Philosophy of Mathematics Discussion Group — Marco Panza's Paper

0:00 I think that the legitimacy of this paper, especially in relation to the people who are in question, who are interested in mathematics, and all the themes that I have been talking about, that is to say, my impression was that there was a demand for discussion around the themes of Jean-Michel Fleury, Ivan Palli, and Jean-Jacques Palli. And I think that... Thank you for your attention.

2:30 Thank you very much for your time. Well, I hate to say it, but Marco should reflect that Newton started giving the Lectiones Arithmeticae to a room which consisted of one person and a dog, and I think by the third lecture he was speaking to a completely empty room, presumably from somebody who was taking the notes. So, he's in very good company. And I was once at a seminar in Florence, given by Lord Fear, which I think was probably one of the most astonishing talks I'd ever listened to, on Midsummer's Day in 1998, and there were precisely three people in the room. Quality audience!

5:00 It's not a problem, it's not obvious, it's a problem, it's a problem. If you had written my text, it wouldn't be there, the volumes would be there. That's it, yes. No, no, but... Yes, yes, you're right. But are there other copies? Is there a copy here? Uh, why? Do you have a text? I don't have a text. I can share it with you. Oh, okay. It's easier to share. I'll get a copy from Mark afterwards. Well, that's what I was wondering. Thank you for watching this video.

7:30 So, I repeat, what I said, the idea was not so much to have a discussion on my paper, which I currently send to the Mathematical Society of the University of Paris. It was a pretext. What I wanted was to put on the occasion of a periodical meeting, which is a question of the year, to give means to a group, for example, the UAS. The idea is that, before discussing these issues, we have to make a structured seminar to be able to follow our will. It's not a seminar to talk about Paris, but to talk about other people's ideas. We have to have an effective evidence to adhere to. So the idea is to test this awareness and then evaluate the possibility and propose discussions.

10:00 Thank you for your attention and see you in the next lecture. I came here before 11 o'clock, and so I had to put the bikes around at any time. And already 11 o'clock for him, it's... I didn't think it would help at all, but... Already 11 o'clock for him, it was the limit. No, the last time I went, I think it was a... it was a... it was quite a moment. I also had to go, it was 11 o'clock, and the other time it was not all that interesting, I had to get up at 11 o'clock, I had to get up at 1 o'clock, I had to get up at 1 o'clock. Of course, I will not expose you to the totality of the things contained in this article. For now, I will try, of course, to answer certain themes, certain arguments, and in general, the inspiration behind it. The idea is, in fact, to say in practice the following, is that there is an essential difference Between the two answers that have been given, is it true that the answer given by the nobilities is not the same as the nobilities of the idea that the...

12:30 But the idea of mathematics is that you have to convey, count, and so the idea is, is there a way in which the study of natural numbers, is it something more that conveys thoughts? Well, also, as he said, it is a way of inducing, how do we do, in fact, the idea of the frontier, how do we cross the frontier, what is the composition of the world here? The good proposition is to cross the barrier, I'm trying to say that it is not necessary to cross this barrier to give an answer to the question of the question of the question of the question of the question of the question of the question of the question of the question of the question of the question of the question of the question of the question of the question Therefore, it is necessary to explain what the language of Benacerate is, or it is possible to explain it. So there is something that is also the original response of the article, which is the reinterpretation of the language of Benacerate in the form of a general reference,

15:00 the general history of the literary partisan. So what is the language of Benacerate? The idea of the language of Benacerate is as follows. The idea is, when we do mathematical physics, we have two constraints, one of which is to give a semantic value that I never knew how to call it, and the other one is to give a semantic value that I never knew how to call it, The idea is that to see the semantics of the assertions of the semantics and to see the physics of the semantics, we have to use the assistance of the assertions of the semantics. If true, it is necessary to explain the existence of a mathematical perception, that is to say, an art of mathematical knowledge. So, the idea of the Seraphim is that these two concepts do not exist. Why do they not exist? Taking the cases of the men who left the original version of the dilemma to be presented, There is also an introduction to the subject. What is the subject of the lecture? The subject of the lecture is very simple. The subject of the lecture is very simple. The subject of the lecture is very simple. The subject of the lecture is very simple. The subject of the lecture is very simple. The subject of the lecture is very simple. The subject of the lecture is very simple. The subject of the lecture is very simple. The subject of the lecture is very simple.

