Ontological status of math. objects in Descartes / Le statut d'imagination dans les ecrit math. de Descartes

1:00:00 So precisely, on the one hand, in relation to this, at the end of 1638, Descartes abandons the project of publishing this treaty and finally it is the short notes of Florent de Bonne who, as a commentary, will constitute for him the best appendix to his geometry, in addition to the commentary of Chouten, and so he makes this choice of no longer considering...

1:02:30 This introduction to geometry does not appear in the edition of 1649. The principles of Amathéseus Universalis by Schreuthen appear in the second edition of 1659. Relatively to this choice of Descartes, which clearly appears in the correspondence, we have a letter from November 15, 1638 where he says that he no longer considers publishing it. Relatively to dictation, we also have elements in the correspondence where we see that there are faults that appear in the introduction to geometry. So, I wanted to ask you this question because relatively to my mind, you have discovered them. Which would result from the collation between this manuscript and the manuscript of Hanhoff which is incomplete and the manuscript of the British Library which would allow to bring out elements of the participation of Descartes outside of the fact that of course the examples that appear are not innocent examples and that we know that Descartes is behind all this but after knowing in what appears what is his part, do you have additional elements on this? All of this is difficult to know if these examples make sense. In Calais, there are the same problems dealt with by the profession and the same problems dealt with by the letters. It's funny. It's still a project that I've been working on for years.

1:05:00 It is important to note that Cartesian methods can solve problems better than competing methods. This is the danger of this test. Besides, it shows the fertility. It works better by using letters than by using business methods or other methods. Let's just say that in relation to... Yes, I think there is another question. I will try to show you the singularity of Descartes in this scene. All the senses of the geometry of the master of his science. I mean, the text, the letter, the sermon, can only be understood as singular. The first acceptable concept in mathematics is a criterion that Descartes has certainly never completely advanced.

1:07:30 This is indeed specific to the concepts of the past generations. The rule exists in the past generations. But this concept of the past generations was already completely Carthaginian. The subject of this lecture is also important because it is the moment of the first reception of geometry. However, in geometry there are two criteria that correspond and Descartes is the first who knows the criteria. Historically, he was the one who had abandoned geometry and who had apparently abandoned it. And I think you confirm more what I say than you confirm it, for a very simple reason. It is that what he says in the 38th century on geometry speaks of what he did not do in geometry. That's why we can't call it geometry because there is no theory of movement, et cetera, et cetera, et cetera. Ah yes, yes, I think the text is... I mean, there are two things. There are two things in the field of movement. You can really be an operator in geometry, in all kinds of things, and there is the movement as an object of the culture of mathematics. ... by a statute of imagination in the field of mathematics.

1:10:00 ... in the field of mathematics. It is through imagination that I would like to discuss some of these texts, among which I would like to discuss geometry. More precisely, the question here is whether or not geometry causes what I call a decline. The role played by the imagination within the mathematics of Descartes, thus marking his eyes in the evolution of his thought in this domain. This could appear as a point of detail, but it is in fact a question already traditional in modern commentary on geometry, and which in addition seems to me to be, so to speak, strategic in relation to the discussion of the evolution of the theory of imagination of Descartes throughout his general philosophy. This discussion has been polarized by two interpretative tendencies, which are, for the better, I would call here, continuous and discontinuous. Speaking of ruptures, seizures or eclipses, by locating them, by regulating them, in the late 1630s or in the early 1637s, the participants of the discontinuous interpretation point out all of them. In short, a fundamental transformation of Cartesian theory of imagination, or at least its evolution, which, according to them, would have taken place between regular and mathematical texts, essays, meditations, etc.

