Philosophy of category theory — Part 2

0:00 Thank you for your attention and see you in the next lecture. The objects are not only objects, but also a natural isomorphism between two fanters, namely the fanter who takes a fanter, because there you have an acronym, the fanter is an object, the other is an acronym, and who makes the evaluation of the fanter on the object, in which case it will appear. And other factors that have a factor and an object are the fixation of the whole of a natural transformation from a covariant factor to a magical factor. This is what I hate the most when I say that I see very clearly here the connection between the whole and the body of the human being. Well, and that, and that's what I want to say, the plan, I translated embedding into plungement, the plungement of Iosella is therefore, well, there are two plungements of Iosella, and there may be a... These are the two factors that make up the category, in the category of the most valuable aspects of the definitions, of the definitions that we are used to. And, as a result of the corollary and the removal of the two factors, these two factors are not evident. Well, I don't know if you can see it, but it's a little bit like that, it's a little bit like that, it's a little bit like that, it's a little bit like that. We want to talk about this change of any kind of theory in this professor's theory.

2:30 Main, the only trouble is that the Pantacategory Category, as you see in my claim, the only trouble is that the Pantacategory A used here is not legitimate for C-large. If you employ universes instead of classes, the Category S form. I would have said that in the terminology of the concept, in the category of ensembles, it doesn't stay close. Well, that's clear. That's why I say that in Portuguese there is no longer the category of ensembles, but there is no longer the category of ensembles in a universe. But on the other hand, Glenn says, we don't have this category of larger ensembles. The demonstration of these statements remains the same, that is to say, the fact of having universes, we cannot be at the heart of the thing because the demonstration does not really serve us. It is even an exercise in itself. I do not say do not do it, but I say do it, it is not difficult to see that it is a demonstration. Peter Mann had spoken to several students of the theory of quantum mechanics, and I will give you an explanation of this, and then I will go to you. The restrictions employed seem mathematically unnatural and irrelevant. Though bordering on the territory of the paradox, it is felt that the notions and constructions as the categories or structures of design or the categories or functions between these categories have evolved naturally from ordinary mathematics and do not have the contrived look of the spearticicelli. The pathologies of the paradoxes. Thus, it might be hoped to find a way which gives them a more direct path.

5:00 When you hit that embedding of the category C, this is what I do, comparing the input of the category C of Pantus from the dual of C into the category of the ensemble, it will play a major role. As we know, that the familiarity of Comte poses some logical problems. To be on the safe side, they are reducing the assumptions of sometimes C is small, or we are not looking really at the familiarity of all sets, but at the sets of given, usually unspecified universe, U. Yet, everyone knows, and this is the basis of C's claim. As soon as U is big enough, the properties of this family, which he is called and faithful, do not depend on U, and the universe must be fully formed. The framework of universe adopted, say, in SGA is perfectly consistent, assuming that the But the remarks on the arena I'm getting show that it's not quite satisfactory, the frame of view, and does not reflect the way we work with Haydn's, which lays on the plane of thought, the sense for man technically. Of course, there are criticisms of this approach, but I would like to finish by going in the direction of musical analysis a little bit by the discussion between Thomas and Englert on this idea of a particular and specific intuition.

7:30 According to Kieferman, McLean's impersonal communication expressed the view that mathematicians are well known to have very different intuitions and these may be strongly affected by training. Kieferman's reply, I believe our experience demonstrates the psychological priority of the general concept of operational correction with respect to structural notions such as group category, etc. I realize that Burgess and Kenkel with you are so at home in their subject that they find it more natural to think in categorical rather than set theoretical terms, but I would liken this to not needing to hear once one has learned to compose music. Well, in my opinion, this is a difficult comparison, and I think that I had at the very beginning of this conference, a bit chaotic, I admit, I had this citation from Thomas Noll's review of your book, where he says, precisely, if the community accepts the use If we use mathematics, then it will have such and such effects. And I think that's the epistemological problem you're going to have to ask us, the question, to find out if our sense of math, technique, is really not communicable, communicable in the world without the knowledge of mathematics. To encourage people to try to understand the geometry of mathematics. So yes, I hope it will be useful for you. Thank you very much.

