Isham & Doering's topos-theoretic views & structure of physical theories

0:00 You want to talk about categories? No, no, because I don't know what I'm talking about. I'm not sure what I'm talking about. It's not bad. It's not bad. Yes, but... I want to talk about the space-time. Space-time and the space-time. Hello, as I said in the announcement, there is this great text that concerns our study, very close to Hicham and Andreas Doring, which was just released on Archive. It was, I think, the 15th March, but already after, I discovered that before, on the blog N category café, it was already Reaction de Baez who had this Archive in the 3rd March. But, anyway, it was also very, very systematic. I believe it's a little bit like a synthesis of what did Hicham And, well, what I propose is that, now, I can read the first part, and it was, let's say, the other, the first part, in fact, it's a sort of a summary of the other part, it's to be clear what they want to do, after, of course, you have to look at what really happens. and I can just present the idea and maybe just remind me of what it is that it is that it is

2:30 Algeab writing, I'm sorry if some people know me better than me but I just give a sort of definition. Well, the text, just for motivation, I think this title, Topos Foundation, is really very important in this text because, of course, it would be interesting what they think about the physicians, but it's not the approach of physicians, it's also different from this text from what we have seen before, because before we looked at the test of using the language of categories in the physique, so we could reduce the mathematical apparatus in the relativities and mechanically in the language of categories and look at what we could give. But here in fact it's another thing, because if you want this project also logicist in the sense of the beginning of the 20th century of Hilbert. In some sense, we can look at this kind of reprise of this great program of Hilbert of axiomotization of physics, but it's not really that, because, actually, the point, it's changing the logic. It's-is that the difficulties that we have in the physics theory, and more importantly in the gravitation of the quantia, and the difficulties of the foundation, where the foundation is comprised as something that concerns the logic. This is a sort of presupposed of the base of Islam, which defines all the projects. But then there is a more special motivation, a very important motivation here, it's sort of a position, a position that he calls realist, his position to interpret instrumentalist, and that's what he says, and in fact, there is almost nothing about relativity in all the texts, When I talk about physics, it's pretty much mechanical here. It's to say that all this idea of giving sense of relativity through categories,

5:00 we don't find it at all in this text, but maybe it will be in the fourth part. a little bit. And it might be interesting to join this other, disons, chain of calculations and visits with that. Well, here we say that, disons, in cosmology, Dans le cas de cosmologie, l'absence d'observateurs extérieurs fait inapproprié l'interprétation de Copenhagen. C'est-à-dire, pour lui, ça c'est une importante motivation qu'il faut faire, disons, théorie réaliste. Et ce qu'il appelle réaliste, dans le cas classique, comme il dit, il y a ici trois points. He said that, first, the idea of propriety of the system is meaningful and reasonable and representable in the theory. The second point, that the propositions are submitted to a logic boolean. It's the classic case, I'll remind you. Well, there is even a very general assumption that the human thought, at a certain moment, at a certain sense of bullying already. And at the end point, there is an space of micro-etats, and if we specify these micro-etats, that defines the state of all systems. That's what is not the case in... That's the three principles? The three principles are called the classic realism. Classics, so, non-quantics. Yes, yes. And, effectively, we replace it, it's a-t-il, when it comes to the base, it's a realism, there's no idea of observatory who intervene, etc., etc., but it's a little bit, it's to say, it's to say that the concept of the value of quantity physical is still usable, even these values, and this is, let's say, the change of the pattern, it's not real, but maybe another object of It's the first point.

7:30 What they call the neo-realism, it will be a modification of the classical realism, an arrangement, on can say. Because after, they always try to show in each stage that all what they say works in a classic case, but it will only be a particular case. What you just said, So the idea is to replace the quantities that take real values to the quantities that take real values in other things. Yes, it's one of the motivation. But the problem is that when we do a measure, we obtain a real number. But that is of course very disputed. Well, at the end, I'm not at all with what you said, because I think that this story of reality is not a motivation. Yes, but it doesn't have to do a number of rationales with a certain... It's in this sense-là. And a plage would be more difficult. The plage would be more difficult. So the idea is to enter this plage in the definition of a result of a measure. So, in fact, if we deconstruct a little these papers, what they present at the beginning as motivation, is actually the last element in the chain, and not at all the beginning. The point of the start, and as usual, the point of the start, there are two problems that exist in the world of physics for 21 months or even more. It's the history of the norm, and it's the history of the composition, Qu'est-ce qu'un système qui a des sous-systèmes ? Donc ça, ce sont des problèmes classiques de la logique quantique. Enfin, voilà, il invente une espèce de solution qui, évidemment, qui est motivée après, a priori, par cette introduction au néoréalisme versus au sensmentalisme, mais qui commence par, pour lui, au niveau formel, par deux problèmes à résoudre. So how can we get the number of them? Because before, in logic quantic, we didn't know. And then the second problem, the theorem of solar and all of which is in logic quantic, is a bit... They leave the place to that. And then the second problem, the second motivation of the fond, it is how to reconcile everything that exists now in the world of physics, which is the 36

10:00 current of the instrumentalism, the instrumentalism variés, with the realism that Hicham should not support. And so, in fact, since they solve the problem of the real, it is, well, since that he finds a framework for the real and the composition and all that, he begins to say that our motivation is the reality. In fact, for me, it goes in the opposite direction. Maybe you have to have a reason, but what I would like to do is just give his presentation superficially to the text. Yes, at the end. After, maybe you have to have a reason. But I would like to write. Yes, for him, he gives the description. Just for the quantum logic, he said at a certain moment that... There he defined the realism. No, but the instrumentalism. He doesn't give a definition of instrumentalism. No, he doesn't give a definition. But it's just the idea that there is an observator... There is something, somewhere, where he says... If there is an equation, it doesn't exist, it doesn't exist. It doesn't exist, it doesn't exist. Ah, oui oui. Il y a un peu de mesure qui donne quelque chose qui se rapporte à cet objet. C'est où il donne... Ah voilà, page 11, premier papier de la France. Il dit dans le cas classique à quoi s'agit... Page 11, il dit à quoi s'agit dans le cas classique, le instrumentalisme réalisé. Un instrumentalisme ça serait celle de Copenhague. No, no, but it's not Kopenhag versus Einstein. Historically, we can say that it's a rapprochement, but instrumentalist is much more general than Kopenhag. Instrumentalist, for him, he gives a definition. Yes, it is realist, but it is not real. Realist is to say that the physical quantity has certain values if it is in certain symbols of values. And that is realist, it is to say that it is really a value. And instrumentalist, if we measure A, then it is just that it is just that we replace

12:30 in the sense of logic, propositions assertives, or propositions conditionnelles. Ah, but that's really the position of Copenhague. That's to say, the thing has to be... Yes, yes, yes. Copenhague is more than that. Well, but perhaps... Yes, but... Yes, but there are other things in Copenhague. But anyway, I think it's true that here, Yes, it's motivation, but it's too generalist and maybe not too important. And all of that, in terms of philosophic, metaphysical, classic, or, disons, neo-classism? Non, is it possible? Non, justement, at that point, it's not possible. Non, enfin, disons, côté ontologie, il n'y a rien de présupposé. Justement, il pose la question de savoir si on peut dire quelque chose à propos de l'ontologie. Il présuppose soit une ontologie d'avance. No, absolutely not. Justement, the realism, which for him is a position of ontology, is a position of ontology, and he is put in question on a little... I want to say that he gives a sense to the phrase, this exists, in a deterministic context. They have a value. But at the same time, the value is this. This is this, and this is what they define in a simple way. So they want to define the objects also. That means that no ontology is supposed to be. And they give a development of this, let's say, an ontology more general? But what he said is that this approach, even in terms of naive, realist, is kind of tenable, in condition that we can relax, generalize this, disons, presupposive, concerning Valerie Boulien, etc. Notamment, on dit le suivant, c'est-à-dire qu'on n'a pas vraiment vraiment toutes ces démarches instrumentalistes pour sauver le formalisme kinky, etc. On ne peut plus maintenir ce réalisme dans le sens simple, naïf, pas vraiment bien du tout précis.