1:12:30 In my research of Cartesian texts according to their chronology, I try to demonstrate, and not only to suppose, the opposite thesis. It seems to count among its compatriots Pierre Coucou, Jean Laporte, Lesley Beck and Jean-Marie Bechard. According to which, the Cartesian theory of imagination, under the different great approaches presented in each text, develops itself in three coherent and unitary ways throughout its philosophical course, without turning to major transformations. At the same time, the approaches of the Cartesian theory of imagination are different from each other. It is therefore a matter of showing that the underlying theory is in essence at least the same. In today's discussion, I could not follow the GQT-CTS filter of mathematics experts to be able to help us understand the status of imagination. Allow me to start my discussion with my SRS, examining the problems in the geometry of the 19th century. And I will invoke punctually the inner and outer texts of my argumentation in parentheses. In the first moment, I will try to demonstrate that the imagination cannot be eliminated from the mathematical program of geometry. In a second moment, I will say a word about the nature of this imagination that geometry seems to have for itself. The final form written by his text, the essay, was developed after the completion of the essay. It can be found approximately between the beginning of 1637, its general conception, the arrangement of the materials in three books, and the final writing of the whole in view of the application. It should be remembered, however, that the essay was not conceived of a jet and that, in the long run, its composition consisted of organizing and articulating the material that the characters already conceived.

1:15:00 If the definitive writing took place between 1635 and 1637, it has a degree of mathematical work within Descartes. In particular, the solution to the so-called Parus problem was solved in 5 or 6 weeks of the end of 1631, and sent to Volus, who posed the problem, in January 1832, and a second one. The classification of the curves in the sketch would be shown in 1637. In addition to this integration of specific tables, we can see in the geometry a resumption and a culmination of all the cycles of mathematical research that had occupied Descartes in the winter of his career up to now. This cycle shows how the lecture of mathematical theory during the period studied by the French starts with the development of proportional comparses and a certain geometric algebra whose clues already appear at the scene of the collaboration with Bergman initiated in November 1718 to November 1719. and whose deployment seems to extend to an algebraic degree that would be almost close to 628. And where, on a first trial of methodical systematization, an HEP, in the degree of the universal maldecyse, a precise ratio between the present doctrine of geometry and the interior mathematics of the case, Controversial society of the past, the unremarkable of proximity, the other cultural affirmations of the 1937s to re-establish the manifest human. At the same time, we see that the program and the horizon of geometry are more restricted than that of universalism, which is a regularity. Geometry does not intend to establish or found a science of the physical world. It seems to be only a trial of a particular branch of pure mathematics. The letter is always present in one way or another in its past, in its texts,

1:17:30 either as a source of methodological inspiration, or as a model of certain knowledge, or as a foundation of physical education. As the subject of many punctual discussions that spread throughout the lecture, Descartes' mathematics goes far beyond what we have ever seen before. Whatever it may be, these essays constitute, if not the only one, at least the main work to be done in the field of pure mathematics. This is a field, according to Descartes, a field by excellence in which imagination can and must help our development. Examine what geometry tells us or indicates about imagination. There is nothing unusual about it. It would be useless, however, to seek in geometry an explicit theory of imagination, which, as well as the other two essays of the 1637, does not come from intrigue. So there is no theory of imagination in geometry. This does not prevent us from exploring, in the non-technical exam, a context that is still very technical, some elements capable of revealing the status of the imagination in mathematical mathematics as it is thought. Before entering this discussion, let us very quickly, even too quickly, recall how the imagination appeared in the previous mathematical texts. This was done in the lexicon of Descartes' Mathematical Discussion, in a vague way, in 1619, in the very important Latin letter of Declan, the romanticist, and then in a more precise way, in a virtual passage of written connotations. However, I will not read the text, I have read the text, but I will not talk about it. The aesthetic of imagination and the theoretical justification of its use in the field of mathematics will only appear in the regulars. Of course, it would not be possible to address here the most complex status of imagination in this context. It is important to mention at least two points that will be reflected in the second part of the lecture. The first point is that imagination is directly conserved by Cartesian or traditional mathematics.

1:20:00 Second point, it appears at least as an important auxiliary element of the mathematical or metamathematical method that Descartes proposes in response to certain critiques among others. For example, the use of the effect of imagination in the fourth rule. By recalling these mathematical studies and explaining their dissatisfaction with mathematics and geometry as they had been practiced so far, Eckhart Tolle said that, in fact, nothing is more vain than to study abstract numbers and imaginary figures to the point of seeming to want to attempt to know the same things as they are. They apply to this superficial demonstration, which we find more often by chance than by know-how, which are a resource of the eyes, of the imagination, more than that of the listener, at least to get used to, in a way, to use his reason. We find this in the lessons, in the 14 and 16 rules, which appear in the second part of the lecture, whose name is addressed to geometry. The imaginary figures in which they are concerned are Bagatelle, These superficial demonstrations would rather mobilize the eyes, the imagination, and the hearing, and we are more used to using our eyes. This criticism of the excessive weight of the imagination and the eyes in the geometric demonstrations as we practiced them, reinforces a hierarchical position already present in rule 3, on one side the hearing, on the other the sense and the imagination.

