Generality & C19th analysis (contd.)

0:00 No, no, I understand. What I mean is that I feel that the Arctic, in fact, has a structure that is interesting, that works well, but the problem is really the presentation of this structure. The only point that I see in the structure, we still haven't really solved this question, is rather the appearance of Kohlschild. I find that it doesn't go at all in there. I don't think there are enough generalities in terms of sounds, it's very interesting, but there is another sound that has been used very often, which is rigor, and which is read right away. So it seems to me that implicitly we take back the texts that are usually commented in the context of rigor in analysis. La Branche, Cauchy, Mérité, Jordan. And you don't forget that. But I feel that this structure is very important in your article because the texts that you are going to draw in the conclusion, at the beginning it was particularly more visual. We all agree that it will go into the production. And that especially, what I asked the question from the beginning, you did not answer that. That is to say, if in your painting you oppose the generality derived from simplicity and the generality derived from rigor, There is something that we don't understand because all these authors claim to be left-handed, it's very strange. But it's not the same rigor, precisely. He puts left-handed here. It's very clear that he doesn't want to put it there. No, no, no, he doesn't put left-handed neither left nor right-handed. No, he said it was the transition. We will put it in the middle. In any case, it does not explain to us why the generality derived from rigor is not that of left-handed. This is absolutely explicit in your article. And we did not understand very well what was the link between generality and rigor. So, if I can put it to you. I find that he wrote a quote from Borel, which is a thesis from Borel, absolutely extraordinary in relation to this question. I don't know if Borel is right, but I find Borel to be really, really interesting. And I find, and I understand from a general point of view, that often you write a quote and you let the reader marvel at your quote. While you could exploit the reason for which these questions came out, because there are lots and lots of things in these questions, and there you will see the idea of the genre. When we write, the questions are there, and we talk about them, while... Consequently, we had to give ourselves the functions in a certain way and borrow them, and in the end, by giving ourselves the functions in a certain way and borrowing them, we didn't really have much more than what Cauchy had written, considering that the functions were there and that we borrowed them. So Schample-Borrel has a... He doesn't answer my question. The question is that Renaud says in the first part of his article, which is such, that with the emergence of rigorous decisions such as Abel and Huxley...

2:30 This is the so-called rigorous practice, not the classical one. Yes, but you make a link. You say that here we see a certain... There is a link historically. They are the same who do this and who enlarge a balance of rigor. Yes, but Cauchy, where does he go? I don't understand. Because Cauchy also enlarges the table of rigor and yet... Yes, but for example, in Borel's quote, Cauchy, for example, like Lagrange, treats functions that are present. So in this sense, Cauchy is still a child of Lagrange. The goal is not to make a general theory of the general function, it is to make a general theory of the usual functions. In what you just said at the moment, when you said first point, what is a general function? Second point, what does a function do in general? That is to say, the distinction between the singular case and the general case. It seems to me that precisely the work of Cauchy is there. It's important to make a clear distinction between the singular cases and the general cases. I think that if there is a sense to place Cauchy here, it is precisely here, because he has a very specific point of view compared to his predecessors. He has a very specific point of view about Lagrange, which he is also reproached by the English at the same time that they are in the continuity of Lagrange to know that for them the distinction between singular case and general case not only does not interest them but they find that it is the anti-generality since it is a return to the multiplicity of cases as they say, so I find that for me it would not be at all a question of leaving Cauchy aside. Give him all his position in terms of position and change of point of view. Maybe that's why I was talking about what Cauchy does in relation to that, that is to say, a comprehensive analysis. Yes, that's right. Okay, there was David who wanted to say something. What bothers me about the way in which you present Cauchy is that,

