Infinitesimal geometry & Hermann Weyl on continuity (contd.)

0:00 I'm going to give you a French text. I'm going to give you a French text. I will try to show you what the idea is, so it's easy to just stop there. I will help you with the restricted procedures, in the sense of this constraint analysis proposed by Ray. There is therefore a stratification at a level. There are ensembles starting from the given categories and the fundamental properties. So we start from the given objects, and by using this principle, we build first-level ensembles, then second-level ensembles, etc. First-level ensembles are those that correspond to the properties and fundamental relationships, not the first extensions. Second-level ensembles are those that are obtained from the previous ones. All of these are defined by mathematical processes, i.e. the mathematical processes, i.e. by the construction of a judgmental diagram by means of these principles and so on. There is a fundamental problem which is related to the stratification of the ideal objects. This problem appears in full light when we consider the question of the construction of relations.

2:30 By using these six numerous principles, we can build a judgmental scheme or an R-relation in which certain classes, which are related to the categories of the whole of the first level, are filled by the IA, i.e. a relation, we can generate another relation by this very particular arrangement, which is the arrangement of the distances IA to Z of the relevant category such as One of x, y, z and two. If instead of... here I have x, y, z with three empty places which refer to the category of the objects given. But here we have all the title in the small page. You can very well have... I don't know if I... You can very well have that. This simple example. If from that, we read the 6th scheme, the 6th principle, that is to say the filling and the differential. X and Y refer to the category of the given objects. X refers to the category of the unidimensional sets of objects of the basic category. The question is what is the status of this relationship? Suppose that we generate from a relationship There are two places, R of x and y, because this is analogous to this, since we have filled the empty space with X. The question that is asked is what is the relation of this type of relation to the primitive relation. This R relation that I have just constructed is such that there are only empty spaces left

5:00 that relate to the fundamental categories of the objects that are given. The reference to unidimensional ensembles. A second-level relationship, R, cannot be realized or a first-level relationship that has a certain property. And so we can see here the conceptual difference between relationships of this type and fundamental relationships. Conceptually, an existential quantification on unidimensional ensembles. A second-level relationship, we go to the first-level relationship. Valid, we are forced to see that it is a relationship of a different kind. We touch here the fundamental logic fact, the fundamental logic reason that justifies the construction. That is to say, we cannot ignore this difference between relationships built in this way and fundamental relationships. This difference that leads to the concepts that are involved in the construction of the relationship. These two relationships, these two types of relationships appear as relationships that link the objects of the basic domain. So there is an idea to schematize or dramatize this conceptual point. If we talk about objects, objects that are given, for example antinaturals, we have relations that have a meaning and a determined extension, these are the fundamental relations. And on this base, we can build, we can erect a system, a system of, an architecture of abstract objects, of derivative relations.

7:30 What matters to Weil is to make right this difference. These derived relations do not have the same logical status as fundamental relations. Obliging this difference necessarily leads to circularities. You see the problem. If we neglect this difference, that is to say if we flatten all the relations as if, like a pie of cake that would collapse, all the relations are on the same level, on a level of equality, then we fall into the vicious circle of Albanics. Let's admit a perfectly free use of the enumerated construction principles. And in particular the principle of existential fulfillment, then we generate judgmental diagrams that do not translate any objective connection. So you see there, it's very clear. From this idea of stratification, we have a diffraction, we have two schools. Either we neglect the stratification, it is the voice of classical analysis for Weyl, and it is linked to the vicious circle, illustrated by the demonstration of superior board theorists. If we understand, all these relationships are on the same logic plane. On the other hand, we could very well build a hierarchical analysis in this way. But then, we come to a second drawback, which is that it would be a perfectly logical analysis whose structural logic would be perfectly... Let's build an architecture of judgment that would not translate into any objective connection, that is to say that we counteract the epistemological reason, the epistemological requirement that we mentioned earlier, that is to say, we would have a rigorous one, but which would not be in the rigorous logical constructions of judgment, in the symbolic constructions of mathematics, but it would be, this construction would be sprinkled, laden with non-fulfilled judgments. So you see, the path that I choose... It consists of avoiding these two axes. This is what he calls the restricted procedure.

