Concrete & abstract, representation theorems

0:00 We are in the same boat with this little... ...concrete and abstract things, in general, and then he tells me... I also have two theorems that concern, I think, these subjects, the theorem of Kehl and the theorem of Strauss, and then he tells me about Yama Yoneda. These are abstract questions. The first, in mathematics, is... These concrete distinctions, in my opinion, are less obvious than the same distinctions in the physical world of material objects because normally in concrete we think, even in the etymology of concrete words, that is to say solid, hard, that is to say normally we speak the concrete object like the table, the chair, the cocoon. Versus the abstract concepts, like tables in general, like the types of objects or the concepts of tables or something like that, which are not necessarily found in the physical world.

2:30 But of course, these great subjects, which I will not go into too much, also apparently the theories of substance, Aristotle, We don't really use abstract terminology, but there is something similar, when we talk about substances, there is a lot of work, and I sympathize with this point of view, which is the main example of what substances are, organisms, living things, things like that. It's a bit concrete, but then when we talk about all kinds of things, it becomes abstract. Okay, it's just a little remark, but in mathematics, apparently, we find points comparable to the table. There is no distinction between things that are material and concrete. But still, there are several reasons. There are several reasons to use this distinction in mathematics, but on the other hand, when I prepared this conference, I thought a little, it's not absolutely obvious that, let's say, this... When we think about how we can form the waves, we talk about abstract and concrete mathematics, it is not obvious that it is always the same distinction, really, because there are several examples when we can imagine that it is the examples of abstract and concrete, but still it is the question that we must try to answer together, if it is really the same type of distinction. But let's say the case may be elementary but still very important. It's something like that. Let's say we have a theorem that says it's theorem number 4. I don't know why the first public opinion that in the isolated triangle, the angles at the base are the same. But how do we expose, how do we expose this theory?

5:00 Really, I find myself at ProRios. I have already said that this community is more or less linked to the clinicians of Saint-Pierre-Siel, more Saint-Pierre-Siel. Maybe the text is more important for math history, because almost all the experts in ancient mathematics come from this text, where he quotes the history of mathematics that is lost. But still, this text is very important. But also, in the sort of analysis of mathematicians, of mathematicians, and really, maybe I shouldn't talk much about that now, but in the analysis, it's really its structure of diagrams. It's much more complicated than what they say today, they have the thesis, the thesis, and that's it. He really distinguished six parts, and a poor apothecary is one of these six parts. There is no so-called binarity between a man and a poor man, which is more complicated, but I may not be able to talk much about it. Or perhaps I could, if you want. But just the beginning of this analysis, it begins. When we compare Pouclus' text to Pouclus', it is a perfect analysis, i.e. the condition of the text is perfect. He says that there is, except in Monsaigne, it is just in the general term that I say, in the training of the scholars, the angles of the basis are the same. The second part is important. What do we do? Or ABC will be between Venus and the Sun. And what's going on here? It's a kind of logical paradox.

7:30 We proved this for ABC, but the theory is not on ABC. All these are related to the solar triangles, so we can say that we can repeat the same reasoning, but we can't do it without the problem. What's going on here? The same example, but in my opinion it's an interesting one. I just want to say, of course, we can look at this ABC here as a variable. We can say that ABC is not a concrete term, but rather a short term term. We can take another term. There are also some small problems here because maybe in this case it is simple, but in other cases we have to look at different cases, maybe we have to give the wrong element for different cases, but not here. But we can still look at ABC as, let's say, a variable.

10:00 But it's not the variable in the modern sense of mathematics, it's rather the variable in the logic sense, that is to say, it's the number that can be applied to a different object, something like that. And that's also interesting, this notion of a variable, which is not at all the variable in the sense of Newton, when he rather thinks of... Mathematical object entities that change. Here, it's not necessarily. We can also try to look at CBC as a singular entity but that changes. But it's not necessarily. The most common point is just to say that it is variable that can take different values in the set. But anyway, this is the first time that we can use this very concrete distinction in mathematics to give us an interesting Platonist theory. This is a very clear distinction because it also distinguishes between ideas or forms, between material things and then the mathematical object in a certain sense between the two. This mathematical world between the idea and the material object, but also in the mathematical object itself, it makes the same kind of distinction between the two. Mathematical objects closer to the idea, in the most general essays, something like triangle, isolates, not concrete, abstract, and distinguishes from concrete mathematical objects, something like triangle, ABC. It is also interesting the role of, what is called, naming. This is a kind of mechanism of calculation and dilution in the norm and it becomes a kind of comprimation in a human, in a human. A lot of things that are really interesting here. And just a little remark that really it was a problem for Kant, exactly, this kind of thing,

12:30 There is a distinction between the so-called sympathetic judgment and the analytical judgment. The idea was not always very clear, but it was like that. The analytical judgment... It is implied by definition, by the concept itself. If I say that two radii are equal, it is analytical because it is in the definition. But if I say something like that, it is already subjective. Even if I say that 2 plus 3 equals 5, it is already subjective because it is not in the concept of 2. Why is it related to this abstract-concrete problem? Because it is the role of imagination that allows us to construct something pseudo-concrete in mathematics. There are all the theories, which are called aesthetics and which have nothing to do with the perception side. How can this kind of game of homogenization have the value of an objective? How can we make a theorem that is true in some objective sense, not in some one-to-one or one-to-one, but in some sense that is ideal for everyone? How does this combine, let's say, the role of imagination and with certain forms of logic? And that's why there is transcendental logic and non-formal logic that allows me to just say something that would have been said by the professor at his school because it was not explained by definitions, but something like that. Okay, it's just a little remark. Is this the first place in mathematics that we are talking about in this abstract concrete discussion?

