From Weyl's analysis of problem of space to gauge principles & structures in relativistic quantum physics 1923–1929

0:00 I see you now with my... ...the moment, Mr Smith, and said, we don't see him, so... ...are you ready? Thank you. Thank you. I feel a little nervous in doing this talk, because it's a sandwich between two talks on one and one, basically. And the rest of the talks on either side of it are equal degrees associated, as far as I can see, and pretty well technical. One of the things I want to talk about, and I want to kind of say it was interesting, is something that was also happening in our period, which also covers mathematics, physics, and philosophy in this period, but it's more in the spirit of popularisation, more in the spirit of authorities, I think what that would mean, reaching out to the general groups, not speaking to them in a particularly technical way. And I think one of the questions we might consider behind the time is the nature of this authority. What is it that enables certain people to speak on behalf of our community? And what is going on when they do? One of the things that's very clear about what's happening with Enrique is that he is portrayed, represented, and marketed as an authority on

2:30 things that he is in fact not an expert in. So he's taken out in ways which are not a a simple reflection of the command of ideas that he, in fact, possesses. So that's one of the things that intrigues him. And another thing that intrigues me is that he's somebody who, in a way, is in a dialogue with Poitier, and that's also interesting. He's perhaps less well-known to who I don't know, so here are some facts about him. He comes to fame in Italy as an Italian algebraic geometer. He's one of the great Italian algebraic geometers. He works with Casper Loewon near the proper great surfaces, roughly the theory that's just until either side of 1900. and in this, where he is Hutchison-Topp as a leading editor. And when he gets his first position, which is in Bologna, in 1894, quite the other hand, he gives a course in progressive geometry. This course was written by a publisher, one of the people who particularly likes it, is Felix Klein. He described it as a writer of Wigthwares and said he wrote for the the 23-volume trunk of a kind of survey of all of the mathematical scientists, and in 1897 he asked to write on the principles of geometry. And there's something interesting about this. Klein mustn't shuffle down the corridor around Hilbert. He's got somebody else, and the reason is that Klein is much more in sympathy with the overall approach to geometry that Enriquez has than the one that Hilbert seems to be. There's some questions of chronology here too, but there's a sympathy between, a collective sympathy between Enriquez and Klein. So he starts to write that article, which is eventually published

5:00 in 1907 as early as 1897 and is on his way to becoming the person who speaks for the Italian approach to geometry and for contemporary ideas in geometry and we'll see in a minute that his own take on geometry is selective let's say that's not a criticism everybody's approach is selective for the in comparison to Pancoré and Pancoré's conventionalism, and then I'll move on and discuss things that we find. Rico was writing about in his book, Problems of Science, other than this contrast of Pancoré. He has a long discussion of what it is to know things. What is a knowledge claim in a way of time? What are you doing when you say you know something? What are axiomatics, as an approach to mathematics, what's the general philosophy underlying all this? became interested in philosophical questions. They arose from a request from the students, almost as soon as we got to go on here, the students said, who could they get from him please, a clear formulation of the basic ideas of Georgian because they knew that this was a topic that was being much written about in the early time. And in the course of the So I think I was only satisfactorily what the maximatic formulation of geometry might be, but not very simply what the philosophy of geometry might be, and what kind of critique of an axiomatic approach should you have. I'll then end with some remarks about the reception of all of the tourism workers. So there's a bit about this. Well, I just reminded you of some dates. problem of science is 1906, and the American vision is quite instantly later after the French and after the German negotiations. So the fact that it's why we translate it is also interesting. There's a lot going on this time, and I'm sure most of you know this will be a little brief about the buzz and excitement of the most popular world at the time. But we have, both in mathematics and philosophy, the growth of international conferences. So here is either mathematics or philosophy, take your pick, presenting itself in an international arena,

7:30 largely, I think, if you want to know, despite the simplification, as a collection of national teams. It's not quite the World Cup, but people unexpectedly don't turn up in any great numbers in Paris in 1900 for mathematics, but they came from philosophy. So, you have the dates here, you know the stories, as well as I do, the mathematicians at the first international conference in Zurich in 1897, and then they followed in Paris, and then there in Rome, which is very important in all of this period. That's how it's been, I suppose. And then finally in Cambridge, England, the final part period. Meanwhile, philosophers are also wandering around the four-year intervals, going to the rather more attractive intellectual centers of Europe, and for instance, they choose to go to Romania in 1912. But Henry Christ is the president of the International Congress of Philosophy. So that's not bad for someone's reputations in Park, Ashbrae, and Chiang. As the geometry you saw, the story you tell, I'm not going to argue with Peter Thor, says there's a real weighting of interest in geometry, stimulated by Marx's passion book of 1882, in which he endeavours to say that geometry should be conducted in the following fashion. You should go to reality and abstract a few concepts from it, state them, stop doing that, and make them out of formal deductions, purely from the things you have defined. In this fashion you might get geometry, for instance, for biology extensions of the finite second column line. Well that's not a lot of geometry, so you go back to reality and you make some more abstractions from it, and then you state them for instance. I still need man to make small deductions. So there should be quite clear separation between the process of abstracting and then the entirely formal process of deducing. So that, in my opinion, is passion.

10:00 There's a story covered by the dots here, which brought up which, yes, they released it all but onwards, and into those dots if you expand them, you may or may not get a large collection of Italians, depending on which the dot you click on, and depending on which authors in recent stories we've been reading with a really great deal of what we've done in the last, not so many years this summer, it's still in progress, the paper and the current are perhaps through Black Science as well, and Redame and others, precisely on what the Italians have been getting up to, and we'll be looking at that quite shortly. I wanted the course of this talk to make comparison between what Ennis Naples said, a very interesting paper in Osiris in 1936, and what Henriquez had said before, Naples's paper was I think called Geometry and the Origins of Modern Logic, some title like that. And the theme of the paper is that in geometry, especially in axiomatic projective plane, you have a view of it. And you often see this in the old text. You see columns like this. And here you have statements about points, lines, lines, and the other column you have statements about lines leading in points. And view of it in the plain objective geometry allows you to replace points by lines, you write the current and all of this stuff, and says Nagel, that's very interesting, because on the one side you have intuition, which are points on the line. On the other hand, you have a geometry in which the line is the basic. And nobody has an intuition like that. So this formal trick of duality, Nagel argued, promoted an axiomatic or abstract view of in which intuition plays a diminutive role, and greater weight is placed on the formal conductive side of things. And I won't come back to what Nagor said in the light of what Enrico has said. Okay, here's a dot. You click upon a dot, and you find this in the 1890s. You know, Fahler, for instance, is trying to write down a set of axioms for projective plane geometry. If you have three points in a line, the standard construction involved in the thing is a complete quadrilateral,

12:30 which gives you a fourth point in a line for the uniqueness of the fourth harmonic point, and the cross ratio of these four points is minus one, or is it basic stuff in projective geometry? And Father said, I can know that there is a fourth point. Could he have a show machine? Which there is not a fourth online point. Yes, he could. And he produces the famous Father, find out the show machine has seven points and seven lines. And the Guardian, I was at the heart of you, and I'm quite excited to do this, and I'm looking at Father and I pick up the same impression. One of the things Italians are doing here is they're saying, well, you do need this, I think, because look, there's this terrible thing out there in the garden. this accident, it can't get into the house, it's not good. And then you press on in this fashion, frightening yourself with some other geometry, with a deplorable property, and then you ban it by another accident, and in this way you might have an axiomatic description of objective geometry, the one you want, real objective geometry. Okay? So there is no saying, well, go out and get gone. They're saying, don't do it. Stay in the house where nice and comfortable and warm and you understand things. There's crazy stuff out there, right now, not the garden and the streets aren't safe, but please don't go there. There isn't a research imperative which said, no, what else would you do? I wonder if you just make some axiomatic statements, what are we making? Why don't you know it so much like the Italians? Bernaysi, I mentioned, for a watch which even Italians tell me is fairly seriously unreadable, in which he gives a description of geography, which you have a non-Archimedean, so the Magdalene's, and the Geometry of the Black, kind of seems to be the only coming in that will actually be there. And here you have a set of piano, perhaps the most famous of them all, who shows that the axioms of the projected geometry, only sensible set of axioms in objective journalism, in three languages, all the MacIntyre gives you Desargo's theory, essentially with the proof that Desargo, you play two triangles in a plane, which are in the spectrum, and there's a lot of line, there's a common line of the two planes, which are two triangles in a plane, okay? Whereas, in tradition two, Bernardo isn't sure, he doesn't say it can't be done, he just says it doesn't look like it. The first kind of as far as I know of you, Pilgrim is improved by an American astronomer, and that's the

15:00 one you will see in most editions of Pilgrim's print novel. The very abstract gentleman of your life is Mario Chieri. I was lucky enough to meet Martin Soto was working on Pieri. She was very keen on Pieri. Pieri, she describes as a mathematician she would like to take home. And they were telling us that she was ingenious. I'm not sure how many of them, but that also is a category of them if you go around the room, at which a mathematician would you most like to take home. This one is spoken for. Okay, you can't compare it to the power of the matter of Pieri. But with this sort of machine statement, He wished to give us axiomatic treatment for real objective geometry in this area, letting him do a different treatment for complex objective geometry, which is in a purely deductive and abstract manner, independent of any physical interpretation of the premise, and so on in this situation. In other words, unlike Patch, who is trying to say, look guys, we're talking about a real thing, I just want to be sure when we're abstracting and when we're reducing. Pierre says, we're just producing. I'm not interested in going back and doing this kind of real-world stuff. I'm really interested in writing about some rules and drawing a consequence as well. And this is four or five years before Herbert starts taking the credit for saying this kind of thing. so there's certainly something going on in all this Italian context and that is the context in which Enrico comes from you've described for instance how clear he goes about doing all of this kind of thing the lady is in France really quite popular it's taken out in France And from Kuturak, it passes to Whitehead, when he actually empires the 20th geometry in 1906. And from Kuturak, it passes to J. W. Young, who reaches an American audience. So in that day, these people are quite popular. And of course, at the Congress of Philosophy in 1900, you have the famous meeting between Russell and Mayer, which is what so excites Bertrand Russell, and in a certain sense turns him into Bertrand Russell that we all know about.

