X. So I can define it in an obvious way. F star sigma, a pre-stack on Y. So you can tell me how much is the category, what is the category F star sigma of Z. You have Ft that goes from C to Y in Cx, so that will be the sigma category of Ft to D. I have to tell you how much the restrictions are worth. So if I have V1 in V2, how much is R? I'll take my notes, maybe. If I have U of V1 in V2, I have to tell you what the restriction is. If I have an image of Ft of V2 and an image of Ft of V1, I will put R of Ft of U. Ft of U is Ft of V1, Ft of V2 is Ft of U.

35:00 And for C, the same, I'm not going to write it down. So you have a natural notion of a direct image of Brecht. So now we're going to talk about pre-stacks, so you have two pre-stacks, sigma 1, sigma 2, and pre-stacks on X, with restriction mortals, R1, I put it at the bottom, finally the pre-stacks R1U, R2U, and then there are also C1UV and C2UV, so a pre-stack is given by two categories, Restriction functors, etc. And two types of functors. So what is a Pestak's functor? Phi, sigma 2. It's the thing we think of. For every u, it's the given. For every u, it's a functor. Sigma 1, 2u. It's a functor. Phi, 2u. Sigma 1, 2u. No, sigma 2, 2u. And then it has to move. Thank you very much for your attention.

37:30 So, we don't ask for this commutative diagram, but we ask for an isomorphism of a functor from the component that is here to the component that is here, and in general, it is written with a double arrow, it is a functor morphism, which is an isomorphism. If there is a dispute, it should be an isomorphism. So, it's an isomorphism of Φ2U2 composed with R1U, isomorphism of functor R2U2 composed with Φ3U. So, it's not over yet. We need to make sure that it's compatible with the CUVs. So, what does it mean compatible? So, here we have the chance with intersections of U1, U2, U3. I don't know if you remember at the beginning of the course, A functor morphine was not very complicated. There is only one condition to check. There are two categories. There is only one functor morphine. Here it is the same. A functor morphine, you will not need to take four or one, it is enough to take three. So when you have U1, U2, U3, U and V.

40:00 I think there are some errors in my text. I could change the meaning. No, no, no, no, sorry. Let's go. No, no, no, no, sorry. There I had, if I go from 1, there I had 2 in 1. So if I continue, here I have sigma 1, 2, 3, r, v, 1. Here I have sigma 3, sigma 2, sorry. In this case, I will have an amortization of the vector phi v, but I could also skip the middle line. So I would have sigma of u1, sigma of u2, sigma of u1, sigma of u3, sigma of u2, sigma of u3. So, what do we ask? When I do this, I identify R1U composed of R1V. Here I have R1 of VU and here I have R2 of VU. And this is not quite the same thing as this, there are C-UV amortizers between the two. So what we ask is that... Φ composed with C1 plus V is equal to C2 plus V, composed with Φb, composed with, it's a bit dark on the diagram, Φ , well, it wasn't that bad.

42:30 For the students, it was quite good. That's the limit of the intellectual capacity. When we stop working here for a month, we can't do anything. That's why we have to do three pre-stacks to understand the two pre-stacks. So, now we have defined what a pre-stack function is. Now we will define what a pre-stack function morphism is. You have sigma1 and sigma2, two pre-stacks. What is a pre-stack functor morphism? The simpler we go, the simpler it becomes. A pre-stack functor is simpler than a pre-stack. For a pre-stack, you only need three openings. For a pre-stack, you only need four openings. To define a pre-stack, you had to look at what was happening with four openings. For a pre-stack functor, you had to look at three openings. For a morphing of pre-tax foncteurs, I think we'll need two open ones. So, what is theta? For all u, it's the data of theta of u that goes from phi of u to phi of 2 of u. So for phi of u and phi of 2 of u, they're foncteurs. So this is a morphing of foncteurs. Foncteurs of categories, once we've fixed them. And what do we ask for theta of u? If we go into U2 by U, the diagram that is missing is U1 by U1 by U1 by U1 by U1 by U1 by U1 by U1 by U1 by U1 by U1 by U1 by U1 by U1 by U1 by U1 by U1 by U1 by U1 by U1 by U1 by U1 by U1 by U1 by U1 by U1 by U1 by U1 by U1 by U1 by U1 by U1 by U1 by U1 by U1 by U1 by U1 by U1 by U1 by U1 by U1 by U1 by U1 by U1 by U1 by U1 by U1 by U1 by U1 by U1 by U1 by U1 by U1 by U1 by U1 by U1 by U1 by U1 by U1 by U1 by U1 by U1 by U1 by U1 by U1 by U1 by U1 by U1 by U1 by U

45:00 So, once we have the notion of pre-stack morphism, we can talk about the equivalence of pre-stack. We will say that Φ is an equivalent of pre-stack. And we can see that phi of sigma 1 in sigma 2 is an equivalence of pre-stack, that's what's important, My daughter has an equivalence of category. There is nothing to verify as compatibility. So if you have a pre-stack counter that has an equivalence of category on each cover, then it is an equivalence of pre-stack. I don't know if you found this answer. Yes, yes, I knew it because I'm a bit lost. That's a morphine of morphine.

47:30 What is that? These are the morphisms of functors, so it is an equality of morphisms of functors, not functors, but morphisms of functors. So, next week, I will define the equivalent of, in fact I will not do the two protective limits, but I will simply do the cores, the two cores, and that's enough for what we need. Obviously, we will not do the example. The art of mathematics is a phenomenal phenomenon of the principles of mathematics and of their existence, in the sense that it was written during the practice of mathematics. What he is saying at this moment is that, for it to work, there must be a corpus of results sufficiently established in a serious way so that, when we get there, The analyst, therefore the mathematician, can say, okay, we are in this category of mathematical results. One of the first was the data of time. So the problem is to know what we can consider to be a debatable data, a premise on which the mathematician has the right to argue. So it is on this question that Callasen works. And they are trying to find out what we can put in this definition.

