Euclid & Aristotle

0:00 I will explain that, and parallel, there are very large discussions to know how the generality works in a book like the Elements of Clyde. Why? There are several areas in which it manifests. but the one who interested me is the question of the theory of proportions. Why there is a discussion? Because in the first lecture, there are two domains of objects which are totally separate, which are the one part the grandeur, the objects geométriques, with a place where it is already where it is, but it poses a certain number of problems, which is the Livre V, and so apparently a place of objects which could be called grandeur, even if we don't know anything, it poses already a problem at this level, The second domain is the number, and each one has a theory of proportion which works for him, and he has no circulation to the deductive between the two domains. And there where it becomes interesting, it's that in the book of 10, all of a sudden, we encourage, in the same proportion, to make a difference between grandeur and number. It's for the definition of commensurability, and it's absolutely essential. If we don't have that, we can't define what is a grandeur commensurable, so we can't write the book 10 because the book 10 is the classification of the grandeurs incommensurables and there is obviously a criteria to know if they are commensurables or commensurables but it's not the definition that you call arithmetic the definition is independent of the arithmetic and after it is the criteria it's not the definition the definition is totally independent the definition is independent, it's the criteria in fact that will allow you to discriminate two classes of the grandeurs Yes, because the definition is just saying that it's a common measure, it's a common measure, it's not a common measure, it's not a common measure, so it's a common measure, in fact, it's going to be given by... And so what I tried to show is that... So these are two mysteries, there is a mystery from Aristotle who has a lot of tortured the commentators during the Middle Ages, because they are all beaten up there, and then there is a mystery from Euclid who is to know that... Entre nous soit dit, avec ce degré de clarté, ça c'est le mystère sur lequel les commentateurs, etc. Je pense que cette phrase n'est pas dans les deux premières pages de l'article. Je te le dis comme ça.

2:30 C'est parce que tout de suite, en général, quand on expose un article, on dit toujours les phrases nécessaires et elles ne sont pas toujours dans l'écrit. Well, in fact, when I read the article, because I didn't completely finish it, I saw that I didn't have at all the criteria that we had said for the introductions. And so, in fact, I think it would be, in any way, an introduction that explains what I'm trying to explain. Like I said, an introduction is not done. But I wanted to do it. Because in fact, my introduction is very long. And quite problematic. And already a little technical. So it's true that for me, it's the two messages that we have to do. D'accord, bon. I've passed a few words after two pages. I've sorted this word analytics, all the translations, etc., etc. Well, I just finished the presentation by saying that there are these two messages that are parallel, and my idea, which I try to defend, is that in fact, it's clear if we approach each other. It's to say that we look at a little bit how the generality works in the elements of Euclide, and we don't try to plaquer a concept in fact, we take it with its aspect a little bit mysterious, and we look at what Aristotle said on the math and we see that it's called. That's what I've tried to show. It's to say that when we take all the texts that Aristotle writes on the catalogue, they have something quite interesting, and that in a certain number of these texts, there is one that I could add, we have the impression that Aristotle considers this question as a fact. what we call the katholou in mathematics, something with what he should do. And so it's even more interesting because, visiblement, it reinforces the indices that it's not simply liable to the difference of its interpretation. For the people who don't speak Greek, katholou, it's one of the terms by which Aristotle revoit in general. It's a term Greek. That's often called universel, but in fact, on the traduit by universel because katolou is also the term used to design the ideas platonician so they call ou idées ou katolou and so it has given naissance to another querelle which is the querelle des universaux that means what literally? so that means kat-holon so olon is really being in a totalité and kat-holon is a sort of reinforcement katolou du olon so there is no reason really to couper the champ du holon du chant du katolo et donc ça veut dire ce qui embrasse dans un tout et après

