Mathematiques et individuation

0:00 Transcription by CastingWords and the contradictions. Two examples of the theory of the theory are the important factors of the history of mathematics from the beginning of the 17th century and the beginning of the 20th century. The polimic of 1900 and the action of the choice are the 4th of the 1st. The 2nd concern the representation of the very ancient, the representation of the données in mathematics mathématiques, une problématique que j'ai ailleurs appelée dans notre texte la dialectique de l'indéterminé. Elle impliqua celle-ci d'abord l'œuvre de François Viette, avant de devenir une polémique célèbre et vigoureuse entre Frege et Russell sur le statut énigmatique de la variante. Alors, j'en viens d'abord à Cantor, j'en viens à mon premier sujet, à savoir l'action du choix, et Cantor, au début du XXe siècle, la théorie cantorienne of the infinite, in order to share both a statue objectal and a écriture, an aleph, to the infinite, meaning not only to the infinite number of those who have been involved in the two centuries mathématiques previous, but also to the infinite number of the term of Orel. In this context, the question is raised by Cantor of the comparability of Cartier is at the origin of the question of the action. It is clear if you can compare the two cardinals. It is clear that... That is the comparison of the cardinals. And I am interested in the order of the ordinate. Or it is clear that the arithmetic of the ordinate is more subtle and richer than the cardinals. The two ensembles of the cardinal must have different ordinaux, and they must be divided by the study. So, prolonging its démarche in the initial cardinal, Cantor would be able to seize the collection of all the ordinaux,

2:30 all the ordinaux, all the captains, at the same time, and also to be able to define with precision the cardinaux transfini plus grand. And for this fact, he had to constate that he had to be limited to the collection of certain ordinate, the ordinate of the same ensemble that he called ordinate. It is therefore that he gave the first definition of a good order, and in terms of modernity, we say that the couple E to the inférieur or equal is an ensemble bien ordinate, if inférieur or equal is a relation of ordinate E, that all parts of E contain a little bit of E. It results that an ensemble is totally forbidden, I suppose, because all the elements are completely comparable. The bonheur is therefore a total relation, a type of privilégié, analogiquement comparable to the nature of the ensemble. This is something very simple. The protagonist is called Zermelow. It is in this context that in four years of intervalles, 1904 and 1908, Ernest Zermelow made two texts visant, it was a man who liked to régler the problem, visant to règle definitively certain questions cantoriennes, and in particular, what was in suspense, to have the comparability of the others, and the others. He declared, in effect, Zermelow declared, in effect, the result of the following, So, all of a sudden, all of a sudden, perhaps, will be written. According to Zermelow, they have always a way, on an ensemble quelconque, to build a total structure of the total, analog to the entire. We will all of a sudden note that for the case of the R, the R, the R and the R and the Conti, so it will be necessary to find another order than JL. However, if this property had been established, the question of the comparability of Cardulo will result in aussitôt. In 1904, it appears to be a letter from Zermelow to Hilbert. In his exposé limineur fondamental, which I expose here in terms of slightly more modern, Zermelow asks him something, he asks him something, he asks him the result, which we call today the principle of choice. He asks him an ensemble. Qu'à tout ce ensemble, grand M' de grand M soit associé à un niveau quelconque, petit M'1, appartenant à grand M' lui-même, que l'on pourra appeler élément distingué, pas plus petit, élément distingué, ceci dans une correspondance, il y a ma particulier, entre l'ensemble P de M, de toutes les parties de M et l'ensemble M.

5:00 These correspondences being accorded, Zermelow is then in a measure of control, in the end of the article, he is a good order associated with M, in the first stage, and in the first stage, the comparability of cardio, second stage, which is the objective. I conclude on this point. If we admit the choice of choice, we have a good order. Well, when we expose today the question, when we say, it's a subject mathematician-analyst, I'm not a mathematician, I would say a subject of the mundane. The question of the question, is to have the ability to choose an element in each ensemble non-vite. We have to answer the answer obliges and obviously there is no choice. If the question is there is no choice, no choice, no choice, no choice, no choice, we understand that at the same time the question is the question of the subject. So, this problem was developed, you will see, I'm going to say the subject, it was developed of a remarkable correspondence, which was long studied, I don't know about it, but it was largely studied on the name of the five letters. The five letters, it's an ensemble of correspondence that were exchanged four French mathematicians français célèbres qui étaient Baird, Horel, Lebeck et Edvin, et leurs correspondances ont été publiées dans le plus plein de la Société Mathématique de France de 1905. Alors ça se passe sur fond d'antinomie cantorienne et sur fond d'antinomie cantorienne, les quatre protagonistes vont discuter à propos des problématiques de choix et des correspondances to Asia. In any case, there was not only these five letters. There was a volume of correspondence and an ensemble of articles on the same subject that the specialties of the science. I don't know about it. It marks an important number of reflections of an eminent mathématician in front of one of the first byproducts, of the recent theory, that is, the eventuality of an infinite trans-finity non-délombrable of choice. It is not-délombrable because the question of the subject of choice in the case of the non-délombrable's pose intuitively quite a bit different. So on the four protagonists, I have made an article that I will not develop here.