17:30 He makes his very first acquaintance with his mathematical modules. So these titles, on such a section, are known in their day, well respected in their day, for an axiomatic approach to geometry, familiar geometry. It's another story, to say, to what extent they've been or been animated research imperative, and you'd like to argue in the archive paper I mentioned that it was a self-limiting program in that sense. But in the sense of axiomizing geometry, the Italians are earlier than the will see as and in this context we find projected and he becomes the kind of spokesman for this sort of activity. So here are some possible words from the book and I have now a puzzled in my mind, I'm really not sure how to talk about problems of science. The problem, I think, is this. There are numerous occasions in the history of mathematics, or in history of philosophy, or whatever, where, in a certain sense, you don't have a problem. That is to say, the words of undoubtedly difficulty and importance and volubly describe is by describing accurately. There's no need to discuss the question of its merits. We might choose to give this or that detail but we're talking about works which in some sense still commend themselves to us, commend themselves to the audience, because of the quality of the audience. I'm really not sure the problems of science should be resurrected. My impression of it as a book, by the way, is that it lives in a certain limber, that reasonably good university libraries will have it on their shelves, but I doubt when it was last taken out and read. Now, this is an interesting category of things, because there's something a little strange going on with Poitier. You can now, in English, get the complete set of Poitier's essays, edited by the next Stephen Jay Gould. Now, this tells us, I think, something about the

20:00 reputation that Stephen Jay Gould had, deservedly it might be a charming person, but nonetheless he's not an expert in the history of philosophy. He doesn't have a particular view on conventionalism. I wouldn't want to particularly rely, I suppose, on his judgment about Poitier's views on Newton's second law, which was something kind of very road-marked in 1904. It's simply the case that Zimdehul established himself as your favorite popularizer of science. And therefore, in marketing terms, the right person to represent the collective mind gray. And the collective mind gray, in English, is frequently cited by all sorts of people doing philosophy and science courses at a very elementary level. It's established as a classic. And a lot of people push off and read it. And I might be a little grumpy here and say, cite it in the original English. and I can make this joke because I'm such a monologue I have to go everywhere and speak my own native tongue because I can't speak yours and I do apologise for that but nonetheless it's something strange that Poincare is quoted in English as if you wrote it in English people don't feel even the need to go back and look in the original French it's positive I don't like the noise of it to be honest somehow it needs to become a classic I mean we can discuss how this happened someone was writing a full-length biographie of Parter Ray, I don't want to see a discussion about Parter Ray floated up to this level, wherever anything remains. Enrique is not in such an elevator, because you should add these in the middle, you may or may not ever have rated, okay, and I'm not sure if I want to stand up here and say that if you haven't read, you should rush out and do so now, okay, you certainly shouldn't rush, but maybe you shouldn't even go, it's not such a wonderful book, it's interesting is that it is a major book, in its head. It is a very interesting historical document about what is going on. And these are the kind of buzzwords in the book. He's very concerned about symbols and their meanings. So we have here a certain line in the familiar struggle between syntax and semantics. He's very concerned about a philosophical problem of science, which, as he sees it, is how is it that talk about science, concepts, make

22:30 any sense at all? Now, he puts all of these questions, is, oh, here's perhaps the most significant opportunity about, he's still going to realise, he wants to say that the statements you make science are true, and the objects are there, and they don't distinguish themselves from quantarelli in the moment on that great planet. But his concern is saying that these knowledge claims are just correct, that the statements are true, then what is knowledge? Knowledge of him is something that is acquired by a historical process, actually two. One is the historical process of the species that we belong to, and the other is each individual one of us. And in a way, that is most interesting, in a way, I think often that research people are when they write about these issues, because it's very concerning to say the way which knowledge is acquired doesn't resemble very clearly the logical order which you might write down in articles. So, for example, it's perfectly reasonable done there in New York City to say that geometry because it deals with some, but not all the things we talk about when we do it mechanics. But in the course of acquiring our geometrical model, we need to do geometry first and then do mechanics. There's a two-way process between these two, so, geometry and mechanics. And that's ongoing now, as Enrique was right to say, in his view, there's a marked distinction to be drawn between the logical presentation of the subject and the thing that actually happens, The historical and social process by which knowledge is acquired. So here's how he distinguishes himself from Wackley, and I'm not concerned in this talk to particularly the defenders, which I'm not going to try and answer the questions about Wackley's view, but I'm not going to answer the defendants. He defines Wackley really as a transcendental, because whenever he disagrees, He, anyway, sees himself as some kind of a realist, he's kind of wrong, and he's been a transcendental person. He regards it as transcendental to say that some process of approximation has to be taken to the limit. So instead of getting to know things better and better, which is going to be all evil

25:00 offered you as a way of knowledge, he says some people claim that you can go all the way to the end of that process of approximation and know the thing in itself. He regards this as transcendental, he regards the religion, and he doesn't have any sympathy for what is he sort of a contrary position to that, which is that if that's what monage is, then we can't have monage either, because we can't go all the way. You want to say we can have it, but we've not involved in the completed infinite process of any kind. I come out of the comparison between him and Poincaré, I'm not aware of what he said that Poincaré ever replied to this, I would love to be able to say it, but here is somebody at an authority of Poincaré, certainly as an mathematician, engaging in Poincaré's conventionalism about nature and space, and Poincaré thought, I'll start out again, my He did engage in arguments and traditions at this stage, and I'm not aware, I'm not going wrong yet, but I'm not aware that one forever requires Henriques on this matter of nature-space. The crucial thing that Henriques can extract from his argument about nature-space is that we have to say that it is homogenous, but the homogeneity of space is important. Do you recall a quite great argument? I think it's brought down to a two-dimensional plate, and you have these little creatures crawling around in the plate, an unknown, and they, in all the dimensions, was cool as they move outward, so they shrink. And you adjust the rate of cooling, the function of the radius, in such a way that we, looking at the plate, You can see there are short pictures of non-fiction, but they can't. And Frank Rowe says, well, that's an awesome solution. Suppose you want to measure something on real space, and you use light rays. Well, you have a choice. You can either say space is Euclidean, and light rays are strange things, or you can say light rays are strange, and space is rather strange. But you can't decide on any other conventional ground which choice to measure. point here, which you can say, that's physics, that's geometry, here's what I'm making the decision. And then people have said, no. Take my first example of our heat. What do

27:30 we know about it? We can generate. We can heat things up. It's our experience that some things expand more than others. It's our experience that some things expand in rates that even depend on the temperature, which they are. We can do all sorts of things with heat. So, So, in practical ways not, it's ten inches to suggest that we have this example of heat. If we really imagine some force in the universe that stormed in our straight lines, we reliably would eventually be a physical regressive, in a situation where we can duplicate that force in the laboratory, and we can say, aha, as it might be, he doesn't give this example, it might be a gravity, and we think, yes, we can bend the rays of light too. So now that we understand what about experiments telling us, or should we ever fail to produce a force which could do that straight line, that our experiments with straight lines in the space that it looks like we're going to do, we would have to say that that's a fact about geometry. So we can distinguish between physics and geometry. It's about things that we control, the model, and the stride. And if we can't control the models, we describe the force. We have to say it's geometrical, and then we have not conventional, but a logical. We use the distinguishing. Well, can I use one hypothesis or another? One is geometrical and one is just... So here we have a sort of summary of the view. Space has to be a real thing for a couple of reasons that we are discussing. What it is is space to be real and it's the totality of all spatial relations that we can discuss. So here we've got microbe as a mere nominalist, because microbe is good beyond spatial relations, to talk about space and reproductions. So that's the distinction that we make here. And then, instead of this absolute independence, which microbe says, actually, consistently, we have here an approximate philosophy, an approximate philosophy, in which we can make precise discussions of things.

30:00 So, I hope we all have the question of knowledge claims. I could put in a situation where Enrique was saying that we can tell geometry, physics, we can say that some statements are part of the real world. Well, so what are we doing when making these claims? Well, fundamentally, his take on this, and he worries from Julius Klein, I'll just speak, is that knowledge claims reduce claims a lot of the space. Reduce claims about memory. And he gives two examples, one of which is the Isosceles Triangle of the Eran, and the other of which is the 30s, a kind of angle sounds. I say claims in the sense that everybody who's period thought. space, you measure the angle of the sun to some triangle or quadrilateral. If you get a non-muginian answer, I'm not going to say it's a non-muginian nature of space. But I think Chris also said, what does it mean to say that? Yes, that's what he's trying to make sure it's true. And following five, he said, what it means is, if you tell me the lengths of size are equal, and they're equal to within plus or minus 0.100%, then I will tell you that the animals are equal to within plus or minus whatever it is, population and that sort of thing. And that's all Indians. And if you live in a world where 0.100% is hopelessly inaccurate, but not 100%, whatever it is, and I will do the same sort of population, I will tell you, that's also the angle, it means that, the sense that it's true is that measurements will show the angle type within a certain area, and that's what the statement ultimately means. So again, you see this kind of approximative statement, which is meaning to many, and you begin to see, of course, at this point, that Rukwuzi is not at all in the spirit of his fellow talents. This is not some statement about, oh, well, we make some axioms, and then we deduce, and there's also this trial, you know, the sphere of such and such, and this formalized body of knowledge, and then perhaps we have a set of philosophical procedures which we follow, which turn the axiomatic

