Identité et Categorification

0:00 Bon, aujourd'hui, j'ai le plaisir d'accueillir Andréa Gastel, qui travaille à l'école normale supérieure, who will talk about the issues that are in the category and the problem of identity, the problem of identity and identification. There were also some who received email from the webinar, there was a site web on the archive on which the text is beyond, The text is on the top of what Ray will say today. It's a very interesting text, very varied. In English? Yes, it's in English. That is rather good English. Mike White did it somewhere. Yeah, I think he checked it. I don't know. I don't know. It's a code access. Yes, it's just... The code access? Oh, no, it's the number access. No, no, it's access, it's just address. It's the address. What's the name? Abstra. Abstra, yeah. Abstra, yeah. It's a page. Click on it for... Well, well, the title of this presentation and the plan of this exposé, I started discussing the problem of the densities and the equivalences in mathematics in general with the naïve, simple examples.

2:30 Après je parle un peu de certains autos, certains gens qui ont travaillé sur cette question, c'est vrai que c'est le plus important qu'on figure dans ce discours. Après aussi je touche sur Peter Gitch, sa théorie d'identité relative et même je parle de Platon tout un petit peu. So, after I discuss the notion of class in the language, because it's important to me. After I consider it, not really theory, but the approach alternative to the identity mathématiques, which depends on the notion of transformation. It is to say that if we start to think about identity in a more or less logical way, standard, that would be a little something. The logician might not be for us as identity in a strict sense, but I believe that it is important. Finally, I started to talk about the theory of categories, I will introduce the notion of catégories in a simple way, almost naive. I will show you that in the notion of catégories, these two approaches really coexistent. We can look at the notion of catégories as a synthesis of these two approaches. And then I will discuss this question of the diversity and equality in general. We just want to see how this general problem is changed in the theory of the category. But then I'll discuss a little more specific question where the question of the identity has already been put in question. and it's the theory of the category Fubre, after the article of Jean Bunabo in 1985, where they discuss the motion of equality in the category. And the categories are purely... If I make this prediction, I think it's a high category, It is a category. And it starts with the most ancient article of Benabou,

5:00 on the big category. It was in the 1960s. After, at the end, I did a little bit like this, not at all, but I decided to talk about mathématiques and reclutiniennes as an alternative to mathématiques ténisiennes and démocratiennes. I will explain what it is, and then I will give some conclusions. Bon, on commence juste avec une question toute naïve et simple, disons si on écrit quelque chose comme 3 égale 3, est-ce que c'est la même 3 de la vérité ou c'est deux nombres différents ? And yes, it's a question too bête, which we'll see in the Worldwide Interoperability, which we'll pose seriously. And what's interesting is that, of course, it's something of nature. I thought there was something like the three copies. But then we can ask what copies of what? Is it original for all these copies? Or maybe it's just original copies. It's original. And it's really an idea that I want to try to do in a way more precise. I want to justify the two. We can think of the mathematics as the families. But after, you have to explain what is the families. I want to show you the notion of class. It's not the same thing. It's other things. mathématiques. Aussi, quand je commençais à penser à ces questions, je pensais qu'en fait pour mathématiques, c'est de aucune importance cette question, au moins dans ce niveau-là, parce que c'est important pour les mathématiques où de nombreux sont égaux ou non, et est-ce qu'ils sont la même équivalent, identiques ou non, c'est juste une question which is not mathematical, it is to say that we can respond but it is not the same thing in the theory of category you can see, this approach liberal, completely liberal, it becomes more difficult

7:30 in the theory of category, it is not impossible, but this question there is, it is totally exterior of mathematics, it is a sort of of a philosopher, but in mathematics, more developed, more contemporary, it becomes really a question of mathematics. Exemple similaire, if I take two ensembles and I make something like a carré cartesian, that is, the pair of all the elements of the ensemble art. Also, the way to think of this thing is something like two copies of the ensemble art. When I talk about copies, I would like to say, when I talk about copies, I would like to say, what is the copies? It is never explained in a way. But in fact, for the ensembles, of course, we can give a classic answer, precise answer. And the answer is that no, there is no copy. It is still the same thing, yes. If we take a few cases, let's say, in terms of the theory axiomatic of the ensembles, there is a response, in fact, which does not leave place for this idea of the copy, because here we are going to define this pair, like this ensemble there, it is to say ensemble, this single tone of A, yes, ensemble, which is the sole element of A, and this pair AB. This one can define as a pair, and after, from the sections of the same line, we can prove that if we have a pair AB and CD, Igou, oui, alors A égale C et B égale D. C'est ça qu'il faut prouver. Mais j'ai juste montré, à quelque sens, quand même, ce n'est pas suffisant, ce n'est pas très bonne réponse, mais disons, dans le sens, selon cette définition, disons, si on regarde this is the perle A, A, A, A, diagonal. What is that? We have, in fact, something called double singleton. We have to take singleton and singleton. And, well, it works at the formule, but I just want to signal that,

10:00 well, obviously, the difference between A and singleton of A, it is something that is something that happens to the intuition. When we think together, we explain to the students, to the students, to the students, it's something like an ensemble of chairs, etc. And then there is a difference between the single tone and the element of the single tone which is an intuition, and even more things like that. There is a difference between the form and the precise way of representing things and the way that we think in mathematics. And it's not... it's dangerous in some sense. It's to say that we have a strict theory official, of one side, and this thought less formal, we always know what we said, which is in another sense. It's to say that it doesn't work well with mathematics, with, let's say, mathématiques, pensée mathématiques, réelle construction mathématiques et ça bien sûr ce n'est pas la situation... c'était suffisant. Il se ricord d'ailleurs, la création de la mise sur le nom de la scène des cinq biotons n'a pas été sans mal, chez Schröder et Pierce, It has not been sent bad, Mr. Schroeder, it has not been able to say the difference. It was a charmian in 1880, but it has not been able to say the difference. I know how to explain it. It's just to think about it as a choice. It's a little bit like a pool. It is to think about it. But for that, it is necessary to take something like the universe, an ensemble from the beginning. So we can think about this ensemble as a choice, in fact. An ensemble vide, it's not just anything, it's not metaphysic, but it's like a choice. When we come to the supermarket, we don't pass anything. But here, this double symbol, it's already difficult. But I just want to signal this intuition that is important in the practice, I don't know, I don't know, analyze, functionally, or whatever, there is this collage between intuition and work, if you can say, with these sorts of fundamentals.