5:00 even though I understand, because there is the referring to, describing, and sampling. I had understood that, unlike Karine who... I saw it in the interruption. It seemed to me that we have a rhetoric that is put in place with and with, but ultimately, what does this rhetoric refer to? How do we relate to these general functions that we want to have, that we want to catch? So we describe them in three ways, we write them in three ways. I understood that, it didn't bother me so much. But the problem is that afterwards, when we go to a class, in fact, the rhetoric is the same. That is to say that the rhetoric that is there is the same as that of the authors, that is to say that this citation there That is to say that when I was trying to give all the rigor to the methods that we were talking about in the lecture, in a way that would never go back to the origin of the generality of the algebra, that is something that we had already encountered in the first part. And as you had explicitly said in the first part that, I read it to you, that the researchers were going with a new, practical, mathematical... All of these elements that accompany a value that was that of rigor, well, it seems to me that... When you start reading the paragraph in the front, you say to yourself, well, here, he comes back to the front, why didn't he put the front part in the front? Well, no, it seems to me that, precisely, he has a foot on both sides and that it's annoying for the table. It's annoying for the tableau, that is to say, you should say more clearly that in fact Cauchy makes the tableau stuck and makes the polarity stuck, it's not true in fact, because that's not what you say, you don't say that he makes the tableau stuck, you say that in fact, in the end, he is still quite close to the big picture, that's what you say, but in fact what you say... In the case of Cauchy, there are two distinct times, there is the general aspect of the algebra and... Mathematics as an answer to a general question, once we have rejected the general question. On the one hand, he is less from Lagrange, from the Jordanian side, and on the other hand, he is close to Lagrange. The first point is that you develop it quite quickly. That is to say, we could believe that he is very far from Lagrange, but in fact, he is close to it. And in fact, you develop quite little the fact that he is still in the rhetoric of the writer, etc. No, no, but I can tell you that I agree with you. And then there is a problem of space, but this is a problem of structure. I think we have to cut... Yes, that is to say that Cauchy could serve as a transition for the third part. It is already the synthesis that begins with Cauchy. That is to say, you made your polarity and then you say, but wait, things are more complicated.

7:30 Because your polarity has a slightly healthy side, you see. In a third part I will show the... Let's go back, look, we have the rhetoric of the first two, and Morel who is at the top of the two, saying I'm going to find Cauchy. It would have been much clearer, it would have been much clearer, the polarity 1, 2, and then 3, roughly, what? Yes, exactly, and it is necessary to come back to work with Hegelian. But we're going to put the article of Aurélien right after that of Renaud. If you want to leave it there, I will be able to see these three things, work in the future. And in my opinion, I think it would be good to... But why work in the future? He simply says, wait, I introduce the distractions that I will need. That's a rhetoric problem, that is, it is poorly presented, that is to say that it is... If you were to say that we have a rhetorical call to action strategy, and we are trying to catch general opinions, but now, how do we do to catch them? At the beginning, it's a rhetorical call to action, and everything we have at the beginning is declarations. Yes, but what we have in the text is essentially declarations of abortion. We believe you, we believe you in words. And in fact, here you are going to go through the description of the practice itself. The rhetoric is not your profession, is it? No, of course not. There is one thing, it's that when I cross words and so on, it's true that it's a general question and I ask it, it's not a real question, it's that I rely on the fact that, basically, I'm talking about texts that I don't know. I know that there is the no and that I can't do that, but... The idea is that I take an hyper-classical corpus and therefore in a sense there are a lot of points that are said in two lines that would deserve three pages of presentation if we were to present them to people who do not know that, but it depends on the fact that everyone has heard it. It is for a reader who basically knows the history of mathematics and classical analysis. No, but that, in the introduction, is via the history of mathematics, of mathematics. That's what we support.

10:00 I'm going back to x, y, z, z, z, z, z, z, z, z, z, z, z, z, z, z, z, z, z, z, z, z, z, z, z, z, z, z, z, So, the first part is about the two practices, and the second part, because in the second part there is still time to distinguish, because if I put Borel with, on the occasion of the commentary, the introduction of a more general concept of mathematics, it makes it a bit... I don't know where the UDL is written, except in the UDL dictionary. I've been looking for it since December. Why is there nothing about the human culture? It's just a question that appears in a paper. And then you take up the citation by saying UDL. I even wrote to you, I didn't see it, but it's my thing. But I think that if you want the opposition between a general theory of the conceptual and a general theory of the conceptual, it's not a good idea. No, page 21, look, it's like I'm finishing part 2. Page 21, the last paragraph of the part 2 of this map, despite being radically opposed to the legitimacy of the A-level, which is the algebra. It seems to us that Lagrange and Cauchy have at least this in common. They study the general behavior, i.e. with the exception of the space of the variable, for what we will later call visual functions. So the agreements are the fact that this idea is important, and at the same time, you as a reader, you have more retained the rapprochement between Cauchy and Lagrange than the evident disjunction of algebra.

