Philosophy of category theory (contd.)

0:00 The use and applicability of mathematics is an essential part of what I'm asking myself, and that is the applicability of a natural theory. So the theory of mathematics in different contexts, and more generally than in mathematics, is the theory of mathematics itself. At this point, I think it should be interesting to hear what you have to say about mathematics. First, I distinguish between uses as tools and uses as objects. It's very simple. Now I use my glasses because I use them to see well. Now I use them as objects, I look at them, I ask myself if they are clean, and I don't know. So that's the difference. Thank you very much. In the activity of mathematicians, this science manifests itself in our life, among other things in the current situation, after having solved a problem, eventually a long-term problem, we move on to the clarification of concepts in addition to a solution that is more clear. I think that there was a theorem, the theorem of Lefchet's quantities, which was first demonstrated in terms of a numerical variant, in topology. They are not yet called algebraic methodologies, but by the introduction of the concept of group tomology and homomorphism, this demonstration has been changed and simplified a lot. And more comprehensively, of course, the result is the same as before, not really a stone or a new substance, but we have changed the concept and that's what I call objects, we use them, we focus on what was the tool as an object. It is important to re-work and analyze conceptually, without trying to demonstrate by the results, but simply by a clarification. It is a possible situation where this distinction exists.

2:30 Obviously, the conceptual plan originally introduced in addition to such a clarification, that is to say, in addition to such a clarification, as the concept of a logic and a logic to study the object. Obviously, at its turn, the medium has difficult problems, and if people find their solutions in a conceptual plan, we can raise them. When you go from plan to plan, you come up with new problems. And that's how the history of mathematics comes about. Maths and cohomology may disappear from the scientific practice of the scientific field in favor of new concepts, it is important to note that we will see later that the community is greatly affected by the problems and also by the concepts, by the examples on which we work. The second distinction is the distinction between reasonable use and pathological use. Well, no. Yes, I will talk about pathological use, but the predominant distinction is to take those between simply correct use and reasonable or pertinent use, if you will. And pathologies, they are also correct, but not pertinent or reasonable. If you have a formal definition... If you now have a formal definition of the concept of a group, then you use the concept of a group in accordance with the formal definition correctly. But it's not sure that you use it as reasonably. It depends on the group. Is it an interesting group or is it a group just introduced for such and such group? For example, something like that. It is not quite the intention of the concept of the group. This use there shows that we have not really managed to put into the formal education what we could say. But it can change. We can want to use it later. We didn't originally want to talk about that. That's why we have the feeling that it's a bit confusing.

5:00 Now, we have a problem that's a bit clearer. What are the criteria for a use that is important to the unconscious? That's important. That's what I wanted to ask you. I still have to ask myself that question. In my opinion, as I said, it works a lot by language, by learning, by a training, an introduction, I would even say an initialization, an initialization, so the way of thinking, of using the concepts of the pieces of language, but... So, this question is perhaps not a possible answer, in the sense that these criteria are not valid, but they arise for historians who must, in the face of historical facts, wonder about the way in which reasonable and interesting uses have been chosen. It is interesting that we can produce vaccines. It also arises for the philosopher. Well, perhaps the historian and the philosopher are the same person. The philosopher has the task of understanding the choices of actions that have been made in his discussion on mathematical action. What is a mathematical theory? So, naively, it is a collection of results of a method around a notion or a concept. For example, a method that is the best of the game, and so on. But the mathematical logic or the theory of the demonstration is not the same. She says that the theory is the totality of the denunciations that can be derived from certain actions. And why does she say that? Because we can, in the theory of the demonstration, solve the problem of the consistency of the non-contradiction. And for that, I am sure to have everything that is denunciated. But now the problem is always which ones are interesting, so this formal definition of what is a theory is not enough to answer this question, because it is not the distinction between interesting and reductionism.

