Le paradox dans les mathematiques de Pascal

0:00 In this way, we discover a power of thought. I hope you will admire the arsenal of pedagogical documents, which I did not use myself, but I will still mark some articulations of my plan on the board. I know that it is always difficult, but I will not say a first part, but in the first part there are several parts. The first part of my lecture is about the real use of the past, by searching for it. I don't think we find it, at least it's not, but it's a moment in his life. I'm going to make a kind of quick intro that will highlight this constant search for the paradox. I'm going to distinguish two ways in which the past always presents itself. It is a paradox in form, formulated, or it is a situation, a problem, which is a paradox, which is also present. At this point, I think that there is, at the beginning, to understand the origin, leaning, the art which was the first master of Pascal, or of Euclideanism, has been followed.

2:30 The essay for Euclideanism, which is very magnificent. These terms are so original in their own right, attached to him and held to resonate, according to his feelings, with an activity... In both cases, it starts exactly the same way, by defining the order... So I ask you, what is an order of law? That's what I'm waiting for. And so, the two, because an order of law can mean two things. Either they are competing laws, they meet at the same point, and these two types of law, which for the common sense, are to have common properties, common properties, and the definition is given by the law against the law.

5:00 There are many books on this subject, but the main difference between them is that they start by defining a kind of right-wing order, the one that points, the point of the point, and another kind of right-wing order, which is supposed to be parallelistic, because there is only one. He highlights even more the parallelism. The first paragraph of the essay for the students. He will define this order in this paper. He says how many of them are parallel? It's the same thing. Brutalism. There is only one example of a parallel. There is Ecker in this paper. It's a whole mathematical experiment of the relationship between the appearance and the reality. What is a parallel? Also, is there a link between the researches of Pascal who values the notion of the word and the perspective is individual since we are faced with the representation of things and the representation is different from the researches.

7:30 It is deformed. In a book, the genetic way shows that, if you talk about the encounter of an existence, one could almost film them. We are not quite there in the paradox, but we are very close to it. Thus, one, which is that of the ancients, who were taught by his father, who inspired him, he did not produce at all, in the sense that I indicate, which is in particular the notion of essential analysis, resulting in the fact that it is impossible to arrive at the terms of a new approach. The use of the perspective is quite characteristic.

10:00 For example, he speaks of a trunk to speak of a tree trunk, he speaks of a tree trunk to speak of a tree trunk to speak of a tree trunk to speak of a tree trunk to speak of a tree trunk to speak of a tree trunk to speak of a tree trunk to speak of a tree trunk to speak of a tree trunk to speak of a tree trunk to speak of a tree trunk to speak of a tree trunk to speak of a tree trunk to speak of a tree trunk to speak of a tree trunk. So, I take another example. I am happy to point it out because if I develop each example, we will not have finished space and life. Here, we have a few years, we will arrive at the year 1648, and another friend of Pascal, a mathematician, Pascal receives a letter in which he explains a number of problems, problems that have been posed to him by his research on distant, distant life. And at that moment, the reflections that touch at the same time on the concept of space. And what is remarkable with Pascal is that to build his theory of the polygene,

12:30 He wanted to enrich his conception of mathematics. He wanted to specify his conception of mathematics in mathematics, that is to say, for Descartes, the space of Pekin, and that's what we are in a field where space is different from what fills it. And it is evident, as the model, third example, in terms of research, you can't calculate probabilities, it's not quite what it leads to. It's even more than a paradox. It's the union of two words that are the opposite of geometry, which is something perfectly regular and ordinal, and randomness, which is perfectly ordinal. Oh, there is a geometry of randomness. That was a huge discovery. It was not a discovery made in one go. It's the culmination of reflections and of conceiving the idea that laws can... At the same time, this research on the laws of the laws of the laws of the laws of the laws of the laws of the laws of the laws of the laws of the laws of the laws of the laws of the laws of the laws of the laws of the laws of the laws of the laws of the laws of