12:30 I have another example. Two points. Here, we can maybe suggest that there is a sort of evidence intuitive when it's the same point, when it's the same point, when it's the same point, when it's the same point. But in fact, I believe that it's not the case in fact, because there is a very important case when two points coincident. Two points coincident. And then the question is, is this sort of exotic case that we can not take seriously? I think that no, because it depends on the idea of the congruence. Yes, the two things are congruence when it coincides. And in relation to congruence, it's something absolutely of the base of, let's say, geometry prédienne, or perhaps the geometry, it's-is that we can't do geometry without congruence, but in this sense, it's very important. And it's a bit the same situation if we saw something like that, signifie, est-ce que c'est vraiment la même chose, ou il coïncide, ou est-ce que c'est un bon ? Aussi on ne peut normalement juste pas répondre à ces questions, faire schématrie sans répondre normalement, mais quand même on ne peut pas penser les points dans la même manière qu'on peut penser de particules, or two atoms, metaphysic, if you want, because there is a very important difference, that the mathematical points are not impenetrables. It is also, just to remark in parenthesis, that we can't, I believe, think about it as an idealization, an idealization of an experience spatial habitually, it is not at all the same thing. our idea of objects spatially penetrated. And that, of course, change the regime of density, that is to say, the character of density of particles, classical, normal, yes, and the point, they are different in this point. Just here, even for this idea of coincidence, At the point of view, we can consider two lines, the points on the line, after two lines, they cross, and again, are these two points that coincide, or is it the same point?

15:00 And what happens here? Well, just to pose the problem, There is a case where we design the points identified at different angles. When we design the projective, for example, we say that the two points diametrically opposable are identified. So it's really the same points, although we can't draw them at a moment. And yet it's the same, they are really identified. Or they are equal. Pardon? Or they are equal. It's the same, in the real space project. Yes, I can't remember. But here, it's to say that we can't forget this intuition that we have, and that's to say that we make this identification in a more abstract way. It's not... We can also draw it in volumes, but in this case, we can see that they will be wrong. But it can also coincide with points, in infinity, in the same way. No, but it's not the same. It's not the same. It's the same point. Yes, so the presentations are different. It's not the same. Ah, it's the same. Well, now I just wanted to mention, to develop this question, because it is in what we call metaphysics analytics, which is what we teach in all the English and English departments, in all the philosophies, which is called Types and Token. It comes from a few remarks from Peirce. Franchement, I don't have a source original, but it's very, very common. And it's sort of a way to answer these sort of questions. which is, if a letter A is one thing, like in 24 letters in alphabet latin, then if I write like that and there are three letters A, how did that work? But the idea that this token is something like an object précis here, like something written or written, like speech act, something like that, and type, it's always a question to explain what is this type, but it's a bit clear, at least.

17:30 But here, I think that it doesn't work for this sort of problem that I just showed you, And it doesn't work because there are some explicative forces of this distinction when at the level of talking, at the level of the individual, there is no problem. And all the problems are already... What are six letters? Why is it one letter when we can do it several times? But when this identity, at the bottom, is really clear and fixed, we can apply that. And this problem that I touched on point or number, it's exactly the problem of the individual. It's not a problem where the object is abstract, it's a problem of individuality. I don't know, I don't know, I don't know. Just because if I speak analytical, philosophic and analytical, I can hear how to say, well, it's all, I don't know, but I don't know, I don't know, but here it doesn't work. Well, now, in the second part, I want to mention Frey, who was exactly the question, because it was the same formulation in the context of the myth. Well, I chose everything original, I don't have to read it in French. Well, just to say that it's very important, this problem of equality, or identity. In fact, in English traductions, we find identity in place of lighting, not equality, which, in fact, is justifiable at the end. Because it's true, I think it's true that Frege, in fact, he thinks that equality mathematiques is an identity. It's a bit of a project to do. But at the beginning, we see that he takes things like mathematics, he doesn't make a debut. That he does later as a decision. And we see some very important steps. How do I say this? I think it's important. The likelihood of equality comes not only in the numbers. The notion of equality does not come from the numbers. It's a little bit more general, which should already be presumed before we talk about equality of numbers.

20:00 Dara scheint zu folgen, dass es nicht, sollte denken, dass es nicht für diesen Fall besonders erklärt werden darf. C'est-à-dire, es ne faut pas essayer de comprendre qu'est-ce que c'est l'égalité à partir des nombres, il faut juste comprendre de manière plus générale. Man sollte denken, dass der Begriff der Drainheit schon vorher verstanden, il faut definier, before, because it was equal to, and that then from him, and the term for the number, the number must be equal to the number and the number and the number must be equal to the number. So we have this general notion, pre-establish a little bit of equality, we have the definition of the number, because the number is called the number in German, in French, in English, in French, in English, there is no common distinction between ordinary and ordinary, but in German it is already something in the language. It is to say, from just this notion of equality, it was established before, we had a number, and then the two gave us a question of whether the numbers are equal or not. Well, it's to say that we don't have something like a specific thing for a number. And that, I believe, is very important as a thesis, as a hypothesis. Because it's exactly what I want to say in theory, theory, etc., what we can think as an alternative to identity, is rather to do something else and look at, say, something specific to the context given. Ah, there is a second citation. In fact, here, it is already for the work of Dardiv, the Grundgeset, it is already talking about identity. Identity is such a very precise

22:30 C'est-à-dire, c'est responsable qu'il y a des sortes différentes de l'identité. L'identité, c'est l'identité pour tout. Bon, on ne peut pas peut-être dire que c'est la même chose que Gleick, qu'il n'a pas l'identité ici, mais apparemment que oui. Au moins, il n'a jamais, à ma connaissance, Frege essaye de faire quelque chose comme distinction entre égalité et concontentité. C'est-à-dire qu'on peut bien penser que pour lui c'est la même chose. Bon, c'est-à-dire, Frege, il co-analyse cette situation que je présentais à Dubiu et il trouve ça scandaleuse, It's not possible, it's something that we say mathematics, it's science strict, but it's totally vague, and it's really important to make some decisions. What is it? And from that, we say mathematics. Sinon, tout ça, ça reste pas science, n'importe quoi. Et bon, aussi je peux mentionner cette phrase très connue, Mark Twain, « No entity without identity ». Pas entité sans identité, oui. Maintenant, en fait, dans ma citation ici en France, je reçois une sorte d'aider policier dans cette phrase que je n'ai jamais reconnue avant, you don't have a passport, you don't exist. But, well, and after all, we can say that, but yes, what can we do, if we don't know if it's the same thing or not, or if we don't I don't want to resonate in a precise way at all. But just for a moment to say, very, very, very quickly, and I can't really justify this lecture of Platon that I made in my thesis, but I know that for me, because in fact, in Platon, the identity, in the sense strict, is something that matches with what he calls the ideas. Because in several contexts, it's almost like a definition for Platon, in several contexts,