7:30 So, when it comes to mathematics, it was traditionally... The traditional approach sees in these observations that there is a hierarchical and conceptual hierarchy, a tool of objects and objects that have problems, and so on. And it sees a hierarchical and conceptual hierarchy that is more and more abstract. The idea is that the consecutive operation of this hierarchy and the abstraction leads to a reductionist epistemology. The only way of justification is a regression to the basic level, because the only way to guarantee the constitution of an abstract object is to complete it. It is a plan on the basic level that comes under one form or another, it is a particular capacity that leads to a knowledge that is intuition. In agreement with this, in the public of classical mathematics, the validity of mathematical dimensions is established by deducing them to a certain number of basic axioms, e.g. the two axioms you saw in the previous slides. Intuition intervenes either in the access to axiom content, as in Friedrich, or in the methods of demonstration of non-contradiction, as in Hitler, or in the current absence of contradiction, as in Chico Bagni. If we accept it as a whole, then in the existence of mathematical objects, it manages to be reduced to a whole. This is what we call ontology. But what understanding, what answer to the question of the criteria, of the choices that this traditional approach can give us? Analyzing a mathematical demonstration in a reductionist way, it comes down to cutting it into elementary parts. By replacing propositions based on complex objects with propositions based on simpler objects, with such an approach, propositions themselves become more complex. It may be easier to grasp the formal truth of propositions by the truth of the components and the principle of compositionality, but it becomes more difficult to grasp the meaning of the components.

10:00 I am talking here about a very complex hypothesis. Even if there are mathematics or mathematics, we all understand one, except that it's true. And, well, regarding the deductions, we can make the same observation. We take the same situation, the strategy where it declines. So the reductionist approach does not explain and does not increase our comprehension of the choices we make. The conceptual transitions, this is a point that can certainly be explained, I think, in the theory of categories. They do not provide a hierarchy of stratified levels, because there are categories of categories, there are categories of categories, there are categories of categories, there are categories of categories, there are categories of categories, there are categories of categories, there are categories of categories, there are categories of categories, there are categories of categories, there are categories of categories, there are categories of categories, there are categories of categories, there are categories of categories, there are categories of categories, there are categories of categories, there are categories of categories, there are categories of categories, The main difficulty is to find a philosophical point of view that allows us to understand this situation. Due to the absence of a stratified hierarchical level, a reductionist approach is not able to find a justification for the constitution of objects in relation to their structure. We need an approach that manages to justify the constitution of objects directly at a level. I was talking about articulation, but I'm not talking about articulation in the French sense of the word, but a pronunciation, a compilation of the ideal object, which is not, and which makes us understand that the approach is to reduce the concept of object-tool to a simple abstract. I will now present the concept of adjussion in a lecture on Charles Hamilton's Curse. It seems to me that pragmatism, precisely, of course, pragmatism, I will limit... It's quite dangerous, because if we say I'm a pragmatist, everyone thinks I'm a genius and so on.

12:30 It's a shame, I'm from Corsica. It seems to me that Corsican pragmatism can provide us with means to tackle biological questions. Perth is a new evaluation of the concept of intuition by opposing the traditional approach of the law in the intuition of particular capacities in relation to knowledge, a approach created, for example, by Miquel, of the Cartesian distance of the intuition that we know. In what follows, I present the alternative conception of intuition found in Perth's poetry according to a general interpretation proposed by Perth. In a way, they extend the epistemologies of Kahn, who for knowledge of objects depends on the subject of the means by which he uses them. And the task of epistemologies is to predict these means in order to make them present. It is about the critique of the present as a means of thinking. Goethe changes a little... This idea can no longer be predicted by reason, it does not seem to be a real way of critical knowledge, but it speaks of the assumption of the basis of pragmatic philosophy, that is to say all knowledge, and here I translate nation by knowledge in French, so that we do not have an omission in which it does not exist. Signs are used in the media. What is a sign? A red light is a sign for an object that has a shape. The red light is still there. It was a sign. I don't know. I don't know. I don't know. All knowledge is determined by the preceding knowledge.

