Category theory foundations, structure & structuralism / discussion René Guitart et al

0:00 And this new notion is a little ambiguous because, of course, there is this idea of theoretical theory. There are people like Thomas who say that all this interpretation is something that has not really an epistemic value. It's a bit the same thing that if I go down on a tableau to make my mathematical reasoning and geometry more easily. In the same way, I can think my point as point. Or, let's say, geometry projective where I think point as right. It's a difference. It's something completely subjective. and at least in the final result, we don't have to at all this kind of thing, it's something that remains out of the cadre of what we can call really objective. On peut regarder peut-être les choses un peu autrement, que quand même ça reste quelque chose d'absolument indispensable, cette histoire d'interprétation, on ne peut pas vraiment faire le turisme à l'interprétation. and from where we also have this different version of structuralism, where we say that we only have structures, it is a structure. We also say that there is a structure that is always materialized, but this materialization can change, but we can never absolutely embarrass ourselves. this materialization. Well, now what I want to propose, what I want to say is that this solution that Hilbert has proposed, a solution to a problem really epistemological problem, which appeared perhaps in the most remarkable way in the geometry, with the invention of the geometry non-coldian, etc. This solution, of course, for the XXXe siècle, would be something very important, but today, I believe, we can do better.

2:30 On peut faire mieux et même revenir, si on revient à cette situation de début XXe siècle, on voit que cette solution d'Helbert, il est disons très limité. Well, we can find this opinion that, let's say, the approach of Hilbert, it gave the only way to think about what is the geometry non-clédient. It is to say that it allowed to think in a more abstract way. And in this case, we can build a geometry, a bachelor's degree, etc. Well, I think that, of course, it gives one way to think about this kind of thing, but not the only way. Because, in fact, this example of a projective duality point and line right, In fact, this example is very, very specific. And it doesn't come at all if we think, let's say, even geometry grémanienne. Why? Because what we have as an interpretation in this case, in general, is not at the level that is not at all reversible. Même si on prend le cas classique traditionnel de Beltrani, qui a trouvé ce qu'on appelle aujourd'hui le modèle de géométrie euclédienne, non, c'est le cas de Batshevsky dans la géométrie euclédienne, Of course, it's not an isomorphism, I've never found an isomorphism between two geometries, because it would mean that it's the same geometries. Not at all, it would be a plungement, and in addition, not really a plungement, because he's wrong Beltrani. It doesn't work like Hubert showed very soon after the paper of Beltran had been published. It doesn't work locally, it's a proposition. There is a singularity on the pseudo-sphere that we can't embrace. There is no other surface in the plurian, tell que les juridiques sont vraiment lignes droites dans le géométrie Lovatschewsk.

5:00 C'est-à-dire, ce qui me semble vraiment important dans, disons, ce progrès de mathématiques, cette idée même, disons, d'interprétation d'une théorie dans l'autre, disons, un objet dans l'autre objet. And that is something that is, to my opinion, very important, very important in mathematics in the 19th century. And of course, it was Hilbert. It gives a response, it gives a It allows us to say what is the interpretation, but it does not allow us to say it in a general way. Because in a general way, it's not enough to look at these kinds of reversibles. Of course, Hilbert comprend in some sense that. If he says, I give a arithmetic model of, I don't know, I don't know if I find an isomorphism. He doesn't say that geometry is arithmetic, of course, but he always thinks, if I understand it, he thinks, if I understand it, on arithmetic, after we build something in arithmetic, we think of something in a way, say, isolated, and then we do an isomorphism. As far as general approach, it's just to say, well, there is arithmetic and theory, there is other theory, geometry, and then there is an interpretation. We can also reduce the geometry in terms of arithmetic and arithmetic in terms of arithmetic. And even... That's something that Hilbert apparently doesn't have to do. We don't have to do it at all. Even if it exists these two inverses, it doesn't mean that it becomes reversible. because for this interpretation in the sense inverse, it's reversible. If I have a theory A and a theory B, the same thing that there is this morphism,