12:30 So, if you want, I wrote down, let's talk about a pleasure in Lagrange. It brings us back to the topic. And then I put the second point, Poisson-Cauchy and augmented reality. But you'll see that in March. Ah yes, but that, in Cauchy, the notion of... No, Poisson-Cauchy is what I told you. Okay. So, in Lagrange, general behavior. There is a general theory, but which cannot make the economy of a distinction, of a discussion, of a general case, a particular case, even if the discussion is a particular case. As you have explained it, it is at the numerical level, but as you have explained it, I think that it will be very interesting, it is necessary to dispense the generality at the level of the form of the requirements. I think that the generality you just said is even more general because we can talk about generality in ensembles. And not only that, it's not only the size of the functions, but it's also the size of the functions in relation to the functions in relation to the functions in relation to the functions in relation to the functions in relation to the functions in relation to the functions in relation to the functions At the end of the lecture, you will see the density but with examples and without looking at how we could use the density variables in relation to the equations. We feel that it would be better to try to find an example of how to use density variables in relation to the equations versus Borel and Cicely. Can you show us that there are different ways of...

15:00 That's your answer. The article is there. Yes, very good. Of course, I wrote that before I wanted to explain it there. But now I'm going to write the last version. But it's better if you want me to tell you... At the beginning, you have to say... At the beginning, you have to say... I'll come back to you. ... to classify and then try to explain it. So, you can come back to me if you want. Well, I just showed you that we would have a measurement... On that, I'm going to tell you something. There is no measurement, but... There is also the dimension and the force. There you go. Dimension and force measurement. So, I'll take the microphone again. I'll uniform myself to tell you that it's stronger. You see, if you send it to the computer... You can prove the need for different objectives, to measure the magnitude in different ways. In the version that touches on the technical aspects, I refer to the density of the material. I agree with Mr. Romain. At the beginning of my lecture, I was convinced by the expression object domain. In the lecture on mathematics, when I read the object domain, I read the domain of... The domain source of the potato function. Yes, so I didn't understand anything. That's funny. Yes, I understood, I said to myself, but what is this? And it's only after that I understood that... So, if you want, I'll just tell you that when you think of a question... No, no, no, no. It doesn't bother you? No, no, no. No, it's funny. Thank you very much for your time, and I hope to see you again in the next lecture. To be clear, when we expand the domain of objects, we release the contrived objects and we walk in the opposite direction. But you could simply talk about the domain of functions. No, that's obvious. Because the domain of functions are two different things. Yes, they are domain of functions. Listen, I found that the expression object-domain is not the same. I'm sorry, but I'm sure I'm in the wrong place. The field of function is still in the short-circuit.

17:30 I don't have a problem with that. I've come back further and further. You've come back further and further, I understand. I read it in... It's a commentary on the situation of Poincaré, so you can't... You just read the quotation of Poincaré, so you understand what he means. Poincaré says that we distinguish two fields, one without limits, the other more restricted. The first is that of function in general, the second that of functionality. No, but I was sure that I had the right words, I'm sorry. Don't doubt it. I'm absolutely ready to endorse... I agree with you, but maybe you would read it again now, having forgotten your lecture yesterday, that it would not be the same thing if you were very attentive to the point square situation. But when you go to the station, you go with what? That's it, you will determine the word. But I find that domain of objects is easier for me. That is in the case of hand functions. You see, quite simply, it's a small thing. If there is a simple way of defining it in a way, of course. When you're going to translate, we distinguish two domains, one without limits and the other without limits. Do you mean domain or object domain? Subtitles by the Amara.org community There is Michael who is still in the room to talk about the text. Exactly. I've been so completely lost in what I've been thinking about here. Is this the only copy?