25:00 he says that the idea is something else, I don't know, the idea of beauty is something else, or something like that. And my idea of the lecture, it's to take that in a precise way, not just like a phrase to reinforce something, but in a precise way and almost logical, c'est-à-dire ces définitions presque. Qu'est-ce que ces idées ? Alors pour objets matériels, vous voyez qu'ils sont, il y a aussi la copie des idées, et aussi je peux remarquer que paroles de copie en mathématiques c'est parfaitement platonicien, pas dans le sens platoniste, objets mathématiques existent aussi, mais dans le sens aussi profond platoniste, ces idées de copie. And in the material object, there is no complete identity. In fact, there is no more why, because it changes, and the identity is always something that is the same. The material object is never the same, it changes, it does not exist, it emerges, it will disappear, etc. There is something like analogies, similarities here, which is why we can't do real science on the subject matter, it's the Platon. On the subject matter, the subject matter is found between the two, and here there is something which is more weak than the densities, but more weak than the analogies and the similitudes that we find in the material, For the number, it would be exactly the equality. And this, in the book of the metaphysics of Aristotle, where he criticizes this theory, he shows this idea of the number material, and the number mathematical. The number material is like an idea of number, it's not an object mathematical. mathématiques. Et il dit qu'il n'y a pas d'édition, multiplication, il n'y a pas d'opération avec les nombres idéaux. Ils sont toujours là. Et par contre, le nombre mathématique, il existe comme plusieurs copies multiples de ce nombre idéal qui ne sont pas des objets mathématiques. Et pour les nombres, ça sera geométriques ce sera encore au autre niveau on peut plus voir que l'arithmétique c'est

27:30 donc là c'est à dire s'il y a quelque chose de spéciale si on peut penser je ne sais pas qu'un point sur autre sorte d'équivalence qui aussi affaiblit dans quelques manières cette densité qui n'existe qu'une idée et aussi curieusement que en fait cette phrase the name of the coin, he had an accord with the two, he had the same intention, if you will, that Freudier, yes, nothing works without identity, but also, for Platon, Platon could be d'accord with this phrase, because Platon, something doesn't exist, something doesn't exist. But even Platon said that even if you don't have a passport, you don't exist, you don't exist, but you don't exist, you don't exist, you don't exist, you don't exist, you don't exist, something like that. Bon, c'est juste, je ne peux pas parler à Platon en manière sérieuse maintenant, mais c'est juste pour donner une idée de comment on peut contester cette idée de frégué qu'il faut prendre identité comme une sorte de notion universelle et fixe de départ, comment on peut essayer de penser autre chose autrement. Now, I'm going to show you what he does in his work, and what I'm interested in now is a theory called Definition by Abstraction. It's his theory of abstraction, which he develops to find a definition of numbers. The idea is that there is an ensemble of three elements, and there are several elements. What is it that they share? What is the property that is commune for all the three elements? There is not really this simple property. simple. Il y a la relation, venaire et bien courant, il aussi cite Kanton, de tous ses travaux, qu'on peut faire relation d'équivalence, oui, par correspondance au niveau bi, au niveau de l'ensemble, mais quand même c'est équivalent, c'est pas prédicat, c'est prédicat de deux places, il n'y a pas quelque chose de prédicat.

30:00 And here he tries to build something like a number from this. But for example, he tries to define a concept of direction. Well, I'm going to say in French, yes, but here, in fact, it's interesting, because he says something really a little mysterious. He says that, well, we have a judgment on the right A is parallel to the right B, or a symbol, etc., etc., like that, and we take it as equal. because it's not a big deal. In fact, we just put this sign parallel to a sign of a big deal. And we obtain, in this way, the idea of direction. And then, I said, it's something, I don't know, it's something, I don't know, I don't know, I don't understand. And we partageons, we coupons the continue of this judgment, it's a different way than before. and we obtain in this way a new concept. Yes, it's not really clear, because the interpretation of the media, when it's possible, it's possible to define something like a function, a direction, which is a direction of A equal direction of B, and if they are also, they are parallel, like that. But here, maybe they think something else, because they also are concerned about defining it as an object, not just as, say, a predicate, but as an object, because they think about the number, the number is an object, and how, from this equivalent relation between the ensembles,

32:30 on peut construire quelque chose comme objet, comme nombre qui est intéressant. Bon, en fait, lui-même, il laisse cette idée, il dit ça ne marche pas, je veux revenir sur ce qu'il propose enfin, mais je veux aussi remarquer qu'en fait, apparemment, c'est quelque chose qui n'est pas utilisablement dans le métier magnétique. Même cet exemple simple, ce n'est pas un bon exemple, en fait. Because, for my own sense, we never use this question of direction. Yes, yes. This is projectile. Yes, sir, sir. The direction, it emerges from the key. It emerges from the equivalence. It's a question from the ensemble. Yes, sir. Yes, sir. And it's not that if I say that... But if I say that it's just always, we can always take family line, that would serve the same way that this direction? That's to say that we don't have this abstraction, just we take family, and that's not? I think that you are not aware of the difference between family and family. Yes, yes, you are right. Yes, maybe, but... Now, I just want to mention this theory a little bit bizarre of Peter Gitch, which is called Relative Identity. It's just something like that. We can say that x is the same as y, but we can't say it in an absolute way. It's authentic. We should always say something like something. And here, there are certain predicates on the table, and it is possible that X is the same Greek as A, and different from the Greek as B. For example, Michael Gorbachev is the same person today, but he is not the same president of the URSS, because he is no president of the URSS.

