Philosophy of category theory

0:00 It is important to note that this is a very rare period of time, taking into account all the measures of the magnetic and musical unity of life, and that life in this world is very remarkable. I don't want to take up too much of your time, but I'd like to ask you to tell us a little bit about the last few years. This is a magnetic day, a day that has been prolonged for a certain period of time. Finally, a seminar on mathematics and physics related to other disciplines focused on the challenges posed by the two fields of mathematics and physics in general. And so we will have an exhibition on the structure of the universe and the concept of the language of the Ionida in music. And we will also accept to resume, as we have done in the past, a musical perspective after the introduction of the Ionida. It will also present its view on the reality of the world around us. Perhaps one thing I would like to say is that this is the idea of a community of young people, not a community of myths. In the months of January, Thomas Nolte will come in the morning and he will also spend time in the committee, doing the reflections, and during the day, he will talk about the conditions, with which he will be able to come to an agreement. I advise you to write down your agenda, because it is exceptional. You will see these two researchers. Thank you for your introduction and for the invitation. I am very happy to be here today and I would like to thank everyone who is interested in this topic. Precisely, before the invitation that I would like to send you in a few minutes, I did not know that there was a municipal thesis with the means of mathematical theory.

2:30 I am first a historian of mathematics who is also interested in the philosophy of mathematics and I have written a thesis on the theory of math. These are the topics that have been brought up during this last historical emergence and which have had an important influence on the historical development of this field. I have a little problem with my computer that I will try to resolve quickly, but I can still start with you. It's normal that you see the computer because I have two files. So, yes, it's a little bit of text. And I wanted to say that... I have a few remarks to make an apological mention of nature, a little apological, so that you do not expect too much of this intervention for your purposes. So, I have already said that I am not at all, I was not at all aware of the existence of these activities, Thank you very much for having me here. I hope to learn a lot from you today. I am not really an expert on the category theory as it is practiced today. I have only written a thesis on history and the period I am interested in. It ends in 1970, with the departure of Brotelvic from the world of mathematics, but not a computer, because since it was plugged in there, it no longer finds its normal mode of...

5:00 It's there, isn't it? Yes, yes, it's there. No, no, that's good. That's there. I think so. So... Perhaps there are some of you who are also interested in the theory of categories, and I would like to make a remark. I'm going to talk more about his philosophy today, and although the historical information, maybe 80% of this book, there are still 20% on philosophy, and that's what I'm trying to talk about today, because I see there maybe the greatest interest for the group. Of course, my interpretation of the type that I will present will feed the historical study, especially based on a diverse discussion that has been conducted on the problems of the technological sphere, which is what I had already said earlier. So, I consider us... With my contribution today as already a possible response, I will elaborate on the original question that you suggested to me, Moreno and Bernarda, to know what is the philosophy of mathematical methods in musical analysis. This is the question you asked me in the first email. Rather, I see the situation roughly as follows. I will explain my answer to the following question. What is the philosophy of category theory? And we will see together if this answer will also answer another question, which is not excluded, because you will see that it is even a thing that I have not shown to feed, I will not show to feed this discussion of possible elements that I say, naively, the reviews of the Topos of Music, for example, and other texts for, because I did not have the time to read them. Otherwise, I still tried to hold on a little longer. And if I had, if I had spoken well, I would not have understood what was possible, but I took the advantage to explain it to myself and to explain it to myself in a good way.

7:30 So ... That's what I was. Yes, but we have to connect. Yes, of course, I had tried because it was not good when it was connected. Thank you for your attention. Thank you for your attention. Thank you very much for your attention and I hope to see you again in the next lecture.