5:00 il y a d'autres termes qui sont le koinon mais koinon ça veut dire commun mais c'est la même idée on sait pas vraiment on sait pas du tout l'exprime autrement et l'autre terme que j'avais mentionné c'est le katapantos où là ça veut dire selon le tout à nouveau and it is very difficult to get to the distinguishing. It is a very difficult to think about the all, in considering that the all angles of the elements are going to take one by one. Yes, for the catapentos. Do you mean that this catolou in the sense of Aristotle, is it possible to encadre the grandeur and the number? The question is to know, what would it say to a universal or general point of view in mathematics? And what I'm trying to show is that what Aristotle said, so I'm not developing the arguments, I'm just explaining the general strategy, but what Aristotle said is that there is not an expression separate from this generality. Well, he brings a comparison zoologic, because I haven't seen it in the chapter 4 of the second analytics. For the catapentos, right? Yes, it's that. Well, I have remarks and questions that have touched on the subject of the zoological thing. So just to answer your question, the idea is to confront the two types of texts. And in these texts, Aristotle will insist on a certain aspect. He will insist on the fact that the catelou is something that is not univoque. Never, in fact. So it works well in mathematics where the notions like the unity cannot be univoques between the numbers and the grandeur, which has no expressions separate. And there is another aspect that I totally left aside because it was very long, but which is quite interesting, it is that they refuse systematically all what is giving a status ontology autonome in relation. I touch a word, but that is something that is quite important, if you want, but I, on this point precisely, excuse me, I was troubled by the way you articulated this thing that you have to remember, with the sort that you make a commensurable in your solution. Because commensurable, it's still a relationship. Two things are commensurable

7:30 Do you articulate what you just said about the relationship, and what you said in the beginning of your article, with the fact that, finally, what you said about the universe, it implies something related to a relationship. Because if you want to talk about the convention, you say, I don't want to talk about the Yeah. But then, we almost got the impression that it becomes an attribute of a grandeur. Ah, no. That's never an attribute of a grandeur. But it's true. Karim comment on the front. Formellement, it's true that it's a bit lourd, but... Revenir systematically the fact that it's a relationship and that there are two terms... Yeah. Because I made an example for a question of style. I made a note saying, I said, I will say, sometimes, as a mensurable. This note, I totally agree with you. And my intuition is that this abbreviation that you have made, in my opinion, in the second part of your article, it has led you to treat mensurable as an attribution and more like a relation. it's to say that a relationship is an attribute for Aristotle it's an attribute of what? but it's part of the general class of what we want to attribute so the question is not there, the question is to say is it a status ontology autonome? and that I don't think I've ever seen it the idea is to say that precisely when we compare numbers and grandeurs as the comparison is done at a relationship she doesn't have a status separate and so she doesn't have the status of a substance, so of a domain of objects. So that's not clear? I have the impression that in the last part, I'm surprised that you show me that I'm wrong, but in any case, I tell you that an author who has read attentively this presentation, is that in the last part, you say that we can separate the grandeur from some who are commensurable, and those who are commensurables, we can treat them like they don't. So, it's a thing that you attribute to the greater.

10:00 Ah, in this sense? Yes, in this sense. Well, it's not very problematic. I would say that it's not very problematic, but I would like you to explain it. You say that we can't isolate the greater than commensurables because as it's a relationship, it can't be attributed to a class of objects. It's not in the case, it's not in the case, but it's the same problem as parallel. And I even marked in the notes that it's parallel. Yes, it's exactly the same problem, it's to say that at this moment, you can't talk more about the parallel. You know what I'm saying? Or you can take the parallel droids, it's a sense, you know? And for example, they are opposed to the right, I don't know, or what you're saying. The question is, is that what you predict the thing is a domain of objects? And that's the answer, it's no. The right thing is not a domain of objects. The domain of objects, it's the right. But you have the right to isolate an interior, a class, depending on whether they satisfy the relationship. I don't think there's any contradiction between these two aspects. It seems... I put a lot of notes on the bar, if you have a copy of the notes, as I often do, you'll see, at some moments, I'm thinking about how to read it, how to read it. D'accord. Well, I just explained the general strategy, but then I don't want to go into the details. For the second book, is it really, I don't remember, in EFCLIT, is that, of course, we can think of nature, but it's really necessary to look at this generality. Why is it possible to form all these problems in-measurables as a problem purely geometric? Of course, it's in relation to the numbers, etc. But it's a little... That's what I'm trying to explain. So the challenge is to give a classification of grandeur, more than commensurable, irrational in fact. So it's still a bad thing. And in fact, when we try to describe this strategy, the strategy of Clit is to say that there is a type of... There is a type of... There is a type of... There is a type of... There is a type of... There is a type of... There is a type of... It's that we build on a large scale which are not necessarily commensurable, and there, obviously, we will have a class of large scale