7:30 it doesn't mean that they are reversible. Because for them to be reversible, EG égale quelque chose comme, j'ai déjà écrit dans le terme de catégorie bien sûr, EGF égale un B, quelque chose comme ça. C'est-à-dire qu'il faut deux conditions dépendantes qui doivent être réalisées pour que ça marchait. Et en fait, bon, on peut dire que ce n'est pas grand-chose, c'est juste rémarre et généralisation, mais en fait, à mon avis, ça change complètement toute, disons, idéologie, quoi. situation like that. There are two theories that are produced in another. Does it mean that they are in some sense identically and that you can imagine at the top of a sort of theory, I don't know, formels, schémas, which in some sense is made by these two? No, no, that doesn't exist. If you want, we can find the concrete. We can't say that A, in some way, is the same thing as B. Do you not say that within A and within B, there is a structure that is the same, but A does not reduce this structure? Well, that's a good question, but what do we call structure now, what do we call structure? Justement, this would be identical to this form. Yes, but I think, of course... Limit to this structure, then A and B are the same if we don't take into account that this structure. Do you point of view of this structure, it's the same? Yes, I agree, that's a notion of standard structure. Maybe for our discussion, I propose a definition of what is structure. It's something that defines an isomorphism. I believe that is also general. But what I want to say exactly is that it doesn't matter at all to manage this situation with geometry, because here I said they are not isomorphs exactly, they are not isomorphs.

10:00 And if they are not isomorphs, I don't know if this notion of structure, in this sense, I don't know if it would be useful here or not. Maybe not. Of course, we can say that A is the same kind of structure and other kind of structure, but I don't know how it works. And now I've already touched on the theory of category. You know, this idea of category and structure, and in some ways, there are categories that support structuralism. I don't think about it at all. Why? The question is, is category, yes? Is it a structure? Or no? And I believe that we can give a negative question. Why? Well, if I say, structure is something defined by an isomorphism. It is to say that we have to think of a category of something, or an object of something which is perhaps replaced by an object, and have a structure, let's say, isomorph. But it's well known that the notion of isomorphism doesn't have a sense for the categories. It's even in the mind of the elephant, for example. If we take an example of something like a category of ensemble, or a category of group, this kind of category that McLean, in 1945, there is no such thing like that. Apparently there is a problem with the theory of ensembles, because when we talk about the theory of ensembles, in some sense that we don't know, we have to talk about all the ensembles. But when we do that, apparently there is no sense to say that there is all the ensembles here, all the ensembles for the second time. And if we look at the category of the ensemble, the first category of the first one, there is perhaps not something else to the ensemble, but it's not possible. It's possible, it's identifiable. C'est-à-dire, à mon avis, catégorie, ce n'est pas du tout une structure, et je crois qu'on peut dire par contre que catégorie, c'est quelque chose de plus général que structure.

12:30 C'est-à-dire qu'on ne regarde plus les catégories comme espèce de structure. S'il y a problème avec cette vision, toute cette idée d'abstrait nonsense, etc., cette image des théories des catégories comme trop abstrait, je crois, c'est exactement cette image qui vient dans ce cadre quand on regarde les catégories comme structure. Because as a structure, it's very weak, it's almost very weak, it's a notion that's not interesting at all. However, if we look at category, something more general than structure, it's to say, if we look at category and structure a little bit at the same level, and more general, in which sense? If I say that structure is something that defines isomorphism, then I say that category is something that defines morphism, It's not a good way to say it, I don't want to say it, but it's important that there is a whole kind of morphism, not only morphism, and it really opens port to richness conceptuels very important, as we know today. but what is important here is that we take actions that everyone knows, which are very weak, and we work with the morphism. From this point of view, I believe that if we understand the theory in this way, there, there is no reason to say that it gives support to structuralism. Not at all. But this vision suggests a more general sense of structuralism, but maybe it doesn't need to call this name, I think. Now, the question, how can we use it in a physique, let's say. That is something that I haven't really worked on.

15:00 Well, maybe to start approaching this question, we need to understand what is this structuralism in physics, in relativity, for example. It is not structuralism in the sense of Hilbert, but I believe there is a sort of noyau commun and this is noyew, because in physics, we also have an isomorphism, like transformation, transformation between the systems of the coordinates, but it's not an isomorphism, it's just an assembly, it's an isomorphism well defined. And maybe we can say something like that, but at least it works well to the general relativity. In general, it's complicated, I believe. In the general relativity, every coordinate system gives a certain model of space-temps. And then we have several coordinate systems, we know how to produce them, and we know what they share, between the internet or the system of coordonnées, let's say the notion of interval, etc. That is something that we can say that doesn't depend on the system of coordonnées. What do you think about physics in this moment? The relativity is, let's say, just restrain, but not the theory of mathematics compared to physics. From the moment when you realize completely It's not to say that we are just looking for the things that are real and mental, but how do we do it? What is your notion of realisation at this moment? But the notion of realisation... The culturalists don't work very well. It's not very satisfying. You have a notion, even in a domain restreint? But I think that if we have a fixed coordinate system, we do the measurements, then we obtain the numbers. But the physics does not reduce the theory of physics. This is a part of important, of course, but it is not just that.