50:00 So here is the continuation of the citation. So, regarding the law of theta and its foundation, there is the analysis of theta, on which ends the discovery of the properties and the definition of the premises, which are the basics of mathematics. We have previously said that the knowledge available is indeed necessary for the completion of the art of analysis. These are the notions called known limits. So, it will give a name. In the end, the known at the beginning is something that will resemble, that we leave to the data that is more or less similar, knowing that the treatment that is done at the level is a little more profound, insofar as its art, the art that is currently describing the analysis of the synthesis, It is both a prologue art, but also a method of discovery. It goes beyond the content of what had been done by the known. So, what is the known? The link with Euclid's data is very clear. The Euclid's book, made up of data, understands many notions of the known, which are placed here. So, I cut a little, other known notions are not contained in this book and that we have not found in any book. So, he clearly announces the project. He is telling us that what he is going to do is not just develop something that would already prosper in the world, but something new, more than he has not found. What does the known give us? It gives us a definition, in this case, that the theories contain all the necessary theory. So we say here, the known in general terms is what does not change, because everything that changes, well, in its nature, the change does not have a definite or natural reality. If it does not have a determined and assignable reality, be it its essence, it is not just that it is known, because it is possible that all this component of it changes from what it has been.

52:30 The thing will only be known if it is fixed, according to a single state, which is its essence, which is it. Let's just pay a little attention to the idea of the change that is behind it, which is not the geometric change. What is the key point of this first definition on which he will return? The known is associated with the essence, so there is a ontological will to define the material beings. This association between the known, the research of the known concerning the essence, will put into practice a deep reflection on the nature of the material, since it is reflecting on what the essence is. So there is a total adequacy between the two, and he makes a classification that, in the analysis and synthesis, finally takes up the classification of the two. That is to say, he will say that there are things that are known by magnitude, things that are known by shape, which corresponds to the value of species, and things that are known by position. The thing that will interest us the most for the notion of space is what concerns position. In this first book, he evokes the known of position. And here is what he tells us, simply. So the known of position is the one whose position does not change. There is a variability. As for what is the position, it is the situation. And the situation is established in relation to a thing posed. The position is in the body, in the surface, in the line, and in the point. So here, you make the line and you make the geometry, simply placed in relation to their dimensions. The position in the body is divided in two ways, either it can be relative to a fixed thing, or it can be relative to a moving thing.

55:00 The first set of definitions that it gives... All these terms correspond to what is done by others. That is to say, what is the position? It is simply the relationship of maintenance of a geometric figure, whether it is the right, the left or the right, with other fixed points. If they are mobile, it is assumed that the whole is moving in the same direction. We do the following calculation. So there is a position at the back in relation to fixed things and in relation to mobile things. So the first is the position relative to fixed objects, so the point, the position that is absolutely known is the one in which the position is relative to a point or fixed points, and the one that does not move but moves, so it is something that is very ... the possibility of studying objects that are very stable. This is geometry, figures, but it will of course go a little further. So, if we say that in what is known as a mobile thing, it will be the one whose distance, at any point of this mobile thing, is the same distance, which does not change. And if the thing moves, the point moves by its movement, as the center of the circle, or the distance at each time of the circumference of the circle, is the same distance, which does not change. However, if the circle moves, its center moves with it, as is the center of a sphere, etc., or the center of a cone, and this demands many things. Why do I give you this first element, which is ultimately relatively elementary? That is, if we have a circle that moves, as the distance of the center is defined as the distance to the point of the circle, the center that is there will be the center that is there. The whole of the figure is transformed with these properties. This element is important. Why? Because it introduces the idea that the elements it is going to work on are distances. So, the idea of distance is an element of geography. However, in this first categorization of knowns, knowns in relation to fixed things, in relation to mobile things,

57:30 The notion of mobility is still relatively unique, that is to say that we have something that displaces itself, or at best the whole of the points are displacing themselves in the same way. This work will continue precisely in a treaty called the Econimity. It returns to something that he had simply started to pose in the Mathematical Analyse, and he returns to it on a whole law that he had announced in the Mathematical Analyse. So he returns to this idea of collusion. During his introduction to this treatise, he talks about the idea of science, and with a relatively Aristotelian vocabulary, although he is not, he says, for example, that there is science in action and science in power, but that the important thing is not that it is in action or in power, the important thing is that there is knowledge, no matter whether it is read or not by someone. For geometry, he says it's the same thing, that is to say that there are both known in art and known in power, but whatever, what is important is that there is something invariant in it, and that's what the geometrist must look for. Here is the end of a quote that we have put on Transparent, he says, there is a cycle for all this, therefore, in relation to this discussion on science, the known is, in fact, any notion that admits change, whether it is believed or not by someone who takes it. So, there is something invariant, an existence of knowledge. So, how does all this translate into the work of Lazen? He is going to make a new classification, a much more complex one.