12:30 which is commensurable as carrous, while the large scale they are not commensurable. So, there, we have really need to know. And the problem is that when we manipulate these large scale, we want to see if it's stable or not, when we add the difference. And so, we need to know in which case, when we use this type of large scale, we can bring, yes, to a large scale commensurable at the start. And there, the criteria is arithmetic. of the proposition 9, if I remember. If it's a number of square, then yes, of course, it's a number. It's a number of commensurables. Yes, it's formulaic like that. So we really need that, not only to distinguish the case commensurable from the case commensurable. It's a bit more complicated. In fact, it's a bit more complicated. It's a bit more complicated. There's something that I explained Very, very quickly. Yes, I did it very quickly. I told you, it was a matter of thinking, but... I've been working on the internet, I've been working on the internet, etc., etc., and I've just found my job, so I imagine... It's like always, there's a sort of problem which is to know in what extent we get into the technical aspects, knowing that, all of a sudden, who doesn't care about these technical aspects it's not at all sure that it is clear. You have two grandeurs who are not comfortable, you take the square, and you have the square as comfortable. Yes, that's what I said, but after... I know very well that it's easy to explain, and I know very well that it's what we're talking about about the TTT. So, to say, what we're talking about about the TTT, the problem evoked by the TTT without even, if you want, I can't even say this thing. I have no idea. I have no idea. What is it? Well, I have no idea. Well, I have no idea. I have no idea. Because in fact, the TETET is in a way of doing another solution. I have said that it was a comparable strategy to the TETET, because in the details, TETET is not like that. What's interesting about T-Tetet is that T-Tetet is comparing numbers, it's explicit. And among the numbers, they separate those who are rectangles, those who are carrés. And then, there will be a strategy that allows, in passing by the carrés, to arrive at...

15:00 There are really a lot of and big debates to know if it's a solution which is rather comparable to the CLID or rather opposed to the CLID. Maybe I can remove the allusion of the TETETE which serves, I don't know what, I don't know. There is still a link that is systematically the commonsur ability in puissance. Yes, it's that the link, it's that the comparison. It seems like there is no difference. You don't say anything, but if you ask the question... I think it's all that I've said. The TETETE is 939. On the page, non page 1. Note 39. With the notes, it works. Note 39. Note 39. Note 39. Non, non, non, c'est en vrai. Si, si. Note 38. Oui, oui, c'est là que ça... Moi, c'est 38. Mais on n'a pas la même impression. T'as combien de pages de texte ? 22, I think. Do you want to see the 22nd? No, but it was the conclusion. Well, if it's the 22nd conclusion, we don't have the same impression. Or the impression. But it's weird that there's a note. Yeah, I'll add it because I often refer to one page to the other. What I say about T.E.T.E.T.E., I can remove it, because if you read the paragraph, I say the least possible. I say that the authors, the commentators, do a very great usage of the text of T.E.T.E.T.E.T.E. You can put it in notes. Yes, I can put it in notes. It's true that it's better to remove the text from the principal. But I don't want to know if it's the same thing or not the same thing. No, I don't want to.