17:30 The subject of the lecture is very simple. The subject of the lecture is very simple. The subject of the lecture is very simple. So, the idea is that if we want to have a schematic that is similar to the schematic that we have in the natural sciences, we must first guarantee that the conditions of the physics of the independent sciences, the conditions of the natural sciences, are the same as those of the physics of the natural sciences. Why? It is not because we often say that the physics of the natural sciences is the same as the physics of the natural sciences. It is true. So, if we want the conditions of the physics of the natural sciences to be the same as those of the physics of the natural sciences, we should be able to do so. So, we have to be careful with our students because we want them to be able to discover a semantics that is a natural pronunciation of the semantics in an independent domain that is structured in such a way that our students will be able to understand it. On the other hand, if we do that, it is difficult to know how to access it. A knowledge is a real, justifiable opinion, and this justification is based on an access that makes it possible to understand it. It is not simply to have knowledge that makes it possible to understand it. It is not simply to have knowledge that makes it possible to understand it. It is not simply to have knowledge that makes it possible to understand it. It is not simply to have knowledge that makes it possible to understand it. For those of you who are interested in learning more about Evalu.it, the way in which it is structured in a domain of interest, how I can access it, how I can have access to it, if I have the opportunity, if I do not have the opportunity, the way in which it is structured, I can tell you, is the way of education. I know that Evalu.it is a genealogy of science. It is obvious that in the case of mathematics, the definition is not linked to the idea of science.

20:00 So, this is a good example of a good example of a good example of a good example of a good example of a good example of a good example of a good example of a good example All of these terms may be related to the question of the epistemology of mathematics in relation to which we are able to recognize that the things that we are searching for in mathematics are in place or not. So, I think that the same dilemma is presented in general, not only in relation to mathematics, but more in general. In general, we can talk about the same type of answers because the answer to the question of an interest is always to be something that implies, that contains as a condition, part and parcel, a characterization of an interest, a definition of an interest. The answer seems to be clear. It is possible that the numbers are not exactly the same. We can have a certain density in the case of the mathematical equation of the game, of the design of the game, which is a number. We cannot answer this question, because it is possible that the numbers are not exactly the same.

22:30 For example, it is possible that the general logic is not exactly the same. So, we must say that under what conditions is the mathematical equation of the game? So, the answer to the question is that the answer is a general consequence of the real material, a mathematical question, perhaps, and that... So, the criticism is that... It can be a new version of the question. So, the answer is that I don't know how to answer it. So, the answer is that I don't know how to answer it.

25:00 They give answers, but physics is not only about that. Because these answers are not feasible. In both cases, there is a difficulty in the fact that they give the answer. And the concept is this. In the first article of the Congress on the Derivatives of Logistics in General, my first... The logical answer, I don't know if I should also consider it, I don't know if it's already there, the data is not there. Maybe I will review it. First, a first thesis is... Why do the dilemmas of Benaferrat seem to be in a good place? In general, we interpret, we have to give a schematic and we have to give, to give a schematic,

27:30 we have to suppose the existence of a quantum system, to give a schematic or a systematic system, the access. And this thing seems to be contradictory. So what is the response strategy? The way we interpret it is defined, not in relation to their existence. These are all referents of family terms within an assertion that is accepted. Now, this response, this response strategy is also that of being an object. To be able to make it reliable, we must first say what it is.