17:30 You remain in all the considerations of physics, you remain in the domain of mathematics. I don't see where the physics thinks that she distinguishes from mathematics. I propose this analogy, but maybe it's still more than an analogy. On can say that... Just the question is how can we really interpret, say, relativism restrain in a way structuralist. And I think the answer is like that. We have theory, which is exactly invariant to a relationship between coordinate systems. It is to say that all the laws, objectives, etc. It allows to be transformed in a reverse way in other coordinate systems. However, the coordinate system gives us exactly a sort of rapport with the reality, that allows us to do experiments, observations, etc. And the theories, it remains, let's say, invariant about that. I think Eddington, he said, point of view, point of view of no body empathic. It's not the point of view of God, let's say, like in the Newton's physics, but it's something that is shared by all the points of view, and the point of view is to understand the system coordonné, the system of measures fixed, shared by all this system possible, and even not actual, of course, where we can attribute a system which is not realised in the physical sense. And in this sense, I think it's a bit of a logique. In the general relativity, I think it's in fact, at the same time, we talk about it for the general relativity, but I think it's not going well from this point of view because exactly where there are observers, say, who are not equal, who are not in the same situation. observatores dans le trou noir, ce n'est pas la même, on voit autre chose que dans les régions pleines d'espace-temps, et c'est-à-dire ici ça change un peu déjà, je crois, mais ce que peut-être

20:00 on peut faire avec la théorie du que dans la physique, c'est changer encore plus, c'est avoir relativity, not in the sense of the norm of empathic, the equality of reference, but rather of a system, maybe not trivial, of transformation of this point of view difference between these kinds of traductions. I want to make another remark, just that I believe this point of view structuralist He is unstable in some sense. Why? Because he can transform himself in something that I wanted to propose, or he retombe very easily in his point of view classical. I think it's going to go through math. Because for Hilbert, it was very important, this idea of alternative interpretation. But, when we start the algorithm, he says, we take three sorts of things. But then at this moment, of course, we can maybe redefine our ontology, redefine our research subject. If we say now that our subject of research, ontology, it's not the points, the lines, etc., etc., it's a sort of, I don't know, an object object without real importance epistemic, real value epistemic. And in place, we think of something like a pure thing. And then, well, we have to say what is a thing. And then we do the theory ensemble. He said, well, at this moment, mathematics is more than two lines, something like that, there are more than two ensembles. And ensemble, how can you think? As an ensemble, you can think as points, as lignes, as cheminées, I don't know. we consider that we are not in this situation, and we can envision a constitution, a theory of the object that we want. Well, from the metaphysics, it's not fair. It's a coincidence. And if we consider the category, even the category figurative, as a realization partielle of this point of view, it's not fair.

22:30 It's not fair. On ne sait pas faire simplement parce qu'il n'y a rien de plus indéterminé que la métaphysique. Il faudrait choisir un sens précis de la métaphysique. Chaque fois que les théoriciens de la connaissance, même issues de la physique fondamentale et de la même travail, se précipitent sur la métaphysique, ils choisissent bizarrement une métaphysique déjà constituée généralement, qui est depuis longtemps obsolète. I don't think all the metaphysics, until the past, are not usable. Well, I don't think about the metaphysics, especially. But I have just this image of what we call the theory formel, a theory more abstract. It can be compensated, let's say, in terms... It's not conceptual, I don't think about the metaphysics. It can be compensated by defining the objects. If we say that the mathematical object, the real object is not something that we had before, but it's an ensemble, it's exactly a little bit in the same way. Because at this moment, we can say that there are the ensembles, this point of view, say, fregian, in some sense, is reconstituted in this moment. The problem is that the intelligence form is metaphysically determined in the theory of the category. Probably not, I don't know, I haven't read the text from this point of view. It's metaphysically determined and the two are probably quite equivalents to this point of view. We can figure out what we want and with the arbitrage the more pure. On se repose en définitive sur l'expérience, mais d'une manière qu'il faut préciser. L'expérience, c'est aussi une interprétation. Oui, mais la situation est extrêmement complexe. Elle n'est pas facilement. Je pense que peut-être que dans ce cadre-là, ce qui fait la spécificité de la physique, c'est l'existence d'une intuition cognitive. Yes, but if we do physics, we refuse the notion of evolution. We are going to get on the philosophical errors of the philosophy of the philosophy. Just a moment, that's sacred.

