Identité et Categorification (contd.)

0:00 RG fois H dans le R fois RG, et ça doit être aussi l'isomorphisme. Et aussi les conditions normales d'identité, de morphisme d'identité, sont remplacées aussi par quelque chose. C'est un peu comme dans le cas de l'équivalence des pétigories. It is just a place where they were here, and they just wrote it, and they just wrote it. Some of them wrote it. Est-ce qu'une bi-catégorie est une catégorie ? No, there are certain categories, and more than a lot of categories, in this construction. There are all categories Ohm, there are categories Ohm, where Ohm is an object. But at the base, all these categories of the etage are good at the categories. It's a bit similar to the fibrillation, because here we have something very beautiful to fider on the level of work, but more or less, on affaiblit things. On affaiblit, on n'a plus une cathédrale. Yes, that's it. No, it's not an associative. It doesn't exist. But that also could use the work of Stachef in geometry. Well, I'm almost finished. Well, now this idea could be prolonged, three, four categories like that. It's not too difficult to think about the geometry of the geometry. It's more than just a metaphor. There are very few things with geometry simpletique. I don't know, I don't know, I don't know, I don't know, I don't know, I don't know, I don't know. And what's interesting is this category, I talked about it in the beginning, not big categories, but two categories exactes. to define what is n categories exact. It's a sort of definition inductive qu'il faut

2:30 construire. Par contre, les gens encore sont intéressés dans la catégorie faible, même aujourd'hui on parle de n categories, on n'ajoute pas faible parce que c'est toujours plutôt faible. Plutôt il faut dire exact si on parle d'autre chose. Et là-bas, si If I understand it, there are 10 different definitions, and in fact, even that, we try to answer the questions of whether they are equivalent or not. Because these questions, there are sort of intuitions that we talk about the same thing, but in fact, very few results precise. I know that Mackay, in Canada, is working on that. And why? Because that notion of equivalence is going to be used. Apparently, there is no more equivalence universe that we can respond. Because each notion of the next category has an equivalence. and which equivalence should be used to compare these definitions. It's not at all evident. And I suppose that, in fact, I know that it's a bit risky, maybe, to talk about things about mathematics without really having a lot of time, but it seems to me that there is a problem with this idea It's common, we always talk about the class of objects, and then we try to make an effaiblment. Well, I think it's interesting to me, but it's something that's not very natural, just to start with some ideas of very strong identity and not very explicit, and then try to to reduce things. To my opinion, the most natural project is to just start with, let's say, a sort of identity zero-null, and just for the last five minutes I use it to explain what it is, and to try to build an identity, but it doesn't make a difference.

5:00 But what I want to say is that in the case of N category, this idea of the meta-niveau where everything goes well, it doesn't work. Because at the moment, if there is 2, 3, 4, it will be meta, it doesn't work. Here, we don't have the meta-niveau, it's to say that we have a refuge in a few levels where everything is normal, everything goes well, You see, this notion, why it's really interesting, of N-category, because it forces really to change the idea. Even with the bicategory, the category is like that, you can have some construction, and everything goes well at the beginning. But if not, you really need to do something. Just to give an example. Well, I can't give an example of N-category, but it's an example of N-category. which continues this space of symmetry, which I've already talked about. I can say R-group, I don't think it's useful at the mathematics, but for the illustration, it's not a problem. When we talk about symmetry, we talk about an automorphism, an automorphism, an automorphism, etc. In other words, there is the automorphism. And again, it is interesting that, except for the exception of NK6, or something like that, we are always talking about the same thing. The automorphism of permutations, it's the same, it's to say that we can permute the permutations exactly in the same way that we permute the things of the past. And this is something, an example that gives hope for the people like Bayes, that the N category could be gérable, because Bayes calls stabilization. On peut espérer qu'à quelques niveaux les choses pas... que la complexité pas émerge de manière absolument incontrôlable. Mais aussi pour... on peut expliquer qu'est-ce qu'est-ce qu'il y a ici. Regardez, on peut choisir, disons, quelques N et dire c'est niveau métal, et ici on a, je sais pas, groupe normal, et dire que tous les autres groupes