15:00 but he said that he had to do the next thing, he had to replace Boolean by Heitzing, and now I want to remind you what it is, and he also says that there is no tier excluded. And the third point, and it's also in the definition of this neorrealism, that in the topos, all this will be interpreted in the topos, and there is the object which is called the object of state object, which in fact, after, it's something that I don't understand too well, because after it identifies this state object with the true value object, with the object Identify. Identify. Yes, he said that it's a bit less than one. Yes, we'll see. Yes, it's a bit strange. the position, but the sous-objet. The state is not the values. It is the same thing. The state is the state of the state, the state is the state of the state. They consider the opposite of the one, after there is this object of state and there is another object which is the object of the value of the quantité. But the object of value, the object of the value, The object Omega, which is defined by the galaxy, which is not the same thing. Well, we'll see. I don't think I'm wrong. But of any way... There is no one, there is no one in the first one. For the first two characters, the sub-objects, are in the same way. There are three objects, in fact. Classically, it's a different object. So it's the space of the etats, the space of the etats. The real is the space of the values. And 0,1 is the space of the values of the real. But after, there is a generalization of these three things. But he doesn't say that it's three things, he doesn't say that it's three things. In other words, he doesn't say that later. But of course, the subject matter of objects will intervene. But after, of course, there is a complication, deconstruction. At the beginning, he explains the motivation.

17:30 which is the physical quantity. It's for him exactly this morphism, in the topos, from the object that he calls the object of state to the object that he calls the object of value. This morphism, it's page 3. If it's the object of state, we make an index which we don't do, and if we value it, we create R for real, but it's exactly the index that could be another thing. And that... L'objet etat, it's going to be an etat quantic, is it that? An element in the space of Hilbert? No, but that's the case. Ah, it's the entire system. Sigma, it's the entire system. And R, it's... After all these discussions, I don't want to talk about it. It's a part of the system. It's a function of the system. Why do we do not want to do it? Wait, wait, wait. Wait, wait, wait. No, but... I'm asking you to know what to say. For the moment, we don't say anything. Because after, of course, part two, three, we make examples, we make the constructions. And for the moment... No, but page 18... Yes, but wait, it's already another thing. Wait, wait, wait, wait. That's the classical physics. That's the classical physics. It's associated with the value of an observable. Now, in the quantum physics, it's not true. Because an observable becomes an operator on sigma. So at the moment, this is the physics classique, no? Well, I think that this is not defined by its points. It's a bit of motivation. This morphism, normally, is not defined by its points. But it's totally general for him. There is no sense.

20:00 These are symbols. There is no sense transgressive, no sense cognitive, it can be any constructions. There is something symbolical called sigma, which we call ETA, and then another thing called Valeur. And then ETA or ETA? No, it's not an ensemble, it's a collection of important things. But after he does another tour, it's important, because in fact, as I already said, in this ensemble, all the logicists, what do they do? They start with two steps. The first step is to calculate the simple and crucial, formel, just in the sense of the most formel, and then he seeks to interpret it in an antopos. And that gives him, let's say, all these issues. This is the first step, and then he introduces, he engages other languages, the languages superior, and he does the same thing. And that, it allows us to do much more things. In fact, the theory is probably more observable than the moment, which is associated with each Etat, an Etat being a member of an object. What is the object? Page 9, in fact, in precision, page 9. Peut-être que j'ai lu même un anglais pour éviter cela. D'accord ? La logique quantique se formule au moins à quelques langages différents. Yes, we can talk about everything. We can talk about the spaces of the earth, we can talk about the energy, there are different languages. So in each language, we can have a definition of what is observable. What is interesting, is that all that is common is a common space that is an object in certain categories and then put in a little coin a classic example for those who know In fact, for quantum logic, I don't remember the page, but I don't agree with this program of quantum logic.

22:30 He says in the notes that it would be interesting to compare our approach to quantum logic, but he doesn't do that in a way explicit. but what he says is that when he talks about logics, he puts the word logics into the virgule, and he says that it is not logical for him, there is a structure that is not distributive for him, if I understand how it is called logics, on the other hand, we can interpret it in topos, that is to say in logics intuition, not in logics classiques, they show that it But he said that we can do it with the logic intuitionist. Well, it's not also an isomorphic interpretation, it's a bit more complicated, but it's still a bit better to be found in the topos, with the internal language intuitionist. He said that the topos are completely different, but of course, Yes, of course. Of course, of all that they say for intuition, it's not possible. You have to go to the topos, but in any case, it's not possible to go to the topos. Yes, but if we talk about the topos without mentioning the topos, we're going to go to the generalization. You're going to talk, Andrea. Yes, I'm going to remind you, but maybe just before, in page 9, I'll do the things more clear. I can read in English? Okay, there are three points here, I believe. Physical quantity A is represented by an arrow. Excellent. It's the same thing I've written here. There is a vector of a system. and there is also a field of language. No, representation. Yes, representation, yes. Physical point is that the language is fixed. The language is fixed or not? The system is fixed. After, we present that in a topos. All right, so it's by a physical language? All right, so it's by a physical language, not by a physical language. It's not by a physical language, with which one can choose a class or a class or a class? He says that in the classic case, it would be classical state space, and it would be the real number.

25:00 But, effectively, we look at the case of the general, okay? Okay, second point, the proposition on the system S, that is, in this formal language that we are defined, there are two languages, the language of zero, composition and language of l'ordre in Portuguese, these propositions are represented by the sub-object of this thing. And they form an algebraic writing, like in Portuguese, that I will show you all. It's analog to each ensemble in Kabula. And the third point, while the top is analog of a state, and the brackets, truth object has been specified, these propositions are assigned truth values in the hiding logic associated with the global elements of the sub-object classifier. Topos. Topos, dans lesquels tout ça passe. D'accord? Voilà. C'est un simple projet. Et bon, peut-être maintenant, j'essaie de mettre clair pourquoi l'agip Viking et pourquoi Topos. Qu'est-ce que c'est tout ça? Bon. Maintenant, la définition de Topos que je vais donner, ce n'est pas la définition, je crois, la plus simple, mais peut-être pas la plus économique, and economic that we can do to be able to make a more stable and then to give other things, but that is not possible. The pose is just a category that has all the limits. I remember what are the limits. In the category, we talk about the limits of diagram, C'est-à-dire on prend n'importe quel diagramme, alors on forme un cône. Oui, truc en cône, c'est-à-dire on choisit autre objet ici, on forme tous les flers vers chaque objet de notre diagramme, The diagram is called a cone, and then we say that this cone has a completely universal.

27:30 It is to say that when we have another cone on the same diagram... ABC is any object? Yes, it is no matter of any diagram. We say it is finished. What's the name of the diagram? The diagram is the number of objects with the number of flags. There is no condition of the clue. There are several flags between A and C? Yes, yes, yes. Or 0? Yes, yes, yes. Yes, yes, yes. Yes, yes, yes. Normally, I don't know why we talk about the diagram, it's to say there is morphism chosen, but in fact it's not possible. Any diagram, a priori? Yes, you can have several flags here. Ok, ok. And then? No problem. And then you form a cone, you take another object, a bit of tissue, from such a way. And then you take all of those here. Yes, there are also several. Yes, you have several cones on the same, but it can exist. It can not exist, of course. It can exist one cone that has its universal properties. that at the moment when you have something like that, you have the only only flèches, and that's why all these entities are so indispensable, you have the only flèches also. And the case in particular, for example, it's the product that we already talked about. Produits, it will be an object like this, it will be just a diagram of objects, just an object. So, you have the projection here, and at the moment you have the projection here, or if you have a projection, you have a point. That's what it's called the property universe. Between the point and AB? Yes, the point, it's not the same. What is the morphism that you have written between C and AB? It's something, it's a category that counts. It's a category with the morphism. C'est un morphisme entre C et AB, mais n'importe lequel. Non, c'est important qu'elle est unique. Ah, elle est unique, d'accord. En fait, je vais annoncer des logiques de premier ordre.