25:00 And that's why we can see the probability of a point of view structuralist. In the sense of where, for even if there was a sense in the notion of the events, the point in an instant, then here is the structure to this notion. But this notion, it is given by the intuition, by the experience. In fact, in fact, we don't know what it is. It's to say that all the work of the physics, it's just to identify what it is really, this intuition that we have. And obviously, it's the metaphysics, but we don't know it, but we don't know it. But the structure structure is to say that I can talk about the point in the space-temps, then I look for the structure that there is. But also, I believe in the interpretation of the general relativity, we see a bit the same thing. At the moment, when we start working without coordinates, it seems very nice. Exactly, I think it's a bit of a point of view that Marc left the defense, that at this moment, it's a little bit of an embarrassment of this idea of production. We have a sort of object fixed, there is no need to talk about this idea of transformation of an object in the other, But we need to define this object in a way too abstract. We can say that we can define this object as a diffumartism or something like that. When we find the right identity for this object, we have no need to think about this object as multiple, or transform into another object. We need to fix these objects. I think that, of course, it's not to have a different coordinate, because it's just a technique that is hard, and we can do better otherwise, but I think that exactly this formalism of the theory of category they replace, in that sense, this function, exactly. And they allow, say, transformation, and not by this notion, but in their formalism, in their transformation of the coordonnées,

27:30 Yes, of course. I remember that the question of Mr. I was in agreement with that. It's that for me, it's a structural issue. I have the impression that there are a certain kind of things that are mêlées in everything that you put on the table, and the difficulty. There is one part of the question of axioms, and the other part of the structuralism. It's not all the same thing as the formalism. Well, on the question of the axiomatic, I've heard a conference there a few days ago, it's been a long time ago, on the theory of the djeb, the science of the time pure. It's quite interesting. And at the time of this conference, I feel like this is an element of debate on the first part, that we could explain, that Hilbert is at Frege, that Peacock is at Hamilton, which is a fraction of that, Hilbert is at Frege, equal Peacock is at Hamilton. So, you can't have several interpretations. First, you have to have an intuition, and that it is unique, that you have to write, with which you have to write, but you can't be able to write like that. The same alternative is between Peacock and Hamilton. Hamilton Peacock is an algebraic, algebraic anglaise in the first grade, who is the first to say that we can be authorized to give any action, or whatever, very formalized in terms of modern terms. And then Hamilton, who is not satisfied with that, he says that we should talk about the brain. Okay, we will do the calculations, we will develop the changes formally, intuitive base is produced. The intuition base is the time, and the gene must be based on the intuition of the time. In the description, it is the sign of the gene. So, we find something that is semblable. This is the first remark, but it is also an interesting remark, because it means that this debate

30:00 is not going to start with Frege. It is already there, in the end of the gene. And then, the second point, maybe, is this idea of the intuitive, of the intuitive, I would say, purely functionally. Like you said, Kassirer talks about that in the substance and the function. well, he will opt completely for the function of the function, the fact that we will be able to define the function at the detriment of the substance, but not at the detriment of the intuition. So it's complicated, because the substance is tangible of the object, there is also there is a confusion, to say that this object, mathematics, or even if we have the idea intuitive of the substance that it is done. That, for example, if you want to put it, it's not the idea intuitive of the substance that counts, it's the idea intuitive of the function. So there, you know, the intuitive side can move to the work of the literature, almost. There is no solution. There is no solution. There is no solution between the intuition, which is like the intuition of the things, which are the things of the world, and then the intuition, where we have the work that we make on it. Or the structuralism, I think that the structuralism that emerged in the years after Hilbert, that is the structuralism that comes from the feminist, the Russian, and who is imported into the mathematics, that is the structuralism, it takes first that in charge. It is already after the systematic, it is the charge of the fact that we will try to take note of the intuition of the functions. It's not that it's formal, it's a bit complicated because we talk about formalism russes and linguistics, but on the point of the mathématiques and the structuralism in mathematics, I think that we could have to decode. is that really it is a formality that is purely detached from the intuition and that it begins? I don't think so. Because the challenge is to have to write