7:30 They are defined as isomorphism-près, and they are defined as isomorphism-près. They are defined as isomorphism-près, because they form the same group. But, apparently, it has no sense, because, finally, it is the same group here. There is no other thing. Well, the last point that I do, is what I call the mathematics pro-tunicien-domain-christian. Well, I don't have the time to talk about the pro-tunicien-domain. Just, I want to signal that sort of the idea of the mathematics pro-tunicien-domain-domain who was actually known by Aristote in the antiquity, but never developed. And it's that Cantor can justify this idea of a line, maybe, as an ensemble of points. Aristote, we found several times that he rejected this idea. With Cantor, we did that. But there is always this idea of points, in the two cases, even the two modes of generation, in the case of the mission that can be generated by movement of point, and here by sort of repetition. But when I talk about it, I think of people who do a little bit of a noise that were really motivated to do something dynamic, which ensemble are statics and categorical, so it all goes. But I want to try to develop this idea in a bit more systematic. mathématiques, et une piste que j'utilise, c'est le fait qu'en fait, chez Euclid, on ne trouve pas une définition de point, mais deux, vous connaissez ? Parce que la première définition dit que point c'est l'objet sans partie, comme un thème mathématique, après il définit une ligne, c'est la langueur sans largeur, mais la troisième définition, he said that the point is a borderline. And then he repeated it with the lines and with the surface.

10:00 The surface is a borderline. And Aristotle also discussed in several places. Leandre said that it's the order of the earth, which is natural, to pass the point in other directions. and the inverse is the order of explanation, it's not something like that. I can't remember... Yes, I'm finished. I'm finished, I'm finished, I'm finished. Yes, yes. Yes, maybe I don't have time to explain what it is. Bon, ok, j'ai juste résumé que ça me donne un peu l'idée qu'on peut plutôt que passer de quelque chose qu'on pointe, parce que c'est quand même aussi étonnant que dans cette N catégorie, on parfaitement reproduit ce pattern très classique, On commence des points, on fait des lignes, etc. Et aussi, aujourd'hui, on a appris à faire continue infinitivement, on fait une dimension. L'ombre infinie de diminution. Par contre, on n'a jamais réussi à faire l'infini de l'autre côté. Si on commence par un pro, on finit par un point. On ne sait pas où aller le point. And I think we can start by points, we can start by something that I call the flux, and look at how we can say, a group of flux, and build something that looks like a category, but in a different way. C'est-à-dire, pas de départir des points, mais... Bon, mais je n'ai pas du temps à appliquer ça, mais peut-être bon, parce que ce n'est pas quelque chose en fait achevé, et juste pour conclusion. Maintenant, conclusion est ça, quand même, même quand je commençais ces recherches, j'étais un peu... j'ai eu plus d'enthousiasme, je pensais vraiment there are some solutions in the theory of category. Now, I think there is no solution, let's say, which is already there in the category. But the fact that

12:30 all the problems with an entity are more pertinent, we can't just play with the context and just always be precise. It's more difficult in the theoretical theory than in the theoretical theory, it's-is that it really helps to solve the problem. It suggests in some way the idea of the relative identity, but not exactly in the sense of Gitch, but in the view. And the relative identity multiple, I think it's important that we have several notions and not just one. C'est-à-dire que ce n'est pas la peine de prendre un comme quelque chose de base avant même de faire mathématiques. Je crois que ça doit être fait dans la mathématique à partir de telles données. Et aussi je pense que cet idée de partir des notions de classe et après essayer d'effablir l'identité, But maybe these projects will be reversed, but it is necessary to just build an identity progressively rather than affaiblit. Even if we don't know exactly what we affaiblit, rather than affaiblit. It makes me think a little bit about social science, I think that's why we talk a lot about identity, It's something that's built, and I don't see why we can do some mathematical work to do these things. Thank you. Thank you very much. Any questions, any comments? I don't know, you've never had the time to talk about what you heard by Fou at the end, but... No, it's me who is... No, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no. But there is an article created, which came to appear on the web, which discusses about categories and physics, and what he says is that the problem of the physics theory today comes essentially from the notion of point and continuity, because it's there that there are all the differences, the singularities, etc. Every person who knows how to do a new physics they know how to get rid of the points. And what they crave, it's that obviously just here, the categories have not been used in the physics, but rather the notion of the ensemble.

