Generality & C19th analysis

0:00 I think we still have a lot of time, because I think it's time to wrap up the lecture. So, speaking of which, I have already done a lecture in English. Sometimes, and I don't know what to say. Well, I don't know what to say. Ah, I don't know. I don't know. So, I learned English very well, and in order to improve my English, I copied Times and Context, which I adapted a little bit, so I have a huge file. It has absolutely no interest, except for the preface. It describes terms that have already been used in books, so it can serve as a framework. Maybe, what I prefer is to include it in a journal, in a book, in a journal, if there is nothing specific in the book, in the structure of the terms. And if it's not, maybe we can decide on the date of publication of the two editions, with the standards. And you will see the details of the format of the two papers. Can I show you? I have the proofs. Do you want me to show you what it looks like? Because, in fact, I have never... You have to be confident. Yes. I have never checked. If I take an hour, it would not have been nothing. But if I take an hour, I will show you what I have.

2:30 I asked myself, since I am an expert in the field of quantum mechanics, if it makes sense to have a proposal for a library of orthodoxy and mathematics. Thank you very much for your time. In fact, the real reason why I thought about it is that I have received a handbook that has been evaluated recently, so it helped me to have this idea. Thank you. You know, Karine, I already work on a handbook on space and time. I would be very happy to do a seminar on this subject with Professor Peter Montferrat. I think that the rainbow was more familiar. I received a set of presentations with dignified resumes from all parties, BLG, English, French. Each one took the time to have his or her own space. We are not talking about subjects that are related, I don't see how we can not talk about them. So I feel that it is still a fairly close format of what we are talking about in fact. So what I tell myself is that maybe, well, I don't know what you are going to say. But why, when we are going to discuss the introduction, one of the things I said earlier is to try to see how we can organize a lot of things and maybe we could ask each of you to make a summary of 10 lines to try to prepare a presentation of the volume.

5:00 Do you agree? In the framework of the book, there is no constraint like you, it's not a maximum. No, but what we can do, because I don't know what to do. Generally speaking, in the book, it's 20 days plus an intro. What is the plan to change it to make it more interesting? For the people who are affected. Just the authors of the book, not all of them. Because the people who are affected. Yes, the people who are affected. Yes, the people who are affected. And what we can do is, if you want, Emilie is in contact with someone, I am in contact with someone else, so we can both ask if there are standards. The series of Cambridge University Press has been called the companion series, companion A, which is quite the same format as the handbook. It's a very well-known series. Oxford, because these two series are well-known. Yes, but also for Cambridge. It's curious, because normally this series in Cambridge is called the Companion Act. It's called the Cambridge Handbook of Literacy. Cambridge Handbook of Literacy. But we can start with Oxford, because you already have people in contact with Oxford.

7:30 I recently read my book, a book by Marcus J. Winfield. This is the new book by Barton Frasen. They are allies in this, really. And so, this book in general... It will be in the same spirit. I don't know, I don't know. What is the last book by Samprati? By Samprati. It is very pragmatic. Perche, Perche? On the pragmatic that the instrument is necessary for mathematics. And also the theme of the presentation is in the title itself. Let's say it's an idea that came to us because we saw it growing up when we said we were going to get rid of it, but it seemed to me that it was maybe a... So if you want, we write to see if there is a size. If there is a size? A size? Yes, if they impose articles of 25 pages maximum. In that case, yes. How many pages maximum? No, maybe. If there are. If there are rules. So, if you want, I feel that... I don't know what you think, but maybe, on the 7th of November, we can prepare a presentation of this, due to what Emily and I have received as a return from the University of Oxford, which is also a way of preparing the presentation to see how it will go in the future. And the seven themes, especially if you remember them, we can also, on this bar, we can fix the deadline and determine the norms definitively. So you thought of Springer, but you prefer to start with... That is to say, if you want Springer, I don't see that you have to determine it.