10:00 And here I'm not going to give you an answer, I'm going to ask you to discuss this with me. I would like to know if we are looking for a justification for this practice of applying mathematical methods in a physical way, a methodological method, a reflection on pertinence, or something else. I would like to know. And then, of course, the more precise question is, what are the categorical methods? I have a question to which I can perhaps answer more easily, that is, is my philosophy of the category theory appropriate to the philosophy of the categorical and analytical methods? So, the preliminary question is the one that I will certainly answer. I am certain. What is my philosophy of the category theory? So, on the subject of what I tried to draw from what I had in my material on the first question, I quote in a passage of the review by Thomas Knoll of Thomas Knoll, It is me who stressed, in this poem, perhaps the passages in Italian, that it is difficult to communicate metaphysical knowledge by means of clear conceptual rules. Apparently, this is what we are looking for in the mathematics of the world. And, I'm sorry, at the end, Noll cites an example and says, Eligibility of mathematical investigations into music in general.

12:30 My question to myself is that I have tried to answer the question of whether theory is a clear conceptual universe, because if we hope to find a solution to this problem, then we have to find a solution to this conceptual universe. I continue. In fact, Thomas Mulligan, insights into music can be convincingly communicated through mathematics, and the underlying mathematical facts cannot effectively be paraphrased in a non-mathematical language. Discussion of the deepening of category theory, how to translate the intuition of mathematicians, who have the impression that they are, in spite of apparent problems, in appearance, mathematicians. There are often people who do not have this mathematical knowledge, this knowledge. And here, the more concrete version of the dictation of Noam is... He clearly says to the virtual reader of this book that one must have knowledge of mathematics, or at least a general education in mathematics. But he ends with the following sentence, but the more it becomes accepted, and here I have read it, that insights into music can be completely communicated through mathematics, and that the underlying mathematical facts cannot effectively be paraphrased in non-mathematical language,

15:00 the easier it becomes for the community to digest the whole diagram. It's interesting because he explicitly says that this community of theorists is not at all open to this approach. And precisely, in my opinion, it is because they lack mathematical knowledge. I will show you how, in my book, I have tried to serve these differences of knowledge, of intuition. I already give you a term that will come back in the future, it is the practical commensal. It is a technical common sense, which is not common to all of us, but common to those among us. I'm going to briefly introduce you to the structures, first of all, I have something to say, and then I'll introduce you quickly as a teacher. It is on purpose that I speak first on geography, and then I present what I do very briefly on the general object, and the large simple section will be what we will do most of the time on this project. I try to present you, to present you well, but as briefly as possible on my approach. All of this was first applied to a general epistemological position, perhaps applied to mathematics, perhaps to something else, and it was also applied to geometry, but it was inspired by the fundamental problems of geometry. Gerhard Reitzmann, who was the director of the Archive in Nancy, was the one who told me about it.

17:30 The set is rather the place of discussion, not at all, that I will present to you. So, structure. This word structure, we will find it in Moreno, in his essay, and he talked a lot about different structuralism, if you will, structuralism of the particle, structuralism of the piaget. Obviously, if I come to the current, we are going to have an analysis of the structures for the pièce de Gaulais. So here, in the title of the piece, the word structure appears. We are going to take a look at it. And again, I tried to understand a little what are the central structures here by the non-written reviews that Matsula exploits, especially categories of set-valued punctures for description and investigation of musical structures. I do research on musical structures. And the other reason, which is also clear to me, is that there is a mathematical basis of musical structures, it's operationalized as categorical semantics for the problems we've got in Haftel. There is not only monoliths, there is logic, there is theory. It is interesting that the category theory also intervenes in the theoretical level of mathematics in the semantics of the program. In the beginning, I tried to say that I knew very little about mathematical theories such as the genealogy of the Serpent Valley, or the genealogy of the 60s, I don't know anything at all about the application of programs in semantics, but it's not a big deal, I hope!

20:00 I'm not going to try to give an answer now, what is a musical structure, I'd rather like to look at the answer from both sides, what is a mathematical structure, that's the question. It's interesting... Moreno also told me that you have a tendency to see your problem in the eyes of the phenomenology of the Serbs. But it seems to me that this is not the result of the reading of your book, in which the references to Gorbaki are much more numerous, which is the reason, but if the references to Gorbaki in my book are numerous, it is not at all because I agree with Gorbaki. And because I try to have a position in relation to that of Bourbaki, that is to say the technique of which is the Bourbaki, of which is a Bourbaki, and so on. So, for me, the philosophical position that I develop myself ... The book is mostly inspired by Peirce, Poincaré and Wittgenstein, and a certain degree of pragmatism as well. But before I present it, I would like to briefly present my view of Bourbaki. So, does Bourbaki have a specific position? And this is a question to be asked, because he has asked me several questions at the Quarry, and there is not even much I would like to know about it.