10:00 What I propose is that we first extend, since we have an hour, the subject of the method. I remind you that we will continue this afternoon with the exchange at IECAM, I remind you the part of three hours. If you don't mind, I propose that we focus on the more philosophical aspect now, knowing that, on the other hand, other aspects will naturally be extended at IECAM. I remind you that the formal name is Manu Phi here, whereas there is a Phi that is not... It's Manu X, the acronym. So, we don't accept the Phi. Manu X. Yes, so let's put that back. So, I have a general question, a bit general, before moving on to Charles, who will also... Why... Thank you for your attention. I don't know, I don't understand. Me too, I don't understand. What's the question? No, no, no, it's a new question. It's a new question. Ah, Thomas Knoll, yes, Knoll. Ah, sorry. Knoll, it's not... Sorry. I'm sorry. No, no, you deserve to be praised. It's important. Yes, yes, I... Thank you. So a general question on the subject. Why does a historian like you choose a philosophy like this one?

12:30 And where does it come from? In my opinion, it is very interesting to have this clarity in the decision of philosophical orientation. But the question is, why the choice of this philosophy? Is it adjusted to your desired object? There is a problem with this category, considering that in the end it would be the best adjusted. In particular, on the field of category theory, there is no such thing as a choice, but rather a plurality. And it's the same. So why this choice, this hierarchy, is it of interest to you? And secondly, we can see that it had one of the consequences. So this is part of the internal debate. They are the chiefs of our lectures, but they guide us very well into a problem of communication, but they guide us again into a problem of communication. Then, on hierarchization, on the point of the hierarchies, when I thought about Dockmann, on the idea of the absolute, he developed a lot of things that are very parallel to each other, that have an impact on the categories, so that's something that I want to talk about. At least, MacLean is also an author. I find that the way of thinking is at least very much linked to this problem of hyper-economy, hyper-economy, hyper-economy, hyper-economy, hyper-economy, hyper-economy, hyper-economy, hyper-economy, hyper-economy, hyper-economy, hyper-economy, hyper-economy, hyper-economy, hyper-economy, hyper-economy, hyper-economy, hyper-economy, hyper-economy, hyper-economy, hyper-economy, hyper-economy, hyper-economy, hyper-economy, hyper-economy, hyper-economy, hyper-economy, hyper-economy, hyper-economy, hyper-economy, hyper-economy, hyper-economy.

15:00 Thank you very much for being here, knowing that the party is gone, it's a pity, because it would have been interesting to see you with him, because, of course, it's the people of our country who are central, and we know that... I would like to ask you if you could give us a little bit of information about the theory of quantum mechanics. And then there was also the notion of intuition. Of course, we are talking about the theory of intuition. And there, in the note that I read in your book, which is very precise, there are things about the theory of quantum mechanics and the theory of quantum physics. In addition to these, there are many other fields, such as physics, geometry, algebra, mathematics, physics theory, algebra, algebra theory, algebra theory, algebra theory, algebra theory, algebra theory, algebra theory, algebra theory, algebra theory, algebra theory, algebra theory, algebra theory, algebra theory, algebra theory, algebra theory, algebra theory, algebra theory, There you go. And not only that, but there are many others. There's a few others. There's Bachelard, Weill. You can read about them if you want. It's simply to better understand, at the start, the I didn't take them apart on purpose, in the sense that I have this rebuttal. On the other hand, the concept of thematization is a theme that belongs to a certain stage.