15:00 I mean, I don't know. For us, they don't talk about concrete abstracts. Yes, that's why I said that. The difference between concrete and abstract is not... Honestly, it's always interesting. But if we want to study seriously these questions, we have to look at all the translations of the terms, the translations of the terms. How do they come together, these concrete, abstract terms? I didn't prepare my talk to do these things, but obviously it touches on that and in, let's say, a kind of jargon, modern, philosophical jargon, it will be rather abstract, concrete, which suits best, in my opinion, at least, this kind of distinction. Yes, it's questionable. Yes, what you said. What? There is a presumption... There are a lot of empirics in our country who give a sense to the concrete and abstract that does not exist in our country. I would never have known your perspective. Yes, perhaps. The notions of abstraction... You are much more in the style of Aristotle at this moment. Aristotle, we can find the distinction between the concrete and the abstract when we go from the first substance, Socrates, this man. There is no concrete, abstract term, I forget how to say it in Greek, but it is Aristote who has studied it. Concrete is the second substance, that is to say, man. Which is an abstract concept in relation to the concrete individual, to the concrete individual that we are. But what I mean is true, but let's say that the words in Latin for abstract, it's really absolutely literary translation of the Greek term that I forgot, which is Aristoteles. No, but yes. I forgot. But on the other hand, to my knowledge ... There is no such thing as the equivalent of concrete, it is something that came much later in the medieval traditions, I imagine. Okay, you have to look at all that, I don't know, it's not the same. But it's always a question of community, philosophy, how we can apply it. Maybe we can use morphism to specify that, but it's always a little bit of cooking.

17:30 Yes, but we have to try. But on the other hand, Aristotle does not give us good theories of, let's say, the philosophy of mathematics. However, Proclus, in this neoplatinistic framework, where he tried to incorporate the rest of them, he really gave an analysis of this problem, and that's why I'm talking about him. For us, it's not because I have a lot of experience in Penrose, but in any case, for me, it seems to be the same. It's a bit something similar when we talk about, I don't know, tables of this kind and general concepts, something apparently similar when I talk about triangles, right? And I think that in general, it's a lot of logic. So the operation that Marcus named, it's the term ecstasy. It means that it's the transition from the idea of the triangle, in the platonic sense, to the schema, what we call the schema, that is the figure or the diagram. And simply, this is not at all the passage of an abstract concept, because the idea is not an abstract concept, it is the very essence, it is the very reality of the trinity, it is the being of the trinity that is, how to say, in concrete terms, it is, I mean, in the figure there is more singularity than in the idea. In fact, the idea of the film is to create the idea, in fact, it has all its reality of the idea, and that's what we're talking about, we're talking about the singularity of the film.

20:00 There is no need for observation of the singularity of the figure for any reason. We only take what is universal and, finally, independent of the singularity. In reality, it's a bit more complicated than that, yes, it's true, the image is like that, the main image is that there is a sort of upper part between the things in the world that is in the top, in the top is the idea, and here the material things, and here perhaps there is something at the bottom that is called matter. Also, there is here a sort of, first of all because there were several of them, even though it is an invention of a neoplatonist, it is not a platonist, there is something like a hand, it is a sort of tri, maybe, but here there are the mathematical things. But what I'm interested in, rather than this global structure, is a more local structure, a structure of mathematical objects. And it's what distinguishes, it's also a bit, sort of, clarified by the corresponding faculties of the mind. It's called nous, something like that, the mind, here it's something like dialoia, meaning reasoning, maybe, here it's opinion, so that's it. But in mathematics itself, it is something that Koch adds, a deeper analysis than that of a standard mathematician. He says that even in mathematics, there are the two sides of mathematics.

22:30 There is a side that is closer to the idea, it is the side of reasoning, but there is also the lower side, which is the side of... They are called Fantasia, Imagination, Imagination, Imagination, Imagination, Imagination, Imagination, Imagination, Imagination, Imagination, Imagination, Imagination, Imagination, Imagination, Imagination, Imagination, Imagination, Imagination, Imagination, Imagination, Imagination, Imagination, Imagination, Imagination, Imagination, Imagination, Imagination, Imagination, Imagination, Imagination, Imagination, Imagination, Imagination, Imagination, Imagination, Imagination, Imagination, Imagination, Imagination, Imagination, Imagination, Imagination, Imagination, Imagination, Imagination, Imagination, Imagination, Imagination, Imagination, Imagination, Imagination, Imagination, Imagination, Imagination, Imagination, Imagination, Imagination, Imagination, Imagination, Imagination, Imagination, Imagination, Imagination, Imagination, Imagination, Imagination, Imagination, Imagination, Imagination, Imagination, Imagination, Imagination, Imagination, Imag Theoretical, mathematical, but on the other hand, we can say in contemporary Japan, constructive. And that's exactly what's happening here. There are no more triangles, let's say. You would be right to not talk about the abstract, but today we can say something abstract, general, and on the other hand there is only the concrete individual, which is the construction of imagination. Yes, but the theoretical is not made up of the concrete character, that is to say the particular, of the figure. It is concluded that only the theoretical terms are included in the concept. Yes and no, yes and no, because in the theorem we have both, we have announced the theoretical terms, that's it, announced, announced, which is the general side, you are right, but afterwards we always have this constructive side, because I said, okay, the isosceles triangle, now, but afterwards I said, either the isosceles triangle... There is nothing more or less, it is not at all enunciated, it is the same thing, it is a complete structure, let's say, and this complete structure, it always has, at least in geometry, its side. Yes, but it's true that in Brookloos, this second side is subdued in some way, in this higher side, it's true, but still it's always there.

25:00 Yes, but construction, we say we can, we can, we can ... There are certain rights and certain points, but under this general possibility, it is not in a particular case. No, no, it's also no, because I've already talked about it, but I'm a little scared to talk about it again. But it's interesting all this, because we've already talked about this distinction between postulate and action in the CLI. There is no action. No, there are actions. Or Koenig and Neuer. Koenig and Neuer. Ah yes, that's what you call an action. Yes, and it's Aristotle who calls it an action. Yes, that's right. But it's also the same kind of distinction. This Koenig and Neuer action concerns the general side, but also the postulates concern only the constructive side. And what's interesting, even in the real text, You can't find any grammatical form to denounce the postulates. It doesn't say that you can do it, it doesn't say that. There is an infinitive form of verbs. We say that we do this, but it's not imperative, it's infinitive. It's a sort of primitive operation or action. Okay, okay, I'll stop, we can keep going because otherwise we'll never get to the second million. Okay, okay, that's the first point. Maybe after we can discuss and say that it's not the case, that it's the opposite, but my intuition says that it is, and maybe it's something very basic because it's everywhere in the subjects we have this kind of... The problem is that in mathematics everything is not as obvious as it is in the physical world. We think in a similar way, but it's not exactly that. The second part of my introduction.

