Lattice Theory, C20th Logic & Algebra, Category Theory — Part 2

0:00 Thank you for your attention. This is not the excuse I wanted to give you. We go back here to the third excuse, which is the interlude. The interlude is the third excuse. From the third excuse, the same problem can be linked to the construction of non-coordinated extensions of structures that were not obviously an extension. The non-coordinated interlude, like the third excuse, is only an example that, from an epistemological point of view, can be crushed by all other fundamental properties, structurally speaking, but which do not pass through an extension. Quite simply, because the extension was not aimed at, because we can make extensions that are conservative, which is why, at the time of creation, the extension was not aimed at the telepathy, it was not aimed at... What is important, however, is to see that there are things that go in the opposite direction, because we cannot exclude them. After the third phase, which is the permanent phase of Symbolism, it is directly operative, as we have seen in this example of Joseph Bosch, whose only job was to teach Einstein, to develop the algorithms, which were developed before him and which were also developed by Rosenblum, and thus to arrive at the same structure. These are things that can appear at any time when a number of mathematicians or mathematicians are in a state where they dare not, no one would dare, in general, that x bar is 0, that x is 1, that x is 1, that x is 1, that x is 1, that x is 1, that x is 1, that x is 1, that x is 1, that x is 1, that x is 1, that x is 1, that x is 1, that x is 1, that x is 1. On the legitimacy of the co-products compared to the co-products, we all know that it is the same thing, there is the legitimacy, okay, but is it nevertheless after this progression?

2:30 The cohomology does not remain fundamentally something else, let's say, the idea that it is a product, and the rest, it is not good for you to consider the category and to oppose the categories of mathematics together, for example, since we do not have exactly the same. The cohomology category? Yes, the cohomology category. So, I think, I think that, to answer this question, I think that what you are saying is very cultural. Cultural. Cultural. Cultural. It's that, I, from my experience, I have been able to make a group project, a foreign project, a complicated project, a foreign project. However, this is a direct product of the group that I made by the students. From the moment that I understood, at the time, not only at the moment, but at the end, that what I was looking for at the time was the composition, the composition is a profoundly different convention. I'm not talking about promotion, it's a cultural matter. I really think that composition is a convention. This is exactly what we are talking about here. From this moment on, don't attempt an element-by-element provocation. It happens, because if you try to provocate an element-by-element provocation, the operation is not effective at all. And if you try to explain it clearly, that's why it doesn't work. Neurological progress in the 40s and 50s showed that provocation was a very common operation. I did not work in biology, so people who work in biology will say that it is an operation, but even more so than me, it is a completely natural operation. That said, I tend to think that something was hidden from us by the weight of the traditions that made the construction of the Cartesian formula together, it was really an operation. That was the concrete, the tangible, the simple, you could easily explain that to children.

5:00 Explained by a program, it's complicated, but I don't think there are any differences in nature, not even from the point of view, there are no differences in nature. Maybe it's because I wanted to see. Wait, there's another question. The other question is, on the subject of... I received your proposal because it is a bit ambiguous because you said at each step that the mathematician is focused on the fact that this third term is the r or the tenth term. But it seems a bit ambiguous to me in my proposal because... These are the objects, let's say, of the work you are working on, which present these properties. But the way of thinking of the mathematicians, the logic that they use, is not the same. It's because the subject is either, let's say, good or bad. And that's why it's a bit ambiguous. You're right, but it's also true. You're right, but it's true that my first choice was good. Because if we consider that logic is a function of the mind and not of the body, it is clear that it is the logic of the mind, which is well explained, and even if a mathematician does not understand it, it is clear that it is the logic of the mind. This is a commentary that seems to me to be relevant. And this is where, in our opinion, the accused attitudes seem to be complementary to the element. In other words, we consider all the elements that do not accept the possibility of consideration. So, we can continue to consider all the elements that do not accept the possibility of consideration. It is important to understand that this whole system does not have any possible ingenuity.

7:30 So we have to react quickly and consider what are the elements of the class 5. When they are not three, it makes sense, because if we continue to think in a linear way, it makes sense, but it is not the same. So, in 1925, we worked on this. This is the formulation that, in fact, we have today. In 1925, we worked on this. This is constantly in practice, and not in the books of science. This is in practice, because this is the school of mathematics that I worked on, in the name of a thinker, of an agent, but not interesting in the whole of science. If you don't understand what I'm saying, you can ask me if it has any impact on your personal life. Oh, I don't think so. It doesn't affect me. You can try to say, yes, there is a thought, but to have the thought to relate to the thought at the time, etc. If you don't know anything, tell me. I don't take it for granted. I will give you an example to calculate the number of reflections on the subject. It's an article that I'm not going to quote you here. It's an article that I put in my scientific journal. We can make complicated, propositional calculations. With a final logic and a future logic. Because you have, for example, a distribution of equivalence, the equivalence is regulated on the two degrees. There, no one wants to say, perhaps, that you are making a mistake, because what would be the meaning is to take you by the hand, or do you think, perhaps, that in the end there are logic to work with? You have to do it. But you can continue to calculate with a final logic, to calculate with, for example, the...