10:00 So it's paragraph 1, paragraph 6, the restricted procedure. It seems natural to restrict the explanation of the concept of existence to the basic categories, and consequently, during the iteration of the mathematical process, to use only the two principles of fulfillment 5 and 6, to use only the two principles of fulfillment 5 and 6, only for places where they refer to a basic category. The construction of a kind of formal system for which the five and six principles, individual fulfillment and existential fulfillment, are limited to the basic category. And it is this clause that makes all symbolic construction not lose contact with this ground of the elements given in mathematics, objects and relationships. This clause is enough to guarantee that the judgment schemes that we could build are not purely linguistic, but have some content. And if we try to look at it more precisely, to give an overview of the way in which we put these principles into practice, we can feel that this procedure is restricted. You see, it's very clear, I insist. Every time we intervene here, we authorize, here in the case of the pre-principle 6, the clause that there is only for basic categories of objects, so it is an element of restriction, it is an extremely strong restriction of the biological law. We can see that this restricted procedure leads to the set of functional complexes in a group and will increase the set of functional complexes made between the objects of the basic category, so that the passage to the second level and to the higher level does not add new sets and functional complexes of this species. This is logically very elegant. We can see that by imposing this condition, this restriction, well...

12:30 We live the stratification of substance, that is, we do not add a set of functional complexes to those that are already there. When we follow the restrained procedure, we no longer need to distinguish the different levels, since the level at which a set is located is already determined by the type to which it belongs. For example, a three-dimensional set in which the first two quasimeters belong to the basic category, the latter to the one-dimensional set of objects of a certain basic category, must be arranged in the second group. There are six principles, the principle of substitution and the principle of iteration, and you will see very concretely how things work. I introduce the natural concept of function. We start from a relationship of five places, R, U, V, X, Y, Z, which we distinguish in fact in two categories. The independent empty places, U, V, and the empty places, excuse me, there is an error, the dependent empty places. If we replace independent empty spaces by objects of the appropriate categories, we obtain from R a new relation to two empty spaces, which correspond to a two-dimensional set. You can see what we're doing, it's very simple. We start from a relation where all these classes refer to the objects of the basic category. This is equivalent to the notion of parameters in mathematics. If we fix x, y, z, we obtain a relation here, a relation at two places, u and v. If we give it a name, we give it the name of X, Y, Z. So here, a parenthesis is missing.

15:00 And we can see that we go from A to A, and we have a kind of construction, a mathematical process, a construction of the ideal object. Since we give this object, which presents itself as a function, we can express the same relation by saying that A and B belong to the set that corresponds to this ideal object. As you can see, this is something that is very much related to algebraic manipulations, this is abstract nonsense, and we see that through symbolic manipulations, in fact, we manage to introduce a subject-type object. What is fundamental is that all these superior types of objects must be rooted in the relations that concern the objects of the global category. And we must always be able to bend or bring the abstract objects back to these relations. So from there, there are two principles. Using this notion, there are two additional principles, the principle of substitution. If we take two relations, for example, a relation with five places and a relation with three places, and in this relation with three places, we consider that U is a empty space that belongs to the category of two-dimensional objects whose empty spaces belong themselves to the categories associated with the empty spaces U and B, that is, to the objects at the bottom. At this point, we can make the following substitution. That is to say, to generate the function which is, you see, f, which is derived from the relation at the end, and this function will be substituted in the relation e. But what matters here, I accelerate, what matters is that at each step in this conceptual construction, this symbolic construction, at each step, there is the e. The engendering mechanism must be perfectly clear and governed by the risk process. This is what guarantees that conceptual constructions have a sensible meaning. We will see how this works in a few minutes. The principle of iteration, I will go through it very quickly, consists of using this process from time to time.