1:22:30 There is also an inspiration from the horizon. It seems to refer implicitly to a science in which the audience interprets the phenomenon. This is, in effect, the presentation of the universalist mathesis, a general science that explains everything that is possible to research from the field of measurement without assigning any particular material to it. These principles would be the basis of several particular mathematical disciplines, from which they differ, however, because they present a degree of greater generality and because they embody better than the others the requirements of the method. As we all know, mathematical mathematics is a difficult and very difficult question in the work of the texts. With remarkable contributions to the dossier, I would not dare to comment on this issue at all. It should be noted that we can identify the universalist mathematics in the methodological program exposed in the rules, in its set, and that it preserves in the imagination an important role in its treatment of the law of the universe. The discussion is complex, I summarized it drastically, I wrote it with a little more attention in my thesis, I don't want to go into too much detail here. What we can say is that rule 4 of the subject of the role of imagination would be less important than the one reserved by the geometry of the ancients since this one has been criticized exactly to lean too much on the imagination of the universalist mathematics and appears to be a fraction of its fault.

1:25:00 We will see further the significant passage of the last rule 14, the second part of the lecture, Descartes associates imagination with the application of the principles that seem to be those of mathematics and mathematics in the field of mathematical and physical objects submitted to the other and susceptible to measurement. In this sense, we will also see that imagination brings an important help to the knowledge provided by mathematics and mathematics. Although the fourth rule does not concern me, its help in the field of imagination will become clearer in the second part of the rule, which is limited to a very clearly defined field of objects in the Sinus rule. It is a field of objects, corporeal, that have something to do with the corporeal. I will quote a very important passage and explain an idea. A demarcation that has remained fundamentally the same throughout Descartes' course, which finds echo in several of his texts. This is a very famous passage from the 12th century, where Descartes says that if the listener asks the question that there is nothing corporeal or that resembles corporeal, he cannot receive any help from these faculties, that is to say, the imagination, the sense, the memory. On the contrary, in order for an academic lecture to be a good one, it is necessary to distance the senses and spread the imagination as much as possible, with all the attention of the senses. But, if listening to an exam subject is proposed and can be brought to the body, it is necessary to inform the idea of the imagination with as many distinctions as possible. And in order to achieve this, it is the thing itself that presents this idea that it is appropriate to go into the external senses. The rest of rule 12, the rest of rule 12, will specify this idea. The second part of the rule will put it in place in the treatment of perfectly understood questions, that is, mathematics, or susceptible to be treated mathematically.

1:27:30 In any case, it should be noted that even in this second part of the rule, listening never loses its primacy as the main responsible for knowledge. This, in my opinion, is a very important point that should not be forgotten. Even when humanization plays a bigger role and is more visible in the context of the lectures, listening always keeps its main or only part of the knowledge. Why recognize it? Because of an anecdote in the Catechism of the Research of Youth in Logical, Geometric and Algebraic Analysis which rephrases what is present in Rule 4 while presenting a second version. The main defect of the analysis of ancient geometries is that they are always so restricted in the consideration of figures that they cannot exercise their understanding without practicing a lot of imagination. This is a reproach that does not echo your criticisms of traditional geometry, which already appeared in the Karna rule. The formulation introduces nuances that seem to be less hostile to the imagination and does not exactly oppose the understanding. In fact, its target is not the imagination itself, but the bad use that ancient geometers made, whose method ended up overloading it by responding too much to the potential of the problem. As we try to avoid these defects in the analysis of the ancients and also the other defects found in the logic of modern algebra, Descartes says that he has developed another method that is able to take advantage of the advantages of each of these titles.