15:00 By the way, I'm mainly talking about the texts in the volume entitled Anticartesian Texts, translated by Joseph Chenu, who is my colleague and professor in Paris. But I'm going to use a few quotes in English because I didn't have time to read this volume to prepare for the conference. Well, I understood that Peirce was the one who said that the knowledge of the past is the same as the knowledge of the past. I think that there is an evolution of the individual first, and also of the species, if you will. The means of perception that we have, and so on, determine, of course, our knowledge. According to Peirce, the distinction between the world of things as such is the world of the afferent, the afferent, the afferent, the afferent, the afferent, the afferent, the afferent, the afferent, the afferent, the afferent, the afferent, the afferent, the afferent, the afferent, the afferent, the afferent, the afferent, the afferent, The position of Kerr is therefore a very small realism for which all kinds of meaning are provisional. The information concerning the existence of objects independently of the means used are automatically obtained. So the position of Ni affirms or denies that the object per se exists, but we cannot talk about this question. When there is more than immediate knowledge, then what is the concept of intuition? All the cognitive faculties we know are relative and consequently their products are relations.

17:30 If you don't understand what I'm saying now, but if you want to understand it, it might be worth it to read it in the next session. I hope so. So, let's read it. The reply to the argument that there must be a first is quite right. It acts on the argument. It's a definite argument. There is no doubt about it. In retracing our way from our conclusions to premises, or from determined positions to those which determine them, we finally reach the all-places, the point beyond which the consciousness in the determined position is more liberally than in the position which determines it. All places, all cases, évidemment, wie die Tiefenstufe gesagt werden, und exemplarmäßig wie die Demonstration, wie die Tiefenstufe, wie die Gefahr. It's a conviction rather than a theorem. Now, what does it mean? The consciousness of a determined condition is more liberalized than the condition of a student. I'll give you two examples. He talks about the end of the 19th century. There were still some results. In physiology, not like the generation of Kant and others, but which has greatly influenced the philosophy of mind and physiology at the end of the 19th century with the great contributions of Hercule de Carles. For example, the fact that the third dimension of space is inferred, I don't know how to say it in French, so we conclude it, not consciously, but on the rhythm there are two dimensions, and the third one, you have to add it, and the eye, the eyes, and the brain are able to do it.

20:00 Either, I don't know, I'm not a physiologist, either it's by training, or it's even already a genetic training, that's what I don't know anymore. What's important for him, what's important for him is that the cognition, one determines the other, the two dimensions. There are two dimensions to the routine. Our consciousness of this knowledge is lower than that of the three dimensions. For us, this is our state of mind. It is not a quantum state. It is not Earthly. No, no, the teleology teaches us that... As far as our perception is concerned, this is not the stage to learn something, but for us, it is the second example, you know, the brain is in a position to neutralize the effects of mathematics and mathematics. Once again, the complete image with the neutralization of mathematics and mathematics. We have already talked about this, but I'm going to talk a little bit more about it in an hour or so. So, I give you a second citation, if you have these loops that you can't go down, down, down, stop at something below, but that there are situations, levels where the inner level is less intuitive than the upper level, then, and that's what I'm going to try to show you in history.

22:30 We can very well have the impression that what is determined in a certain sense by something else is still more intuitive than what is intended to be determined. I call this a level exchange. The basic and the thought behind this economy is to say, for example, that people together do not feel good, that you are not in agreement, that you are not in agreement, that you are not in agreement, that you are not in agreement, that you are not in agreement, that you are not in agreement, that you are not in agreement, that you are not in agreement, that you are not in agreement, that you are not in agreement, I also think that mathematics is interesting, because, yes, it is a very complex field, but its composition of arrows is more intuitive than the one of social law. It determines the direction of your determination. But it is a situation that I put in context. We can use another historical example of mathematics to see that what we have seen is perhaps not the same as what we have seen at the time of Frege, before the contradictions between the ancients of mathematics, certainly before we learned that the action of choice is independent of others. We could still believe that one day we would have a demonstration either of the solution or of its content, but now, let's see, it's not possible.