32:30 the functions on the basis of the intuition that we have these functions. Not in the air, So this is the second kind of remark that I would like to say, because it is very complex. And then after, on the role of the author of the notion of the model, on could say a bit, but... On the role of the category, there are two or three things that we can decide. Well, first of all, your example of the tableau, to make you feel the problem of the F, which would be the common form which does not exist, which would be in A and B. Let's take an example very simple. We take the type of object, the type of object, an object with a point on the subject. It's not an object, it's an object. A, it's an object, and B, it's another object. they are isomorphs, because it is enough to imagine a translation, to take the right and to put it on the other, in the point where it is, in the translation, in one sense, or in the other. But the question we have at this moment is, is there a right point in a sort of canon, or no, there is not, because each time there is one, in the context of ensemble, in fact, we could start to think there is one, because we could, as a point, take an ensemble of the people in an ensemble which is the same. for the concrete description of the objects, that leads us to this. And that leads us to the possibility of analyzing the objects of the types of structures. So it's all a little bit of a remark on the fact that it's a shape, and it's in the science of biology, On voit bien la différence entre la similitude entre deux objets vivants, deux organismes par exemple, il y a deux bras, deux jambes, puis d'autres similitudes. Et puis la conclusion qu'on va dire, ah ben ils ont un ancêtre commun. Ce n'est pas la même chose. Il faut trouver une sorte d'origine à la similitude. Alors effectivement, dans le point de vue catégoriste, en un certain sens, quand il y a de la similitude, It's not obligated to have an origin to this similar issue. We abandon the question of origin. It's interesting, this example of the right-wing, for example.

35:00 In this case, I think we can say that yes, there is no right-wing, there is no right-wing. And what you can say, effectively, that there are several, is that at the moment you think, it's not just the structure of the right point, it's that there would be something else that distinguishes. And so, finally, the point would be... Yes, but the problem is that in this actualization, it's not just the actualization of this form, in this case, because it's an actualization, it's not just the actualization, it's the actualization plus other things. And the question is how to identify this other thing, because if it's just the actualization, then there is no one. The problem is that when we actualize, when we project in a system, and there, in the case, when we do it in the future, it's because, in fact, it is plonged in a geométrie, by ailleurs. At this moment-là, there is this one and there is this one, but it means that we introduce something else. And the question is, well, let's see what it is, if it is something else. And so, it's not that, well, at this moment-là, we introduce F in something bigger, on the Just the first question about maximization. Of course, if we talk about structuralism, it's something more large. If we find structuralism in relativity, for example, there is no maximization. However, this particular case of structuralism, on peut dire qui appuie beaucoup sur la axiomatisation. Par contre, c'est tout à fait compréhensible, je crois. Pourquoi ? Parce que, comme je l'ai dit, pour Hilbert, si on ne considère plus, disons, notre demande des objets comme quelque chose de définitif, comme la base, si on ne considère plus, disons, la géométrie ukrainienne aussi, a notion of space, something that is really something that can be done in which we can appeal to other theories, then what remains? And the answer of Hilbert is that it remains logical. And from where all of which we can say, I'm not sure, we can say logicism, but in the sense that it is also weak, perhaps. It's not at all this idea of all

37:30 to reduce the logic, not necessarily, as we find it, but the idea is that the logic remains a sort of base of everything. Yes, but what is the logic? Yes, but for Albert, these problems don't exist. But today, yes, it doesn't work. And for, if you want, if you want to look at AHB, not just as an object theorist, but as a theory, maybe this idea of the logic is something that we see here, something very weak, but still that is for all things. But what I want to say about this, let's say, a debate traditionnel between some kind of essentialism, or maybe maybe intuitive, which is the importance of intuition, of objects, etc. And, for my book, I believe that this vision that I propose is really a kind of third one. This intuition, let's say, in the traditional version, maybe a little vulgar, it's just to say that A and B, in themselves, they are already defined. We have a good definition of the point, of the line, and then we can do other things. However, this idea formalist, it says, ah, no, it's not at all, it's a sort of place vide, it's not serious, it's serious, it's a structure that creates a relationship between us. But now, if we just don't try to reduce this... If we don't look at structure, we look at category, in the sense of what I'm saying, if we don't consider these objects as replaceable in a reverse way, substitution, it's an idea of substitution by motion, it's always something reversible. And if we don't think about that, we can't address this object, we can't say... Like I've already said, I think it's a sort of really false idea, that we can't figure out and replace an object by another object. This just doesn't work because...