10:00 It's just that... And also, they are always so expensive. It's a question of, for example, 75 euros versus 150 euros. When did you come up with this idea? There will be constraints. Generally, a handbook is something that is supposed to serve a student. And so it has to be accessible. There are constraints on the content of the whole pedagogy and the total coverage of the question. We can try, we can try. Listen, sincerely, I have contributed, I have evaluated nowhere in French and Swiss, I have had, in the last two months, the editors and all the senators who tell me what to write and what not to write. All I can say is that it is practical, practical, practical. You can not lower the level. No, no, we're not talking about all of that, we're talking about the strategic presentation, because we have a certain number of texts that are quite difficult to access, we know that, we have to discuss between us, so we have to see to what extent it fits into a handbook project or not. I would be delighted if it could fit into a project, but you'll see how I'm going to react. So, you've conceived in the spirit of a handbook? No, but if you want, the problem, I think that it must be shown and understood by a certain number of people. The subject of the lecture is also related to the subject of mathematics and physics, but I think the subject is the format that we have today, because the format that we have today, I don't see it being a real book. How many articles do we have? Thank you for your attention.

12:30 Do you think there is anything else to be said about these movements? Absolutely. So, I propose, one of the aspects is that you are asking for a number of questions to be answered. By the way, if you could centralize them and verify that they are already in place, that would be great. And when we talk about production, we talk about this, and Emilie and I, we are going to talk about this separately, the two editors separately, which is a way, in my opinion, quite interesting to keep the project. And so we will disseminate this information if they have received the project. You have to send the resumes to us and send a mail to ask for the resumes to all of us. When the 4th year is over, you put them together. You can explain the reason why we came here. For the editor, right? For the editor, yes. No, no, I'm sorry I'm late, I had to ask Renaud to... Think about proposing books like a book for Oxford University College.

15:00 That's it. So, if we go there, we can submit the results and do it in any way. We will submit the results. You can use them with me. Submit an exam with us, give us a review, and we will move forward with the second lecture, and then we will submit a paper. If you know that we are going to submit a paper, tell us in the comments if you want, otherwise we will look for you. Okay? So, let's go. So, I'm not going to start a debate, but I'm going to let David speak. I don't have much to say, I think it's really good, it's interesting, the only thing I have questions about is the Kochi side, because I think it's going a little fast, if you want. And on the branch, I think it's going well too. No, on the branch I understood well, I don't think there is only one point, but the point to be addressed. I understood the opposition very well, the idea of using genetic solutions based on simple elements. I don't know. But then you do... On the branch, the table with the eight... Yes, yes. And then you move on to Cauchy. And I understood that it demonstrated that Cauchy is still in this style. The branchian, except that he substituted it. The idea is that... He is facing the same challenge of generalities. And so, first of all, if you want, I found that there were not many elements in the third term, but in fact, I did not quite understand why. I just want to believe that continuity will play the same role as the genetic description in terms of form.

17:30 But it's not easy. The form in the voice is written. I didn't understand what the form and the genesis of the description were. And then, on the other hand, it seemed to me that it came to emit a kind of small grain of sand in something that Tessier had put in place in the first two parts. That is to say, the idea that with the appearance of rigor, with a culmination that is imposed, a new practice would be put in place, which would not be that of Lagrange. I find it a bit embarrassing to put Cauchy on the side of Lagrange, we are a little bit in trouble. I found Cauchy to fit very badly in the table, because if we take your opposition table between Lagrange and Jordan, which is clear, for me it was not clear. And in general, the idea that you put in the table that generality derives from rigor, in opposition to generality that derives from simplicity. I understand the generality that derives from simplicity, but I don't understand the generality that derives from rigour. I saw that there was... Do you understand, but do you think that the generality that derives from simplicity is treated between the two? Apparently, yes. I thought I understood that one would exhibit the generality of the notion by giving it a simple form, and that it would be considered as an element of basic principle. For the introduction, I think it's not exactly what I was saying below, it's always the fact that the way of exploring the system is to give the basic elements and the basic rules of construction. As soon as we give something that is at the level of the basic elements, it's good for everything, that's what I'm trying to say. I think it's rather... That's what I understood. Okay. The simplest example to illustrate such a problem, such a difficulty, such a theorem, is the choice of the simplest, which guarantees that what we are going to say is applied later in the most general way.