32:30 statement of the forms of the statement of the world. These are not being there for. What he is saying is that dealing with meaningful objects. So we then get this sort of striking argument as to the Well, it has a sense that Gauss regards certain things, which is true, quite a parallel axiom. Here we have to quote Gauss, from Sartorius, regarding geometry as a logical structure, only in case, in the American sense of theory, only when the theory parallel is conceded as axiom. And I read this and I thought, you know, how I missed something. You know, I was expecting a name also. You know, Gauss has these things about the nature of geometry and he was giving a kick. Cresson and the Great Spurgeon, we have exactly a complete disagreement here. And really because he was never constituted of geometry, it has a character of homological acid, and all the definitions, fundamental entities, are homologically defective. So, stop. But anyway, let's have a good view that they are homologically defective because they make assumptions that I am real. And the geometry is basically concealed by a physically We could have different systems of postulants forming various hypotheses, and they would expect different physical hypotheses. In fact, on the basis of these different physical hypotheses, we could even say that some geometries were right and some geometries were wrong. There's two ways that can happen. It might be more accurate. You discover that something is not the case. For him, it would be the case that you measure the atoms and find it more and more accurate. You know, they just weren't turning out to be another negative reason. This was for him, you know. But there is another example for Fletcher, one called Philippe Klein's Space Force,

35:00 geographies on spaces which are not topologically about the world three. These are going to be wrong too, but they do express certain kinds of physics. If you like, the physics are called a science fiction book, in which, not only if you do In terms of our meaning, this wouldn't be expressed in the pantheon as we expect. There's nothing wrong with that. But geometry is not a thing, a formalized abstract thing. It's talk about meanings. And we're all listening to the suspicious philosophy called meaning-finicists. You only get a finite number of things said to you in your life, and you only get a finite number of things said to you about straight line. You get a finite set of pretty good ones to fill up, from which you get a pretty good sense of all straight lining, but it can always be improved by another statement added to this list of finite things that you've heard about straight lining. So your concept was a little bit blurry, a little bit capable of extra refinement. we could always learn a little more. So, definitions of him come right at the end. So, you know, sent him a book from today, and he would have no definition, no problem at all, the definition of a left quasi-witch walk. And it would differ from the class of witch walks and being left and only quasi. And he would have a bunch of definitions, which eventually would collapse into terms that are meaningful. But at the more advanced levels of mathematics, these are general definitions for me. This new word you've never met before precisely means this other stuff. So you might find what an abelian group is by finding what a group is, by finding what it is to be communicative, all that kind of stuff. But eventually some term like multiplication or combination or somewhere in that will happen. you're going to fail and you're going to end up talking about meaning instead. There's quite a lot to what Ben-Ridquist did, and I don't have time to describe it here. Here's just a list of topics, some of the things he put in the book of 1912, from a set of essays he wrote for the journal, the founder of Scienca, the start of the

37:30 20th century. So it's an interesting collection of things that shows how broadly now he's setting out a storm. Enrico is presenting himself very much as quite a range of topics as unauthorical. It reaches additional authority by being an expert on the great geometry, by being a very lucid writer, by being lucid, genuinely smart. But you don't train enough for a geometry and thereby acquire authority as well. somehow there's an extra process involved that allows you to become an Italian expert or an expert at least on Hayden which is different from the process allows you to make an article and see if it is published in Marshall. I'd like to say The talk has been helped us do so much more about health office than I know, but I was just hoping that I haven't got my ass case already before I came in and put things on it that David will have to correct me about. Helmholtz is the authority, if you like, scientist, that, I read, was most identified with, and he sees himself as responding to Helmholtz's insight into the office of a cosmological field in the service of science. And he goes directly from Helmholtz's example to his question of of epistemology and biology and scientific questions, or to the epistemological problems, as how the folks have suggested it would. Now, I don't want to get too involved in what this entailed for every person, because, again, it's a problem that I raised at the start. in the end I don't think I want to argue that the problem of science is a major I don't think that's an important for us I think the important for us is capturing something that was

40:00 it would be too similar to it going to be played well for that particular audience but it somehow captures a feeling which is congenial to a number So it would probably be reasonably easy to take some of the philosophical decisions and say that they are, in some sense, unsatisfactory. But to do that, I think you missed the point. The point is, here is someone writing a book called Problems of Science, which firmly addresses issues in philosophy, and to which it's been led by his own diversity as a person, and And I write his own ability to have a different branch of science, that's what it is, a very hard and recognized branch of mathematics. In fact, the science in this book is pretty woeful. Even in 1906, I think, there's no mention of all of it. There's very little science. One of the ways in which it differs from Pointero. So, in some way, one quite very writes about scientific issues, he gives you that impression that he's got his hands dirty, at least theoretically dirty, okay, he's somehow engaged with the issues in the way that scientists do this, I question about what research is, how to formulate it here, what we know, what we don't know, you know, who becomes, as you know, for the French reading community and authority on electromagnetic theory. Turn to Pantor A for advice about, you know, this experiment between the early, which turns out to be wrong. You turn to Pantor A for a good expression of the ideas of Helmholtz or Maxwell or in good, clear, mathematical terms. And this, I think, shows up in the popular essays. Somehow you have a feeling that Pantor A a great person, though, who speaks, and you don't get that impression. The science, it seems to me, in this book, all it's called problem is now, is second-hand science. I wouldn't want to say that because members of his family seem to be reticulous and aggressive, but I do have this feeling that the science is a little, and there's a little distance here in Henry Crose. What you get, however, is a good deal of philosophy at various times.

42:30 He fancies himself as a certain kind of positivist. There are already other kinds of positivists up and running by about 1908, and he isn't one of them. And he is often concerned to protect himself as somewhat opposed capitalism, and he does so in a variety of topics. So again, lengthy discussions about physics, you get what it is or isn't to be a capitalism or physics, in biology, historical positivism, sociological positivism, a lot of this kind of... If you like intellectual talk, and I hope you can just fire it, is a book that is aiming to present these issues, and it used to be very popularized in the form, presented to a wide audience but without demeaning or trivializing the subject. I'm busy reviewing a popular mass book at the present, and I'm painting it, and apparently we have to spend pages and pages and pages to tell people how to differentiate, I think I think it's 3T squared plus 5T, right, in this book. This is a book about acclaim and about the problems, OK? If anyone wants to win a million dollars, it's just knock off in that case, okay? This is a popular book on it, and apparently, we are getting a relationship on it. I just hope the book sells well to some 15-year-old genius, OK? Because otherwise, I can't see what a popular item of enterprise is in this book, because somehow, everything's been leached out in order to be comprehensive. You don't get that with any words. In English words, a real attempt to become pensible about creating heavy weight problems, not discussing about what reality is, not discussing about what facts are. None of which you find in front of the way. As far as I remember in front of the way, some of these terms are not used or they're taken for granted. Apparently they have a common sense meaning, whereas in English words you get quite a discussion of what any of those words mean. And obviously, it's quite long. It's a few pages or something. I haven't got a plan going for it. So I think just one. One, it is to be objective or subjective. It's the progress that we make in science simply arbitrary.

45:00 How many things come about? How many distinguish between the objective and subjective? And this leads into the discussion of logic. And logic, in his view, is probably not what any of his Italian logicians would have recognized in this dispute convention between him and the others. the logic for any of the words includes discussions proof but also because it's a positive it also discusses the method of discovery that part of it and for him an inescapable part, he is not somebody who says, yeah, you've got this really weird business, you've got this ideas that some guys are going to learn, apparently don't, you know, but it doesn't make any sense. I mean, psychology is mush, okay, so let's just do the logical proof, I mean, let's clear that, it's hard to understand, okay. His view is that no, you've got to make that distinction. It is neither true to the growth of the subject, nor to the next growth of the subject, and And, in fact, what's going on, in some sense, is not logic in this formalised area, but some question of the nature of the human mind. My hypothesis is that any recruitment today is as likely to be found in an orthodoxy, psych department, as in a mouth department. If you wander around between the two and it's not only in psychology or whatever, somehow when we come to know things is part of how we can organize them in a logical fashion. And this does include enormous discussion of a topic which you also find intimately and quite hooray. It's probably having its heydays at this time. I'll give you one brief mention of it Knowledge is supposed to come in different kinds according to the sense organ which requires it So you have a sense of touch and you have motion and you have sight and each of these would give you

47:30 for instance a different definition of straight line Straight line are things we can see if you allow possible trouble or unaffected on a straight line, if you take a solid body and rotate it, the simplest thing is to rotate by an axis. The thing you see, the particle travels on, the axis around the body rotates, are all, we want to say, the same thing. A straight line which you have detected in one way or another, and our ability to do geometry involves our putting together these different types of straight line and reference as one thing. So there's a very lengthy discussion of different types of knowledge derived from different types of sense experience and how these are fitted together into the jobs that we present to have, which can of course always be revised and improved. It's not surprising that geometry took a long time to come along because it has two concepts of straight-lining, one more objective, two more lines look like, and one more tactile to do with geometry, finite-straight-lining axes. So you've got somehow two physical geometries having to be brought together and reconciled before you can sort out what is going on with the nature of infinitely non-straight lines. I promise you that not every bit of a problem of science makes clear and unspeakable sense, and you may well think this is one of those bits. But again, this is a view about how knowledge required by the individual and knowledge required by the species, and the nature of geometry is an ongoing process of acquisition of knowledge. Now, I promise you a reference agreement would be so with that. And we could spell it. It goes back to people, but you can take all the lines into one space and consider it as a point of a new space. So all the lines in a three-dimensional space will form a new four-dimensional space or whatever. And this is one of the things that have become very important to these types. The idea that you can take the several pieces and for the space, the space of the orbit system. And with each this, this is the point, the

