Renormalisation (contd.)

0:00 Thank you. I thank Jocelyn and the organizers for inviting me to speak. I thank them and I curse them a little at the same time because it forces me to talk about a subject that is not strictly the center of my thesis, because my thesis is about the relationship between probabilities and beliefs. And today I will talk about the relationship between probability and time, which is a little different. So I don't necessarily have answers to all the questions you are going to ask or ask me. But I'll try to do my best. The question is about the relationship of probability with time and causality in Reichenbach. For a simple reason, as we will see, he defines the direction of time by simply resorting to probability, and by defining causality in a probabilistic way. Basically, everything is based, as in general in Reichenbach, on probability. And I will see why and how he operates that. So, just as a preliminary, I will start by recalling two or three facts about Aschenbach, because not everyone necessarily knows the name or even the existence or the main works. Aschenbach is known as the Berliner representative of logical empiricism. I'm sorry for those who already know that very well, but I prefer to put things in order from the start. He was born in Hambourg the same year as Carnap in 1891. He died in exile in Los Angeles in 1853, due to Nazism, in which he had to flee in 1833 when he was a professor in Berlin. His thesis, his first public work, deals with probabilities and his last one as well. So it is really the center of his concept.

2:30 I simply mention that he was one of the first five Einstein students in Berlin on the theory of general relativity and that he wrote quite a few books and articles on relativity, so he is one of the main specialists of this question in Germany in the 20th century. He became a professor of physics and not in the philosophy department but in the physics department in Berlin in 1926. He was exiled in 1933 in Istanbul and in 1938 in Los Angeles. This is just to remind you of the general context of the course of his work. The texts that will interest us the most today, I will remind you of them later. The problem is that I did not have time to make a polycopy of the bibliography because my computer crashed this afternoon. But I will remind you of the dates later. Today, attention will be paid to an article from 1925 called The Causal Structure of the World and the Difference Between Past and Future. It is in the 1920s that he mainly dealt with the direction of time, with his latest work from 1953, published in 1956, called Direction of Time, which is a posthumous work. In the 1920s, he started to think about time and the relationship with probability, and he also finished on this subject, since it is the last work he left us. By contrast, and this is the last recapitulative sheet on Rauschenbach, by contrast with the stricto sensu circle of Vienna, Rauschenbach not only lived in Berlin, but defended the theses that were for the most part heterodox or fought by the circle of Vienna. First of all, it may be the main characteristic, but it will not interest us today, well, it will interest us to a small degree only, he defended realism in the philosophy of science, in epistemology in general, while realism was considered a false problem by the circuit of Vienne. He defended a principle of probabilistic verification and not strict. He was also a probabilist, which was not the case for all logical employees in the early 1930s.

5:00 He defended a certain justification for induction, while it was there again considered by some in Vienna as a false problem. And then the last thing, and I will finish on this, because it will be a difficulty of the approach he proposes. His entire work is anti-Foundationalist in a certain sense, in the sense that there is no certain knowledge on which knowledge can be based to define itself. Whether as a result of knowledge or as the basis of knowledge, there are only probabilities, so never any certainty. Defining the direction of time in this context is all the more difficult to do, obviously. Why did Reichenbach, from the 1920s to the 1920s, formulate a new theory of time? The main goal was to fill the gap he saw between the efforts that had been made to strictly define space, already at Helmholtz and many others in the 19th century, and the little effort that had been made to strictly define time in an axiomatic way. So, first of all, we had to fill this gap. And then, we had to respond to the novelties generated by Einstein's theory of restricted relativity, of which he is, let's say, the philosophical spokesman and the defender too, the defender, the defender convinced on many occasions. In order to construct a serious theory of time, axiomatic, with definitions and theorems, as we do for space, we had to distinguish two types of things. First, what was considered factual to measure time and to indicate its order, and then what was considered only conventional, since, as you know, Einstein, in his first writings in any case, had been interpreted in a very conventionalist way.

7:30 So conventionalism had gained a lot of echo and it was necessary to situate precisely in relation to that. And then secondly, and this is a very characteristic of the entire epistemology of Rauschenbach as well, it was necessary to distinguish in a rigorous and definitive way between those which raised subjective considerations and objective considerations on time. And today I will only deal with the objective definition of time at Reichenbach, and not the relationship between the two, which is also a question that he asks himself later in his 1953 book on the direction of time. This is to situate the general frame of his question. The main difference with many other authors on these theories of space and time is that for Reichenbach, Time is a priority concept on the space one because we can give all the characteristics of space by relying on temporal characteristics. So when we talk about space-time, he affirms, we do not put space and time on the same plane. The fourth dimension of space-time is a separate dimension and more fundamental than the first three. There is not a strict parallelism between the two. This is why studying his philosophy of time is, in a way, going straight to the heart of his doctrine and not to the secondary consequences. What does Achenbach support about time exactly? First of all, he recognizes that there are conventional aspects in the definition that we have of time and the determination of its duration. Similarly, in the definition of spatial relations, certain determinations are less conventional, such as unit of measure, etc. But there is also an undeniable reality of time, and it is on this... Attention to these two conciliatory aspects that his theory will have to state. How can time be defined both conventionally and how can it possess an indisputable reality, which he calls objective?

10:00 By doing this, not only does he... They differ from a conventional or purely subjectivist approach to time, but they also differ from a so-called idealist or irrealist approach to time such as the one that had been, according to his own words, in any case, defended by Kant, in the doctrine of the ideality of time, namely that it is the epistemic subject that is the founder of the temporal condition. For Aschenbach, this is not true. The most important result of the considerations he develops, he says himself, is the objectivity of the properties of space and therefore of time since we define the properties of space by that of time. The reality of space and time would be the irrefutable consequence of our analyzes. There is a problem in Raffin-Bart to see exactly what reality means in relation to objectivity. I'm not sure that he himself has seen it very clearly. In any case, what is clear to me is that I have not seen it very, very clearly. If you manage to enlighten me on this, it will help me. Regarding this tension between conventionalism and realism, this is why it opposes itself. It is that in the choice of a unit of time, of course, but also in the choice of the fixation of the order of time, We are forced to proceed by a definition of coordination, a definition of parcoordination, it depends on how we want to translate it, zurdungsdefinition or coordinative definition. A definition of coordination is a definition that coordinates, as its name suggests, a purely formal symbol with, let's say, empirical objects. Or if not objects, at least with meanings. All of these have an empirical reference. So, on the one hand, there are paranormal things, and on the other hand, there are empirical realities, if you will. The junction between the two is operated by a definition, and this definition itself is precisely unconventional. That is to say that one could, in many cases, define, for the unity of time it is quite obvious, but even for congruence, for example,