27:30 It's all about algebra, in the historical sense, I can jump to the other era. And I would like to start by talking about the notion of the group, the algebraic group. In the course of trying to solve the problem, which is really a very simple problem, we can perhaps say that the whole history of algebra up to Galois was the main problem, of course there were many different things, but it's just the point of view a little simpler on algebra, but still, it was just like a technique of... The first equation in the second order was written by the Arabs. I don't think it was written by the Arabs. I can't participate in this discussion if it was written by the Arabs. The third equation, but what does it mean by solution, that is to say general solution in general cases, when we have a polynomial of order n equal to zero and we have to find the roots, the roots as some functions also, some functions, let's say, explicit of the professor. This is the problem. It was solved by the Arabs in the case of Decree 1 and Decree 2, but it is also not trivial because in the course of this, the negative numbers were invented, even with the first order. With the second order, we can say the irrational numbers. The third world, it is also important because there is a lot of the third world because of the complex numbers that happened.

30:00 Because it was, that's it. Listen, this was done at the beginning of the 16th century by Cardano. Yes, by Cardano, but after that there was no progress, let's say. And of course, let's say in the 18th century, I believe that most of the serious mathematicians already had this feeling that the answer is rather negative, but really, we prove that this negative result was achieved by Heberis Goloa and ... Or maybe a little earlier by Niels Adel on the basis of this little paper of hers. It seems good. There are no general formulas. And I don't have time, of course, to expose now the theory of Galois, but the idea of Galois theory is like that. If we take the equation of number 2, like that, and I can still take here... So, we have the elements here in the form of 101, if it's not null. After that, it's the known effect in the elementary, which is called Terrain-de-Villiers, that if x1 and x2 are the roots, we have what we have. And what we observe is that the two expressions are symmetrical. This is the idea of Baudouin, to show that it is not possible to… Yes, and of course, with another degree or two, it will always be symmetrical, it is also the idea of Baudouin. It doesn't work if this degree is more than 4. There is no general formula for these kinds of permutations.

32:30 But it's just the beginning. There is another important idea in colloquial theory. Historically, it was the reason for forming permutation groups. And what are my percutations? That is to say, we have given a set of things, a set of things. I always think of finite things. And after, maybe we can discuss this passage from finite to infinite, which is really not at all trivial. But now I'm talking about finite things, or at least here it's not finite. And permutations are rare. Let's say I write the order of this and then I put the same elements in the other order, but here we can also say that there is a kind of bijection between permutations rather than a concrete bijection of a simple number. And the way to write this normally is just to write the natural numbers here, it's like indexes. And here, we just... there are some changes in this index. Alpha 1, alpha 2, alpha n, and this permutation that we apply to a concrete ensemble just changes. Then it changes, okay? That's what permutation is. It's just that I'm going to go into more detail because I want to come back to the K-Lit theory. Okay, but then, I didn't really prepare to make a very precise story, which is of course very similar to the group's, but then, when the idea of the groups was considered, and I can give a standard definition of a globalist, that is to say, there are some... All in all, I have elements with operations, that is, each two elements, there is the third which corresponds to the two and is unique.

35:00 There are operations, and it is important that I note that there is a special element called unity, such as if I write operations like that, for any element. There is also the inverse, we can say an inversible operation in the sense that for each element A, there is an element A minus 1, minus 1, 2, 2, and let's say why, sorry, forget the parameters, that's it, that's it, that's it, that's it, that's it, that's it, that's it, that's it. The most important development of these groups was, of course, the application in geometry, because the people like Felix Klein or Peter Skellig and others thought that all geometric transformations could be thought of as a group, and with the direction of the group which is just composition, that is to say, we make a transformation after the others and it becomes something too. Yes, yes, it's just a movement. I make a movement like this, then another movement, and this will be the third movement. That is, it makes a... and of course, Felix Klein, who proposed his program in 1872... It is clear that if we think about this group of possible transformations in the space given by geometrics, it contains the most important formations of this kind of geometry, let's say.

37:30 And then it was all the work of Sophos Lee who developed this idea in a more concrete way, because it is a continuous group and normally it is ... But also the example that we can find in permutations is, let's say, the anatomical solids, each anatomical solid has a large symmetry and each large symmetry is a kind of subgroup of permutations because there are some permutations that are... All of these are possible, but not all of them are possible. Of course, we can also find groups everywhere. We can talk about groups of addition of whole numbers, but not only positive ones, but also negative ones. We can talk about multiplication groups and things like that. The notion of the abstract group, something like that, I don't know yet, in the history of mathematics science, it's not always easy to find out who is the first, where, in which text, this is the first time that this notion of the abstract group has come about. But I would like to say something about Van der Paarden's work, Modern Algebra, from the 1930s, when it was, I don't know if it's still in the group's memory. Abstract was the first time to go there or not, but the whole idea that really algebra is much more than just a kind of technique to solve equations or maybe manipulate something else, but it is an object of research, it was thanks to Van der Waal who gave all the advice, not only from the group. I think there are a few formalized things that already exist.