15:00 Consciousness is bound to accept the fact that you can make an indefinite mistake, and that the Geometric Paradox forces you to rationally contradict the essential experience. And we can be sure that the soul and the mind reason justly, so much so that it is one of the best examples of paradoxes, because there are paradoxes in which there is a logical distinction, which is always very useful. There is a distinction between the proposed meaning and the divided meaning. This means that there are certain propositions in which the two halves are not true. It is impossible to separate them. This expression has no meaning if we separate the two words. It completely changes the meaning if we don't have the two words together. In which it is said, for example, if the most happy person is you, when you have a hunger and thirst for life, you will be slaughtered. I would say that this is not a real paradox. At least it is an adequate and sufficient paradox, because having a hunger and thirst for life is a certain state, and being slaughtered is another state. But it does not happen at the same time. So there is something consequential. So at that moment, it is a form of atheism.

17:30 There are many other themes that could be dealt with, but there has always been a reflection on figure numbers, for example a square is a figure number, or a number that can be represented by geometric figures, and there are, I don't know, here the square and the cube, there are triangular numbers, triangular numbers, the construction of these numbers is substandard, and for example, There is a remark in Pascal's book in which he points out that we cannot reproduce this with points by making a square of points. We can never make another square of points that is double, whereas if we build a square, we can very well build one that is double. It's a way to bring out another aspect, which is the distinction between continuous and discontinuous magnitudes. Continuous magnitudes are these lines, and discontinuous magnitudes are these lines. Every time we translate a discontinuous magnitude by a number, we end up with a number source. So, that's another subject. The infinitesimally large and the infinitesimally small are important things, and the infinitesimally large and the infinitesimally small are also notions of mathematics.

20:00 The only case is that they do not correspond to the experience that each of us has had. The domain of the human is alien to the infinitesimally large and the infinitesimally small. We will guess, then, that there is an extreme hot, an extreme cold. Extreme values are not accepted. And of course, not enough to make it possible to say that there is a place in the world where this is possible, where this is possible, where this is possible, where this is possible, where this is possible, where this is possible, where this is possible, where this is possible, where this is possible, where this is possible. The most important idea that Pascal came up with, especially when he was asked a question, was to know if the smallest number in numbers is 1 or 0. There must be a sort of confusion between these two terms in the field of zero and it is from the moment when the indivisible is identified with the zero that it is not the case, I think, but the moment when the indivisible is identified with the zero.

22:30 So here are a number of examples that are destined to make an inventory, a limited and incomplete inventory of this present, which shows in any case its presence in the world of science. So now, my second part would be the foundation of the paradox. So how do we know that the geocritical, the theoretical narrative, lends itself to the good God, if so? The answer is obvious because Pascal seems to have an idea which is enough. So, what I'm going to do is first of all, I'm going to produce a problem which is the notion of evidence. Here again, I'm going to think that Pascal is talking about a cartesian evidence, that this is the basis of mathematics. Anyway, in Pascal's case, this notion of evidence is not absent. It is an occasional evidence, not because Alta is not an evidence simply in the fact that men feel themselves, men are all similar, they fit into the same language, and therefore on a given question, they carry the same judgments, carry the same judgments. The axioms are simply the notions on which humanity encounters itself. What compares to the axiom is not the axiom of values, but simply values related to humanity.

25:00 But this is not the essential point. The paradox and the evidence are relative to the principles. There are two things in reasoning. There are principles and there is human reasoning. And that in a spirit, all parts of the evidence, on the contrary, Pascal is already at this stage, and the principle is being built, or at least it can be built, in this case, from a return of contradictions. Man only naturally knows errors, and for him to find the truth by eliminating the error is to find himself in the presence of situations already and this is applied as well to the sciences, to the mathematical sciences, the sciences of the future, the science of the future, it is the experience, the experience that will be judged whether or not it is true or not. We deduce an absurd of an experience that leads us to consider the hypothesis as an absurd.