1:30:00 He presents this method as a double proportion. He does not refer to the mathematics of the Swiss university, but I think that fundamentally we can give credit to most of the commentators. There is even a resemblance of details in the story that is made in the lecture. I will not go into details. It is the fact that in both cases, we have the critics in the imagination and we also have its maintenance in the Cartesian method. The experience presented in the lecture was presented in a text edited in 1935 and 1937. That is to say, there was a gap between the writing of Hegel and the writing of the second part of the lecture. If it had been questioned, if it had been abandoned or made its position in relation to the approach of imagination, it would have been... These changes find a translation or a symptom in the second version of the text that makes the problem. It's nothing. It's nothing. It doesn't film it. It doesn't authorize imagination. It doesn't film what I was saying about the goal of imagination in the second part of the dictionary. The first part of the text that makes the problem of universalism is in the second part of the dictionary.

1:32:30 The interval of several years that separates the writing of the text from the position of the cases, in this point, in any case, has not changed. Cartesian, in the regular, doubled in the lower, has acquired certain uses of imagination. He has, however, announced his competition in the methods he proposes to replace them with the positive ones. By admitting that geometry at the same time exposes a theory and exercises a practice of mathematics, he returns to the interpreter on the one hand to ask what is the goal attributed to imagination in this theory, The third question is to measure its effective use in this practice. This is a question of other topics. It is the same here for the needs of analysis. These two tasks touch each other and complete each other. The results of one must help to verify and confirm those of the other. For the first task, the question of the background concerns the particular way of articulating geometry and algebra in the mathematical program that I am exposing. Since the most important image for geometry is based on the spatial representation of these lines and figures made for algebra, whose processes and operations seem to make an economic representation of a new representation, the determination of its status in an essay is conditioned by the interpretation of concrete relationships between these two branches of mathematics. Thus, those who see in the test the reduction of geometry to algebra almost always tend to consider that it excludes the imagination of the knowledge of mathematics. On the one hand, in the opposite pole of the test of algebra, the contents that do not have such an absolute reduction tend to allow, or at least allow us to do so, the permanence of the concept and the epistemic function of the imagination.

1:35:00 For the second task, that is to say the examination of the effective use, A good starting point is an examination of the vocabulary of the products of imagination present in the essays. Throughout these three books, we find 17 occupations of the theme of imagination and numerous geometric figures printed on these pages. We see that the verb to imagine almost always refers to the mental actions carried out in the consideration of the lines, properties, and geometric operations that the text is explaining, often with recent letters. We will also see how these actions reveal the functioning of an imagination that we will have to qualify in the formation of lines and figures conceived by the spirit without provenance of the senses. The conjugation of geometry and algebra unilaterally crosses the three books of the Essence. At the very beginning of the book, Descartes begins to put it in place by discussing the full geometric problems in which construction only employs the circle of the right to which it can be reduced. Geogeogeogeogeogeogeogeogeogeoge The process explained by Descartes basically includes three stages. The first one is the initial construction, that is to say the geometry of the problem. The second one corresponds to its algebraic translation of its treatment by equations. And the third one corresponds to the final re-translation of the solution in the form of a geometric representation, which will be the construction of the problem.

1:37:30 By suppressing the spatial representation of the geometric problem, the generalization to which they are subjected in the second stage tends to eliminate the function of imagination in its approach to human beings. This has allowed the interpretation according to which geometry obscures the imagination of mathematical knowledge. As I have already pointed out, and I will always do so. In order to develop considerable arguments, it is important, however, to say that the attempt is very far from reducing geometry to the general. The concept of agilization lies only at one point among others in the mathematical method described in the book The First Stage of Geometry. This method does not at all support the spatial representation of the problem of geometry, for which the imagination always plays an important role. I find it very important. What I want to say is that the cartesian approach of the so-called problem of Papus to a strategically presented solution by Descartes is similar. The superiority of this method on the subject of mathematical physics is that it occupies more than half of book 1 and extends to book 2, where it reappears after an excursus on the subject of book 1, after having announced its approach, signaled the blockade, the insufficiency of this book, the colonnus papus, and which organizes the demonstration of its solution in a moment. First, the initial geometric representation of the problem by attributing the problem to page 342. Second, the translation and processing of the problem into algebraic terms by using equations in which the geometric line is symbolized by letters. Third, the description of the geometric construction of the solution based on the result of the notation with the help of the rule. The 16th allows us to perceive that the algebraic treatment of the coordinate of a moment 2 does not eliminate, in a sense, the course of imagination in moments 1 and 3, in which the spatial representation of n and u appears effectively in the text or is restricted by it, presenting a certain heterogeneity.