17:30 What is the relationship between the analysis of mathematics and physics? Which is the finite relation? If we need a short word, I would call the relations in question, the relations that are generated by the constraint procedure, finite relations. So, finite, how do we translate them? Largely translated by finitivist. What poses a problem in the sense that the Finitists already intervene a sense in the logic of Hilbert, the senses are different, so the Finitists pose a problem. The English authors here propose to translate by something that would correspond to the idea of ​​this restriction of the Restraint Procedure. To find a translation, it corresponds very precisely, these are the relations obtained by the Restraint Procedure. So, we can propose delimitations. We can see very clearly what is at stake in the case of fractions. In the case of fractions, it is particularly clear. We can clearly see what the fraction object is. The fraction object, the sign of a fraction symbolizes, so there is a whole philosophy of what is magnetic symbolism. The sign of a fraction symbolizes the operation composed, in the sense that two fractions are equal when the operations of these delimitations lead to the number of results for each vector a. In the case of fractions, we see that wherever they serve to measure magnitudes, fractions are present as multiplicators. Multiplicators of magnitudes. And by taking the inverse operation of division and by comparing the two, we see that we have something like a kind of operating system that is captured by symbolism. We can clearly see how to obtain it from a relationship. This is how Weyl constructs rational numbers and real numbers to form a specific definition.

20:00 All constructions are rooted in the whole numbers in terms of properties and relationships, which are even constrained by the recent process. Weyl has indeed delimited or finite the properties of basic relationships as well as those that are derived All of these are derived from the application of the logic principles of construction, i.e. the general principles, 1 to 4, the principles of completion, restricted to the basic categories, and finally the principles of substitution and iteration. And we can see that with this whole collection of principles, we have always been able to specify... Logical principles that allow us to extend what is particularly clear for fractions. In the case of fractions, it is the paradigmatic case where we can see here that the symbol for a fraction captures an operating chain. And that is the direct idea. It is necessary to build, to authorize logical constructions as long as they allow operating chains to exist. To propagate, to rise in the architecture of symbolic constructions, to avoid the dead wood, the dead wood of expressions that would simply be linguistic, spartan, without real content. So we can see what is happening in the fractions. We extend this to the rational number. In a very simple way, here too, we start from a relationship and we determine. This corresponds to very common things in the fields of mathematics and mathematics and psychology. Where the things where Weyl stands out from its predecessors is obviously when it comes to defining real numbers. Weyl can then define real numbers as open, clean and rational initial segments of emissions. Real numbers are therefore ensembles.

22:30 There are 4 dimensions of an entire number with certain properties. Segmental missions, in fact, we must not authorize the totality of the real law, we are talking about the rational, the colloquial. These are the segmental missions of a rational which has certain properties, which is built exactly in the same way as what has been indicated for fractions, that is to say, from a relationship of 4 empty places between entire numbers. In this perspective, the concept of real number is presented as a delimited relationship. What matters to Ryle is, of course, to build a concept of real number that corresponds to a delimited relationship. In the sense that we have seen earlier. And in this way, that's what matters, is that if we attach ourselves to this concept of real number, we can talk about a totality of real number. The whole of all real numbers is a notion that makes no sense in the framework of this restricted analysis, and we can therefore constitute a new category of available objects, real numbers, called MN. Now we come to the point that interests us, which is that this totality of objects is not the real law of classical analysis. And this appears very clearly, precisely, Because this conception, to which the restricted process leads, leads to a deep shift in analysis. With the definition that is given of real numbers and the sequels of real numbers, we see that the words there are all and each appear only relative to the natural number. That's the idea, to ensure that the quantifications are always referred to the available objects, And in this way, we see that some theorems of the analysis are maintained and others must be abandoned. What is maintained is, for example, a sequence of real intervals in which the size of the middle and the width of the edge determine the real number. We can extract a real number in this way, in the restricted sense. Among the conserved theorems, a monotonous growing sequence bordered by real numbers converges towards a real number.

