Line geometry to relativity theory (contd.)

0:00 The idea is to obtain a complete classification of all these cubics, and later this Rosenberg and Sainz models, so it gives you an idea at least of how physics is thought in a sense. In this context of rigid body motion and projected geometry, he criticizes in a very interesting way also his teacher Fluker, who, using line geometry in connection with rigid body motion, says that Fluker somehow implies that there is a causal relation between the forces acting on a rigid body and the infinitesimal motions, but has long studied the connection between these two. It's not a causal relation, it's a reciprocal one, from a geometric standpoint, so maybe you can understand better what this means and what this is. In order to actually study rigid body mechanics, you have to know things like the mass, the center of gravity, the inertia, and things like this. These things don't enter into the line jar. Geometrically, you can study purely on the basis of directions in which they move.

2:30 That's essentially something like a gradient. That's not hard to understand. Well, a first-degree line complex just means that at each point I have a bunch of lines that all lie in a plane. And like this one here, it's not in a complex. It's idea of polar lines. In any physical idea, well, the null lines, the ones that actually belong to the system, are ones along which is no turning point. If it does have a term, there will be a second line of the plane. Distinguish the directions of a field taking place in space. Of course, it's not a vector field or something like that. It's like a gradient. Then you have models for these various motions that take place. A model where you have an axis along which you get one of these null lines.

5:00 The tilt of the null plane varies with the distance. Is it finally a helix? Is that clear? I have a system of forces that I can use the X, Y, Z and L, M, N to describe the sum of these transformational and rotational forces, rotational and transformational components, and seek such co-effects, and so you have some obvious way of linking these things together, and the question is, do they transform under all of these?

7:30 It's very elegant, it's simple, very simple, very, very complicated things otherwise, and back in 1871, but he's not satisfied, you know, when he goes to Wikipedia that he's really got into a lot of it, and so he writes an essay and distributes it to Abraham and all these other people, because we need to really look at the clients that are after getting at these quantities that aren't inversions, they're not... And so, therefore, you have to distinguish between two types according to the theory of the transformation properly to keep up the same. There's something different going on that one must distinguish between. And that has to more or less have clarity about what are the invariants when you're talking about it. But you wanted to do that surface, of course. Where you can find a lot of this stuff, you're probably not recognizing. But they ought to know about geometry. And then you get a whole bunch of this exactly the same. Actually, what is the detail? He's read a famous book about elementary mathematics from a higher school. We want to know what happened. He publishes this book, and here he dedicates it to Norway.

10:00 We have to thank Engelman Shepard for writing his books. And then he writes an essay about Klein, which is really scandalized. And then, so the theory is, I am not a people of, he writes these down well, but he didn't want them to go down well, he was just tired of what was experiencing in the problems in Germany, and so he wanted to acknowledge that he, allies, French.

12:30 I think that Lee's main point was that Klein talks about having an invariant theory of continuous groups with no idea of how to produce one other than maybe a very classical theory. For Lee, it was developing a very, very general theory of groups with an attendant invariant theory, so he did quite a bit of work on the differential invariants. So, in fact, his points are certainly correct, but I suppose Klein tried to make various distinctions about what was going on in his work and Lee's, These are some of the things which were his work, and in Germany he talked about it if Klein had some. So this is my last thing along here, but after his death in 1899, then came the Klein, which has never really been written. And this was to be published in the collecting works of the editors in 1960s to keep it out.

15:00 ...Klein's rule and the direction in which Lee was taking group theoretic techniques in geometry. Well, the papers I flashed up in the candidate that must be dead, also that must be talked about this stuff a lot in their weekly ventures in the field, on some level or other. It is a bit odd if something maybe, say, in that lecture in 1910, he's really trying to call... So, in a way, it looks very old-fashioned, but Klein has to stay there. Now, when he writes in the second volume of his mathematics, then I think he's much more intuitive. There he's pointing out that one of the things that Minkowski did that was terribly important was to try to write down the math for the form so you didn't have to do anything because you knew by way of the set-it-up that you were going to Minkowski there. Of course, as I mentioned, in 1916 he was beginning to get a lot of stuff. In 1910 he had a very good idea of what was going on. The first one is about the energy problem in general relativity. So, to which extent Klein was able to, let's say, try to resolve this problem.