30:00 C'est-à-dire, pour A et B données, oui, A, 3B, c'est, on l'appelle produit, ou limite, car spéciale de limite de ces diagrammes qui sont juste deux abjets séparés, si, pour n'importe quel C est de la même catégorie, d'accord, so that there is also a morphism here, then there is only one, only one, only one, the morphism of C.A.A. You can see that if we think about the cartesian of an ensemble, it works. Does it mean that there is only one of the categories such as a morphism of R.A. and a morphism of C.A.A.? Yes, of course, because of this definition, it depends on the morphism that defines the morphism. If you have a second morphism like that, then you have the morphism in two senses, which you give in one direction, which is composed of one entity. That means that there is an object equipped with two morphisms. and which is unique to have an extreme object. But on the other hand, what is the property in plus? Well, the property in plus, it's just that among all the objects that have an amorphalism, there is one that is, in a way, a maximum, universal, universal, universal. Yes, I want to show you that it's maximal. In the beginning, there was one that was A x B with A and B, and then there was one that was maximum. C, with an emphasis on the verb. In reality, this other is in a sense included in the first. It is categorized. It factorises the verb A to B, as the triangle on the side commutes. It is not equal. Well, that's just the case. Now, in fact, this notion is also a very strong limit, because, for example, the terminal object, which is an object that there is a single flash of any other object here, this would be a limit of the diagram 8, for example. Now, if we reverse the flash, we obtain the notion of co-limits,

32:30 We are also reproduced, of course, in the particular case. And here, it will be the initial object. D'accord? Another, just for example, which is what we call pullback, is the limit of the diagram of this formula. If we have A, B, C. The limit of this diagram, if it exists, it would be B, which is called B. It's to say that we have a problem like this, and if I take any other problem, which is also common in the same way, well, I don't have a flage of composition, then there is a unique problem here. It's just a little complicated diagram, but the same notion of the limit. It's very important. In fact, it's enough for Topos. I say that Topos has all the limits and co-limits. It's a bit too much to ask, because we can ask if there is an object terminal, after Pullback, and after we can do the zone, but that I don't do it, ok? Topos is a category where there is all limits and co-limits, all the infinite. What is the infinite infinite? This is the first point. The second point. If there is an object exponentiel or an expression, and in the case of the ensemble, it will be for two ensembles, A and B. It will be the ensemble of functions, from A to B, and this is of course the fundamental property of the ensemble, that all these functions form still an ensemble. What we call exponentiation. Exponentiation. Exponentiation. Exponentiation. D'accord. And if we want to define it with a flash, it will still be a universal property. If we define it like this, it is to say, in a simple way, we can think of the ensemble of all the functions of A to D. And now, the proprietor of the following is that I take a certain value of A, and I take a certain function, and normally I obtain a certain value of B.

35:00 D'accord ? Il y a la fonction, il y a l'argument, et j'obtiens valeur. Voilà. Et maintenant, il faut aussi trouver la propriété universelle, c'est-à-dire pour nous qu'on obtient l'objet B, tel que, ici croix A, ici il y a une flèche, tel que... So if there is a flash here, then there is only flash that makes the diagram interpretive here. And this thing, if there exists, of course, is called exponentiation. It is to say that in the topos, we say that it is to limit, to limit plus expression for each pair of objects. And this is what we call Cartesian clause category. Cartesian clause category. This is something that is very interesting to the category of the ensemble. But the last point of definition is that there is what we call the object of vérité, the object of the vérité. Maybe before I finish the definition of the topos, it's better to explain why the algebraic thing was not. or is it obvious for everyone? No, I don't know. Well, I remember the notion of monomorphism. Yes, in the category, monomorphism is what? It's just morphism, B, A, B, M, with the following. If we have two other morphisms here, If we have Rg, then we have Fm. I create a composition of values geométriques. If we have Fm equal Gm, then it implies F equal Gm. Well, we can also verify that it corresponds to the notion of application surjective.

37:30 That means that M is a monomorphism? Yes, it's a definition of monomorphism. If we reverse the reflection, we obtain the notion of epimorphism. And here, in fact, there is a difference with the ensemble, because epimorphism is not a thing surjective, It's not the same thing. There is this duality that we don't find in the case of ensemble. We don't find it because it's a little bit of ensemble, but this notion of application surjective, it's not exactly... Of course, each application surjective is an epimorphism, but not inverse. And if, for example, there is morphism, which is at the same time monomorphism and epimorphism, it's not necessarily isomorphism. That's the difference. The analogies have a grand sense. It's kind of limit. What is monomorphism? And now the idea that monomorphism, in some sense, represents a part of it. Yes, like so, ensemble. The intuition is clear, if we are ensemble, we can always project it in some way, the application surjective. Hicham is used, of course, in the classic case. So, where do we come from? It comes from an ensemble of parts, an ensemble of elements, and for inclusion, a structure boolean. But here, what do we have here? This is a simple question. If we try to think about the part B B as monomorphism. Of course, there is also something important in the concept of this part, we think with two objects. To say that there is a monomorphism, there is a part of it. There is already a part of this part as an object independent. In the case of an ensemble, we tend to think that part is already in a way. If we give B, we give all the parties. In fact, here it is explicit that it is not the case. But now, we are going to look at if it really ressemble something like a particle.

40:00 And now, we can see the following. If we have, say, this is our object, and if we have two monomorphisms, A and, say, I call C, it will be M and R. D'accord? And I try to make a morphism here, because it is common. What do I obtain? By this property, I see that if this leaf exists, it is unique. because if there are two, they are equal. It's to say that there are only flèches here. So I can have the only flèches in the same way, and that will make the two objects isomorph, because each flèche is unique. And that means what? that the sub-objects of objects, of objects, form a category, or object of this form-là, and these categories are other things than a part of the part. A part of the part, if we do the difference, if we do not identify objects isomorphous, it will be a part of the part. And this is a good thing. If we try to think of it as a part, we can also think of it as a normal part. It works well with the intuition. So you can re-enoncer the thing? If we have two monomorphies on the same object, And I think of this one, if it exists, it doesn't exist, but if there is a one here, it is unique. And if there is another one, unique in the sense of the name, if there is also unique, So, it's the isomorphism, the proposition of the two. For each, for each, for each.

42:30 But it's-to-all that our objects are ordained, partially ordained, or pre-ordained, if we can make a difference between objects isomorphism. They are pre-ordained. And that, it corresponds with... The objects that are ordained, are all those that there is a monomorphism verb. Yes, yes, yes, all objects of B are preuves of M, like the part of the object. Ah, they call them objects of B because they are not? Yes, yes, I can say... Ah, I can do this here for you to show... Yes, but in fact, rather than the object, it is not just these objects, but it is objects with a flèche. It's a flèche, if you want, but a flèche... You can't talk to the flèches without the domain or the predominant. D'accord. D'accord? It's to say that it works well in the classic case. D'accord? It's reasonable to think about it. But now, is it still a difference? Because in the classic case, we know that there is a structure boolean. D'accord? Excuse me, I'll try to repeat it. There, it means that all objects in your category are supposed to have a monomorphism verb. No, no, not at all. In fact, the word sous-object is an abuse of language. No, no, I haven't heard the word sous-object. The word sous-object is an abuse of language. It's the fletch, in fact. Yes, the fletch? It's not the object, the object itself... Yes, the object, it can have a fletch. The sous-object? The subject is a flech, but when we say flech, we have a codement. We can't separate the flech from the other. That's a role. Yes, of course. It's still a bit more complicated because it's not exactly the flech, it's the flech. Yes, it's true. But if you want to pass, it's the pre-order. If you talk about pre-order, it's not the way to talk about... If you talk about pre-order, it's not the way to talk about it. And in this case, what is the situation? The situation of B, in relation to the pre-Homme, is that... Well, it's an object. It's an identity. Well, it's an object. But, to the moment, it's all more or less a reprise of the classic situation, of the neurology. But now, it's a little different.

45:00 The definition is very general, in any category, it is to say that we can't say more that there is a high order here in the case general. But if we think that these are the case of categories with limits, co-limits and exponentations, it is to say that it is a core of this category, It changes a little bit. Because what is... I don't have other limits for the moment. I also need terminal and initial. What will be initial in the pre-order? It will be the lowest element, minimal, minimal. Minimal, maximal, it will be terminal. And then for each object given, the product in this sense that I have done earlier, it will be the SUP and the product will be the INF. It is to say that it is a FISO, TRI. d'accord mais maintenant question est ce que c'est trahi boolean et apparemment on n'a aucun idée comment on peut définir complément il n'y a pas de quelque chose pour complément rien du tout mais on peut introduire chose qui s'appelle pseudo complément et qui signifie les trucs On dit que bon, si c'est a, maintenant je parle en termes de triï, d'accord ? Si c'est un élément de triï, avoir, on s'appelle pseudo-complément, si seulement si pour chaque autre élément x, on a que x c'est plus petit que... x plus petit que a, si seulement si, on a inf. Je fais inf comme ça, d'accord? x et a bar égale 0. 0, c'est l'élément minimal. Oui, même je peux écrire pas égal pour ne pas utiliser ce symbole. which is always small. Yes, it's a form of just to have an identity as a symbol.

47:30 That's what we call the pseudo-compliment. And the definition of the process, we can define, we can replace S0 by an element of B. That's what we call the pseudo-compliment. And, well, the case of the sub-complements classic, it's the topology, if we take the open, it's to say, in the sense of assemblies, it's just a complement-assemblies, but the sub-complements, it's the interrogeté of the sub-assemblies. And, of course, the normal complément is the sub-complements. Why are these words of the pseudo? So if I look like this, 1 here, I look the most simple possible. D'accord. That's a complement now, yes. That's to say that we have sub 1 and inf 0. But if I take, for example, x here, we see that it's also issue of complement. That's to say that it explains the terminology. It's something more general. But we can, by the way, have an example. I'll tell you because it's the same term. It's issue of complement, but not the complement. Now, if we replace here, these conditions here, by... Yes, it is to say that we make a pseudo-compliant relative to certain elements. So, we call it an implication. Maybe we should also talk about the subject of implication. I don't know how to use this terminology. Because it is something that resembles an implication classique. D'accord? Well, I wrote like this.