40:00 Well, in fact, it's already the case here. If A and B are only the instantiations of F, then there is not A and B. There is no one. It is to say that we can distinguish A and B, and I will put a little bit between them, that if A and B are attached, not even from the point of view intuitive, to something different. the difference? Maybe I can say it in this way. I propose a definition. I propose a definition. But in this case, we refer to the visual perception of the image. We are obligated to include that in something that is prudent, otherwise A and B. I propose to make the distinction the following. There are concepts which are I can say concept-form, and other concepts, which are concepts-categories. The definition is the following. Concept-form, it's those that all instances, all examples of concepts, are isomorphs in a certain sense. It's to say there is an isomorphism, or also a naturel. If we think almost all examples of mathematics, something that we form, something that we know, Well, of course, it's a concept-form, in this sense, yes, because all the circles are in the same way, even if the circles are topological, it's also always in the same way. D'accord? The number, if we think the number as something like, I don't know, class, equivalence, ensemble, fini, original, it's always a concept-form. However, if we take something like an ensemble, which is like a group, it would be a concept of a little bit. Because not all the ensembles are together. D'accord? Voilà. Et ça, la différence, en fait, c'est ce que je veux faire ici. Et cette idée de l'abstraction dans le sens ontophrédien, qu'on peut, disons, remplacer Iso-morphs, pas objets abstraits qui représentent ces traces, ça ne marche que pour concept faux et pas pour concept faux. Ce qu'on peut faire avec catégorie, c'est le bien interpréter dans l'autre catégorie, trouver un mot foncteur, construire.

42:30 Mais il n'y a pas plus cette idée qu'on s'embarrasse d'objets, qu'on fait abstraction. I will also try to link all this a little bit to the question of the Herménitique, because of the importance of production and translation. But of course, it's also the one that we call it. But what's interesting about Herménitique is that, exactly at the time when the people, like Dilday, appuyer these subjects, It was just that Jean Congluté did not know that it was a mathematics at the time, at his own time, that he allowed to say, to give this idea of Herminiti something specific for science M1 and against science trick, it was just not true at all. What I think is particularly interesting, from the point of view of the physical approach, is that there is a possibility to do the metaphysics with the metaphysics. I don't know if it could go to you, but I mean, from the point of view of the physical, the physics considered as metaphysics, which is what it is really, what is the intuition, what is the intuition when we have the intuition and how we do it, etc. It's evident that the form, with these things, it's not a chance to do it, because we always get to the structure. if it was something with the same structure, it would be the same thing. the idea is to be able to say okay, we identify the forms on the object that we have access by the conscience and by the physical it is not that there is something that is this world this realization this intuition that we have it is different for each one but in any case there is a possibility of surpassing isomorphism, a structure simple qui dérealiserait totalement l'omarchisme. Mais on va voir comment faire. Ça reste à un niveau excessivement vague. Ah oui, non, non, c'est pas... C'est même pas manipulable. C'est-à-dire que c'est un cadre conceptuel ? Ah non, oui, oui, pardon.

45:00 Oui, non, mais... Well, I have a remark, a little bit more general, but that is a question, it is that, well, I'm going to say that with you, it is quite a bit different. It is clear that the mathematical, the maximization is not exactly the same as the mathematical, in mathematics and in physics, it's not the same thing, or the problem of the approach and the information on the national level, it's not the same thing. So, in physics, there is a book that is interesting, for example, of Patrick Malin, made by her and by her family at Oxford, where you can see in detail how we have evolved I think there is a lot of difference between the 60s and the 20s and the 20s and the 20s, there is a lot of difference. But to come back to this question, there is a need to be exercised on the notion of object. The structuralism, in general, has a pretty modest finality, I would say, compared to the fundamentalities of the physical theory. The structuralism, in general, is something that analyzes the theory. So, it's an approach to talk about the physical theory, not the physical theory. It's to say that the notion of the relationship between the concrete and the abstract is very often évacuée. what is in the physical experience I have one electron and so by the principle of exclusion another electron which is obviously different the physical theory does not have an object concret it is a bit the same problem with the famous moving now the moment present

47:30 which is executed by the theory of physics, and there are other things that are executed. So the notion of object in theory physics, viewed by the structuralism in physics, is already a notion of object quite abstract. And it is at this moment that we start to make the parallel between the structuralism in mathematics and the structuralism in physics, but viewed with an angle already quite particular. a lot of things. I think it's something that allows us to learn a lot about the theory, the structure of the theory. So the structuralism in physics is the structure of the theory. So all of this, in fact, I'm rather interested in this argument that I don't want to talk about the subject of the subject matter, but we need to try to make a distinction. And so, it's a very interesting passage. It really implica a passage of the category. I think that, for example, the reality of the content, the usage of the category, it's an important part. I don't know about this, but I don't know about it. However, it is very important that when we say physical, we say here on the physical, we have a very particular vision of what we see in physics. So we talk about the physical theory with the scientific theory and the reference infinite to the concrete, almost to the phenomenology, and it is a clear. and it is on this level that it happens. Yes, but it is one of the possibilities that I speak rather than before. But in terms of application, it would be really a bit liable to this epistemology after the physical work? After, what exists, the structuralism and physical, This is not only the problem with the humanization and the Uber, but there are plenty of things in our lives. And the approach structuralists, for example, is that sometimes they are not structuralists, but other than other things, apart from all the relationnels, which also can be placed on structuralism, all the operationalists, we can say, well, it's not bad, it's an object of structure,