50:00 space of all circles and the plane is an example of what it is. And these new spaces, the plane might not be, are one thing might want to have it. Another is geometrical and ultimately meaningful, because you could always decount the term back to their original meaning. So But don't say this is a point in some Grasmanian, it becomes plain force-based or something like that. So you're always meeting there underneath, okay? So, as I warn you, the most familiar definitions are not definitions at all, they're just gestural. They're pointy, vague in the direction of something that you acquire a monoclonal in its ongoing way, whereas the more obscure definitions are, if they have genuine definitions, So a point in this Grasmarion is, by definition, So we have a super populace that implicitly defines the objects it refers to, although, of course, the process of dealing with them has become much more abstract. So in that sense, I think it gets very close to what Nagel said, and Nagel never referred to any of his words, that we have ongoing research in geometry pushing is called a much more abstract form of reason and methods for dealing with abstract spaces which really are defined and the unnecessary, abstract, and logical. Whereas, he specifically disagrees with some of the Italians, my uncle by name, and others He thinks it is morally wrong to do mathematics of a purely abstract kind, that you really shouldn't be trying to do geometry in which points, lines, planes are implicitly defined by some axiom, without any reference to the real world, there's no intention ever to like the real world that kind of for him arid formalism he thinks you actually shouldn't be doing this it's a moral question that this kind of mathematics is somehow firmly of limitaries, or germany, or limitaries. Interesting thing is, in the Italian context,

52:30 Enrico's house has another context in which he's interested in. DTC's one for me to pick up is the American Reception. It's impossible to pick up the British Reception. And the British aren't interested in discussing these issues in journals that survive. And I think they wouldn't be discussing them in journals. My suspicion is he bumped into the other person capable of reading this in the quality of Trinity and had a little chat. So that's it. Nothing survives. So the Americans, I think mine is an exception, but the Americans have a number of journals where they are trying to relate to their nascent community of politicians, as it might be, So there's a lot of bustle, so you may want to communicate with something in Chicago, and therefore you want to write it down and publish it, whereas you can go to the London National Society by train and chat with the three other people whose opinion value. The British intellectual scene is much harder, if not even possible, for me to get a measure of, but the American one has a number of resources for things, and you probably know It's like being by Gover, or so you can't can. There's a lot of open-the-core publications reused by Gover. Open-the-core, in the Journal of the Manalists, edited by Paul Paris, are a huge source of articles, not only by big names, but by people responding to these names. So you can get a good sense of the debate there. So the products of science today very well to an American audience. And the most comprehensive study of a concept which modern science is built. There's minds, philosophical journals, so they like the philosophy. They don't notice that there's very modern science in this thing. But we have here, you know, a grasp of modern science, traditional philosophy, psychology, which is rarely found in one life. That's true. There's a lot of people doing bits of this, and it's probably a long chance about one of the best people to be doing it all at once. And so it was translated into the journal, in two parts, in French, where quite correctly, the translation of the second theoretical part introduces the feeling sort of intermediate between Martin-Pain-Pere and the point about the community. This is also Martin, when he published some of his essays in French translation, he says there's no common examples. It's very, very close to what Martin-Pere wrote about. So the translator

55:00 is certainly right, but Martin Comparais close at this point, and I think it's usually that there is something there too, and then the ancient shepherd probably calls for an English translation, and that's what he will produce. Professor Kent Royce is the translator of it, possibly the Josiah was ready to see, I think it's the person who pushes for it. Let me see if I keep them on board with the crab, too. The first time was a senior professor at Harvard. He was a book shepherd. He met a request at the Philosopher's Congress in 1908. That's when they had the idea of trying to get his book into English. The idea was that she would translate it that has the advantage of, I've read, very deep in modern technological problems, and as a book for modern German, the NERFA representative of a great Italian school project, they have And that's because we're interesting, because whilst Enriquez is extremely using what he called logic, it's not true what one would look like as a notable representative of a great Italian school of logic. Right? He's rather opposed to what we could do. But somehow Enriquez has got himself up there, and he's obviously not about to give or No peonings in here, no symbols, kind of that kind of stuff which the Italians are doing. You might think that they really need to behold themselves in this period. It's something that, you know, kind of sort of interests very easily in how the Italians project themselves successfully. Whereas, by the latest Enriquez, certainly did project themselves very successfully. I mean, apparently, at least this influential American audience has, you know, unrepresentable. Well, this is the sort of footnote. He has some good voices and some of his troubles, and he becomes very interested in quite a

57:30 and axiomatic style of mathematics as the 20th century gets underway and is appreciated by these characters. This could be a little hard to find. It's a very important video, extremely important video. Rather heavily disparaged in fiction and scientific biography, which quite possibly at this stage is showing its age in this particular film. Many of you in the audience will know more about it He is one of the other people who goes over the whole territory. Philosophy, language, mathematics, other works. And he's an extraordinary person. The works are in the use of very own merit. It's the nature of psychology in the theory, and in its complicated relations with philosophy, especially in the German environment. So Wundt has views, and it's, I think, a resource in this stuff about tactile knowledge, physical knowledge, etc. And then Wundt wants to argue that, oh, I can, we can have a philosophy which allows us to discover other generations. But, Kant is nonetheless somehow right, that we are somehow Euclidean beings. Okay, so this is a little bit less, right, I mean, he wants to sort of please everybody's show, he's read everything. But he said, I was saying, we're going to have a warm source for Enrico's, because the crops are on time. This talk is somehow improper at that time. And this is by way of getting a walk off. Conclusion. what's this really interesting about Enrugu race is also in things of great interest to me about the period the period up to 1914 a heyday popular exclusion major figures and they are not talking down doing work they may have enough sense not to But they are not going to leave out the typical stuff either. They can talk about it somehow. They will. And I think this is rare. And it's very, I think, indicative of how good these people were at it.

1:00:00 That Poincare is widely circulated in English, French and German to this day. And N-smart is not that far behind. And I'm looking at our popular essays, and they write their own terms for their own merits, because you can engage people with philosophical debates arising from what they said. It so happens upon Greene S. Marker, of course, philosophy figures in hard science, hard on having, but somehow they also wrote essays. We still consider it had to be effective. to be word-free in front of students, word-free in our stuff, word-white in front of us. And it may not be quite in their class. I've tried this to actually what we did, and what I want to suggest is that, as you know, one might wander around pursuing a view of the change in mathematics at this period, and why it becomes much more abstract in regards to other social and political movements are going on at the same time. Why does painting become more abstract? Why do you suddenly get Schoenberg? Why do we suddenly have Millwood? There's something I'm stalking in all of this that is more broad than just the writings of Macrae or of Mark or of any of this. There's something in the theory which makes these people say they have interesting things to say, which makes an audience go, yes, it is interesting, which has to do with the fact that the experts themselves know there are significant problems, pipe rate, or so as a wide area of things, but all these people know, if you think of geometry, it's interesting because it's said, even the experts, I'm not sure about the nature of geometry or the nature of physical space, that has a claim on the audience's attention, apparently, and it's dealt with when you know perfectly well that the mathematics is alive and interesting so the general public is right to pick up on the institute so you have, I suggest, a very significant moment in the institute of science at which these figures reach out in very energetic and very far away to the general audience with a question about in the end, probably on the course and perhaps on the cap questions about the nature of knowledge and that might be part of the story

1:02:30 I just wanted to ask you to explain, I didn't understand this conflict between Gauss and I didn't understand whether, firstly, what was meant by saying that Geometry was a peckin until you added the last peckin, and what Gauss meant, and secondly, whether you just understood what Gauss meant when he said that, when he disagreed. What he's disagreeing is a statement, not by Geist, but by Sartorius. And Sartorius' claim is that Gauss believed that geometry would make clear axiomatic sense once you've decided the parallel postulate. So let's say anyone in three fields at that point, you can somehow endorse the parallel postulate, or you have two versions of the Parallel Postulate, and then two others you click in the Donovan. What's not true is that Gauss says the paracross is true or false. Either way, once you've made an axiom about the . You will have a geometry that says which Gauss guard is axiomatic. And you take the same thing, you take the class of the parallel of the particular classification you prefer, and then you've got a pretty good axiomatic system. And that's what Enrico says you haven't got. And you haven't got it because the basic definitions, all the others that you took on for, are not, to the standards of a magician, definitions. Right? Typically a definition, and I made that facetious remark on a letter phrase I wrote for, right? I mean, you know, commuted with an ethereum ring, you might put it in itself, or 12 clauses, or the abstract words the students have never met before. In fact, those things are definitions. But when you're finding it down a straight line, and you don't have definitions, you are now gesturing towards things which we know about in the way that human beings know about things. Well, that isn't to be, it can't be encapsulated in axioms. But this is an odd view, because typically, you know, going back to Aristotle,

1:05:00 You say, actually, hey guys, I've got to start somewhere. You can't define this and that. You get into the reverse. There come a point, sort of, I'm stopping you. I'd rather understand you, actually. I might think if you don't understand. Now, here, on this point, you're a crazy person. So this is kind of proving an undefined term. And then we call some common meaning. It's an easy straight sign, angles, stuff like that. in the open page of YouTube, he says, they're not definitions. So what Gauss said is actually fundamentally wrong. So when you say, as we sometimes do, that these guys have got an axiomal approach to geometry, Enrico says, no. Partly aware, of course, that they're coming to very weird things to get axiomal approach. But partly because he thinks they're wrong. Well, there's a number of Gary, by 19 axioms, Enrico and Sarthas, a set of axioms. They codify what you've already meant. He must say, don't forget that. So that's the screen about what an axiom is. Axiom is a high level thing to do, which definition is a high level thing to do. Meaning is, on the other hand, fundamental. That was very sweeping and wonderful, especially, I think, because this genre of writing hasn't really gotten all that much attention, and you've shown, of course, that there's tremendous interest in it. Two of the figures that pop up in your story there are Klein and Wilbert in different ways. And I was thinking, I've heard more about Enriquez now, or about this type of writing in general. You could call it semi-popular writing or something. that people like Klein and Huber don't seem to be really engaged in that sort of thing with respect to geometry and so forth and that although obviously there's sympathy between Klein and Enriquez, it seems as though there's a profound disagreement about certain things unless, particularly when you come to the end and you say Enriquez has a moral stance about someone doing or whatever it is, I mean, he seems to have a very, very odd take that Klein certainly