22:30 But I present to you first the main reasons for... There are very clear affirmations, I'm talking about this here, in hypopathic texts, by the text of one of the segments. There is the introduction to the book of Ensemble, which seems to indicate the difference between the two. Mathematical architecture, Sinigour-Watting, in a volume produced in 1948 under the direction of François Le Lyonnais. I don't know who it is. The Grand Courant de l'Ambassadeur. The Grand Courant, exactly. Then there is the conference, Foundations of Mathematics for the Work in the Petition, also Sinigour-Watting. Well, some articles, including Léonie Cartan, concern the problem of mathematics, which I haven't had the chance to see, but are in what is called the Rosé Review, and it's not easy to read. So, that's it. I'm going to say a few things at the end, and I'm going to take a little bit of your time, but there are still some points that we need to take into account. Do you really take the problems seriously? Because it's pronounced, the theoretical problems, and you don't automatically say that in this field, in mathematics, you respect them. In fact, do we really consider this as a solution? If we agree that such and such is the solution of the problem, why not necessarily do it again? Then, do we show the ethics of the position? Are the different ingredients coherent?

25:00 First, we have... What we could call the hypothetical position of the deductive of property. And I give you two quotes, the first on the screen, the second is longer, you will see it later. So, the subject of the following problem, Hilbert had the idea of ​​demonstrating the non-contradiction of mathematics. And Goethe has demonstrated that it does not demonstrate itself. And what was Bourbaki's reaction to this? Because when we don't have a demonstration of our concentration, we can perhaps try to persuade ourselves in a different way. And Bayl was interested in his conference at the level of Bourbaki. The absence of contradictions in mathematics as a whole or in any given branch appears as an empirical fact rather than as a metaphorical principle. The more a given branch has been developed, the less likely it becomes that contradictions may be affected in its further development. Et j'ai envie de vous lire le passage correspondant de l'introduction de l'intelligence sur les ensembles. We are still a little far from the center of all this, but I still have quite a lot of time, so I can do that. So, Obaki asks the question, what happens if we have achieved a momentarily contradiction? So, can we acquire the certainty that this will never happen? Observing that the meta-mathematical can be proposed to examine the problems of non-contradiction by its own method, the formal demonstration. A theory that is non-contradictory is in fact to say that it contains a correct formal demonstration leading to the conclusion of zero. However, metamathematics can be sought by a method of reasoning bound to mathematics to understand the structure of the formalized text supposed to be written in order to finally demonstrate the impossibility of an academic lecture.

27:30 In fact, we have given such demonstrations for some partial formalized languages, not as rich as the one we propose to introduce, but rather rich enough to be able to write a good part of mathematics. We can ask ourselves if this is true. What we have shown is true, because if mathematics had been more controversial, some of its applications in the natural world, such as mathematics or geometry, would have been more popular. It would be necessary to escape this dilemma, where the non-contradiction of a formalized language could be demonstrated by formalized reasons in a less rich language, and therefore carry a line of evidence, except for a student theorem. On the contrary, in the demonstrations of relative language contradictions, i.e. those that establish the language contradiction of theory by supposing another theory, for example that of quantum mechanics, the mathematical part of reasoning is so simple that it does not seem at all possible to put it to the end without announcing at all the rational laws of our practice. We will see later that this is obviously written before we ask ourselves the question of how to delegate them to each other. Since the universe of mathematical theory is now logically attached to the theory of the ensemble, it is then that any contradictions encountered in one of these theories would give rise to contradictions in the theory of the ensemble themselves. This is obviously not an argument that we can use in the interpretation of the theory of the ensemble. However, for 40 years, we have formulated the actions of this theory with enough precision. And when applied to draw the consequences from the most varied fields, mathematics, we have never encountered a contradiction. It is based on hope that it will never happen. What we were hoping for is that the observed contradiction would be inherent to the principles even that we put at the base of the theory of thought. We talk about the idea that the theory of the ensemble is counterproductive because we have not yet encountered contradictions and we are interested in a demonstration of relative consistency, of non-contradictory consistency.