1:00:00 So, here is the classification. So the whole book is based on this classification, which is therefore defined in several ways. It starts with a distinction that is purely Aristotelian, between discrete quantity and continuous quantity, knowing that even if it details a certain number of known properties of discrete quantities, its treatment will only apply to continuous quantity. Because for the same reason, time and weight will not be the object of the following sequence. So, some examples, for example, of knowns. Let's take the line. This is an important point because it starts from the line to define the whole of this geometry. It does not start from the point. In fact, it does not even start from the space. The fundamental object is the line. When the line can be known according to several things, its length, its depth, but we can know its size, its magnitude, and possibly a position that can be in relation to a point of view, a point of view, etc. So, the point in there, if you look closely, it does not appear here, but it can appear somewhere. The point is defined simply as the extremity of the line, and the known about the extremity of the line, which is the point, is composed of two notions, and it is at this moment that the point is defined. One is the relation to its center, that is to say that it is not divisible, it is encyclopedic, it is innovative, and the other is that it is not possible to define it. And the other is its position, that is, its distance to another point or to other points in the imagination, if this distance or that distance does not change. A very strong return to the notion of distance, which, as it is the basis of this theory, requires a specific treatment, which obviously does not escape.

1:02:30 To stay for a moment on the point, and to complete the table on the possibility of movement and motion. In the known, what are the possibilities for the position of the point? The point can be known, the point itself is fixed, and the points existing in the imagination are also fixed. So this is the first situation of the analysis of the synthesis. The points are fixed, the point on which we work is fixed. The second thing is that the points existing in the imagination, the points of reference, are fixed, while the point of known position is mobile around a fixed point. So we see this idea that a point that we would only know the distance to another point can be considered a point of known position. The third is actually the second of the analysis and the synthesis, which are the points that move in the same direction. So there is an additional possibility of determining something in relation to points of mobility. It is not necessarily the same. So, introduction, in a way, of the movement as a premise. What about the geometry of our daily life? How will the figure be defined? The figure of the line, so the string will take the line, is the one that constitutes its essence.

1:05:00 And here it will go to the particular case of the right. What is the right? These are two points. And the fact of being the shortest distance. So it defines the right as being a geodesic of space in space. So this distance will need to be precise because, as we said earlier, it allows both to define the figures and to define the positions and the directions. So... The right line is the distance between the two ends, provided that this distance is shorter than the distance, which constitutes its essence. These are the two ends because they are the limits of the distance between them. If we require that the distance is shorter, this distance will be the right line. Redefining the distance. So the question that can arise is why does it take so much time to... To define something that could be obvious for these students. In fact, what is very clear about it is that there is an infinity of different figures between two portions that we could call distances. That is to say that it is between two points. There are tables that can be called distances. And if we say that distance is simply something that is defined by something that is between the two points, we must actually add something more, otherwise we do not have a university and it may pose problems to define the specific properties. He takes the example of the circular line, which is the second part of our portion. The circular line is also the distance between these two entities. It is a tree. There may be between these two entities many circular lines of different sizes. Allazen is perfectly clear with spherical geometry, so he knows perfectly well that if we plot on a sphere, the data of the two points does not allow us to decide precisely the distance between these two points if we do not add the conditions above. So, the right line is the shortest of the lines that join the two points, and here we come back to a point on which I will go into a little more detail later.

1:07:30 But since the form of the linearity is lost in the imagination, the figure of the linearity does not differ from one right to the other and that it does not change, the right of magnitude is therefore the one for which the shortest distance of its extremity does not change. So, how can we find a way to base these theories? They move in a kind of loop. The definition of something new means that it is placed in an abstraction and imagination, and it says that in imagination this is possible, that is, I can imagine my straight line. So there is a slide towards something that is no longer a space of space. So this definition is extended to all figures. That is to say that at the same time there is the figure and its definition by distances which are in these kinds of imaginary entities which have the property of being different. The many examples below are only additional examples of this. I think that for the theme of psychology, this is an element that shows a very long range of theories. On the place, what is the work of Aladdin? On the place, in this book, which is philosophical-mathematical, he does not want to try to put into question theories, the definition of the place. He says, for example, that the place is the answer that we give to the one who gets rid of the place of an object.

1:10:00 The place is the answer to the human vision, quite simply. But, according to him, there is something that is necessary to be said, which is that if you can start with the object of disagreement, it is the place of the body where the distances do not exceed the distances of this body. This is the notion that we must seek. In other words, simply answering where do I find it or where do I find myself in a conference room is not sufficient, in any case, it cannot be difficult to answer in any case. What is difficult is to define the place, therefore, as close as possible to the core. Obviously, it resonates very much, and it will be the case, as a question that has to be answered by a mathematician. So that's exactly what he does, that is, he goes back to his way. A philosopher could certainly contest this, but he recalls in his own way what he considers to be the place where geometry exists. So, the entirety of the quote is very clear, but let's say that for the whole body, there are two things that can be found in the middle. One is the surface enveloping the body, that is to say the surface of the air enveloping the body which is in the air, the surface of the water enveloping the body which is in the water, and the surface of the whole body within which there is a body which is in it. This is why it is called one of the two groups of equations. The other notion is the beam. The imaginary is filled by the body. If indeed the body moves from the position in which it is located, the enveloping surface in which it is located, it can be imagined, liquid, without body in it, even if it has the guarantee that there is at the end of the body different from the one that was in it. So by position, I mean one of the places precisely in which each is called by convention of the place. These are imaged distances without matter in between points or degrees belonging to the space-time space-time space, and this characterizes the other two. These two, as I said, are one, that of Aristotle, the other, as I said, that of the Pythagorean theorem. How to contradict these theories? The first one to which I am attacking And certainly, in the most virulent way, the other, as we can see, is not, the theist is less aggressive, even if it is rejected.