1:07:30 never shared. He cited the lectures by their employees, and in those lectures, of course, that's, I think, the first place where he throws out this fundamental duality that's very, very important to him and to Google, but I think they share the view that there's There's pure mathematics, and there's approximations, and of course this whole discussion is really all about approximations, because it's all about which geometries and so forth apply in nature and how all of that works. And there, of course, you could find all kinds of consensus. And of course, you would have said that the roots of geometry are empirical, but that's not what I'm doing here. I'm setting up, you know, a purely scientific study. So to me, Klein and Gilbert are really very much more about inner geometric work and so forth. They're very much aware of pure mathematics. And for Klein, he's a duality. That, of course, is far away from all the questions that seem to come up in the new I think that kind has been appropriate here for selectively, not entirely fairly, and certainly not in a sense of the spirit of what kind has been describing. It's used by Mucrose to get in this discussion of groups of course by a number of people this time, and was mentioned briefly yesterday, this idea that what you have is solid bodies. We could put them on a very heavy diet and they become just two-dimensional, and we could really put them on a very heavy diet and they start becoming one-dimensional, but somehow they're just very, very, very thin kind of ultra-anorexic versions of good psychotics, okay, and so you get what you, and then we would just like to say, look, we never meet we really never meet a line in real life, right it's all a little bit of a thick it's a little bit of a wire, really, it's a tube sometimes, but, you know, we could

1:10:00 and then it can't go over the limit, because that's transcendental here, so it has to have this approximation with that argument, but I don't know if you just give a good example of that, simply this, you also disagree with Christ, because Henrietta wants to say that it's very informed that I require knowledge of life. We actually don't need thick things, we tend to learn from thick, but they're thick, and that's in the nature of knowledge, nothing goes up to this philosophical ramble. It seems interesting, this is a bunch of great junk, after all, who had all this important How in the world does he get this moral stance that somehow those things he's talking about? Does he really believe that it's grounded in something? Oh, yeah, he does, absolutely. He was one of these terrifying people who, apparently, if you walk back from the poetry and ask him some questions, So you're dealing with the contract algebra, so you've got four real dimensions and people need to let you see in four real dimensions, right? God, that's what that means. What about 18? So, you know, well, yeah, okay, I agree. But somehow he thinks that it's ultimately, you know, you've got a lot of science to go where it goes, but it's hard. You could trace back core and inner core and meaning. And then you build up weird and weird things for good reasons that are mainly accessible to practice. but he really does want to say, and he's good-sized about this, but if it's just formal, you shouldn't be doing it, and why not, it's just going to know what it is, right? Yeah, it's a little bit similar about this reality concept or the pragmatic aspects of Enrique's thought. I've noticed in one of your footnotes that there was given biases American connection reference to William James. Yes. And so it is, one can imagine that there are also connections to Charles Sanders' purse and to a reader nowadays of the philosophical discussion of that time, it is so strange, I feel it, so strange that it's a completely lack of imagination in Poincaré's work of the importance, for example, of discrimination.

1:12:30 structures. At a time when already a lot of work was going on by electrical engineers making relatively advanced geometry of circuits and other things. There was a lot of work going on and some appeal for a reader in this year, 2002, which Perth has, is that he clearly was aware that the relation of geometry to reality is more complex than this terrible, tiring and old-fashioned discussion which everybody thought was finished at some time in the middle of the 19th century, and I'm curious whether Enriquez was aware of this that geometry had many other aspects, there was a discrete geometry emerging of great importance in electrical theory. It's such a long time since I've read the principle of the culture of the essay that I couldn't say that there is no passing mention here and there. In what I read in Perry's talk, I only saw a curious consensus between him and the Italians, otherwise it's agreed to, that these finite things are not interesting. And I think it's difficult to underestimate how important for this whole milieu, which is not a subtle question. It's not done and dusted. As you know, Frege somehow can never come to accept it. Frege is also someone who starts from meanings. So that, the question of the dimensionality of space These things are somehow a big deal for them. So, let me not say it again, because I've never seen it. I've never seen a record when you were first. I'm not the other way around. Anybody on the planet where everything first worked like that? You know?

1:15:00 But the finiteness of these things, Everything you say is somehow true, but it happened on parallel to planet ours, it happened here, in the way, and that's what you're pointing to, because we think it didn't happen. They, you know, the neighbouring that it was took planet geometry more seriously sooner than we did. But these guys, so far as I remember, it doesn't get... It's a good idea for you to take it with you. Not only for your machine, but knowledge, but that's what I said. Touch. Yeah, you mentioned this form is, I think, this strong value judgment of our, the non-value judgment of the form of the Portuguese mathematics, and the misunderstanding of the English taking you as a representative. How did the more formalist members of the Italian room, the Piannus and Piannus, react on this? I don't see them reacting on this, except for he has a great movement by love. It's not such a friend, but it's mentioned as a friend. Maybe they are in Ireland. My friend says blah, blah, blah. Then you find them starting to say that this guy is a misunderstanding. But it does seem to be the case that the Italians seem to think of themselves as formalising And when you've done that, you walk away. You were just trying to tidy it up. They don't even necessarily think you would do anything new or better, which is sort of sacred or rather sacred. If you do tidy it up, sacred or rather sacred, and this is the tension in the Italian community. People who have been attracted to the message think, I want an immunopathic on the research. I'm going to do, we did out-break services. And those guys go in that technical direction and succeed. And then you get the ones more attractive, Piano's view, which is that let's eliminate misleading problems

1:17:30 in natural language. And then you've got buildings, which apparently they walk away from. Nothing new gets done. So they can push Piano. Piano, you see, He thinks you want to axiomatise real claim and free narrative. Because we know what claim and free narrative spaces are. Anything else is just string of coordinates. It's not to be axiomatised. He's got to push beyond him and show you to axiomatise. End of sense. and then they never push the button and start the mocha and drive the machine it just sits there in a nice, highly touch up and they do something else or nothing, God of course gets more into the learning language and various educational reforms pierre faith or what and then what they were doing is It's no longer done by them, whether it's done well by postulationists in America or whether you think Russell was the intellectual descendant, those are all other questions, but they are going to be inspired any longer by what they tell you to, because they tell you to don't do this. So there isn't even much of an effect, there isn't even much of an argument between the forwardists and people like New Yorkers. I haven't found it and read it. I know when I speak to people like New Yorkers, they say, oh, Jeremy, go and read 300 pages of a review stills. Could I understand that you could say that an active and a great job of this What do you create a disagreement in the Reapers, but accept him as a spokesman for them, even if you create a discussion? Custod Robo is certainly no excuse, because they're good friends. Although they couldn't accept it. I mean, for all I know, you know, if I know, you've got page 10 of the problems of science and put it down in his gun. You may be thinking, oh, what a Reapers? I have no equation. I wouldn't want to say if you ask me if Enrico spoke something to all of those people that all of those people were endorsed from the science plus or minus a little I couldn't say Enrico has made a new career for himself

1:20:00 you mentioned the term hypothetically deducted in the context of I think the conning of this term is pretty interesting as an indicator so to speak of what is going on in mathematics at this time. I found it first in Rosenberger, Geschichte der Physik in 1882. I would be interested in the origin of this term. Now, did Enriquez pick it up from Gehry, that would be one question, and the other question, was it Enriquezman made it popular, so you mentioned negative, it became quite popular later in the philosophy of science, in logic and empiricism. Do you think that Enriquezman played a role in the spreading of this term? Question 1, did Enrico's hear from Pierre answer yes? Question 3, did Enrico's play a role answer yes? Question 2, much harder to say. Genuinely don't know the number of other people in this area. You can almost say in some sense, in any case Pat has a view, maybe the phrase isn't there, but once you have a phrase you can say, ah, homeless upon the surface. So I can't answer now, especially if one can physically try to be fascinating, I'll capture the problem and ask you for the details. I don't know. I've always thought there's a more general view that is disseminated through various thoughts. and I want to work with the Kazeera, and I don't know if I can tell you to realise that I was wrong, but I've never, I haven't done enough work to know how this style encapsulates immediately in this particular phrase, acquire the, with disseminating as it was, with the success that it had. It's a fascinating question. I can't say I can't give an answer to it.

1:22:30 The style of hypothetical deductive reasoning the methods very explicitly talked about in the 17th century without the world. I would call Newton's method, for example, axiomatic deductive, axiomatic in the traditional sense. But in the tradition, yeah. Remarks about it, I think, instead of very... But you know, I'm a notorious begriffsgeschichte. I believe that the uprise of new notions indicates something happening. I think this notion was coined in the context of mathematics, not so much in the application of mathematics to physics as in Huygens' problem. I think Robert is on to something, and I still want to say this is a kind of sense, that there was at least a new salient, disability, sensuality, in this period. And you can go back into what you call it. I mean, in some sense, any process of thinking of saying, well, what else is going on here? You think about it, then you do something, and you make some claim about what these things mean we do. These things, you know, we've probably been doing since we became grammaticals, right? But, it's presented here as a new phrase, it does have a kind of new sparkle, clarity or something, clarity, as a new sort of colour about it, that it would be interesting to document in this period, right, whilst of course you're obviously right that it's hard to imagine anybody doing science and not thinking of themselves making hypotheses and making clear they are about what a deductive system is now, about what the relation between doing deductive work is and hypothesizing or abstracting. There's a new level of detail. That level of detail, I think, changes the way these terms work and operate. We want to get down

1:25:00 Well, my dear question was, I take it, if I understood you correctly, Enricus was accepting the idea that in abstract and inapplicable geometry it was reasonable to think of the postulate such as defining the primitive theorem. But in real geometry, it wasn't. And this was part of the distinction between the two. And therefore, this method of implicit definition is kind of the immoral kind of geometry. Now, that makes me wonder that in Planck-Ré, for example, you see not only the use of this method of implicit definition, but a strong philosophical argument in favor of it there's a strong philosophical attack on the very idea that we truly do understand the primitive terms independently of the absence a challenge to people like Russell for example tell me what you mean by these primitive terms without involving the absence in some way and in the context of mechanics tell me what a force is without being able to accept the law It's part of Picardy's philosophical dialectic that Henry Crosse simply missed one. Well, I don't know. I think Anne Lawrence is a part of it. Oh, I'm going to go to that. I'll be a great, I'm going to go to that. Picardy knows what numbers are, and I think it's absurd to be derived from the fact that we know everything is wrong. Henry Crosse said the same thing about points in the life. The implicit definitions of more complicated things are not going to be unpacked eventually as abstract things, but as things that you know about from what you're going to do in the beginning. It certainly gives you a little personal. I'll share with you guys if you can feed this over time. And that was kind of a... Can you call it... ...in the beginning of a... ...concert law. ...a promise, another and more pass.

