L'imagination transcendentale en mathematiques (contd.) / Q&A

0:00 I don't know what you said earlier. Yes, I agree with what did you say? On essaie de faire des mathématiques une petite machine à fabriquer des vérités éternelles, et on se trompe. Ce sont des vérités contingentes qui ont un sens à cette époque. Même si c'est le point de vue précédemment comme absolument vrai. Ça, je ne suis pas d'accord, justement. Oui, justement ? Non, mais je... Le mode d'émergence des mathématiques est contingente, c'est d'accord ? Que l'objectivité mathématique elle-même et sa valeur soit de vérité, disons, soit contingente, ça je ne suis pas du tout d'accord ? Non, c'est la valeur de vérité, ce qui fait que les mots sont intéressants. Ah mais ça c'est autre chose, c'est un jugement de valeur que tu portes, c'est différent ? In 20 years, in 20 years, we have to interpret the concept of objectivity, which is something that we have to do with. There is a debate that is also there-dessus. Is that history, I mean, is that history of mathematics is simply to do an ontology? At the end, if there was a strong content, objective of the annonces, we would be happy to do an anthology. I did not say that in an anthology, because recently, we published a great anthology called Sur les épaules des géants, which is comprised of the texts fondateurs of Galilee, of Newton, etc. Or, Francis Balibar a fait dans un compte-rendu la remarque très forte qu'effectivement, donner ce qui est une pure anthologie, c'est-à-dire vraiment sans aucun commentaire, sauf quelques petites introductions attribuées à Hawking, qu'on peut lire dans n'importe quel dictionnaire biographique, scientifique, est-ce qu'on a fait, je veux dire, et c'est des textes très très forts, les primes qui pientent de Clyde et d'autres, c'est des textes très très forts, Est-ce qu'ils ont vraiment une réalité si forte qu'on puisse se contenter de les livrer sous forme d'anthologie ? Alors autrement dit, est-ce que le rôle de l'histoire des mathématiques ou des sciences générales,

2:30 c'est simplement de faire une collection, telles les belles lettres par exemple, de faire une collection de textes classiques avec l'établissement le plus sérieux au sens philologique du texte, mais en se privant des commentaires ? Autrement dit, quel est le rôle du commentaire ? C'est là que je voulais dire. Well, in the perspective of May, what is the role of the commentaire? Well, he-même, it is ambiguous because he gives two interpretations, two lectures of the same verse, if I say, of Clyde, and it is clear that these two lectures do not have all the same sense. And if we take, for example, the Principia of Newton and the lecture that makes Maxwell, which is a very wonderful book, we have the impression that a century later or a century and a half later, we have a very good commentaire, but it doesn't seem like a commentaire, it doesn't seem like a commentaire, it doesn't seem like a commentaire. And, of course, the same text, the same text, as it evolves in the authors, it takes a different point, and, of course, Maxwell is extremely loyal to Newton, and, of course, when we talk about text, we see all the progress. So, for example, when the replacement of the argument purely geométriques is frappant, there are figures of a geometry of complexion, incroyable, with 20 lines, 30 lines, 40 lines, and then the argument much more algebraic that Maxwell can give a century later, and so, when we talk about the content, I mean, the idea of being sociologized, Je déteste autant que vous. Je vois aussi bien les méfaits, mais je veux dire, la notion du texte qui porte ça, c'est plutôt la question de la compréhension, dans l'imagination. It's the rule of comprehension. There is the concept, there is the schématism, which is represented by an image, and these images are more or less adequate. Effectivement, the problem that we asked earlier, is that there are modes of representation of numbers that do not be manipulated by the complicated numbers. When we write 1, 0, 1, 0, 3, we don't represent a number, we don't represent a chain of operations.