25:00 I don't know where to hear real numbers in this restricted sense. But there are theorems that are separated. The theorem of the upper bound, for example. And so, this is where we come to physics. Finally, not to physics. It is that it continues to completely remediate the concept of the continuity of functions. Because the continuity of a function, no matter what we do, is not an unlimited property in the traditional concept of traditional mathematics. We see here that this classification, in fact, the conceptual moment that is attached to this classification is the data of a closed totality, ideally closed, which is not in fact available. Where is the problem? The continuity of a function is not an unlimited probability, that is, to decide whether a function defined by means of our principles is continuous or not, we must respect the totality of natural numbers, but also the totality of the ensembles, more precisely the four-dimensional ensembles of natural numbers derived from the combined application of these principles, with any degree of complication. If we take these principles of definition as an open system, That is to say, if we reserve the extensions by adjunctions of new principles, the question of knowing if a given fraction, a given function, excuse me, if a given function continues, will also remain open. That is the fundamental point. In fact, Pfeil calls hyperanalysis the system we obtain if we add the real numbers to the whole numbers. These are the basic categories to which the fill-in principles can be attributed. If you remember the 6th principle, this one with the star, there is a y such as u of x, y of t.

27:30 Here we authorize the formation of a judgmental diagram where y could be sent to real numbers. By authorizing principle 6, the principle of completion, in fact we build, we considerably enrich the domain of real words, and so it is in this sense that Weil poses the problem of the continuity of a function. If we assume the principle is limited, the procedure is limited, the system of principles, we see it as an open system. We cannot answer definitively the question of whether a function is continuous or not. So, I'm going to go to... She understands the importance of this restricted analysis for physics. In the paragraph 6, she proposes a kind of historical recapitulation. The concept of function has a double basis. In the first place, what leads to it are natural dependencies that dominate the natural universe and which consist on the one hand when the states of things and the properties of real things vary in time, which is the variable of independence par excellence. On the other hand, in the causal relations between action and consequence. In the second place, an entirely distinct root of this is found in arithmetic and algebraic operations. As a result, traditional analysis is vaguely represented by a function, a formal expression of independent variables measuring the finite number of applications of the four operations and the few transcendental relations. This is well known. Look at these two sources of the concept. This is what we need. The point of confluence of the two sources of the function is the concept of physical law. It consists in this that in a physical law, a relationship of dependence given by nature All of these are represented in the form of a function constructed in a purely conceptual and arithmetic way.

30:00 The laws of the string theory are the first important example. The evolution of mathematics in modern times led to the realization that the principles of construction specifically algebraic, apart from traditional analysis, are much too narrow. As much for the physical and logical reconstruction of the analysis as for the role that should be assumed by the concept of function in the knowledge of the laws, laws of the processus quaternary. The principles of general logical construction must replace these general principles. We clearly see the purpose of this reconstructivism. It is a matter of substituting the old conceptual sharpness of the arithmetico-algebraic means. These arithmetico-algebraic operations were the symbolic resources from which we tried to express natural dependencies, useful to characterize the laws of physics. But these conceptual arithmetico-algebraic resources turned out to be too narrow. It is therefore necessary to substitute these resources for a set of much more powerful resources, namely general logic principles. But the goal is always the characterization of natural dependencies, these general logic principles. Obviously, this is what was done in the 19th century, at the end of the 19th century, in the 19th century. Substitution of general principles. But they must not be too general, because if they are too general, we build traditional analysis, which is actually full of dead wood, to put it that way. In other words, non-reputable judgment schemes, in other words, the idea of characterizing very accurately, in a more precise and exact way, these principles of construction that will allow precisely to do, that will serve the project of physics. So, I will end there, intuitive content and mathematical content. To accept the gap or the divorce between the intuitive continuum and the arithmetic continuum. Between the intuitive continuum and the mathematical continuum. There is a gap. This gap is patent.

32:30 We rely on the analysis of Husserl in the IBM. The distinction between the phenomenal time and the objective time. This is the well-known phenological analysis of the retention-protention structure. In the great tradition of German psychology, it is Karl Stuhl. The characteristic example is the clock hands. In fact, every moment is not an isolated point. The elements of time, as experienced ... Intentional elements are not isolated elements. If we deconstruct the flow of experience, as we know it, into an element, into an isolated point, we cannot reconstruct it continuously. Each element captures in its structure, encloses in its structure, what precedes it and opens up to what follows. In the astronomical experience of the entire clock, we do not hear in the same way the second couple, the first, the third. In a certain way, each sensation has this structure of retention and protension which means that the prior stages of intuition disappear but are retained in the present. And we draw the following lesson, we bring them together, you see, this phenological analysis of the intuitive continuum is closely linked to what we said at the very beginning concerning moments as points of transition or points of surface as two-dimensional elements of unigenerism. Time phenomenology analysts take the following conclusions, the following consequences. An isolated point in time has no independence. In other words, it is by itself a pure nothingness. It does not exist as a point of transition. And on the other hand, it is included in the essence of time to allow nothing to be fixed exactly, but only in an approachable way.