1:27:30 Of course. Yes, sir. So, we look forward to the second part of the... ...in the same line. Right, a bit of continuation almost. I started in 2021 and this talk will follow up the development of why it's an understanding of the relationship between geometry and physics and concepts of matter. from the first step into what he calls an analysis of this out-of-links and analysis of the problem in space to his work later in the British on case structure of the computer mechanics on the computer application. And although it's a rough short phase, it's nine years, about nine or ten years It happens so much that it is of course a very important advantage. To good number one, first of all, it will be, I think it's some sort of longitudinal section again, through this highly important decade plus some year from the invention of relativity theory to relativistic equality theory. And the conception of matter, or the perception of matter, the concept of matter, in particular in my, by the mind, the change, every human, and that time it's a very useful concentration reflection of what was going on in the whole community of mathematics, physics, and physics. And I am going to concentrate on why and make only some side references to the common community. And, of course, this part will have to introduce a little bit more of mathematics than the last one, where I could just indicate a little bit, and so maybe it will be less sexy. It's at least an interesting story. Okay, so I could just give another review on the whole talk, which I did today, I could resume already, the story, what I will tell you here, is that, I mean, that one was in that story, told in the last talk,

1:30:00 a strong information of John McFee Fields and Mecca, what I call the reductionist one of 1918. And starting from that one, he had a phase of doubt, which I talked about yesterday. And during that phase of doubt, I only indicated already that in the period of 1920-21, he started to do the analysis of the space problem. And that's an, apparently, it's an intermediate step of understanding the basic structure of the uniformity of geometry with a view towards physics. So that's an intermediate step. So here you will find a short story from the strong identification claim of geometry created matter via A or via the mathematical analysis of the space problem. two worlds, what I would describe as an open conceptual frame for a comprehensive symbolic representation of quantum metastructures, at the time at the end of the transit via the Dirac field, a unifying, again, electromagnetic system and the geometry, plus the new wave metastructures of the Dirac field. And in that time, things were very complicated to quantified stuff that started in the late 29th century as well, I don't have a discussion before, but we do have this development doing that again. Let's start, I will have a short overlap just with this transparency, I remind you of this In this transparency, that's the indication of what was purely infinitesimal geometry of 1918, which lay at the basis of the presumed of the equation itself from 1918 and 1918. Then we have this idea that we have a purely infinitesimal metric by the conformal metric plus a benchmark. Okay, now, just a reminder about that. And so that was the modification of Riemannian geometry, which was founded at first on very

1:32:30 speculative but strong philosophical thoughts and linked the better understanding of the And purely institutional structure of the matrix were tailored in his mind to the semantical law to explain matter in 1890. Now I started a thought where this strong semantical thing was really bad in, let's say, about 1920. So there was reason. Just to make this short, he continued geometry was good, although he started to doubt the direct motivation in relationship to the meta-structure of the being, they were violent. So he had good reason to think another time on the, let's say, philosophical motivations of his newly invented geometry from a point view which was more general, perhaps deeper, at least more general, than the more direct belief in meta structures. And in the same letter which I quoted yesterday, at the end of 1913, December 1928, explained to Felix Klein that he had now completely given up the mere approach to meta, he reported from some changes in the fourth edition of outside criteria, and one of the points was that he had started to investigate the problem of space, as Romm would name it, with new theoretical methods. And he had started and he claimed the result which was not very proven So, it was superficially, it was at the same time, and I think by the motivational structure which one can start, I just don't understand, one sees the different levels of the university vision, it's quite clear, that it was not just a change.

1:35:00 He, of course, picked up a long tradition of the problem of space. It's done in particular in the 19th century, and he was quite clear that this old conception of pre-mobility of written bodies, it would make any sense any longer if one considers geometry from the point of view of general relativity, so one had to pose the problem of space in a new way, taking into account purely in the point of view of motion. I will now jump into his presentation of his analysis and for that he developed the idea that one should not, of course, no longer understand finite motion, so one had to understand what he would call motion in the infinite, infinite. And how could one understand, let's say, common rules in a very general way from the point of view of this pure infinite solution Okay, he assumed that in order to do somehow a very general congruence geometry from the point of view of infinite dimensional differential geometry, one had somehow to signal out in each infinitesimal neighborhood of each point, related to the same potential plate, which represents the infinite isomal rotations. He called them a rotation. I write them as gp in each point p of the manifold. And in our language they would operate on the tangent space of the manifold in Weissberg was, they were going to be . All of these groups, that was a mistake,

1:37:30 but they wanted to make conceptually sense. All these groups, they are little different points, but they should be isomorphic. And all of them should be isomorphic to a connect to Glitzak group of the special individual group. I mean, it has posthumated that the determinant should be changed by following consideration. That's not very remarkable. And then there were a larger group, the general India group, which he called the similarity group. That was just the normalizer in the general India group. And that's why he used the normalizer. And he said, OK, that was very intuitive, actually. from point to point, one can't be sure that perhaps what he called the orientation of the group on his GPs linked to the different points is changed, which means if one compares this infinitesimal congruence operations, they may, from a certain point of view, they may be changed by similarities if one changes the point. And if one wants to, one can really understand it as classical congruence classical similarities in the complex of its only geometry by the day of 1918. Completely makes sense. So it's really a generalization, a general structure, and a lighting of the thing which he had developed. So, in order to introduce it, one first has to introduce these infinite rotation routes, which are linked to a similar route, And then one really knows in each point what the congruence should be. In order to have a complete concept of infinitesimal congruence, one has to have some symbolism to characterise displacement. One goes from one quarter there, not finite, to the infinitesimal level. That means there should be coordinate displacement given and that from the symbolical technique of the time that she had been at the big verb already from the point of the affine connection

1:40:00 without, depending on the metric in the manifold, that was quite clear to him that the most was to take just any general linear connection. OK, that's now for those who are just a little different analogy. I mean, for those who don't know that much about it, in each point, this is a matrix, if you want to. It's a three-index system of real numbers indexed by three systems which run out of the intervention of the metaphor. And then, so transformation. That's the linear connection. And then he said, if it should make conceptually sense such a congruent displacement, can it be just dethroned by just one linear displacement? If you know what congruent displacement are in this point and in the infinitesimal other point and we have one quadrant displacement, it's quite clear that here we can just add another infinitesimal rotation, as you could say, and that would give also something which could be understood as a quadrant displacement. So that means if one has a linear connection and that's an infinitesimal rotation, which would be a differential for values in the least algebra of the equation rule. Then one gets another linear correction, and in effect, they seem to have the same displacement structure, so it's likely more because it is the last of this which is the then he said what we have done to know has been just some analytical part, analysis of process what infinitesimal motion could be.

1:42:30 And then he continued, if one wants to have real, let's say, useful conception, one had to add some synthetic principle. And this synthetic principle, in fact, were two principles. the principle of freedom and the principle of uniqueness of affine connection. And this principle of freedom, I'd like to describe, I mean... The question is, how many different covariance displacements can we choose once the group is specified in the general linear group? If you have a very small group, there are not very many rotations, if you have very large ones, there are other restrictions. So, it's a question how many displacements we do have if we have looked for a certain group. And to characterize good groups, one should have a synthetic principle. The principle demands, that's the principle of freedom, that choosing one point, the group is such, that each, all the variables in this linear connection at the point, which are cross-legged, or which is, are, as we as a natural, space, can be realized as in concrete spaces. As he said, the group shouldn't be too small, so that demand of freedom, and by the way, he gave two parallels to this demand of freedom, the first one was that that's the only thing one could have, what was the mobility of finite bodies in the classical context, that was one parallel, the other one was the social one, which I can't, a little bit back to a little bit later. And then we add another postulate, which was . These displacements were given by equivalence classes of linear connections, let's say. Because you can add a rotation in each part, an infinitesimal rotation. Then the question is, may it happen that in such an equivalence

1:45:00 class lies an affine connection. For the differential geometry, it's clear you have a connection which is portion three. For those who don't know the differential geometry, it's a particularly good type of connection to the geometry. So, and what it was, the rule should be such that in each of these equivalence classes, first there is an affine connection and then there is only one. So there is exactly one affine connection compatible with this infinite thermocombus structure. That's the posture which I have written as posture as well, and I used to see it, because it's much stronger. Okay. So, so far. gave a reason for this type of structure, which I hope could give a quote. In his Barcelona lectures of 1923, he explained that these postulates could be understood in certain relationship to what happens in society, namely for a good constitution. A good constitution allows for the members of the state, of the society, the largest possible of freedom. So, that would be the possible freedom. But that's not sufficient for a good constitution. A good constitution has to act somehow a coherence condition that under all possible activities chosen by the members of the society or the state, there is somehow a coherent common rule which is realized by these different choices of the different members of the society or the state.