27:30 Yes, we deduce, we conclude an absurd manifest of the negation of the theory. In finding the principles that are founded, that are in the... but in the classical points, moreover, of mathematical reflection. In my reasoning, I have to say that I will never have the time to go through the infinite number of points, because someone has an answer to this theory, and he says that we should not consider the points, but rather a square of points. If, at some point, we divide reality into a continuous totality, then in fact what counts is infinity. And of course, the relationship between physics and mathematics. With the idea that for a phase of evidence, a principle is built from the reflection, and therefore from the data on which it is based.

30:00 The second point of zero and infinity, the fact that zero and infinity are part of knowledge, that is to say that at this moment the mind is faced with the reality that it calls to master. First of all, there was often an allusion to these things, because the rules of physics were often the same. So, the infinity, notice that the number is a number, it is part of the infinity of infinity. The number is the real infinity. In any case, it was the infinity of infinity. I'm afraid to talk about infinity. It's a name that we don't change. But how is it possible that a name that has been used for a long time does not change its meaning? And that's the bottom line. And if there is a paradox, it's because there is one. There is also a paradox in physics. I know some people who, when they are told that from zero, you write from zero, it's obvious. But they know people who don't know it.

32:30 And we find, of course, the discussions between the languages of the chapters. One of the passages that is most interesting in the commentary on calculus is where there is this sentence, which is one of the most striking, which is quoted in an exhibition, too much truth is at stake. It is at stake in a very strong sense. This word, you know, in the 17th century, So, it's all about zero and atypics. What I notice even more essential in my book is the spirit of zero. The spirit of atypics. So, here we are talking about the two. The heart. The condiments. You know the texts, you know, the difference between the spirit of life and death. What distinguishes the spirit of life and death? It is important to note that both do not follow the same principles, and therefore it is good to note that the spirit resonates in the same way. So the sequence of reasons is the same, whether the spirit is the same or not, it is a point that has been completely missed by many philosophers. The difference is not in the principles. In the principles... The principles of the spirit of the spirit of the spirit of the spirit of the spirit of the spirit of the spirit of the spirit of the spirit of the spirit

35:00 So, this definition of the geometry of the spirit of geometry is considered to be a kind of a world that is different. We are in a different world. So, the geometry of the spirit is in the spirit that is born. It starts from the common sense. It starts from the experience that everyone has of other people. And it is from this discovery of the expert in the field of experiments that he can build on this theory. And it is only the theory of what the geometric spirit, which starts from reality, which is relatively easy to understand, which is not easy to understand, that leads to the truth. On the contrary, the spirit of physics is in the presence of a multitude of observations. These observations have so many principles that it is not extremely difficult. I believe that this restitution is essential to geometry, and what Pascal says about geometry, which we can see here in geometry, is that it belongs to geometry and its univariate. And finally, the last point is that to elaborate, as we have already said, by starting to elaborate, we can use mathematics.

37:30 The fact is that the knowledge of mathematics and the knowledge of man are not born from the process of the same language. They result from the same language, from the same theory. But this problem is very good for an example of the 1660s, in which Pascal says that geometry is not a job, but a job for man. A non-standard professor isolates the other professors because the non-standard professor is the one who is surprised by the semblances of the other professors. I think there is a second reason for this. It is an excellent argument. It is a non-standard professor and he deserves to be estimated much more. I would like Pascal to see if this is a project of physics or of honesty. I would like Pascal to see if this is a project of physics or of honesty. I would like Pascal to see if this is a project of physics or of honesty. I would like Pascal to see if this is a project of physics or of honesty. I would like Pascal to see if this is a project of physics or of honesty. I would like Pascal to see if this is a project of physics or of honesty. I would like Pascal to see if this is a project of physics or of honesty. I would like Pascal to see if this is a project of physics or of honesty. I would like Pascal to see if this is a project of physics or of honesty. I would like Pascal to see if this is a project of physics or of honesty. I would like Pascal to see if this is a project of physics or of honesty. The idea is that we have an image of reality and the will is driven by an ethical image of reality, and each one of these figures corresponds to the possibilities of the domain of the human being and the domain where the rhetoric is, where the class, the class, the rhetoric, the art, the community, etc. And so we find, for example, the sketch in the background, between the rhetoric and the human being.