40:00 It's true, I would say that for Mr. Gogurou, it seems that we can make him go back to the Gogurou treatise of Jordan, which he called Theos. I mean, I've heard of them already being used, but I don't think it's the same thing. Okay, but what I find interesting about this story is that it's a kind of standard vision that I've learned myself. There is a kind of development of abstract concretes. There is the concrete example for the notation of equations, then there is the other example for geometry, then Van der Waal and Nino van der Zerden combine abstract notions of groups and it makes a group. That is to say, we have... This is an abstract notion, and here we can say abstract because there are no other conclusions other than the one I just said. But still, this is a bit of an image. But what I mean now is that the image is a bit... We can look at it like that, but there is another side of history that, in my opinion, is very important. This is exactly the theory of Kehle, who says that... In some sense, and in the sense that I want to specify, but the sense is so strong anyway, there are no other groups than group-group permutations, that is to say, and it's interesting, I just want to demonstrate this theory, but even the idea is interesting, there are all the examples of groups, there is after this idea of ​​generalization, generalization, on another level, more abstract, more advanced, but after it becomes that, let's say, In a very simple sense also necessary, but the precise formulation of course is isomorphism. Each group, any one, is isomorphic with a few groups of two notations, groups of four notations.

42:30 And of course this notion of isomorphism is important. But obviously it is the sense in which one can say the same group. For the group, it's isomorphism, but it's almost like, if we think of chair permutations or apple permutations, we say ok, in mathematics it's the same thing, but why is it the same thing? Because it's isomorphic, that is, we have to replace the elements, we end up on the same structure, we feel clear, and that's exactly what's happening with the groups, because... I also take the finished cases, but for the finished passages, we say that there are no difficulties, let's say there are no other techniques, it will not be the difference between the two, but here there are no other techniques, but also I think that this passage in these cases is not at all very good, it's very simple, but in the book of mathematics we say that it is the same thing, the problem is finished. But let's say these difficulties with infinity are general difficulties, not specific. But the idea is that we have a group with an infinite element and I fix an order. It's just an element. Then I take some green elements, which is one of these. And then I form all the products in the form of P, A, I. It happens that I get the same elements, of course, because it's a group, but in some other order. I don't know exactly which is the same, but what I discovered is that he was born in St. Petersburg in the year 4831. It's the same here, it's the same here.

45:00 No, no, no, he died in 95, 95, 95, 95, 95, 95, 95, 95, 95, 95, 95, 95, 95, 95, 95, 95, 95, 95, 95, 95, 95, 95, 95, 95, 95, 95, 95, 95, 95, 95, 95, 95, 95, 95, 95, 95, 95, 95, 95, 95, 95, 95, 95, 95, 95. We don't use these notions, we don't use them. Ok, and what happens here is only this change of order, that is to say, we have a permutation here. That is to say, we can, by fixing some order of the elements, we can make a correspondence between the elements for B and the permutations. And then we just have to show that permutation is homomorphism, which is almost obvious here, because, let's say, we take a permutation like this, and it corresponds to some C elements. But then we observe that BC corresponds exactly to the product of permutations, oppositions, which is exactly... It's a bit philosophical because there is almost no construction, there is no technical argument here.

47:30 Almost analytical, if you will, but also the results of everything, because, just to repeat, there are several groups in geometry, in topology, or whatever, but then we show that all this is only permutation. That is to say, this idea of the law of the past, in some way, that's all. But of course, the difference is with infinity, because, let's say, in geometry, it's almost always, except for the cases, let's say, of the absolutists or the Venetians, or something like that, where the cases are normally of infinite groups. Well, it's not so much possible. Of course, we can also talk about permutations in infinite times, of course, but let's say it's at least a place where we can, even if it's a bit of usual difficulty with infinity, it's obviously not more than that. It's not easy to imagine, but for the proof it's a bit weird. The proof is... Yes, the proof is there. Yes, but it's interesting, exactly, it's a bit primitive, it's not a big deal, but it's a bit why it's interesting from a philosophical point of view, exactly, because it's not a big deal, it's something almost analytical, almost in the group concept, but... At the organisational level of mathematics, it is very important because knowing this vision that there is a notion that we can abstract, there are several cases. I say it is a question. I say that it is just a question, it is, let's say, a case, I mean, very special, something that can be said badly. These are the cases that in some sense... It contains everything. And what is also interesting is the notion of the whole.

50:00 Because we talked about concrete in the sense of the whole. It is a permutation of the whole. That is to say, it is a case of the whole. It is not a case of something else. It is interesting. And what is not clear to me is whether there is a link, It's not an illogical analogy to say that this concrete is abstract in the sense of a triangle and Bocuse says that it's not exactly that. If it's the same abstract concrete here, it's not easy. It's not easy, but on the other hand, it's a bit the same ideology because we think, OK, we have... Another example is the ideology of model theory. If we think of Hilbert, we have an axiom list and a model. It's a bit the same thing. We have something abstract like a system and we have a model. But it's the same thing here, but in a different way. It's not the system of logical actions, it's all the same. We have this abstract group notion and we have a model of this concrete structure. But afterwards, it's a certain model that... Key terms include all the generalities. I don't want to talk about it, but there is something about Hegel. He talked about something universal, concrete. He expressed it in a paradoxical way, but maybe we can illustrate by mathematics what he thought. There is something concrete, stronger than Hegel. There is a whole theory of abstract and concrete. Yes, yes, yes, I forgot. I was not very good at math, but this kind of things... There are some people who are good at math.

52:30 OK. The other example is a bit similar. It concerns the algebra of noise. The algebra of the ball, because we can also use the algebra of the ball as an abstract structure, and really in several ways, not in a single way, we can think of the algebra of the ball as a core or ring. Yes, we can think like others, but I always prefer this tree, I don't know, for me it's a little visible, I can watch and look at the tree of Ome and say, is it Bouléan or not? It's like that, it's like that. I have already explained this, but I will repeat it very slowly. We have the partial order, which has, on a diagram, its relationship, which is reflexive, antisymmetric, and transitive, and let's say... I want to talk about something that is more or less in this sense of the word, what is it? Then, let's say, I have a few elements like that, maybe. And here, I always, it's just my vision, my design, it's the lowest and the lowest. Normally, it's the elements that are important. I will talk about the two elements together, but not necessarily. And then, it becomes a triad here for each two elements. Each two elements for A and B. There is a pole above the concept of its minimal, which is called a joint.