50:00 and here I write for each mix I write for each mix It doesn't have implications as an element. Or, in fact, we can perfectly do it. Because we can also see that if A or B, it will be this case-là, in the Cavoulians. D'accord? Well, it's simple. So now, observation that in the topos, if we take on a tri of these objects, if we take on an exponentiation, it checks exactly these conditions. Exponiation A, B. This is a generation that gives a pseudo-compliment and implication inthusianism. This is all to say that we have created this object, which is not boolean, but which is called fighting, which corresponds to a logic inthusianism. There are several axiomotizations possible, etc. But that I don't touch. What is Abar-B that you have written? Abar-B that you have written in relation with Abar-B? It's in the Kaboulian. In Kaboulian, it's the implication A-B. If you take this thing in Kaboulian, this is the implication that you have. It's a very special case. This is x and a, not x and a bar. And here it is. No, no, here it is. Here? No, no, on the top. On the top. It's not x, a bar.

52:30 No, no, on the top. There's no bar. No, no, no. But here I don't write the bar because there's no bar. Here I define what it is. This one, it's one element. There's no bar. There's three here. A, B, and A, and B. There's no bar. Well, that's the logical example I think, as I said, it's open. You should write this thing that you wrote in the middle. You should write it for the exponentiel, because that's the right definition of the exponentiel. What you gave you earlier, it's too complicated. What do you think? Well, you can write it. Yes, we can write it as a definition, of course, but what I wrote earlier, it gives an ensemble function. It's the one that gives an ensemble function. Yes, it's the case, but I don't know. Question didactique. That's the good idea. That's the one that has a sense. But then you say that it could be universal, here. No, not at all. It's an junction. It's natural. You can say that if you vary A and B, it varies on the contrary. And that's the same thing as what you said earlier. But here it's all on Trédit, it's very simple, and here it's very general. Well, it's to say that all this Pudet, in some sense, Hicham, to come back to him, which means that instead of taking together some basic logic, we replace it by some topos where we have a certain... When I say that the algebraic or logic intuition is too general,

55:00 because there are several of them, of course. And then, it's to say that it should be a good choice, but it will always be an enthussianist, always an enthussianist. And what he says is that it will work for the mechanics of an enthussianist, even if the Italian doesn't work. It's to say that it's not something much more bizarre than an enthussianist. It's a little bit of an enthussianist. Well, I don't know how to verify the details. It's just a remark that I want to make. I think it's supposed to be like what he does, because he says that all physics classical physics is based on the ensemble, or something like that. But at my opinion, it's not reasonable, but it's an answer as strong as it's not obvious. There are plenty of physics who do it without thinking of the ensemble. It is to say that we can give this argument that there is no thinking of the ensemble. Yes, we can catch it. But it is not always obvious. because if we really analyze, I don't know, but I don't know, it's already done in the conclusion rather than assembly. There are also other things, but it's a little special for something about the mathematics, not really here, but in fact, it's too hard to say Because in fact, what we have in the logic is that we have to do something like algebra class. It's something that remonte to the boule. It's to say that we don't really have to do a theory ensemble at all. And more than that, when we start to do a theory ensemble in an axiomatic way, it becomes very ambiguous because we already have certain notions of class or something things that form a gene boolean to make a semantic, and then add a whole notion of ensemble, all the axiom at the same time, and then say, here, we interpret it, and that works. And of course, I just finished this idea of this column. In fact, it depends on what

57:30 we take as semantic. We have a standard model. Excuse me, there are two origins of the topos. So we are really in the cadre. The second origin is the topology of the topology where there are points. But the topology of the 100 points is really what has given the essence of the idea of the topology. Yes, perhaps. But in fact, I want to talk about the 3rd essence. The 3rd, the 3rd. The 3rd is used by Richard. not these two that you want to... Of course, it was just 100 points, it was important. I was talking about the origin of the ideas. Yes, of course, but for him, he doesn't use... Well, now, perhaps this moment that I'm going to do, I believe, it was important to read, that the notion of topos, which is also complicated, like construction, because here, as an example, we can say that the example The most simple and historically important is, let's say, pre-fessor. Or, if we take an space topological space, in fact, we can take any particular category, and we have this thing with each object in an ensemble. And we make the fonctor control. This is a definition of pre-physos, it is something that is an important category here. I have a hope to say that we have to put the formula, but I have to put the condense. And then there is a way to define what would be the morphism of this object. And then we have an example of the topos which is important, but in fact it is that we can actually define what is the topos in a more general way, in a more simple way, which is called the topos elementaire. Et pour ça, pour toutes ces trois choses que je dis limites, collimites et sémonitions, il faut juste ajouter ce qu'on a appelé

1:00:00 objets de valeur de vérité. Et ça, c'est au classificateur de sous-objets. And this is the next question, I just want to give my motivation here. When we assemble this, we have an ensemble A, we have an ensemble S here. On peut faire les choses tout d'être dans l'ensemble de deux éléments en zéro et envoyer tout ce que dans S dans A et tout ce que pas dans Z dans zéro. D'accord. Comment maintenant on répand un cas plus général de séquence fonction ? On a sous-objet de A comme ça. D'accord. Now we have an object that is perhaps more complicated than zero. Here we have an object terminal, here it is single tone, and we do this diagram. This will be a point. We call it a point. I just remember the morphism of two objects terminal, which is in any particular object. We ask that this thing is called back. After, we can prove that if this thing exists, it is also unique. like in the other case, but I don't want to do it. And we can give examples very simple where it's not zero, but other things. For example, if we do an ensemble, we do something like a monohid, something that acts on an ensemble, I don't know, one of the elements in the other, it's like that, yes. We can think about it as an ensemble. but in a way discreet. So, the object of the vérité will be like this, 0-1-1-1.

1:02:30 It's a little like the stage, or it's true, certainly. there is always a point that is called true. That's in the definition. But from that, we can see the difference. We can see that it will be true in the third part. It will be true in the third part. Something like that. But it's important that all these examples of topos who was motivated by topology and something complicated, they admit this problem. And this is a very strong discovery. It gives a rapport between the logics and the theory of topology. And of course, it's the other aspect of the theory of topology, which is what we call the topology. And now, just a few words for the logics. And, of course, there is a structure of the writing on each object that we can interpret. We can say that we can construct, we can construct, we can construct, and we can construct an object that is supposed to be a model. And that's what he does, in fact, at the beginning. But in fact, there is something a little more interesting, because we can try to find a rapport, we can say, closer to the logic and this construction. And in fact, we can, first, we can think each other, because there are several objects, and that we can interpret it as a typified logic, the structure of the heighting, but there are several. This is the type. This is the idea of the logics typified. This is what Bell did in his book. This is what Bell did. And then, there is this question, because there is all this ideology which is legal. The interpretation that we make is purely formal, etc. and then we make a model, we say that the topos is a good model for that. But also there is this idea of internal logic, and that in my opinion it is just a change of attitude, if you want. Because internal logic is to say, instead of building a logique

1:05:00 like this, and then searching a model in the topos, we can rather, I don't know, in a way to build a topos and look at the logic that colles with it. Also, from the topos, we can do this kind of lightness. I, for the moment, I don't like that. But exactly, it colles with the topos. Another approach. I don't know, I don't know, maybe you can correct me. There is a category syntactical, that is to say, rather than starting in a standard, in a way recursive, to construct the formula, we can already, at the beginning, do this kind of writing as a category. And then, of course, it gives certain categories as a category of formula, and then look at what are the categories of phoncto, of this syntactical category in, let's say, an ensemble or in other things. And this is something that I mentioned earlier, that, in my opinion, it changes so deeply all this kind of tasking, I don't know, the theory and the interpretation. I think it's a lot and in my opinion, it's important. And what did Hecham say, he uses his language interne, but still not too much. He always writes things at the beginning, and then he begins to represent them in the top of the domain. a rather elegant way of summarizing what is involved in constructing a theory of physics is that we are translating the language LS the system is not a local language, the topos, it is not a local language, the topos, it is the idea, if I have it, it is a language a little bit independent, ok, after it describes with his language, let's say, the system, it is a language to describe the system, after

1:07:30 And then we take another topic which is a topic of logic, which allows us to give the value. And then we look at this translation, interpretation of the one in the other. And I think that, in fact, it might be critical in a general way, but a little more categorical way of thinking about it, It's rather, rather than talking about translation as something achieved, think about categories of morphism. It's to say there are several translations, so there are several theories with the two topos. And look at what happens at the category of morphism between the two things. And also, maybe my last remark, I will finish. After, of course, there is an interesting problem that comes from what we need to do with the language and what else to do with the interpretation. And that, I think, even in the classical case, we can ask the same question. In fact, it's a bit, if you want, in general, it's just a little bit more. But it's always a different way to think about conjunctions and disjonctions, in terms of intuitive. Because in terms of categories, tri-I, liens, analysis formals, it's something completely asymmetric. But it's not at all, because when you say, and I have faith, so like that, we can immediately interpret it in a way ontological way, that there is something like two facts which are objective, which are in the world. But if I say, or if I have faith, or if I have faith, we will always think about something epistemological, There is no real object that corresponds to a disjonction, but there is an incertitude of description. And this is something that is very profound in intuition, in habitudes, but of course it does not correspond to the formalism, because we don't find it at all.