40:00 The last point I would like to make is about the evaporation of problems that appear and that is the one that is the rule, the one that provides the starting point of the analyzes that will allow the construction of solutions. But the fact that it does not lead to results that poses a problem that does not resolve it. All of these problems, which are initiated by Pascal, will be in the search for a balance that will be obtained. There are perhaps two or three ideas that I would like to share. I think that there is, in Pascal, the frequent affirmation that there is a unity of things, a harmony of things.

42:30 There are a number of points that we find in the book. First, unity, the numbers of spaces. We have already talked about this earlier. It is a point of mathematics, but it also insists on the fact that numbers imitate space, And here appears a notion that is capitalistic and pascalic, and that is the symbolic symbol of the world, the numbers and the spaces, the cymic nature, and in particular the numbers and the spaces are in correspondence with each other, where there is the possibility of coinciding the three of the researches, the sound of the mirror, the relationship of the two. So, the idea of order is one of the major ideas of Pascal, and we saw that this notion of order appeared from the first reflections that I proposed on the paradox. It appeared in the acts and ordinances of law, and Pascal also says order of law, so he sends the word order from his first trial for the eunuchs, from the age of 16. In the spirit of our work, there is a word that contradicts the spirit.

45:00 But the work as Pascal conceives it is not completely foreign to the paradox. In the sense that, especially in the famous fragment of the Three Orders, the fragment of the Three Orders has a lot of various correspondences. The idea of man is in tact, says Pascal, established again by the paradox. There is a force of intelligence that no longer becomes the capacity of love, which is the order of the heart. There is a kind of impossibility of crossing the interval between the two. And despite all this, but at the same time, there is a figuration, each body is figured by the past order, that is to say that there is a harmonic relationship that finds the greatness among us, and so much so that, symbolically, the greatness of the past order refers to the greatness of the present order, for example, Archimedes. The great scientist was also a prince. Prince of Syracuse, Prince of Syracuse, Prince of Hachette, Prince of Leuven, and, of course, Prince of Hachette.

47:30 So it is a harmony that establishes itself within a universe. It is a harmony, a pyramid, a univision within a universe. Mathematics as the creation of a universe. I'll come back to a remark I made earlier. Two scientists create exactly like an artist, a creator of a universe. But they do it by other means. The universe created by this universe is a different universe than the universe created by that universe. Mathematics is an opening to the world, an analog opening to this world. To conclude, I would like to ask you, what is geometry and what is space? It is, it seems to me, one of the greatest, the greatest, the greatest, the greatest, the greatest, the greatest, the greatest, the greatest, the greatest, the greatest, the greatest, the greatest, the greatest, the greatest, the greatest, the greatest, the greatest, the greatest, the greatest, the greatest,

50:00 All of this is subject to a system of genomes. But this geometry is a way of understanding, a way of humanizing things, because the geometric spirit is a form of the human being, it is a projection of the human being. But that doesn't mean that this image that the geometry shows us, doesn't mean that there is nothing else. And precisely, geometry itself invites us, if not by the main reason. First of all, it is that reasonings are never finished. Science can never be finished because it continues. This is the reason why there is no progress. There is progress because in the field of mathematics, which is extended, in the field of science, there is a culmination of that. But this culmination of that knowledge remains a context that meets all the needs of all the disciplines that we have and that we want to see in the future. And so on and so forth. And so on and so forth. And so on and so forth. And so on and so forth. And so on and so forth. And so on and so forth. And so on and so forth. And so on and so forth. And so on and so forth. And so on and so forth.

52:30 And so on and so forth. I hope that you did not expect me to be a senior professor of mathematics at Pascal, but I can give you five minutes. There are a lot of possibilities to avoid paradoxes or solutions. I would like to know if there are any conflicts between Pascal and Newton. Pascal is an amazing mathematician. He is a great mathematician. He is a great mathematician. He is a great mathematician. I would like to know if there are any conflicts between Pascal and Newton. In this case, the relations between the two are not the same. It is the same. It is the same. It is the same.