5:00 It means to take 3, multiply by 10, and again multiply by 3, and again multiply by 10, and again multiply by 3. That's what it means to write about 103. It doesn't mean a number, it doesn't mean a chain of operations. And the factorial, as I said earlier, is the same thing. So, the imagination is not to describe images or images, we don't do any operations or we don't do any images, but we describe the images that we could do. Like Michel Lazar said, in a bit complicated, there are demonstrations that we do, there are those that we describe. I thought that the remark was very pertinent at the end of an article very complicated of Michel Lazar. démontrer, j'ai fait que décrire une démonstration, et au prix Ronaka ce serait un peu la même chose, enfin des démonstrations très compliquées, il est douteux si on les fait ou si on les décrit simplement. Donc là, c'est tout le sens du, je veux dire, c'est tout le problème du sens du texte en lui-même. Je sais qu'on ne va pas le résoudre de soi à cet instant, je veux dire, je veux quand même le mettre comme une frontière à cette affirmation of the research of content. The research of content, I'm all right. And that there is an objectivity of content, I'm all right, but it's important to be aware of the limits, simply. And that the imagination, it's also the, in the concepts, it's the possibility, but since the beginning, I would say, Clyde, il dit, considérons par exemple pour la démonstration de produit, considérons tous les longs premiers jusqu'à telle limite, prenons-en le produit, ce n'est pas une opération qu'on fait, c'est une opération mentale, une opération qu'on décrit comme une possibilité mentale, avec la possibilité de l'incarner dans des exemples explicites. and the demonstration, we have the impression that the demonstration works if we are persuaded that in each instance in particular, what is written as formalism, as a general construction, will give us something true. There are other arguments. I mean, I don't want to do this at this view of the history of mathematics, but to justify the concept and to imagine conceptually, I mean, there are limits. I'm not saying that it's not extremely important to do it, but you need to be aware of the limits.

7:30 Yes, but in a certain way, the problem at the bottom is, is that the fidelity, at the historical event, is important or not? I'm not sure at all that it is important to be faithful to the historicity, let's say. What is important is what we say and what it means, what it means, what it means, what it means. I'm not a fetishist. If the client wanted to say something, very well, he said it. It's the same for the philosophy. If you read a philosophy, what is interesting is what it means. It's not what the author has been trying to say. We can gloss on what he has been trying to say. But what is important is the sense of what we say today. Yes, yes. The schéma, say, intellectuel, the content of pensée that is behind. If we read Kant, it's not for knowing what Kant thinks, it's for s'aider de Kant to think about ourselves. And the mathematics text, it's a bit for that. It's also for s'aider to think about the problems. It's not necessarily to know what Galilée thought about Galilée. In other words, it's impossible. It's impossible. It's why I'm even trying to come back to things that, at the moment, I'm very nervous, like the way to practice the history of the science at the Dieudonnet, which was very charged to presuppose a little bit on the signification, etc. But, of the other hand, the advantage of a practice like that, very chargé de présupposés méthodologiques c'est qu'elle apporte quelque chose elle permet aux gens qui lisent cette histoire-là de penser avec elle, d'avoir une certaine intuition d'un mouvement historique qui chez Dieudonné se concrétisait par la pensée structurelle qui a apporté quelque chose inévitablement cette approche-là historique il y a des gens qui ont lu Deudonné qui ont tiré quelque chose sur l'évolution de la pensée qui n'auraient pas tiré de quelqu'un qui aurait pris les textes un par un at the time they would pass in such a context. Yes, we have to know, and we don't know, it's not, it's not, it's not, it's not, it's not, it's not, it's not, it's not. Well, I've been watching a lot, I've been watching a lot, I've been watching a lot, but if it's a classic chaos, it's because the novel,

10:00 by a new culture interactive d'Alder, who has already been traversed a few years ago, and he was wondering why he could regret it. Lui, Arexia, he was born. That's true. For them, it was the most important part of the world. The classics, the electorate classics, and that's why they don't have the classics. I don't think so. Thank you. The last 20 minutes of the seminar will now be re-recorded from the website in order to ensure that we got all of the passage of the changeover. He was talking about ontology and the beginning of the question and answer session, including the question from Cartier. this is starting from one minute from one hour 20 minutes and 39 seconds into the entire lecture including the question and answer session leaving just under just under 20 minutes to run. Il faut saisir bien votre pensée peut-être. Je voulais revenir sur ta lecture de Veil au sujet de Heath et sur ce qui est réproche. Dans un certain sens, je suis tout à fait d'accord avec ce qu'il dit Bernard Tessier, il va passer loin.