12:30 one could define spatial or temporal congruence. All of this in a completely different way than the way we do it, i.e. by resorting to a completely different empirical method than the one we use. Of course, we could use as a timeline either the master of Paris or the vibrations of I don't know which atom to define the time unit. The convention is quite visible there, but the convention is also at work in the definition of the congruence of time. Concepts or formal symbols that appear in the theory of time in any type of empirical reality, a priori. Of course, a posteriori, there are only certain definitions of coordination that will be applicable. And this is where the empirical nature of its determination of time comes into play. All definitions of coordination will not be applicable or fertile. And he will show that the definition of coordination that he chooses, he will actually choose two different, a first by referring only to causal relations in the world and the second by referring to probability relations in the world, he will show precisely that one of the two definitions is better than the other, is richer than the other. But a priori, any of these two definitions can be used, and above all, other definitions may be used as well. So it appears very conventional, and at the same time, once the convention is defined, the application of this convention is a purely empirical question. So the two aspects, both convention and empiricality of time, are... They are all conciliated by the theory of the coordinative definition, namely, we conventionally define the relation between a symbol and an empirical object, but then we show that reality obeys and respects the definitions we have given. Thanks to this distinction between What is really of the empirical order, that is to say the application of a definition, and what is of the conventional order in the determination of the characteristics of time, on this point, like on all the others, Rachenbach manages to operate what he always tries to do, that is to say to separate the objective and subjective aspects.

15:00 This is exactly what he also seeks to do for the concept of probability, on which I will only return at the end of the presentation. It is about time that it is about doing this, that is to say isolating the objective core of the subjective assertions that are added to it. So the central problem now, you have understood it well, is how to justify the reality of time, or what he also calls the objectivity of time. What he uses to justify this is to go through two different concepts, but which are related, the concept of cause and the concept of probability, and more precisely, the concept of probabilistic cause. We will see exactly what it means. The purpose of my presentation is simply to explain why I am starting to explain it, and above all, how he takes it to do that, and to indicate only at the end, in conclusion, some difficulties. And so, with what tweezers do you have to take your theory? And this is where my lack of knowledge of the subject is revealed, that is to say that I have no revolutionary idea about the value of his theory, but some remarks to make. On the other hand, I know that for about 50 years, since his last book, Many authors have focused on them, so maybe you know more about them than I do. My goal today is to focus on her conception in the 1920s and 1920s, and since she did not vary in her foundations until her last book, it is the heart of the theory of Reichenbach. Regarding, on the other hand, the later developments, I would say almost nothing today. To be more precise, now, let's see... Which thesis is Rashin-Barr going to defend exactly? First thesis, the objective order of time is based on causality relations. We can therefore define a purely causal theory of time, and the cause being defined in a purely objective way by Rashin-Barr,

17:30 the order of time, suddenly, obtains an objective quotient. This is the first strong thesis that we will have to explain. Second thesis, causality can only be justified by resorting to probability relations that allow it to be applied. We can only strictly apply causality relations if we assume complementary probability relations. For the moment, it may seem... A little bit abstract for you, but I will explain in detail why it is necessary to do this and what it means. Thirdly, only probability allows to confer an objective sense to the present, unlike the simultaneous. That is to say that the causal theory of time allows to define an order of time and a direction of time, but does not allow to define the privileged direction of time, does not allow to explain why time always goes from the past to the future. We manage to give all the phenomena, but we do not know why they must always go in the same direction. In order to define a privileged direction in time, we have to go through probability relations. Causality does not allow it to itself. That's what we'll see next. Sorry, you can interrupt me at any time. Fourthly, the conclusion of these first three tests is that we can only build a complete theory of time by resorting to a probabilistic causality, that is to say, a concept that calls for both causality and probability. If we don't do that, we can't give an objective meaning to the present. And lastly, as everything is based on probability, what we will ultimately have to do is give a truly objective content to probability. This is possible thanks to the frequentist conception of probabilities, in which the probabilities are reduced to the frequency of the arrival of events. So we start from a fairly simple thesis on the fact that temporal relations can be given by

20:00 These are causal relationships, and in order to appreciate what causal and time theories mean, we have to go through the probability concept, and the probability concept itself is based on frequency ratios. I'm just announcing a few difficulties, if you want to reflect while I talk about the kind of criticism that can be made against the theory of Reichenbach. Or maybe I'll see that at the end. It's really the conclusion. The exhibition plan will be as follows. First of all, I'm going to expose his most famous conception of the order of time, namely the fact that time rests on causality. This conception is first exposed in 1921 in a small article and secondly in 1924 in a book, as I don't have time to do the bibliography, in a book called Axiomatic of the relativistic theory of space-time, which has been translated into English, but not into French. Chapter 24. We will then see why it is necessary to complete the concept of causality by probability relations, which will allow us in a third time to define not only a time order, but also a privileged direction of time. All of this by making the difference between the past and the future, thanks to probabilities. It is in one last time that I will try to see how the probability itself allows to confer time to such an objective because it is defined in an objective way in a frequentist conception. Is that clear? No? I could not make a bibliography, but I still indicate the main dates of the works to which I refer. His thesis, published in 1914 and published in 1915, is called the concept of probability for the mathematical representation of reality, which was supposed to be supported by Nathorpe and then finally was supported by someone else because it was not possible in Marbourg, but basically the direction of his thesis is a critical neocantism.

22:30 Concerning causality, the one I just mentioned, from 1924, axiomatic of the relativistic theory of space-time, the text that deepens this first approach to time through the introduction of the concept of probable relationship is the 1925 text, The causal structure of the world and the difference between past and future, which... Gerhard Schoenberg has a privileged status among his other writings. He constantly returns to the fact that he had anticipated the lessons of quantum mechanics. For him, it is extremely important that this text be published in 1925 and not in 1926 or 1927 because he showed the way, according to him, to the relations of Heisenberg and to the limitation of... Let's say the certainty we can have for the prediction of any event. This is therefore the central text on which I will rely. In 1928, he makes a kind of philosophical summary of his first theses, in the book which is today the most famous by Rachenbach on the question, on the philosophy of space-time theory. I'll leave this one aside, and I'll point out, just for memory, because I won't report it in a fundamental way today, his posthumous work, The Direction of Time, in which he comes back to this question by developing it in a formal way, in a much more complex way, and so, on the one hand, it goes beyond me a little bit, and I think it was difficult to make it happen today in an hour. That's why I won't report it. Is this the logic of the last 30 years? Yes. In fact, the idea of a probabilistic logic is already present in him from the end of the 20s, it is certain. But his axiomatic of probabilistic logic is 1932. Is it a trivalent logic? A trivalent logic is a particular case of its logic, which takes continuous values ​​between 0 and 1. So then, if you put thresholds where you want, you have three values, undetermined, true and false. It is strongly based on the Warsaw School, to do that, on the Poles, but it is true that it dates from the early 1930s.

