The continuous & the discrete & set-theoretical thinking (contd.)

0:00 The project of the acquisition of mathematics was something like mainstream, we can say, but we can really see two very different sides of what we can call mathematics, and from a contemporary point of view, it is perhaps clearer, because one thing, of course, is all these things together, it is also a kind of generalization of the notion of language, but on the other hand, it is exactly this... From today's point of view, we can say that there is an algebraic side to arithmetic. I was also thinking a lot about Leo Kohl and the development of the rise of mathematics. Maybe he is not going to be in history either, but it is the beginning of history. For instance, maestros is not at all the same as Debeckin and not the same as Koneka. It's some kind of intermediate product which you accept things to some extent, but it's not... In a way, at some point I had to label it somehow, but I was calling it a semi-constructivist. You accept the real numbers, but not much more than that. And from there you want to reconstruct them. I don't know if I prefer to point out the relationship between the two fields, arithmetic and geometry, and I point out the analogies that are preferred between the two contexts. I would like to tell you that there is a need for general knowledge in your studies or only in mathematics. For example, there is a physical work where you need to be an analogist. Yes, it is necessary to see if it is possible to see that it is necessary to do the analogies, if they are necessary of the time.

2:30 If there is a context for a debate, what do you think? I don't really know what to answer. Actually, there are two questions. On the question of physics... For example, Gauss makes works on physics that are closely related to his point of view on mathematics. Do you think that there is some relationship between the need to do analogies in mathematics? I don't see what could be given, what could be the frequency of the analogy and how the analogy could intervene. Mathematics and geometry in the framework of the physical research of chaos. But maybe we need to think about it. I don't know. Maybe it's a way of seeing it. I don't know. I ask the question again to all of you. I don't see... How does it work? We don't want to answer the question of the problems of the celestial dynamics. When we answer the question of the problems of the transcendent numbers, etc., it will say that we can't have these kinds of problems, that is to say, understand the public question from a problem of the numbers. It could be... Okay, yes, that's very interesting. Yes, it will say that. There is no meaning of the constraint of the physical problems of the celestial forces from the questions of the world, because we respond to a question that is a mathematical question, which is the difference between the names of the properties of the properties of the properties of the properties of the properties of the properties of the properties of the properties of the properties of the properties of the properties of the properties of the properties of the properties of the properties of the properties of the properties of the properties of the properties of the properties of the properties of the properties of the properties of This is the question of the formation of science, precisely, for the foundation, for the system, and for the acceleration of the process. What is the purpose of all this? It is because in the works of the Congares, when we admit that we are in the middle, for example,

5:00 we start to have the problem of knowing if we have an ensemble of components at the bottom, for example, on the Congares question, etc. And, in fact, the stability, if you put the two of them... All of this can be explained by the resolution of these problems, which are the problems of knowing if we agree or not. So, he says, we can't have these kinds of questions, we can't make an analogy between physical and non-physical. It wouldn't have any physical implications. Yes, that's right, we wouldn't have any physical implications. So, we can't put the resolution of these problems there. So there you go. The only thing that can go in that direction is the very particular status of the square method, the relationship between the square method and the methodology, the measurement theory in astronomy. There is an analogy which is very profound. It says that nature proceeds as we would proceed by using the square method. There would be something that I would answer. It's the only thing I see that I don't understand. Yes, so I'm trying to answer the question that we ask ourselves. It's about the question of the extraction, the abstraction of geometry, to specify the disciplinary principle in which it is, when we ask a question that we don't understand. It's a tradition to ask the question. So the question I asked you five years ago is the same. It means that in five years I will not be able to answer. No, no, no. It's him, not me. If we take the particular case of the passage of circular functions, I mean, of circles, to the line of Skaf and the way in which the integrals are built. I don't know if we can say that we are dealing with an excursion process or an attraction of the human being. I have the impression, and I already had this impression at the time, that it is rather a form of capture or introduction of the genesis which is different from what it was,