30:00 Obviously, the term family terms, family terms, family terms, are not referents. What we must have is an assertion of the logic of the term. In addition to these things, we manage to show that all of our ideas are related to the national axiom of the concept. For example, for the relation between concepts, for example, for the territory of a complex system, for the relations between a complex system, if we have a logic of an order with two action patterns, the number of a concept is equal to the number of our concepts, and so we introduce, thanks to this, we introduce a new language that operates on concepts, the way of representing a continuum. Well, that's the general form, we could give a particular form to this.

32:30 The 19th part, which is the 6th formula. Excuse me, the last part of the formula, which is the 8th, etc. For each P, for each P, the number of P, which is equal to the number of P, the number of P, defines, in my opinion, a class. It defines this concept and then produces an object.

35:00 And we can use the notion of an asset of a relationship to define it. At this moment, we don't have only the concept, but the concept itself. All the objects that represent the condition of being in the ancestral relationship of the detector condition with the zero-determinism of the zero-determinism of the general conception form a sub-assembly of the elements that are related to the millennial performance of the language of science. These are the terms that I am going to use because they are very important for the idea. The criticism that is made by non-physicists to the criticism is very hard. When you say that objects are characterized, for example, in the natural world,

37:30 In this way, you do not do anything else than to explain the subject necessarily explained in mathematics. Me, we, in our definition, the number of concepts is increasing. The concepts that are denoted by the term, let's say quickly, for example, language, the pronunciation of the predicate, the product of the north, let's say, of the north. So the concepts that are denoted by the predicate, that is, the objects that are denoted by the predicate. When we are talking about the subject of the lecture, we are referring to the subject of the lecture, which is the subject of the lecture, which is the subject of the lecture, which is the subject of the lecture, which is the subject

40:00 This structure is coherent, which leads to a very momentous distinction, and is enabled to find a compelling argument for it, in Shapiro's discussion. What advantage does it provide for the simplest possible conservative response to the nature of dilemma? The coherent characterization at once communicates the nature of the structure and gives an example, and this is an example, The answers to these questions are as follows. The answers to these questions are as follows. The answers to these questions are as follows. The answers to these questions are as follows. The answers to these questions are as follows. The answers to these questions are as follows. The answers to these questions are as follows. The answers to these questions are as follows. The answers to these questions are as follows. The answers to these questions are as follows. The answers to these questions are as follows. The answers to these questions are as follows. The answers to these questions are as follows. The answers to these questions are as follows. I will give you at the beginning of the page that we have a mathematical expression that follows the A and the P, in the case of an M.

42:30 This means that the conditions of the A are in certain conditions C. And the terms A, prefer, prefer, prefer, prefer, prefer, prefer, prefer, prefer, prefer, prefer, prefer, prefer. I think that this answer is the key to the conclusion that we have observed during the lecture and that the truth is there.

45:00 The question is whether we can give answers in other ways. That is, whether we can give answers if, in fact, we take into account what we want. We can give answers if we do not take into account the differences. Simply, these are not answers. These answers may not be very interesting. If we had a good mathematical theory, it would be a coincidence, because apparently it is a coincidence that we interpret it. In fact, we have these answers. But in fact, there is more, there is more than one answer, not only one answer. In this general answer that you give, do you have a fixed reference, that is to say, in the case of this general answer, do you have the number of these objects of the necklace or is it anything that verifies certain predictions?

47:30 What is the general answer? But when you say now, we don't need structural or structural to answer the question, etc. We have this answer that goes in fact for both, to analyze the logic of physics. But in this general answer, is there a fixed reference for the numbers? Or do you have only a description? In this general answer, I accept the idea that I called the article the reality. And that's how I saw it. The definition of the definition of an academic lecture before fixing the definition of the definition of the definition before fixing the definition of the definition of the definition of the definition of the definition of the definition of the definition of the definition If it refers, it is an object. So it is an object that simply refers. The reference refers to a reference. So, at this point, obviously, I have not created the object, I have created this one. It is simply that, in the course of speech, the problem of organization is that we are not the ones who analyze. No, no, no, no. I don't want to continue with the concerts because I want to go quickly now. I just want to say that I don't want to talk about the various arguments that you have made, but I don't want to talk about them for another six minutes.