1:12:30 If we consider that the place is the surface enveloping a solid, the surface, sorry, the outer body enveloping the solid. Allenden takes a number of examples, you could call them experiences of thought, but which are of a geometric nature. So, for example, if we cut the middle of the cube, if we follow the geometry, it would be the surfaces of the cube, but the surfaces inside the body of the envelope. For example, if the cube was in the air, it would be a square surface of the air that determines the surface. So, at the same time, it means very well... If this is the place, they propose, for example, to cut the tube according to a parallel plane, and they glue the pieces together. What happens when they do that? Here we had an object, so if we calculate the surface, there are 6 times the area on the left side. If we glue in this way, we find 1, 2, 3, 4 faces and a half-face. So, we find ourselves in a situation where this object, by a kind of equipping of the mind, if we follow the... At the same time, the definition of space in a statistician, we would say that here we have a place of a certain size and here we would finally have the object, it has not changed the substance of the object in the meantime, but there would be a larger place. For Adazen, it is a contradiction, he does not see how we can say that there is a specific place if we can build a larger place from the point of view of the statistician. So let's take multiple examples to manage several counter-examples. The example here is simply the example of an outermost.

1:15:00 Let's imagine an outermost, so the surface of the outermost is fixed. It can envelope, depending on how we see it, it can envelope more or less large bodies. So how to say that this outermost would be the place of something? The same applies to a solid machine. If we make holes in it to remove material, there will be less material in the solid, yet the surface will increase just as each time. We will add, for example, the surface that is half the size of the solid. The last example that I will show you is that if we take a malleable material, such as wax, we can form a different one. There are different figures with the same number of mathematicians whose surfaces will be fundamentally different with a limit case that has shown that there is a limit case. So this is anti-analytical physics. Then he attacks the thesis according to which the place could be the goal. So here is what he says, he will rather reject it, he will rather remodel it. So, when we say that the imaginary is filled by the body, an ambiguity is present in the case when we say that the city does not exist in the universe. If we say that the place of the body is the city, it means necessarily that the place of the body is something that does not exist. And yet the body exists, and every existing body is in a place. If therefore what is in a place exists, its place exists, it means necessarily that the city exists. In France, this is a negative affirmation in the mouth of those who say that the city does not exist. So, for him, the biggest difficulty of the place is to postulate that it does not exist. This is where it can cause problems. So, how to go beyond? And is it possible to think of the place in a different way to solve all these paradoxes? Not paradoxes, but contradictions. Here is what he proposes.

1:17:30 So, he says the place of the bodies is only the distances on which the distances of the bodies are superimposed, with which they are united and to which the figure that is in the imagination is similar. There is therefore a distinction of two things, the real figure, if we can, and in which we can imagine a set of distances. And the same thing in an imaginary solid where we see that the imagination is the place of mathematics. It's not the most appropriate term, but it's the place or the geometry in which I never think. So, from this definition of the place of the body, since this is the last part of the lecture, Since everything we have shown has been clarified, then the place of the body is the distances which, in the imagination, are a void without matter, equal to the body of a figure similar to the body's body, which we want to demonstrate in this lecture. This is a small situation. So we have a figure, we imagine inside distances. These distances, we imagine others in the same space as the space of the imagination. And this will be the best agreement when there is a univocal relationship between the two, at a distance of time associated, etc. So that's why the definition is the definition of the place according to the law. So, in order not to make a comment directly after this, I propose to leave this comment blank. I have 40 minutes. Ah ok, that's fine, it's going to be fine. It's going to be fine, I'll finish in 2 minutes. We just have to look at what we have said, how these successes are close, and how they are not. Ryan, since he has had enough time, is going to project something that is an idea of movement in geometry. So that's going to be Ryan's critique. Not that this idea seems to him to be unrecognizable.

1:20:00 According to him, it is not part of the premises of geometry, so as such it cannot be used. Perhaps she did not see well in the zine that precisely this side of the zine was to put the movement in the premises of geometry, since Ralliard refers to Aristotle by considering that the movement does not have its place since the figure is not the matter of movement. The second thing is that Rayam rejects the idea that the place can be of a mathematical nature. What he criticizes here is that Alhazen does things, mathematical treatments, on something that should not be allowed. So the most interesting criticism that I like to mention about existence is perhaps that of Agassiz, who is a philosopher and who... The author takes the trouble of writing a treatise that opens and closes at the same time, which is quite remarkable. I still read what he says because I think it's interesting. What prompted me to compose this treatise after many books full of self-explanatory proofs is that it is a book on the place, where he thinks that the place is the empty existence and that it is the enveloping surface. His proposal for this book is not of his level and it would not be appropriate to attribute to this one the perfection of its eminence. If this book is attributed to an illiterate man, it allows me to justify it, because we fear for the truth if an illiterate man sometimes defeats us. I like to say that this is the rest of the world. In the name of humanity, which has not examined it. So he takes, indeed, all the... Here, he could explain how to do it. He tries, so he rejects each time, the theses of Stalin. Perhaps one of the most important points here is that in the label formula, if the imaginary distance is that the right is a length over a width and the one that is a number, we understand that the place is a right. There is no width, and he says before this and after that the body fills it, and therefore the body which has three dimensions, sorry, length, width and depth, how can it fill by a single distance which is the right, a depth or a width?