20:00 We're not going to make a photocopy. Okay, no, it's not a copy. This is the copy with the equations. Okay, okay, it's not a copy. This is bad. Yes, yes, yes. No, no, no, later, later. Not now. No, I'm sorry, it's been totally possible to follow everything that's going on even without the text. Peut-être plus facile. It's logically wrong if there is no... It's not a priori. A priori, yes, precisely. What? A priori, yes, precisely. If it works, it's because there is a context that has not been explained and that allows it to be just in this particular situation. But a priori, it's logically wrong. What I simply want to say is that you cannot make a general diagnosis that... If you want to mark it, the midi logic that there must be post-analysis attractions since the usual mode of reasoning is useful, we pass from special to partial. But I said if you want that in a book we will think about what is the generality, how there are people who can elaborate systems where we could pass from special to general without making a different demonstration and all that. If you want, you should at least point out that logic and history are two different things. Logically, he's talking about logic, he's not talking about history. Yes, but it's not history. What I'm trying to say, if you want, is that... No, but it's history that you're talking about. No, we've never been objective.

22:30 We have a way of reasoning, which is that certain propositions that have a different meaning... Predicate logic, you can talk about a particular case and if you don't make any assumptions about it, you can go from the particular case to a universally quantified proposition. That's an example of, you know, you're reasoning about a triangle, you don't make any specific assumptions about the triangle, so, in other words, that particular triangle can stand for all the triangles, and that's codified in predicate logic. Yes, but it is not a particular number. The term of induction is cheap, but who knows who is going to reject the projective geometry and consider it by saying that it is only an induction, it is also cheap, right? So in fact, at this moment, there are a certain number of people who consider that in the absolute it is impossible for a reasoning to be correct if it goes from the special to the general. But, for me, there is something that is extremely interesting in relation to generalities. It is that there are practices that allow us to go beyond traditional practices. All of these terms allow us to pass from the special to the general, and they are correct. Because there is a context. Indeed, with what you say, it would be a pity not to show the richness behind it. However, in Kochi, in analysis, I did not see that at all. But no, I'm not saying that... I'm not talking about mathematics in general. I'm simply saying that...

25:00 No, but if Couchy had the same arguments, both to reject a reasoning of a spatial passage in general, in analysis, which would be one of the elements of his rigor, and that he would have the same reproaches to the projective geometry, it would be super interesting. I haven't seen it in years. I don't know the text. I'm not criticizing the fact that before him... We can draw general conclusions or particular cases. Maybe he will think about it, but I don't know. I'm mixing two articles. You're talking about scientific fiction. When did I learn about scientific fiction? When did I learn about scientific fiction? If this is really happening, it's really interesting. But I don't have much to say about it. No, but what I really want, if you want, Abel says... It's a sad way of thinking about the special general. Right? Well, you insist by saying that logic tells us that there must be a question because it's a question. It seems to me that one of the quicknesses that we introduce in our studies is to show that the states-states and the parties in general are not all cut out for each other and that there are modes of analysis. This is not a general idea, it is quite transparent, we say the same thing about Aristotle, and the general comparison is not incompetent. I thought you were saying that the triangle is thinner than the other one. No, but in the example of the general triangle, in the example that you take, you have to understand that there is a possibility that the case that we take represents a class of data. Otherwise, we are stuck. So, if you want to say that analytical functions do not represent all functions, you are right. Or algebraic functions, because... He's right. Even from your point of view, he's right. No, no, but what I'm saying is, well, I think of things from a different point of view. That is to say, we can take the first sentence... Everything is based on geometry. Everything is based on geometry. Where we use only one relation, only the relation to the other. All of this can be established on a particular figure and be constructed on a particular figure within your profession.

27:30 Is it a particular figure? No, a particular figure, a circle. And it is based on all the conics. Ah, yes, of course. Because the circle is part of the conic family. In what I repeat, there is a context that is not explicit but which is subordinate to the fact that it allows to pass from the particular... Special in general, if there wasn't this implicit context, it wouldn't work. I think that we are supposed to have advanced a little bit in our research and I think that the way of taking mathematics... I would like to say in two sentences that in general mathematics is not complicated, everyone knows how it works. Evidently, I think that it seems to be a bit wrong. Yes, I think that... It's Abel who speaks, not you. No, because Abel doesn't speak logic. Mathematical truth cannot rely on induction. I don't think that I put it, so amusingly, in Abel's mouth. No, no, no, that Abel speaks, that's for sure. But I think it must be at your place that I found some things, some doshis that made me think of... It seems to me that page 19, if you see page 19 in what you say about Cauchy, you have all the criticism of Cauchy compared to you. In other words, in terms of methods, I tried to give all the answers I could to the algebraic analysis, which I am very proud of in France, in order to generalize myself, to reason the generalities of the objects. The generalities of the objects, in fact, is what people... But that's what he says at the beginning of the algebraic analysis, it has nothing to do with France. It's the beginning of the algebra analysis. But he's going to take the same arguments apparently as the resonant arguments. When Bosley presents his works to the audience, it's Gauchy who is the reporter. It's exactly the same mathematics. You know what I'm talking about, right? Yes, I know what you're talking about. It's exactly the same arguments that will come out. That is to say, it's exactly... It's induction. And it's only induction. And I can't do induction. You see, there is a really interesting articulation with the generality that you're talking about