7:30 I'm going to focus on the essential. I'm going to study one completely, which is Lebesgue. And I will give you an indication on Adama. I'm not going to discuss the position passionnante of your own. Lebeck. On the fact of the protagonist, Lebeck is the one whose preoccupations of philosophy mathematiques are the most apparent. He seeks to get the aspects contingent of the debate, this debate on the choice, to provide a signification philosophically to terms of the enjeu. Objet défini, principle of choice mathematiques, existence of an object mathematic. in your comments, which are mainly here. I will first invoke, as it is insistent on the Jeanne de Beek, this question ontologique, raised by the correspondence of Zermelow and of what I called earlier the object and the subject of the choice. These two terms are very nice. Could we demonstrate the existence of a mathematical being without the definition of it? It's the Beek who speaks. Then, of course, the personality of his auteur question pourrait sembler paradoxale ou provocatrice. A son interrogation, Lebel répondra toujours évidemment par la méritie. Mais la question repose vraiment à rebondir aussitôt en d'autres. Qu'est-ce que définir ? Qu'est-ce qu'une existence mathématique ? Et s'y ajoute, inspiré par Zermelo, qu'est-ce qu'un choix ? La réponse de Lebel, que je vous donnerai un peu plus longuement tout à l'heure, est en substance la suivante. Définir un objet mathématique, c'est essentiellement pour lui le Citation est de lui, de pouvoir nommer, c'est-à-dire le représenter symboliquement comme, je dirais, un objet laïglis discerna, que sa représentation permette de l'individualiser. Et pour que soit possible une telle symbolisation, le seul moyen de connaître de l'objet est une propriété caractéristique qui le cerme et l'individualise. Voici une nouvelle sur ce sujet. So, I would like to say, on the convention that I indicated, that is, on the demonstrates the existence of a math method in the definition, and I say, it's not so much now, if this convention is universellement admissible, but I must say that I often employ the word existence in another sense. And I say, for example, I will leave the transparent, I will start with what is transparent, to say that in these conditions, the existence of a mathematical object is exclusively liable to its status of being differentiated.

10:00 It can therefore come out of a predicative of an object, by an exterior and global judgment, which is important to attribution of existence on him, but of the use of and clear of a characteristic individualized property of each object. This is what Lebel says in the following, as a method usual from Cantor, to prove the existence of certain families of non in showing that their complementer is not a reference to the entire because the elements verify a propriety non-universe. Or, he said, So, mis in general and without any precaution, this method allows well to establish abstractly what is the number satisfying to a condition, but not concrètement the way to actually know one. It is clear that Lebeck project, here in this concept, it is very clear that Lebeck, This is evident a subject, a mathématic, implicit, calculator, in which Germain will be recognized. He will therefore substitute, if it is possible, the demonstration of the constructions of constructions to the proposed by others. And there it is remarkable because there is a certain number, I think there was an in-existence, he tried to take this work with the success of the U.S., and to provide a demonstration of an existence constructively. For Lebeck, so, an existence mathematical is authentic, i.e. non illusory, the term is not illusory, if, in any way they are primitively given, we can substitute an existence constructively to the form of the system. An existence mathematical is, for Lebeck, an object differentiated, but that only, it is not necessary for him, and it would also be also very limitative, to demand of course a procedure of calculation, like the Croninger." A brief note on this one. The other side of the contrary of the non-constructions, nor constructions, nor susceptibles to become, the Bay trouve a model contestable in the elements produced by the functions of choice of the Zermelo which are not, in spite of their appearance ontology, to individualize.