1:22:30 So the problem, the problem is that, that is to say, in what this distance allows to define something which at the beginning is three-dimensional. The criticism is deep. I will pass to Philippe. So, where there may be an incomprehension on the part of Bernard Lévy, it is precisely what he says about this possibility of superposing. So, when you have a figure with three dimensions, it can not be superposed to a figure with three dimensions as it is. It is a classical phenomenon of the impenetrability of others. This cannot be done neither in the imagination nor in the existing. This is why the geometries do not think, do not suppose, do not allow it, do not use it in any of their models or their demands. If it was possible to imagine them, it would be ironic to mention them, but as this is not possible in their imagination, fortunately, it is a little bit in the practice of mathematics and philosophy and it may be in this that they did not see that the Treaty of Lausanne was certainly not of this kind. I will conclude. What happened in it is that, I will not go into it, I will not go into it, but... There is in a way an invention of space, in any case a separation of geometry, the implementation of an autonomous geometry. That is to say, in Hallazen, the geometry detaches itself from the senses to obtain something that is... Which are no longer constrained by the rules of the world of the senses, in particular this impenetrability of bodies which is true in the case of the senses, in the case of geometry, must be able to be surpassed, and it is perhaps, in other words, that the theory on the wall is correct. And we are simply on a phase of Adapten which is at the end of the analysis of the synthesis, which is perhaps a bit his approach. Science is no longer a field of study and this field is the summit to which we rise and to which we attract the spirit of those who research with it in order to reach it or to achieve it.

1:25:00 Pierre? Yes, it seems that one of the points of awareness of this question, and it appeared when Hazen criticized The definition of space by using the notion of void, it seems that this question of the void is really present in his study. Did he exactly take a stand against Aristotle on the existence of the void or not? Is it clear about that? What he does, from what I can understand, is that what he is looking for in Dane... It is to demonstrate that the geometer does not need to be part of this debate, that is to say that for the geometer, it is not necessary to know if life exists or not. Because he can place himself on another plane, which is the plane of imagination, of imaginary existence, which is not the book. What he is trying to say is not the book, because the book of Le Bonheur is already something three-dimensional. It is in this sense that Albert Dali's critique is perhaps the most enlightening. So I think that, on the geometric part, what he says in the... In the middle, it is precisely that the geometer does not take part. After that, what he says in his writing in cosmology, I do not know enough for mathematics. There is an interest in what he thinks is really useful in the world in general. In any case, for the geometric part, it seems to me that it is quite clear. What he is looking for, precisely, is the little quote, I do not know if it is a myth, but the geometer does not have to take part because he is not there. Because he is doing something else. On this point, there is no place where he speaks of the question of the place of the last sphere. He does not. And it is deliberate to set aside this question, which is the fundamental question of the heritage. But I think the question has an idea, that is, as long as it does not place itself on the same plane, it does not take a position either. I understand the logic, but precisely for the harmony of everything, this distinction between geometry and physics could be reduced in a spirit. How does he respond to this?

1:27:30 He does not respond, not that I know, but it will indeed be a... Because to separate geometry and physics, the place of geometry and the place of physics, it's a very stable position when we talk about things of this world here or that world there. But that's exactly what Albaquer will say. That is to say, he will say, beware, beware of the theory of Alaten and D'Arceveuse. Because he says, what could you forbid? Geometrists have to imagine distances as large as they want. So, as the place is defined by distances, there is a risk of exceeding the place of the place and having something that could contain everything at the same time and that would even be infinite. And that, Albert Dali says it explicitly in his thesis, he says how? How can we conceive such a thing? There is a deep problem, a problem of the law of everything. And, as Bernard Davy says, he thinks that, in the future, he really puts the idea of the law in danger by his theorists. Because, what prevents, when the law is only in mathematics, from extending it as much as we want, to do the debate of the law, etc. That's what he tells me. Does he read well Aladrén? Is that what Aladrén says? I don't know. I don't know if Aladrén would go so far as to take part in this debate. I noticed that he doesn't seem to separate geometry from kinematics. Because he talks about mobile paint at the same time. There is also a difference in distance between a certain number of mobile imaginary points, and also in the definition of the place, he imagines two successive moments since he takes an object and then he imagines that we remove this object or that we put it in another place and he reverses the imaginary volume that this object left when we passed it.

1:30:00 I thought that since Euclid, we had made a more precise separation between what is geometry of a port and what is cinematic of another port, which makes the time intervene. The work on the movements is from the start distant from the cinematic in the sense of the time intervening. The justification of the treaty is that these are geomagnetic transformations in the current sense of the word. That is to say that there is no idea of time, and precisely, there is no idea of successive instances either. That is to say that when he takes, and this is what he wants to demonstrate, which is perhaps the deepest, that is to say that when he takes a land that does not move here, there is not really a successive instance because It's not all simultaneous, but what he needs is the permanence of the structure of the elements. That is to say, there is a kind of continuous deformation. But this continuous deformation, these relations between the geometry, there are not always, there are zoonotists, for example, Time does not intervene as such, it is not cinematic, it is geometry with movement, therefore with the transformations of the movement. When we do homothesis and we have to infer the properties of one from the other, it is because we have a figure that transforms by homothesis, time does not intervene in it and we have the properties on the elements of the intersection. If you don't know all these mechanics, I don't know to what extent they separate the two things. In any case, in the case of geometry, it may be the same as geometry. Just to make sure that we understand well, if you say it a little abruptly, what would make the difference between the notion of place in Alhazen and the notion of volume, the old notion of y, stereo, what would it be?

1:32:30 The possibility of mathematicalization. A kind of analysis of the volume from the notion of the position of a point, the distance of the points to all the points of the volume, a kind of analysis of convexity. And what is interesting is that it is mathematical. That is to say that the problem of the stereo is that it is posed in parentheses. And it's like, for example, the principle of the philoponics, it is that we are obliged to postulate the distances from the beginning of something of this dimension. Which is not operational on the mathematical field. Not at all operational, that's for sure. So, the treatment he makes and the new premises he poses are mathematical-type premises. By the way, it's Sistar who hosted it, it's Contemporary. Well, it's not Contemporary, but it's Contor... it's commentator... No, it's precisely that there is something... The ultimate development of this type of relationship is the one of the units where everything is defined from the relationship. This is the first step of this approach where the geometric object is no longer defined by itself but by its relationships. In my opinion, this is the first step of this approach. So to be in relation with the previous exposés, for example with the exposé of Klaus Volcker the other day, could we say that there is an emergence of this notion of a proper space before, if you would, much earlier than the beginning of the 18th century? The idea is that there is creation at this moment of a space for mathematics. It's not a simple abstraction as it is for philosophers. It's a mathematical data that supports mathematical processing. So it would be interesting to see if these two traditions are independent. What are the relations? Are they completely independent? Are these texts known in the tradition? What is the problem of transmission?