30:00 and the generality that will be expressed about Charles, about Primoz, etc. So I have two things. First of all, this articulation, I agree that it's quite interesting, but I didn't know how to read it. When I read your article, there, I didn't know what I was talking about. So there, of course, now that you've explained that, I think it's quite interesting. And second thing... This is in part 2, and the other one is in part 1. It's because they don't talk about the same thing. Here, it's really a question of linearity and validity. It's really the need, aside from any functional equality, to describe, without being in the rigor of a book, to describe the domain of linearity. It's a whole other thing than saying, do I work with a general function or do I only look at particular functions? That's what I call it. It's really a first type of work on generalities. Here we are really in the distribution of functions between general functions and places where I can't talk about them in the same way. So that's still my articulation in two parts. Mr. Citan, I don't agree with you. There are two different articulations in the article. I'll have to choose one. Or I'll have to be very clear about the points of the tables. What you just said about the reposition to Cauchy, with the citations, I didn't understand what the reposition was. Because I don't have the context at all. Thank you very much for your time. And to distinguish where she does what she does in general and where other people need to talk about it.

32:30 In any case, compared to Cauchy, there are, at the end of the day, the passages of Constantinople, the imaginary expressions, Lecassave, the induction. There is really the whole set of thematics that are really exactly the thematics of his position in the concept. In other words, I have... In my drawers, there is an article on Kochi by Aurore François and I do not see it in the discussion. On the other hand, from a general point of view, if you will, it is really interesting to see that the different subjects are concretely articulated. No, no, but I do not know the part of the article, but I will read it. Very exactly, in fact, I do not know if you will remember because it is a bit far away. In a projective geometry, we learn about continuity principles, and in fact, Gauchy criticized continuity principles as an induction that has nothing to do with it, etc., and here, in relation to the general expressions, the analytic version of this continuity principle comes into play, which is subordinate to this citation. It is all the more important to... To mark the distinction between Cauchy and Consuelo on this point that in some texts we assimilate them, it may be that there is a continuity in both cases. Ah yes, there is a continuity. We study the continuity, one in geometry and the other in analysis. In a certain sense, for us, the argument of relativity and algebra in analysis is a great achievement. We, as architects, are not satisfied with these terms, even though we have a connection to the projective geometry, because there are two cases and two names. At one point, Cauchy sees both of them based on the field of age. So, in the same critique, the argument is the same for the field of age. So, for us, on the one hand, there is the projective geometry, and on the other hand, there is the objective geometry.

35:00 In fact, if you want to... How do you think people think? Is there a form of epiphytology in the world of science? Yes, there is a historical form of epiphytology. Is there a general form of epiphytology? Does this general form of epiphytology work with people? Are there underlying hypotheses about space? On which are based the algebraic equations, it is enough to take the form of the geometric equations that we have made specific to the algebraic equations to find the geometric equation of this generality. In fact, what is a link between the two? Yes, when you see that there is a link between the two. Thank you for your attention and see you in the next lecture. There are more rich examples for the density than examples. At the same time, the interest of the example of Q in R is that it allows to illustrate the variety of points of view. Suddenly, I look with the density and I see that it is the general case. Suddenly, I look with the measurement and I realize that it is a case of division. It illustrates the relativity of the concept. And then, for the dimension, I also have an example that is square because I do not know what I am talking about. It is an argument of dimension and density. Fincairé critiques des raisonnements des effets Klein du type dans l'égalité des dimensions dans les espaces de paramètres. Il montre qu'il y a une injection, une application univoque injective entre deux espaces de paramètres dans les mêmes dimensions et qui s'est pliée pour Klein, alors que Fincairé dit qu'il y a beaucoup de travail et qu'il manque un élément...