12:30 And in this story, there is an aspect of the transcendental constitution, which is to put in evidence to get to this intuition pure in which habite the mathematician. And in this transcendental constitution, there is, of course, the but, what is important, not only in a social story, but really in the cadre of concepts placed in place, but also in the premise. At this subject, my lecture of Heath was really... I was surprised when I read that, in 1908, that the first theorem of the first book of Cliff was not demonstrated. On prend a segment, on construit a triangle pilotaire, and it says that, since there is no point, it presents alternative, évidemment, une preuve dans une axiomatique hibbertienne. Et c'est dangereux dans le sens de pousser au maximum ce qu'il veille aussi dit, sortons la structure conceptuelle. Et dans ce cas-là, il prétend que l'axiomatique hibbertienne, avec aussi le continu en la Cantor, soit la façon implicite. Ce qu'il y a derrière dans l'approche de Hocklid, évidemment, c'est un autre concept de continu, qui est là où les points sont aux extrêmes but they are built by intersections of lines, and in fact, we obtain a point. This should be inherent in an analysis. Do you think that Weyl would be able to do this side? Well, Weyl, it's good that Gilbert has given several traumatizations of the geometry euclidian, so he would not be a gross error like that en renvoyant Euclide au continu des réels, des choses comme ça. Tu es donc d'accord que derrière la construction... Non, mais ce n'est pas parce que la construction a une certaine objectivité qu'il est facile de rendre compte, ou en tout cas qu'on peut la décrire de façon convenue, telle qu'on la décrirait dans un manuel aujourd'hui. Mais tu es d'accord que de toute façon il y a une certaine objectivité That's right, it's a construction of objectivity, but it has its contingency. It depends on what we call contingency. It's a story, in a sense not banal, not psychological, not sociological, but it's an

15:00 story of a contingency. There is a continuum. Yes, I agree. But it's a story non-tempore, so it's... La difficulté, c'est de saisir le fait que toute constitution est contingente, autrement elle ne serait pas là, et ça n'empêche qu'elle soit objective, justement pour ça, et qu'elle soit efficace, pour cette même raison. Bon, évidemment, de toute façon, il faut toujours tout nuancer, donc, bien entendu you have to answer both on the form and on the form. The message that I wanted to pass is that you have to prioritize the objectivity of the knowledge, the dimension objective of the knowledge, when we do a type of analysis which is based on mathematical objects, which is the form we do. Because if we do not do that, on very quickly in the n'importe what. I agree with what you said, but you know, I don't want to confuse the knowledge, the progress of mathematics, there is no history objective. Because if you take Euclide, I agree with you also, but if you take Euclide, There are lectures of Glead who are almost contradictoires. If you take Fowler, the role that he gives to Glead, the role central, if you talk about the fractions continue, it's almost contradictoire with the analysis of Dunway. I would encourage you to go to the extreme, and I will go to the article of the Nets that you said, and tap on the sociology of the history, the sociology of the history, according to what I'm sure with you, but I think it's difficult to reinterpret it. on dirait que si on t'écoute on avait un peu l'impression qu'il y avait une histoire absolue, qu'il y avait une histoire des mathématiques absolue, pas représentée par vrai parce que je ne veux pas polémiquer surtout ici, mais il n'y a pas d'histoire absolue ça n'existe pas ça l'histoire ça ne peut pas exister à mesure où c'est une science humaine, ça ne peut pas exister je crois que ça se mêle dans le cas que tu as traité ça se mêle avec la polémique

17:30 du, disons l'universalité, la non-existence du miracle grec et le tiers-mondisme en mathématiques alors ça se mêle un petit peu là parce qu'il a voulu relativiser mais je crois qu'on peut avoir deux lectures différentes par exemple même du traité de Clyde non mais sans doute le problème en fait c'est je pense It depends on the level of precision of the interpretation that we want to give, the level of precision of the description. But if there are mathematics who are formalized in a relatively precise context, where the concepts are not yet crystallized, etc., which is what happens necessarily. Of course, we have a risk of trying to have a very precise in which the concepts must be defined. We have a risk of determination, say, of the sense or of the port of this theory. So there is a certain methodological prudence. In this sense, there is no definitive interpretation which can be given. to not succeed to give a perfect content to propositions of the type of content or any type of math or the formalization has not about. So that is true. In this sense there is no absolute results obtained by the history. that does not make the dynamic aspect in the constitution of the knowledge that is what is interesting. The fact of searching to determine the enonces in attaching to a context axiomatic or format very precise is not necessarily that which is interesting. We are not forced to search to give an essence Au-delà de ce qui est dit, on peut se contenter de les lire et de penser le sens, le contenu de ce qui est écrit, sans surdéterminer ensuite cette signification-là.