20:00 In a model, we can use classical mathematics, that is to say, if we manage to get to certain levels of mathematics. We have obtained the maximum of generalities, well, the analysis in the 18th century is not in fact in the 17th century. The analysis in the 18th century makes a lot of sense. Yes, but there is still a fairly strong sense that I find in the general. No, I'm talking about a paradigm that has to emerge. Yes, but we see a lot of people talking about analysis. But that doesn't mean they're talking about the same thing. They're talking about the same thing. It's a huge problem. What is analysis? And I may be wrong, but I'm going to say it myself, without even complaining, because I've talked about it in a lot of lectures, that analysis is not only the interpretation, On the other hand, it is certain that it is a paradigm that emerges as a discourse, and we recognize it as being a certain general discourse, which is what I wanted to say, when he described it, he referred to it, I recognized it anyway. So my problem was not there, really, we can discuss it later. In my case, it was rather Cauchy who entered the painting. I didn't see how Cauchy entered the painting. I saw what you wanted to say. I'm just explaining the fact that Cauchy doesn't enter exactly here or there, and it's impossible for him to enter from the side. It's normal for him to enter the painting. But in fact, it's not really like that that you present it. You say, we might think that he's more modern, but in fact he's more like Lagrange. That's what I understood. And I found that to be unclear, I think, the problem with mathematics. In fact, even when we were discussing it, I was saying, we were in a situation of the end of the square, etc. Again, again a little on the regard and... But in fact, you say a lot of things. It's not the same, it's not the same, it's the same, the same.

22:30 But what I miss a lot, it's directly related to what you say, to clarify what she's doing, what she's saying, for example. What I say, for example, is that it's a problem, because... I think we all have a background in life on Cauchy and the rule of law and so on. And sometimes there is, sometimes there is, but maybe it makes a difference. Here, I take the passage of Cauchy. In fact, in Cauchy, I have two stages. I start with one, which is the classical reading, the rejection of the generality of the algebra, the so-called Niveau and Niveau V. And then, in a second stage, People pull each other's hair to see if it's ours or not, and so on. So here I'm analyzing the concept of continuity, not even by terms of content, but by the function of the concept of continuity that I analyze as a function of argument of generality. But what I don't like about it is... For me, if you want, you can put the generalities of the two approaches, but for me, you have not elaborated neither how I am now, nor what the generalities are for me. That is to say, you make a hypothesis that it is Parmenion who plays a role in relation to the subject of law, but I have not been convinced. In Kochi, there are elements that allow us to expand our understanding. Of course, because there are two things, there is one in the effects. Plus, to quantify the length of the lecture, that is to say, on the other hand, he also talks about what is the general conception of Kochi. That is to say, if you want, the question is, there are two ways to understand the general idea. In a way, yes. But, for this reason, we need to know the arguments that show that, at Cochise, there is the will of the consumer, the consumer in general, and that this conception of the society, it reads the form of what you have in your hand, which is marked in a beautiful hand, and for me, it's not the same.

25:00 I would just like, for your information, to read what I have found on your book. So, it's in a very particular context. In short, it is in a context where one wants to work on a theory, one wants to improve the analytical theory of quantum mechanics. But when he says... So here you are abandoning the discussion on quantum mechanics? Yes, of course. I just came to look for the... Ah yes, you have to answer David later, excuse me. So in fact, in short, before this lecture... We have completed a few propositions for mathematics, and we have drawn the Perfection consists of employing as many possible principles as possible and to bring out of these principles all the realities that can be understood by the single force of humanity. For me, this is a different conception than the one you described and which I refer to in the book of history. It is a different conception which is to identify the things that can be understood in principle. And they are more or less made in the sense that by simple transformations, we will generate a whole field, and that we have in the mechanics of the air, in the treatments of the general geometry, in the mechanics of the branches, for me that's what the mechanics of the branches are, you can find the principles in Martin Luther and Karl Marx and so on, so it's really, it's to say it's not a marginal conception of the world. I just want to point out that there are several different concepts. Yes, I agree with that. I think that both are incompatible. Because in fact, you have the analysis of the principles of one thing, the analysis of the principles of another, etc. You will see in different ways that the analysis is not incompatible. The question is whether we consider the analysis as going back to a certain number of pre-established truths, whether they are constructive or not. And the more we have reached the degree of generality, the more we have these things, the more we can solve them.