1:35:00 There are still holes in my work at this moment. We have very precise, very serious work like this. It expands a little. The very idea of space as it is, in the end, is not really questioned, even by Alria. It is not questioned. And he does not question the idea that there may be a proper space for mathematics in which we can work. What he questions is simply the idea of movements that seem a bit subordinate to each other, etc. So there is an achievement at that moment. Autonomy, so we were talking about a de-automation of space, and that's Atiyah, and what's curious is that we find it later as a premise in the geometrical treatises, such as Pascal's treatise, for example, in Geometry, it's the science of expansion. I say it like that because I don't really know what happened between the two, I'm not yet completely clear with that, but there is work to be done. Thank you for your attention. Thank you for your attention. So, we are making a leap of almost four centuries and Philippe Lombard

1:37:30 We will also talk about the evolution of space in the Renaissance. When he asks fundamental questions on the invention of geometry, of space, things like that, the rails tell you that it's a question of perspective, but anyway that's what it's about. And if I have this quote, there is another reason, it is extracted from Science and Hypothesis. I always like this edition of Science and Hypothesis. I have a very nice drawing in the cover, which is named Fredman and Briggs. This was one of the most beautiful historical examples of a whole collection of drawings. So, there is a question of the invention of space in a way to pose the problem geometrically between the two systems. This is what has been done, and this is what will be done just after my presentation by Goddard on the hostility of the ages. And in relation to what happened there, I would rather put you on the other side. That is to say, to try to explore the idea that space was not invented geometrically by geometrists, but was invented from that time on for pictorial reasons and for language problems.

1:40:00 So everyone knows the principle of perspective, so our works are not missing. This is in fact the principle of the window of Alberti, that is to say that if you have a decor, the lines, a circle here, it's just the painter's eye and the principle of the painting as I showed you earlier is to put a window between the decor and the painter and to put on this window what the painter sees of the decor. The painting is supposed to represent reality through this rule, which we can call a convention of language, which we can call a geometric rule, and that's what the principle would be, in fact, which is the principal principle, which is the painting of the Renaissance, from the 15th century, more or less, and which is described in various forms. Here, it is the juror around Janssen who indicates the painting, with the representation here quite empirical. There is obviously geometry in it, but what I would like to explore is the idea that it is not the geometry that has made things work, and historically it is not really the geometry that has made things work, it will come little by little, it will come in parallel, it will come afterwards, but this is what, at the time of the invention, at the time when this language of pictorial works was imposed, of the so-called realistic type, with perspective, we will look at it. In a way, the first thing to be noticed is that, at the level of painting, the perspective fits into a precise era, in fact, it has an apogee around the 16th century, but it has, at the level of the language of the plural, a before. I took a few examples, there is one that we cannot deprive ourselves of, it is this miniature which represents the saddle of a plane. And it is very terrible because the painting is really placed above and when we say look at the painting to look at the photograph, there are some who even lay down for two seconds. This cultural language has a space conception that will obviously be revolutionized with perspective, but not necessarily for geometric reasons. At the aesthetic level, we find very beautiful things that do not have realism in photography.

1:42:30 And we find more or less avid antecedents, more or less successful, to the perspective. If you go to the Panteprini studies, you see, you already have a representation of the volumes of space. If you look for the lines of fall, there you get something like that. That is to say, it is, in quotation marks, false. The painter did not break his head to find a geometric rule behind it. And you will know, there are still embryos of the perspective. This one, which is a miniature of the previous one, if you look at the tiling under the character, you can follow the tiling rays, but there are objects above, you see, when you look at the tiling that starts at the bottom, you see that, and above you find more tiles. That is to say that the realism even of the number of tiles does not pose a problem. Here it is the same, we find, we already have a little perspective, that is to say if you look at the tiling at the bottom, You have lines that are going to come together to support the flow lines, but just next to it, there is another point of flow for connections that are supposed to be in the same direction. All that is embryos before the path. In this one, which is quite typical and is a bit of a pain in the ass, if you follow the flow lines... If you follow the building on the right, you have a bit of anything. If you follow the one on the left, you have a bit of anything too. But if you look at the height of the doors, you see that this door that I just indicated there, we see the bedroom on the left, while we see the building on the right. It's the same on the left buildings. You have this inner bedroom at the bottom, while on the top floor, it's the other side of the window that you see. All this to say that the preoccupation of the volume is not a priority of the most classical illustrator, there are already rules practically, this one has a symmetry and a geometry, it is the fish's fin that is said, so you see we are well ahead. At the level of the problems of realism, it's the same, you have this and you have this one which is magnificent, I like this one a lot because there is really a realism of the baltaquin there.