25:00 The goal of his theory is to define the temporal order in a purely objective way by resorting to the concept of causality, which will be defined in a purely empirical and purely objective way. Again, on this subject, as on other subjects, he often insists on his precursor character. His first article on the question is from 1921. These articles are then discussed and criticized by Court-Levin and Carnap between 1921 and 1925, but let's say that it is in 1921 that the first appearance of this theory is mentioned. The general principle of the relationship between time and causality is to say that we can define the temporal order by distinguishing cause and effect in a non-temporal way. By simply defining the temporal order in such a way that an effect may only be posterior to its cause. If we have effects and causes and we define the relation of posteriority between the effect and a cause, then we have a temporal relation defined by causality in a non-circular way. So that's the general principle. Let's see how he takes it exactly. First of all, the fact that... Jean, you already have a remark. How could we consider ourselves as antipersonals? We'll get back to that. Scholarly. The fact of associating a time order with a causal order is a definition by coordination, as I said earlier. And here I simply recall what the definition of coordination means. The two steps will show, on the one hand, that this relationship can be carried out in a conventional way and, on the other hand, that it can be justified empirically.

27:30 So why do we want to link time to causality? For a fairly simple reason, we can define causality in a strictly empirical or objective way. The two terms are not always very well distinguished in his case, namely as the transmission of a mark or a signal between two events. I will come back to this, but his causal theory is a theory of a transmission of a mark. The advantage of this causal relationship is that it is not only purely objective and that it does not suppose time to be defined, and therefore it is not circular, but secondly it also establishes a continuous order. It allows us to account for the order of time by the reals. Here is how he defines the signal in the following way. A signal or a cause is a physical phenomenon that propagates from a real point P. Real point is defined in a precise way in 1924. Basically, it means a material point that we forget it has a mass. So don't focus too much. Let's say a point P. It is a phenomenon that propagates from a point P to another point P' If I apply a mark to this phenomenon, this mark can also be observed in P'. The best example is to take a ball, I don't know what, to make a chalk mark on it and to see that this mark is transmitted when we find it in any other place. Well, no, obviously it's not... Sorry, I don't want to use the word space here, otherwise it would be too much of a circle. We find this mark in another event. In 1924, yes, I specify that because sometimes we wonder a little bit if his vocabulary is wrong, he uses the term signal and the term cause is used later, rather in 1925 and 1928, but basically it's the same thing, a cause is the transmission, well, we can assimilate the transmission of a signal with the transmission of a cause since it is each time the transmission of a mark. That's it. That's why there is no problem, even if the vocabulary varies.

30:00 By signal, we can, in the first two axioms of this theory, hear as well a transmission of light as a transmission of any other signal, which is apparently more valid for axioms after axioms number 2. But for now, I will limit myself to axioms number 1. These are the three axes of number 1, 1, 1, 1, 2, 1, 3. So for now, any signal will do the job. And to make things more intuitive, let's imagine that in P a signal is sent, a mark is made at a certain event E1. This signal is received, or this mark is received, by the event P'. But here, we are not there to observe it, so we cannot really say what is happening, it can simply be modified. We call it in this case S-star. And we find it in P, where we can observe it, and it is the event E2. So we have introduced a mark on E1 and we find this mark on E2 when it returns. If the mark can be recognized, then we are talking about the causal connection between E1 and E2. We can define the order of time in a strictly objective way in the following way. Two events E1 and E2 taking place in P. E2 is said to be after E1 if we can choose a signal train, a signal, such that the departure coincides with E1 and the return with E2. E1 is then said to be prior to E2. Conventional and by coordination, the order of time, i.e. posterior and anterior, and the causal order, the transmission of a signal. So that's the definition. Now we need to know how it can be applied to reality by axioms. So how can axioms apply this definition? First axiom, axiom of order. There is no signal train such that the departure and return coincide in P.

32:30 This simply means that there is no temporarily closed universe line. This also means that we cannot encounter an after-self. The return to the future is not possible. This relation, which is assumed by axiom number 1-1, is transitive and asymmetric. If E1 is greater than E2 and E2 is greater than E3, then E1 is greater than E3. So we have a causal relationship that allows us to define the temporal order in a very intuitive way. The 1928 formulation is a little clearer, that's why I point it out. If E1 is the cause of E2, then a small variation in E1 is associated with a small variation in E2, whereas small variations in E2 are not associated with variations in E1. It is precisely the fact that this relationship is asymmetrical. That is to say, if we note the variation with a small star, we can observe very well E1 and E2, E1 and E2 with variations, E1 without variations and E2 with variations, but never in this order. We will not observe a variation in E1 that is not followed by a variation in E2, if E1 and E2 are related causally. There are two other actions, number 1, 1.1 and 1.2.1.2.3, which specify this question a little bit. They are important for the theory, but not for my presentation today, I mention them in passing. For any couple of events E1 and E2, there is a signal such that the departure coincides with E1 or E2 and the return with E1 or E2, that is to say that we can always compare the temporal ratio of events. And 3, action of the power, the events in P constitute a linear continuum and therefore we can represent this order by the order of the reals. It's a theory that works quite well, that seems solid. So, of course, yes, well, just to be precise, do the axioms have anything to do with it? I only did here transcribe these axioms, as they are really given in the text. You can see very well that they are absolutely not formal in the sense of mathematics, they are expressed in a fairly common language where all the terms are not necessarily extremely well defined, etc.

35:00 In fact, it is possible to make them in a strictly formal language, strictly satisfying from the point of view of the ensemblist mathematics, but Rachenbart did not do it himself. It was done a little later by Andreas Kammler in the notes. If you are interested, I mention it to you, but it's in passing. Volume 3 of the complete works of Reichenbach in German, where all these axioms are noted in a formally, apparently impeccable language. Probably by density, yes. Between the two events, there is always a third or a fourth one. Yes, yes, I think so, because he often discusses Weyl. So, a priori, he read that. Yes, there is an analysis that is fine, but it is bad in relation to quantum mechanics, because there is a lack of space. If you don't dig deeper, there is a gap, which is a problem. Yes, there is a problem, but it is bad in relation to quantum mechanics, because there is a gap, which is a problem. There, I can't tell you because I read the thing and in fact it goes straight to something else, after announcing the axiom, it always goes straight to something else. I can't answer you too much. We'll have to see if this discussion is really at Reichenbach's or if it's not.