37:30 Il y a un truc qu'on voit. Qu'est-ce que tu as fait, enfin pourquoi la dimension c'est différent, soit de la mesure qu'on a pris ? Well, all the North sides are null, it's at the same time an empty interior that is null for a museum. Yes, but what is it different and why is it not covered either by the museum or the museum itself? That is to say, there are several things. There is a part that I did not give in the exhibition and that I did not take back in the article, which are passages of square points. In the Great Memoir on Fuchsian functions, there is an explicit passage on the question of the general and the universal and why the arguments of dimension are not sufficient. He wants to show that certain classes of functions allow to integrate certain classes of differential equations. And these classes themselves are arranged in parallel. So in the first time, he looks that in most cases it is not the same dimension. It's not the same number of parameters. I'm sure that it's not possible for this function space to be equidistant. And in the list, and in the list... If you want, you have made allusions to dimensions, such as density, dimension of the distance, etc. I did not understand what you were introducing exactly as an additional thing. I think that the reasoning E, for us, is not only a logistic value, but an usual one. It depends on two independent parameters. The other depends on three independent parameters. It cannot be put in the past. So, Poincaré, in his two lists of family of objects, shows that in almost all cases, it's not the same dimension, so it's not even worth trying to figure out why it's all the same, and then he says, in one case, in some cases, the two families of objects depend on the same number of parameters, and then he tells us, of course it's not enough.

40:00 The first argument is that, in general, equations are not integrable by their functions, because the functions' spaces are the spaces of the entire equation. And there is a case where A is the right dimension, and so he says, here we can no longer settle for these families of arguments, and I'm going to develop a whole new method, and there he shows the topology of the two spaces of parfaits, to prove that an open and closed application is in fact... In 80 pages, in order to simplify this space of parameters, he invented an argument that no one had invented before him. But to present the organization that we have here, he presents it in terms of two waves of general arguments. An argument that was very roughly determined in 1842, because when you write the thing, you can see that there are parameters in it. You can count the parameters. But in fact they are not independent. So he spent 30 pages studying how many independent parameters it depends on. And so he made the first type of general argument that shows that in general these equations are not resolvable. The solutions are not equivalent to the value of these functions. In the case where the question remains, it has not been eliminated by the first wave of cremations. You have to go further. And here we go again for 80 pages of study with other techniques. So we see the articulation between two-dimensional arguments, which are in the process of the first arguments of fluidity and fluidity, and the second generation of general arguments. So I think that there you, with Borel, you describe well what the mediums have to say. I think it would really be a pain in the ass to talk about these things. So the part that I sweetened, that I did in the exposé, but that I sweetened there, because I got to 29 pages on the square point, is that honestly, I was also limited to things, even in general mathematical calculus, all those who have a bit of a history of analysis, everyone didn't tell me that there was a problem to try to find out, Integral, Riemann, everyone didn't know how to talk about it. So there you go, that means that at the end, there will be a third part. We can see much more difficultly in mathematical physics than in mathematics.

42:30 You can tell us what you want to do. You can tell us what you want to do. You can tell us what you want to do. You can tell us what you want to do. You can tell us what you want to do. You can tell us what you want to do. You can tell us what you want to do. You can tell us what you want to do. You can tell us what you want to do. You can tell us what you want to do. You can tell us what you want to do. No, no, it's simply the notes on the article that you're talking about. Well, I don't know what you think of it. I think it's still a very interesting aspect. What I say with my hands, does it appeal to you? Especially when you say, from the introduction. And the article is beautiful. Well, that's what we said at the beginning of the class. That's what we thought at the beginning of the class. Oh yes. And in fact, it all comes to an end, but we can't really work on it. We can do it at Borel. Yes, but I don't think there's anything to do with it. So, I have a little idea here that I hadn't thought of before, but it's because we talked about Marko, and we talked about his exposés as well as others that we did here on Lagrange and the extensions of Lagrange. I feel that there is a form of generality embedded. What you are not talking about is the algorithmic aspect that Marco has mentioned several times. The notion of form in Lagrange is very nourishing, even if it is not as explicit as it is. But this is how it goes from one function to another, from one coefficient to another, etc. There is a whole underlying algorithmic work. I doubt that you are going to start developing on this, because it would take a lot out of you. But since you are talking about mathematical practice... I think it was really part of the mathematical practices that nourished Lagrange's work, and not talking about it at all is a bit of a shame, so maybe I can make a note or something like that.