20:00 Effectivement, le problème sur la description du nombre comme une collection d'unités qui réapparaît tout au long de l'histoire des mathématiques, si on cherche à lui donner un contenu extrêmement précis en ordre de la théorie des ensembles, on est bien embêté. Parce que comment est-ce qu'on va faire pour définir l'unité ? Comment est-ce qu'on va faire pour définir ce que c'est qu'une multiplicité ? Rousseau a essayé de faire ça dans la philosophie de l'arithmétique. philosophy of arithmetic is to think philosophically and rigorously these concepts that have been used for centuries, what is it an unit, what is it an multiplicity, how can we describe it philosophically. But in terms of mathematics, we can't do it in a way satisfying, without using formalisms which go beyond what was thought by the author of an So I have a little impression that most of these debates on the history of the science are brought by the desire to interpret or determine the text. Which is not necessarily a good thing. mathématiques et la machine à fabriquer modalités éternelles, et on se prend. Ce sont modalités contingentes qui ont un sens à cette époque, même si ce sont absolument vraies. Ça, je ne suis pas d'accord, justement. Le mode d'émergence des mathématiques est contingente, c'est d'accord. Que l'objectivité mathématique elle-même et sa valeur soient I don't agree with that. Well, it's a value of a value, but it's interesting. But that's another thing, it's a judgment of the value that you port, it's different. Thank you.

22:30 There is a debate that is also there-dessus. Is that history, I mean, is it just simply an ontology? At the end, if there was a strong content, objective of the annoncer, we would be happy to do an anthology. I did not say that at all, because recently, we just published a great anthology called Sur les épaules des géants, which is a copy of the texts fondateurs of Galilee, of Newton, etc. Or, Francis Balibar a fait dans un compte-rendu la remarque très forte que, effectivement, donner ce qui est une pure anthologie, c'est-à-dire vraiment sans aucun commentaire, sauf quelques petites introductions attribuées à Hawking, qu'on peut lire dans n'importe quel dictionnaire biographique, scientifique, de ça. Est-ce qu'on a fait, je veux dire, et c'est des textes très très forts, les primes Kipia de Kline et d'autres, c'est des textes très très forts, Do they really have a strong reality that we can be contented of them to deliver them in the form of anthology? Or is it the role of the history of mathematics or in general, it's simply to do a collection, such as the Belles Letters, for example, to do a collection of texts classic, with the establishment of the most serious, in the sense of the philology of the text, but by depriving the commentaries. What is the role of the commentaries? That's not what I wanted to say. And in the establishment, in the perspective of May, what is the role of the commentaire? Well, he himself is ambiguous because he gives two interpreters, he gives two lectures of the same verses, if I say, of Clyde, and it is clear that these two lectures have not at all the same sense. And if we take, for example, the Principia of Newton and the lecture qu'on a fait Maxwell, qui est un petit ouvrage absolument merveilleux, on a l'impression à la fois qu'un siècle plus tard, ou un siècle et demi plus tard, on a un très bon commentaire, mais qu'il ne se veut pas comme un commentaire, qu'il se veut comme une réécriture, enfin, qu'il se présente comme une réécriture, mais pas comme un commentaire. Et bien entendu, je veux dire, le même texte, au fur et à mesure qu'il évolue dans les auteurs, il prend un poids différent, And yet Maxwell is extremely loyal to Newton, and yet, when we talk about text, we see all the progress.