27:30 Yes, but the problem is, you know, that's what I was talking about before. It's our status, what you just said, these things. Because these black things, they can be simple things. These black things, they can be the things of the world. And from there, the nature, the generality, the same, the same, the same. But in the case that Renaud analyzes, in the citation he gives on why these forms are general, and that we should not yet take care of the particular values ​​that we find in the antithesis, there we are still in the case of a simple form. The value is not given by the frequency. It corresponds to the citation of... What I mean is that I have only one formula, and by applying particular values to certain letters, I will have the particular cases that will be in it. But I have only one formula that contains the whole theory. That's how I hear it. So maybe I'm out of tune because it's also the reading that the English do in the first half of the 19th century, i.e. science in general, if the concept is not part of it, they are, for this reason, against Cauchy, who wants to put the multiplicity back there and with the principle that everything is... There is a function other than a certain form, because as we said, there is on the one hand the formal form, which is the rest of the form, and then there is what it gives, so we apply some particular values that are two completely different points, which is the great distinction. And indeed, I think that the form that represents the function, because it is not the function but it represents the function, this form is the general principle from which we can define the whole theory. What I wanted to say was that, if you want, when we talk about mathematics, we run the risk of wanting to put things in other people's hands. I would like to ask you to show me the continuity of Cauchy.

30:00 As a problem of generality, there is a logic that is not enough to be commented in the text, there is a simplicity of perspective, so in fact I would have tried to comment everything in terms of generality, and not because it comes from the text. I'm sure you're right, but what I mean is that they are not in the text. And in the organization of definitions, theorems, etc. Then the question is, according to your remark, where is it? Is it the theme or is it the thing? This article is a giant brainstorming on... On you? No, on me. Yes, but I've never read it, I can't remember. No, but you ask me, do you not take the thing that is related to mathematics? No, what I read is... The definitions of the enemy and the activity of the enemy are the tools to build the general direction of the economy. I don't know if you understand what I'm saying. It seems to me that precisely the one who wants the general direction, who aims to achieve the general direction, and sometimes his enemy is that the enemy and the enemy are two sectors and that's why he wants the general direction. And if you want to know what he wants to do, what he wants to do in the economy. This is a demonstration of what the tools for mathematics are. What is the nature of generality?

32:30 I have a suggestion. It seems to me that when I made this table, this table is themed... There are differences between these two people, which detaches the discussion and which makes the level more abstract and which is not often a discussion of the grand jury. And so my suggestion is that from this point of view, write to me in a second article. And you work this opposition by trying to treat philosophically all this project. From a historical point of view, Gauchy is the essential piece to understand the articulation. Because here I present two slightly frozen tools, and for the needs of comparison I freeze them. It raises the question of how can we pass from two epistemological universes so strongly structured. Still, we need to know. And precisely, to show Cauchy, it allows us to give elements that are not only historical. There is another perception. That is to say, if we look at the texts written in this period, some follow this conception, others too. And the students of mathematics, they have both, because both of them, even because they are in propositions, are essential for the pre-religion. The students of mathematics today? In what period?

35:00 What period? That of before or today, in March? Um... From a historical point of view... Just because this one and this one is apophysiac, so you have some texts written in this tradition, some texts written in this tradition, but the students who are coming up have access to both kinds of texts. Because Jordan's treatise was written in the early 1880s. And Lagrange in the 1800s. So I'm not sure any generation of students ever had the opportunity to learn analysis in both kinds of textbooks. And Jordan's textbook was not very much taught. It's a difficult question. But maybe mathematicians could have access to both types of trees and enjoy both and use both. But maybe from the point of view of this essay, it's better to present this polarity and this polarity without complicating it by introducing Cauchy as the middle term. It's more instructive to leave the two poles clear. But historically distorted, unfortunately. If the two poles are clear in this part of the essay, do you really think Cauchy makes it less clear? What I think about this problem is that there are a lot of extremely interesting things in these parts, but for me, I tell you how I think, there is a problem in the dimensions. I give you an example on the page where I want to talk about dimensions. Jordan appears with the table. That is, in fact, you take all the other genres, and so on, etc., but in fact, Jorvan, he appears with the characters. It is true, if you want, that to elaborate a contrast, we wonder if Jorvan is substituted with others,