37:30 And if it is, I'd have to work on it, but I haven't done it. But it's true that there's a problem there. Anyway, this question is linked to what I was going to say, which is that these actions are not... All of these are mentioned in a formally perfect language, but these consciences are evolved by Aschenbach in a certain way, because what he wanted to give was an axiomatic that he called, I don't know if it's like Weim, I don't think so, but constructive, that is to say that he started from an axiomatic that starts from terms that are immediately intuitive and then define intuitive terms. There are a very small number of propositions to account for the philosophical and essential content of the theory and not all the formal subtleties. This is what I wanted to do explicitly, according to Kamla in any case. For Kamla, what Rachenbach did in 1924 in this article was closer to Euclid than to Hilbert. That is to say, we started from things considered intuitive and we are strictly developed instead of starting from purely formal and syntaxical relationships to then coordinate them to empirical things. In this book, it is less true for other axioms than Dunn-Reichenbach, for example for probabilities. But in 1924, it is his project. I think we also have to connect these weaknesses of mathematics with the project and the context in which its project was set up in 1924, which was also a defense of the theory of relativity in the face of the attacks it faced by many philosophers of the time. That is also why I think it had a language that was not necessarily perfect. I have a question about mathematics. It is true that the idea of the teaching profession is in the mainstream. How does it get rid of the problem of the conventional character of the simultaneous? Well, it's not a particular problem. That is to say that the simultaneity is conventional, but from the moment we have a point that we give ourselves, we have cones, cosons on both sides.

40:00 Some phenomena will not be orderly compared to us, but others will be, so there will be a determination about some phenomena and a determination about others. All events are not necessarily determinable in an order for me, but only the events that are in a cone of causality. That's all, there is no bigger problem than that. It's unfair, our desire and intuition that we can link all the phenomena in a temporary way to each other in relation to our referential, but that's all, it simply limits that. I mean, I think, it doesn't pose any particular problem. By the way, he discusses this question a little later. So, it's a bit imperfect, formally, but basically, We still have a strong theory of the relationship between time and causality. But there are two shortcomings, and these two shortcomings will be covered the following year, in 1925, in his article on the difference between the past and the future. There, he's already starting to get late. How much time do I have, actually? You have 22 minutes maximum. Oh, yes. It's hard because he knows I don't want to talk anymore. Two limitations. First of all, this theory calls for a theory of strict causality, and at Aschenbach there is no theory of strict causality possible and applicable if it is not complemented, if it is not correlated with a theory of probability. So we must see how we can reconcile a purely causal order and a purely probable order of things in this theory. Secondly, the present is not characterized by the relation of causality. The present, unlike the future and the past, is not here distinguishable from the simple simultaneous.

42:30 These two questions will be resolved in the article of 1925 that I mentioned earlier. And radical as well, as mentioned by Sherry Schoenbach, probability relations can replace the relations of strict and causality for any description of phenomena. By causality, we mean strict causality, that is, the fact that a phenomenon or an event is strictly involved in another. As soon as we have A, we have B. It is this relationship that must be weakened. And if we weaken it, then we will also have to review the theory of time. What is the probability of conditional probabilities? Otherwise, there is asymmetry, we cannot talk about sub-conditional probabilities. We will determine the order of time by probabilities that are not symmetrical. We will see. This is the general idea. But then you will see that, obviously, there is an asymmetry in probabilities. This replacement of the strict causality by probability is both necessary because it allows us to realize the articulation between probability and causality which is necessary in any rigorous epistemology, i.e. which wants to be applicable, and advantageous because it allows us to answer the question of maintenance, so it answers the limitation of number 1 and number 2. This article, published in 1925, is important, in particular because it is anterior to the formulation of the relationship between Heisenberg and Heisenberg. And it allows Reichenbach to say that he has anticipated a little the path of Heisenberg's relations, because at the same time, Heisenberg's theories confirm the well-founded of his theory. So why, first, a probabilistic framework for causality? Because strict causality is not applicable if we do not postulate a continuous probability distribution for the factors he calls disturbing. They are sometimes called perturbing, sometimes remaining, I don't know if we would say perturbing or perturbing in French, in this case, well, the idea is simple, we can only say that B is implicated by A, strictly speaking, if we neglect the factors that could disturb this relationship and make it not valid at all times,

45:00 especially because we do not know with infinite precision the state of A or the state of B. In addition to the causality relation, there is a probability relation between the different factors that could come to disturb this relation. And it is only in an ideal way that we can talk about strict causality, that is to say by abstracting perturbation relations that can occur between two events. Strictly speaking, as soon as we state a causality relationship, we have to state that there is also a probability relationship on the perturbatory factors that could disturb this isolated relationship between A and B. There are two knowledge principles, which he calls the causal connection principle and the probable distribution principle, which are indissociable. This is already the 1914 thesis, but in 1925 he goes a little further and defines In fact, there is a third principle that synthesizes these two first principles, a principle that does not have a name, but that we can call the principle of the probabilistic cause, where the two principles are linked. A probabilistic connection exists between the cause and the facts. That is to say that we still accept the concept of cause and effect, but we also accept to characterize them as... If A exists, it determines B by following the laws of probability, i.e. it determines B with regularity, and yet we can still call it a cause. In fact, this distinction between the two principles and a unified principle is purely conventional. We can either continue to work with the two different principles, knowing that we can never dissociate them in the applicability of a relationship, or we work with only one principle, and that is the one that will be privileged. The strict causality then appears as a limit or a probabilistic cause, All of this in an absolutely regular way, i.e. with a degree of probability of 1. And it is here that he shows in a note that he had already announced that the relationship of causality was not necessarily always necessary and that it was therefore compatible with Heisenberg's relations.