12:30 So, in a form even more psychological way, this question of the individualization is the substance of a public remark, always of Lebeck, in the subject of the correspondence of Zermelo, I will give you a transparent report of Borrell in the letter 5. So Borel, Borel wrote this, and Borel wrote Adama, Borel wrote Adama, for to tell you what he said, Leveig. And it's very interesting. I would like to first sign up an interesting remark, made by M. Leveig at the Société Mathématique de France. The 4th May, how M. Zermelot can be assured that in his own reasoning, he speaks of the same choice of the distinguishing element, because he doesn't characterise in nothing for himself. It is not possible to be in the way possible, it is to be coherent with himself. This is a paradox révélatrice, if you will, of the place qu'accorde Debeck to a possible subject of choice. But we understand now how the disruption of the notion of choice has been so central in these polémiques. Jusqu'à... you have to see what there was before, in 1905. Jusqu'à Zermelot, in fact, and at the notable exception of the calculation of the probability, the term of choice would be a part of the vocabulary mathematics subject to discussion. S'il s'agissait of how we calculate the probability of choice on an ensemble fini and deliverable, a procedure of nomination of an element distinguishing was always effective and often implicit. On would like to say the little bit, but in the end of the day, in the end of the day. In other words, there was no time to ask the question which would then appear as a matter of metaphysics, of the subject of the choice. When, after Cantor and Zermelow, the reference could be considered as a transhumanity actual non-dénomable, Alors se fit jour sur certains, dont le bègle, la nécessité d'une distinction de droit entre l'être et le faire. L'être, un choix d'objet est possible dans l'ensemble, et le faire, on peut choisir et construire un autre. Je cite ici le bègle d'avoir faim de l'être 3, qui est absolument impliquable. L'alimentation de l'être est impliquable, mais difficile de la contredire. Je dirais quelle était la solution historique. Vous ne pouvez quand même pas en rester là.

15:00 Deveille, la vie est trop courte. Je dis avec vous ce transparent. Faire un choix, ce peut être écrire ou nommer les éléments choisis. Faire une infinité de choix, ce ne peut être écrire ou nommer les éléments choisis un à un. La vie est trop courte. Il faut donc dire ce que c'est faire. On entend par là en général que c'est nommer la loi qui définit ces éléments choisis. But this law is for me, and for Mr. Adamart, also indispensable whether it be an infinite number or not. If it is not a infinite number, it is still... When I listen to a law, and I continue, when I listen to a law definition of an infinite trans-finity of choice, I have to be very defiant, because I have never heard of the same law, but I know a law definition of infinite number of choice. And this is super, because I have to manage the future, but it is maybe not an infinite number. Choisir c'est donc, pour Leveig, nommer ou décrire. Comme il le dit plus loin cependant, il reconnaît que d'autres en puissent donner une autre définition. Alors je cite ici Leveig. Il est vrai, dit-il, que j'emploie le mot choisir dans le sens de nommer, c'est ce que j'explique tout à l'heure, et qu'il suffit peut-être pour le raisonnement de M. Zermelot que choisir signifie en CA. Alors ce n'est pas... Il semble tendre une perche à Zermelot mais ce n'est qu'en apparence advantageous pour l'intéressé de Zermelow. D'une part, Zermelow n'aurait jamais accepté d'être ainsi mécaniquement enfermé dans une contrainte d'ordre psychologique, à savoir celle de tous vos pensées comme un élément dont il ne veut pas sûr qu'il puisse sortir, et surtout, d'autre part, Zermelow lui-même ne pourrait garantir que Zermelow pense toujours au même niveau. Bon, donc voilà quelle était la condition de... Alors j'ai dit, je vais brièvement traiter celle d'Adamars et celle d'Adamars méritera So for the objective of the conference today, it's the one that is developed, but for the philosophy mathématiques, there are a lot of positions of philosophy possible, and that is also the one that is also passionate about it. According to Adamart in a position idealistic, the question of the alternative is not possible. we have to accept also completely, without restrictions or reserves, the action of the choice in its form the most general, because there were the actions of the choice in the form of the number of others, etc. The choice transfinis exists. What they create exists in the same type that the other objects mathematically defined. This position, which is given to the majority of mathematicians today,