1:47:30 And otherwise it wouldn't be a good constitution. And then this coherence, that's not my work, that's the meaning, the coherence of the condition which a good constitution should contain is in the geometrical context is given by the posterior uniqueness of effect. The unique effect connection makes the link between the different choices and the different points. What's the essence of space? What's the essence of space in the social picture? Yeah, the essence of space. That's it. That's it. It's the interrelationship between individual. The individual are the choices of the parallel displacement in each point. and the essence of the space and the huge space education is only achieved if first there is the largest possible freedom of choice in the different localizations plus a certain And then the second question is that the formal structure is such that just one divine connection or on the social level, if everybody, every member of society chooses a certain way to act in the frame of the constitution, the constitution should be such as that some common good is real. It doesn't break apart. And so you feel that we're in Reckon 23 and that we're under the pressure of highly conflictive class struggles and pushes and so on in the German society. And you'll find it in other of these articles on the Foundation of Mathematics, you'll find a direct, let's say, influence, at least on the rhetoric, of the Christ-Phochet revolution quite clearly. And here he strides again with, let's say, movements, motions inside society, motions

1:50:00 on the level of physical space, question of liberation, breaking apart from the classical structure of space, but liberate would be greatest even possible under certain conditions, a certain coherent condition, shouldn't we say? OK, that's that, I'm going to do that. So that's only the principles. Then, of course, mathematics start. And I only use a very, very short glance of mathematics. And from the conditions given in this hypnotic principle, one can derive easily certain consequences for the rules which are compatible with these principles. It starts with a dimensional condition and so on. I don't know the details, never mind. That's quite easy to derive the conditions for the lemma. And then there comes a second part, which is really difficult. Now we are on the level of algebra. We have certain algebraic conditions for the rule, which are derived from these principles. Principle-treater, principle-treater. Then we are inside algebra. And there are certain conditions of a linear and broad group, which can go ahead with the postulants. And then he, at first, in step-by-step investigations, starting with dimension 2, going to dimension 3, and so on and so on, started to prove that these algebraic conditions are satisfied exactly The group is a generalized orthogonal group with signature P, Q. P is damage of positive side, Q damage is negative side. Non-negative, P plus Q is damage of positive side. And that was another work. And here it comes in after a question of repetition of Lea and Elgigrath, and he finds Gata's work and stuff like that. That's very important, because now we are back to generalize our formula groups and plugging in generalize our formula groups and these infinitesimal rotations,

1:52:30 one can, with a little bit of work, in fact, one of the history used in the non-determinant case, invariant theory, that was explained by the experts, and everyone can see that the conformant structure can be reconstructed, which gives it proof. So from this data, in the end, one can reconstruct the violent infinitesimal Jogodri, die Pythagoreische Metrik, as we would like to express at this time, on an info. So, beautiful. You have done a conceptual analytical work, added some synthetic principles, modified by some relationship to the classical problem of space, modified by some imperial thought inspired by by a question of social coherence, and derive a very general structure of interactive cohorts which characterizes the essence of space in your spectrum, that's a question, and arrive on it that is satisfied only by groups which allow you to reconstruct a binary metric. Now, what is important is that in this world, you have not only this group, but for the geometrical structure, the rotation group, was sitting inside the similarity group. So, I just have to indicate, if you translate it to modern terms, you wouldn't have to use And then it's quite important that you would have to use a background structure, a bundle, a principle bundle, which doesn't need to be all feminine, but the similarity. And then you start the associate bundle and you reduce the framework, not from starting from a principle bundle, you go over to a bundle which is more or more principle. The structure in the group is the similarities, and the fibers are the rotations. And then you have a group extension. You have a larger structure group, then you operate in the integratism of the neighbors. The fibers are the smaller group than the structure group.

1:55:00 So it's hidden in it by its conceptual analysis, there's already some two-letter-proof construction. And just to indicate, Cartoon had the problem and it became a phase of differential analysis, but they forgot about this two-letter structure. Cartoon reduced to just one group, at least the Cartoon It's difficult to read the file, but it's more than it is. Okay. So let's talk about the analysis of the space problem. Now I will make some very short remarks. It was finished in 1923, exactly the same year where the last edition of alongside interior appeared. While the honeycomb into the representation theory of the B groups, this relationship to the important representation theory, via Rauch, maybe more just one, and as Tom Hawkins explained, just a minor motivation for him to take up the work of the B groups, the one. And he came back to these questions only after the new quantum mechanics. Now, I will just very shortly outline what happened with White's Gage idea from a side of physicists. Here I just remind the original Gage idea of 1918. And then in the time between 2007, I came at that moment yesterday in the discussion, there were several attempts by physicists to transfer it, pick up the general idea but modify it to make it better than it. And the first one was in 1922 by Andrew Schweringer and we had that in the discussion yesterday. And in 26, 27, it was picked up by Fock and London, already on the basis of the Neukölln theory, but let's say in the Calusa, what would we later call Calusa Klein, because it was not a classical Calusa one, but moving away from Shethro. So the interesting point is that at least Foghman, I think, in London, did not know.

1:57:30 So they didn't know that it was a natural possibility. I think about the . Weil did not have read the poem to Schrodinger 1922. It will be granted to Fitz London's work of 1927, when he gave his lecture called on . And we find it in the first edition of group . We find an interesting remark, which is no longer to define the second edition of that one, not to be found in the English translation, because that didn't create the moral group. There, the background is just to say a very simple idea, again, if I forget about the fact I can do it. Reduced to some essential core is the consideration one has a wave function, which is a complex function. One could just try to modify the phase by a phase. And essentially it should contain the same information. Now, when there is an electromagnetic field with a potential fire, as in the other theory, and now you consider the Hamilton operator of a central electromagnetic field, which means that this momentum operates the alpha, in the classical trivial extent, added to the potential function for, let's say, an electron, in the more description, rotating around the center. This real momentum scheme is, if one just formally modifies this momentum, which is given by a similar relation to the perfect core function, formally in such a way, making up this term of a potential, then this operator, Hamilton an operator turns out to be what was considered to be a Schrodinger approach to the motion of an electron of a charged particle in the separatist.

2:00:00 And now, the interesting thing, if one changes these phase factors, that means one has somehow a related transformation of this contentious, which is formally identical with what I had to do if I let the gate consider as a gate transfer formation in divided geometry. That was seen by, that was, this Roman law was seen by London, and that was, let's say, stripping off the five dimensional stuff, that was essentially the law. And what I gave here, he was a close to how Biden reported about the largest idea in the first edition of the material real fund mechanic, and then he said, that it's very convincing, not completely conceptually, but very convincing, and then he said, yeah, I don't know if you want to determine the combination. Here is a brief. If one changed the training of function by the phase factor and the potential by this modification which was formally identical with the gage transformation of the OEC. This is formally identical with what the OEC had produced by purely speculative reasons to unify reputation and electricity. And now he said, I'm not strong now that electricity is not capable of reputation, but to meditate. But of course it wasn't just a surprisingly technical effect. The question was whether there couldn't be a more better understandable concept of a reason for it, or practically, or however you put it, in practical learning, that is falling in the space, something like that, it's completely unclear how this could be related to relativity. That's not non-relatical sense.

2:02:30 And his genius idea was essentially relativistic. So it really was a surprise that he didn't control. During his letter, at the end of his letter, in January 28, Diderot published his new approach to the electron, a relativistic one. And the year before, at least he discussed with Niels Bohr and told him that he was doing a new approach for a statistic. What happened? He talked about the context in our mind, so there is something to make it. Was it? It's not. It's not. It's not. It's not this year. Okay, in 28, he published his new approach to the, let's say, generalizing education in the relativistic form, and just, I don't know, I could give a short, very, very short I mean, Paulier described it was essential that Dirac changed to a four-dimensional values of the wave function, so four components, complex value function, now on the Klosky space that we use. And this C4 value function has to satisfy a certain equation, a famous Dirac equation, and important in it are certain matrices, the Dirac matrices, which are linked, just a statement, very shortly, to the matrix, to the Minkowski entry of classical value, of special value. And the interesting thing is for Derek it was quite clear that that was a good relativistic characterization and it was studied by and by people that there was some representation

2:05:00 But I have to at least claim the point that if I get such a four-dimensional direct quantity as my own life state, the direct cause, the direct cause. If one has a direct cause, such a function, then one can relate this by the direct matrices to variables real values, four real values, which can then be a vector, just by applying the corresponding solution form. And then, one gets a vector and there are four matrices, and under a transformation of these sectors, they correspond on the level of the corresponding Dirac quantities, just a special linear transformation of the four-dimensional complex steps. So what we find is a relationship between the SL1-3, the Lorentz group, and the SL4-C at this moment. I realized that this was not a, that was a refusal representation. I don't want to know too much that it would be that by good theoretical reasons, belief that real physics must be a very useful representation and therefore change to two slightly different two-damages, Dirac quantities, Dirac quantities. It's not that necessary to be very detailed on that point. What is the problem that the direct quantity, and especially when you find two or four or whatever representation, is related to the direct quantity? Putting it from a modern term, once followed by a physical context, the universal covering of the loaded group. Now, the question was immediately for why and for other actors at the time, could one relate

2:07:30 a theoretical equation somewhat of general relativity and then there is a great difficulty in the world. General relativity was considered necessary to be a generic covariate, which means one had to represent the general number of group on the tendons based somehow on these quantities. And it was quite the immediate, if I started the group, from the group, theoretical point, that this relationship could not be extended to the general group. It was immediately by John von Neumann in early 2008 in an article, and was stated just like that by Weyel in the book, the version of this lecture course. So if I really wanted to relate the Dirac equations to general relativity, there was a deep concept of problems. The Dirac quantities, other people said a little bit later, talked about spinners. other people like Fox, let's talk about semiconductors. They just were different types of objects, if not they were only invariant or covariant under a smaller drone. And there are, I mean, now we have a game of the late 20s of different approaches to unify a few theories, and Einstein came into a game that was paralleled, This is parallelism, which would allow to integrate the theory of creation just by a ground form of treatment online. People, not like Schneider, other people who attempted it, but while an independent, relatively independent of a flubbing in Fock in Leningrad, had more or less the same idea. I here concentrate only on why, but I should say, in the same year in 29, more or less the passing core, the protective core, was developed by And the idea used the method of all phenomenal trains on the metaphor, which, just to explain very shortly, if on that differential geometry, you will, of course, have to somehow correct the tangent space for the infinitesimal displacement, or whether you would like to express it, usually you will do it by taking derivation And then we have a standard basis frame in the tangent space, as we would say, or in the infinite zone of Lagoon, as Bayesian scale.