37:30 sometimes between the others, and sometimes between the others, and if Jorvan is not substituted with others... Is there anything that surprises you when I say that it's Jean-Paul who chose to do this? Yes, I commented on Labelle, Éricley, Riemann, Bois-Raymond and Hubert. It's true. And all of a sudden, it's Jean-Paul who's going to appear. Yes, that's right. So that's it. That's in two or three explanations. All right. I don't know if you want to repeat it. Yes, and his Hebrew treatise was emblematic of that. Because the others, it's still the question of emergence and evolution. In relation to these ideas, there is another part of my article, in my opinion, which is an obligatory article, because all this is the idea that there is a correlation between the form of the TV and the fact that we have to make a general image. That is to say, in fact, somewhere in point 6, you slide, you bend. There are many examples in the article that the context genre is linked to the question of the generality of the universe. If you look at sports, you can see that there is a lot of sports. There, it goes very quickly. And in addition, it appears in the table at a place... There is no such thing as a dominant unit and it will have to comment on the next page relatively quickly. Can you explain this difference? What is the difference between the two? In any case, the two are very similar. That's why I was wondering how you call this difference. It reminds me more of Klein's reflections on Jantin's theory. Klein says it's more for mathematicians than for physicists. Excuse me, I have an idea of what you're talking about.

40:00 There, there's a command of the order, which appears well. It's from 1990. Okay. It appears that treaties of the highest rigor, the peak of rigor and abstraction, may be more intended for mathematicians than for physicists. So it's true. Here, it's a point that's been sketched. The question of... If you want to keep this problem as well, in my opinion, you can give an overview on the subject saying one day I will write an article on it, but the master, the master of the thesis, does not at all have the freedom to elaborate, etc. Thank you for watching this video, see you in the next one. I don't want to... I thought that Marie-Jo supported me in the attempt to develop what I tried to explain. If you want, for me, the idea of talking about the human being as someone who is close to his genetic evolution is problematic. And that's why I would like you to send it to Marco in the press with the form and the quantity of the lecture. And... So what I'm going to say now, if you want, is when you say the description... I have the impression that you refer to the series, so what do you refer to when you talk about the general structure of mathematics? Because at La Grande, I still find it impossible that, I don't know if it's possible or not, since we have expanded the significance of our subject to be quantified by the general structure of mathematics.

42:30 So how do you catch that in a general structure? That's it, that's it. We use basic elements and training methods. I'm trying to catch up with the object field. I'm trying to catch up with the object field. I'm trying to catch up with the object field. Yes, but it's very... well, I don't know, you don't have to take what they were saying to the letter because they can say it in any way. Well, Belge too, he says it in any way and you can read it in a very modern way by saying it's any way, but in fact it's not true. Concretely, their any way is always based on operations... We can't take their texts by the letter. We can't hear them as we say them today. It's not possible. It's a point to be really precise each time when we cite them. It's the big one. Page 13. The question I ask you is whether you agree with the genetic description of the question. If we specify, as he just said, what it means, yes, we have to... Genetically, we have to say that we have real words and that we arrange them with a simple word. No, it's not quite that. It's more the form in question. What I understood is rather what you said in the... Well, I don't agree with what you said earlier, namely, there is a form... No, but from a general point of view, it's like that. There is a general form that gives... In all cases, I hear two different things. The genetic aspect is not the same as... There is the genetic aspect, the way of perceiving the universe at home, the definition of the paint, which is that it is regular, and there is also the strategy to do something with these things, which is a universal form, knowing that the question of its universality plays two roles. It is universal in its form and it is general in the sense that it works almost everywhere.

45:00 I agree with the architecture. I absolutely agree with the architecture. I absolutely agree with the fact that this is genetic. I have a problem with genetics in the range because I have the impression that, for me, if you want, it means that there is a barbed wire, and it is through the combination of these barbed wires that we will have the object in question. I'm not sure that in the branch it's limited like that, rapidly. I don't think it's limited because we don't know the list of rules or assemblage rules. It's a very vague way of... well, yes, it's vague. I... listen... It's not a definition... Yes, but I'm telling you that there's a genetic aspect to what I'm talking about. It's true that I don't feel... I'm opposed to mathematical mathematics. So, somehow in the context of... The question is, how can we give ourselves to these objects? In the history of quantities formed in any way of quantities, what is important is the quantities, that is, we have well the bricks that we are going to fill in by the letter, we do not know exactly yet where they will be placed, which explains well, in fact, it is the universal form that ultimately we will also constrain, but the concept of departure is that we have quantities, we combine them with each other, and that's what we find. The universal form is the series, and the genetic form is the formula, it's x-1, root factor of x-b. That's it. The formulas do what you want, and we treat them, because in fact... The operations to be done... The operations to be done to remind us that I've read... That's possible. I've also read a little... But I'm sure. And I can... That's it, to be more precise, there are things that... So, when we come back, it's a general question that the professors of the Secondary School of Mathematics are quite absent, because they have a lot of time for themselves. Is it normal for them to come back?