7:30 but it is still a form, it is a very specific way, but it is a way of introducing the genesis. And I was saying that, for example, even when we parameterize the integral, the search for the right parameter... It's still a way of grasping geometry. Of course it's concentrated, but it continues to act in this concentration, and hence this idea that what you call schematic mathematics, on the one hand, it has a completely... In the sense that it specifies the construction of a discipline that has developed as such, but at the same time, it means that in the word schematism itself, it is in a certain way to have reworked geometry, as it existed all the time. So this problem arises exactly today, exactly in these periods. That is, if we take the number of theoreticians, they exist, I don't know, maybe less, six or seven ways. There are disciplines or sub-disciplines in the geometry of the world, there are people who know the geometry of the world, they are experts in the geometry of the world, that means that we think, we work in this field that is that the world has a certain form of geometry. Okay, but can't we consider that... The question shifts a bit, or the accent shifts a bit, which makes it more difficult to know if arithmetic preserves the tutelle, first of all, because what is, in my perspective, a kind of interiorization, of the absorption of geometry in the systems of the medium. In other words, if you don't want to say that the systems of the universe become of the universe, in a certain way, that's what science is, for Kronecker, it's a way of subjecting the disciplines of mathematics to the royal science which is arithmetic. But in this subjection, geometry is not in rest. No, no, no, not at all. Yes, we do not do arithmetic from nothing. By the way, we see it as just a remark, almost in the form of a goutave. In the course of Berlin, the lessons of Berlin, when he can give an arithmetic theorem, there is an example of a magnificent arithmetic theorem that gives the arithmetic tetragony of physics.

10:00 We are at the heart of his life. It is a matter of discerning what there is of arithmetic in the formula. So it is not a matter of closing doors and windows and ... Of course, if we could extend, we could study this specificity of geometry, this form of geometry, which has all sorts of characteristics, in one of the cases, there are all sorts of forms, which are under the command of iron mathematics, but which are discernible. Thank you for your attention. These are essential developments, essential connections, essential elements of development, I don't know how to put it, of mathematics, which is something that is deep and essential, characteristic, important, phylogenetically relevant, of mathematics. Through a phycological analysis, it is known to the analytical phycologist that through a phycological analysis of what is relevant, that is to say, I often We don't want to reduce the research of lost time, of course, but if we want to use lost time as an example of linguistics, if we want to use it as an example of linguistics, then the reduction of the French syntax is possible, but it is much more logical.

20:00 And not just to base, it's not necessarily against, it's just that we can also think about that, it's a possibility that has not been really exploited, it is not really wrong, at least as far as I know, but we can also use logical concepts to, not necessarily against something, it's just that we can also use logic for that. I think that's a little bit what, it's not really posed or… No, no, no, but it's a position, an idea of a mathematical system. The idea of a global theory. And we have never taken this idea seriously, not a single time in history.

25:00 I think it's related to the question that was asked by the signator, because what is frustrating, extremely frustrating in this gesture is that we are opposed to a theory of signification which, I agree, is totally cadenated, but which was cadenated because it had the idea that it was the only theory of signification that was possible. But we do not have a theory of signification to propose. And in this current You see, in this gesture there, which consists of saying that there is not in the theories that this level of transparancy, intra-theoretical, I think that everyone can agree on that, but after that, there are 100,000 theories of signification that can explain what is described here, in particular, and it is the same thing, I think, that was not very clear when it was exposed. For example, everything that has been exposed in the texts that are there is explained very well by a pragmatic theory. That is to say that analogies, etc., the extra-theoretical sense, it corresponds well to the analytical tradition, first syntaxic phase, second semantic phase, third pragmatic phase, everyone today is in this parabola on the side of the analytical, that is to say, we got it wrong. We should not be terribly syntactic. Semantic is not enough either. We need other criteria of meaning that are not only... That's not what Wander was saying, that's what Sébastien was saying. No, I agree. What I mean is that... I'm not saying that there is a pragmatic theory. There is no pragmatic theory. There are a thousand theories that will fit into this paradigm that consists of saying, be careful, and that's what's frustrating, is that in fact... For the moment, the description we have is purely open, that is to say, in fact, we do not know, and I think it's something where it would be interesting to have your opinion, we do not know if, under this richness of meaning, there is something of the kind of content, are we able to say there is something of the kind of content, are we able to say there is something of the kind of content, are we able to say there is something of the kind of content, are we able to say there is something We have a theory that is sufficient to say that it is more than pragmatic effects, you see, there is something of the content that we can identify because we have a theory that allows us to say that, or is it that all we have for the moment is that in the facts, in the facts of the way in which the meaning works, it does not work at the moment. You see, that's what I think is important to me.