17:30 which I'm trying to explain why, was at the time, however, quite radical. This is what we're going to do when Adamart is opposed to Borrell. You don't have a citation. Conclued again on this point, for example, the question existentielle can be here to the subject of the choice. And on this point, Adamart will accomplish an astonishing work of explanation by syntax. Each of the four positions are present on the action of the choice. In fact, he will affect a different form of a different form, grammatically speaking. For him, in fact, we must use the form passive and write l'ordination est-elle possible? Forme qui exclut, of course, all the subjects. He refuses even the impersonal of the on. Il n'y en a pas de crainte, dit-il, c'est-à-dire inexplicable, de crainte, dit-il avec sincérité, d'avoir à penser ce que c'est que ça te prend. Et sur ce même point, il prête à Baird l'intention profondément détestable, selon lui, de l'emploi de la forme active, à savoir, pouvons-nous abandonner ? Je mets en avant directement la proposition de ce sujet, donc à la marne. Voilà. Qu'est-ce qui s'est passé après 1904 ? Je vais terminer cette première partie, il y a deux parties dont on est exposé. Après 1904, l'axiome du choix fait l'objet d'un important traitement, tant logique que mathématique. Voici quelques brefs films conducteurs au sujet de l'axiome qui fait aujourd'hui consultantiellement partie du paysage quotidien. Je vais tenter d'expliquer de quelle façon. Le mathématicien contemporain n'est plus bien en effet troublé par ses polonies depuis le début du siècle. Alors, j'évoque statut et usage de l'axiome. Alors, c'est des problèmes de nature différente. Le statut de l'axiome est une question de nature logique et se résume principalement à l'examen and independent of the action relative to those actions in the usual models of the theory of the ensembles. The question of the usage is more than in mathematics. In some pieces of theory mathematics, it's the working mathematician, it's the theory of mathematics, the analysis functionally with the theory of Van Banner, or the representation of the object of Goole with the theory of the idea of W.R. 1. On a la situation suivante. On a besoin, pour démontrer un résultat important dans la théorie, on a besoin de l'action de Zorn, qui est une forme équivalente de l'action du choix.

20:00 Mais souvent, on a besoin de quelque chose qui est un peu plus faible. Simplement, il suffirait qu'on ait ça. Alors, l'action du choix entraîne le ça, qu'on va appeler la propriété T. Well, in this configuration, the efforts are held in two different directions. The property of T is-t-il demonstrable without action or the choice? Well, that would be good. And more generally, which mathématiques obtain-t-on without action? Well, that was long enough studied. Mathematics is a good choice. That was the first sense. The second sense, the action of the choice is-t-il a consequence of T? logique entre deux propriétés. On ne peut pas faire mieux. Et donc, ça veut dire que dans la théorie concernée, eh bien, l'action du choix sous sa forme initiale est le résultat nécessaire à obtenir ce qu'on veut. Bon. Je conclue là-dessus pour dire que la problématique précédente a montré l'impossibilité chez certains sujets, tel de vrai, de se satisfaire d'une existence abstraite de l'objet d'un choix. and his argument constructivist is, in my sense, very strong. When he asks the question, can we define, can we define, or this impossibility of satisfaction is exactly coextensive at the present of a subject mathématician of choice. In these conditions, what is happening to this problem? of the choice, the response to the fact of the mathematicians from 1904 had been either the description explicit of an axiom, which we put in the axiom, like the others, although it is not like the others, or the possibility of this non-inscription. But it was not even though that his inscription exclut all the virtuality of a subject mathematician as it was perceived until now. The problem was in fact the nature of the philosophy, and, in order to be resolved, it will have at least been approximately circumstantial. It will be symbolized in an action that we could have added to all the pieces. My second example, so I will terminate with this question of the fact of the choice, I will be all about to explain it. My second example is located at the root of the symbol, and it is much more ancient and more fundamental.

22:30 It appears to be a contradiction inaugural. Well, since the antiquity, the two categories epistémologicals, the données and the requis, have accompanied all questions revêtant the form of a problem. These two categories grecques, the catégorisation had been reprised by l'école. I'm going to expose, at least two examples, the state of the idea before Viet, either in geometry, or in the category. This presentation is absolutely simple and banal. I don't think that I will not learn anything about the text, but it is indispensable for the argument that follows. I will put a transparent on a problem of geometry before Viette. If I don't have a transparent problem, Let's go with the code. For geométrie, I'm not yet. I'm going to say, because of the difference between the two grandeurs and the rapport, you can find these greats. The greats are represented here in the bottom. The rapport is not a number, it's a ensemble of two numbers, that's to say a ensemble of two greats, which are represented on the table. So you can see with me, the inscription in line means resolution in line and resolution geométriques, and so A a or a number equal to the difference. On observe a typiquement rhétorique, suivi... On observe a typiquement rhétorique, suivi d'une représentation du donné par des quillus. Like in Euclid, the representations in the text of the way and the letters of different cultures, such as D or BC, are not operative, but purely designative, and serve only to identify tell or telle donnée, or telle configuration intermédiaire construite. I will not enter here in the discussion between the symbolization and the representation, the letters such as D or BC are inscrites in the context of a grammaire, so it's simply designative and holographic. The solution comes with an instruction rhetorically explained.