35:00 And in fact, these two ideas are central to understand the analysis they propose. The idea of a point of transition which, in the case of surfaces, corresponds to these sub-elements that are the points of surfaces. Something that has, in a certain way, that does not break the texture of the surface, and on the other hand, the insistence on this notion of a mathematical determination, close and not exact. Any attempt, any theory. Mathematics of the continuum must take into account the following facts. The conclusion of the problem. First, exhibiting an individual point is impossible. Points are not individuals. They cannot be characterized by properties. The continuum of real numbers in the sense of classical analysis consisted on the contrary in individual points. And so we understand the legitimacy, the epistemological justification of this restricted analysis, which is the first point. Secondly, what Weyl calls hyperanalysis, that is to say the system that is obtained by adding real numbers to whole numbers as a basic category, this hyperanalysis is such that there is more of a set of real numbers than in restricted analysis. The propositions that are valid in abstract analysis, in the sense of Weyl, are no longer valid in hyperanalysis. So what matters is precisely that, for Weyl, analysis alone, in its own sense, in the sense of abstract analysis, and not hyperanalysis, provides a theory that can be used for continuous and possible dependencies between continuous and floating. The true geometry of continuity is only treated analytically, i.e. by developing analysis as a pure theory of waves and applying these theorems to geometry thanks to the principle of transfer contained in the concept of coordinates.

37:30 Here, the hyper-array is unusable. I will end here. Just a word to leave time for questions. In our construction of analysis, a theory of continuity, which, beyond its logical correction, must be able to be justified rationally in the same way as a theoretical one. This rational justification here, this requirement, this essential epistemological reason, is precisely that of an appropriation of objective content, which is what guides the construction. What distinguishes the Riemann geometry from the Euclidean geometry is the substitution from the point of view of contact actions to the point of view of distant actions. These are well-known things. The Riemann geometry is in some ways a Euclidean formulation that satisfies the spirit of continuity. But it takes a much more general character by this formulation. The project of a pure infinitesimal geometry consists in a certain way to change this movement and to perfect what Riemann had started by getting rid of the geometry of the last residue of the geometry of distance action, of this principle of distance action, that is to say, of all

40:00 This abstract concept is in fact embarrassing for all of us, namely the fact that, in the development, we can compare lengths that are different from each other. This leads to the elaboration of the theory of the system of gauge, which is true, but I do not have the time to develop it, I do not have the time to enter into it. These are the same epistemological reasons as those that prevailed in the rewinding of mathematics. The last stage of Wahl remains in the light of the theory of Einstein. An examination of mathematical principles examines that the consequences of the theory of Einstein were necessary. The author of the present book, Hermann Wahl, nevertheless notes that the geometric theory of Einstein realized only a part of the ideal of a perfect geomathic geometry. There is one last element of the global geometry to be eliminated, which always comes from its past. In fact, Riemann assumes that we can compare from one to the other the measurements of two linear elements located at two different places. The possibility of such a comparison at a distance cannot be accepted in a true geometric geometry. We can see precisely that, through these systems of gauges, a comparison is made of this concept at a distance. In these years, 1918, we notice this extraordinary productivity of Weyl, which, in a certain way, follows a simple philosophical intuition,