35:00 But this is one of the examples that he played, Peter Gitch also, it was something that one of his disciples told me that he was very concerned about the trinity in theology, it was to try to explain it from his theory. And normally it was never seen in his theory. Also, people noticed that in fact, maybe we could introduce something like that, If I write here x, y, y, I can say that it's not the same A, not the same A, not the same B, but you have to say that it's the same x and it's already the absolute sense. If Mikhail Berbachev, ok sir. J'utilise un mot propre pour identifier déjà le sens apparemment absolu. Mais ce qui m'a connu, c'est qu'en fait, il était aussi motivé par la même sorte d'exemple que Freire. Pour avancer cette théorie, any equivalence relation can be used to specify criteria of relative identity the procedure is common enough in mathematics, for example there is certain equivalence relation between order of five of integers by virtue of each, we may say that x and y, though distinct ordered pairs are one at the same rational number c'est-à-dire c'est la même type d'exemple vous pouvez aussi prendre le ligne They are different from the line, but the same as the direction. But it's an analysis different. And the difference is what? There is a sort of parsimonie ontological, we can say. There is no new objects here, but there is this diversification of notions of identity. Yes, apparently, he proposed no formalism, sauf l'Egypte de Prémy-Avourg normal. Apparemment, ça ne marche pas bien là-bas. The absolute identity, the theory, it is normal, regardless of procedure is unrigorous, but on a relative identity, it is fully rigorous.

37:30 Justifier this sort of pensée contre le projet d'être faillite. I don't think that Bayes and Doerr was motivated by Gitch, but there is a resemblance, it's remarkable. Well, now I'm going to talk about it. In fact, he is disclosed by this abstraction, and I don't want to talk about it because Igor, you're right, he's always concerned about the point of view intention and extension, It's something that I don't want to discuss today. And I saw his solution, in the simplified way of Rasset, in the principle of mathematics. And Rasset said that it's a bit the same thing. We have all this, let's say, an ensemble of three elements. For example, we just take everything, and look at these objects. It's not the way to do abstractions, inventing things, just in class, and all of that. And this is the number of solutions that Haas propose. And it's a little different from what Freire propose, but it's still very close. So, here we don't think about this problem of intensity and functionalities. In fact, Rassel, in the same way, we have discovered that these sort of classes are not good, there is a paradox, that is to say that from a class of this type, we can build paradoxes, the paradoxes of race, and the solution which, I believe, just today, is well regarded as a not final solution. The solution of work is to make this distinction between an ensemble where we exclude the universe of an ensemble, this sort of ensemble which is for paradoxes,

40:00 And, on the other hand, this is what we call the classes propres. The main difference between classes propres is that the classes propres cannot be an element of other classes. I just want to make a few remarks about this solution and why it seems to me a bit problematic and why it doesn't respond to these problems that I posed at the beginning, in my opinion. In fact, there is this notion of individual, very interesting, in fact, it's individual in the sense of logic, which Bernays introduced with his distinction between classes propres ensemble et être individu pour lui, c'est être élément de classe. Oui, oui, si, c'est un peu comme ça, politique, être individu, c'est être élément de classe. Pourquoi être individu, c'est être élément de classe? Il n'explique pas les termes. Mais moi, je crois que les raisons, ça a de propriété, d'action, d'extensionnalité. Because the classes, as an ensemble, perhaps it's also the most important characteristic, it's the multiplicity, which is something that is defined, determined by these elements. Well, and that's how Bernays wrote the class. There is no quantification on classes, it's Z, it's an ensemble. And it's a formula for this reason. He can't fix the classes because he can't fix the classification on the classes. And now, just a few remarks about this notion of extensionality. Why? Because of where comes this sort of evidence where we can say that, yes, the two ensembles are the same elements? I just wonder that there is no sense to say that, except that the identity of the elements is evident or at least less problematic. Because otherwise, what do we do? We can answer the questions when the two ensembles are the same, they are the same, they are the same.

42:30 But then we reduce this problem to the problem of the number of questions, maybe the infinite number of questions, of which elements are the same. It's a consensus, except this identity of elements in some sense. In fact, in Zermel-Frenck, for example, it depends on the foundation. Because in action-foundation, if we take the elements of the ensemble, the elements of the elements, etc. After the number is finished, we fall into the ensemble. This is also counterintuitive, in fact. But here, I think it answers the question. If we think an individual is something with an evident or simple identity, it is to be an element of class, in condition that the extensionality works, it is enough to look at the element as an individual. this is an explanation, but what I propose is that in terms of things like a class of ensemble or a class of ensemble equivalent to each other, it's just a fact. Not a fact in the absolute sense, but the problem which was a lot discussed, it was a problem of On a toujours pensé que le problème avec les classes, pourquoi il n'en marche pas avec les classes comme avec l'ensemble ? On peut dire que c'est trop grand, trop grand. La classe propre c'est plus grand que tous les autres. Même il n'y a pas aucune justification précise de cette idée, sauf pour ne pas être l'élément... Mais ce n'est pas une justification. Et moi je crois que c'est peut-être fausse cette idée, absolument fausse. And this is the case that, in fact, we can't look at, let's say, all the assembles of three elements, something like that, as individuals. And the problem with this multiplicity is that they are not individuals, I propose this as an hypothesis, just because these elements are not extensionnels, because these elements are not individuals. That's what I propose, what I want to say, what I want to develop in terms...

45:00 That's why I think the solution of class is not good. Even if we don't have a paradox with classes, they are free, but there is this very, very, very hypothesis that the class is individual and the class is extension. And that, I think, when I say it's not true, it's not something that I believe in the absolute reality of a mathematical concept. In fact, it means two things. We can try to do something like that, maybe it might be coherent. I'm not sure, but it's not explained. paradoxes, which causes paradoxes, which causes paradoxes, which is the problem there. Maybe not, but in any case it is clear that there are other things. That it doesn't matter with, let's say, mathematics traditional, it is that, let's say, we talk about the class of all the circles on a certain level. It's not what we thought, we never thought about it until the 21st century. to add something. It's not like... Even if we can find, let's say, the development of this idea at Proclus, even at Karls, I believe. There is something to make an object mathématique individual. It's not given gratuitly. But, we can say that it's traditional, on s'en fiche parce que ce n'est pas important, il faut tout changer d'idée, mais c'est peut-être plus important qu'avec mathématiques contemporaines, même dans ces applications physiques, ça ne régordait pas mieux. Et aussi, regardez ce qu'il donne à Bernays comme action, que tant qu'on a donné certains prédicats, We can't say that we have an ensemble of elements that verify this predicate. There is an action in the same way. The same thing is A or Z action. There is already an ensemble and then we can choose an ensemble that verify the predicate.