47:30 I have a very concrete suggestion. And it is important, especially if we are going to send these books to Oxford, that I think they will be mentioned. Proofs and Reputations, the book by Mark Atouche, which is very similar to this one. You have to put it, you have to say a lot of things on it, and you have to put it in your head. You have to say that you are fascinating in a certain way. That's the most important thing in my work. So it's really something that I have behind me. So, I go back to what I wanted to say earlier, on the level of... That's how I felt, with the names that you gave me, I think you have a note of what you have to do, of what you have to do, if you want. You start by introducing... Oh, that's good! I don't know how you experienced the introduction, but for example, if you want, the introduction, I'm not talking about the second part. Okay. And if you want, I experienced it for once. Usually, I explain to people what they put in the introduction and what they put in the introduction. But here, I found that what you put in the introduction, it could, from time to time, lead to an introduction. Because I find that you come back, if you want, to a motivation. In conclusion, there may be a lot of precision in the introduction, so you start with Abel, you continue with Dionysius who, by working on the generality of the general, will work on what the function logic is, then below you paint...

50:00 By saying that there are three ways to send a lecture, and by starting with a list of three approaches, sending one, writing one, and analyzing one, then... I don't see the three lines of the approach that you have introduced to follow up. That is to say, I didn't tell you that you were able to distinguish these things. And in a way, I didn't see that you were able to distinguish these things because you interrupt yourself. You introduced distinctions, and I don't see how you can go back. Surely, if you want, you can work on them in a way that I don't see. I would like to say that, as a teacher, I thought to myself, why did he introduce that? I don't see how he can go back. It was at this moment that you introduced the different things. There are a lot of things that can go back. Then you went back to the opposition... You went back to Lagrange. You look back on the graph, I didn't understand what I was doing in the economy of the article, the paragraph that indicated two mathematical practices, I tell you it's good, I tell you the truth, I despise you, and I didn't understand well what the paragraph that you put in the article is. On this subject, you negotiate in the middle, then you start in the third part, which is the general ecology versus the algebraic generality. So, in the general ecology versus the algebraic generality, I know that it makes a great difference. In general ecology, you talked about it three pages before saying, well, that's why I like your thing. That is to say, if you want, you say, and it's a good principle, it's a good principle. You talk about Poincaré and then you talk about topology. It's three times less than ten.

52:30 And you show an opposition. We show too much that we don't see things in general. In fact, we can do a lot of types of concepts, of generalities, which are different types of tools, of scientific tools. It's absolutely fascinating. But how? If you want to order a prologue in your article, you see, I don't see... The construction of the article, we have the impression that it goes in all directions, there are chronological returns... It boils ideas, there are lots of things, it's exciting, okay, but I miss... This is the first part of the lecture, and then I'll go on to the second part, because now that I've done this part, I can go on to that part. This is the first part of the lecture, and then I'll go on to the second part, because now that I've done this part, I can go on to that part. Transition or restructuring? Maybe not restructuring, but we'll discuss it later. I'll tell you what's going to be in the introduction, which I haven't written yet. But I agree with you, it's a bit of art. But usually it's funny because... It's normal, it's normal. But what we tell people is that the introduction, you have to write it at the end. So that's what you did. Well, you have to go back and forth. It's not like she moved it. No, but I like it. So you're going to explain to us and I'm going to take notes. I'm listening to you. No, I'm going to do it, I'm going to do it, go ahead, I'm listening to you. So, the contract was the term that I was interested in because, for my thesis, I didn't have the need to say that, I didn't have the need for this background, and so it was a big brainstorming with, at the beginning... At the beginning of my exploration of the lexical, I looked at the texts I had read, the films I could see emerging, with the word generality, or the word lexical-immediate. And so, once I had my first research, I had a lot of documents, and in fact, for me, in my head, it was organized in three steps. Okay, three steps!