50:00 there are plenty of things that we can call structuralism. So, there are a lot of research on the structure of the theory that have nothing to do with the problem of Hilbert. The problem of Hilbert, well, it's Michael... Yes, yes. But then there are many other developments structuralist where you can study the structure of the theory and physics that are not in the universe. In fact, how do I see how we can serve this epistemological discussion at the work of physicists? For example, in the last session, we looked at his article where we tried to build something like psychosal, etc. And then, what the author said, instead of writing relations with order, between events, I wrote my thesis and I obtained a category, etc. And I just thought, well, there is still more tension. Maybe it's a good thing, I don't critique the technique, but I suggest to make more tension because of some way we can deduce relations in morphism, but in fact it's not innocent what you mean. Morphism is not the same thing as relations. And the difference is that if we write something like that, then of course, we think that there are, let's say, some great ensembles, I don't know, B, B, the man, D, O, O, D, even when we find the product C, A, something like that, D, O, A, and B, it's-to-all, there's this kind of relationship that subsiste where we substitute A and B and then we see it's true or false. Finally, we always look at what we think about in the real values. However, if we have something like a morphism of this sort, it's something like that. We can't replace A.V. by something, we don't have a priori this idea of function in the values of truth.

52:30 It's an object mathematical. It's to say this traduction, which is quite a lot of change in class. But in fact, it's speculative in my case, but I believe that in a more physical sense. That's it. Because it's a transformation. It's to say that it's a transmission of information, like a measure, or something very concrete. So here, it's a relationship, it's an abstract, which is not immediately interpreted in a physical way. But after that, there is a very serious problem, which is perhaps even one of the most serious problems, which is the return of concrete in the language of the theory of physics. And it is not at all obvious that it really worked, because the theory of physics is something that is something that is referred to in the last place, and it's just for that we can understand that the concrete phenomenon, the concrete singular has been replaced by the abstract repeatable. And the return of the singular in this point, Sur ce plan-là, c'est problématique. Est-ce qu'on a été quantifié par exemple au travail dans des cadres où il s'agit de des choses singulières ? Je vois ce que tu veux dire, mais je crois qu'il y a en fait une réponse. Because, when I say here it's Raybeck-Encray, in fact, we can say that all this story of the isomorphism is already dead, and also in these cases. But, at the same time, we can have... No, but I want to say that, for example, if my theory operates with some things like A-Pledge Bay, if I want to say that my theory is physical, it's that maybe not tomorrow, but after tomorrow, I will tell you how to do their experience, and see if there is a test or a test, and there, it works, it works very well, because I developed my theory with,

55:00 it works, but the problem is that, too far, I'm going to tell you, it's not very evident, how do I tell you, not only one experiment, but two, the same thing? Well, in fact, there is also a question that is behind, for example, on the moment What is the physical? At the moment when it's the physical, we're going to do it. But if I understand, if we do cosmology, it's a new one, cosmology scientific, not just metaphysics, then also this scheme of substitution, apparently, it doesn't apply. If we really... It's true. It's true. I think it's not all at all certain that in the future of the physics, we need at a moment to concretize what is A and what is B. It could be, in all case, and it seems that the evolution of the physics could point out in this sense-like, it could be that ultimately that the concrete, that is the relation between a and a and a, that there is no need to concrétiser, or that there is no need to concrétude, and that the concrete, that is the relation, in which case, the physical could be reduced, it could be nothing else than an étude. No, there we are, we are talking about this kind of relationship, because the physical, as well as well, like a theory and relational. Like a theory, there is a relation, and there is at least two interpretations of the mechanics of that. However, it is not the same, it is always the same, with the sense of relation a little bit like the correlation information, the correlation between the rapport and the relation. However, here, what is changing is also our idea of singularity of A&B.