1:10:00 And just last thing, similar to what might be important for his party 4, where, as Alexei said, he tries to make this idea of system and system system, that is to enrich the four levels. It's a bit of the same idea, how to manage the system. He introduced a category monoidal, which is already here. But, look at this idea, there is something like everything, all the world, the universe, and in fact, he refers to this idea. He said, for example, that there is the only real system, the only one, and it's the universe. And on the other hand, you can think, I don't know, I think there is an element maximal and minimal. But what I want to say is that, at the intuitive level, there is an element minimal. You can think of an atom, in terms of metaphysics, something like a small part, but we don't have an element minimally. We introduce something like a void, but, say, a void, it doesn't exist. Yes, yes, yes, there are some difficulties. At least there is no symmetry in our, I don't know, in our nature, in our spirit, which I suppose, I suppose, and to my opinion, Yes, for this reason, there is no more attention to Hicham with this notion of the universe, like Karl has already said in a few remarks, he said, but why are we talking about the universe? There is no reason why there is something like an universe, and it is not super clear, it is just that this kind of closure, we have no reason to suppose, in fact. Of course, it comes to discussion of infinite space, etc. Well, it's just a little bit for the project. Just a bit of a summary. Well, the first thing I just explained is that I just explained. The second part, he works with this language of order zero, a calculus professional and intuitionist, and tries to apply his mechanics quantic. The third part, he tries to do the same thing, but much more elaborated It's already on the language interne, in the sense of the language of the superior, and in the fourth part, it's a little bit separated, it's a little bit different, but it starts with this system and system, how we can produce the two systems, etc.

1:12:30 J'ai lu tout le cas. Enfin, je vous présente après trois minutes, si vous me permettez, un point de vue, je crois, assez différent de ces papiers. Justement, comme a dit André, pour offrir ce qu'il a dit à la fin, si on se pose la question de voir si l'approche Hicham est la bonne, et puis peut-être il faut aller plus loin au niveau des catégories ou autre. Mais avant de se poser ce genre de questions et de commencer à faire la critique, il faut voir ce que cela apporte au nouveau par rapport au langage tout à fait ordinaire et standard de la physique antique. Et Hicham et Doreen sont tout à fait conscients de ce problème. or is this approach that allows us to go a little bit further? And that is a question where opinions can diverge, but they think that we can go a little bit further. And I'm going to explain in two words why. So that is really the added value of this approach. Après, l'histoire de tout reformulait dans un autre langage. C'est très beau, mais est-ce que c'est juste une reformulation de plus ou est-ce que ça ajoute quelque chose de scientifique ? So, what this adds is exactly on the two questions that have been posed in the other other than the other one, it is a very long time, it is the question that has been posed in the quantia, which is not only in the theory of the algebra, but also of the fundamental applied to the quantum mechanics, it's the question of the number of reals. And the second question, it's the question of the system. What does it mean to be a system of a system? For example, the paper 3, it's the answer to the question of the number of reals. The paper 4, it's the answer to the problem of the system.

1:15:00 And the paper 1 and 2, it's an enormous introduction, with plenty of formalism, with a lot of philosophy in it. It's an introduction that allows to introduce this reformulation of all the theories which, at the end, allows to make a path to more towards the number and a path to more towards the problem of this system. So, what are these two paths? The path to the solution, let's say, to the problem of the number, is that they are not able to solve it, but in the context of the Algebra von Neumann and the Logical Science, we did not even know how to go further than the number of numeric chosen. So we did not see the formalism, the language to consider the complex and equatorial at the same level that other things. And what were these other things? We didn't see it at all. So that, for him, and it's why, for example, in the conclusion of the paper 3, he talks, and it's interesting, it's really interesting, he says, we have all formulated, yes, but this page, for those who look at it, it's the page 34 of the paper 3, he says, yes, we have all formulated, but it's not very interesting, but we have supposed the question in terms of sense that we could not even ask before. This question had no language in which we had to formulate. After, he doesn't give a response. He said, yes, I'm finished on the number he said, yes, so we have supposed the question, we have seen that we could perhaps do other that the real, but it remains to see if we can really or not. If we can really, for example, take the topos monoidal and do some physics with this other topos that has nothing to do with the ensemble, or the object intrinsic of the values, which is called the one that is very early on, the object which contains in him the values, would not be liable to the number of things.

1:17:30 So that is the first step that they succeed. What would that be? Like a question? That is the main question. Is there a physical that is not a physical in the topos of the ensemble? Because, after, well, another element that he brings, of course, it's the paper 2, the designization, the designization, it's that he shows that what is instrumental, what can be formulated in the topos quantic, that means in the topos where the objects are from the preface, or from the sous-algebra of Boulain, algebra abelienne de l'algèbre de von Neumann, que cela peut être interprété, représenté en fait de façon pour vrai, représenté comme un certain instrumentalisme dans l'autopos classique, dans l'autopos des ensembles. Donc ça, bon, ça, vraiment, c'est pas une énorme avancée parce que, on le savait à peu près, ce qui est nouveau pour moi, c'est the papers 3 and 4. And so the papers 4, it's a bit the same thing. The question about the system has been asked for a long time. When you talk about the system, I don't understand. Before we talk about the catégories, how is the problem of the real? Okay, okay. There is a way to solve the problem of the real. Why the number real in physics? Because we have an object with which we measure. But it's not a rule. And so on this rule, we put, as it was said at the beginning, we put numbers, so we measure things to a number rational, and we take the limit. So, the number rational, and we take the limit because our instruments of measure become more fin, so we do the physics with this notion of limit. But now, can we remove that? Yes, but... That's already an interpretation. That's the interpretation instrumentals, where the real number comes in physics, because it's lié to the way to measure, to another way to measure it.

1:20:00 Because if not, it's not a problem at all. If you have a square of 1, the diagonal of the root of 2, it's really real. And if, at the contrary, the approximation that you do, you always measure a rational... But you're a physique. Yes, you're a physique. But in physics, the reals are supposed to exist. Yes, in theory, the reals are supposed to be part of the theory. A grandeur is supposed to have a real. Yes, yes, but now, I'm not finished, but now, his argument, his argument is that we look at the case of a system fermé. That, he said it a little in the beginning, and then he came back to the introductions and conclusions to these themes, but never in the text. When we are in a system fermé, like the universe, so we do the cosmology... Seul system fermé. There is no reason to think that we can come with our little dream and measure it with whatever it is. There are, because we can measure things in the universe. Non mais on les mesure de l'interne mais justement ce n'est pas du tout la même approche instrumentale où on vient avec la règle et on dit bon cette quantité est égale à une valeur racine de 2. I'm not sure with that. Just an example, it's a little naïf, but it's done by Mandelbrot. It's to say that you make the measurements of 1000, and you make it more small, and it's just supposed to have a limit. It's what you're supposed to do, you're supposed to do mathematically. It's not justifying, there's no experimental facts. Experimentally, it works a little bit, at a certain limit. But even the readings of the book, it gives you... You see, it's an analysis which is useful, which is useful,

1:22:30 but maybe you can find others who are better in this situation. I think it's not a physical, but it's very simple. Just to say, I have the impression that it's not a fundamental problem. Ah, yes, yes, for him it's a problem. It's the beginning of page 1 of the first paper. To get rid of the continuum. To get rid of the continuum. But it's not the same thing. For me, it's not the same thing. Non, il y a les problèmes de continuum en formulation fondamentale, disons, de la physique quantique, il y a deux problèmes de continuum. Il y a le nombre réel comme coefficient, donc il y a la transformation des états, donc le nombre réel comme coefficient et le nombre réel comme valeur. Il y en a deux. Pour mobilité, troisième. Oui, il y a les probabilités, bon, oui, d'accord. Comme l'espace-temps. Oui, mais pour un système, oui. But it's clear that there are several places where you have the problem of the continuum. Let's say, there are the non-reel, the non-completion, as a coefficient of operators, so it's something that brings us to the state of the continuum. The space of the state is a vectorial space on the real space. Yes, the space of the universe. So, there is the continuity there, first. Second, there is the values of the observance, which are the number of n. The values of the observance. The third thing, there is the probabilities, which are the numbers between 0 and 1. It's all? Yes, at least it's all. Yes, it's all. Yes, it's all. Yes, it's all. Yes, the theory quantic, when the standard standard, it's all. There's a real part of it. I'm sure. There's a real part of it. I can't see these things. But just... The same thing, this analysis shows that if you do something like that, it puts a constraint very, very strong on the possible theory.