55:00 It is the same. It is the same. It is the same. It is the same. It is the same. It is the same. It is the same. It is the same Please do not exaggerate the fact that he met Pascal, but unfortunately he is a re-Roderval. In this re-Roderval, he took a step forward, and he had a discussion with him. For whatever reason, for the first time in his life, we come to an agreement. There's another one in front of us, it's not here, it's over there, it's over there, it's over there, it's over there, it's over there, it's over there, it's over there, it's over there, it's over there, it's over there, it's over there, it's over there, it's over there... Thank you for your attention.

57:30 Or, or, or, or, or, or, or, or. Oh, it's okay, it's all right, it's all right, I won't say too much. I think, uh, we should start with you. Yeah, it's... I can't really disturb him. It's okay, it's all right. All of them do not seem to be of equal interest. The arithmetic triangle in itself is a bit numerical, which has a particular interest. The numerical orders, which we will talk about in a moment, about figure numbers,

1:00:00 is still something that is very ancient. The combinations, Pascal does not innovate much. The two parts that have attracted the most attention are the combinations. And when we read texts written by Marie-Claude Schultz, it is in general on which people respond directly. It is the Potestatum Rubens Camus Summa, i.e. the Pascal's little treatise on the sum of numerical powers. And of course, what was asked earlier, the subtext on the parties where the geometry is at fault. I would like to consider the treaty in its entirety. From a literary point of view, it is not obvious that considering a mathematical treaty from a literary point of view is a fruitful business, but I think it is at least the case in the texts of this time because for people like Pascal, as for someone like Descartes, Writing a mathematical treaty is first and foremost writing a book, in the same sense that it is writing a book that is composed of an apology of the Christian religion or provincial religions, for example. The balance of the hearts and the weight of the mass of the air, that is to say that these are texts in which the literary form is not indifferent, where it is significant. The treatise on mathematics is a text which, the first one, was entirely in Latin. Pascal printed the entire text, it was ready, and the book is not a mathematical sum from Saint Vincent,

1:02:30 there is not even a dedication, in fact, a large book with the narrative of the two impressions. These two impressions, as you can see, are parallel, but there have been changes that have been made. The first change, the first version is composed in a certain way. The first treatment is the study of the arithmetic triangle itself, that is, the relationship between the numbers placed in the cells and the numbers placed in vertical rows. Following the numeri figurati, there is a treaty on the products of continuous numbers, for example 4x5x6x7x8. Pascal dedicates to it a general resolution of numerical powers, a treaty on combinations, a treaty potestatum numerica unsumma,

1:05:00 And then there is the digitalization of numbers. The second edition of the lecture is not only about the language, but also about the plan. That is to say, we see a use for combinations, which are not new. In fact, they are parts of the numerical order treaties and combinations that have been detached from the interspersed order. The use to find the powers of binomys and apothos, that is, it is a very brief one, which shows that the bases of the arithmetic triangle give, if you will, are now interspersed between, interspersed between the arithmetic triangle of nations, the cells, a cell, the number that is in a cell of the arithmetic triangle is given by the addition.

1:07:30 As a consequence, we obtain the number of a cell in the arithmetic triangle by taking the addition of the one that is immediately above and the one that is immediately to the left. In the French version, they inverse these two construction methods and the generation of the arithmetic triangle. All of this will become what is done by the upper cell and the quasi-left cell, and the fact that we can obtain a cell by summing up the cells of the upper stage will become a secondary property, what he calls a consequence. And, naturally, this leads to a modification of the whole of his treatise, which appears, the triangle-arithmetic treatise, with consequences and concentrates.