25:00 The solution is to play in parallel, to play in parallel, to play in parallel, and to play in parallel. This procedure supposed to be reproduced by all the lecturers was, in fact, the QEPI, the mode convenable of the resolution of the problem of the law. At the moment of the interpretation, because there was an interpretation, because there was an interpretation, because there was an interpretation, and there was an interpretation. I would like to say the evidence. At the moment of the interpretation, one point, however, became essential. The data should be considered as well. The fragment of the statement of the beginning, given that the difference between two grandeurs, pi, etc. was, without context, interpreted as the fact that this difference was one, fixed, but arbitrary. The two terms, arbitrarily and fixed, would legitimately be considered as compulsory, debouchant sur une forme de contradiction entre le singulier, la grandeur est une et fixée, et le général, elle est quelconque et peut prendre toutes ses valeurs à l'intérieur d'un certain champ. La question donc de l'arbitraire et du singulier dans les données, si problématique à exprimer réphoriquement, comment une grandeur peut-elle être à la fois fixe et une grandeur ? Vous pouvez ainsi recevoir une expression adéquate tout simplement par la sanction de la figure géométrique. the problem posed, being conventionally transcribed by this type of representation, and very quickly implicitly. Ainsi, it is convened, implicitly, no plus optimist, that if the figures were supposed to evaluate universellement, their spécificities were not in the middle of the day. In these conditions, the procedure of resolution is a ported, i.e. a suite of instructions rhetorically explained, to the parallel, to the right, to the left, to the left, to the left, etc., the suite of instructions rhetorically explained, should be similarly The result could then be eriged in a theory of geometry. By the use of the capacity reconnus to the geometry, and to represent a universe universe that transcends, which, in this example, would have had to do a procedure inquisitorial, a procedure of research, trouvée de grandeur, which is an annoncé qui laisse penser that we would have to be an annoncé inquisitorial, se constituent ainsi en une méthode générale visant à l'établissement d'une vérité générale. Et c'est bien cette faculté conventionnellement accordée de Fidu qui aura permis l'assomption et la contradiction précédemment évoquées

27:30 entre le 1 et le 1. Donc la contradiction est assumée dans le registre symbolique de Fidu. Ceci est somme toute de l'âge qui comprenait la résolution de la géométrie dans l'Antiquité et au Moyen-Âge. If you have the time, I will tell you earlier my question, that this continues to cause problems to the young apprentice of our time. This is a subject of the act. There are some of them who refuse. The interpretation of the figures, I conclude on this point, I would say that the interpretation of the figures by the geomers has been built since the antiquity, on a convention in general, which is the figure of its singularity to offer value universe. But, in fact, it is to say that two readers, traing in different fields different figures from a man enoncé rhétorique, were supposed to effect two representations concreta, contingent, of a man abstract, extérieur, or two geomètres, as well as the top of the text. In such a condition, it was then evident a perception necessary of objects geométriques idéaux, in the same time that the subject abstract, also idéal, of the knowledge genuines. This was the question of the geometry. I'm going to pass on the transparent, which is also banal. plus le calcul avant-vient. C'est la résolution d'une équation commune. Une équation commune, c'est ce qu'on a appelé l'équation du second degré. Le terme équation du second degré n'a pas le sens qu'après les 4. La hiérarchie des puissances. Donc, la résolution par le calcul d'une équation commune en termes modernes 2x2 plus 2x égale 32. Dans l'algémorale, il y a un ouvrage qui a fait d'art 472 d'une vingtaine d'années avant-vient. The representation of the data is explicit in the form of the figures, 2, 12, 32, and the requis of the unknown is also represented by the sign of the primitive, which I call the creux, and the creux of the, to examine the pertinence of the system of the one and the one, which is pertinent to certain aspects of the other. This is not the subject of this experiment. Montbelli établie d'autre part d'une preuve en nombre, c'est possible. Une preuve en nombre, et la preuve en nombre présente par rapport aux démonstrations géométriques