47:30 That is to say that we cannot always predict with absolute certainty the succession of phenomena. So, ontologically, this principle has a correlation which is that the universe is a universe of probabilities. Each step in the course of the events is an NCDD. By the way, this poses a problem on which I would simply come to the conclusion that the distinction between what is epistemic order, cognitive order and ontological order at Reichenbach is sometimes very difficult to distinguish. The fact that we have modelled all the events by simply calling for probability correlations is directly correlated with the fact that the events themselves are in probability correlations, and therefore are dice rolls. So there is a real problem, I think. How many phases are there? As much as you want, it's not a question. There are more than one, let's say. That's not Einsteinian at all. It's as anti-Einsteinian as possible. You said that he wanted to defend Einstein in relation to... Yes, but not on probabilities, precisely. Rauschenbach is the prototype of the radical probabilist, for whom there is no certain knowledge to be sought, there is nothing at all. When did Einstein's sentence come from? It's a little later, isn't it? It's his letter with Born, it's in 1927, isn't it? The god who doesn't play the dice. In 1931 or 1932. Even earlier, in 1931 or 1932. So, in fact, we forget that in reality it was, by contrast, but already present in 1925, God constantly played the dice in the universe. Maybe Einstein was very brave. Maybe. But at the same time, I think that the image of a dice launcher is so common that it has been found for millennia, well, for hundreds of years. So, how to reform the theory of time, once we have lost this strict causality that served us at the beginning to formulate it? I'll see how we get there. First, I'll show you how to solve the problem of maintenance by calling for probabilities.

50:00 The present is not the same as the simultaneity, the present is what is possibly simultaneous for a subject, what is eligible as simultaneous for a subject, whereas what is neither the present nor the potential present, the present is eligible, that is the future, it is the past, and the past is not eligible as potentially simultaneous. It is this distinction between simultaneous legibility and non-legibility that allows us to distinguish between the past and the future, and thus allows us to define the present, the present is the threshold that crosses the universe by going from an undefined state to a defined state. This is the moment when the eligible simultaneous becomes ineligible as a potential simultaneous. Is it written in German or in English? This is in 24, so it's in German. It starts to be written in English from 36. The future is what is eligible as a potentially expected event, and the past is the difference with that. Why does probability give meaning to this? What is probable and what is undetermined is the state of elective simultaneity while what is past is the defined state and so it is what is no longer simply probable, which can be determined in a much more solid way than only in a probable way. Probability allows us to give an objective meaning to the undefined character of the future and therefore a defined meaning to the concept of the present. I have already answered this question. There is a certain relativity in simultaneity, but it does not pose any particular problem in Reichenbach.

52:30 How to replace the concept of causality in the definition of a time order? We use for this the probabilistic implication that I noted with an arrow here because the sign that he uses is the sign of the implication with a bar in the middle, it is not in powerpoint, so I used an arrow like that. And it's obvious that if A appears, then B appears with a certain regularity. We can already see that probabilities are directly linked to frequencies. For him, frequency is already presupposed in this definition of probability. All laws of nature then have the form A implies B, and the method to characterize the order of time and its direction, i.e. its topology, We will start by characterizing in a probabilistic way the laws that are already known as laws of succession over time. We will set out a criterion and this criterion will be reciprocally considered as what defines the succession over time for all the laws. This is the acme of this lecture. It is the explanation of the tip-off at Raschenbach. Everything is based on this, on a tip of a fork. When we talk about a law of nature where events are probable causes, we hear this. If we have two events A and B, then we have C, in a probable way. And conversely, we don't have C when we have only A, and we don't have C when we have only B. For example, if A and B are the impact of a billiard ball and it is the collision of these two billiard balls, then when I give a blow to each of the billiard balls, I can have a collision, but when I give a blow to a single billiard ball, I can't at all define the probability of a collision, obviously, because I have to draw a second ball. So it's extremely intuitive and it's this diagram that expresses it.

55:00 A monolature in which an order of time is given is a monolature where we need two events to predict a third one, but where the prediction of the third one is impossible when we have only one of the two events that allow us to predict the third one. It's this asymmetry, and here we find the... What was the question again? It was you, I think, sir. We find the asymmetry in the probability relation that allows us to base the temporality. We only have probability relations here, each time, and we define them asymmetrically, that is to say that it is not because... So obviously, if we have A and B that imply C, we also have C that implies A and B, but what matters in this case is that we cannot have... We can predict C if we have only one of the events. On the other hand, the other relations are quite trivial. Well, not really, because these two are also interesting. That is to say that between two events that allow us to predict a third one, we cannot necessarily define, in general terms, a probability relation. To do this, we have to resort to another type of event. On the other hand, when we have the facts, we can obviously predict the partial causes. Not predict, but reverse the partial causes. So, without having introduced time at the beginning in this diagram, we find it. Time goes by like this. We can predict C if we have A and B, but not if we have only A or B. The model of the probabilistic definition of temporality in Reichenbach. On the other hand, there are forks that are in salt. Sorry. That's just what I said. There are forks that are... I'll just finish on that.

57:30 It's the formulation of what I just said intuitively. If the application of probability is valid only in one direction, This is where it is asymmetric, i.e. it goes from C to A but not from A to C, then the antecedent is the event temporarily after. But it was missing maybe, but that's exactly what I wanted to say. If the relation goes from C to A but not from A to C, then we call A after C, of course. What does it mean exactly? In the previous transparent, we just called C imputable. Yes, C implies A and C implies B, but what we don't have is A implies C. The negation comes from this whole parenthesis. I'll show you the diagram. We can't predict C when we only have A, so we have A implies C, which is not valid. Same for B, of course. On the other hand, we can very well reverse the causes from the effects, but what is asymmetrical, where temporality is introduced, is that we can very well have this direction and we cannot have this one, because we initially defined the prediction of C as being strictly linked to the conjunction of A and B. We cannot dissociate between the two to predict C. So the relationship of probabilistic causality is determined in an asymmetrical way and thus allows to define a time-passing order. I point out that we did not use the concept of mark here for the transmission of causality. And here I also simply remind you that it is a C and not a D. At first glance, the question is solved. One last thing to point out is that all forks are not good forks. There are forks in salt, I think that's how we translate saddle, right?