55:00 The problem is that in these three boxes, it turned out that I was wrong. No, no, no, no. Borel. Borel. Borel is great. So there was... You saw three pieces. This table is not quite standard. Does that make sense? That's a good instruction. So that was actually my idea of probe. You make probes with a tool that is not your usual. You have to put a diverging thread that does not have the usual line. You have to put it in the right place. That seems to me to be a good example. So I get three coherent families, and in addition there is a chronological-conceptual articulation that seems to be free, the three moments are not just three steps, they are articulated conceptually, and there is... And how are these three steps articulated chronologically and so on? In fact, what I present in 1 is the displacement of what I present in 2, what I present in 3 is the synthesis, the displacement in 2. I don't understand why Lagrange was able to do this, because for me, if you will, there is the famous polarity, Lagrange, no, it's the famous polarity, Lagrange, what you say about the research, the object, the examination, the demonstration, for me it was part of it. I think that when they were trying to show polarity, it could not be uninteresting to describe it in terms of curves.

57:30 After that, you can approach it with a little inclination. How do we place Spock or Spock? There are people who have tried to be from the Jordanian side, but my thesis is that they come out of the German side, but then I understand that the third part is in the third part. So there you go, I understand, as much in part 1, we can see that we are talking about it, it is quite in line with the normal, in the normal agreement. Ah yes, no, you told me I was going to stop for... Yes, you are talking to introduce distinctions that I don't want to work with. And in particular, if you want, you have a certain number of remarks. If you want, you have a certain number of very interesting remarks, pure and conscientious examples. And examples that are very difficult to eliminate. That is to say, the ones that work to mark that there is actually something outside. These examples work in many different ways. If you want, Alain, you can refer me to some of your other examples. At least, it could be a conclusion where you can talk about what has been developed to make a conclusion on three different modes and the functioning of the examples, etc. So, let me tell you why I... In these three cases, we will not describe the entities, it is in part 1, because it is still three modes that we find in this specific practice, it is really the practice that I call the practice, the thread, the thread of Abel and Jordan, the rigorous practice of the general function as we want to do. These two mathematical practices are in some way interlinked. The big one is on the form part. The big two is on the first part with a second piece on Cauchy to explain why you think that Cauchy, from the point of view of generality, is quite wrong. And the big three is on what you call the synthesis. It's the synthesis. That's it? I didn't understand that. The two mathematical practices are between the Grange and Gauchy. Yes, but how?

1:00:00 The Grange and Jordan. If we read that in the article, we know that it is wrong. The two practices are not between the Grange and Gauchy. No, no, no, no, I do not agree. It is the Grange and Jordan. What I want to tell you is that one part of the polarity, the great part of the polarity is the proof of three microbes. The small pole of the big 2 of polarity is the second pole, and the small 2 of the big 2 is the presentation of polarity. And the 3 is to say, without saying, that it will be located close to the polarity. Me, if you want, I don't know what you were doing in your mathematical practice, but there, at this point, where are you? I understand, it's related to the genesis of the article. Because at the beginning, it was not an article on two mathematical practices. At the beginning, these are questions of different... I have listed what I would like to do in the slide, questions of generality. And it turns out that, in fact, the first and the second are linked to two different questions of generality. The first is related to what is a general function and in what people have tried to make the analysis more rigorous by saying that they had to give demonstrations. These are the examples of the types of functions with which they are defined. This was the first case in my corpus. The second case was a very different question. It was, what does a function do in general? How to distinguish between regular behavior and general behavior? So, did anyone notice this second question? And in writing, it's maybe more the fact that these two are different. It's something that bothered me, in fact, is that there is an apparent structure and there is a hidden structure. There is an apparent structure, in fact, it's juxtaposed. That bothered me a little too. It's juxtaposed because there are three ways to approach generality. You advance them as you go. There is a first one, then there is a second one. And we understand that it is not a dependent of the first one because it is articulated in a problem. This is not very right, but it is a problem of rhetoric. It seems to me that this is quite clear, your way of presenting it, but it may not be very straight forward, that is to say that finally...

1:02:30 I was the one who got involved in the marches, I don't know if it's true or not. At the end of the day, the polarity is ultimately more interesting than the idea of reading. When the list emerges, it is mainly the two practices. On the other hand, it does not at all solve the problem. I find you a little severe on the subject. It's not a seniority. If you want, I'm a lecturer, if you want. I say absolutely word by word, I try to see that I understand absolutely everything, etc.