17:30 I think at a moment, it should be the lecteur sage that if there is no effort to take the definition of the 5th, he doesn't understand what you're saying. I'm pretty d'accord with what you're saying. So you think it should be done with what? The definition of mensurable is to put in the text and take a half-page to explain it clearly. I don't know the definition of Edoxian, the reports, there are some people who know it, but because it's a volume of generality, there are a lot of actors who don't know what it is. When you send a note to the page in which you don't have a definition of division, in which you don't understand, you don't understand it. In fact, when you don't understand it, you don't understand it. In fact, for all to say, it was in the text. It was first in the text, and then I was basing in notes because I thought that it was going to ask for others. I thought that it was terrible. Well, it's like I wrote it, I put the definition in the text, I also put the demonstration with the parallel process of Antifereze. Well, dear friend, Hiss in 1, the translation that you put in the page 1, not only it's not Hiss, but in addition, it's missing... It's not a translation of Ys, because it's on the site of Perseus, so it's marked that it's Ys, and I don't have to verify that it's Ys, and I'm not going to verify that it's Ys, and I'm not going to verify it. Well, that's why I got to see it. It's totally wrong. Because it's not that it's not YS, but for example, I'll give you an example. In the 10.3, it's just that AB and CD are not commensurables. I know it's a detail, but... They are not what? Commensurables. Well, I don't think that it's in the original text. And it's not him who wrote it. I'm going to get out my stuff. It's given two granders commensurables, you can find their most common measure. Ah, yeah. Well, I'll have to fix it. It's a detail. I'll have to fix it. And if you want, for 7-2. Yeah.

20:00 Well, if you want, if you want, they're not the first one. Yeah. Well, it's not the text. It's not the text. Yeah, I'll have to fix it. because in the middle there was a problem, because there was a problem in the middle there was a problem, because in the middle of the text there were six names A, B, C, D, which were the first one compared to the other, the first one. I told myself that I was not in the middle of this hypothesis, I was reporting the text and I was in the middle of it. It's totally paresseux. But you told me that I didn't have a problem. I didn't have the courage to copy it. I'm going to copy it. But you can't know the difference there. Well, the figures are not. No, the figures are not. Just here, just here. Oh yeah, thank you. No, it's okay. You know, when he says, It's me who took them out. Yes, I think. I don't think it's Supercellus. It seems like the text of Percellus is not a figure. I don't know. I have a figure, but I took them out. But I can put them out. So, before I put them in the text. I thought it was good. I thought it was good. And so, I've been basing in the notes the most part of the definitions and demonstrations. But I can't remember that, there's no problem if you think. I think that the demonstration is perfect. First, because we have a place where it is easy to find. and then because at this point you can put it in a page in parallel, you can report it systematically, especially if you read it every time you cite it, I think it's good. No, it's good, it's good. It's better to make a text like that. There are people who know it, you can see it. Sinon, moi je vais te dire un truc, je n'avais jamais vraiment plongé dans le Général Selon Aristote. Donc comme je ne comprenais rien les deux premières pages, j'ai sorti les secondes analytiques et j'ai quand même un problème.

22:30 Je suis sûre qu'il y a des commentateurs qui font des interpréhensions profondément différentes. There's something I haven't understood, if you want, when you say, on the first page, The epistemic, to translate it as science, is it not anachronique? Is it not just knowledge? Well, I'll take the standard translation. There is a term for knowledge and there is a term for knowledge. It's a diagnosis. I think it would be interesting to look at the history of this translation. I think if you look at the translation... The most interesting thing is that Aristote is the first to be able to define something as epistemicone. It's not just epistemic, he wants to define a knowledge as epistemicone. Yes, but for me it's justified, if you want, it's the real knowledge. Ah, no, because there are the demonstrations epistemicone. So it's not just epistemicone. I think it's interesting to see the translation of Aristote before the 19th century. My intuition says that there was a shift to the 19th century. Because science or science anciens du terme, it's not science. Yes, of course. But if you want to use... It has nothing to do with the actuality. And today there is this polysémie. The science has kept the science anciens also. Yes, but you can see that there is a sacred... It's true, it can induce an error. It's a little averti, but it's a little averti. I think it's more sensitive to the use of science and the derivation. But it's not the same thing. If you talk about science or the terms of derivation, then there's a problem. Of course. But it's not at all the same sense that the science is used in the way the most important, especially when you talk about the history of science. It's impossible. Otherwise, you wouldn't have introduced a savoir, for example. Well, that's not true. That's not true. But it shows that there are people who consider that there are different types of savoirs, certain sources of science. And I think, if you want to, in any case, if you prefer the point of view, that to translate Epistéry by Science in the 13th century,