30:00 Now, in terms of the categories, you know, what we did, what Grossetnik did for some reasons, was to introduce the concept of the universe. There is a wide range of concepts in the world. In the universe, there is a set of components. If you have two sets of components, x and y belong to each other. x and y belong to each other, so a set formed by two belongs to each other. If a set belongs to U, then its set of parts also belongs to U, and finally, if you have a set of an X that belongs to E, an X that belongs to O, then the union of all the elements, of all the parts of the X belongs to A. In other words, it is intended to have direct products like this. So the axiom is that everything belongs to such and such a universe. And this is also equivalent to another axiom on cardinals that are highly inaccessible. I will not present it here. And there is now a result. Mathematical myth, but which does not at all fit into what Beauvois said, which is that it is so simple that it seems impossible to put it into a relative term, this non-competitive term. There is therefore a process to be done to see that this is not relatively consistent with Hermelot-Prinkel.

32:30 Just by supposing the amount of the addiction of the brain in the brain, or if it is a source of the amount of the addiction of the brain in the brain, plus its action, it is independent. We cannot demonstrate the amount of the addiction of the brain in the brain. You may not have heard of this example, but I will explain it a little later. Paul Cohen, when he collaborates with the SGA, the Seminar on Mathematics and Mathematics, But at the time, Oubre-Boitier argued on adopting the category theory for the elements of mathematics. They could have said no, we adopt it because it is not in accordance with our position. We do not want to be stronger than that. We do not want to be stronger without demonstration of the arguments that exist. In reality, they may be of different order, but for me, it's a bit of a question of whether or not they take this philosophical solution seriously. So... The bourgeoisie has never been... Yes, but... I don't know. I don't know. I don't know. I don't know. Yes, okay. How can I say it? It's a matter of terms. Thank you. Now, the second remark is about what we call structuralism. Structuralism means mathematics. It's the science of structures.

35:00 The objects of mathematics are structures, and we can't do without them. If we place ourselves here at the point of naivety and do not address the debilitating questions, neither philosophical nor mathematical, according to the problems of nature, the verses are a material object. In this new conception, mathematical structures become, frankly speaking, the only objects of mathematics, as we can see in the two articles. The concept of the whole, but define the concept of structure. For Bakhti, we give a mathematical definition, but I ask myself a lot the question of the relevance of the formal definition of the relationship between the intended model and the theory of which it is based. And I don't think that this definition really manages to define well what mathematicians call a structure. It may be a definition of what mathematicians call a structure on a set, but we don't have to talk about a set of mathematical terms. I find this very interesting. This is the most elaborate version of the 4th chapter of the book, which was published in 1957, and I'm not going to talk about this one because it's too long, but I'll call it.

37:30 Yes, but the pretensions are the same, in fact. Yes, that's it. Yes, that's it. Here, it goes through the examples. Here, it goes through the 1 to n, very generally. So, you have, for example, three different sets of FDFG, and they say that by... We can form others by taking their ensembles, by using the product of one of these ensembles, and we have 12 new ensembles. On the joints of the three ensembles, we can start over on these 15 ensembles of the same operations. In general, we say that the ingredient of the ensemble obtained by this process is part of the scale of the ensembles behind or below the objects. There are three sets of this scale, R, X, Y, Z, a relation between the elements of each of these sets. R is defined as a part of an element of a set of parts. He gave a certain number of elements of a set of a scale, a relation between elements of these sets, the application of parts of certain elements. In the last slide, we come back to the idea of a single element of one of the sets of scales, because the pre-2 is still in a scale. In a general way, considering a set of m of a scale, whose base is, for example, 0.3% of Rg, we have a certain number of probabilities. All of these are precisely balanced in an element of M, and T is the intersection of the M-parts defined by its properties. It is said that an element sigma of T is defined as being a structure of the species T. For example, we could see the structure of O, which is the whole of the chain of O, and the relationship that obviously supports the action. That's it. My criticism, well, not my criticism, but a criticism, is that there is a very strong distinction between the two.