1:10:00 The promotion of uses has a meaning. These uses, well, I think we can see a reaction that is quite frequent with Pascal, which is the fact that the major idea is often imposed on Pascal afterwards. So I gave you an example that has become almost canonical. This is a famous thought fragment, the so-called tyrannical fragment, which you can see here. And for those who would not easily read Pascal's writing, I tried to give the transcription just below. What is interesting is that we realize that the thought has not radically changed during the corrections, but we see in Pascal a key concept appear abruptly. We see the key concept appear, the concept that crystallizes everything. The first version, the corruption of nature, appears to desire universal and out of its order domination. And Pascal then wrote that the deep meaning of his thought is concentrated. And this is where the idea of ​​tyranny appears, the idea of ​​tyranny that we see that tyranny consists of universal and out of its order.

1:12:30 What is the purpose of these uses? They are, contrary to what one might think, not identical. The use of figurative numbers is relatively simple. Pascal is happy to say that figurative numbers, triangular numbers, pyramidal numbers, are precisely, as by chance, those which are in the arithmetic triangle and therefore all the properties of figurative numbers will be directly deduced by simple transposition of the numbers which are in the arithmetic triangle. You will see that the transposition is direct. Combinations are a bit complicated. Pascal is obliged to give a certain number of properties that are not related to the arithmetic triangle, but to the doctrine of combinations, which are well-known properties. I put these properties in page 2. A number is not combined in a smaller... Any number is combined only once in its equal, the unit is combined in any number, as many times as it contains a unit. In other words, Pascal is obliged to start from a certain number of principles of the theory of combinations, and it is only then that, by a proposition, he states that he attaches combinations to the arithmetic triangle. The sum of the cells of any parallel row equals the number of combinations of the exponent of the row. To find the number of combinations in a number by consulting one of the cells of the triangle, Pascal is only an interpreter.

1:15:00 On the other hand, on the rule of the games, the first thing to do is that the money that the players put into the game does not belong to them anymore because they have left it. In other words, he starts from a principle that is a paradoxical principle. That is to say that my money is no longer mine, and it is from this paradoxical principle that all the parts will be deduced under two different forms, first in the form of the recurrence on a tree, which is now classical, and then, second point, we can determine the parts by consulting the arithmetic triangle in the case of at least two players, and Pascal is obliged to deduce other paradoxical principles. If we interrupt, we must not at all take into account what... Already won, but only in terms of the number of particles. There is also a sense of the binominal.

1:17:30 This is an extremely disappointing use, because Pascal... ...numeration of numerical powers.

1:30:00 The works of our long prayers, the encounter of world events,

1:37:30 and Pascal's questions, at the same time as an intervention of a vulgarizer, and, therefore, I take the sense of these contradicting arguments. So, I would not be pleased with the value of scientific skills... It should be noted that this is precisely what Pascal is trying to do. It is not by chance that he leaves a part of his treatise in Latin. It is the part that relates, to put it in an expression, to the cuisines, the summations of powers, or more precisely to what is purely operative. The criteria of divisibility of numbers, the summation of numerical powers, it does not interest... The concrete view of the correspondences that may exist between arithmetic triangles, then combinations, parts, etc. This is something that the honest man can think about, that he can deepen and that will be for him the object, the pretext or the cause,

1:40:00 as Pasteur says, in the geometric spirit, of reflections that go better than all geometry. In other words, the learning, the discovery of these subject of geometry, or to insist on it, of geometry, of mathematics, we would say, is at the same time to progress in the field of mathematics and at the same time it is, for example, in Pascal's case, the search for brevity. The research of the Brieft, which is a quality that mathematicians, I think, appreciate, I remind you of what Eribon wrote in front of the same cursus, nothing superfluous that brings disgust, nor anything difficult. The point is that the best method of teaching science is the one in which the Brieft is combined with ease. But it is not easy to be able to obtain one and the other, mainly in mathematics. It is also a quality of the world's rhetoric and it is very significant that we find in Pascal the same conclusion in a text which is a little, it is sir what I had to tell you before entering into the matter and I could have perhaps put in less space if I had worked more in the 16th provincial.