30:00 des avantages méthodologiques incontestables. D'abord la représentation de l'inconnu en tant que tel. Une représentation X-Ti en tant que tel. Bon, et puis les diverses pratiques important, habituel, que vous connaissez du calcul. Mode hypothétique du raisonnement. Si 2x2 plus 3 est égal à 5, alors 4x2, etc. Même si c'est difficile, c'est bien ça qui est en jeu. Dans une procédure analytique, l'automaticité du calcul, vous avez une certaine automaticité du calcul, toute procédure qui avait été inexistante dans la révolution juridique dans laquelle il y avait 6, 5, il y avait, pour passer d'une ligne à une autre, ceci demandait un minimum d'invention. L'affichage de la solution par Bombélie, donc la dernière ligne, la chose est égale à deux. L'affichage de la solution par Bombélie est ici évidemment non pas une suite d'instructions comme dans le cas en ligne, mais évidemment un nombre explicité. Bon, la mise en regard des problèmes en ligne avec ceux en nombre découvre donc des profondes différences entre les conceptions des géomètres du XVIIe siècle. La figure géométrique avait sans doute une singularité, mais celle-ci était postulée non signifiante. In terms of the calculations, the concept of the 16th century and the other, the value of the calculation was entirely to what was represented by the chiffres, i.e. by the signes necessarily interpreted by the number. This is profoundly banal. The invention of Viette came from the fact to introduce a symbolique by which we could continue to use. The considerable advantage of the preuves in the number is in conservating the universal character of the enunciations and solutions, particularly the consideration of the donnée as a quelconque, which is to say the arbitraire and fixer, which had been the real privilege of the geometry. And of course, I've said, I've said a lot, but it's not only because it's a figure, it's because it was consubstantial at the point of a figure, a convention for the truth of the geometry. Now, I'm going to talk about an écriture symbolique of the Viet, in first of all that for representing, I'm not in the calculation, and for representing the inconnu in the calculation, l'inconnu in the calculation, a primitive non-chiffed was necessary precisely because it was inconnu. It was used to n'importe what, except a chiffre. And there were several solutions, in particular, the Italian system, which we used to use

32:30 and also the letters that are designated as the unknown. The Demarche of Viet a conduit à introduire alors des lettres pour représenter aussi le donné. It is possible to employe des lettres de types alphabétiques différents selon la nature du représenté, donc voyelle pour les inconnus et consomme pour les données. So, in coincidence, this was also posed in the comments of many problems. In the article on the origin of this distinction, it is clear that in general, in a problem, there is less of an inconnu than the domain. So, if you decide to choose the alphabet, it is a consideration of a good sense that Viette does not write. But, by the way, there were some interpretations on the alphabet synaptic. I'll pass. So, here is the position of Ville. For illustrating this démarche, I'm going to take an example dans lequel il commence son hennétique premier. Soit donnée la différence de deux côtés B, l'agrégé, c'est la somme, la somme de ces deux-là, D, qu'il faut trouver les côtés. Voilà. Soit le moindre côté A, le majeur, etc. Et vous avez un calcul, je vous laisse suivre, et le résultat c'est A sera égal à D-B sur 2. Suite, il saute à la ligne et il prend une incensation numérique. Le résultat terminal que communique Viet est alors en substance, si on appelle B la différence et dé la somme des deux côtés recherchés, alors le plus petit côté voudra D-B sur 2. Ainsi en a-t-il été le tout premier exemple historique d'un problème résolu dans le cadre d'une effectuée symbolique purement littérale. resolution purement exprimée, Viet a conclu par une égalité symbolique, c'est, il faut le noter, c'est la première fois, des moins des sur deux, valable quelles que soient les interprétations de BPD, ce que tout le XVIIe siècle, à la suite de l'Aïnis, a appelé un canon d'abord et une formule ensuite. Le terme de canon est le premier qui a été employé, l'Aïnis disait déclarer établir l'état du PPD. La visée initiale de Viet était claire,