55:00 Maybe we can say reunions, reunions and here what the myth tells me may be encounters. Yes, but if I talk about union and intersection, it would be an assembly language, and it's a bit of an idea that I can... And it's important for the Stone theorem, that it's not just the whole, it's just the structure of the abstract, of the theory, okay? And it becomes a boolean if this aberration, we can think of it as an aberration, it is used to observe... To say that it is Boolean, Greek Boolean, there is a third thing that is really the strongest, we can say, it is something called a complement, it is another element like that, except the bottom, the bottom, and that will always be the bottom, and that will be the bottom. The existence of this element, the top and the bottom, is not the same for all countries. But if it exists completely, it can be called a commercial bond, which I am not discussing. Is it only for the Fini, if it is for both of them, of course for the Fini, or is there also the remedy of... In the end, I think it's a very important point, but I'm not going to discuss it right now, I'm going to discuss it when the case is over. The question is, is it Boolean or not? I don't think so, because there are no complements.

57:30 Okay, so these are the theories, and the notions are also rich and abstract. In the sense, let's say, it was very important in the history of algebra to consider that today it seems almost trivial but at the time it was really a great advance to understand, let's say, equivalence and the language of rings and of strings, it's the same thing, but like two algebraic structures, also different, and then it escalates during the homicidal period. By the way, it was Charles Peirce who proposed a philosopher for the representation of the tubular tree. And after the theorems, which I prove that for the calvinists, and in this case there are really infinite calvinists, they also differ essentially. We can also give a lot of examples, and I want to talk about logic, about logical operations, about conjunctions, disjunctions, for Reunion and Rampont, other examples of Boolean structures. We can also talk about class algebras, that is to say sub-assemblies of data. We can also find other things, and there are a lot of examples of structures. But the theory says that, indeed, all the algebra of the ball is still isomorphic for only the algebra of classes, that is to say for sub-assemblies of assemblies. I think we have 10 minutes. Yes, anyway, I can start. After you are in a hurry, you can go back to the lecture.

1:00:00 No, but I don't think I'll be able to answer all of them. Ok, I'll answer them. Yes, but I'm not going to start with the demonstration. I mean, we have to... Ok, ok. Ok, why are finite cases so different here? Because all finite cases are atomic cases. Any theory, or even any other frontier, an atom is an element such that it is close to the background, to the zero, that is to say, an element that exists, such that there is no element that is between the two, that is to say, it is not an emission, it is something, an atom. There are two elements which are larger than the bottom, but they do not exist, they are between the two. This is called an atom. Here the triad is finished. This is called an atom. The triad is called atomic if each element has a few atoms below it. And it is obvious that if the theory is defined, it is always in two words, because otherwise it will be the number of defined elements under each given element. And that makes a lot of difference. And after the theorems, they just say that there is isomorphism. Okay, maybe I can leave that as an exercise. There is an asymorphism, as simple as in the case of the theorem of Kehl, between the elements of tri, of A, and the set of atoms, I don't know what to say, y perhaps, two parts, y of A is the set of atoms as they are contained in the A, contained in the sense that we dissolve the A.

1:02:30 There is very simple isomorphism, as in the case of... This theorem was proved by Talski before the Stone theorem in 1935. And for infinite cases, it's much more complicated. We can't use the same construction, of course, because it will be more atomic. But still, the theory of these two tasks, which is Kalafini, tells me that it is all the sub-assemblies, the assemblies of the atoms that give us the polar structure like this, that is to say all the atoms on the atom. We can represent all the triples by the autos, that is to say, to distinguish the triples by the eyes, we can never have something like that, but it will always be possible for each. But I can leave it as an exercise. But that too is... Ok, I'll leave it as an exercise and next time we'll see why there are homomorphisms between this theory and, of course, the homomorphisms that send this orderly relationship of the theory into the relationship of inclusion of all atoms. This is the same thing as a general structure, which is indeed a generalization.

1:05:00 Historically, the first case, if we look at the work of Boulle, is that he thought of logic, but he looked at the classes of the day. And that's where he found it. This structure is called Boolean. After that, it was all about abstraction. We thought of the Boolean structure in an abstract, algebraic way in several ways. But historically, it wasn't after that. In those cases, it was really after that. After that, it's a bit like the Stone's theorem, which shows that in all those cases... All this generalization is a bit... not really a generalization because all the structures are already in the structures. Except that in the case of infinity, it's not necessarily all the sub-assemblies of an ensemble of an atom. Because here, if we look at the set of atoms, and then it's all the sub-assemblies of an atom. There is a set of atoms that gives us this given algebra. But if it's infinite, it's not everything, it's a sub-set. Okay, and now with categories, the theories of categories, we still have the same hope. The categories we have... I already mentioned that we called it an absurd nonsense, that is to say something super-abstruse which is really, we saw that Monsieur André Ketigari was obsessed with it in the sense that it applies to everything, I say anything, whether it is groups, ensembles, topological spaces. But then it becomes a bit of a question. Is it possible to find some theorems like this that explain us and from the point of view of foundations it will be something that we can understand, that explains that it is really this kind of false generalization that we then fall into together. The same thing we have in the theories of Key, we fall under the ensembles, it shows us the importance of the ensembles, if we don't have the ensembles, it's all the same.