25:00 So, there, there is... I don't know, I don't know what to say, but... But now, we must agree with all the prejudices which we actually have when we end up in the study of philosophy. These prejudices are not to be expelled by a maxim, but they are things which does not occur to us, can be questioned. We do not even have the idea that there is a question. For us, being intuitive does not mean being true, it means that we do not feel like we are. This is the important thing about the CART and the CURSE. For the CART, what is intuitive is true and there are many reasons for it. I started this lecture by talking about a certain type of use, the use of a tool as a reasonable use. Now I'm going to talk to you about the use of mathematics. So I'm here with five parents. I still have to take a pragmatic approach. I'm going to start with this one. First of all, it supports the fact that there is no knowledge of mathematics independently. Thus, ontology coincides with epistemology because we cannot talk about it, or we cannot do it as an anthropomorphic medium,

27:30 we cannot talk about it independently of their understanding. This point of view leads to the rejection of an absolute intuition. It is clear that the intuitive nature of a lecture depends on the situation in which it is presented. Thus, the intuitive nature of a lecture is a particular way of using a language. The intuitive usage of a lecture is an usage in which the truth of the lecture, in other words, the truth of the reality of the usage, is not in question. If we consider the use of a concept rather than a principle, we have a possibility, which is not questioned in the use of a principle. We consider that either the correct or reasonable character of the use of a principle, in mathematics, the correct character of a principle is necessary, but it is not sufficient for the choice of the use of a principle. For example, the philosopher who seeks to understand the choices that are made must rather take care of the reasonable character. However, as we have seen, a use is called reasonable when its question is not asked in the interlocutor. So, what we call a pathological use, when we feel that it is not in accordance with our intentions, we force ourselves to question it. The criteria of the language game are mainly negative. So, an intuitive use of a concept can only take place during a reasonable use in accordance with our intentions. And I'm talking about what I call a technical common sense. The criteria for validity and for the reasonable nature of the use of what I call the common sense, the language system, terminology that I explained was that if we ask the user of the language to question the usually intuitive uses, he will call the common sense. However, the question of the criteria depends on the context and the knowledge. This observation gives the possibility of a justification in relation to a common sense of a technical level. In this perspective, the objects of a later stage are not simple observations, let's say, we can say on this question, of the original objects. We claim the position of the physicists, but these are the very theories of these original objects. I will show you this, for example, in a moment. Because for the expert, such a theory, originally conceived as a tool, In other words, to become an object itself. So, contrary to the electional epistemologies, according to which intuition intervenes at the last level of regression, to establish the truth and the basic principles, according to pragmatic epistemologies, intuition intervenes at each level, because the criteria for use are questioned on one level and not on the other. This is exactly the difference between the use of an object and the use of an object.

30:00 There is a lot of discussion about this. I hope that we will do it in the future, but I will try to do it myself in a philosophical discussion about this approach, and it would be necessary to examine the relationship that it has with other approaches and philosophies. This kind of discussion can be discussed here at the Marienplanet, as an example and as a way of observing the world. The principle according to which knowledge is obtained by examining its acquisition mode leads to the non-separation of the capacities of founding the knowledge of the world and guiding the knowledge of its acquisition in other ways. Opposé à Einstein, je pars avec cette distinction abondamment employée dans toutes les sciences quant à quelques slides et à quelques subprimes, mais spécifiquement en contexte de découverte, contexte de justification, c'est une distinction que je ne fais plus vraiment parce que pour moi la découverte va ensemble avec la justification, c'est ce que je voulais expliquer. Maintenant j'appuie, j'essaie d'appuyer ça. I will not go into the subject of theory yet. The position of the immeasurable source is relevant, in my opinion, to the philosophical concept of the theory of the Catholic Church. I would like to underline this information on three aspects. Illegitimate constructions, the presence of two types of the Catholic Church, and the object of the theory. If we answer the question, what is the object of the theory of the Catholic Church, it will be there that we will find the answer. There are also a number of other fields of study, such as mathematics, geometry, algebra, mathematics, physics, physics, physics, physics, physics, physics, physics, physics, physics, physics, physics, physics, physics, physics, physics, physics, physics, physics, physics, physics, physics, physics,