1:40:00 In the field of geometry, in the case of the construction of ovals, the function of spatial representation conceived in the imagination in the formation of lines and figures is clear. Also clear, it is very reflective of the imagination for the exposition in the form of geometric figures intertwined in the test. These functions, however, are eliminated when the algebraic treatment of problems by equations comes to the fore. It happens when Descartes classes the courses accepted in geometry, that he takes the problem of Papus and presents the method of construction of the normals, that is to say, of the right terms, in one of their forms. At this moment, the figures between the medias and the texts seem boring, as well as the vocabulary of the imagination, which is almost always used in a technical sense. In these more technical occurrences, the verb to imagine helps us to specify the status of the imagination in the essay. Interestingly, these terms remain almost always in the formulas where they appear. Our task is not only to design the lines for a game in cartesian education, but also the description of these lines themselves. It refers to particular lines that he already imagines written in one way or another. For example, I cite the line EC, which I imagine to be written by the intersection of the rule GL and the multilinear plan CLKL. In the other passage, we see the line for C and E, which I imagine is described by the description of the CKM parabola and the rule GM. So, Descartes does not only imagine the lines, but also the processes themselves, from which they are, or may be, taken or constructed.

1:42:30 There are several ways in which we imagine the description of a curve, no matter how many of them there are in geometry, This is the only concept of the relationship between all these points of a curve line and those of a straight line, in a very simple way. It is easy to find also the relationship that they have with all the other points and give them to us. Then, we need to know the diameters, the expressions, the centers, the other lines, the points to which each curve line will have a more specific relationship. Or, more simply, all the others, and thus to imagine various means to describe them and to choose the easiest ones. I have not talked about all this except the short book that can be described on a surface, because it is so flat, but it is easy to relate what I have said to all those that we could imagine being formed by the regular movement of points or bodies in a three-dimensional space. The regularity of the situation presented in this passage is one of the few hypotheses according to which the formulation has resulted in the state of Cartesian intelligence. Everything happens as if it were the case that the object of our imagination did not belong to the resulting line of the construction operation. We also understand the construction operation itself. This nuance in the formulation appeared only once in the rule of 18, in a particular passage about The number of geometric operations of the lines and arithmetic operations of addition, subtraction, multiplication and division. Descartes said that we must present these operations in front of the imagination so that it examines them and even makes them appear in the mirror.

1:45:00 These two combinations presented the operations themselves. So this line that appeared only once in the 18th century, more vague, It seems to put us ahead here in geometry, more than in the rules of Clemence and Seize, where there were also passages in the same direction. A function that we wanted to qualify as the producers of imagination. It refers, of course, to a certain standard of interpretation of imagination, which already has a considerable history, which has been renewed more recently by the study of Jean-Luc Marion, in comparison between Descartes and Keynes, By the development made by Christophe Couriot in the U.S. I explored the productive dimension of Cartesian imagination and an academic who is a great disciple of the ancient imagination, the technology of ancient mathematics. I will not go into this discussion because it shows the representation of the lines of the digital world. We cannot find the image already drawn in our eyes, since it is the condition of possibility itself. We could not draw it on paper if we had not precisely imagined it at the beginning. What L'Alliance tells us further, it seems to suggest that the imaginative presentation of the lines itself depends on our ability to imagine, the procedures through which they can be written or built, The third book opens a wide range of principles according to which, in each genetic problem, we must always choose the simplest line that can be solved by avoiding as much the too simple lines as the too proposed ones. To explain how we can avoid these undesirable excesses, Descartes creates a super long algebraic structure on the equations.