25:00 it takes a different connotation. Well, I... There's a tradition of Latin, a tradition of Latin... Yes, but it's science. It's not at all what we call science. Not at all. And the number of people who make the conclusion is quite enormous. That's why I think it deserves... I can explain it in notes, if you want. But I think it's the same for all the words you want to find Aristote. Logique, it doesn't mean logique. Métaphysique, it doesn't mean metaphysique. Yes, it's a tradition conventionally, but it's a tradition. So we're coincés on practically all the words. But I can put it on science if you think it's an error. Well, if you want to talk about scientific knowledge, science... Well, I'll tell you, I'm going to go into an instant. And, well, there's a thing that I can see very well on page 1. Catapentos, Catauto, and Catou. If you want, here it has to look like three things that are on the same plan, while in fact, the first two are also under the last one. It is to say that the Catelou... If you want to look at the paragraph 73, because I have to do it like I didn't know, it says that we don't want to say what we want to say not attributed to all subjects, but I have the impression that in your article, you showed that Cap-Telous was something that was above the first one and that... That's what I didn't understand, I didn't understand why this idea of reciprocity that's something that's in the last phrase? That's for the moment, that's the paraphrase of Aristotle. He gives three criteria. After, there are discussions, I'll never finish, to know what that means. He gives three criteria, and these criteria s'empilent the ones on the other. So, it must be catapentos, that's sure, but there are all sorts of things that are catapentos, because it just means a general énoncé. And so, he is not necessarily scientific. justifié. It is not necessarily in the demonstration. Then, there is absolutely an essential line between the subject and the predicate. But there is an essential line and a little

27:30 maladroit. The term is by-soi. But the problem is that there is an essential line between the subject and the predicate. It is absolutely an essential line to the essence of the human. but it's not... it's not... we don't have to reciprocate the essence of the man with the fact that he works because I don't know, the singe can move or I don't know... I'm so happy because thanks to your article I've understood everything that's not a good idea so it's not a good idea but there is also a word for a good idea which is the prop but it's that the idea it's to say that visiblement he tries to he tries to affine his thing so he says it should be katapanto, it should be for all it should be for all, but it doesn't matter if in plus we have a reciprocity between the attribute and the subject, so really a property characteristic of the essence at this moment, it's catholic and then we say the term universel does not say that at all we are totally d'accord it's interesting absolutely and it's what I started to say but they use it also for the ideas, and they use it also for the chapter of the following. So, there, we understand a little bit, even if there is a lot of literature on the question, as you can imagine, because all the commentators médiévaux have tried to know how it works, and the three are not all together, etc. It's a terrible thing, but we can see that it works well. There, where it becomes a little bit, and we don't understand nothing, it's the chapter of the following. It's to say that there are three ways to make the catholic, and there are two examples that do not correspond to what he just said, Because these examples do not correspond to things of reciprocity, or to places where we can even predict a subject. It doesn't pertain to the modes of prediction of a subject. In the first case, he's the one. Yes, but to go back to the first paragraph, I think there is an explanation for that. Because if you don't know how to do the transition in itself to the universe, You want to give me examples, because it works well? Exactly! Exemple! Because after you give me examples, but it doesn't work. I don't know why, because it doesn't work. Exemple, I've marked it! Because there, frankly, I don't understand. Ok, ok, ok. But no, I'm sorry. I'm sorry, I'm sorry. I've senti two productions, which I also started to get out of my shoes,