40:00 The concept of an algebraic geometry is not only consistent with what he claimed to be the development of the motion in his head, but more importantly, the concept of an algebraic geometry is not only consistent with what he claimed to be the development of the motion in his head, but more importantly, the concept of an algebraic geometry is not only consistent with what he claimed to be the development of the motion in his head, but more importantly, the concept of an algebraic geometry is not only consistent with what he claimed to be the development of the motion in his head, but more importantly, the concept of an algebraic geometry is not only consistent with what he claimed to be the development of the motion in his head, but more importantly, the concept of an algebraic geometry is not only consistent with what he claimed to be the development of the motion in his head, but more importantly, the concept of an algebraic geometry is not only consistent with what he claimed to be the development of the motion in his head, but more importantly, the concept of an algebraic geometry is not only consistent with what We will soon be able to do this without always going back to a set of subsections, because if you take the theory of variability, for example, you forget the schemas that are maintained by the literature without having to repeat the schemas that are maintained by the space of topology, geometry, and physics. I think that the word structure in mathematics is not necessarily formative. I have the feeling that it is rather a game of language in the sense of Wittgenstein. When we ask mathematicians in training what a structure is, using practical examples, they will at some point be able to talk about it reasonably without having any evidence. So, before we do the presentation... And finally, what I say in the book, it is a historical and philosophical study of the theory of mathematics. This theory has not only led to important mathematical applications, but also led to philosophical debates on this new historical-philosophical approach. The theory has been introduced into algebraic topology, a mathematical discipline that, going back to Poincaré, implements algebraic objects such as cohomologies, in the study of topological spaces. In the context of Anne d'Arméglès, who was the one who introduced topology, it was a question of studying the operation of applying a space on another along a continuous function.

42:30 And the effects of this operation on topological groups. Here, the theory of categories is used especially to express these tests that all have quite high results. I think this is something descriptive of the solution of the categories. It is good to be able to explain, but the results could have been better formulated. In return, the use of the theory is more decisive than in two other decisions. The homological algebra and the algebraic algebra, in particular through the components of the homological algebra, is used as the topological algebra of cohomology, although associated this time with an algebraic object rather than a topological space. You can study the cohomological modules of an algebra. In 1956, Henri Cartan and Samuel Hall were given the general procedure to build such a group in a module category. Brodelic made this procedure applicable to a more general category of categories, a category of characteristics only in the category of categories with generators. Here, the theory of category is used to deduce, not only to estimate, because Rotterdijk relies on it in his demonstrations. This success is then deduced internally by the theory of category as a subject of research, unlike finance, which follows the first terms of the law. Thus, it was possible to make an essential progress in the treatment of certain problems in algebraic geometry, the ideas that deal with algebraic geometry in algebraic mathematics. However, algebraic geometry maintains intimate links with the theory of numbers, which is one of the most common problems in mathematics, as well as the last theoretical one. The demonstration of these theorems, completed in 1913 in Hawaii, is the result of the concept developed by the Institute of Laboratories in the 1960s. Here, the theory of the category, then, is to build objects and presents itself, in a certain way, as an alternative to the theory of the whole,

45:00 We can't talk about all these successes without talking about the many debates that have arisen over the history of the categories, and in this book we can't ignore the historical descriptions of these debates that are new to us. There is still a lot to talk about. First of all, there was a debate in your country, we talked about it earlier. We also realized that the application of the tool in the state of the world leads to problems concerning the realization of the surface. This is a debate that took place in your country. Is the collection of objects in the prism, is it a set of classes or something else? It is precisely this clarification that causes the structural problem. By the way, it's a big problem to move historically, I have a problem, there were no applications of exercise operations on categories according to the So, their problem is that they are influenced by a distinction between the classes of the students in the class of the students in the class of the students in the class of the students. In the course of the lecture, we discussed the problems of the tendency to change the concept of taking mathematics better, and in particular, our work to consider the theory of mathematics itself as a fundamental of mathematics.