1:47:30 The context in which the arithmetic notion and imaginary roots appear. This is a technological evolution of the four in relation to the mathematicians who have preceded it. He calls here imaginary some roots of public equations. Mathematicians, physicists, and others. But the first difference between the imaginary roots of the three is that the four oppose the real roots in a formulation that seems to take them as the result of an imaginary operation. Otherwise, both the real roots and the false ones are not always real, but sometimes only imaginary. In other words, we can always imagine as many derivatives as possible in each equation, but there is sometimes no principle that corresponds to the one we imagine. If we can imagine 3 in this one, there is at least one real one. If we can imagine 3 in this equation, there is at least one real one, which is 2, In the way I have just explained, we would know one or the other imaginary. This characterization of imaginary roots seems to entail a paradox. By naming them imaginary, there is an improvement that we can imagine them, and when we let them see each other, they invite us to suppose, at least, an eventual effectiveness of imagination in the domain of mathematics. As we have already seen in one of his previous texts, The Letter with Man, from 1909, he has once again, in a later text, However, despite the apparent convergence of these three texts in their common suggestion of an identical imagination,

1:50:00 it is necessary to admit that it remains rather vague in all three. As well as in the other two texts, Descartes does not go into this field of geography to develop the suggestion as we would like and to explain in detail what exactly would be the operation of the analysis of the question. By imagining them, can we imagine a quantity or an algebraic sign? We arrive here at the paradox by saying that we can imagine these roots of four important animals as a quantity of an information. However, as he will say a little later, it is not that a corresponding quantity of imaginary roots deprives us of a geometric representation of problems involving equations in which these roots appear. And everything makes us think that it is precisely because of this deprivation of a geometric representation that Descartes excludes the imaginary roots of the field of objects or of the arsenal of algebraic instruments, i.e. his geometry. This is the important point. In the consideration of a particular case, he affirms that the geometric problems in the corresponding equation, which comprise imaginary roots, cannot be constructed. A root is not true in these two last equations. We know that the four equations of this process are imaginary, that the problem for which we are faced is full of its nature and that it cannot be built in any other way because of the given quantities of the three systems. The possibility of geometrically constructing problems in which equations are composed of imaginary roots prevents Descartes from accepting them as a treatable object or as an operating instrument within the mathematical method proposed in geometry. Could we thus understand the exclusion of imaginary roots as a symptom of the exclusion of the imagination of the 2003 forum? Not at all. It is rather the opposite. If these roots cannot be accepted in Cartesian geometry, it is not because they concern arithmetic imagination, but because they hinder the geometric representation, i.e. imaginative, of the curves that should have corresponded to the equations containing these roots.

1:52:30 Soon qualified as imaginary, these roots end up becoming unimaginable, i.e. unfigurable in terms of geometry, the equations that contain them. There is a psychological note of Vincent Carreau on the difference between the imaginary and the imaginative adjectives in Descartes. This has been partially reflected in the book I wrote in collaboration with Félix Rousseau about the article on imaginary spaces. It's a shame that Carrefour, in his text, did not refer to geometry, he was content to refer the discussion of geometry to a passage by Robert Amogat, because I think his scheme of the distinction between imaginary and imaginable, in some cases, is very similar to the exam of geometry, which could be considered an additional confirmation of geometry. In short, the roots are excluded. Not because it reinforces the roles of the imagination, but because it prevents the intervention of the methods of this auxiliary faculty. Exclusion shows us that the imagination has to keep an inevitable function in the methodological program of geometry. Thus interpreted, it appears to us as an indication of the relevance of the imagination, not of its elimination, despite its validity. This example of geometry allows us to articulate the answer to the question of faith on the role of the imagination of the mathematical theory that the essay exposes, and the answer to the question of faith on the effective use of imagination in its respective part, among the questions of earlier. On the one hand, it does not contradict a certain theoretical Cartesian design of reducing excessive dependence, criticized in the rules 4, 16 and in the second part of the discourse. The analysis of the ancients presented in relation to the imagination.

1:55:00 The lecture on algebra and its theory of equations exactly corresponds to this drawing, providing the appropriate instrument to describe this dependence. On the other hand, our research helps us to re-activate the conclusion according to which Descartes wanted to eliminate the imagination of geometry in general, or to treat geometry in particular. In fact, the examination of geometry is aimed not only at the permanence of certain imaginative functions in the geometric processes of construction and representation of courses, of written or written cases, answers to the question of history, but also concrete answers to the imagination in the expository processes that effectively use the answers to the question of facts. Since in his discussion Descartes does not take part in the definition nor in the characterization of the imaginative functions provided in the theory of mathematics and operating in his exercise, present in the set, he returns to the decree to characterize them. He also returns to specify, if yes or no, the permanence of these functions that range the Cartesian drawing and reduce the dependence of geometry on imagination. Based on the way in the book II, the operations of the imagination, the description of the lines, I tried to show that this operation concerns the production function of the imagination because it consists of mentally building lines in which the representation does not come from our senses but has been produced by our own mind, even on the parts. If we do not completely embrace these interpretative paths, we can find in the Cartesian texts very clear extensions between an imagination that is, for example, identified in the memory of the Regulae, a famous passage from the 12th century, and the imagination that is identified in conceiving, in the same rule, something of the order of the intervention of the viscognosces on the glen.