17:30 The level of my work has changed a lot. Thank you for your attention. Yes, but that's another thing, it's not the whole thing. Well, whatever. For McLean, I'm young, I haven't read all three of them yet. I've read all three of them, but I still have a lot of things to read. It would be important to have a philosopher, because MacLean has written it recently, to give the minimum of knowledge of mathematics, a philosopher would be supposed to help you to develop the universe, to give you the minimum of knowledge of mathematics, to help you develop the universe, to give you the minimum of knowledge of mathematics, to help you develop the universe. I know he doesn't do that all the time in his book. No, no, it's a book. Yes, it's a book. Yes, it's a book. Yes, it's a book. Yes, it's a book. Yes, it's a book. Yes, it's a book. Yes, it's a book. Yes, it's a book. Yes, it's a book. Yes, it's a book. I have a question, two small remarks. The question is about structuralism. You talked about the structuralism of Bourbaki, but today people like Hellman, for example, when they talk about structuralism, mathematics, I don't talk about structuralism, a little more broadly, usually we start with Hilbert, with John Hilbert of Gundladen, of geometry.

20:00 And the question is, do you see the difference between, let's say, Hilbert structuralism, if you approve this idea that it is structuralism, with Bourbaki structuralism. And I think this question touches on another remark you made. You said that a structure is not necessarily a structure on a whole. Of course, we agree that it is not necessarily an ensemble in the sense of Cermé-Rofrenkel, etc., because it is restrictive. On the other hand, we can still talk about it as a class, even when we talk about Grundlagen and Hilbert, we can say that there are things of a certain type, and these things, at least we can think of them as ensembles, or as classes, ensembles in the broad sense, not in the... The axiomatic theory sense. That's the main question. And a little remark when you say levir. I think you're right. You said it, but it interests me a lot in this levir approach, the foundation, let's say, to put little bricks as a foundation, because he started with, how do you say, as a good student of Tarski, as a very classic approach in this sense, that is to say, he wrote the first theory. Yes, but if I'm not mistaken, he also attended the task seminar in Berkley, just the first year before he returned to Kalambe. He wrote the theory of the first order, a bit like the theory of Sermier-Lefrenkel, but instead of taking this relation of epsilon, he describes... And then he changed his mind a bit, I don't know if you saw his 2003 article, where he criticizes a bit his foundations, he says these speculative things about Russell, Frege, etc., etc., and in fact he says things that are a bit closer to your position, I think, because he talks about foundations in the sense... He makes a strong link between mathematics and education.

22:30 You already have a bit of the stuff on the left. I'm trying to join this foundation since 1960. Oh yeah? Oh yeah. It's interesting because you have a little bit of that. Yes, yes, I know. The last remark on external-internal, you mentioned that we describe objects in an external way in terms of categories, I just remind you that Bill LaVere is very against this point of view, he says, in fact, you know him well. But in fact, the questions really interest me. It's the first question. I was asking you about the relationship between, say, Bourbaki, Hilbert, and the notion of ensemble in this whole affair. Thank you. So, you also talked about the most recent study, Hellman. No, it's Hellman. When he writes, he starts with Hilbert. That's what it's called. I haven't read it yet. It's terrible, but well, not yet, of course, but when they say that we can get rid of these genres, it's not at all true, it's true, because they put things like, I'm not talking about a group, in general here I'm talking only about substitution groups, and so they ignore these terms, so it's a bit of a question, are we going to find something different? For the mathematicians, I want to make a real judgment on the subject. Now, Hilbert. I don't know exactly what you call, or what these people call, the structural, of course, the epistemologies, but of course, I have said in my conference that there is a difference in terms of your values. In Hilbert, the intuition intervenes in the metamathematics, in the methods used, in the methods used in the demonstration of the intuition, in contrast to the formal methods. In Gorbaki, intuition intervenes, of course, in the empirical facts of the language of the professor.