30:00 because they were completely different from them. And then I thought, we are bad. Okay, I'll try it. Yes, I'll try it. Well, another point, I'm not only taking the text original for this first page, but for the page number 2. And here, on the page number 2, first, I understood that the way the examples were reported to errors devait faire l'objet d'une littérature fleuve et que les commentateurs n'étaient pas d'accord les uns avec les autres sur quel exemple se rapportait à quel cas. Alors dans ce contexte, sincèrement, je n'ai pas compris, et ça a été mon point de départ, je suis partie du texte original parce que je ne comprenais pas la chose suivante, je ne comprenais pas en quoi ton exemple des lignes parallèles se rapportait à la question des particuliers. Like I didn't understand, I went to see my traductions. I understood that it was the bordel. And I wondered if I had really needed the first error because it seemed like the first error was much less important in what you do than the second. It seemed like... If I understood the example of the parallel, there are people who are attributing to the third error. The example of the triangle isocelles, there are people who were describing the first error. I did not understand why the parallel lines were the first error. I said frankly, sincerely, I did not understand. I had the impression that the example of the triangle isocelles would be the first error. But I am prepared to imagine there are hundreds of commentators that are in the first error. You do not need to tell me. to say that I found the most clear, it's the second one, and it's precisely the one that you have to do. It seemed to me that there was an error that consisted of an ensemble of things, something that vaut for a bigger ensemble, and that this error, it seemed to me that it afflees

32:30 in the end of your article, it is to say that there is only the second one that is essential for you, and in the same time, at the end, I have the impression that the third error is... In any case, if you can explain why in parallel it is liable... C'est facile, c'est ce qu'on a dit tout à l'heure, mais c'est vrai que là je ne l'explique pas, mais je l'avais fait un peu exprès, c'est-à-dire qu'il y avait un côté de vous replonger dans pourquoi les gens n'ont rien compris, c'est-à-dire, c'est peut-être pas très adroit, tu me diras, mais... Mais c'est réussi. Why did the people say what is this catalogue? That's what I wanted to revive. And it's a bit of a conclusion, that's what I wanted to show. It doesn't mean that people ask what it is, because it's extremely funny. But the way I understand it, it's what we said earlier. Here, the problem for Aristotle, is that in fact, we can't use the parallelism as an attribute of a subject. It's not possible. And so, we don't have a way to characterise it outside the particulars. So, the particulars are the right parallel. The parallelism, it doesn't exist. Wait, but that, if you want, for me, it's not what's been marked. It's not what's been marked. Because now, you talk about 50 angles who are couped. So the problem is that we have to find a characteristic of parallelism, and we don't know very well how to manage it compared to the criteria that were given earlier. And so we're going to plant. That's what he explains. So why we're going to plant it, according to him? Because we don't know how to characterize the property outside of the particulars. Otherwise, we would have no problem. Well, this point, sincerely, I swear that it's not complex, and I'm sure that it's very clear. I'm sure that it's very clear. The fact that we can't talk about the parallelism in particular, it's a very clear information for the interpretation of the text. All the time, knowing that there are other interpretations that relate to this error. Yes, I can remove it. Because, finally, everything that was on the relationship, I removed it. No, no, no, we're going to put it on the couch. Yes, we're going to put it on the couch. That's what you just said. That's the question of the subjectivation of the relationship.

35:00 Yes, because for what we're going to do at the end, it's not easy. But you explain what you just said. So the idea is that the domain of the object is the right? No, there's no domain of the object, it's that the problem. There is no domain of objects, in fact. You talk about all the things that are true, but you don't talk about the same thing. What I want to say is that parallel is an attribute of one right. Like commensurable, it's not an attribute of one right. It's a relationship between two right and two right. So, there is a sort of generality for us, of that, the theory general of relations, and we took the idea of the theory general of these relations, but when Aristotle comment this type of relations, he says that there is a default of generality because... No, no, what he says is that we risk to be planted because, just like, we don't have the tools clairs, and he gives us an example, for example, I define the parallelism in saying that it doesn't cut, that the sequence doesn't cut at angle right, and as in fact I just Yes, this one is not very clear, because there are angles in this situation, and so... There is a same sequence or two parallel, okay? It's a couple with the same angle. Okay? Suppose that I have a definition of parallelism, but instead of saying the same angle, I say with the same angle right. Ah yes, but you didn't give the same angle? I don't give the same angle? I don't give the same angle? I don't give the same angle? Yes, I didn't understand the example. because I didn't understand if it was to say that everyone had an angle altern-altern ego, I didn't understand. I don't understand. If you want to comment, I don't understand, but I only do paraphrase the text of Aristotle. He is like that, if you want. Like you know, everyone doesn't interpret it like that. So you understand that it's not comprehensive, otherwise we wouldn't have a bad commentator. I understood that Aristotle said that if we define the parallelism of the right, part de droite sont parallèles si et seulement si parce qu'une droite est perpendiculaire à l'une n'avait pas l'heure là on a on n'a pas la définition générale du parallélisme on a une bonne définition tellement elle loupe un des critères de la généralité une bonne définition et parce que tu dis donc finalement on loupe la généralité parce qu'en fait pour n'importe quel angle ça doit être vrai et donc identifier dans la définition des propriétés équivalentes dont l'énoncé n'est pas suffisamment général It was a force of me.