42:30 which is at the same time very profound and simple, and which Weyl actually leads on several fronts. Based on the fall of analytics and the... sorry to interrupt if there are any reactions. I have two questions that may be related. The first concerns the question of symbols or symbolism. It seemed to me that at least twice we had more or less identified concepts and symbols. You have used the expressions symbolic construction and concept construction, and the two appeared a little in the same way, so this is my first question, could you clarify a little this relationship between symbol and concept at Weill? And my second question, it is related to the first text you mentioned, the book Le Continu. We have come back several times to the point that we should not consider a point as an individual. And I wondered if all the construction that Vaille does of n-dimensional ensembles, et cetera, still solves that, because my feeling is that if we consider an ensemble, the notion of ensemble, whether it is this restricted notion or another, A set is a collection of individuals, so I don't really see how that solves this question. But that's not the point of this question, it's not in this... Yes, I understand that afterwards there is a construction of the reals, but doesn't this construction still deserve this general framework that still seems to me to be a set? That's it, it's a bit vague, my question. Yes, certainly, certainly. I will first answer some questions about the relationship between symbols and concepts.

45:00 It seems to me that precisely one of the concerns, one of the demands that we had in the past is not to have in mathematics the use of symbols, of symbolism. There is a kind of epistemological requirement, a kind of epistemological nature, which consists precisely of closely associating conceptual construction with symbolic construction. In fact, this symbolic construction is like a kind of ample garment that we wear. How can we restrict the symbolic construction to the real judgment? How to ensure that symbolic constructions are annihilated by the judgment of one another, are enervated by the judgment of one another. And that is the requirement. And what he proposes in the continuum allows to answer clearly and explicitly to this problem of epistemological nature. So what do the resources of logic use? The logic of Göttingen and the theory of ensembles is absolutely obvious. He says it explicitly. He says, I was born in the tradition of Cantor and Debeckin and I took a path alone. What does he say? No. I took a path alone by being guided by this external requirement of the principle of continuity.

47:30 It is necessary, starting from these conceptual resources of theory of ensembles, to add something to it. I would like to add a principle of constraint, which is that theoretical concepts do not emancipate, do not separate from the content of the object. Yes, I think that's it. I had already said that. It was just in relation to the idea of concept. I don't see too much. In fact, what I see is that there is symbolism and, let's say, the objective contents, which are either the fundamental objects or the constructions of the ideal objects. And I don't see where it's located. I mean, when we use the word concept, how does it articulate? But I don't know if it's fundamental to understand this, or its thought. We could summarize everything that is required of the system of physics, as a man would say, with a word, think concretely. He says it like that. It's about, in mathematics and physics, thinking concretely. That is to say, trying to get rid of abstract concepts that veil. The abstract concept can be as much the one that sharpens classical analysis as the concept of long-distance comparison, in the Germanic geometry. In both cases, it is a question of refactoring the conceptual sharpening in order to get rid of these abstractions. As for the precise status of the concept, I don't know if there is a specific theory of concepts that gives its meaning. The center of gravity is rather in the construction principles. There are six construction principles, it seems. There is one thing that I don't quite understand. I think I have understood. I don't know if I understand it in an exact way or if I understand it too much.

50:00 What Veil wanted to say, when he talks about surfaces, for example, I think he wants to say first that we have to use the notion of an object and that, at the base, the corners of a surface, for example, are the elements of an object, and in a mathematical language, what he then expresses is that In addition to this whole structure, we have to combine this whole structure, and what he does is to say that he puts topology first, then he puts a different structure. I think that's what he did. Of course, in the construction of the realities that he does, he insists on the structure of order and the topological structure that can be built from the work on the whole of the realities. So that's what I think I understood. Now what I did not understand ... This is where the construction of the real that you have written differs from what the mathematician does. What are the differences between the reals of Hermann Weyl and the reals of the reals of the mathematicians who work with him? Is his set of reals smaller or larger? And I didn't understand what you call the superior Borm theorem. Is it a theorem or is it an element that serves as the definition of a theorem?