1:07:30 Does it happen in the theories of categories? That's the question. But what is also important in the theories of categories is that the same idea of, let's say, representation, I think, is... It becomes somewhat problematic, yes, because yes, in the theorem of Kehler, we say there is isomorphism, but in the etymologies we have already always asked what is isomorphism, how many isomorphisms, there are perhaps more than 15 isomorphisms, there are several isomorphisms, and it is, let's say, this notion of isomorphism in the etymologies is very important, yes, exactly, because… It's important, in a way, to help theoretical theorists. We can't just talk about models in a light way like that. We must always specify what are the good things, what are the bad things, and so on. And that's what I give to Ludwig, who explains to us that really, there are some things that don't give... There is no false result for this, which trivializes things a little bit, but still there is a kind of particular result, I would say, where the majority is. In fact, I am very interested in agreeing to the announcement. Today, the lecture is not very complicated, but it is not clear to you if you are interested in it. I will start with a few references on universal projects. This will give me a pretext to talk about my father and the natural transformation,

1:10:00 So, we already know more or less what is a problem of a person, I think that no one, well, I've never seen in literature someone who was lucky enough to give a definition of what it is. It's a bit of a notion that, I think we could, well, there is a sort of definition, but in general, it's a notion that is left in the back, you know, and that... And presented by these examples, the archetype of the universal problem is the definition of the universe, for example. For example, a product, we have already said in the category C, if we have two objects A and B, a product C3 is an object X. The morphism is such that if we consider any object with a morphism, we have a single arrow, a single morphism. Let's say A and B are vectorized in a new way through A. It's something that doesn't always exist, but it does exist sometimes. What does it mean? It means that we have a certain sense. Let's say we can say things like this. For example, if we can see y, we can see x. We can consider, for example, the category formed by these objects y1.

1:12:30 That is to say, an object y2c minus 2x. Like this. I say that it forms a category by considering it as a morphism. A morphism like this, which is equal to Y. And in this sense, X is if there is a final object of this category. And so, as a final object, a unique idea is a unique isomorphism. So dancing is a term you can have at any time, but not dancing. So, what do we do? It's... It's... One of the great powers of the Gregera method is to formalize this kind of construction which consists... I think it's the best way to describe the object X not by... By constructing from two realities, as we usually do in ensembles, by saying that the theory of ensembles means that I have an axiom of the pair which, from two ensembles, gives me the ensemble of the pairs of the ordered groups, in a certain way, what we do is that we finally describe what the morphisms look like. An object of any kind, what I'm saying is that we write completely in the category, in the category, I don't know who it is, we write completely what look like the morphisms of a regular object. What is it? By, the diagram shows that the morphisms of a regular object are the same as the morphisms.

1:15:00 I think this is a universal project. It consists of describing an object, and so what we are saying is that this property determines x. So these big x's have some kind of external point of view. But it's produced, it's combined in the sense of... Yes, all of it. So this definition, of course, is to intervene in something that was not present in the diagram, which is the ensemble. So, it will be a little bit everywhere in what I'm going to tell you. The idea is that, since the definition, we adopted categories. We have said that morphism is a matrix of different types of ensembles, and therefore, in our definition, ensembles play a role, and therefore, we will singularize the category of ensembles in particular, so it may not be too clear. But I think that by definition, it would be useless to define the categories by saying that the morphism of x is an object of a certain topos and therefore we could evacuate it if we wanted the notion of an ensemble like that, if we did not want to fabricate the theory of an ensemble before.

1:17:30 Or we are always here to reason in terms of the diagram, which is completely different. I don't know, but I think it's very interesting. Especially since the idea of Yoneda was often, because we didn't talk much about it, it was created on the basis of the idea of the ensembles that we have. So I want to see a different point of view. The point that I don't understand? How can we say that this is unique? You just defined this arrow as a kind of product of the arrow on the side, but why do you say that it is unique? It is not at all clear that this angle is the origin of the arrow. But if I had a military uniform, it is not at all clear that I have anything to determine a movement that exists. In fact, it will be one of the immediate consequences of Yoneza that a thing is defined by a formula of this type. All of this is entwined by this formula of cycles, x-verdons, if they exist, with an x-morphism. So, one of the things that Dylan de Joneda says is that we can completely characterize an object by its morphisms, if we are explicit. What do the morphisms of these objects look like? So we know how to describe the object.

1:20:00 It's better that way. It's also an expression, it's in a category. In the category, it's said. Well, that's to say, I think we can have a description in a way or another of that. That is to say, we look at an object as something. And as something, it's in a certain category. If we want to put more structure on it, we can add the catacombs. When we come back, the wood is in the small pieces together. Yes, that's right. It's not a good job. I'm going to do it. So, I just want to... I really want to formalize in a more general way what it is when we talk about the product, we talk in a more general sense about the image. So for that, what did we say a little informally? We said, a priori, we give ourselves a certain number of objects, a certain number of arrows between them, and then we have the cones, the arrows of nature, and the diagram of the universe. And we defined in the same way if there is the limit of the diagram as being, if we can call it, the terminal object in the category of the cones. And knowing that the cones are based on categories, we always know that if I have the cones, they are not a morphism between them.

1:22:30 There was a problem with terminology. What is a cone? I had said cone. For me, it's not what we said at the beginning. A cone is just an object with branches. It's not a force of gravity. Yes, that's right. The reality is if it exists. This is the last object, so it's the last one. So, we have a very interesting explanation of what a diagram is in general. Because, a priori, it's not the last one. Because we would be asking ourselves too much about how to indicate the motor arrows of the diagram. I think this is a very important question. And I think it's amusing to see that. We can say in general that a diagram is just to give a certain category of heroes which is a kind of graph. A normal diagram category C is just a pointer of a certain category of heroes. The form of the diagram is the data of the object. There are several arrows. What do you call a diagram? I think we can't give a more general sense of the diagram than that.

1:25:00 What does it mean? What is a function in general? For each object x, I give myself an object alpha of x, and for every arrow in x, I give myself a morphine of alpha of x or alpha of y. The only thing to ask for, in a reasonable sense, is that it be compatible with computation. The second key is to have composed arrows, and on the other side, to have composed arrows. So, I would like to call it S.A.H.E. and S.A.G.E. And so, it sends identity on identity. What does it mean? It means that we just want to keep the structures that we put in a certain position. I think that this is really the notion that emerged at the base of the theory of categories. One of the occupations of algebraic topology is that people had invariants for their topological space, which is called algebraic structure in general, and they wanted to formalize this notion of algebraic invariant of a space.

1:27:30 Fundamentally, as we have already talked about, there is a pointy space, which is also a group of point-x-based lacets. There is a point x inside, but I can define the lacets. I have a class of equivalence, a notion of equivalence between two axes, which consists in saying that two axes are different from each other in the motor. So here, we can continually deform the array. So, on all the equivalence classes of the set A to X, we have a group structure, which consists of... well, the other element would be the constant set. And if we have two sets, the composed set, it consists first of doing the first and then the second. And so on.