1:57:30 This is a completely different kind of operation, a kind of commemoration of the imagination that he also accomplished. This image reappears in a passage of the Sixth Meditation. I will try to quote it. Did you quote both of them? Absolutely. No, no. In this meditation, for example, the quantum of imagination differs only from the quantum of reflection in that the spirit, in conceiving, turns in some way towards itself and considers what ideas it has in itself. But the imaginer turns towards the body and considers something that is in accordance with the idea that he has formed of himself or that he has received by the senses, that is, in accordance with an intellect. In this case, we could risk the hypothesis of interpretation, in which verification at a high technical level would be a task for a specialist in Cartesian mathematics. According to such a hypothesis, geometry may avoid the gap, rather than reducing the presence of a function. In the heart of the architect of imagination, who would have preserved an alteration of the previous geometric tradition, she is redefining this function, i.e. transforming it into a production of mathematical representations, especially independent of the senses, which would be a new articulation of the operations of the mind, installed by the intellectual discovery of the art of algebra, supported by the internal analysis of geometry. To be attested by the analyzers of geography, this permanence of imagination, which is defined here as producer, in its methodological arsenal appears consistent with the operations of Descartes himself, both in the lectures and in the subsequent texts. By telling in the second part of the lecture his project of a mathematical method, which is able to combine the best of the geometric analysis of the ancients and the modern algebra, Descartes announces

2:00:00 The suppression of each one's faults through the other and not the suppression of one against the other. In addition, we must not forget the numerous declarations of Descartes in several texts later in 1637 on the utility of imagination in the field of mathematics or in the knowledge of the objects in question. This is curious because the interpreters according to which geometry would have eliminated or almost eliminated imagination In the field of mathematics, there is always this silence, these many texts. There are a good thirty texts that designate the representation of curves and lines by the word to imagine. There are five or six texts in which the function of imagination is explicitly described within mathematics. I will explain quickly two or three of these texts. For example, he will say in a letter to Mersenne 39, the practice of the mind that helps more in mathematics, namely imagination. This is 39. Or else, in the 5th meditation, he says, I distinctly imagine this quantity that philosophers call continuously the continuous principle, through the extension to a longer, larger, deeper, which has this quantity, or the thing to which we attribute. In my sixth meditation, when I imagine a triangle or the culture I use in my imagination, when I think of something corporal, I am used to imagine many other things other than this corporal nature that is the object of geometry, or the alphabet of 43. The bodies, that is to say the extensions, the figures, the movements, can also be known by hearing alone, but much better by hearing than by imagination, it is the alphabet of 43. We have the greatest pain to imagine even a polygon of 7 or 8 sides.

2:02:30 The author, that is to say himself, who is quite gifted when it comes to imagination, who has long cultivated this talent, can imagine them quite distinctly. Others cannot as well. Not that the author would be so strong in occupying his mind to imagine, if it happens that some of them in mathematics It is not possible for an engineer to succeed less well in physics than in these kinds of questions. It is not possible for an engineer to succeed less well in physics than in these kinds of questions. It is not possible for an engineer to succeed less well in physics than in these kinds of questions. It is not possible for an engineer to succeed less well in physics than in these kinds of questions. It is not possible for an engineer to succeed less well in physics than in these kinds of questions. It is not possible for an engineer to succeed less well in physics than in these kinds of questions. It is not possible for an engineer to succeed less well in physics than in these kinds of questions. It is not possible for an engineer to succeed less well in physics than in these kinds of questions. It is not possible for an engineer to succeed less well in physics than in these kinds of questions. It is not possible for an engineer to succeed less well in physics than in these kinds of questions. Thank you for your attention.