15:00 And his theory, I haven't read the paper and I haven't read it yet, but his argument is that if we get rid of the notion of the ensemble of catégories, they would naturally arrive at descriptions spatio-temporelles, which would not be able to intervene at the point of time, and that could solve the problem of the physical. I was wondering if you were talking about flou, is it in the same sense, or not? But yes, it's a bit of a bad idea, because I saw something like Lysmoline, even in his But at the end, at the end, I think that the notion of the standard and the category may be changed. Because we always start with the notion of class, which is more general than ensemble, but this notion of identity is already there. And it's to say, maybe, I don't know, maybe I could try, maybe it's good that I haven't talked about it because it's still a little bit, but I think that we can change the concept of categories for the project, maybe through the 6th category. Lui, il dit qu'avec les catégories, on pourrait arriver à une nouvelle notion. Mais il y a deux niveaux de questions, je crois. Parce que c'est vrai que, disons, quand on prend l'objet des catégories données, on n'a pas obligé de penser l'objet comme ensemble. Même on peut définir qu'est-ce que c'est un point dans l'objet donné. Et ça, ça nous donne une sorte de liberté de cette, je ne sais pas, l'hypothèse ensembliste, si vous voulez. But it's at the local level. However, at the global level, as long as we define the category and class of objects, we are still in the same way. We have no need to define what is the object. And even if I have understood, we can only define the object by certain flashes, which are the flashes of identity. So we have no need to define the nature of an object But it's not important, but if we talk about class, maybe if we don't think that the object is in a form of ensemble,

17:30 then there are many good results, like the M. Ioneda, etc. If we talk about the category large, there are less things that we can do with the category large, It's the problem. But still, it's always in class. And I... I don't know. I think maybe in a category, it's an obstacle. In the theory of the category, at the local level, it works, but at the global level, it works. I would like to read this article. I can't even say that when we introduce classes in 1880, it's just to make sure the problems of the genre. It's a big issue, but we say, well, there are paradoxes, but in fact, in reality, all the théories of the world are always perfect in our society. It's a bit more of an ensemble, if you could say, something that sounds like an universe, etc. It's a limit to an ensemble, an ensemble, but not an ensemble. You know, you were able to take that absolutely in the middle of the question of the anthropologist. Yes, the question of the anthropologist is that, we do not need to focus on the question of the classes, the role of the classes in theory of the categories, because this role is important that if we imagine that there would be a category from the matrix in which we would have the all. Or, there is no theory at all that is presumed by the theory. In the contrary, it is a practice local. We only have friends and friends with certain properties, etc. In the historical texts, in everything we have written, yes, we have the theory of all groups, or the theory of all groups. In reality, in the collection, we have to take the theory of all groups. The theory that we demonstrate in the background of the group is related to a collection, form an ensemble, an ensemble, which is a particular group.

20:00 If we look in detail each theory that we demonstrate, it's a real theory. It's an ensemble, it's a good idea. And its cohérence depends only on the cohérence of the classical theory. So if it's an effect of everything that makes the problem, then it's an effect of everything. In fact, it's not a problem. Well, that's a first point. Well, I've been frappé by a argument that you have given, which is that when we consider the category 2 of the group, then all are present. But actually, at this point, they are all two distincts, and you have even supposed to be indiscerning. Or, this is not the practice. In leaving the interpretation, the consideration of the category of all groups, like the category of all groups, and all groups of all groups of all groups, And then we arrive at the results where, by definition, two objects of this new category are necessarily distincts. The problem is evacuated like that. I can't answer that. I want to conclude that I have a first remark about the fact that, about this question of the world, of the cataclysmic, paradoxical, and of course. In fact, we can remember that already, when we were talking about the world, we had a little bit of a problem with the question of the paradoxes, because the paradoxes are not the fact of considering the classes, but the fact that everyone wants to consider a class who will be the Tout. It's also the fact that we want to have a Tout, which makes paradoxes. Well, if I consider it a particular case, it's an ensemble ensemble, it's not an ensemble, but if I don't suppose that there is a lot of time, it can be a lot of time. But let's say that we can dissociate the fact that we consider a large ensemble, a case, from the fact that we pretend that there is an universel, and it's true that this question of identity that you talk about here, is completely confronted with this question of pretending to be a tout. We see very well in the difference between identity and identity, which is equal to the difference.