1:00:00 In which we start the characteristic relationship of the forks in reverse point, that is to say the saddle fork, it is the disjunction of A or B. If we have the right to predict B from the disjunction... If we have the right to predict C from the conjunction of A or B, then all other predictions are possible. If we have the right to define the probability of the appearance of C from the conjunction of A or B, then we have all other possible relations. So this fork, characterized by this relation where we have a C, is not a fork that allows to define a time. An order of time. On the other hand, it is extremely useful to say to Schindler, because it allows us to predict the probability of causes. We have any effect, we can predict the probability of causes. If we have B, we can predict C. And if we have B, we can even predict A. So it's an extremely rich relationship of probabilistic causality. It's extremely rich, but it does not define an order of time. For example, if the temperature rises, then we can predict that the oven has been lit. And we can also predict that the steak we put on it will grill. So we can predict both causes and effects. And C acts here as a common cause, so it has two effects. It does not allow to distinguish a direction and that only the point fork distinguishes a direction of time. So here is roughly the conclusion. The past is therefore objectively determined what it can be inferred from a simple partial effect. It is the fork in salt. We can objectively determine the whole past because we can predict, or rather, reverse c from a or b. However, the future is objectively undetermined, because if all the partial causes are not given, then it cannot be determined, we cannot predict it. We need the conjunction of A and B to predict C, we cannot predict it from either A or B only,

1:02:30 while we can very well, in the scheme of the fork in salt, predict C from A or B. And this is where the difference between past and future is objectively founded, because of the difference between these two types of forks. And it is founded without resorting to anything other than a simple probability relation. And these probability relations themselves, I don't have much time to say, but they are founded by something else, objectively, namely by frequency ratios. Conclusion. You see, Jocelyn, how are you? Good. Conclusion on this part. Because there is a last part to which I wanted to come, but which I will sacrifice. Probability relations allow you to do a lot of things. They justify the application of strict causality. They define a probabilistic causality, as we have seen, they allow us to objectively define the present by difference with the simple simultaneous and therefore they allow us to differentiate between the past and the future, which was not possible in the deterministic hypothesis, and they allow us to objectively define the notion of the direction, the unidirectionality of time. The demonstration would not be entirely complete if I did not give the fundamental theory of probability as the frequency. I give them linearly, simply because it's nice to have probabilities and objective relationships that allow us to define a time order, but if there are no probabilities that are the same objective, then in this case we have not gained anything. Assigning an objective value to the probability comes back to Aginbar to define the probability in a strictly frequentist way. Frequentism In addition, the position that gives a definition of coordination to probability, which is this one, in relation to probability, expresses the limit of the frequency, if it exists, with which an event of a certain type occurs in an infinite sequence of reference events. A frequentist position reduces the probability to a simple discount of the number of events

1:05:00 All of these things are part of a set of events, and as this calculation is absolutely certain, it never suffers from contestation, if we count a number of things among another number of things, then the probability relation is given in a purely objective way. The big problem, of course, is to know how to determine these limits, and if there are no subjective elements that intervene in determining these limits. But let's make it short, Rauschenbach believes he succeeded in giving an induction theory that provides the limits of frequencies that are probabilities without ever referring to any belief or principle a priori, and therefore claims to have founded a purely objectivist theory of probabilities. This is Frequentism. There is a problem with Frequentism, and then I conclude that Frequencies are Frequences and are therefore defined only for generic types of events on infinite classes of events. However, for the moment, we have characterized the temporal relationship as singular events. So there is a real problem to know if we can really be both frequentist and define the temporal relationship, the direction of time, by probabilities that, a priori, are only valid for infinite sequence of events. So, well, I will not try to discuss this story too much, we have to go through different concepts. Conclusion to which I would like to come if the title does not play a trick on me. These are just a few indications of things that seem to me, not banal, but particularly delicate or difficult in his theory. So here I absolutely do not claim to say anything interesting about the thing or to have taken into account in a way...

1:07:30 I have been informed of all the critics and all the re-formulations of this theory, especially by two main authors, Wesley Salmon and Bas Van Fransen. Here are the conclusions to which I came in a really naive way. First of all, the argument on determinism seems to me to be difficult to hold on to, namely that, ah yes, I have barely devoted a moment, so it's ... well, let's say that for Aschenbach, we are justified to do the economy, to eliminate the principle of determinism, the hypothesis of determinism, because if we are determinists, then we have no more justification to act. Well, it seems quite questionable to me, even though it also seemed to Schlich, but since I haven't talked about it so much, I'm not going to talk about it. What I have approached as a difficulty is the uncertainty, as always with Reichenbach, between the ontological aspects and the epistemic aspects. As much to talk about relations of probabilities that have a certain degree, as to say that the events themselves are determined causes a fairly strong friction on its distinction between what is the order of our modelization of the Real and what is the Real itself, while it has no scruple to talk about the Real as such, that is ... If he were, I don't know, a sort of consequent Kantian, he would say, well, what you don't hear in reality is the way we, basically, model something. But he doesn't take this approach, at least not systematically, and so we are often lost between what he calls objective as epistemic and as ontological. So, how can we determine probabilities for singular events, which was the means by which I managed to give a theory of time, while we are purely frequentist?

1:10:00 Well, that's a question I don't want to talk about. And then, last thing, I'm not really sure about that, but it has always caused me a little problem. To determine the values ​​of probabilities in frequentism, we have to resort to induction. In other words, two sub-regions. But induction, if it must allow us to determine probabilities, and it is these probabilities that allow us to determine the order of time, then it must be done independently of the order of time. Therefore, unless I am mistaken, it would be necessary to think of an induction that is totally independent of the temporality. And that, it seems to me... I don't see the point of having a non-temporal induction. Maybe it's just a basic problem that I haven't thought about enough. That was the conclusion. What is the notion of event? It's a problem that Karmla raises when he tries to make the axiomatic of the thing, because there is no axiom of the existence of events or things like that. Here again, you have reached the limits of my knowledge, but I do not believe that the thing is defined in a very rigorous way. I have not come across rigorous definitions of this in a seminar. That does not mean that there is none, but I have not seen any, and I did not have the impression that they were discussing this precisely. It seems to me that you are not sure of the steps you are taking. I do not think so. What is the notion of topology? Because at the beginning it means that I am witnessing something and that this thing is here, so it is after. No, but at the beginning you just have to hear the fact that a mark is imposed on an event.

1:12:30 And in return, the fact that this mark is observed. First of all, we must not imagine space. We must simply imagine that we have in front of our eyes a collection of events that some wear brands and others do not. We initially define the fact that we have introduced a brand and that we find it in another of these events. We introduce it on E1 and we find it in E2. But there is no need to represent ourselves more than that, a real path, a real light, or whatever. We simply have four relations between E1 and E2. I arrived earlier, I won't get there too easily, but E1 star E2, E2 star E1, etc. We simply have that, and we call it departure, the fact of having found a mark on E1. Well, I don't know. I don't see the problem here, actually. Is there a front and a back? No. No, no, no. You have your petanque ball and there is a piece of chalk on it, a sparadrap. That's it. It's your mark. We don't care if you introduced it or not. It's there. It's there and then... Not then. It's there and then we have another event where it's not there. There is no need to have imagined making the mark, we just need to see what is on it. We have introduced a small thing. A signal is just the fact of signaling something. We do not necessarily have a number of pockets. There is not necessarily a temporal relationship there, I think. We do not imagine a transfer or ... So the notion of time, if I understood correctly, time, we haven't talked about it yet.