50:00 I see three possible answers that should be given and I continue these three answers to be successful. The first answer that I see, in fact I do not know if the first one is already going, the first answer is, the answer works like this. The Frege theorem, the Frege theorem is the theorem that says that we can derive... This is a revision of the definition of natural rules of space. So the theory of the rules shows that natural numbers are not limited by individual constants, but by the actions of the actors. The Escher theory is a theory of concepts. The modernist predicatives, the constant monadic predicatives of concepts, the natural numbers can be identified as concepts.

52:30 These concepts are causal. Natural numbers are real objects. So why is a natural human being obligated? Because he is obligated, because we have guaranteed, by being logisticians, the fact that we do not want to go into detail, the fact that we can listen to mathematics essentially, and this response has some chances of being comprehensive to the extent in which we have a conception of what is a concept, therefore a predicate of logic. So, we have moved the problems of the perception of the idea that one concept is nothing else than being what is denoted by an individual point of view, but not by an individual point of view.

55:00 This is the first answer. The second answer is that we have reduced the uniformity of the distinction between two concepts. So, first of all, it is a very complicated problem to be able to guarantee the accuracy of the analysis, and even if we have guaranteed that, it is not... Why would the analysis guarantee the fact that it would be an object, not a specific object, but an essential object? In fact, the only condition for which we could say that the analysis of the scientific analysis guarantees the fact that our interest is real, The independent is non-stipulant. It does not depend on the non-stipulant. It is independent of the non-stipulant. So the fact that the genetics derives from the principle for which it responds,

57:30 the number of A is equal to the number of T, is equal to the number of Q, which is also guaranteed by the public, is the most probable. It is because there is only one number, and the number of T is independent. So the validity of the statement depends on the conditions of the independent of the genetics. Even if the idea of truth is simply reduced to a parameter, we guarantee a certain idea of the goal of the objective, because it is independent. First of all, there is indeed a concept in which the numerology is independent of the logic, and nothing comes from it. To say that it is independent, absolutely. To say that a mathematical theory, some actions, some theorems of a mathematical theory,

1:00:00 because there are logical conditions, Second, independent facts are the facts of the concepts, the great part of the relevant concepts for which, specifically, the mathematical logic. So that's what I don't say that there is a reason for the principle, because I could not answer that question, but I do not believe in the literature.

1:02:30 The very last sentence of an allusion to physics is the way in which the... The three questions are based on the observation that, in fact, mathematics, as well as natural mathematics, inverts the relationship between truth and falsehood. We will say that we have truth because we have truth, and not that we have truth because we have falsehood. Apparently, if we look at the status of the reality of the reference of letters in the case of mathematics, we can still often have a presentation of natural elements in the case of mathematics. That's why it's important that, yes, because our self-discourse, in which our distinction is marked, yes, because our truth with mathematics, in fact, precedes...

1:05:00 All of these are directly related to the dynamics of physics and therefore the rigidity of the dynamics of physics is what imposes the value and what confers the rigidity of quantum mechanics. At this point, in this argument, the conceptual distinction of objects begins to fluctuate and to be limited to the conceptual distinction of objects at the same time and the constraint that imposes the value. They all fit into one analytical discourse that was written, so it's still a true theory, despite the inversions. And finally, at the same time, as I said, our way of asking the French people is to give them the truth of the subject, because all the truths that the French people tell us, in fact, are truths in all aspects. Behind it, it's just an adagio... Not an adagio, it's an adagio. It's an adagio. No, no, but I understand, I understand, I understand, I understand, I understand, I understand, I understand, I understand, I understand, I understand, I understand, I understand, I understand, I understand, I understand, I understand, I understand, I understand, I understand, I understand, I understand, I understand, I understand, I understand, I understand, I understand, I understand, And in the same space, the science of science and physics coincide with the ambition of Frege, who presented the hypothesis in such a way that the fact that the hypothesis covers the dimensions of the universe is one of the three answers. So it's a good idea to rephrase the question. Let's rephrase the question with the first part of the question. The first point is that we respect science and physics as it is, because we confuse the two. It is an art, it is an artist, it is an element, a fundamental ingredient of theory.