1:25:00 It's not... The fact of using the real? Of course, because at a level structure, it's really... I don't want to change that. I'm going to ask you, because as they sound good, for the next session, maybe we can fix it. In 15 days, no? We are the 22nd? In 15 days, it's the lundi. In 15 days, it's the 5th avril. Ah, no. Jeudi 5th avril. Jeudi 5th avril. Jeudi 5th. I'm going to announce it's a month of May, the 15th and the 22nd of May. It's John Bell. Yes, he did it. He did a seminar the 10th May, John Bell. Yes, but he did a seminar the 15th and the 22nd of May. Yes, of course. Yes, I think. So, the 5th April. He will be there. Here it's the 10th of April. The 5th of April. Pardon, I was going to say the 5th of April. Then there are some things. Then, yes, the 10th of May also. It's the 10th of John Bell? Yes, the 10th of May is John Bell. But I propose that we have a category of 3th of May. 3 May, 10 May. 3 May, and 10 May, Séminaire de Jeune May. Excuse ? Oui, donc j'ai dit quoi, le 5 Avril ? Ah, du voyage au 5 Avril. Bon, bah c'est dommage. Ensuite, c'est nous qui sommes en voyage, parce qu'il y a une école, et ensuite le 3 mai, le 3 mai, et le 10 mai, le séminaire de Belge. Vous étiez les physiciens, non ? Oui, madame, oui. Je voulais simplement poser une question, c'est un ignorant. Est-ce qu'on peut parler de spectre continu, ou est-ce qu'on peut se passer ?

1:27:30 No, not at all, there are a lot of operators who have a continuous spectrum. Yes, no, no, but attention. Disque-continue-discontinue, it would be different. At this moment-là, it would be different. The spectrum is often the real. Disons non-discriminate. Yes, yes, yes, yes, yes. There are some continuous spectrum operators in physicality, but is it indispensable experimentally? Ah, bon, c'est ça ? Pour résumer la technique d'hydrogène, si, aujourd'hui, ça devrait être compliqué quand même. Non, à l'intérieur des théories qui marchent aujourd'hui, ça fait partie naturelle de sa théorie. Mais est-ce qu'on peut en imaginer des raisons un peu a priori pourquoi ça ne peut être que ça, pourquoi on ne pourrait jamais faire to be able to have an experience with the spectrum of the X-ray, I think there is no reason to say that there is absolutely to have an experience with the spectrum of the X-ray, to have an experience with the spectrum of X-ray. I have never done that. I work with a musicist, but I have no idea. Yes, I agree, I think, because what they change, are the spectrum of the operators, I would say, interne, in a way. They have no space-temps. For Alain Cohn, the space-temps is always a variety, a continuous. The space interne, in theory, would be a group of lives in a variety, and this is what Alain Cohn replaced as a geométrie in two ways. But in the quantum physics, if you take the object of the impulsion, the object of the impulsion, it has a continuous spectrum. That means that the impulse of an electron will not take any value quantified, but any value. But the continuous is opposed to quantified? Continued? No, no, no. It is opposed to discret, if you want. Because in the theory quantified, there are some operators who are... Well, the position has a continuous spectrum. Yes, but the position has a little bit bizarre. Yes, but it's not clear. Yes, I repeat, the impulse is a quantum operator which has a spectrum of continuous energy. But the energy is a quantum operator which has a spectrum of discrete energy.

1:30:00 So, for the moment, the two exist. And if we wanted, for example, to do a quantum theory with an impulse operator which is discrete, I think we could do it, but it doesn't matter. The discreet continue, the body of the vieille, they don't want to be hypocritical. Yes, the discreet continue, but the spectre is an ensemble of values, and so it's the same, so it's the same. We can't take the theory of the operative liners, limit the operative of the discreet, and only do with the theory of the operative liners the body of the vieille. That doesn't work. We can only do that. We can only do that. But not with these tools. I think we need to do more than with the operators. We need to go further than the operators. Well, I'll see you soon. In fact, I believe there is no contradiction. It's completely compatible But I will tell you that for the 5th of January, there is the Kohn Fest, the week of the Lancôme, and if there are people who want to go to the exposition... Ah, to the U.S.? No, it's already... Yes, it's the Francais, it's the U.S.P. Yes, there are a lot of things. Yes, I think it's a mess. Yes, I think it's a mess. Yes, I think it's a mess. Yes, I think it's a mess. But on the other hand, I think it's a mess. But on the other hand, I'm interested in discussing more. I don't know if... This paper? Yes, but there are four. Yes, yes. Yes, what I do is just a little introduction to the first part. Maybe you can present it after. Yes, I don't know, I'm going to write it like that. But there are at least four things that we can do. The first thing was the introduction. Then, it follows a bit the structure of the paper. The second thing is important for me, I don't know what you think, but I don't know what you think.

1:32:30 It's the story of this Daseinization, which, in fact, is a bridge between what they call... it's a bridge between what they call... between what they call the neo-realism and the instrumentalism. So, it's an argument that we can do both at the same time. And that's new, in all the world of physics, that has never been done. That's new. The third thing, it's the story of the Nombray. The object-valeur is something that makes part of this topos. The object-valeur is not external, not external, but the object-valeur which is part of the topos and which can vary. And the fourth thing, the subsystems. The subsystems are very passionate for me because it's one of the biggest problems of the development of the physical. How do you formulate this problem? The formula is what corresponds to the notion of subsystems. What is the meaning of the system? Because, to formulate this problem in two words, all the approach of Hicham, from the beginning, is something that depends on the system and the theory. So there are two indices always, theory and system. We start by saying there is a system and we do a theory. So there is the topos, which corresponds to AS, and the content of this topos, which corresponds to the theory of quantum theory, or the theory of classical theory, or I don't know what else. What he does in the fourth part is a bit of a question. If we fix the theory, so the theory is fixed, let's take a look at the theory quantic. Now, let's look at all the systems with their topos, with their language quantic, LS, LS1, LS2, LS3, etc. The particular system is, I don't know, an electron, an atom... Ah, no, the system for me is a fundamental notion. Yes, it can be...

1:35:00 Yes, but for example, this is an abstract notion, but yes, it can be 3000 electrons. For each electron, he has his language. We fix his language in such a way that all languages correspond to the theory quantic. So for each system, he has his language LSI which corresponds to the theory quantic. the same object, the same object, the same object. Now, let's look at the category of all these opposites. So we have for each system, individually, but one system is not liable to another for the moment. The theory quantic is individually for each system. Now, how do we read them? This is a question that has been asked in a different approach. different. But how do we read it? For me, the answer is well known. If a system is represented by a space of Hilbert, which is the whole of the states of this system... Yes, but why the dimensions multiply? Why, for example, what is there, which means that we can multiply the dimensions? You see, there is a system of dimensions 2 and another of the system 3. You say that the system is described by What does it make us the right to say? I'd say that it's a postulat and... Ah, no, no, no, no. The question is to see how that happens. He doesn't respond to the dimensions, obviously not. But, for example, in the standard approach, it's one postulat, but if it's a postulat, if it's a postulat, we reconstructly reconstructed almost practically the theory of science. So, in fact, this postulat, with him, really, it produces the half of the result, or even more. The postulat of the multiplicity? Yes, yes, yes. It's a little more than that, but it's the question of dimensions. Yes, but I understand. But at my opinion, maybe after, I don't want to enter the TT, but at the beginning, it's a little, how to say, it's optional. You see, when you replace this simple notion of party, paromorphism, etc., at my opinion, it should be prevented from this sort of intuition brute,

1:37:30 like a big universe, and then a little party, you know. It should be something more intelligent. For me, it's actually a postulat, but I would say it's one of the postulats, and I would say even the postulat fundamental of the physical quantity. And it is natural. So, you can put it in question, but it's to say that you put it in question the physical quantity. Exactly, it's the first phrase of Hicham, you see, paragraph 1 page 1, he says, it's perhaps iconoclaste, but if we do something else, we do something else that the theoretical. It works very well. There are a lot of things that do not work in the quantum physics, but in any case, it works very well, and the modern interpretations of the quantum physics, in terms of decohérence, etc., are fondées, exploit completely this principle and justify it. Yes, but... But, you know, but... But, you know... But, you know... But, you know... Yes, I agree, but it's already something that is so natural, too. I don't think so. Today it's become natural, because we are in it, but from the point of view of the logic. Yes, it's not necessarily because of everything, what we're talking about, it's produced, let's say, in assemblies, artisan, normal, and if you're a little more flexible, you don't cut everything. But, Mark, it's not just that you'll see, I hope you'll be exposed to Jeff Boob, and you will see that in fact the history of the dimensions which is multiplied is not at all a unique, universal, fundamental or whatever because there are theories that reproduce there are theories that are called convex sets there are many theories that exactly who try to take a different product than the Cartesian or the Dimension and reproduce a lot of properties of the quantum physics. And it works very well. You produce all sorts of things with a panorama of products that you can choose. You can choose a product of the other or the other. So the people study a bit the landscape of the product of the standard And the standard product is not the only one. Obviously, the theory quantic like we know, works with this product.