47:30 But for classes, Sperleys, I think that we can do it. We have a predicate and we have automatically the classes of objects. And this is something that I don't want to do the research of the notion of externalities in history, but I don't think about how we think externalities, I don't know, because it is important to add that if we think about the example of unicorns, for example, extensit, extension. Extension est vide, oui, mais il est vide pourquoi ? Ce n'est pas quelque chose, je ne sais pas, dans certaines circonstances physiques, c'est-à-dire définition d'extension ajoute quelque chose en plus des concepts, oui, et même si on pense quelque chose comme concept de l'homme, par exemple, tous les êtres humains. Mais aussi, c'est quelque chose de variable. On peut parler de l'être humain qui existe. Mais bon, on peut dire que ça n'a rien de voir avec mathématiques, mais pour mathématiques, c'est donner des extensions comme sorte de fonction de concept. C'est-à-dire, il faut... Ce sont des données supplémentaires extensions. Ce n'est pas quelque chose which is defined in a unique and automatic, from the concepts given. In fact, in the logics category, I can't talk about that today, there is a very natural way to define with Griffiso what is its extension as a function of the concepts, which may be variable. Oui, finalement, je dis qu'une famille d'objets s'était déformée, peut-être quand même bien autre chose que classe, juste pour montrer que, à mon avis, ce n'est pas une solution de problème. Now, I want to talk about what I consider as an alternative of an identity mathematics. And that, in fact, depends on some ideas, on can say metaphysics, but in a sense very naive.

50:00 Something like that, is that I am today the same person as here? but it's also, in fact, it's been discussed in the comment of the analytical community. There is a philosopher who says that no, and in fact it's the majority today who says that no, and he says that there is to take this ontology 4D, and many people refer to a relativity rather special, He said that in space-temps, the real entity is rather an event, and here is another event, and today is another event. I think it's the contrary in physicality. In relativity, a person or an observator is a line of universe. So it's a person at every instant. It's a succession at every instant. Yes, I also think it's a reference. From the point of view relativist, the answer is yes, I'm the same person here and today. Yes, I think it's true. Yes, exactly. If you never look at the general relativist, it's true. It's the universe. But, I just wanted to give an idea. But, on this other type of non-identity, which is also an English, It's called periodurance, it's to say that when we look at it, it's like I eat, I eat, and this one here, it's like two parts of my body in the space-temps. But what's that periodurance, it's to say that there is something like a substance, in the sense of the restitution, who can survive from here to today, or even change the property. This idea is also difficult at the point of view of identity, because there is a sort of appearance, or at least contradiction with the law of Leibniz, because the same properties are different, It is to say that we can no longer identify a thing by property, here I have a property that I have to say. But here, when we think about this idea, it is a little bit easier. In mathematics, it gives us an idea of transformation. If we take this example of the circle, for example, instead of thinking about the class of all the circles, but it's just a circle, but it moves, it changes the size, and in some sense, it replaces all the circles of the class of the name.

52:30 But what's interesting is that, yes, we can do the same thing with the ensemble, yes, we can think about this correspondence, not like relation, in the sense of logic, yes, there are two things, something like transformation, but like transformation, that one, ensemble, devient un autre. On peut identifier dans ce sens-là, dans le sens de substance. Ce qui est intéressant, on peut dire, bon, c'est aucune importance mathématique, c'est une sorte de jeu de métaphysique, mais en fait, il y a une sorte de conséquence mathématique, mais c'est fort, de ce changement métaphysique. Because, in this way, what we find is the group of information. When we identify the things, the ensemble in this way, we identify a lot of correspondents, but not all. Because the rest is the group of symmetry, the group of permutations. And it's something that comes from this novel metaphysics, if you want. And, well, I have a little bit of my time, I don't think that everyone knows what is the group, I hope. And what I want to do now, just, at the beginning, I want to say that in fact, we still use, let's say, identity or equality normal to work with this sort of thing. Yes, if I just write an alteration in the group A, B, B, C, E, I have to have this equality. It's not that it's still there. But there is an other sense of identity, if you want to. also in the sense of the form of this idea that there is one thing which is transformed, but in the sense of the form, there is also a permutation identical. For three elements, we have three factors real, six permutations, and among them, there is a permutation speciale which is defined by its properties, which is a permutation of density. C'est-à-dire, on voit ici que le sens d'identité, on peut... Il y a deux choses ici, oui, qui sont en quelque rapport avec l'idée d'identité.

55:00 C'est l'égalité qui est un problème, bien sûr, mais en plus cette idée de transformation identique. Bon, ça c'est quelque chose de nouveau. Et ici, j'ai fait juste un tableau pour montrer quelque chose comme l'analogie, disons. In fact, I believe that this is also interesting, even if I don't know the exact sense of what I'm talking about here, between relations and transformations, because many of the things are the same way, let's say normal. In the case of relation, here we are in place by transformation. Disons, on écrit ces flèches au niveau de signes d'équivalence comme relation, on a, pour associativité, on a, non, c'est vrai, non, les transitivités. Merci bien sûr. On a l'idée de composition. Encore il y a réflexivité de cette relation. On a ici cette transformation identique que je parlais tout à l'heure. Symmetricité, on a ici dans ces cas-là bien sûr on a toutes les transformations qui sont réversibles. Well, it's at least analogies, at least analogies, but of course it is not perfect, it is not a perfect introduction. All isomorphism defines an equivalent, in saying A is equivalent to A, if there exists an isomorphism. Yes, but how? If there exists, how? And if we talk about an isomorphism, it's something different. in which sense? In which sense? In which sense? In the sense that if there exists isomorphism, we don't show how many, we don't show how many groups, it is to say in which sense? They form a group, they form a group. Forcibly, it isomorphism. Yes, but we don't show how many groups, which group. If I write all the isomorphisms in a in a way explicite, in some sense, it's more information. Relation says that an isomorphism exists. However, this is something like an object, an isomorphism.