1:07:30 The problem for me, the most general problem, is to understand how to understand the enterprise of the liquid, that is to say, that does not belong to the liquid. The problem is that it is their fault. The problem is that it is their fault. There is a total lack of confidence in the subject of mathematics, marked by the object as the object. I'm sorry, but there is a word, in my opinion, in my opinion, I had an element of confusion about the problem, the problem is to know if, I come back to this feeling, is to say that you have to present the truth of the subject in such a way that it finally falls into the box of the whole truth. I would say, deep down, that the information we have to demonstrate will never go beyond. These are rational arrangements of academic studies. So that means that if I present the problem to the students by simply arranging the department in such a way that all the arrangements of the department are related to the subject, that's fine. For me, this intention of the two of you is to improve the academic system.

1:10:00 A department arrangement that fundamentally does not express academic studies, it comes down to denying that... There is a mathematical theory, which would be a truth about an object, because of the fact that it is a part of a theory. This debate, I think, is to find out if there is a truth, if there is simply a categorical arrangement of an object compared to a light object. So, now, it seems to me that... Well, the logic seems to me to be less clear than Tregeu and Lafrette. Tregeu and Lafrette said, we will not have an object, and so I understand badly that there is still in you this claim to have an answer to the question of mathematics which is substantial, as we will see later. Because, in fact, It's not the real idea. The real idea behind the recipes and the content is that mathematics is not a substantial object. The only substantial objects are quantum objects. That's how I understood the idea. So that's a debate that I wanted to see. The last thing I want to say is that if we want to have an authentic object, to my knowledge, there is no technical solution to that. There is only a theoretical solution, which is to have an authentic object. What is an object? Is it an object that is composed because we believe that we have a metaphysics as an object that is justified in certain functional functions? I believe that there is a nuance that I agree with at the base, but these arrangements that I am telling you about can mean a lot of things. I think that at Freyer, it means the introduction of objects of this type, which are not abstract objects, but objects that are dependent, of course. Thank you very much for your time, and I hope to see you again in the future.

1:12:30 I gave you a lot of information in a quarter of an hour, but in fact not much more. So thank you. I think that the true desire of Frege is to be metaphysical. What does that mean? It's a metaphysics, it's a technicality. Frege's technicality, that is to say, in the domain of the particulars,

1:15:00 which is the natural language, exists because

1:35:00 you can define the truth of the ancient universe, the truth of everything. That's why P is true for everything, and then you define the truth of each of the P in terms of demonstrability with the perpendicularity in your theory. And at this point, you can play with the verifunctionality, but the truth for each x P is not the demonstrability of each x P. The demonstrability of each x P is not at all the fact that P is a theorem for each a. So you guarantee that it doesn't mean that for each x P is true.

1:37:30 This means that for each x, p of x. It does not mean that I have demonstrated that for each x, p of x. And for each x, p of x, why for each x, p of x? Well, because by taking into account the individual constants, I can verify that the individual constants are dissipated. So here you have the linear equation, and I have the impression that in many cases, mathematical physics works like this. With, in this case, who worked with Arendt and Lohmann's theory while doing the difference, These are the terms that are already given. These are the ones that refer to an independent existence. Those that are supple and that you have two different theories that you have already accepted. You are an angel. And so the real way to solve the idea of animation is the idea of objects. So, you see, when we...

1:40:00 And that's what always makes me feel bad, not only because mathematicians work with them, but because in themselves they are properties that insist on them, and that's the first time you look at them with all the mathematics, with the probabilities, with the algebraic geometry, with the...