1:40:00 But to understand the role of this product, we study the alternatives, and we see if it reproduces or not the properties of the quantum physics. And in fact, there are a lot of things that reproduce. So, we try to study this postulate. Yes, I'm sure that it's interesting. But for me, if you know, the two problems that he mentioned are not what we call habitually the problems of the internet. It's interesting to put them in cause and see what it does in exchange, but... I don't want to change, I have an impression. I want to reposition it. You want to reposition it. If you want to find something, that means that it could be something else. Yes, exactly. Exactement. Il s'attend aussi secrètement à l'exercice de sélection pour dire que c'est bien pour cette raison qu'il faut sélectionner ce produit sensoriel et pas un autre produit en cartésien. Mais derrière, il dit cette phrase, juste sur la page 1, il dit tout ce qu'il faut dire. Il dit que les propriétés comprennent, enfin les invariants, il dit certain critical mathematical ingredients, mathématiques critiques qu'on présuppose comme ça, a priori on postule sans vraiment réfléchir, sont responsables de pas mal de propriétés spatio-temporelles qu'on déduit après et si on veut comprendre l'origine de ces propriétés, il faut se poser des questions sur les promesses sur les images. En fait, moi, je ne sais pas si cette conjecture peut-être pas justifiée, mais ça me semble c'est I think there were some relativistic considerations in this moment, because you see what we thought before. We can think of this, you know, object of category, I don't know what's in our model, etc. If you want something to replace the coordinate system, I don't know, or the observator, and maybe rather than just thinking about how we can build small blocks, we can also think, in terms of relativism, we can try these things in terms relativism from these objects, or something like that, but like space, let's say, the space is just like a big pot with everything,

1:42:30 and alternative to a system of relations, or something like that. Maybe here this idea relativistic... No, one of the points with the relativity could be this story of the notion generalised of the space of the Etats, Because, in theory relativistic, the Etats are the values of the time, so there is a little bit of an indice temporel, the indice T, that we put in the Etats. And then how do we define an Etat in the theory relativistic? Obviously it's much more complicated. In theory, it's more simple. Yes, it's more simple, but it's not at all an indice T, because there is no indice T. Yes, but the time is... It's an evolution. Yes, an evolution. So, one of the arguments in the introduction, is that, for example, this approach generalization of the notion of the Etat allows to include this generalization which is indispensable for her, for her, for her relative. D'accord. Oui, non, mais... Bon, moi, je crois que ça vaut le coup de travailler encore là-dessus, hein. Cela dit, peut-être quand même pour la prochaine fois, on pourrait faire... Oui, je ne sais pas si... Toi, peut-être une petite prochaine fois, on peut l'élaborer... Non, mais on avait dit qu'on parlerait peut-être de théorie des champs, pour alterner, hein. Oui, mais je ne sais pas qu'après, il faut parler avec lui, tu vois. Well, that's not a bad thing about John Bell. Yes, that's a good point. Yes, before John Bell, perhaps. Yes, the 3rd May, we can do it. And you would be ready to do something? No, I can tell you what's written down, but... Yes, continue. It's still a little bit more time to read it. The problem is that I don't know John Bell, So all that I know about the topos, it's very, very obvious to the knowledge of the mathematicians. The mathematicians are there to help you. So what I'm interested in here is how, finally, we arrive at the responses to the physical questions that are liées to the life.

1:45:00 Well, it's a good question. For the moment, I don't see it at all. But... Yes, what I can do is... If you have other articles that we have read, I can read things... Yes, what I can do is... No, but Hicham, he was always a little like that. Because it's not new, Hicham... It's not that you have to say Hicham, it's that you have to say Hicham. Yes, but Hicham, it was just a reformulation in the language of Topos because of the problems that we knew very well in the standard language, and we wondered why he did that. Why do they re-formule, again once again, everything that we know, the college specular, all kinds of different things, in the language of Topos. And here, it's a bit the same thing. We re-formule, again once again, everything in the language of Topos, and we ask the question at the end, what did it bring to us? And that has perhaps allowed us to have a language where we could pose a question that we could not pose before, but without being able to respond. In fact, it's not only the language of Topos, it's particularly, because it's not like Topos... Yes, it's not a part of the language. It's Topos, it's an element, it's-is Topos logicist, I don't know how to describe it, but... Just a question, the notion of Topos was built from the notion of category or independently? Well, you could have been before. Pardon? You could have been before. Yes, yes. You could have seen Topos as a category. And it was the story that it was good at the beginning of Gretendik, etc. And then, I don't know if Mike talked about this story, that Grotendieck visited Bill Rovir at Bach, it was not that year, and it's Bill who has this view of this reality, that Grotendieck didn't even think about it. What I don't understand is that you wanted to reformulate your own point in a way, Yes, by plage, and that is actually the most important thing to do. But on the other hand, there is a logic in which we don't have to explain it.

1:47:30 I don't see why it corresponds to the movement. There is something that seems to be missing. It's not the same thing to say. I have the impression that this allows us to be related to the observatory. It seems to be something realist. It means that a system has a propriety. It doesn't mean that it is the nature of its propriety. Before, classically, we would say that it is x equals 4, x equals 3, etc. But what he would say is that it is much more complex than to assign a real value in physics. which is an opération semblable, but with an object of value of art, which allows us to relate to the different observers. Yes, but... Yes, but... You see, they're trying to say... They're trying to say... But if they have a very clear remark, I don't know if it's them who have put this remark, which is, for example, Butterfield, who has insisted, is that, of one hand, yes, they want to introduce, and it's what they do from the beginning, they want to introduce this approach neo-realist, which contains in them, in a certain way, quite precisely, the approach instrumentalist. And so, all the notions of observatory company apply to the approach of the instrumentalist. And in the approach of the neo-realization, we call it, there is no need to talk about the observers, the probabilities, etc. So, of the one hand, they say, yes, yes, we have achieved, and then, they say, yes, we have achieved, because they say, we have achieved this approach of neo-realization. But Bohr said that when there is a theory quantic, whatever the language is, there is fundamentally an interpretation instrumentally. It is to say, for them, their construction, except for example, except for the theory, so their construction here is based on the category

1:50:00 of the prefaceo, of the sub-algebra abelian, of the algebra of Freud-Denemann. But, finally, where does it come from? Why do we know that it is the theory quantic, the algebra of Freud-Denemann? It's not because the Jabotronneman has these operators who live in the space of Hilbert. So, in fact, our motivation, the fact that we know that it's the theory quantic, it's because the space of Hilbert with his operators is there. And Bohr said, fundamentally, when there is an space of Hilbert and the operators, it's instrumental. So, they say, we have a problem here. This is page 3 of the paper 2. we have a problem here, because it happens as if we were coming to this interpretation of the neo-realist, but in a way, very low, very profondément in the fundamental, we have supposed the approach instrumentally, in taking the space of Hilbert, at the beginning, but it is because they are a bit too Copenhagen, I don't think so. So, I'm not sure. So, I'm not sure. I'm not sure, because after, they generalize it. the aspiration of Hicham and the idea of Hicham, it's to be able to be an instrument, as an observator. If you place an interpretation style Copenhague, there's no problem. because the interpretation of Copenhagen consists of saying there is no problem, we take the point of view instrumental and we don't ask any questions. So, by definition... No, it's not... I'm sorry, it's not... Because, in the interpretation of Copenhagen, there is no realism. So, you don't ask the question of knowing what are the reals and what can we do. So, the problems arise if you have an interpretation of reality. But today, I believe that no one thinks there is no interpretation of reality. No, there are some people... Yes, but there are two ways to do the same thing. I have the impression that if we generalize enough the approach to oppose, in fact, it's not for surpassing, but for unifying these two approaches. Yes, exactly. So, in fact... But there are two facets of something more vast, and that they are realist or instrumentalist, you need to justify it.