1:30:00 And so on. And so on. And so on. And so on. And so on. And so on. And so on. And so on. Thank you for watching this video. And so we have to do it if we want twice as fast on alpha and twice as fast on beta to be able to travel with both. And so the composite lasso if we make a third lasso, gamma. So clearly it's not going to be exactly the same thing to do alpha-beta, to make the second lasso alpha-beta plus gamma. And you can make an alpha and a beta-gamma ratio. Because in America, an alpha and a beta go together in a group, and a gamma goes together in a group. Whereas in Europe, an alpha goes together in a group, and then a beta-gamma goes together in a group. It's not exactly the same thing. But it's the same thing at one point of view. Because we can deform everything without any problem. The speed is the one that goes together. And so, we have the associativity of free automobiles, and for the same reason, the s is constant, and the p is a neutral element before the free automobiles, because we first do nothing, then a turn, it's the same thing as doing a turn directly, because the deformation takes place. It's a bit funny. And so what I'm saying is that if I take the category of pointy spaces and continuous applications that keep the base point, then I can define a constant on these spaces.

1:32:30 By the way, what does category mean? It means that if I have an object, I can call it a space. So, a topological space with a base point associated with a group by this process. And so if I have a continuous application, for example, A, which sends little x and little y, then I also have a group entry between pi 1 of xx and pi 1 of y, which simply consists of, if I have an alpha lasso, If I have an alphabets, I can associate the alphabets on the direct with the alphabets in the film, i.e. its image, and that means the alphabets in the direct. So you have to see, but it's not the same as the alphabets on the motor at the beginning, even if it's still a motor. It's interesting to see our history, this kind of... The language of Foucault, because, of course, we can say, here is the topological space and we find a group inside it, as you explained, but we can also say that this is an example of a group, this is where we have the general notion of a group and we also build a concrete example in the concrete framework of the topological space. But, the other thing is this change. When you talk about punctors, you can help them, of course, there are groups that help them, but if you want, it's more operational than the notion of punctors, because, in the end, with punctors, you do a lot of things, it's, let's say, it's a more classical vision, it's a kind of metanotion of...

1:35:00 I don't know when there is an explanation or something like that, which is not in mathematics, in principle, it is rather in, I don't know, presupposed logic, it is not thematized. But with puncto, it is a method but in thematic, it really becomes something like a mathematical object, puncto, it is not just an idea. I think it would have been something like... I would like to formalize the notion that has been present for a very long time, but left in the void, which is that of invariants, for example. So, what gives us a pleasant and strong invariant is not only that two isomorphic objects have isomorphic images, but in addition, morphisms also have anaphylactic images. It's important to keep that in mind, because when we talk about things like that, we say that morphisms in a category are even more crucial than objects. It's a great idea for the technique. For example, when he talked about motifs... I don't have anything to do with the objects of the category of motifs, what interests me is to know what is the difference between a morphism of two, a morphism of two varieties. I don't know if I will continue to talk about it, there is not much to say. I think so. Okay. Thank you. I just wanted to finish what I wanted to say about Fock. Why can I call a twisted computer, in the most general sense, I call it a broken diagram.

1:37:30 You have to imagine that the advantage of category theory is that it brings together a lot of notions, especially those of partial order or graph, because we have a continuous theory of categories. There are categories with objects that we don't need to know their names, but the morphisms are small, they are more like arrows, a little bit smaller. For example, if we look at the four categories, I mean that essentially I have three objects, and I say that I have two of them that have morphisms and a third one. I call it... well, a counter of the two C's. It consists of a three-object data, so C and A, and images of the morphism. So there is a morphism of L, and there is a morphism of A. And it's exactly a diagram of the form of the... Well, it's not necessarily, it's maybe one object. It doesn't work either. What if they have only one object? Well, they have the right to have all of them. But that gives me a problem.

1:40:00 I have the right to have all of them. I have the right to have all of them. I have the right to have all of them. I have the right to have all of them. I have the right to have all of them. I have the right to have all of them. I have the right to have all of them. I have the right to have all of them. I have the right to have all of them. And so, indeed, I have particular diagrams that are constant diagrams, that is to say that all the stars associate the same image, y. And so what I say is that I can see a cone as being the data of a... A constant diagram of a constant functor, alpha, dc, and a morphism of its functor towards the functor it gives as its base. I will specify what a functor morphism is, and I will explain it later if I need to. A functor morphism is really a functor, we do the only thing that is evident. So, what is reasonable to do when we give ourselves two filters in a category? What is reasonable to apply to the transformation, well, to the amplification of the filters of energy? It consists of these data to say how we can fill them.

1:42:30 For each object x, we have access to images of... In a natural way, a component between the two will be a morphism between these two things. I'll go into more detail. In the sense that it is compatible with the Torbjörn's structure, so it is compatible with morphisms. So if I have a morphism in Y, in A, then I have a motor of F. All of these are associated with the morphism of quantum mechanics, like that. So, what is the difference between the two? It's the same thing. You will first apply the transformation of f of x to f of y, and you will first apply the momentum of f of x to f of y. The transformation of the momentum? Yes, the natural transformation of the momentum. It's the morphism of the momentum, which I call the step. I think it would be a good idea, systematically, to introduce this theory in this language of object and morphism, but... Of course, we can also say particular cases. It is also a question of categories of categories, of all the categories, of just categories, that is to say, when we talk about categories as objects, in other words, based on morphisms.