1:45:00 The piece of furniture in which the prince, the king, and if you look at the arm, it is in front of the arm, it is in front of the arm. The feet are inside, under the pelvis, but the arm goes beyond for some reasons. This is the before of the perspective and a beautiful day in which we went to the invention or the release of the rules of the perspective. Historically, we can place the genesis. Here we have the Batister of Florence with the construction of Brunelleschi who would have invented a device that would allow him to look through optical illusion and to find the view of the Batister. In this montage where the character looks at an ice in front of him, which sends him the image he has at the top of his drawing, he is supposed to see what is happening and he can find the image behind his mirror. And then, very quickly, we will find representations that are true, that is to say that they become what we consider to be correct to represent the schools. This is one of the first. The point of view is placed very low, but the followings, etc., we will come back to them, are quite correct. And you have the most famous examples. It is the fragilation of the Pyrrhus of the Franciscan that is perfectly just, in quotation marks, at the level of the lines of follow-up and the geometry of the screen. Another paradigmatic example is that of the ideal city, and then the one we found there. So all this is the culture of the Renaissance period that interests me. And after, you know what has become of the perspective in the cultural language. An exhaustive history of the question, but it must be seen that one of the most typical elements is this one, that is to say that the perspective becomes even iconic, in a way, we manage to find this idea of the point of flight, the flight of the rails, in very famous books, so that's the little idea. And then after, realism became very secondary for reasons of choice of painters and language. You have cubism, you have, I'm not going to say anything, but at the level of realism, it's nonsense.

1:47:30 You have the abstract and you have the games, possibly surrealist games, which are not bad. All this to say that the period that interests me at the cultural level is a fairly precise period with a fairly clear advance and then the discovery of rules that will work to give a realistic illusion, let's say photographic, so as not to be afraid of the unknown. And then after, maybe it's because photography has developed, the cultural language has regained its freedom in relation to this ambition. So it's in this period that I'm talking about. So, if we want to look at the rules of perspective and the rules of the representation of space in terms of language, we can look at what happened. So, at the level of the premises, just at the beginning, you had representations that started to be... Relatively realistic, with care to be taken to the presentation. The tiling has a great importance in terms of perspective, both visual and geometric, in the sense that it allows to structure space a little. And one of the first works that we place at the birth of perspective is this one, this annunciation, where the tiling begins to present symptoms. These are the terms that bring it closer to the real perspective. If you follow the lines of this drawing, you begin to have the point of flight of the center. But if you look closer and follow the diagonal of the tiles, the painter did not go so far as to make the tiles pass a diagonal alignment of the points. This is typically one of the embryos of the rules of perspective, and little by little, from... In the 15th century, the exact rule to paint a painting like this emerged, which forced the tiles to be aligned. In other words, if the painter had been a little bit of a precursor, he would have made his scale with his point of flight,

1:50:00 he would have drawn the red diagonal before and he would have placed his horizontal, there would have been no problem. He would have made a little bit more. And that's practically what people are going to do concretely, it's one of the essential rules, is that to have a fair square, you have to pay attention to the diagonal. The interpretation of what is the diagonal, we will see it in a moment, but whatever, in the end, it would have been easy to make it fair, so to speak, at the level of the rules. And that's not the case. The first, and the fundamental treaty, is the Treaty of Advertising, which explains precisely how to do to have the central point of flow right. With Alberti, therefore, we have the explanation of the fact that if you take a sketch, we always have the painter who is in the middle of the painting, with his glass in front of him, and if you have a sketch to represent for the decor of your painting, the sketch will be projected on the glass of the painter's eye in this way, the lines will go in front of you, and you will therefore have a regulation of the horizontals, With the rule of the diagonal, for example. I don't want to insist on mathematics, but basically they knew about that. Well, they knew a little bit, as I said earlier, but basically they didn't know anything. And you see that with this simple situation, you can decode most of the paintings with a central point of leak, that is to say that the window is in front of the painter, or the painter is in front of the line, as you wish, and the central point of leak where the lines of the grid go, as I mentioned, the rails of earlier, They correspond to the projection of the painter's head on the plane of the painting, that is to say, to the projection of the painter's eye on the painting. On the contrary, if you look at the painting, to see it by obeying the rule of the law, you have to place yourself in front of a painting like this, exactly at the perpendicular of the painting, which is in front of the point of view of the painting. Distance is another problem. With Alberti, we already knew at what distance the painter would place his paintings. Basically, these were the rules, and it is with such rules that paintings are made. One of the best examples is the Annunciation of Piero da Francesca. You have exactly all the rules you can find.

1:52:30 Piero da Francesca was one of the mathematicians of the time who tried to justify the rules. The justification of the rules remains the mathematics, the source of the mathematics, which are relatively marginal. What is interesting is the painting. It is not the justification that will change the picture. And you have frescoes like this one, where this time the points of departure may be different, but it is because the fresco presents different paintings. In fact, you look at the first part, then you move on and see the second part, etc. It's not a single painting, it's a lot more than that. That was Italy earlier. If we go back to Northern Europe, we also find at about the same time, about the same rules, but not to my knowledge of treaties that demonstrate things. And the Flemish painters or the painters of Northern Europe were already making representations with points of eight, but not necessarily just. In the sense where aesthetics is taken over and you therefore have convergence rules that are more or less consistent in the tableau, but in an empirical way, if you will, without going as far as the protection of treaties such as Alberti or Piero da Francesca. North Europe also used the same rules. It's really technological. So at the level of this language, if we are interested in the problem of knowing how to use this language, we must understand that the first thing The introduction of this pictorial language is a revolution at the intellectual level. That is to say that before the perspective there is this and after there is that. On the left, there is before and after. The difference is that in the before paintings, the greatness of the characters translates their religious greatness. It is not a question of making a great man next to an important religious figure while Pierrot d'Affranciscan will make a plagiarism of Christ where Christ appears very small in a painting and where in the foreground, and because they are in the foreground and because the rules of the perspective are relative, the men who speak become larger than the personage in front of them. With these two... You have the measure of what can happen as a heretic revolution in terms of cultural representation, and especially in religious matters, and in any case, it is essentially a religious painting.