22:30 Because the fact that things, like you said earlier, the fact that things are equal, a priori, in fact, it relates to an affinity, because it depends on the one and the other. the difference is not effective. In principle, we have to have a number of different points. The difference is not effective at all. It is to say that, for example, if we take the suite of the number, we should be able to see the index and the index they are even. However, for example, they are different, but they are different if they are different, because they are different. It's banal, but it's a good thing, of course, in the context of the humanism. In fact, we refond the same mousse in the context of the humanism. And it's true that we can't confound the theory of the humanism with the theory of the difference. So it's also a question of the two and the first one. The role of the two in the structure. And secondly, the role of an identity, not of an identity, but of a difference. In fact, there are three terms. Identity, the difference and the difference. You have to talk about the identity, but there is the two and the difference which is the role of an annex. That's not my question at all. I choose what is there. If we look at the category of vibrations, we always fix this meta, that it is something that does not answer. It is construed in something, but it does not answer to the question, because there is always this identity with all the same problems. Now, if you think about a category, Here, where the problem is pertinent, you can't do anything else, but here, it gave me this idea that this notion of class, which I really like.

25:00 You see, there is a sort of decalage between this construction all a bit standard from a point absolutely classical or clitian, and this idea of, I don't know, algebra, definition algebraic, from the other side, you see, and they don't work well with each other. And, well, you can say that it's negligible, it's not important, but after... I don't know, but I think that it doesn't work. Because we have always... But you see, it's not clear, it's not the question of the size. When I talk about the size, I see a problem. It's not that it's too gross, it's not at all. But it's rather the idea of an individual. You see? Otherwise, it's not the idea of a point. If you see that, it becomes a general. And that's something I think is a real problem. There is a point on which is very interesting. There are things that we do in physics. There is a certain tendency to be deconstructed compared to certain objects mathématiques traditionnels, like the fibrer, or the ensemble quotient. The provenance totale, an ensemble of elements, a relation of equivalence, an ensemble quotient, And what you're saying is that there is something exagered. It's about making the ensemble quotient. Because it's important to make the ensemble quotient, the ensemble classes and differences. It's something exagered because, as you said, you have to take the family of all the cercles, and then after all the cercles. I agree with that. The point of view of the families, is very different. The cercles, the family of cercles, It seems like there are many families who are in the exploration of the universe. It's a very interesting alternative to what is the construction of an ensemble quotient. I had a conversation with Benika in two or three weeks and he explained that in the physical moment, They are very interested in considering that now, perhaps,

27:30 the idea of fibrer, for example, is perhaps a little bit exagered, but the space total of fibrer does not exist. What exists is the coherence between the different fibers. I'll give you an example of what happens in the view. When we look at the two of the two, each of the two of the two, each of the two of the three, there is a construction, there is a relationship between the two of the two, which is the same, we are not in the question of the identity, but that the view of the ego and the view of the two of the three are the same, in fact, it is not necessary to the theory. The theory can be said that there are two objects that we see here and there, and there are three objects that we see here, but they are reliated by certain things, And we are consenting to say that there is a liaison. Yes, yes. But you don't know where to stop. Because if you think about the mechanics of two to four... I think that these elements are... That's what I'm saying. That's what I'm saying. That's what I'm saying. That's what I'm saying. That's what I'm saying. Yes, but... the analog topology, it's not possible, it's all very physical. Yes, it's true. Yes, it's true. Yes, but we don't have the connection, without having to say, that this connection is a different form of differentials which acts on the vector, which is in a certain space total. We don't forget what it does. There are some objects apart from this, and then there are three. Toi, you talk about... the construction of the ocean, the construction of the ocean, the direction of the ocean. But you can speak, if I understand, of simplification, because there is no need to do this. Well, it's not easy to do this, it's not easy to do this. Yes, and I think that... Yes, exactly, but I don't know if it's the same thing as to say, but when I was talking about classes, even before the category. I think it's just an acceptable simplification,