37:30 I told you, I didn't want to be too much for him. It's because I had no idea what was marked, so I had no idea what was marked, but it's you who talked about the generalities with all the angles... In any case, if you want, you can have an interpretation. It looks perfectly plausible, but in my opinion, it is completely different. It's not obvious that the problem in this definition on the question of saying that I don't transform my relations in a subject. Because the angle right is true for the angle right and for all angles, and there is no reason for the definition of the dissymmetriser in favoring something which is not naturally favorable compared to the other, it's not a thing to refuse to transform my relations in a subject. Exactly, it's not that we are at two different levels, right? No, no, no, no, I'm not sure, it's to say that there is one side what he says, and then there is an interpretation that we can give. So what he says is that we have missed universality. Now the question is, for the reasons you have said, if we have isolated a case rather than a case more general. And all he says is that in this case, the reason for which we have been tromping is because we can't find something on the top of the particulars. And now it is necessary to interpret it. So the way I understand it is that what we can't find out of the particulars, d'un point de vue aristotélicien, c'est précisément les relations. Les relations n'ont le sens que dans les particuliers. C'est ça la théorie d'Aristote. Donc, on ne peut pas faire des attributions sur les relations, et c'est pour ça qu'on s'en mêle les pinceaux, d'après lui, quand on essaie de donner une caractérisation universelle. Cela dit, il pense bien, c'est ça toute l'idée, qu'on peut en donner une. C'est-à-dire qu'il y a bien une bonne manière de caractériser le paralémisme entre ces deux cas. Mais simplement, il veut dire pourquoi est-ce qu'on se trompe. C'est ça, le but du chapitre, to characterize the universality. So the story of not substantializing the relations and all that, it's me who says it, but it's not at all in the text. All that he says in the text is that we can't say something in dehors of the particularity. I think if you want to know what you say, in the text there is that, the paragraph 3 is that, my interpretation is that, it should be pretty articulated. And since there is a lot of interpretation, and you know that you don't know that in saying that there are others, but that you don't know that. There's a flotement in the terminology, you talk about characterised by the fact, and then it's this demonstration.

40:00 We're going to pass from characterisation to demonstration. Well, that's Aristotle. For him, a demonstration, it always depends on an attribution. I don't know. and so we have the right, there is really a conversion possible between characterizing and demonstrating, it is to demonstrate, it is to give an attribution which is catapentos, catoto and catolou but in this case, it doesn't work exactly like that because we don't have a subject, on n'a pas l'intérêt ça je suis d'accord mais donc il mime un peu ce mouvement qu'il avait donné mais peut-être moi je peux l'enlever cet exemple parce que je sais pas s'il faut le développer dès l'introduction d'accord ici tous les trois j'ai l'impression sur le fait que comme tu vas développer une solution qui va se rapporter à le commensurable que le commensurable c'est quand même une nation ouais c'est embêtant il y a un autre truc qui m'a fait penser que ça I think it would be a good thing to keep these three things. I think it's rather the third error, but you can tell me if I'm wrong. Ah, no, it's the second. I think, if I understand the second error, that it excludes the fact that we can do a demonstration by case, by disonction by case. If we have a demonstration by disonction of time, it means that we don't have a real attribution to the subject, that the demonstration doesn't have an attribution to the subject. It seems to me to understand that, but tell me if I'm wrong. In the second error? Yeah. In the second error, the problem is that we have a property that vaut for several objects, I call them objects, he calls them subjects, and there is a rule in the philosophy of Aristotle, it is that we can't pass from a subject from a genre to another, never. So the problem that point in the second, it is that there, we did it, it is absolutely clear, and visible on was right because it was the case to do but we have no idea of the way we can determine