27:30 Because there, I think we are at the moment of asking the question. Well, that's what I understand. We are at the moment of asking the question. So, after, we have to work on the question. First, we have to ask it. And for me, that's what I think. Yes, yes, yes, that's also what I think. That is to say that here, we are at the stage where we ask the question. Yes, but we are not going to, how to say... No, I'm asking Ivan for his opinion. Yes, of course. I'm asking for your opinion. I see a little what Sébastien was saying at the moment. That is to say, I find it very interesting someone who proposes... There are points related to this schematic of continuity of power or the extension of domain, all the resources we have to exclude certain points. And let's add something, the text of Marc Wilson is a text on Frege. There is also a possibility to understand Frege and Russell in it. It's not a text against Frege, on the contrary, it's really a text... I would like to ask you who is going to have lunch at the restaurant next door so I can go there the next day. So can I ask you... Thank you for your attention. I don't want to go to lunch with you. I'm sorry. I'm sorry. I'm sorry. What time do we meet? Three o'clock. Three o'clock. 2, 3, 4, 5, 6, 7, 8, 9, 10, 11,

30:00 Thank you very much for your time. It was very interesting. In what perspective, in general, as to how the history of mathematics should inform the philosophy of mathematics? We'll rejoin you, okay? Yeah, sure. You'll go to the Chinese restaurant? Okay, I'll see you down there. Yeah, I know where it is, don't worry. See you. See you. See you down there? In a while, yeah, sure, sure. Let me look at this. Oh, now I can miss that, see there. ...many times, or reduced to some period or something like that. This is something that is needed. This is a trick. And so now, from this point of view, he is able to reconstruct what is going on with the human surfaces and he even says that this is not at all an introduction of geometry into mathematics, but it's not just, we could understand it as a, there tends to be a proto-secularity kind of understanding when we consider these classes. So the department presenting it in a generative way is not at all an essential part. The essential part is only the way he did it in his dissertation, and it will be the way he does it in the 1857 paper, but that is a minor interpretation. This is because he made up expository reasons. He doesn't want to get involved in too many of these things. He already has enough complexity to discuss, so he decides to present things in a limited language. He says, I like geometry, I'm implying something like that, which already shows that this is not essential.

32:30 What is essential? The essential thing is to understand and analyze the topological properties of these things. Even before the lecture, he has written a manuscript on basic ideas of topology in international manifolds, which is very abstract, much more abstract than anything you can read in the actual literature. So my understanding is that he has this solution himself, which... In a way, he got into complexity and difficulty with manifestation, and he found some way out, not completely rigorous up here, but he found some way out, and along the path, by the way, he comes to realize that now if you have this manifolds with a topological structure, then you can have all kinds of different numerical structures, and so you can develop a dimension of geometry in the way you can do it. So this is my understanding. And I assume that Debekin probably got a good idea of the kind of things I had in mind because I wanted to go into this process for a couple of years. I had a lot of time to discuss with him. But with many other people it was not at all like this, and even after having published in the Mathematical Academy, it takes a lot of work to really go through connecting these things up and reconstructing, so most of the people I suppose didn't pay attention. The role of geometry in Riemann's Heide-Kleinung or not? It's a guess that is aimed. In fact, it may only be a way of speaking because it would be better to find one to make yourself understood. And maybe afterwards, he'll have some remorse or something, it's not the best. He chose a way of speaking. Geometrics, in a sense, is geometric, but not geometric of other geometrics. It is a geometry that is not assemblistic. It is the geometry of places of variable magnitude. It is a geometry in which multiplicities are a primary evidence. So, a geometrics of science that is not at all a geometrics of assemblism. But that's the same thing. He had to say it like that.

35:00 But, what he wants to do... The fundamental question is the question of the mode of donation of a particular functionality and the question of the necessary and sufficient conditions, i.e. minimal, for a particular function to be given. So the alien strategy is what I call the Grenzschutzschutzschutzschutzschutzschutzschutzschutzschutzschutzschutzschutz. Geometrical aspects are ways of presenting the interdependence between the elements of a function. When we have any function, the values it takes at different points are independent from each other. To have access to this function, we need to know the quality of the values. But when we have to do a function of zen, the dead... It's not like that at all. Values are not independent, and so they present this non-independence in two different ways. On the one hand, when we have the real part, values in a domain depend entirely on the values at the top. I have to say, in a way, I don't have a problem with that. It's a view of the whole, but there are many types of dependencies. And secondly, the real part and the imaginary part are not independent. They are not connected. And the degree of connection between one and the other is the degree of connection of the surface. We can choose the number of constants that are introduced when we pass from the real part to the linear part. So there is a reference to a geometry that is not our geometric, but a geoscientific geometry to figure out things that are dependent. So to answer the question, what is the most economical mode of giving a particular function among the functions characterized by an academic thesis? Thank you. Thank you. We're going to take a 10-minute break.