30:00 to say that if there is something about concept, I don't know why, then there is something like totalitarian. Because I believe there are several, it's to say extension, it's a function, it's not defined automatically, and unique. It's something like a function, which is maybe complicated. When you talk about all of an ensemble, it's maybe just something that's not defined. In general, we have the choice, in terms of, finally, to devisser, like we said, to devisser all the details of what we would do, and to devisser all the details of what we would do, or to say that everything can be assembled, to reunified in an object a little more simple, fibrer un ensemble de classes, dans lequel l'objet on aura finalement volontairement oublié une partie de l'information qui nous permet de faire des calculs plus simples. Donc on est tout à fait dans cette idée qu'on réalise ces homophyses canoniques et qu'on les oublie ensuite. Ça c'est effectivement une phase, mais la phase suivante c'est à travers oublier de revenir, c'est-à-dire de revenir à ce qu'on a oublié And again, it's redébillé. And this return, it's a number, I think. In the modern world, it's called the homology, the geomology and the geocomology. It's exactly the movement of this return to the analysis of what has been forgotten, of what has been forgotten in the isomorphism, which we have thought as desigualties. So all that you have said about identity and identity, the analysis of that, the analysis of what is happening there, It's considered, obviously, a cross-terre-écart. It's called the X-Morphs. It's called the homogic. It's certain that we don't understand that the X-Morphs are in position to think about the identity. Because, just simply, it's in this context-là that it's the analysis of the difference between the X-Morphs and the X-Morphs. I'll give you just an example here. Monoïde égale catégorie à un object. Un monoïde, c'est une catégorie à un object. Parce que c'est là, c'est quoi ? Une identité, une c'est une réalité. Regardez pourquoi c'est misieux.

32:30 C'est misieux parce que ce n'est pas une identité. Ce n'est pas la même chose. Non, ça semble la même chose, mais ce n'est pas la même chose. Il y a une espèce de collapse de la société qui est assez sournois. because we are too late to say that since a monoeil is the same thing as a catégorie as an object, so the catégories of monoeil is the same as the catégories as an object. Well, it's not the same. They are not the same. They are almost equivalent to these categories. Just in this MÊME, which is completely... which is completely different. Just to analyze at which point this MÊME is the same, when we look at the categories of objects that seem the same, but still there is not the MÊME that is exactly the same, and it is seen in the MÊME. They are not the mortals, it is very easy to say. They are equivalents. They are equivalents. They are equivalents. And that's another way to show that his identity is more visible than the work of science. André, do you have a question? Ah, well, yes, thank you for your remarks. Well, I would like to add a question that I had to pass. I think, while you were talking, I had very often thought about Leibniz, who brought this question, and the remark that René, who spoke between identity and identity, is susceptible to the process of decision-making, and the other not. And between the two, Leibniz introduced the observable. The observable, it's not in identity, nor in identity. And one of the principles of the philosophy of science is the identity of an indiscernable. You don't have to read it on the paper, but you don't have to read it here. Or, the identity of an indiscernable in a certain sense of indiscernability, it leads to a mechanism with relation to the equivalence and quotientage. Or, if you want to say, if we are indiscernable in a certain sense of a certain criteria,

35:00 then we will consider it in a quotient, explicitly or implicitly. And the one who has followed Leibniz in his reflection on what is it that these objects are distinct, if we can't discern them, is it not the same? Because I can't discern them. Leibniz had said that there was an interdisciplinarity. That's why I said there was an interdisciplinarity, in the sense of what you have analyzed for the rejeter. But the one who has perhaps the best analyzed this, is Frigge, after, when it comes to the script, because there is a function that explains that it is the status of the variable, the letter. He says that if I take x and y and I say that it is real, then it has no sense. He says that it has no sense. Because these objects are not characterized for me. I am incapable of citing a property of x which is not a property of y. On est donc dans la confusion, on a donc à fait, donc le statut de la lettre, tant qu'il est manipulé, implique une inversée d'arbitrité de fait. Donc, si vous voulez, ça c'est un passage, je serais vu, qui n'a pas été très relevé, parce que j'avais voté dans un article, parce que je trouvais ça exceptionnel sur le plan de soins de l'écriture. He's opposed to Russell, who is telling us what is the variable, and he's very embêtised, Russell, because the variable is not a math, it's not a mardi, it's not a mardi, it's not a mardi, etc. But it's something that, at all times, you can have everything. Frego, he's going to go there, to say that. I don't know how to do this, but because it's the question of the indiscernible And it's a question local, concernable, in a sense of view. Well, I think it's supposed to be a surprise, yes, I can't wait for it. There is an identity. It's still an identity, but if I understand it, it's epistemic, it's to say there is really an identity. Yes, it's absolutely epistemic. It's an epistemic concept. We always talk about Nietzsche. The scientific world has lived for four centuries sur la définition de l'ANX de la manuté et de la différence. Une situation, c'est réadapté.

