Philosophy of Mathematics Discussion Group — Marco Panza's Paper (contd.)

0:00 Here is their conceptual simplicity. Yes, that's a way of saying it, but the same thing can be said in another way, which is to say that we have a way of saying it in the same way as in the English language. No, no, no. If you allow me to make a very, very short intervention in English. It seems to me that all these interventions, since the first intervention by Jean-Jacques and Marco, are linked at the same point in connection with the absence of a wall. For a prelable ontology, we know that with the interpretation of mathematics and this point is the point that is posed by Kreisel when he says that the issue with which we have to do is not with the existence of mathematical objects but with the kind of guarantees that we have for the objectivity of mathematical history. It's like a mathematical relationship to replace the hierarchy between chosen objects. Exactly. It's the same thing. The true nature of mathematical physics comes from the totality of mathematical physics. In each mathematical physics, the true presentation of the whole is the whole of the physics of mathematical physics as a whole. And philosophy can reflect all this by showing that there is an idea in all this. It can allow us to have a very good game. We can say that it is a... The capacity of the idea is... It could be... Well, that's a different position. But what I mean is... It's true that it seems intense, but there is an idea that mathematicians find it easy to present the presentation that is effective. It's because mathematicians exist. But that's... Of course, but... I don't have an opinion. Simply, the search for proof comes back to those who want a more comprehensive mathematical concept. Yes, but I mean the same thing you say. I think you mean... We have a huge work between us, philosophers, which is to realize very well. So we can do this by looking at mathematics from the outside, by looking at the immensity of mathematics,

2:30 and trying to have a point of view that globally realizes all this by researching very carefully, which is very difficult, or in a much more modest way, which I see, by realizing a little bit. So, I understand why, in order to realize a problem that is a matter of mathematics, we must, a priori, start from a characterization of what the matters that are in their totality are, rather than address the problems of how we can make projections based on their properties. It is much more natural to address the problems of mathematics by the relations that are established between the different theories, rather than by the means of an ensemble. This is the way to solve the problem, but in history, all the authors did not take a view of the whole at the same time. I mean, most of the authors were local, of the Freudian type. They proposed reductions, etc. And the question is to know why we are going to choose this one. We see the advantages. The advantage is that it is reflective because it is internal to the logic. So it has tools, it has a technique, etc. But it's not, the Oblégia, it's not a view of the whole on mathematics and logical theories. The opposition is rather between whether the description we make agrees with a kind of logic we have, a naive logic, not in the sense of the kind of logic we have. I don't have a view of the whole of mathematics, not at all. It's just that on some names we see, we see a certain behavior of things that we want to learn from others. And the question is, do we want mathematical philosophies that provide a set of objects to meet that or not? That's the question. And it seems to me that this is a very important question. I mean, what you say is a Ukrainian king, no, I agree, I totally agree that it was interesting to say that in all these cases, but as you said, we have to be satisfied. But I don't understand why, I don't understand what you're saying. I don't understand what mathematics is. I study mathematics, I see mathematical objects, I work on them every day, I see them, but I can't talk about mathematical objects, I can't talk about mathematical objects, I can't talk about the history of mathematics, I can't talk about the history of mathematics, I can't talk about the history of mathematics, I can't talk about the history of mathematics, I can't talk about the history of mathematics,

5:00 This is not a promise, it is a promise to realize the mathematics of the past, the mathematics of the history of mathematics, the mathematics of the history of mathematics in relation to the mathematics of the past, because the theories of the mathematics of the present are related to the mathematics of the present. To realize the mathematics of the present, that's why it was in the past. The idea of the Atiyah is an idea of the mathematics of the reality, which is the reason for all the rationality, otherwise we do not understand anything. So we realize the mathematics of the past, but we cannot realize the mathematics of the past. We have to realize the mathematics of the present, which is simple, which is understandable, or we have to use it. I don't see how I could integrate mathematics and physics into a mathematical subject. So the problem I have is, what do I mean by a mathematical subject? How can I characterize the fundamental category that I use when I talk about the past mathematical subjects, in a reasonable way, how do I make it more precise? The goal of this lecture is very, very down-to-earth, to simply try to see more clearly the set of words, the significations of the set of words that I use. I feel that when I do the history of mathematics, I use words as proof, as a demonstration, as an object, as a notion, as a tool to which I give a very vague meaning. I would like to be able to give a very specific meaning. Wait, wait, wait, wait, wait, wait, wait, wait, wait, wait, wait, wait, wait, wait, wait, wait, wait, wait, wait, wait, wait, So, in any case, I still agree with Yannick in the idea that there is a problem with the presentation of the mathematical object. You can say that, I can imagine a hypothesis, for me, even if the law of law is entirely traditional and temporal, nevertheless, the perspectives and all the labyrinths of elaboration that continue in mathematics are part of the maintenance of the mathematical object.