1:55:00 You see a little what is happening. So that's the first thing. The second thing is that, in the end, when we look closely, it is not the geometry that put forward the point of leak, that is to say, gave importance to the point of leak in the paintings. It is the aesthetic, the structure of the painting, the geometry of the painting itself that gives importance, that justifies the importance of the point of escape. The interest of the point of escape is not for reasons of rhyme that go to infinity, it is for internal reasons of the painting. So, first of all, it comes from a certain side, but it makes the gaze converge towards something that is central in the painting. Here is an example of a simple one, and this one is, let's say, profound. In fact, the very structure of the painting makes the clients converge towards a point of fall in the center, and it is in the center that the important thing of the painting takes place, that is to say that it is at the point of fall that we put what is happening and what is being told by the speaker. So, if you have something new, right? And so you have a game on the point of flight that will eventually offer the painter something that will detach itself again from realism. You see, if you take this scale of Bührer, the point of flight is on the right. That is to say, it shows that the viewer is not supposed to look at the paintings as the painter is supposed to have drawn at the realistic level. I explained it earlier in fact. When the painter translates on his glass, if it was a photograph, if he was translating on his glass the scene that describes it, the painter is exactly at the perpendicular to the point of departure in relation to the plane of the painting, so he is really on the right limit of the painting, and if the spectator was supposed to find the geometric unrealism By replacing his eye where the painter put it, or if you prefer, where the object is put, he would have to look at the table on the right. And the point of flight will take up an aesthetic, an aesthetic command for the painter, which will detach itself from the question that justified that the point of flight was there for realistic reasons. It is so true, if you will, that this is a thing with two points of flight.

1:57:30 It's quite rare, in fact. Many paintings have only one point of view and you have two points of view. And if you read the critics' books, most of the commentators will tell you that this is a painting with two points of view. Because there are two points of view. But we are not at all in geometry. Geometry would say that, indeed, the one who keeps the landscape and who made his glass is in front of the glass. What are these straight lines? They don't look like a cross-stitch that goes from front to side, and with a cross-stitch in some way, there are of course two vanishing points. It is clear. It is at this time, let's say, of Piatore, that we begin to have rules that explain that there can be several vanishing points. There can be vanishing points on the entire horizon. But commentators continue to say that it is a painting with two points of view, by confusing the point of view and the vanishing points. Geometry was very poorly mastered at the time. And, last point, in terms of the usefulness of the project, there is a study that would be very interesting to do. It is the importance of the perspective of the projects in the tables that represent the annunciations. The annunciation is a table, in fact, which is very codifiable. You normally have the Virgin, you have God somewhere in the painting if you look at the scene, you have the Holy Spirit, you normally have the Announcement which is often done with a clear card, and there are other elements coded in the Announcements, there is one that we see a little better, it's the palm. The woman between the two characters. Basically, in a table of the Annunciation, the angel who comes to announce the news is always separated from the human woman to whom the news is sent. There are two worlds, in fact, the intellectual world and the temporal world. So the Annunciations are very coded. Here, for example, it is no coincidence that between the two, there is the window between the two characters. This is exactly the logic of an announcement where the two characters must be of two different worlds. I come back to what I said earlier. You find in this announcement the end somewhere, the separation between the two, at least symbolically,

2:00:00 and then you find somewhere the critical point which will serve, aesthetically and unconsciously, the look of God on the whole question. Here, it's the same, in the one of Perona-Franciscan, which becomes very, very classic, you really have the point of flight there, and if you look closely, you have to keep it close enough, and you have to keep the spacing on the ground, in fact, the Archangel and the Virgin do not see each other. They are not one in front of each other. When you look at the geometry of the pillars there, in fact, there is a pillar between the two. Painters, in a way, have obeyed the rules, but despite the illusion of the optics of realism and reality, they are not one or the other, they are separated by the peak. And for example, you have here a bomb of Botticelli, and you have the spirit that comes up there, which still represents God, which is really below. Can we go up there? How do I do? That's it, I escaped. And in this other one, you find, with the following points, the idea of a God who can be behind. Here is one last one with the following point which is directly linked to the God who is watching. And basically, at the level of the language of painting, here are the ingredients. So we can ask ourselves the question of knowing what is the link with the real of this language. All the periods I mentioned, from 1400 to 1600, represent only imaginary cities or sets of scenery. Typical is this, others are almost the same. All sets of scenery are made for the painting but are imaginary. If we try to question the link between perspective and the world, One of the interesting entries is to study the cartographic plans of the time. The representation of cities before the rules of perspective was something like this.

2:02:30 Here, I think you have Jerusalem or Capernaum, when you have Nuremberg, you have representations of cities like this. And then, little by little, came the plans. And the plans of the time were not the plans as they are today, with only the journeys and so on. There was always the representation of the monuments. The first plan of Paris that we know is this one, 1550. I took the one of the city because it's a big plan, quite huge. So if you recognize Notre-Dame on the city line, you're supposed to see that. Can you see it? A little later, you had one in 1552, which is this one with a much more sophisticated representation, This one comes later, it is a bit more rudimentary, it is one of the only plans that we know historically from Paris. This one is already interesting because it is in a very plunging view, completely unreal to the extent that it is imagined by the... You see, from 1615, we have the fleeting perspective, whereas before you had something quite rudimentary. So I took this one a little later to show you some details. So this is the plan, which is the Truchet plan of 1551. And you see, it is completely detailed. So if you look at the cityscape, you have this. If you are wondering about the representation of the left-wing districts, you see how the districts are represented by the houses, the painter, the illustrator does what he wants, he represents the houses of Rennes like this.