10:00 The logic being embodied in the two degrees of belonging, or in the name of the inventor, the degrees of similitude, the degrees of dissimilitude, etc. But in this part, I say, once we have done all this, once we have done all this, once we have done all this, there is nothing left to be done. How do I choose? And then, in this part, I say, in this part, I say, from a certain distance, from a certain approximation, from a certain distance. In other words, from a partition of states, i.e. all the states belonging to a system, to an instrument. A state, an instrument, an object. This is my recent response to something. If we consider, let's take another example, logic, logic, logic. Multi-valence is an application of degrees of equivalence, so I would say that it does not belong, it belongs to one, but it belongs to two, it belongs to two more or less. But for us, every time we have calculated the degrees of equivalence, we have to choose. And so we have to choose the real, this set, bivalent, the closest to the center of the geometry. The closest to the center of the geometry. Thank you. Okay, and your answer is still quite positive as far as we're concerned. Ah yes, that's what I'm talking about. I'm talking about the... the kind of... the kind of... No, but... No, but it's not just about mathematics, it's a... it's a virtuality that allows us to refine the results by taking them into account or by taking them into account. Once we've done the calculations, once we've done the calculations, once we... To conceptualize a situation, if we want to decide, we have to decide, if we want to decide, we have to decide, we have to understand, we have to understand, we have to understand, we have to understand, we have to understand, we have to understand, we have to understand, we have to understand, we have to understand, we have to understand, we have to understand, we have to understand, we have to understand, we have to understand. I see that there is no such a deep problem, I think of this example of space, of space as a vessel that contains everything, and of atoms that contain nothing but contain us in all the other things, and it's practically the same thing.

12:30 It's dual. If we think of A as B, like capitalism or something, there is duality in the sense that we can reverse it. But these three theories even more so include the A part. Yes, but that is to say that we can reverse the arrows. There are some intuitive facts, some debates, that there is one space. Why one? There are several atoms. By the way, it turns out that there is no such duality in our head, even if we can make it a category or something, that there is something that... Anyway, it's just a dilemma. And in fact, we see it differently, as we said, we have distinguished three points at the beginning, yes, how to question them, give them the algebraic pure, but I believe, I agree with you. The fourth, perhaps not the most important, would be geometry, and it works through all these stages. Can you explain it to us? Yes, for example, in the beginning, we can think of a kind of naive geometry. But then, with Marshall Stone, of course, because he has all these topological programs, even in a sentence he says... We have to use topology, something like that, that is to say... No, I don't follow you, because... No, I don't follow you. If you think about the algebras, because that's what it's about. If you think about the algebras, I don't see any kind of genetics. Genetics in the sense that we don't practice it. Cheston, we don't... No, not Cheston. No, no, not at STORM. The only one... No, not at STORM. At STORM, we have the... The chronology of... Yes, of course. The chronology of... The visualisation of the chronology of all the known ideas, and we don't talk about the chronology. The space of STORM is not a very easy object to visualize. And STORM didn't perceive it in any way. Not at all. I would like to ask you why did you choose such a complicated way of topology?

15:00 First of all, we can see that the ideoproblems in the simple case are equivalent to the points. For example, if you look at the whole set of ideoproblems, the ideoproblems are the ideoproblems generated by a prime member. It's the same thing, if you get out of this frame, it's the opposite, we say more with the ideoproblems than we say with the numbers. This is a question of the level of the constitution. But at the level of the topology, why choose the base of the family? It is not at all in the world of genetic examples or of genetic visions, but it is to look at what is happening in the world of the world of the world. What are the first ideas? We are in a world where the first idea is the maximum, it is the same thing. I absolutely agree, but I think, well, I can say it's a hypothesis. There is this naive geometric side in Verne, there is this geometric in the abstract sense. In Verne, in Verne, the representation of geometry does not accompany any calculatory opening. I think we can still try to find the intuitions and find the intuitions of each other. I don't know if they are the same. I don't think so. I don't think so. I don't think so. I don't think so. I don't think so. I don't think so. Thank you for your attention. Wait, I'm not done. The only attempts to geometrically represent the people of the Red Bull are to produce what we call hypercognitions. There are two types of geometry, 0 and 1, 0 and 1 power n, in the geometry of the atrium, considered to fall into the visual range, because otherwise the visualization is wrong.