7:30 And if we say that, what it means is that we will have two levels of the mathematical object. We will have a mathematical object in its internal meaning, in order to translate it into mathematics. This is a level, so we need a certain number of, we need a philosophy to explain what this level of the mathematical object is. And then another level which is the physically considered mathematical object in relation to its validity. The validity to which they give place. So you can say that perhaps the mathematical object, if you consider the system to be the only one of its validity, is the inhabitant of the supposed universe of the theory of ensembles. So you can say that. And you can say that it is sufficient, if you will, for a certain notion of the mathematical object. As for the question of validity, you can say that it is not sufficient when you have another requirement to think of the mathematical object and to think of it as a crystallization point in a game of thought. Although both are articulated at the same time, since the game can never, against a very large mission, which is not surprising, I can tell you that we will have an academic lecture on mathematics and mathematical physics. Yes, yes, it is supposed that we are going to write the game, but my point is exactly the same. I totally agree with you, but I do not see how I can give this account of mathematical practice. It's not a word, it's a kind of object. I have trouble understanding it. Well, we can do a little bit thanks to Vincent who is an assistant. We say that the two researches are necessary, and that's what we are going to implement. What Marco says is that in the extension... In the first one, we can find the second one by the presentation of the part that is. Frankly, it's not easy to find the part that is. Because in the presentation you gave me, it's more like the term itself. That's what I don't understand because it's automatic. No, but you seemed to say that, well, I'm on the other side. The presentation of the whole. First of all, it was done. It was done regularly. etc. So, on the other hand, ensemble presentation is a very bad concept, but we can have at least three or four ways to address the question of ensemble mathematical presentation.

10:00 It presupposes a mathematical philosophy that we can try to validate, verify, and practice on this level. You can have an excellent position on the level of mathematics because, for example, you can say that there are structures and then you can go back. It's a way. It presupposes a concept of structure that is not the same. I'm not going to go into too much detail, but I'm going to go in a different way, because in history, we wanted to do this presentation together in its own way, one way or another, which is a kind of algebraic theory or algebraic algebraic algebraic algebraic algebraic algebraic algebraic algebraic algebraic algebraic algebraic algebraic In the end, Bourbaki, Witten, and Lagrange, Lagrange, Lagrange, Lagrange, Lagrange, Lagrange, Lagrange, Lagrange, Lagrange, Lagrange, Lagrange, Lagrange, Lagrange, Lagrange, Lagrange, Lagrange, So I am far from thinking that mathematical physics is thought to be this way and that it signifies... No! That is to say, it is one of the great parts of mathematics in its essence. It is not the whole thing that mathematical physics is about. It is a philosophy of mathematics within mathematics. It is a rethinking of mathematics in relation to certain constraints.

12:30 For example, methodology is not a rule in relation to mathematics. We have to consider the problem of how to rethink mathematics. We must start from what is the mission, the only mission. From my point of view, it is not even one of the 10 natural methods of mathematics. I call it mathematics. What is mathematics? First of all, it is not so much that it has been a method of mathematics, that it has been described as a method of mathematics with these terms, but it has been intended to take that into account. And in relation to Huckman, the particularity that I have in relation to Huckman, is that this pretension of taking this mission at the expense of mathematics allows us to use the right tools for the F.R. Isn't it a joy to think that we would push this operation with the right tools for the F.R.? Yes, that's a question we can ask ourselves. It is important to know that with these terms there is no contact with the question of ontology. The question is to know when we go into this field, a certain reconstruction of the theory from the language of logic, it induces or it demands its object, it is absolutely obvious, the question is to know if it is compatible. This is a real problem. You say yes, and at the end you say that we can integrate practice into a concept of an object. That's the question. That's pretty much it. And so it's not about saying that there is another concept of an object that we could give you, etc. It's a kind of doubt or suspicion. Apparently these doubts are rather alternative. And when you present them yourself, you present them as alternative. So at the same time you say that they are compatible and that we can do it. It's not going to work. It can't stick. No, but I think you're right. There's a problem with the two entry points, which are better than the analytical ones,