1:42:30 But there is not a very naive and very simple problem, which is that how can we solve arithmetic without already having an idea of the domain of the object of the activity. There, for example, in mathematics, there is a very clear idea that in fact arithmetic would consist of confusing the numbers. Jean-Jacques's remark is that if we do not... We have a lot of trouble understanding why there would be number theory, why to study arithmetic in the sense of number and whole numbers, we would have to intervene more often. That is to say that, visibly, the naivest point is that when we look at mathematics, whole numbers are not only their construction. And there is one, and there is one, and there are many things that their construction by Léonaud etc. does not like at all. So I'm in the celebration of our young people, I'm just in the celebration of our young people, but it's not that far, sir. No, but that I understand. So that's a point. I understand, but the object is not good. The second thing, but let's go back to the first thing I wanted to say, it's really... In these debates on objects, concepts, etc., what strikes me, it's the same, but what is missing is that, for me, a mathematical object is not only a structure, it is a set of networks, it is a set of networks, it is a set of circulations, it is a set of trajectories, and... An object is a certain stage, but in a trajectory, in a multiple range, at different levels of depth, with structures that are more or less complex, with layers. All this aspect, in my opinion, is lost when we understand the reasons for this passion. And this is the reason why, fundamentally, I find in phenomenology a destructive power. I have not been able to reach this level of intangible core, intangible intangible core, intangible intangible intangible intangible intangible intangible intangible intangible intangible intangible intangible intangible intangible intangible intangible intangible intangible intangible intangible intangible intangible

1:45:00 I don't see why this problem has to be solved in such an alternative way. It seems that if we can solve this problem in a good way, it is the solution to the problem of the game. We agree on that. The problem is that reconstruction is given from a geological point of view, which is of no interest to us. And in terms of objects, it seems extremely important, because... But, David, David, he says that it is a fantasy. The truth is that it is a fantasy, but there is a little mathematical theory, which is not the only one, in which...

1:47:30 There is a problem, a very complex one. The last theorem of Fermat is a theorem that can be formulated within a series of calculations. So does that mean that there is a certain inevitability? Can the theorem of Fermat demonstrate this? If we could not demonstrate that this is a theorem, we could not demonstrate it. So, what are the statuses in a theorem? What is the language of the theory? That is to say that the formulation of the two forms of demonstrability can not tell you why one is interesting and not the other, at this moment or whatever. We consume a kind of resistance, we want a resistance to objects, to their results, and there we are going to use all the instruments that exist in the mathematics and we are going to look for the same demonstrations that are not internal. And it seems to me that it's not because the theory of logic describes something, it's because it's a kind of reconstruction of that, it's a kind of reconstruction of logic. So, we must be shocked. I don't know, I don't know. We must be shocked. No, we must be shocked. No, no, we must... Jean-Jacques Spillage, who wrote a novel on telepathy of objects, there is also the telepathy of a certain mode of theory. Theoretical mathematics is not simply the theory of compression and linearity. Theoretical mathematics is based on a set of objects and therefore the notion of a set of objects is dependent on the set of objects that characterizes the own modality of theoretical mathematics.

1:50:00 The remark made by David in recent times indicates the fact that we must be involved in the definition of an object that has the ability to be interesting and not to be ugly. And that this susceptibility must be integrated into the lecture. But I agree, I totally agree that this is exactly what I said about anatomy and mathematics. That it depends. Simply, this notion of interest is difficult to explain. And the conclusion is that it is not so. That is to say, it is interesting. I want to defend you in the sense that It is obvious that all this project of the previous one could have something to do with mathematics, it is obvious, in my opinion, it is not necessarily just a criticism, because of course, I believe in mathematics, in the sense that it is not proper, perhaps we do not need something, we can think of the natural language as ... Primitive notions and then do a lot of things with that, that is to ask a question, which is still a number, and give an answer that may not always go very well with mathematics, it's obvious, but still interesting, which is not trivial and which is a place for discussion, in my opinion, is strong as a form of math. And of course, it's very interesting to try to find the contacts with mathematics, but it's not a surprise, you can't wait to find out what's interesting. But also to learn anyway, you see, that he is sensitive to certain things, he is not a mathematician, he is not a computer.