1:45:00 But why are these cases very important, let's say, and not just categories? Generalization of words is not that simple. It's a bit like in the theory of ensembles, let's say, in the theory of ensembles of words, we just use the small words, let's say, for the elements and the big words for the big letters, for the ensembles, but let's say in the more general theories, for example, in French, we just say there are ensembles... There are relations between the ensembles, right? They are all the same, because if we don't distinguish between, let's say, the object and the special category, we talk about category after category, it's a kind of generalization, because each time we can say there are not only categories and, let's say, the morphisms. It's not exactly the same thing, but it's a bit less the same. We can always look at an object, at something complete, in which there is the same structure as a degree, and there is all the other things, also the exterior, but also the interior. Yes, it is true that it is very flexible. For example, something that we had talked about, The way in which categories appeared as generalizations of groups in two different directions were the monoids, the groupoids, and the groupoids. So here, I have a monoid, so it's just what we can see as a single category. So there is a factor, the monoids in the categories that are included. There is also a category in which all the arrows are isomorphic, so there is a difference there.

1:47:30 What I mean is that the category of groups is the fiber product of the category of humanoids and of the category of groupoids on the category of categories. In other words, what I mean is that this diagram is a full pack. So the group is the enemy of the diagram. Well, it's the same, it's still to be said, but I think it's very amusing, so if you want, you can do it. Why did you say that? What? Why did you say that? Because it's a joke, you wanted to show why it's true, it's true for a political reason. Oh really? Yes, because we can always... Thank you for watching this video. It's like here. If you have read the book of Kastrikin on algebra, there is no mathematical book even on the structure of chapters like that. It's not linear. I also thought it was a kind of structure of our course that can't be linear. I don't know if that's the right way to put it, but I'll try to... It works, it works. It's the limit. It's the limit. When there's a limit, it's the limit. And I think it's... I think it's... I think it's... I think it's... I think it's...

1:50:00 I think it's... I think it's... When we look at this, it's a field in the Ancients' Cafeteria. The line, just below, is the part of the real Carthaginian product, the Cydraic. The whole of the Cydraic is the image of EF, the image of Arrhenius. And it coincides with the image of Arrhenius. For me, to imagine this as one of the things Alain promised, maybe it's compatible, it's very common. For example, for example, for example, for example, for example, for example, for example, for example, for example, for example, for example, for example, for example, for example, for example, for example, for example, for example, for example, for example, for example, for example, I would like to try to develop a sort of philosophical discourse on this abstract, concrete question in general, and I think that in the theory of Pythagoras, there is of course an abstract side to his trade, of course, it is social. I think that there is more to it than there is to it. It's just a matter of taking a little bit of time. Yes, yes, yes, yes. Thank you for your attention. Yes, but I have something to say, it's not too technical at the mathematical level, and the other thing, really three things, the other thing I would like for me if you could continue with the idea, and the third thing, even if it's not very good, how can I say?

1:52:30 Convient with the second is to introduce more systematically the language of Foncteur, because I would like, you know, this idea of, if you have, let's say, a little bit of Foncteur, you can call it Coma. In a much more general way, as I said before, we can look at the object and concretely translate this theory of elementary categories a little bit into the language of Fonctot. To conclude on this aspect, I would like to say that the limits, for example, can be as long as they allow them to be. So, maybe just to quickly explain why there are universal problems that are not written in terms of numbers, that are not expressed quite well in terms of diagrams like that. So maybe talk about it quickly, because it's a fascinating subject, but we can extend it to adjunctions. Well, that's it. No, no, I think that's enough. So, after that, we can talk about how the M. de Nézard solved the problems. But do you still want to create a demon of the past? Well, no, after that. So, in the third point, we will define what is the problem of the present. Yeah. For me, it's the definition of a solution to a problem of a person. It's to say that there is an object X that... I'm going to put a pointer to the ensembles.

1:55:00 And I say that this X object is such that the morphisms towards it are well given by the images of the vectors, as I did for the two products, by saying that it is the vector that produces the A vectors and the B vectors. So it is an example to say things about this vector. In particular, if there is an object that works, it is unique in a very different way, in a full sense. And then, because the MZNN is stronger in this part, it allows us to go in two directions, in a certain direction, from the counters, the morphisms between the counters that appear naturally, to give morphisms between the problems. What we can do in practice is to start with physics. I would like to bring up an article by Bill Wiggy. It was really very early, in 1965, I think, on the category of the category of quantum mathematics, I think that's what it's called, exactly where he introduces it. I can also find this article, it's just an idea of the degree of the function of the data. Yes, and then you go on? So maybe we can talk about objects in real life, because it will be a bit simple, and then he will tell us some things and we will talk about the classifier of these objects, see how Yoneda's theorem allows us to understand his theory, for example because we have a very simple morphism, so there are two sub-objects associated, for example.

1:57:30 This allows us to be transported to a morphism in which we are in contact with the norm. We will have to see if it is a bit different. So, the fact that I am a non-sexist, at least in the United States, I am a non-catastrophist. Because I have learned it, I am content with the story of the genesis of the United States. Well, for those who have given their answers, you were right. Yes, it was also a really great story. It's just that I can't hear you anymore. Can you hear me? Yes, I can hear you. It's just that I can't hear you anymore. We can define what a group is, but instead of defining a group in ensembles, we can define a group in categories. There is an apparently obvious definition, and what is interesting is that it coincides with the other definition that we want to put. And the coincidence of the two is a consequence. After that, if we finish this before the end of the year, what we can do is to think about the theory of models and the theory of topology. We can rethink this idea of modeling. Thank you for your attention. There are two directions. The topology is a little bit of a problem because it requires a little bit of logic to do the geometry and so this notion of the internal logic of the topology, it happened later but the coincidence between the two notions did not have a very good effect.

2:00:00 But is the topology... Thank you for watching this video. That's why it's called the Elephant Experience, because there is a different approach to look at something as a domain of discourse, to take it more logically, but also for the generalization of scientific spaces. It's really something not very concrete in the sense that... Est-ce que tu travailles avec quelqu'un, ton professeur, c'est-à-dire, tu fais de l'éthicorie ou tu fais d'autres? C'est un projet que je n'avais pas l'habitude de travailler avec, mais j'avais l'habitude de travailler avec. Je travaillerais si j'avais du valet, je crois. Par exemple, là, quand j'avais l'habitude de travailler avec, j'avais l'habitude d'aller à l'université, on faisait un cours avec Lili, qui est un élève de moi. The notion of topology is really...