15:00 which are active, between which there can be relations in which you can collaborate or you can be an alternative. And it's true that from this point of view, there are times when I think like you, because I really believe that the bridge must be made, to resonate the thinking of mathematics. You will not be able to interpret mathematics in one way or another, and it cuts off the problem of the bigger international debate. It cuts off the question of mathematical physics. First of all, it is a philosophy that can be completely disrespected. It is a philosophy that can be completely disrespected when there is a debate on mathematics, when there are certain definitions of mathematics. When there are some definitions of mathematics, you have to let go completely. I would like to ask you about your work in the field of humanities. I would like to ask you about your work in the field of humanities. I would like to ask you about your work in the field of humanities. I would like to ask you about your work in the field of humanities. I would like to ask you about your work in the field of humanities. I would like to ask you about your work in the field of humanities. I would like to ask you about your work in the field of humanities. I would like to ask you about your work in the field of humanities. I would like to ask you about your work in the field of humanities. I would like to ask you about your work in the field of humanities. I would like to ask you about your work in the field of humanities. I would like to ask you about your work in the field of humanities. I would like to ask you about your work in the field of humanities. There are three or four articles in the book, including the texts of the two books, the real and the physical.

17:30 I have not seen the texts. No, but it's a different topic. No, but it's a different topic. No, but it's a different topic. No, but it's a different topic. No, but it's a different topic. No, but it's a different topic. No, but it's a different topic. No, but it's a different topic. No, but it's a different topic. No, but it's a different topic. No, but it's a different topic. No, but it's a different topic. What one must ask is what a mathematical number is. One must ask how one can, in general, realize the notion of a mathematical number. So the answer is to realize the notion of a mathematical number, we need a strategy between the two. Then we know the equations and so on and so forth. After that, the projection does not exist. Yes, but in the case of a mathematical number, it has the structure to be taken into account. If it is simply PR2, then PR2 plus, please, if it is PR2 plus... This is a problem for all, not just for us, but for the whole world, which is the most important thing we have to deal with. This is the answer to all the problems that we have to deal with. But it is not the only problem. We have to establish a general schema which is the form of mathematical mathematics, and then we have to give this formula in the case of physics, which is a problem. I see that the good question is not what mathematics is. The good question is what is the strategic attack that we must follow to realize what mathematics is.

20:00 That's what I asked you earlier. In the sense that you think there is a strategy that causes problems, but in fact it is a problem. No, no, no, no, no, I don't think we know what an object is. The answer I want is not that we know what an object is. The answer I want is, no, the answer I want is, what are the variables, if you don't understand, what are the variables that I have to give a value to know what the object A is. To know what a certain object X is, for example, I have to give the value to the variables Z, Y. F, V, I don't know. I want to know what these variables are. Then, in each particular step, I will give a certain value to these variables to define what the particular object is. So my answer is not to know what the whole number is, what the function is, or what it is ... There are not many answers. It depends on different contexts. The whole number for Y is probably something different from the whole number for LR and for UGIL. I agree. But I would like to ask myself a question. And I do not see why this answer, this question, is dependent on a global approach to the totality of mathematics in which this answer is relevant to the application of the links that exist between the analysis of numbers, the theory of the probabilistic of numbers, and complex geometry. It's the way in which, when you apply this pattern to understand a particular object, that this answer is relevant to the definition of these links. But in choice, there are people who are independent. Well, more or less, as a matter of fact, it's a little more mobilizing, but you see that, for example, if you consider that objects are placed in structures, as you said in this example, and so before the invention of the analytical theory of numbers, we had certain structures. And then we have a new one. There will be two options. Either we changed the object, first option, or we were wrong about the object. And we don't want to say either of them. We don't want to say either of them. I don't know, I want to say at the end of the day, I don't want to say neither that we changed the object, nor that we got the wrong answer. No, I don't know, I'm telling you phenomenologically, I don't want to say that we changed the non-branched object with the non-branched mathematical function.