25:00 Now, the notion of the whole in Hilbert's work, I don't know any research on it, but it must have existed. Yes, there are many things. But even a more general class notion, because you're right that if we speak together in a sense... When I say together, I mean collection. Okay, okay. No, but I know that Bernays wrote an article on the concept of structure. I haven't read it yet, but it seems to have been done in adolescence. Just before geography, it was at the beginning of the 20th century, I think. He... I would like to read a remark by a professor of science, I think it is very important, because in the end it is something that is very important for us, because in the end, the approach of mathematical science is a community in the sense of First of all, I would like to say that I am an expert, and in the sense that my technique is one of those things that, at this moment, the world does not yet have a core, would probably allow us to have a future perspective, in a more easy way, on the applications of... This is an example of a whole community, not just the people who do the analysis, but the community of people who do the analysis. First of all, is there a community of music theorists who identify themselves in a mathematical perspective?

27:30 This is the core of our discussion. We are looking for an instructional perspective. In the United States, we have an organization that has just been established that has a lot of exchanges that we have had to try to resolve. I would even like to present here, it's more logical, it's a community tradition, exactly the same that we find in the community philosophy. The idea of a philosophical perspective as it is present, for us, it could really make a difference. Continental math, Chan has done well, he has a lot of Pascal. I don't know if we can link all of this in one sentence, but I don't think we can agree on that, because if we put, for example, the philosophy of quantum physics as a possible approach to the fundamental of quantum mechanics, it would be very difficult for France to put a dialogue with a mathematical perspective like the one of Newton. In other words, I don't think that our differences come initially from the different philosophies of the world. I think, let's say, our work space is basically what place mathematics can have in a theory of music, in a theory of music, knowing that it is a theory of today, of today's music, and not necessarily of today's music. In my opinion, there are three completely different ways to do it, to create a community, which are the three ways to use and to invoke mathematics, because there is no agreement on whether to use mathematics or not.

30:00 So there is a way, let's say, that is rather the American way of music theory which, in my opinion, is of the order of the musicological application, I would say. We take great words from the archives, from the musicological approaches, especially musical and musical. There is another approach that is rather the one initiated by Rimeau and in which a lot of people are involved here, which is rather, let's say, theorizing the materiality of music. In my opinion, the most mathematical and musical objectives are the proofs of mathematics' ability to confront itself with this theoretical and musical matter. And then a third logic, which is a little different, which is, I don't want to go into too much detail, the one of reasoning and mathematics, which is no longer in the types of applications. So a third orientation, let's say, which is perhaps less centered on mathematics in general than on works, on mathematics, on the antitheses of mathematics. I don't know, I don't know. There are three. In my opinion, three quite different ways that try to enter into the framework of our activities, which is absolutely fictitious. They do not have the same spontaneous philosophies, because philosophies are more or less adjusted to each of these enterprises, but we can see that these enterprises have different subjective interests, not philosophical interests, subjective interests, but they have a philosophical attachment, but let's say, roughly, I would say, between a musicologist subjectivity, a rather mathematical subjectivity, and a subjectivity rather of working musicians or musicians. There are a few different philosophies, and we work in this space. So, in my opinion, there is no community between these three than subjectivities. But, on the other hand, there is precisely... I don't know. I don't know. There is strongly... It's just to present you a little bit in which phase you are in and we will say, let's say here at ENL, well, we put rather the emphasis on philosophy, it's less musicological, less, I don't know, an opening maybe of the thing, or a variety of the thing, while the quality is perhaps more centered on the first two modalities, one could say, I don't want to offend anyone. It's just a little conclusion.

32:30 So, the question is, what does it mean to be sceptic on the basis of knowing if sceptics are sceptics and if sceptics diminish the cross-references between sceptics and the cross-references between sceptics and the cross-references between sceptics and the cross-references between sceptics? Thank you very much for your time, and we hope to see you again in the future. The question is, what is the positive and the negative side of each of these three terms, and what is the positive side of each of these three terms, and what is the negative side of each of these three terms, and what is the positive side of each of these three terms,