57:30 And when it's relation, it's a sort of logical thing that says yes or no. It's not the same thing, but it's interesting that there is this idea that we have in some sense. But this is the most important thing that I want to discuss in the categories, because it's exactly what is interesting. like that, can we really replace, let's say, a relationship by something like a structure with the flesh, without a little bit of a base of relationship? Well, this is the point central of the word, it's the continuation. There are a lot of categories, I can't introduce them, but I just want to introduce them in a way very naive, what are these categories. We saw that there are two ways to think about mathematics, like classes, like classes and circles. Another way, I don't know exactly the name, it's well known, it's important for the geometry of the 19th century, with the point of the program, etc. We can just think about an object that can be replaced, but also it was just a remark that I forgot, you can't say that there is an circle on the plane, because it's totally bizarre. And what Klein did, he said that it's not a circle, but something else, it's an space entier. And in fact, it's not always a very good solution. It's a remark that Pierre Cartier said that, in the physical condition, it's not good because it's the idea of the displacement, if we want to apply the theory of the group, we can say that it's all the space that is transformed. It's something bizarre. Of course, we can think of a group who works locally, but it will always be something something, how do we say, what do we say? There are some solutions at home. Here, for me to say something, we just do it.

1:00:00 Even all these big debates ontology against Perdule and Saint Duhle, we do the same. At the same time we say that there is a class of circles, which makes us transform an other. Here it's a bit clear what is this transformation, but it's exactly what we're doing at Geumetrii and Fkridi and Tunaï, because it's something that's very habitable, if I say that there is transformation, transformation, transformation, etc. But here, we can ask all these questions that I did at the beginning, is it the same or not? If it's two things are different, you see? Why this idea of transformation? When again, if something is transformed, there is something that is preserved, or maybe not? That also I will leave as question. But if you want, it's a sort of compromising in some kind of synthesis, I don't know, from these two points of view, disons extrême. Il y a une classe d'objets isolés et il y a une chose qui transforme, qui devient toutes les autres choses, mais on fait juste les deux. Mais il y a une nuance importante qu'il faut quand même distinguer entre transformation d'objets entre autres, comme deux objets différents de and transformation, transformation of objects given as an auto-transformation of them-mêmes. And this is a sort of solution that we have made in mind. For the sacre, it is also evident. Rotation like that, like the same, and other things like the other. But it's sometimes, it's a certain hypothesis, if you will, depends on what they have made, this decision of what is an entity of the sort, in the sense. And this is what we call the category. It's to say, the object which is transformed. Even in these cases, all the transformations are still reversibles. And this is what we call the tropoïdes. And in the case of general, we can also look at transformations which are not reversibles. the ensemble, with all the functions, and this gives us the most general case of the categories.

1:02:30 I'll show you here this diagram, if we take the group, the group, of course, it's a particular case of the categories, with different objects, with all the morphisms, we talk about morphisms of transformation reversible. In fact, there are two generalizations, On peut penser par groupoïde, c'est-à-dire à l'heure de parler d'un objet, on parle de plusieurs objets, et après on s'enlève de cette condition de réversibilité. C'est un groupoïde, un seul objet, c'est ça ? Non, non, groupoïde, c'est catégorie telle que plusieurs objets, mais tous les morphistes sont réversibles, que isomorphisme. Et un groupe, c'est un groupoïde où il y a un seul objet ? and on the other hand we can pass by the monohyde, so we can refuse the reversibility here with the object and then after several objects we get on the same. A monohyde, it's a category with a single object? Monohyde, yes, it's where we get the vector. Ah, oui. Et, juste aussi, on peut dire, on peut regarder comme exemple, comme ça, oui, c'est pas certes, mais l'ensemble. On peut voir, on peut faire de manière abstraite, c'est-à-dire qu'on peut juste prendre quelques classes, quelques morphismes et imposer certaines conditions dans l'effort de l'équation, en fait, qui sont montrées comme diagramme notatif. The diagram mutative, it's only the equation. And from that, we can see what we can define, from this sort of diagram mutative or equation, we can define what it is, just to take an abstract element, but to make this decision in this way, it's maybe not something else. And that's also a sort of changement very important in mathematics that I can't talk about today because it's a bit of the idea of the foundation. How we can, instead of having a fixed calculus, and then give a certain axiom, and then, perhaps, if we take a certain universe, the way we're used to do things, we can try to pass by the theory of category.

1:05:00 That I can't say today, it's very important. Now, I have just a few remarks about what happens with this question of identity in the category. One thing, in fact, I don't think there is a lot of new problems in the world, but it's sure that all these ancient problems, which we have already seen as an example of a naïf, they become less supportable. It's more difficult to be totally tolerant and just think about it. For example, if we talk about groups, we talk about groups, that is groups like objects and the morphism of groups like morphism. And of course, even in all the contexts, we say something like the group S3, the group symmetry of the three elements. Is it one thing or is it several things? But that also depends. Of course, we can talk about something like this type of text, but in some context, of course, we can have two copies or three copies, or even any number of copies. And that doesn't make any problems. However, in the categories, it's not easy like that. Because if we say that we have all the groups there, Well, it is to say that it is responsible in some ways. Of course, it is always natural to look at the object isomorphe, the group isomorphe, as the same thing. But even in the same time, we can't just... we need to remember that, we need to say something about that. And in the situation, in fact, the similares emerge already in the category of what we call abstracts. There is a very simple notion of the object terminal, that is to say, by definition, the object of the category, that there is only one morphism of each object in this object. It is called object terminal. After, immediately, it comes from this definition that if there are two objects terminals, they are not. And of course, I don't know, it's not an exception, but I believe that in all the contexts, we think about one thing.

1:07:30 Or there is a context where, I don't know, it's easy to distinguish between two objects terminals. It's not an object terminal. But it's something that is also natural. But still, it is necessary to say something, to do something. In addition, there are a lot of people who say that the theory of the category shows that we need to replace the identity by the homomorphism. and that also the people, as a philosopher, use this idea for, as a justification, what we call structuralism in mathematics, in fact. The subject of mathematics is structure and not individual. I don't believe it. Because there are things that are quite evident, that the equality is indispensable. To define what is the object terminal, there is to have equality in the sense strict. What I say is that it is unique morphism, but it must be unique in the sense precise, in the sense of semblable. After, we can identify in some senses the object terminal, and in fact, what people say, I just want to take a little bit of attention, they say something, to my opinion, a little bit arrogant, just to say, for example, strictly speaking, the canonical isomorphism is an essence. It is to say, he also talks about situations when he defines something, but then he says, ah, no, it's not a thing, in fact, it's just an isomorphism, but after he said, having realized this is best and the interest of quality to forget. But, on can say it like that, but to my opinion, it's interesting to try to be more precise. On can't always make a joke and say, ah bon, you don't understand, you need to give a few theories. And I think there is no theory here, there is a problem. Also, there is a specific thing for the theory of category. In fact, the notion of homomorphism between categories,