42:30 what is the demonstration we say that it is anonyme it has no name precisely because it regrouped different subjects and then by the demonstration by case if you want that it is another thing it is a problem more general Aristotle has a model of the demonstration which doesn't correspond absolutely not to what there is in mathematics, and so it's very bizarre. For him, it's still a syllogism that s'enchaîne. It's still a model that he gave. And normally it's actually on a model in which all the propositions are attributive. And it's not at all what we find everywhere. Sometimes we find it, but often we don't find it. And he does it as if it was the same thing. But that, we don't know why. He thought that it was a model. So if you want to say what you said about the demonstration by car, we can say it for many other things. If the demonstration by absurdity was authorized, he mentions it. He says that they are a little less weak, but they are still good. He does the demonstration by absurdity for logical cases. And they are validating by the theory of the theory which is a very logical principle for him too. But it's not a syllogism. But I'll tell you something. It's a demonstration by car. If you talk about Aristote simply. It seems to me that, independent of all things, it is secondary compared to your article. It seems to me that in your solution, you are allowed to distinguish the case, and you have a different type of case, and how it is. goupilles dans le traitement généralement. C'est pour ça que je trouve qu'indépendamment du fait que le sujet m'intéresse, c'est un truc qui te concerne. Comment ces critiques rencontrent la question de la démonstration par cas ? Je ne suis pas sûr. Je pense que là, c'est vraiment un truc très particulier. C'est dans le cas où chacun des cas renvoie à un genre. Et ça, ce n'est pas nécessaire. Tu vois, je veux dire, comme ça, il y a un cas très connu qui est la démonstration de la commencionalité de la diagonale Well, it's a demonstration by case. We suppose that it's... I don't know if it's a demonstration by case. No, it's not really a demonstration by case. It's not really a demonstration by case. You suppose that it's rational, but then after it's a pair, it's a pair. It's a kind of a demonstration by case, a pair and a pair. No, it's pretty interesting about the episode where you say that

45:00 there's nothing to do with it. Yeah, it's true. I'll try to think if he gives examples of the elements that I have, but I have the impression that it's not that his problem. If the case is inside the same genre, I don't think that it's the same. I'm going to tell you why I took my question, because to my opinion, a demonstration general, it's a question of my article, or if I don't know what it is, it's a demonstration general, it's a demonstration which vows to each stage for all the elements of the genre. So if you were forced to separate for different stages, you would pass by the paths that didn't establish the attributes in itself. It seemed to me, if you wanted to feel, but after all these areas were established that there was a difficulty for the demonstration in his conception and his demonstration. It's possible. Personne n'a fait ça ? Yes, I can't do it. But I think it's not at all his problem. It's quite clear. You see, the fact that he chose something that doesn't have a name, it seemed to be interpreted as... But when they say that there's no name, it's not an object metaphysic, there's no subject, it's just that we haven't found a good subject? Well, there are discussions to end with the commentators to know what is this story. We don't know anything, we don't know anything, they say that there's no name. I think it's coherent, what I try to explain, if you want, is to show that it's not quite anything, and that the commentators are often fixed on a certain thing without trying to have a view of an ensemble. And when we try to have a view of an ensemble, in fact, I think that it is relatively coherent. I tend to say that all this brings us to the problem that we want absolutely that there is a subject. If there is no subject, in his sense, it is to him. And a subject will be more or less to a genre, in his sense. And if it is not the case, well, he is very embêté. And in fact, it is what we see. But he does not do as if it didn't exist. When there is a generality, but he doesn't know how to attribute it, he expose the problem. Maybe it's related to this ecstasy, you know, when you try to triangle and abc.

47:30 The ecstasy is more for the existence of the object compared to the definition. Yes, but in a case like that, you can't do this ecstasy. Because you can't say abc, it's either number, or... Thank you.