1:15:00 Yes, because he derives from it. We can't remind him of it. In your essay, you have a superposition between determination and predictability. Is it really like that? At one time, especially in Damascus, with a bit of isolation, no one had understood the changes that had happened in the 20th century. I know it very well, there were people who were still in the 20th century. And also the effect of strong determination, which seems to me to be able to immerse oneself in a physical dimension of time. Determination is the equations of mathematics and physics, normally it's like that. And there is also a conclusion, I don't know if it's true or not, but in the equation of perfect gas, there is a causal relation, absolutely without time, there is no time in this equation. Ah yes, but it's not because you define time by resorting to causality that as soon as you have Well, that you need temporal relations as soon as you have causality, yes, wait, yes, in fact, he calls for causality to define the order of time, but to define, but does he say, he doesn't say that, well, yes, it's the same, that we can, that he doesn't say that each phenomenon, no, but simply, yes, no. Does any causal relationship generate a temporal relationship? Yes, in fact, yes. And that, according to what you're saying, wouldn't be good for all equations, is that what you mean? Yes, because there is a causality that is rather epistemic and objective. At the same time, for example, if in the equation there is the perfect gas that you take as an example, it seems to me to work well. You vary T, and so you vary P. You vary the equation itself, but there is no vector.

1:17:30 Ah yes, yes. I understand. On the other hand, it is very difficult without having appreciated the work of the critical systems, namely the dynamical thermodynamics, which pull you too far from what you already know, and to have a good amount of time. Ah, so this is perhaps a reproach that you could no longer make if I had been more competent and if I had seriously read the direction of time of 1953 or 1956. Because on that, maybe Guido, you know that better, right? He extensively refers to micro-statistics, macro-statistics, thermodynamics. So there may be an answer to your question, but here it really reaches the limits of my knowledge. It would be necessary to ask someone competent to do this for both of you. I have a question in the same order as you. Can you say a word about the discipline of mathematics in particular cases? Because the ethics of particular cases... In Frequentism, we could say that it is the time of the experiences, the definition of the state in the laboratory, which is very recent. So it's not necessarily in time, precisely. That's what he defines as the state. In any case, we can see how this is not to be attributed to what is ideally in the state, but then what is the time of the unique ego? In fact, the starting point, in my opinion, is a little bit off-putting in relation to what he is doing, because he is not referring

1:20:00 Fundamentally in time to define the frequency. The repeatable character is not even really important in fact. It is... Yes, that is... We need time, but we need repetition. Repetition, yes, but not repetition in time. Maybe we need repetition. But so... Time is very good, we said it in general. So there is no repetition of the history that we are talking about. Yes, but what do you want to probabilize in there? That's the problem. If you want to give the probability that we are universes, then of course it's not definable, quite simply, because we don't have a class of references of universes in which we could take ... well, we don't do it, the question doesn't arise, I think, quite simply, we don't give it, it doesn't make sense, I think, to ask ourselves the question of knowing what is the probability of the existence of our universe. I have a more specific question. What is the great work of Raoul Schoenbach on probabilities and in which year? Well, it is 35 in German and 49 in English. But there are two. In fact, today we know almost only the English work, which is quite heavily repainted. But at the beginning it was written in 35. Well, published in 35. Written in 35. But here I am placing myself before, of course. Because here, the reflection of probability is independent and it is even in the frequency. Yes, of course. But the fundamental intuitions are already there. Frequentism is there, and the fact that we cannot have a relationship of causes without supposing a relationship of probabilities in the factors that do not determine the cause. But for everything that is the technique, the axioms of probability, the logic of probability, and then the justification of the universal applicability of the frequentist interpretation, then of course we are in the middle of what will happen from the years 1932-1933. It is in 1933 in particular that induction is justified. Before that, there was no justification for induction. We have an axiomatic of probabilities that calls for limits of frequency established by induction, but there is no justification for induction.

1:22:30 So we don't have a solid theory of Reich-Marsch and probability before 1933, but the intuitions are present since 1914. We can't separate them, if we talk about them. No, that was not the point of talking about the work of Reich-Marsch on the probabilities themselves. No, yes. Because there, we have to continue in time. Yes, yes. There is nothing that can be separated. We can't separate the three. We can't separate them. Even in the 1935 edition of Frequencies, I remember that he had still evolved in relation to, at least from the point of view of, von Mises. Ah, but largely, largely. But well before 1935, in fact. There are already 30 texts in which he fundamentally distinguishes himself from von Mises. I have the right to do a little bit of advertising here. We are soon going to publish a collection of texts from Rachenbach, Carnap and others between 1929 and 1936, in which I have translated two of Rachenbach's texts on probability, and in which he already sketched his distinction, his difference with Rachenbach, with von Mises. In my memory, it is more or less the same as the ones that have been noted more automatically compared to the ones of Penrose. I know less about that, but in any case, the difference with Penrose is clear. Basically, it holds to the fact that there is no need for an axiom of irregularity in the probability sequences at Reichenbach. We don't need to assume that the sequences are random. Any type of sequence is possible to probabilize. Secondly, there is no theory of application of probabilities. It's surprising to say this, but von Mises, who was an applied mathematician, does not have a theory of application of probabilities. His theory of probabilities is a much better one. I read it through Rauschenbach, so maybe it's false, but for Rauschenbach and von Mises, He makes a completely abstract theory of probabilities, which is not that of Reichenbach strictly, but above all, he does not try to link the axiomatic with the practical operations that can be done to determine the values ​​of probabilities.

1:25:00 That is to say, there is no reduction theory in von Mises, while Reichenbach gives one. That is why he fundamentally distinguishes himself from him. But that's for later. The important thing is that we can make non-coupled probabilities and that the frequencies of the probabilities will remain the same. There is no need for a little bit of a construction of probabilities or we do not count the advance that will be given. It is simply a class of events. There are events that are more or less probable. And at the same time, there are those who are almost certain, almost certain, that I am not a scientist. I think it was done before Savage, because we already see it at Dofinity in 1937. ... in the first part of his lecture. Well, yes, that's true. We can see the mathematical expressions under the form of relativity. Ah yes, surely, yes. The discussion was held yesterday. But that's true, I wasn't one of the epistemologists. It was the mathematicians who decided to go into it, and fortunately, the theorems that founded a certain approach to mathematics. But my point of view, on the other hand, doesn't have that. I think it's important in a project to base causality on probabilities, to have a foundation of probabilities that is not hyperbolic or hyperbolic, but rather hyperbolic, because at this point we have only seen the sexes between them.