1:10:00 because one can define category and category, or category as an object of a great category. which is called the transformation, the morphism between the categories. And in fact, if we think about two categories, they are morphed, they are morphed in the 50th century, which is reversible. That would be even too much. It's almost like Martin said in his book, it's in the case. And it's also evident, if we talk about something like a category of groups, And I said that all the groups are already there. What is it that there are two or three copies of these categories? That would be true, the categories of these groups. But I said that all the groups are already there. And for this reason, we define the equivalence of these categories in the sense more stable. there is this notion of functeur and also we can define what we call transformation naturel between functeurs transformation of the functeurs between two... this is what you can do by the way there are two functeurs there is something that transforms the function of the function of the function of the function of the function is transformable, isomorphe à l'identité de chaque côté. C'est encore quelque chose de plus faible. Et à mon avis, c'est plutôt, théorique et théorique, c'est de donner l'idée qu'on a besoin de plusieurs notions. On ne peut pas juste faire toutes les choses avec une notion unique de l'identité, on a besoin de plusieurs. rather than to replace everything by theomorphism. But we can say that we already have several. Yes, we have this identity, egalitarian habitually, after the isomorphism, the equivalence of the category. But it's gênant, because it's just in the situation actuelle, that when we take one notion of the cellulite base and even to be precise, something like the identity of the object of the class.

1:12:30 in the sense of Bernays. And then all the others are already built from that. And that, I believe, is something that does not correspond to many of them. Why? It would be interesting to find some cases where we can think about these different identities, if you want, in the sense of Gitch, but who are equal between them, that is to say, some systems of equalities, but not just an equality, like Frey told us, the base, absolutely established before, and then to build other things. You have to think about several entities from the beginning. That's in fact the end of my proposition. But now, just to look at how it is, is there something in the theory of the category that one can think about the realisation of this idea? Because until the moment, it is still But the fact that from that, it's not something mathématique, really. And in fact, just one remark, very brief. In the case of groups, of course, almost every philosopher is mathématic. if I say that in this transformation of identity, it is necessary to understand how identity is in the same way, I don't know, it's just an object, it's just another object. And of course, it's based on equality, and here it's just an identity, it's just an identity, but it's not a real identity. Because it's not something logical, It's an object mathématique, it's not an object logique. But I can't talk a lot now, it's interesting. But with the theory and the category, everything changes. Because, I say, logic and mathématiques are more separate. In fact, if you want to continue the project at the beginning, we can do logic and mathématiques. But in fact, this project, at the beginning of the 20th century, was repris in another way, and in a way more conservative, if you will, by Jean Combrasse-Refregue.

1:15:00 It was used as a methodological logic, not a methodological logic, but it was still repris this idea that it was to take the logic as a base of all the mathematics. And of course there were a lot of people who were not d'accord. Something that is called mathematical, yes, I don't know, I don't know, but also bull. That's why I don't talk about the Christianism. It's something more general than the Christianism. But I think there are a lot of categories that it really gives a sort of project very strong for this sort of idea to put it in place. Because before, at the same time of the UG6, it was a very strong project to reorganize everything with the calculations, the predicates and things like that, but at the same time of mathematics, of course, we just don't think about it and do other things. But here, with the category, there is a project of reorganization of mathematics from other points of view. And in this new case, this sort of argument doesn't happen. We can say, no, it's not something of logical. We know that in the category, we can make a lot of logical. C'est-à-dire, de manière générale, on ne peut pas rejeter cette idée, mais par contre, si on regarde ce qu'il passe, ce qu'on appelle la logique interne de catégorie, ce n'est pas très sophistisant pour le problème que je parle ici. Parce qu'en fait, qu'est-ce que c'est la logique interne de catégorie ? It's to say that we assess certain calculs logiques with categories of données. Of course, depending on the logicist, we can say that these categories are the model of calculs logiques, and these calculs logiques come in the first place, but the categoricals, rather, think of the contrary. But it's not important. In any case, in this logic logic, there are certain relations

1:17:30 of entities, like Sir Neofrage, or by definition, which is part of the logic. There is this sort of equalities that we have in the group of Binabandi, what given, habitué, on peut dire, dans les catégories, et c'est qu'on fait que ça correspond presque à la manière réciproque. Dans les cas simples, pas dans les catégories, on peut faire quelque chose comme l'interprétation, ce qu'on dit en anglais, sound, et sound c'est-à-dire vrai dans le modèle donné, c'est-à-dire dans notre catégorie, because here the model is fixed, in this sense, it is complete, it is to say that they are interchangeable, this identity which comes from the logistic side with the calculus and the probability normal, it is complete. But there is no idea of internalization of identity here, in the sense that it works, let's say, even with the certificate, because the result is that we can express the existences of the universe by the algebraic properties in the categories. It's a result very good. But there is no analogy with the categories. and the other side, and it happens like that. It's a little bit... There are no projects, let's say, in what we call the language interne, the topo and things like that. However, in the domain a little more specific, So, I'll take 10 minutes, okay? There is an essay very important to really talk about this question. And I start in an order where we can talk about historically, because I start with Mr. Petit-Bourri Fubré, who has been able to take the work of Alexander Gratendik, and where he has aborded his question. And from here, the question of the beginning of Benabou is how many theories of categories we use in fact when we talk about theories of categories

1:20:00 in an ensemble, in a naive way. Yes, there is a class of objects, but also we talk about family, monomorphism. On utilise beaucoup les choses à cette manière naïve et il essayait de préciser qu'est-ce qu'exactement on utilise. Et à partir de ça, on prend les catégories des ensembles, ou si on peut prendre quelques topos plus générales, on peut toujours penser pour l'ensemble, and we can just look at the category C, and we define the categories of families of objects, yes, it is to say the objects of this category F, that are just families of objects indexed by certain ensembles in S, like that, and morphisms also families of morphisms. Well, I don't know how to do everything, how to define the composition, but all that is also evident. We start with just one ensemble. We make an indexation of our family by an ensemble. And this is what we obtain, we obtain a category which is called a fiber. Family indexed by the same ensemble, here. But after, there is a supplementary structure, the form, because if we look at the ensemble, this application functions, the ensemble G, ensemble U, here we can also define the same way fibres on G, d'accord? But after, we have, of course, something like morphism between family indexées par l'ensemble différent. C'est-à-dire, si ici j'indicite par J et ici par I, il y a fonction de J à I, je veux bien savoir qu'est-ce que c'est ces X, I ou I sont fonctions de J. Et qu'est-ce que c'est ces morphismes ici ? Bon, on définit aussi ça. Moi, ça c'est aussi simple en fait à faire. Juste je remarque que notre ancien dans une fibre on identifie, on dit que