22:30 I want to say that we enriched it, that we see it in our policy, and I don't want to say either that we were wrong about the non-branched object. After all, I don't have a solution for that, it's a matter of methodology. And so it's not a problem of what is the right structure. It's not that. The real problem is the evolution, the structures, the fact that, for example, piano can be associated with others. That's the problem. And it's not about choosing. You say, as if we had to choose one. Indeed, there you put us at the foot of the wall. We must choose the right structure that is clearly written. Precisely, it is to say, and it seems to me that a philosophy of mathematics that would realize this evolution should be able to say that it is not a problem simply to choose a good structure. We have to understand how these structures interact in such a way that we have the impression that it remains the same object, the whole object. And I'm not saying that I'm wrong, it's a problem of theology. And it's simpler than the question of having a view in the sense of mathematics. Well, it can be done locally. I don't see why. I don't see why this is the subject matter. Of course, if I want a structural model of the object, I have to realize what is the relationship between the object of physics inside the structure, the object of the generator of the structure and the regime, and what exactly is the structure, how to control the relationship between the structure. It's part of the automatic that I put in place. But I don't see why. Thank you for your attention and see you in the next lecture. That's not the point, I think the point is not the point, the point is the way, the gateway by the logic that remains within the theory, which is still a formal language, the objects are still the values ​​that take the variables, is this gateway there is compatible with this type of problem? I did not say anything, they are identities or whatever. That is to say, is it inside a square like that or even inside a square like that? No, but I never said that when I talk about structure,

25:00 you have to wait for the structure of a logical theory formalized in the language of the most radical of the millions. The big problem is to try to give a sort of mathematical theory that is simply large to be able to account for it, to be able to say, for example, that the elements of a cube are called structures. And so, the segments of the elements of a cube are called structural. Of course we have these problems, but these are crucial problems. All in all, in the case of Penrose mathematics, the best way to pass the structure is the logic of Newton. But I don't see that the logic of Newton is the only way to pass the structure. This is the subject of the lecture, but I think it's interesting, it's not at all crazy, we are relaxed for at least a minute. In short, we can think about how to do it. We can think about how to do it very well. I don't know what it is. I've never thought about it. It's orthogonal to all the others. The article by Marko that I'm interested in is whether the philosophies of x and y allow us to see the object as something other than the concept. So here, it's an absolutely opposite proposition. I mean, the real notion of object is the concept, and that's where we're reacting to a materialist philosophy. And I wonder if that would answer the question of the structuralism of the problem. Because, as a concept, it's a concept. I would go back to mathematics. Mathematics is still a very strong subject. I mean, I have rarely heard the example of a philosopher who writes a sentence in which I agree that it is necessary to obey the law.

27:30 I guarantee that if you wish to set the author of a theory of mathematics as a verb, you need to get the concept of a verb. I think he has it. After that, the way in which he thinks of gas, the way in which he thinks of physics, I am very far from saying that it is true. But that he feels that there is a disciplinary religion in the mathematics of physics, that is to say, there is one. You have to realize that gas is a constant object. That is one of the fundamental things in mathematics. I would like him to have the paper and to put it in the discussion of the original. In order to solve this problem, we need a local plan, which is to attack things that produce in the world, as I said, that produce in the world in a specific way, because there are no... Oh, no, it's true. Even the laws of the universe agree. Well, it's great to see you next time. Thank you very much. Thank you for your attention. Thank you for your attention.

30:00 Thank you for your attention. Thank you for watching this video. Thank you for your attention. Thank you for watching this video. Thank you for your attention. And some would be closed under exploration, some would be closed under other operations, but they would not arise in the future. There is nothing perceived by some of these students, very small number of them, but if you want to get a feeling for the mathematics, then we don't need to get a feeling for the mathematics, and we have to get a feeling for the physics.

32:30 Thank you for your attention. Thank you very much for your attention. Thank you for your attention. Well, I'll go through them one at a time. There's only 10, there's only 11, but I can tell you what they are. I'm quite sure. Well, I can go through that with you. Have you got one? No, I've got one. Why is that now? Why is that now? Because I don't know the two things I'm expecting. Thank you very much for your time. Thank you for watching this video.

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