1:52:30 What are the numbers of these objects? What are the numbers of x, y, z, n? That's the challenge. If you consider that an object is what is the reference of certain values or theories... So, the answer is, of course, by definition. That is to say, we have asked them that an object, that is to say, that was constructed in a logical way, and that the entries, their properties, were expressed by this object. That's the whole point. But we don't necessarily want to say that. We want to say that, precisely, but that's a bit the remark of the students, that is to say, this concept, this definition of the subject in its own right, that is to say, we are going to rephrase what this object is. It is a mathematical object in a very particular form of reduction, but we can very well consider that the notion of property in an object is not a property in the world of life. No, but then, let's not confuse it. There are philosophical points and political points. I believe that the point of life is that the numbers here satisfy the needs of the great realm of Thaïma. Only if... We only have properties other than those defined by Penrose. It is an open mathematical problem. We don't know if the denominator of the denominator is independent of the function of the denominator and the denominator of the denominator is independent of the function of the denominator. Can we have a world in which both the function of the denominator and the function of the denominator and another world in which both the function of the denominator and the function of the denominator do not evolve? If this is the case, then of course all the ideas we have about being a member of a progression, which I think is not put into discussion by the mathematical body, are no longer there. This is false. Simply put, numbers are not a characteristic of Penrose's sensations in a complete way.

1:55:00 Because we have societies that hold essential numbers, such as the satisfaction of Penrose's sensations, which are not taken for the five actions. But this is not a problem for physicists. It is a complex mathematical problem, and I do not believe that any conference of physicists will be able to solve it. This is a problem only for mathematicians. So, in other words, can we demonstrate Pernod inside PAD? If that's not the case, it's just a mathematical problem, we assume that it's not true, then, of course, the problem is not that the terms are the ones we talk about in all the theories of the world. The problem is that the terms are the ones we talk about in PAD. And then, PAD is... This is consistent with other theories, and so there is the possibility, mathematics lives from this possibility, to look at the properties of other theories and define them from other theories, because there are links between other theories, these links must be explained. So, here I am explaining and explaining these links. I have explained this possibility of working on objects of the entire universe outside of this theory. So even if we find that the piano is independent of the sound of the piano, this problem remains, because it is a particular case. In fact, in some cases, we would be very surprised to know that the piano is independent of the sound of the piano. These are philosophical problems, they remain. But why is it a philosophical problem that we should not address it like this, by saying that we have a theory that defines objects, and after that, we can refer to other theories, because there are certain links of other nature that have to be explained, and it is this theory that has to be explained, which allows us to speak outside. So this is the problem of transcendent demonstrations. It is a huge philosophical problem. But I do not understand why a philosophical problem would deny the legitimacy of reflection. Analyzation. Analyzation. No, no, it's not that. No, no, it's not that. No, no, it's not that. No, no, it's not that. No, no, it's not that. No, no, it's not that. No, no, it's not that. No, no, it's not that. No, no, it's not that. No, no, it's not that. No, no, it's not that. No, no, it's not that. No, no, it's not that. No, no, it's not that. No, no, it's not that. No, no, it's not that. No, no, it's not that. No, no, it's not that. No, no, it's not that. Philosophically, the problem arises in the same way. That is to say, yes, why not, but why not not? That is to say that we have no argument. There is no argument. And what is good is that if you say that a real concrete mathematical problem incorporates the whole structuring of theories...

1:57:30 So, in this case, if there is a theory, well, if there is an object, a philosophical concept of a mathematical object that poses a theory of mathematical practice, the question is, is a theory of mathematical practice something other than mathematical science? I mean, will there be a way to realize the richness of a mathematical object philosophically, without going through what is a bit of a procedure, which is to say, to have a discipline, whatever it is, that doubles? All of the mathematics to be able to do topography or even the description of the terminology. But for them, it's not the topology of the double. No, it's the topology because in the sense that we will never have...