1:22:30 j'écris cette morphisme là entre les familles, ça s'appelle les catégories fibrèles, catégories des familles. J'ai juste écrit cette morphisme par cette perte pour mentionner en manière this function between the ensembles, and this f is all the findings of our category of départs. And in the sol-phibre, I think this is the same ensemble. Bon, maintenant, il y a certaines propriétés de cette structure qu'on peut observer. Mais maintenant, l'idée est comme ça. On essaie de faire de manière réciproque, c'est-à-dire qu'on parle de quelques catégories, on regarde un fonctionnaire abstrait de quelques catégories in the ensemble, in the autopilot. But then, we try to think about this object of category rather than family, not as an individual fixed, but as a family. And then the question of which property should we be able to verify for this possible thing, to look at the objets de catégorie quelconque, comme les familles. Et, précisément, qu'est-ce qu'on essaie de faire ? Effectivement, on s'intéresse des familles de cette forme-là. Oui, s'il y a une fonction encore dans les ensembles. Ici, dans les ensembles, tout est clair, and what do we do to have an identity here? How do we identify, let's say, the identity of this family? How do we do it? In fact, there is a property which defines this function isomorphism, In fact, it's unique, canonique, and it's morphism. It's called the Structure Cartesian, which is introduced by Alexandre Gautendique,

1:25:00 and which is the result that for each morphism PSY in the category R, tels que c'est V ici, puis vérifier, faire cette triangle commute dans les ensembles, doit exister la sole, la sole matricule, the diagram communs, it should exist. It is something like, in the categories normal, we observe this situation in the ensemble, it is construed in a way explicit, after we forget the interior of our family, we just want family like that, and then we try to that from a certain property. But the idea is that when I talked about it at the beginning, I built a family. Family, in this sense, it's just the ensemble, the classes. And I I can define what is yj by this italian. But now, if I look at this, I look at this point, I can't describe what is this italian. But at the same time, I can define what is the isomorphism. And now, this one is already in the SAS, in general, the category fibrous, that is to say that it is just a function like this, which verify these conditions. But what is interesting, I can now really take this identity in the object, in general, in several different ways. different. Il faut juste choisir cette identité de chaque famille. En fait, ces choix je fais

1:27:30 pour chaque fonction dans les ensembles des bases. C'est-à-dire, ça juste me donne the idea of how the interior of the objects comport from an ensemble of the bases. And this is what Benavut calls the vibration SINDE. And the sense of SINDE, it gives an equality in this category S which, say, doesn't exist. At the beginning, I formed the family, but now I just think about the family and I think about what I can think about it. And from the mathematics, this is evident that there is no response unique. It depends, it depends on the construction. There is the case when there is no solution, there is no solution. solution. Bon, et ce qui est intéressant aussi, disons, pour notre problème, il y a deux choses que je veux remarquer. La première, c'est que cette famille définie comme ça, elles ne sont pas extensionnelles. Extensionnelles, c'est-à-dire qu'elles ne sont pas déterminées par ces éléments, parce que c'est ce que l'élément peut être défini, ça dépend de cette sensation. Dans plusieurs manières différentes, on ne peut pas être défini ici, c'est défini avant, disons, ces éléments. Par contre, disons, pour cette approche quand même, c'est bien limité et ça, en fait, Benabou, en réponse à certaines questions, il y a évidemment, il vraiment souligne à la fin de l'article que EFS est une catégorie normal. Quand même, on bien utilise l'égalité normale, ça ne permet pas cette idée de vraiment dépasser cette idée un peu pré-établie d'identité et l'égalité, et ce qu'on fait, c'est juste qu'on construit à partir de ça quelque chose dedans qui est négo. Il interprète ça à manière, disons, épistémologique, il dit comme quelque chose il y a dedans déjà identité, mais on s'intéresse de savoir comment on peut distinguer l'objet à partir de catégories de bases données, quelque chose comme ça. C'est-à-dire quand même, c'est pas quelque chose qui remplace en manière définitive la notion d'identité, mais avec la notion

1:30:00 d'identité qu'on peut la mettre à théorie, on peut dire, on peut construire cette identité It's really a category of it. Well, the second proposition, is what we call the superior category, where I start with two categories. In fact, I've already given an example. When I was talking about the natural transformation, we introduce morphism between morphisms, which is called two morphisms, and they can be composed in a way, let's say, normal, and that gives a category of all the morphisms between objects A and B, but also they can be composed in their senses, and that is also a category, it is called category and we have a category of morphism between two objects, and morphism, here, are two morphisms. But if we want to be composed in a dite horizontal, it is to say if we have three objects here, yes, we can do these things. And in fact, the example of the two categories, but we didn't talk about the biocategories before Erisman, I believe. But examples were already there from the beginning. Exemples, categories, categories, transformations naturelles, like the morphism. And what's interesting here, look... Well, there are of course things that you need to verify to make it work, which is called the conditions of coherence, which I leave to my side, but the idea that I built from the categories of data, this enrichissement that we say that all of this comes together. This is what is interesting for our problem. We can look at this category, Ohm, it's a category of morphism with two objects fit, if in the category of Ohm, it's a category of art, I have also a category, if we have a product cartesian,

1:32:30 we can build this function, this function-là. It's something more general than just the composition. This composition is a morphism in the category of the base, but we can replace this morphism by category of morphism. You see? And this gives us an idea of the following. We can just think more about this category, just say it's a class. d'accord ? On dit que c'est une classe et il y a aussi une classe d'hémophysme, tout en bas. Mais on ne définit pas ce que c'est vraiment catégorieux. Et tout ce qui passe à ce niveau bas, tout ça se définit à un niveau de deuxième étape, à un premier étape, à travers ça. Et ici ça nous donne la possibilité d'affaiblir les choses. C'est is the first time that we are doing for organic in kami