45:00 But I don't see the link between the general and the embedded. I think the end of the supra-sense is too much for me to talk about. It's embedded in the sense of... how to say it... What Lagrange insists on is the notion of series shapes, etc., but what nourishes the fact that these series shapes are generally conceived is a whole algorithmic work on the passage of coefficients from one to the other. That's his basic material. What I mean is that his notion of series is not a formal notion, it is completely nourished by a whole work of calculations that... Marco has explained this perfectly several times, even in Newton's book, so it goes all the way up to here. I don't know if it's not too bad because there are calculations. I can send you the corresponding texts from Marco. I think it will be good for us to have Marco's article, because I think we need a backup. It's not an a priori formalism. It's a formalism that relies on a relatively inductive practice of passing from the writing of a coefficient to the next coefficient. Algorithms are fascinated by them in the process of calculation. That's it. The generality is not, I mean by that, it is not set a priori in Lagrange as an argument that would be strictly philosophical, it is based on an effective mathematical practice.

47:30 Wouldn't it be better to say that the generality practice is an element of the reason practice? Yes, that's it. If we read Montaigne, it's not the same, but it is. I would say that for reasons of comparison... I take an element of Lagrange that seems to me extremely representative of something and that it is not at all an analysis on the generality of Lagrange. That's it. Okay. There is a reason to... In the generality of the term of physics, there is also a fairly strong thesis that we have never seen before. Mathematics is a recurrence and I mean that... I have not been totally convinced by this thesis without a little... So it depends. If my concept is that there is still the... That doesn't make sense. Because it's still the... We can assume that there is reflexivity in the others, on the classifications, that's exactly the comparison with the classifications of physics. Yes, but the idea is that there is a mathematical tool, all of a sudden, that allows us to... It's a bit like what we saw the other day with... Cateodics? Cateodics. For me, they are exactly the same, they are a theory of what classification is. Yes, but you have to regulate it. You have to regulate it. No, you have to go into biology. Why do you have to go into biology? I don't know. It seems a little too strong to me. The idea that this kind of insecurity is too much material. I know that it's a theory that we often get out of it, but... I know, what David is telling you is that others are going to say it, and then it's going to be wrong. Yes, of course. Yes, yes, no, no, but I... I don't know. No, but I mean, everything we tell you is a reaction of a vector, then it's... I don't know, there's a vector, then it's a vector. But I'm really saying that it's specific to mathematics. You say specific? Yes, yes, yes. Specific to the mathematical science. Specific. You have two hands, don't you? Just before the references. It's a bit strong. I think it's a bit strong, but I don't know if it's very useful. Can I suggest that we stop talking about the second part of the lecture? Can you do a little bit of a remark? It would be interesting to have a little bit of a reduction because otherwise the margins would be a bit...

50:00 Whatever you want, I'll give... It's your campaign? Well then, yes. You know, regarding these elections, I'm confident. I hope so. If I'm not elected... If I'm not elected... So much the better. You see what I'm saying. I'm confident. Regarding these elections, I'm not elected. But wait, but those who want to go to the elections, well, yes, go to the elections, while going to the office... I'm much more interested in the fact that the geometrical tradition on the reception of geostars, we always have a lot to talk about, the debates around them, and the contemporaries. It's a very interesting literature there, with people like Steiner, particularly, who are astonishingly productive. And one can speak to that regent for the first time in a very short time. But because it was a kind of intranational crisis, a normal crisis. The subject of, not the subject of first order philosophical questions, but rather of the issues which seem to be sort of gone far deeper than they seem to be. It's been very interesting. It's happened in the functions that I... Yeah, that would be a good point. It's a point which I think is often made. The outcome of the... Bon, alors, excusez-moi, je reviens. Okay. And one in a lot, a lot of them were, were, were, were, were, were, were, were, were, were, were.