37:30 Elles sont les mêmes, les choses qui peuvent être substitues l'une à l'autre, partout. Salva veritate. Salva veritate. Elles sont différentes celles qui ne le peuvent pas. Ça, c'est encore une autre définition possible de l'identité. Parce que en mathématiques, on fait des substitutions qui ne sont pas du tout salva veritate. but this definition has been a part of the centuries. On voit the citations of the authors that they refer to and they are not quite we quite substitute and present diverse and quite and present. On voit the mathématiques of Cambridge and the posterior, Jewells, par exemple, citer sans arrêt Lenny on this subject. But we can see that It's just that it's a definition of identity that is not the one that you have cited. If it's a definition of a philosopher, if we can substitute it by all that, then it's in the essence of being the same thing. I have a question just on that, because when we listen to it, I have no question. It's about to know if it's a definition of identity, or if it's not a formulation of the identity of identity. It's also a problem. I don't know that it's the principle of the identity of the discernment. Perhaps I've seen it. Perhaps I've seen it. You've seen it. I've seen it in the principle of the discernment of the identity of the discernment. That's to say, I've insisted earlier, the character of the discernment of the discernment. The citation, the definition is a problem, but in a absolute caractere. There is not an affaiblissement, at this point, on the things that are important, an affaiblissement potentiel inscrit in France, but indiscernible, if you will, maybe it is indiscernible at a moment, for an observatory, for a context given, But the end of the philosophy is that when we don't do it, we don't do it. But I think that if we replace E.A.T.E. by a discernment, that's not correct. Because that's really a discernment,

40:00 and that's just... You have all right. You have all right. I'm not sure. And then after, here, we can't even say, then it's the same thing. I would say that the law looks more strong than the other. In fact, the most important thing is the implications of the reciprocal. Because, yes, it's the logic that we can substitute, but it's reciprocal also. And that is something... How can we treat these problems with something that changes? It's the same, but it changes the problem. Yes, it's not a problem. Because they treat something that changes with a definition which is also inscribed in the changements. If you want to substitute everything, it's not a practical concept. But it doesn't work. I don't know if I'm here and today I'm the same, but here I was... You can't substitute yourself. I can't substitute yourself. You can't substitute yourself. I'm here and today, but not here. I'm here for today. You're the whole of all the moments, so you can't say here I was different or identical to today, because you are the same of all the moments that you have lived. Well, then... That's the implication of our mind. Yes, yes, yes. But that's... That's the paradox. But that's the idea that there is no density that is... That's to say I'm not the same here. Yes, because... No, but... The essence, it's not a sense because you, it's not a... You, it's not a... You, it's not a... You, it's not a... It's not a... Who is it? and who he is. I think that, at the same point of the activity, and at the same point of the person, I am not what I am at this moment, but I am everything that I have been through. Yes, absolutely. And that you don't know. I think that my personal personality is that. We are talking about the list in which... There is no joke about it. We are talking about the substance, the essence, the essence, the concept, the past, and the avenir, and there is no accident as an accident since they are inscribed in the concept of the substance. We can't do physical with a lot more concept than that.

42:30 I don't know. The philosophy of the enemy is absolutely exceptional. You remove God if you are tapped, and you replace it by a metaphysic, by a word profane, and you have... and there is no changement. It's not that there is no changement, but it's as if you change it, and there is no changement. It depends on what you call it. I mean... It's what you call it, because it's the same. Just you... But all of this, all of these parallels, they are also liées to an object terminal, There are two of them, but in fact, we don't have no idea. And it's true that we don't have an expression of the object terminal. It's like it was unique. There's no reason to be unique, but in fact, like this. But to respond to these details on the unique, we'll come back to the question of the observation of the cosmology. But the question of the unique is linked to the question of the human. And I like to talk a little bit about Lacan. because on the question of the unique, Lacan, he brought an insistence very interesting. It is true that for defining the unique, it is to pass by two, and it is to say that all the other is the same. All the unique is all the other. All the other is the same, it is a problem. And in the substitution that we talked about earlier, it is a good thing to say, Well, we don't know if we are the same. And when we do the substitution, we see that it is the same. Because we are unique. But we don't know if we are the same. We don't know if we are the same. Thank you very much.