35:00 represent symbolically the data in the calculation as well as the inconnu, which was the first to do. And after that, in fact, the utilization exhaustive of the letters of the alphabet constitue désormais celui-ci, the alphabet, like this reservoir of signes non numériques, interpreted as inconnu or donné, in the counterpoint of the numbers, interpreted by the numbers. At the moment of the interpretation, because there was also an interpretation, at the moment of the interpretation, the division of the consons it was in principle to dicter two different interpretations, the unknown and the other. However, this is the fundamental point, this same definition, comprised by the geometry of his time, the geometry of his time of Viet, does not have a contradiction due to this fact simple that we have examined. In all calculations at this time, the domain was only that was susceptible to a representation explicit by chiffres. Dire comme Viet que la console B, par exemple, représentait une grandeur donnée signifiait donc un principe que par le signe B était représenté un nombre fixe à la valeur connue de l'auteur du texte. Dans ces conditions, cependant, le lecteur n'avait certes pas de connaissance. Comment Viet pouvait-il affirmer que B était le signe d'une donnée ? The objection of the time to such a concept of knowledge and representation are exactly the protagonists of the affair of the action and the choice. How is the author himself certain to think about the same value? How is it not here that we write Lebeck about the Zermelow, but to be coherent with himself? How can Viet be sure that Viet always thinks about the same element puisqu'il ne caractérise en rien pour lui-même. Bien sûr. Cette apparente contradiction et ses objections méthodologiques furent donc autant d'obstacles respectables qui expliquent à mon sens pourquoi, alors qu'il est aujourd'hui éclatant, la représentation du donné par l'être a été un élément décisif dans le développement des mathématiques. Personne n'est pas nécessaire, il faut changer d'âge au moment de l'introduction du symboleïde par l'aïe. Et cette modification n'avait aucunement accompagné at the end of the day of the 13th century. Here, there are obstacles in the methodologies of the interior, but you can't do anything like that. Depends on the definition of Viette

37:30 of the representation of the domain, which was contradicted for the intellectuals, it is indeed to return to its practice to understand the use of the effect. And the example, which is still in the table, is then demonstrative of the real form. Placé in front of the B of the previous calculation, everything that each author can and does only inform, it is that the author says that, in the letter, he is a number whom the author considers the value fixed, and he is therefore not the object for him of a requisitorial procedure, of a procedure of research, and which is the same all along the text. How do I write? There is no more universal knowledge of an entity communicable to all, of interpretation inscrites in the notation of the same, all the lecturers must agree, like all the jurors, all the lecturers must agree on the fact that the letter B vows for a certain number, which the value is supposed to be connue of all, in the same way, in the text. That the system, it's me who speaks here, that the system, which is in reality the representation of a convention on the given, and not the explicitation of this given, must be regarded after all, as a way, in a way, in a way, necessary, It's not necessary, it's a theory that we consider it as necessary, to surpass the contradiction inaugurale. However, this disappearance of the explicit in the symbolization of the domain, which is necessary, between guillemets, was mechanically trained in a second at the moment of the interpretation. Elle est à son tour mécaniquement entraînée en un grand second au moment d'interprétation cette faculté nouvelle de voir considérer se donner comme arbitraire. Si en effet la seule information pourtrie par la lettre comme signe était d'indiquer une convention portant sur la catégorie du représenté lui donner et non d'expliciter sa valeur, alors celle-ci bien que fixée était libre d'être arbitrairement choisie. In other words, the B de l'énoncé de Pierre is the sign of an entity given, in the sense of the definition, but of an entity given. In other words, under the sign, we can be able to find, at the same time, the individual a collection to which he pertains. And this énoncé rhetorical contains an ambiguity remediate, an arbitrary fixation, and therefore a contradiction. And if we can, in the natural language,