32:30 The categorization of all the elements of a concept is an illegitimate notion of the theory of a concept, taken into consideration by the categorists, naturally imposes itself on the studies of pathology. In particular, there is a lack of pathological character in the concepts concerned. I do not think that the concept of the category of all the thinkers refers to something pathological. They do not belong to themselves, because they are made to show that there is a problem with a sum of all the small texts of the sum of which is the sum of all the small texts of which is the sum of all the small texts of which is the sum of all the small texts of which is the sum of all the small texts of which is the sum of all the small texts of which is the sum of all the small texts of which is the sum of all the small texts of which is the sum of all the small texts of which is the sum of all the small texts of which is the sum of all the small texts of which is the sum of all the small texts of which is the sum of all the small texts of which is the sum of all the small texts of which is the sum of all the small texts of which is the sum of all the small texts of which is the sum of all the small texts of which is the sum of all the small texts of which is the sum of all the small texts of which is the sum of all the small texts of which is the This is exactly because of this ideological character, because they are not designed to talk about, to be on the other side of the practicality of the common, that is exactly because of this character. It is reasonable that the categorists are convinced that we can and must use these constructions even though the universes of our time do not know how to justify their existence, and this is what we want, not only in relation to the position of the world, but also in relation to the position that we have here. This affirmation is easily recognized as a result of the observation that by relying on the universes to justify scientific constructions, we would replace them in a reductionist attitude of obscuring by the most obscure. Because, we will see that in the example of the letter of Ibn Edda, doing that with the universe, it does not explain anything, it is not at all to justify, it is even more complicated than it was before.

35:00 In the case of mathematics, we can have something that is normally correct, but that is not completely correct. Because it is not the originality of the words, it is just the translation of these words. This is the case, in my opinion, that researchers have produced in a way that is at least artificial. So, the fact that the hierarchy of laws is not stratified does not pose a problem for experts in the field of mathematics. The eventual inhomogeneity of a concept in relation to the beginning of an ensemble does not prevent the justification of such a law. How can we explain these situations sensibly? To do so, we need to inspect another situation in which a reasonable and appropriate nature of the structure of these equations is faced. For the two types of laws of mathematics. The theme is often the same, the constitution of objects is justified by a sense of how they are created and valid in the way they are created. The use of entities as objects justifies the use of entities as objects, which is clear, the fact that these categories, that the categories that would be pathological in relation to the game of language ... All of these structures are equally reasonable in relation to the applications of quantum mechanics. What do I mean by that? If you introduce a category to a public who doesn't know it yet, but who knows, for example, groups, topological spaces, and some other structures. So, pure mathematics. You will say that a category is typically a collection of objects such as a mirror or a tablet, plus the functions such as the hands of an architect. You will say that the objects are typically structured sets and the arrows are typically functions that respect these structures. But that's one type of category, there are others. There are categories that go in that direction. There are quite a few examples that I have tried to include in the book, but I find it difficult to include them in this original conception.

37:30 Historically, it was not even part of the original conception of Halliburton and Klein. They have, for example, considered... Today, we have a group of mathematicians and physicists, and all the examples of mathematicians and physicists were used not to be in agreement with this intuition, but to define the domains of computers. What is the informal rule to use a category term, i.e. the term which is not a reasonable term? If it was the language game of the set of structures, etc., then the categories after the sub-sub-sub-sub-sub-sub-sub-sub-sub-sub-sub-sub. The so-called structural algorithms should be treated as pathological, so it is not the good criterion. The good criterion is actually located at an even higher technical level. This can already be seen in this context of formal definition, despite the informal nature of the term of the structure. However, we could perfectly develop a formal definition of the concept of the first model, then the models of the objects are put together and we are all conscious of it, it is possible, while we find the model of the current definition that does not accommodate such a definitive definition. The so-called second type of the category is that it is composed of a variety in its turn of different definitions. Moreover, the current definition has been conceived since the 12th century, it is very common, to include also the categories of the second type. There are certain terms and conditions that are very important if we want to limit ourselves to the theory of relativity.