45:00 It's a thing on which I would like to ask, I think, with that I would like to ask the question. Also, on the topic of time and token, it's the idea that in a symbol, in an écriture, we have to deal with an individual. There is an individual, well, an individual, an individual, well, an individual. What is not the case, if we have a discussion about the philosophy and the mathematics, etc. and then it's a difference that we can see in relation to the intuition sensible. And so I wanted to know, well, I don't know if you have any questions about this, but I wanted to know what you could say about the theory of the category and the use of a symbol. I have the impression that the questions, the problems of identity, are also due to that, when we write things, we have to deal with the entities individually, which are the sign. And are these problems not due to this moment? For example, if you do the same thing, it doesn't happen if you write several times the number 3. 3 plus 3 plus 3, je peux dire de toute façon ce que j'ai, c'est le même signe en tant que type, c'est le même. Alors que si je dessine des cercles qui transforme ça dans un autre, il y a une différence de situation. Donc je ne sais pas si tu as des idées par rapport à ça par rapport à l'autre. But I suppose that this idea of a type of token doesn't work out of math. Yes, of course. Because, I mean, of course, we can even try to define in which sense 2 x 3 is the same sign. Yes, there is a resemblance of forms which is maybe even defined in terms of transformation and geometry. But, on the other hand, this idea of the type of token, it's because there are individuals of base, like something written, like, I don't know, the pigment, the molecules created, something like that, and that, I think, it doesn't work. Even, for example, if you think of electronic text, I don't know, PDF,

47:30 or internet, like this article, what is that, like, this... material, there are, I don't know, all the signals that work, It's something that doesn't exist. So, of course, we can give the sense of the idea that it's the same text. And if all of that, of course, we can try to reduce in an event of sexual experience, or like that. It's indépendant to all of that. You can write it in different ways. But there are several levels that I would like to say. It's very evident, but it's very well known in the type of token. There are at least two levels. And I think that the first thing, there are more than two, but it's perhaps trivial, it's also evident, but it's perhaps more evident and more difficult to say in a way, it's to say that there is no level the most low. That is what I believe that changes things. If I say that there is a type of... The idea of my project is a little bit relativised, we can say this relationship type and talk. But it will be already another thing. But even when I talk about classes, I think that it is still not relativised. There is no real theory of relative identity relative dans quelque sens, je ne sais pas, technique. Ça serait intéressant. Je vous ai répondu tout à l'heure par avance à cette question en citant Frege sur une question de la lettre. Parce que vous avez pris comme exemple 3 et 3, ce que Frege a fait démonpring. Pas de problème. Mais la lettre, elle est une, multiple. Et donc, si vous... La lettre est-ce qu'elle est là ? Ce n'est pas la lettre X. La lettre, on a raison, ça dépend du niveau. Si on se place dans le cadre, en contexte, des mathématiques après Viette. Après Viette, dis plus ou moins, parce qu'avant Viette... Si on se place dans la lettre après Viette, la lettre désigne à la fois un objet, mais il désigne également la classe de tous les objets. and Frégue se fait un regal de cette contradiction fondatrice se fait un regal de ça en disant si je prends la lettre X et la lettre Y

50:00 il n'y a aucun moyen de distinguer les propriétés qui appartiennent à X et qui appartiennent pas à Y autrement dit, la variable est un concept qui n'existe pas voilà donc si vous voulez alors je vais continuer là dessus parce que là c'est quand même quelque chose qui m'est cher le rapport avec la théorie des catégories On a souvent dit que sur le plan simplement de la symbolique, un des apports essentiels de la théorie des catégories avait été d'apporter des diagrammes. Alors, d'abord, il y a des catégoriciens qui, je crois, ont récupéré quasiment 100 diagrammes, au moins 100 égrammes publics. Mais il y a une chose qu'on n'a pas vraiment souligné, c'est le fait que quand vous écrivez, par exemple, E equal spectre de B, where E is an espace to be compact, an espace dual, par exemple, un facteur de la catégorie des espaces to be blue, vers la catégorie des espaces to be blue, etc. You consider B as the sign of a variable, with all that, all that the mathematicians have given to you. And that is something that is very tempting, something that is very useful, that we had, for centuries, shown that it was useful, but we haven't done it until now. So why haven't we done it? First, for the interdits related to the category of all the groups, all the groups, all the groups that we have talked about, and that, the first catégoriciens, in 1945, they had nothing to do with this question. And the second, it is that when we used the variables in the analysis, then we would have to do the substitution of a group. Or, before the theory of the catégories, how could we substitute? We could not replace a group by what? Or in the same form, or in the same form, or in the theory of the catégories. We have to say that there are substitutions legit, it's the morphism, and so the fonctions of the transformation of the objects of the morphism. Autrement dit, what was impensable, but what was pensable intérieure, because everyone how the centre of the group varie, how the group varie, in general, what, but varie or how? And on n'a pas le droit d'écrire ça. La théorie des catégories a permis que, sous la lettre, on puisse savoir comment on structure. Ça, ça me paraît être, sur le plan epistémologique, une annoncée considérable. Parce que c'est la démarche de Viette, qu'un simple annonce, qui est reprise ici, arbitraire et fixée, c'est toujours cette dialectique, which is reprise with letters valent for destructuring.