40:00 to support the pronunciation of an enunciation of a contradiction, or to fix this contradiction in the term, on can by the way, and to inscribe it in the symbol language. This is, at this point, my second conclusion. We can then, apply to the calculation, this same dialectic of the singular and the quelconque, constitutive of the geométric and the ancient, and the corollaires that I've declared are then semblablement in the logistics, object idéaux et d'un sujet corrélatif, abstrait de la connaissance mathématique. Qu'est-ce qui s'est passé après ? Je charge vite sur ce point-là. Ceci n'a pas été mis en discussion. Les protagonistes suivants, Valignon, l'hôpital, Euler, qui ont employé tous un mot ambigu et commode qui est le mot variable, une variable, ou employé adjectivement without any inconsistency definition. If you read the definition that we don't have a barrier and constant, you will never be able to do what that is. So, the history has not been there. What happened in the practice? In the practice, we continued to use the system of representation at the end, and so the practice functioned convenably, but the problem was not regained. But it was a point where the mathematicians were revening sur ce qui constituait un des points essentiels de la réculture quotidienne. Et ce temps a été, c'est sans surprise, le temps du début du XXe siècle, et donc une polémique entre Frege et Russell. Alors, Russell considérait, il s'agit de la variable, la contradiction inaugurale a été baptisée dans l'ombre de la variable où personne n'a une définition correcte, et donc au début du XXe siècle, parce qu'il s'agit de fonder les mathématiques, et quand même de donner une définition consistante. Alors, Russell considérait pour sa part le terme de variable D'une part, dans The Principles of Mathematics, il considérait le concept comme central. The variable is, from the formal standpoint, the characteristic notions of mathematics. The est en italique dans son texte. De l'autre côté, plus loin dans le même texte qu'il reconnaissait, it appears, from the above discussion, that the variable is a very complicated logical entity by no means easy to analyze correctly. L'argumentation de Russell, which is very drôle, in this text of 163, revient à l'écarter toute interprétation de type cinématique inscrite dans le terme variable. Sans quoi il ironise, eh bien, le signe N serait un proté mathématique

42:30 qui vaudrait un le lundi, deux ou trois le mardi, et puis ce qu'on veut le dimanche, et ça dépend du week-end, etc. Donc, et la tentative d'assumer rhétorique... Donc, la tentative, néanmoins, il tente d'assumer rhétoriquement la contradiction. He said that a variable sign M is for Russell nor an element fixer nor all the elements, but only the representation of any, any, and a, a, a, y, which is an English language. And the any is one of the indifinals of the theory of the same. We don't know why it is a pirouette, but it's not a way to consider it, but it's his way to consider it. The position of Frege fut au contraire de reconnaître qu'il se trouvait devant une contradiction rhétoriquement irréductible et ne refusait de la dissimuler en la recouvrant d'un terme de vague, si commode soit-il, comme variable. Ainsi se postille à Russell et dit le mieux est donc de ne pas se servir du tout de l'expression de variable, puisque au fond nous ne pouvons pas plus dire de la signe que de sa dénotation qu'il est variable, etc. C'est un point sur Frege va argumenter toujours avec véhémence. Dans un autre article qu'est-ce a function. He returns continually at the charge in the opposed to another traducteur called Schubert. I will give you an extra significant of this text, which some aspects go incontestable to the principle of the identity – I'll come back to my text – to the principle of the identity of the Manishina. This text is absolutely remarkable. There is no one, it says Frege, no one variable. The absence, the contradiction inaudible, which is in discussion in the end, three centuries after this. There is no number of variables. The absence of non-propres or any number of variables is confirmed. This is the book of Diablo. But we do not designate the variables by x, y, z. This is therefore a way to speak in usage. So, these are not non-propres. Like 2 and 3 are non-propres, non-constants. The 2 and 3 are distinctly effective and assignable. But how do we distinguish the variables by x and y. On ne peut pas donner les propriétés que possède x les propriétés différentes que possède y à supposer qu'on n'y associe jamais quelque chose à ces lettres, ce sera la même représentation confuse tous les deux. C'est exactement, il s'inscrit exactement dans la problématique par la médiane. Et on sait qu'il y a x qui nous a donné ? C'est en dessous, là, c'est un vaste x-a-l-function. C'est toujours la traduction française de Claude Hamer. It's an article that's a remarkable article. It's savoureux, on ne pourra pas s'en faire un pire de gaffe avec deux protagonistes. Russell n'est pas tout à fait sans défense, quand même.

45:00 So, I will conclude on this point and then I will come back to my conclusion for three minutes, five minutes. I conclude that the position of Russell has proposed to renvoy the signification of the letter of Viette to an Eni, even if the last one was indefissable, and thus to attempt to assume, in a certain way, the contradiction inaugural and the fact of Viette. La contradiction inaugurale est ainsi à tenter d'assumer d'une certaine façon donc la contradiction inaugurale dans le registre rhétorique. Une position que rejette donc violemment Frege, dequel n'a cependant pas proposé d'interprétation valide dont le registre signifiant a ce signe symbolique qu'est la lettre de Viette. Alors bon, sans entrer dans la technique...