52:30 Well, the most important thing is that it can be a theorem if we define it as a theorem, but it can also be the definition of a theorem. It can be an ingredient that serves to define the rules of the equation. Yes, but there is always more or less a circuit to be reached when you define, depending on which definition you use, you can demonstrate such and such a thing. But what you are saying confirms the fact that the theory of the upper edge is only possible if you apply the classical concept of the non-linear world, which is very good for reasons that are philosophical. So, the realities I am constructing, how do they relate to the real world? He says it very explicitly. Because he thinks it or he maintains it. Yes, he says it very explicitly. If we do not use the restricted procedure, that is, in the case where the sixth principle, the one with the existential fulfillment, if we do not use this sixth principle restricted to the objects of the basic category. That is to say, if we accept, if we use in all its generality, that is to say by being able to use, by accepting an existential classification on an ensemble of whole numbers, for example, the properties of whole numbers, at that moment we build what he calls hyperanalysis, which is in fact a... Does Pi, which has the status of a transcendent real number, is a real number in the sense of Ray? We can always define Pi. You see, Ray is inscribed in a long tradition of constructivism, of mathematics in the 19th century. In the case of Pi, when we open the lessons of the theory of numbers or of Kronecker, the lessons he gave in Berlin,

55:00 The irrational numbers did not exist. And as an example of arithmetic theorems, Cronenberg gives the arithmetic tetragony of the NX. It's the series pi, pi over 4. But all the real numbers, in the sense that mathematicians now understand the real law, are all intervalles with their rationality enclosed. So it seems to me that what Ben constructs cannot be anything other than the whole of the ordinary real numbers. So, yes, the difference comes from that. I don't see... I don't see... Well, maybe he takes it differently to do the construction, but the final result is human, it seems to me. It can't be anything else. Or what he could have is, eventually, to introduce infinite possibilities, and then build what is called standard analysis, standard analysis numbers. Maybe that's what he was thinking, but if that's not the case, I don't see this problem. Other than a real number, in the sense that mathematicians understand it. In what sense? It's a set of inductive orders in which the majority of them want to have a superior boundary. That's the definition. There's only one. But you know the problem, precisely. The starting point is to deny that this theory of Marx is a validity of the entire year. Why? For a very simple reason. You can see that in this construction, which is perfectly conceptual, the choice is unbeatable. It is perfectly clear. In fact, to grasp the look of the whole, the physiognomy of the whole of the demonstration,

57:30 you can see that you start by giving yourself the right to do so. What's the difference between the two? If you start by giving yourself the right to do so, if you give this statement the status of theory, but if on the contrary you use it to define the real numbers, you do not give yourself the right to do so, you give yourself the rational ones, and then you give them what they need, the rational ones, and then you give them what they need, the rational ones, and then you give them what they need, the rational ones, and then you give them what they need, the rational ones, and then you give them what they need, If you assume the existence of a superior band, there is a real number, which is the probability of the superior band, you will not be able to extract this real number, that is to say, you will not be able to specify it in terms of whole numbers. On the contrary, in the case of tetragony, it is not a coincidence that Krahnecker takes this example. Precisely in response to his philosophy of mathematics, in the case of arithmetic tetragony, you can see that you manage to articulate a construction, for example that of the series of the 8, which allows to give a meaning to the symbols, a meaning in terms of entire numbers. The real number that constitutes the superior branch does not make sense in terms of whole numbers, that's the idea. So what is the meaning of defining an entity, an ideal object, in terms of whole numbers or in terms of basic category objects? That's the whole problem. What does it mean? What is the meaning of this rootedness in the objects of the basic category? That's what makes all the beauty of this constructivism. It's about being able to formalize this intuition, because this intuition is part of the corneum of knowledge, and for everyone, these themes of constructivism are in the air. I don't see what it could be otherwise. Maybe it presents the construction in a different way, but...

1:00:00 We can say it differently. If you define a reality... The fundamental question is, if you define a real as a means of a sequence of If you are not able to do this, it means that this real number is not available. In fact, you can talk about it, but it remains sparkly, it remains linguistic. You can talk about a real number, there you can say, there is a real number, a real number which is the upper band, but this real number there ... It's just the words that allow you to make references, linguistic expressions, and it has no content. Why? Because the only content you can give is to exhibit it, to construct it concretely by exhibiting this series of rational intervals in which the bands are specified in terms of the entire number. So you can't extract it from the area of the superior band. You see that? Yes, it's a bit of a... I am a mathematician. I try to understand the writings of other mathematicians from the past in terms of the mathematics that I know and I am not trained to reason in terms of mathematics. I think this is the source of our... I may have noticed, I don't know if it's correct what I mean, the question you ask is to know if the reals of Bayes, if there are ordinary reals, let's say, classical, which are not reals of Bayes. To be able to ask this question, you have to give yourself the classical reals and inside that, look at the reals of Bayes. But for that, you have to use procedures that are rejected by Bayes. It seems to me that there is a problem with the very meaning of the question.