52:30 I think that's not the diagram, the diagram is contingent, it serves as support to the reflection, when it is commutative, it serves to define objects, I think that in a future symbolical, But if I understand well, we find that Mr. F. Ply, he gives a general theory, after we take the patriarchal ABC, we can all prove with ABC, but then we say that the conclusion is true for the case general. And ABC is also in the same time as the patriarchal? There are two things, we're not going to talk about it. Let's say that ABC and Outline have a function purely designative, and not operative. It's a design, it's a designative, it's a design of the letter, the right, which is there, for AB, etc. But what you say is another thing, it's the fact that the figure juridically doesn't have its singularity in our terms or in our terms, and that it vaut for a class of figures, And so, the proprietor is demonstrated on a figure, or for a case of figure. In other words, the figure is arbitrarily fixed. That was the case of the geometry. What he did, he did it in the sense that it works without calculs. Yes, yes. But it's kind of the same thing. It's not the same, because there are differences. Second, there are differences. Because in geometry, there was still a representant. I remember that there was a conference among the professors of mathematics, And I said, well, the story of the figure, there is no problem, we are used to it, we have that in our substance, that the figure has a singularity, but it is not signifiable. The figure is quelconque, by definition, between guillemets. Well, I had a professor of the bibliography who told me, but not at all, my students don't accept this convention. And they said, it's possible that we don't accept the convention. What do we do at this point-là? Well, we don't do it. Well, it's all. We can live. Very well. We have a geometry. We have a geometry. We have a geometry. Is it an aveugle? Is it an aveugle? No, but if you want to use the geometry to demonstrate the proprieties from the figures considered as emblématiques, which has been, in gomming,

55:00 in effacing, in occulting, in the other side the singularity of the figure, which was kind of the problem of the geometry, It's to say, there is a figure, I want to use it for reasoning, but I don't want to use it. I don't want to use it. Dessiner, but not montrer. No, no, no. But is it an aveugle of naissance can do something? Yes, it's true, it's not real. Today, no, no, no, but... Yes, it's not real. It's an essence. I'm sorry. I'm sorry. I'm sorry. It's not real. It's an example. I'm sorry, I'm sorry, I'm sorry, but it's not real. And it's a great idea. It's a great idea. And it's a great idea. It's a great idea. It's a great idea. In fact, I can't respond to the question of the last one. Why I was a little aggressive in class? Because I know I understand the category as a way to think about the concept of mathematics. and class, if you want, it's also a way of thinking, an object mathematical. I think maybe there is some sense that if we look at a category of something, like an ensemble, like that, in fact, we have no need to take the base... I'm agree with you. I'm agree with you. I'm agree with you. on the iniquity of the fact of the distinction between class which has been put there. It has been put there at a moment for contouring the paradoxes that the mathematicians took totally off. But by the way, I don't see what it is replaced. I don't see what it is. No, no, no. It is replaced by the universe that indicates that there is a problem and that it is stable by the community of reunions, intersections, by the parties, etc. And we work localement and then it is stable. The 71 of MacLeod, the category for the working mathematician, he had to react, if you want, because the article 45 was insufficient. And so, he, he said in the book of Botanique, he put it in the book of Botanique, because there, Botanique said, and so, he put it in account this objection, but he did it in a way absolutely rapid, without that it was not important. And the mathematicians will say that it's not important, it's true.

57:30 Well, of course, someone who is a mathematical scientist is always like that. I found very good your expression, because it's supposed to be a logicist. Any other questions, comments? Well, thank you again for the lecture. Thank you very much. That's what I would like to say. And the test... It's not a barren, but it's a barren. I want to show you the barren. How do you feel about the barren? I think it's CRAME. I'm going to show you the web. It's been a week ago. It's been a week ago. It's been a week. Thank you.