1:27:30 The quasi-certain, the quasi-certain is the quasi-certain, the quasi-certain is the quasi-certain, the quasi-certain is the quasi-certain, the quasi-certain is the quasi-certain, the quasi-certain is the quasi-certain. But I remember that you had a course. I mean, if you're talking about a course on a path, you have to, if you please, date it. If you don't, I'll have to repeat myself. I'm already in a much more comfortable place than I was before. Ah yes, it was a place where you were the most proud, that's for sure. It was a very comfortable place. But that's not really the theme I wanted to talk about. I'll even be one day in my office, but it's not a course on the model of action art. I've already started going there. I don't know why I haven't done it yet. You were talking about the question of departure and... Yes, of course. Because one departs from the other. The temporal side appears as a return and a return, or the other side appears as a departure. You introduce a mark, then the other temporal side appears as erasing a mark and erasing a mark, or the other side appears as introducing a mark. The method of the March seems to be good to define the causative connectability of simultaneous events that are not causally connectable. These are always events that are in the past or future of the subject. I do not see how the mark gives direction or distinction to the effects.

1:30:00 I believe that there was an addition to the history of Marx. It's the same thing. It seems to me that there is something more to be said about Marx. First of all, this story of Marx. It intervenes when we try to define the order of time in a strictly causal way, so in 24 or in 28, but not in 25. In the article I discussed the most, we don't need a mark situator, we only need a probability relation. In the context of the brand's epidemiology, which for him is a little less complete, if we take the fact that the symmetry of introducing a brand is to erase a brand, then we would have a reversibility between the two types of causal relations, isn't it? To look at the opposite order, that's what I'm talking about. And these pairs are the pairs of moments that can be controlled, and distinguish these pairs from pairs that cannot be controlled. I think he wants to do both, because he defines the difference between the posterior and the anterior. This is a unique direction that he tries to determine by causality.

1:32:30 We're talking about singular events, there's no need for... There are certain constellations that we see and other constellations that we don't see. So it's... it's something empirical. Yes, it's empirical, yes. There are certain things that we see and other... They say if we have these constellations, as you say, then we can define a causal order and therefore a temporal order. So it's only conditional. That's it. It's when we have that, then... Then we can define that. We could think that there are certain constellations, and then there are also the opposite constellations. No, it's not... They are elements of the same type, but then there are no logarithms. No, I don't think that's what they're doing. When we have that, we draw it out of order. But he doesn't say, if we have order, we must have that, for example. Is that what you want? Is that the difference? No, no. I was just giving an example of the Antecedent. It's not... It's not... You could think that there is a universe, the Antecedent, its constellations, etc. It's not really that. No, it's not always like that. If it's like that, then we're pulling the trigger. But there are many other constellations that we could have, I think. I don't know if I got it right yet, but I think that's it. I have a question. I wanted to make a remark about the definition of the now that comes to mind. Of what? Of the now. Ah, the now. At one point, I said, the now, we're going to come back to it. It is a threshold. It is the only condition for the development of the universe. It seems to me that there is a spatialization. A threshold, always a place.

1:35:00 It is a principle. Spatially, it is a principle. No, no, but here it is an image. We say a threshold to say it is an image. In fact, it means something logical. The moment when what was undefined becomes undefined. The fact that I say threshold, I could have said any other thing. It's just the passage between the undefined and the defined. In the experiment of the sensitive subject that Hegel did on the sensitive subject, consciousness, namely the media, It is a welcome, a non-negative relationship, but at the same time it is a sensitive relationship, and this relationship is determined as an incessant incessance. What I show is that in this dialectic, in the experience that makes the consciousness incessant, which is at the same time implicitly, I show that the way in which we seek to make the consciousness incessant is an experience of contradiction. It can be translated into Hegelian language. What does Heffner say, if you will? Heffner's expression is the contradiction between the two. It's not a contradiction, it's just the transition from one to the other. The contradiction is if the two things are incompatible. They are incompatible because we simply pass from one to the other. Well, I don't know if that's the point you want to raise. If you want, there is a contradiction in the fact that the same term is used to work with the language of the university. The university, the language of the university, is the moment in which the language is used to be said by the fact that the university is sometimes used to say everything by the negation of the negative, but at the same time nothing.

1:37:30 Do you have a question? Because people are starting to get bored, it may be time to finish. The contradiction of the three elements in the succession of the two elements of the question, the first and the second, is that the contradiction is a synchronization of physical terms. Yes, yes, that's the starting point. It's accepted as a starting point. The contradiction of the three elements in the succession of the two elements of the question, the first and the second, is that the contradiction is a synchronization of physical terms. Yes, yes, that's the starting point. It's accepted as a starting point. So the contradiction of the three elements in the succession of the two elements of the question, the first and the second, is that the contradiction is a synchronization of physical terms. And so on, until now, there is only one transition between the two worlds, between the two worlds, between the two worlds, so there is a definitively fixed Atiyah. That's quite reliable. That's pretty much what he's saying, if you want to translate it into Hegelian language, that's about it. I'll come back to the question. I don't know at all if this way of introducing time could possibly be your research track to try to understand what we don't understand today, as far as I know, what this phenomenon could be, very close to the Big Bang, where time would not be distinguished. These are the three high dimensions of space. The four dimensions would be something indistinct. There would not have been this rupture of symmetry yet. Well, yes, that's it. After an epsilon of time, there is space on one side and time on the other, even if they are connected to each other. I don't know if my question is well asked, otherwise there is no probability in cosmology. No, but because we talk about it, maybe, I don't know what we can say about it. I don't know, frankly, these are clues that I've never... This is not confused, we can define spatial relations by using temporal relations.

1:40:00 I think some people, some specialists ask themselves the question, but no one has yet found the answer. And then, in addition, I suppose that today we don't have any more... I don't know, but the current cosmological theories, in my opinion, are very different from what Wieschmer could think of. In particular, the three dimensions, I think there are many more. In string theory, how many dimensions do you put? I would like to thank Alexis for a pleasant discussion.