57:30 because the approach relationally in physics doesn't change anything compared to the possibility of experimenting, for example, in physics, and in quantification. But here, I think it's rather what shows the pertinence of this kind of question, it's that in gravity, not in mechanics, but there is a reality quantique today that exists since 2001 at a time, the debate methodological between people like Saskine and people like that, on the other hand, on the other hand, on the methods of what is scientific, applied to the problem of cosmology, there is no reality quantic. So the problem of the singular, it is also the problem of the intelligence of the theory of the core, etc. So the problem of the singular exists already in a methodological reality quantic. it is there that it can be interesting. Is that the method of the physics applied to an achievement where we can lose the singular It's always a method proper or not. And it's there that this kind of formalism acquires a sense, actually. I think that one of the perspectives, perhaps, from a point of view, is that maybe there is a way to traiter the singular in terms of structure, in the sense of high-stack, I don't know, I don't know, in the sense where the categories are. It's to say that these categories, if I understand well, are singular, just because we can't, because this is not an isomorphism, but they are not duplicates. So maybe there is a way to traiter the singular from the physics, from this point of view. In my opinion, it's maybe a point charnier of the epistemology. It's why people like John Mayer, or even Ukraine,

1:00:00 So, when we have objects like these letters, and then these letters, and then these objects, they are completely different. From a certain point of view, we can consider them as abstractions, that these letters are perhaps the lines of objects in certain types of real. But from the other hand, we can also consider them as concrete objects, and with this way, we can also be alimented. So, after all, is there a difference between, let's say, the difference between, let's say, the difference between, let's say, the nature between this experimentation and this other experimentation, and to take an appareil a little bit more complicated than an internet, for example an ordinateur, and to put in, in lieu of putting in a computer, and then put in a computer, and then put in a computer. Is there a difference between the computer and the computer? No, but there is a difference. There is a difference. If we add a third example, there where it is, for example, there are theories, here, about the universe, imagine the problem of the ecologist, there, we are not at all in the way that we have experience with an apartheid comp, and, well, there are some theories that can be, perhaps, in this kind of thing, there are several, to start by the intervention, starting by the intervention, that it does not allow. So the question is to know how is it going to work, how is it going to work, and how is it going to work with the directives, which is just so much that it's going to be able to do this.

1:02:30 I'm talking about the point of the experiment. Yes, but if we go wrong, effectively, your brain is an apparatus. It's not at the moment where we are going to measure it, but at the moment where we are going to use an article to say to others that we are going to measure it a button and not the button, it's there that it becomes something. So, an experimenter in his lab with his appellate measures, it's something that is very concrete. It means that there could be another observer that would not be observed in the same time, but when this observer comes out of his case and communicates the result that he is now in general, it is there that it would be... But there is now a situation in physics, which is all pretty well, to build a physical theory in which in this case, we are not able to move on. So, what do we do to do with ancient science in trying to know? Or do we move on to an epistemology and methodology? Yes, it's just really, I think, it's not at all, of course, because of the case reversibles, so-called, case of morphism, that has been given such importance, because apparently it's well linked to a question of identity, of repetitiveness, and it's very interesting, We can look at the limits of the language. Well, there is a part that has a seminar on the subject,

1:05:00 which is already in place. Identity, I don't touch a lot today. I believe that even in theory, categorical, as it exists, it's a question that is not managed. It's one of the notions that is not internalized in the sense of the category. There is the category of february, but when we do category, why do we do it like this? We do it like this. In the theory of category, there is no Mayan. In the theory of category, there is no Mayan to do it otherwise. It is necessary to develop it. Well, we have to decide for the next time, in fact, because we have made these changes, we can do it in a week or in three weeks. Maybe in the three weeks, yes. Because there are some who can't say that every second week, there is this Johnny who C'est-à-dire que moi je propose de faire dans les trois semaines alors, d'accord, et Marc, il va revenir, hein ? Un deux ans ? Oui, ça sera en deux, oui. Mais Marc, il revient en le vendre, il m'a dit, oui. Un deux ans ? Un deux ans ? It's fine. It's fine. No, no, no, no... But then we can say 22, and then we can write it again. For the seminar of the Fondement of the Physique, there are three in March. I don't know if you want to receive all of these announcements. If you want to receive all of these announcements, if you want to receive all of these announcements, I don't know if you want to give me a mail. Well, my mail, it's all. Give me a mail, I'll put it on the list. Thank you.

1:07:30 Thank you. It's fine. And there's some time left behind you, how much did you see it. Is that the only place where you could hear it. Thank you. I would please you