2:10:00 And now, one can do, as one has it, on the Lorentz manifold, one has a matrix. And, of course, I can introduce a lot of the normal basis, for example, by Schmitt for normalization, whatever you want from the center basis, you get an orthonormal basis in each poly, which is varying degrees. V0 to V3, I've tried to repeat it. This is political. You have a coordinate frame, which is not alphabetical, and you can introduce any differentially moving alphabet frame. Then they just can be moved to base, moved with respect to the basis, the standard mark, and it's alphabetical. And now a change, one has considered not only a change of coordinate, which if one chooses this way of representation of the vector, it would not affect the representation of the vector, but one has in addition, because the choice of the orthogonal frame is somewhat arbitrary, one could choose any other orthogonal frame. So, in addition to coordinate transformation, one had to choose changes of the frame, which gives, by changing the orthogonal basis against another one, put together orthogonal transformation of the coordinates of the vectors. So that's beautiful. In the case of the Lunar in school, what is the S.O. 1-3, that's our problem, that's our problem, that's our problem in that context. So just by using, let's say, this formal trait, which was well known, part of Weizenberg, The in-language, if I had died in human geography, one could develop a possibility to describe tenuous spaces and different, so well, the stuff in a metrically cold, rely only on a reduced crew operative in the inter-so-way framework, so one of the fortuna, and then, as you've seen the orthogonal Valorius group, that was immediately represented of the spinner spaces, that was the DNA, slightly more methodised. And then I just outlined, it gets a little bit complicated in the details, but I think I can indicate the idea.

2:12:30 If we have the manifold and we have in this such a tangent space, there are two different, I try to indicate, Now, the black one and the blue one, we get, of course, the order of vectors that are connected to these orthogonal frames. And the change of the frame is given by an orthogonal matrix on the coordinates of the vectors. And, as we know, these are represented in a rather natural way in the Dirac, whatever Dirac 5 or the original Dirac one, by the representation in the corresponding spinner space, either two-dimensional or four-dimensional. It's really different than the theoretical. And that's fixed. Which one is preferable? I don't discuss it. And so what one can do to a change of orthogonal frames, one gets a change, a corresponding change in the spinner space by the represented matrix. So that's a concept that's an easy idea. Technically it can be described that it's rather complicated with a lot of indices and stuff like that. And the differential geometry gets interesting again, I just indicated. Here on the Leggitita F1 connection, and one can have very, that's very natural to try to somehow lift the Leggitita connection of the Leggitita structure. And that was done, of course, by Vyre, enough for each other. And then an interesting thing occurred. If one introduces the spinner connection or the spinner covariate differential, as Vyre said at the time, then it wasn't completely determined by the underlying particular connection. It was only determined up to a phase factor, which was varying with the point of the waveform. And so, just to give a short indication of what I did, the covariant differential, by the C or the Lyra quantity, the Spinner, it was basically determined

2:15:00 but only up to an additive term which had its form, which was essentially a differential one-form multiplied by the difference. Mainly measured for the dimension of the respective spinners space, either two-dimensions or whatever. And so, the spinners have intrinsically larger symmetry than just xl2c. And so, one has again found a small group, of what was called a congruence group in the context of the analysis of the space problem and what had been the similarities in the analysis of the space problem were now the slightly enlarged invariance group of the spinners. So in the spinner context that would be basically there. It has a group extension happening here and that length in the end has to make the story now very short. When we look at the connections which describe this covariative differential, then there is a pure state of art, determined by the energy. But to characterize the whole connection, one is separated within one part. And if you want to, this is exactly given by this differential one. And so, if one chooses a different post-formular frame, one gets here the transformation is SR2C, and if one multiplies it with a phase factor, one just needs to re-gauge the differential one-form. And by exactly the formula, is essentially the leaders of the one-dimensional line of the one are identical. So it's no wonder that the image transformation in the world context, which rose here very naturally for Wyatt's book, was formally identical with the one we had came across in the length of the creation. So there was a symmetry extension, or a gauge,

2:17:30 in the prophets and in Wyatt's language, physically, it was very important because it led to very interesting conservation laws and the gauge in the new one led to charge conservation. The symmetry in the SR2 part was not to discuss it well, I don't want to discuss it here, but Biden discussed it very strongly in the core of the state government. And from a structural point of view, one can compare what I did here to what I had done for the mathematical analysis of the space problem. But now, no longer analytic consideration of concepts, no longer synthetic or principle motivated by a certain relationship to classical analysis of space, plus highly interesting and very speculative, sensitive attempts to transfer the motion inside the social space to the physical space, not only these relationships, but just analysis of what the physicist had done from a point of view of mathematical, consensual charges. That's now why I become very sober, compared to what he has done in the beginning of the And now, I perhaps can come to my last part, he, of course, now had reason to rethink the whole development of unified figures in the 20s with our contributions, not only with his own, but Gartner Klein, Einstein, Edmund, all these things. And to be now rather short, but I think it's related to our talk yesterday, in the realm of Bonn Act of 1930, which has been published in German, he discusses this question, reconsidering unified field theories. From the point of view now, we have understood the first twinkling light of quantum physics. It was already twinkling a bit stronger than in the beginning, at 23,

2:20:00 our direct equation that was very interesting for a while. It was twinkling the magnitude larger than before. And then he discussed his understanding of the relationship between matter-structors, joint triangle numbers. And he had reasons to argue with his, let's say, colleague Fock Hilenogra, who had done this as well. And just to be very short on that one, Fock was really, he was driven away by this beautiful Geometric idea and in common work with Ivan Reiko, which he started in his first publication stuff, Ivan Reiko and Fok sure claimed that they had found the quantum geometry. And later on when Fok had become a bit sober in the middle of 29, he no longer laid graves in the more quantitative, but he continued to be quite clear that this was a geometrization of a deer. Which is perfectly easy, but why did not appear? I jump into Ryan. In this Rolf Braun lecture, Okay, Herr Fogg, bezeichnet über die Erleitung der neuen Eichnung, noch alles was da, Fogg described as a geometrialization of a geometrialization of a theory of that. Ich kann Ihnen da nicht zustimmen, einfach nur. And that comes from the following sentence. Mir scheint, dass wir auch eine Geometrisierung herunterrichtet haben, I, from my point of view, have no longer attempted geometrization after we have unified electricity or linked electricity with matter in spite of gravitation. I fear, I fear, that the tendency of the geometry, of which the gravitation follows, through the most important arguments to be touched on the right. I feel that all this business of trying to unify more force field than just gravitation,

2:22:30 very intuitive and justified, was changed the wrong path. That's in now 30. And in other bits of pieces, he says quite clearly that the understanding of matter had drastically been changed by quantum mechanics in the middle of the 20s with the right of the new quantum mechanics, and now the question of matter goes completely different. And he goes on and asks, aha, okay. He discusses the question of OK, we have not to achieve the geometrization of the Dirac theory, in contrast to what Drotkin gives. But, and if he does, he knows that it's easily covered. Perhaps it's not possible when I am feeling that relief from his life. But then he again continues, maybe it is possible. But then it's very difficult. And that's it. The second line was called, which I'm going to do. if one really wants to try the geometry, so must one a naturally ungroup of the geometry and find a natural geometry If I wanted to go on with the geometry, I had to find a natural geometry, geometrization of the Dirac. For example, if you have to go out if you have to use it, if it is possible, I have no idea what to say in a geologian language, and I have here the Christian English translation. If one wanted to draw the geometrization tendency, one had to set out in search of geometrization of the matter field. If one sees here, the electromagnetic field is added as a treatment to the body. It's a Zugang from Selma and the other eye.

2:25:00 I have no idea what kind of jobs that I have. Okay. I have one more minute. That's surprising. Because the first straight up, you should say, I mean, why didn't you accept the possibility? To some degree, the direct equation had been jumped. and just by his own approach, even by his own approach from all. Why did he reject that? So there are several reasons of interpretation. First is, when we are reading it, we have, as a cultural background, the whole development of the 50s and 60s, connection and differential geometry of connections, While I prefer to use covariate derivations, so it was, okay, so it's nuances, but we are educated, we are educated, I mean all of us who have some certain knowledge, we have knowledge of geometry, and you have to consider this type of geometry, which was not yet so we had a tunnel, it was built, that was the first part, but I thought it was obviously Let's say, a new step on the Geographic tradition, that I did. So, one is considered a second factor to understand his strong opposition to Fox, claiming that this one only is the Geographic tradition of the Vienna Field, to consider the community relationship. Why did he one of the greatest propagandists of this early days, the Geographic tradition in the 1950s? And the whole community of physicists at the time was going on in that game, who read these tiny nuances in the later edition of Romf-Zeit-Matthews . And so, it was a point for him to make clear that struck us with the old German translation. Now, there's another game. It was more than particular of Weinschrein's approach at the already from the very beginning, from the conceptual beginning, this new distant paradigm and he thought, even in his responsibility to apply the law, he wrote letters from the New York Times after they proclaimed Einstein is not formed, the world founder, the new

2:27:30 distant paradigm, the theory of why the world is unbeaten. You should know approaches and it's not a real way of doing it and there are other So it was necessary for, let's say, for the broad discourse, so to speak, to make a lyric of the old German translation, let's see. And then I come to the third point, that the question, even if you have a German translation, let's say it's Dürer function itself. In 1923, Jordan and Heisenberg had started to try to contact, to so-called second contact again with him. Things were forgot. It started to be very, very difficult. I don't know why he knew about it. He didn't refer to it. And so one was entering a situation which seemed more or less hopeless already at the time and we find the occasion in five that he followed it and he was quite clear that the thing was completely unclear what would come out of it and so I would relate this one that even the geometrization could do something much more strong and we had to understand somehow these quantum structures that the deep one that one was stochastic or a second quantize for all that we talked about and that's a thing which has been achieved by I want to take you to the next two years, maybe. I don't know. I mean, here it leads to .