40:00 If you always have functions, you don't have to rely on the theory of space and space. It's a matter of action. What is now the object of the theory of relativity? At least in my opinion. What is the interpretation of the theory of relativity? The category theory gives the term object, not the category theory which uses the term object for the members of the first connection. Objects, in the sense of category, are characterized in a completely external and negative way. If we are not obliged to think of an object as a whole, then we cannot read the elements of the object too much, but we have to do something else. The points of continuity, which are more or less the same on the object, will determine the elements of the object. Even in the case of politics, the objects are not the same. This is the external characterization, and there is a possibility to do what I call change of perspective. There is a sense of the term object in which a theory has objects and a theory has objects. What are then the objects? In this sense, the theory of objects, of course. We must be careful not to simply understand the two things denoted by the term object. The use of the term that proposes the theory of the state of the object is formal compared to the use of the term. But there is a common point, as we will see in the examination for the state of the theory of the state of the object, the pragmatic test in which the objects, the theories of Newton, are not a simple abstraction of the theory of the regime, but of the theories of the system. The objects on the outside are the theories on the objects on the outside. The result will be this more extreme thesis. Otherwise, the theory of categories is a typical theory of operations of mathematical physics. I have already insisted on the fact that many categories can make the difference. However, the theory of this type can be viewed as a certain way of finding the theory of the structure in question, favoring the acquisition of information on the structure by the study of the interaction of the instance, type of structure, sex-moment, term of function, and the composition of the structure.

42:30 We are not talking about the theory of groups in the theory of groups, but we have the part of the theory of groups in this category which is expressed by the composition of the structure. It is not by chance that the objects of a particular category help us to find the real object of the category. The objects of the category come from the categories and the category has objects and is therefore multiple. By naming these mathematical terms, we aim to identify them as the object of a theory, in the sense that in mathematics, the relationship can be the object of a theory. In this sense, the concept of algebra can be obtained, and has been obtained historically, by the way of abstraction, the abstraction of the essential properties of a theory. A theory of this concept becomes an array of theories of different categories, and a theory claims that this theory, the theory of this theory. Well, it is possible now that I speak precisely. Earlier I said that when you talk about abstraction, it's a bit slow. I will not... Maybe another discussion. Yes, yes, yes, I can. Listen, I will... I think I said a little... Well, we have another question before we move on to the next slide to finish. We have another question about what I do in this idea with the dichotomy of the second type. So, I think, precisely, because we are at the end of the group. We encounter not only the abstractions, the things that we had at the previous level, but we also encounter things that arise at the same level, through the process, through the language. So, I finish the generalities and I go to the lemma of Yogeda.

45:00 And this will be an object of the category of origin, which is the category with the ancestors, I don't know if you know what I'm talking about, but I don't know if you know what I'm talking about, but I don't know what I'm talking about, but I don't know what I'm talking about, but I don't know what I'm talking about, but I don't know what I'm talking about. How is the y-injection defined in the equation of the value of the alpha concentration? It is the arrow of alpha corresponding to a, a to a, like here. All of this is composed with the identity of two objects A, A7 and A8. I don't know why it doesn't work. It's not that important at the moment. Demonstration, the claim in its own way, is that the problem is political. All it says as a demonstration is to create this diagram. There is also a square, a square, of course, these are all parts, it is a matter of natural transformation, the definition of natural transformation between the counter-offer of the invariant and the operator. And then the little f is just a arrow between two objects that we have discovered at the origin, and not more than that, so that's the demonstration. But that's not what I'm interested in for now. It's the corollary rather. So, let's start with the letter M for the factor B.