25:00 For example, when the replacement of the argument purely geometrical, it's frappant to Newton, there are figures of complex complexity, incroyable, with 20 lines, 30 lines, 40 lines, and then the argument much more algebraic that Maxwell can give in a century later. So it's always, when we talk about the content, I mean, it's, I mean, the idea of the sociological idea, I hate you, it's not a sociological idea, but I mean, the notion of the text, it's rather the question of the comprehension, I mean, in the imagination, the imagination transcendent, it's the role of the comprehension. There's the concept, there's the schématism, which is represented by an image, and these images are more or less adequate. Effectivement, the problem that we asked earlier, is that there are modes of representation of numbers that are not to manipulate the numbers. When we write 1, 0, 1, 0, 3, we don't represent a number, we don't represent a chain of operations. It means to take 3, multiply by 10, and again multiply by 10, and again multiply by 10, and again multiply by 10, and again multiply by 3. That's what means the writing of 103. It doesn't mean a number, it doesn't mean a chain of operations. And the factorial, as I said earlier, is the same thing. So, the imagination, it means that we don't do images or images, it means that we don't do images or we don't do images, but we could do images that we could do. Like Michel Lazar, in a little complicated demonstration of Algev, there are demonstrations that we have, there are those that we describe. You know, the remark was very pertinent, at the end of an article very complicated by Michel Hazard, he said, at the end of the day, he said, I didn't demonstrate, I didn't do that, I didn't describe a demonstration. And for Ironaca, it would be a bit the same thing. It's a very complicated demonstration, but it's doubtful if we do it or if we do it simply. So, it's all the sense, I mean, it's all the sense of the text in itself. I don't want to solve it at this moment. I want to put it like a frontier to this affirmation of the content.

27:30 The content is all right. And the objectivity of content is all right. But you need to be aware of the limits, simply. And the imagination is also, in the concepts, it's the possibility. But from the beginning, Euclide said, for example, for the demonstration, consider all the longs first to the limit. Prenons-en the product. It's not an operation that we do, it's an operation mental. It's an operation that we describe as a mental possibility, with the possibility of the incarnate in examples explicit. And we have the impression that the demonstration works. We are also persuaded that in each instance particular, what we describe as formalism, as a general construction, will be something true. There are other arguments. I don't want to oppose this view of the mathematics, but to defy the concept and to imagine I mean, I want to say that the content concept is that there are limits. I'm not saying that it's not extremely important to do it, but it's important to be aware of the limits. But in a certain way, the question is that the respect to the historical event is important or not? That's our problem. I'm not sure at all that it's important to be faithful at the historicity, let's say. is what we say, what we say, what we say, what we say, what we say, what we say, what we say, what we say, what we say, what we say, what we say, what we say. I'm not a feticist. If Euclide wanted to say something, very well, he thought it was very good. It's the same for the history of philosophy. If you read a philosophical text, what is interesting is what it brings us to. It's not necessarily what the author has been trying to say. We can gloss on what he has been trying to say. but what is important is the sense that we have today. The schéma, let's say, the intellect, the content of the thought that is behind. If we read Kant, it's not for knowing what Kant thinks, it's for pensar, for s'aider de Kant to think about ourselves. And the mathematics text, it's a little for that. it's for to think about the problems it's not for sure to know what Galilee

30:00 thinks in plus it's impossible it's why I'm trying to talk about things that at the moment m'enervaient a lot the way to practice the history of the science of the dieudonnet which was very charged to pressuppose a little bit on the signification etc but from our side the advantage of a practice like that very chargé de presupposes methodological it is that it brings something to people who read this story there to think with them to have a certain intuition of a historical movement which should be given to be concretized by the structure which has brought something inevitable this approach there is historical people who have who had something to retire on the evolution of the thought, who wouldn't have been of someone who took the text one by one who would have said, here, at a time, he passed into a certain context. You know, we have to be aware of it. We have to be aware of it. You know, it's not a fact. It's not a fact. Well, I have been watching a lot. Yes, I have been watching a lot. But I have been watching a lot on the fact that If it is classique, Gauss or Elvide, it is because it is said that by a new lecture attentive, Daltaine, who has already been traversed by a hundred years ago or a hundred years ago, he could perhaps have reached a hundred years ago, and he could have reached a hundred years ago. That is true. For Veil, it is the most important part of the history of the book. Thank you.