1:02:30 We think that the question must be asked from the point of view of the real classics, which is the point of view that Valais rejects. So I don't think we can really answer the question in terms of... I don't know what I'm saying here, but it seems to me that this is the problem. Your answer, please. I think that Valais' contribution is... Valais' contribution which... It is important for the development of the differential geometry theory, but not this one. So, of course, what do you mean by the condition theory? For example, you said that he refused, at a certain point, the comparison of two lengths at a distance. It is simply the point to develop the Riemannian geometry, in which we have a variety that is united. All of this is related to the idea of a human metric and that Weil thought of the variety of a connection, but not of a human metric. Or maybe he already had in his mind the notion of abstract differentiable variety. By the way, at a certain point, you showed us a text where it seems that he already had the concept of a card. Which is the one we use now in the presentation of the Egor-Geshi theory. You are certainly referring to the excerpt from the idea of the Riemann-Schlecher theory. He takes back the Waller-Strauss theory. He puts the Waller-Strauss theory first to join the Riemann theory and to axiomatize a general concept, perfectly general, of the Riemann surface. So it's not a map, it's not quite that, it's not a parameter, it's the idea of, you see, rather in function theory, it's not differential geometry, it's in function theory, the elements, in the theory of control analytics, you see, you take convergence disks and then you explore the complex plane like that in the next few years. But what Riemann had understood, precisely, is that by taking a tour, it was necessary to change the set of definitions so that we could reason in a complete way, rather than a series of definitions.

1:05:00 But what do you think? What do you think of my remarks when you say that I reason as a mathematician and not a theologian? That's not my point of view. No, no, of course not. It's just that my training is such that I reason from things that I know. But I think it's a very good thing to reason as a mathematician, precisely. But Weill himself says that for Weill the two have to be paired. You see, he says it himself. He resolves in terms of mathematics by excellence because he explains the meaning of his whole construction, but he explains that what leads him, what guides him in this construction is what he calls an epistemological order of reason, philosophy. So philosophy is not necessarily something that is absolutely evaluated, something that comes afterwards and says, no, no, no, you can't. The relationship to history poses very deep questions because when we interpret fragments of history by using concepts as they have been stabilized, does it respect or does it block the real conceptual development? This is a real problem that we encounter every time. In a sense, it gives an extraordinary power to be used. These concepts provide clarity to understand historical themes, but it can also, in some cases, dissociate what was said. Yes, I agree with that. I think that when we think about the years that have passed, using the current concepts, Maybe we can better understand some things, but we also have the risk of, let's say, leaving aside the influences that these authors had and which do not formalize in the system that we use now.

1:07:30 So we must obviously be wary of these things and the basic knowledge that goes into it. Thank you. Of course. Would he have received any kind of influence from Peirce? I've never read or found anything like that. But can we consider this way of continuing? Yes, yes, yes, in that sense. Cinecology, etc. and Peirce, it's far-fetched, but isn't there this idea of a kind of... It's an intuition of a sort of intensification deployment. Each point is deployed at a level, in a sort of... Is there something there that would sometimes make you feel bad? The theory of infinitesimality, but we can see the intuition that Shepard... In relation to classical analysis, we add a lot. We are more in the addition than in the restriction. But sometimes when you make certain citations, you do it for nothing. Yes, that's right. It should be asked. It's not cited in any reference in the continuum. There is no reference in the continuum. In round-site mathematics, I don't know. I haven't found any. Does he talk about Hamilton in the continuum? No. Yes, because what he says about fractions is very close, of course. There is no explicit reference in the continuum, indeed, to Hamilton.

1:10:00 But maybe it's because Hamilton, in a way, reopens mathematics in other directions. But it's true, that's the parenthesis. I think we're free. Thank you.