1:52:30 No, it's not that it's a need. One of the products of Hicham is to say that the point of view, the experimentalist, is one point of view, and the instrumentalist is another point of view. But it's already like that. No, no. But he doesn't know what. Well, he knows what is the point of view. On what is common? I'm not sure. I'm not sure. No, it's not the absolute, it's Hicham who says it. Hicham says things like that. If it's that, I'm not sure. No, but before we don't agree... No, no, but there is... But he says it formally. It's not an argument philosophic for him, it's an argument formel. He shows how it works formally. It's... We don't agree with the formalism. No, but we know that... l'interprétation réaliste comme l'interprétation de Copenhague sont en accord avec le formalisme de la vie quantique ? Non, le problème c'est que l'interprétation réaliste qui est en accord avec certaines choses que tu vois, il y a Valentini qui viendra parler en automne c'est en accord mais dès que tu fais la théorie des champs ça devient trop compliquée C'est indépendant de l'interprétation ça ? C'est un réalisme, c'est un réalisme bizarre parce que bon après le réalisme qui n'est pas tout à fait un réalisme einsteinien qui est différent, donc c'est pas le réalisme tel que défini dans le papier EPR, c'est autre chose. Yes, the non-local. Yes, the non-local. But, after, we can't say that there are observables with the values définies. But, just the other part... He wants that, he wants that, he wants that. If we say that there is a system, it will allow you to understand this non-local. Yes, but it's what he says. I think it's necessary to go to these 4 papers to understand that. But in the paper 4, it comes to say that there is a certain pullback that we look at. If we work with the classical theory, this pullback corresponds to something super-classic. And if we look at this pullback that corresponds to this system, if we look at a theory of quantum theory, we see that there is no simple reconstruction in something that relates to the first system and the second system.

1:55:00 So, it's a kind of an intricate concept that we leave in the language of the topos. That's what he says at the end of the path. It's something very precise. Yes, but it can come from a formal way with all the imagination. But if you want, you don't have to do the topos to express the intricate concept. Because, it's something that the theory of the actual, it's very good. But it's the same diagram that is written in a classic domain, it becomes cartesian, it becomes a good projection. And from the other side, it doesn't matter. And it's something that... If you write a composite system in physics quantic, in general, you turn the H to zero, you have a contraction, and you find... Yes, but it's something that, in this approach, for example, in physics quantic, because in physics quantic, you postulate the composition, instead of being postulated, that it comes to the end of the day. In fact, just to say that I think that they always talk about the big novelty, Topos, something like that, but in fact, this idea that it is announced, but maybe it is not quite that all of a matter of fact, that all the classical physics is, I don't know, it's a symbolist, so we can really doubt it, because, finally, the logic and intuition is becoming a lot of avant the theory of Topos, etc. And even the people who, I know, at the mathematics level, I don't know, in physics, but in mathematics, people say that everything that was done before, it's constructive. This way of reasoning, let's say, it's something new, maybe, or not, but of any reason, it's not something that's habitable. and also in physics, but I don't know, even the fact that until the 20th century, the people in physics didn't work that in a continuous function, maybe I'll say a little bit, but I don't think it's true, it's already, it adds to things, it's to say, we can look at, we can find this approach, maybe also classical, but in the sense, if you want to say, And in fact, the fact that we have discovered the logic indigenous in Topos is also astonishing because it shows that the logic indigenous is still something, let's say, natural. That's not something... Yes, that's something. But, I see the things that I'm interested in.

1:57:30 When you describe, your theory quantic is well defined. There are several interpretations. If you want a real interpretation, you are obligated to admit that in reality there are some bizarre objects in the way they are non-local. And it's something that we don't have the habit, that we don't have the intuition. If you want, on the contrary, to describe everything with the classic concepts, which correspond to the position of Copenhagen, you are obligated to sort of paradoxes and things like that, which you are obligated to put down the tapis in saying that there is no question. I have the impression that, finally, l'éthopos is perhaps a new tool to describe the reality. Because today, for example, we talk about particles. We describe the reality. We talk about particles in a point of space-temps. These are notions that we have the habit. It's been centuries that we have talked about. When you think about what is a point and a particle, you realize that it's completely paradoxical. You can't define it. It's a paradox. But we have to take the habit. So, it is almost clear that in the quantum physics, you will be obligated to renounce these kinds of notions. If you talk about particle, you arrive at the consequence that it is at the same two places. So you are obligated to pass to other things. What we have today is these notions of functions of the ion, so non-local, which are perhaps a little difficult to avate. I hope that it works very well. But there is perhaps an other language that is that It's to say the description in terms of space-temps, of ensemble, which are included the ones in the other. And, perhaps the ones in the other. And to pass to another description. And that means that in a century, no one will think in terms of things located in the space-temps, etc. But in terms of... Well, that's not the Minister of Education Nation. Well. But... But... But... But... I have an analogy. There is an analogy that, in fact, is a bit like this. Look at the overall approach in reality quantic, which pose as a fundamental structure, for example, a graph. Everything is in this graph, it's the reality. But then you construct, you reconstruct the variety spatio-temporelle, the notion of particles, and things like that. You don't construct really the variety apart. Yes, yes. You can always plonger. Yes, so it's always, it's never... It's the matrix. Exactly. Yes, exactly. So, in fact, we could say... Well, it's not bad.

2:00:00 Yes, but it's even worse. It's even worse. It's even more automatical. We can't get rid of it. We can't get rid of it. We can't get rid of it. What we reconstruct, are the operators who give the same measures as if there was a variety of light. But then, there is a graph that we believe in reality, fundamental. And then, in fact, we construct a different level, a little postgrading, if we don't do it. And on this level of postgrading, everything happens like in physics, where there is no because everything that has been reconstructed as emergent is really the same. So here he says, instead of talking about errors, we reformulate everything in the language of the topos. Yes, exactly. For the aspect number real, I think that is a very important point, because it's really linked to the aspect hibersian, It's the other side, because the aspect berthian is what allows us to say that the amount of information that is disponible for the observator is finished. And the problem of the number real is that the aspect berthian is that if you have an open space of Hilbert's dimension, it allows us to say that the state of... Oh, yes, that's my thesis. Well, I wrote a thesis. You'll be able to write a thesis. But it seems very naïve that to say that we have an space of universe of dimension finite, it allows to limit the information available to the observatory. It's a little the opposite. We can do it rather the inverse. But for the space of universe, when we see what is the space of universe, because, as you know, Von Lehmann and so on were not happy with this notion. Von Lehmann after three years, after three years, after three years, he renounced. But when we look at what motivated the appearance of the Heubert, there is a crucial, really crucial, of the numeric of an index zero. So it means that it is a complex, and that is something that, until now, has a single motivation which is called the solar theorem which, in fact, depends on another

2:02:30 very charged and very strong. So, finally, we don't know how to motivate sauf postuler l'apparition des normes royales complexes contraignants et c'est un truc qui apparaît à gauche et à droite par exemple dans toutes les approches des algèbres c'est-à-dire le phoneyman et compagnie la question fondamentale c'est pourquoi donc c'est cet algèbre là avec ses coefficients complexes et pas autre chose et on s'aperçoit très vite dans le moment de that to start by saying that we live in an age that comes with almost all the machinery that we want. Justement, for me, it would be an argument in favor of this. Yes, but you know, instead of postulating all the formalism... Well, postulate of hyperposition implies that you have a vectorial space. It's because there's interference. You observe in nature the interference. If you come from the experience, you say, there's interference. If you come from the physics, there's no point of view logicien. But from the point of view logicien, we can do the same segment, but in the other direction. From the point of view logicien, it's fine. We're very far from the physics from the first principles. We're exactly in this case. We're in the program not the physics, but the theory and the physics, from the fundamental principles. And the linguistic principles, in fact. And you know, to say that the principle of superposition is a fundamental principle. Why there exists something rather than nothing? Yes, yes, yes. But if you say that the principle of superposition is a fundamental principle, it's already enormous. Yes, yes, yes, yes. You already obtain a moitié of the result. No, but if you don't have the principle of superposition, you don't have any reason to abandon the physics newtonian. Well, it's not true, but it's why it exists all the work. Historically, it's not true, but it's not true. If you don't observe... Well, you can say that the quantity... Well, for example, with the information information, you can say, okay, I want to do the physics where the quantity of information is available...