17:30 Well, it didn't work. So I don't see a fourth point of view on the atrium of the atrium, on the geometry of the atrium. Thank you for your attention and see you in the next lecture. We can say that the points are more relevant, but on the other hand, yes, I see, it's good, of course, but on the other hand, what we say is that in these cases, yes, we can always think... In other words, a set of parts, but not just all of them, but all of them together. And this, in a certain sense, means a return to the same station. That is, we can think of it as an element, we can think of it as something, we can think of it as being together, but not as being together. But it's interesting for me, what's really going on, that is to say, Hadassah, etc. My second question is the same as the first one. Because it's the theoretical, it's not the subject of today's lecture. It's the theoretical representation of Stone. The theoretical representation of Stone. By what? It must be explained, I can't go on like this. It must be explained what is the logic, what is the crisis, starting and ending together. Of course, he answered. Fini or infini, quelconque, et l'une par rapport à l'action, et isomorph, un sous-agent de nous, un grand parti, un ray of self, un fiber of self, d'un ensemble donné. Il faut voir quel est l'ensemble donné. L'ensemble donné, c'est l'ensemble des idéaux premiers. L'ensemble des idéaux premiers, à nouveau, là, on retrouve des choses très techniques. L'origine de ça, c'est l'origine d'un ensemble émanant. La première fois que j'ai vu ça, je me suis demandé si j'allais aller chercher ça. Since I didn't understand why you were looking for that, I don't have the answer. There is an ambiguity, a multivalence. Because there is the problem of the unicity of the complementary. We don't always have that. We don't have that. We don't have that.

20:00 Yes, of course, that was... That was the deal. We couldn't have that. So, what do we call the complementary? The complementary in the sense that we define it. Now we can define it. So, it was Epstein who talked about the operation of generalized complementation. Posterity didn't follow that. As you may have understood from the example R I gave earlier, the X and Y are rather indicator barriers. So it basically says the weight of the proposant Y in the considered elements. So these are not general explanations or combinations. But in ternary logic, as an example, in ternary logic, this term is also used. So if you have a ternary logic, it is therefore... We consider x as an element of 3 and what are we going to call the negation in this equation? So, we put the negation table as follows. It is 0 and 1 and 2 and 2 and 1 and 0 since we have a circular equation. So, non-zero is 1, non-1 is 0 and non-1 is x. Not for Lukasiewicz. Not for Lukasiewicz. He didn't hesitate to make it appear in the lecture. It's not very normal. What's important is that this corresponds to a circular orientation. Before we talk about the technique of Lukasiewicz, this corresponds to a standard operation on the double pose. The components are x0, x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12, x13, x14, x15, x16, x17, x18, x19, x20, x21, x22, x23, x24, x25, x26, x27, x28, x29, x30, x31, x32, x33, x34, x35, x36, x37, x38, x38, x39, x40, x41, x42, x44, x45, x46, x47, x48, x48, x49, x49, x49, x49, x50, x50, x50, x50, x50, x50, x50, x50, x50, x50, x50, x50, x50, x50, x50, x50, x50, x50, x50, x

22:30 I will not go into details here. The other negation, I will not go into details here. The other negation, I will not go into details here. The other negation, I will not go into details here. The other negation, I will not go into details here. The other negation, I will not go into details here. The other negation, I will not go into details here. If we take b equal to zero, a-x is equal to zero, so it equates to x-a-a-a-star, and a-star is the negation of a-star. A pseudo-complement. It's not an injection, because a star is necessarily represented when we arrive in the center, etc. It's not an injection, that is an injection. So Lukasiewicz had written, in his book, on his own computer, that Lukasiewicz had written x equals t0, t1, t2, and the negation of Lukasiewicz was x, and the negation of t0, c2, was 1. Thank you for your attention. Bosch-Waertz and Kepler-Holland.

25:00 In the sense that they are all included in the application. That is to say, a1x² is equal to b. If, on the whole, we have this equation with all the solutions, we have the matrix, we have the matrix, we have the matrix, we have the matrix, we have the matrix, we have the matrix, we have the matrix, we have the matrix, we have the matrix, we have the matrix, we have the matrix, we have the matrix, we have the matrix, we have the matrix, we have the matrix, we have the matrix, we have the matrix, we have the matrix, we have the matrix, we have the matrix, we have the matrix, we have the matrix, we have the matrix, we have the matrix, we have the matrix, we have the matrix, we have the matrix, we have the matrix, But we have three negations, two, I don't know, plus the one about the flux, the one about the flux, the one about the flux, the one about the flux, the one about the flux, the one about the flux, the one about the flux, the one about the flux, the one about the flux, the one about the flux, the one about the flux, the one about the flux, the one about the flux, the one about the flux, the one about the flux, There too, it is inscribed in the question of science, that is to say, when we say what is left of the question, there is the question of science. The rest leads to a structural reconstruction that depends on the way in which you will consider it as legitimate. If you consider that it is legitimate to deny from the application, then we will choose the solutions for you. If you consider that it is legitimate to deny by reversing the points... I personally consider that it is legitimate on the part of the people to do the things. I had been very surprised, there was a colloquium on the application. When we looked at all the propositions of each other, there was no political agreement on anything at all. But no, I mean, what I'm trying to say is that I work with two operations that I don't have the same level of delegation. One of the operations is Vettel and the other is Starr. Look at how this is composed. I don't have a theory on what is a fact or a negation.

27:30 No, no, I don't have a theory. You wanted to make an evolution, but it's not possible. Quantum logic is another thing. There is no distributivity. We have to see if it is possible to build a structure, a concrete structure that is within the logic of quantum logic, which I don't know. Thank you for your questions.