17:30 My second question is about the joint work between Mulder and Klein on the unified field theory. So, I guess it's trying to unify the electromagnetic field with the gravitational field. Is this work going on the line of Weyl or Karuzak line or something in between? No, no, no. I just had a bit of this meeting, I mean the second question, I had a bit of this meeting when I said, when I thought in the unified field theory, I mean essentially, Kierkegaard wanted a unified field theory, right? And so that means theory of matter and the idea was to find an appropriate branch in... And that was a specific subject that Hilbert had to learn her work on, I would say. She apparently had little knowledge of or interest in, I'm not sure, the typical parts of it, work on differential invariants. Then she did her famous work, which was trying to resolve these energies. She went about that in various steps, so two papers, one on the differential. This is a very interesting observation that someone would see very complicated layers of invariance that basically shows that the enrollments systematically understood within the context of, I can't really give you much more detail than that, it's a highly formal exercise in a way, but he's able to show where they diverge. And then the last thing, I think that's what's interesting about the paper on the manageable form of conservation, and he follows pretty much the Exactly along the lines of Einstein. And there, he writes in that paper, it's quite interesting, he writes in that paper, and then you can see why that paper really fits, about balls, grooves, different types of vectors that you have to distinguish in order to relationship it from the variational, I think, famously well-known way, you know, to decline in general relativity type of identities. To clarify this, did you want to just follow up on the same question or not?

20:00 We just want to say a bit more why, as you say, Klein has done that and templated one problem when he just started to speak physics, which is strange given, say, later kind of recognition that it's right by Klein and so forth. So it's clear that with the Romanian approach you wouldn't have the same, but why is that related to physics, that he just started doing physics here? I'm not sure what you mean when you say engaged. If it was Jorgen, you'd say, he's done it. He abandons the Erlanger program after 1872, he abandons it from about 1872. After 1890, he begins using it, but in a very difficult, much more intense type.

22:30 He does that mainly within the rigid body mechanics and not new physics. So do you mean he just has different fields and allegorally-formed connections between them, so it's knowledge? Yeah, I think, well, maybe what I should say is the following, because it's easy for us to forget this. There is no well-established standard vector analysis. I mean, you know, people like... There's this sort of famous debate about how to best present these things. There's a very complicated picture of what's going on in physics about how you represent these coordinate 3 magnitudes. And so his whole program is saying, well, you've got to look at transformations. And nobody has done it. I have a feeling that here's the question for all of you. Do you know of anyone else at that time who was looking at, carefully looking at, how these things transformed? I don't think it becomes a natural thing in Catan, and people did that for a while. There's formalisms that are designed to do this type of thing, but Klein at least had that goal of trying. I think that's his, that's his big objective. So you showed us this picture where there is transition between two Riemann genus surfaces to a torus. What kind exactly of mathematics he was using to deal with them? To study singularities, what kind of deformation he was using for topology? I think it's what we call the hand-weighting myth. Yeah, because at that time I guess that there was not that much interest in mathematics. I mean, because that particular stuff he does, there is no follow-up on it, I think, no serious follow-up on it, until Hermann Bayer.

25:00 Hermann Bayer then has topology and things like this become then sciences and mathematics. Klein thinks very, very topologically, but he has none of the modern tools. He's describing what ought to work, and this is very, very prominent in a lot of his mathematics. And of course, nowadays people can go back and say what really does work and what doesn't, in the case that they can't be pushed through. These deformations he carries out are just topological transformations, so they're not things like later algebraic geometries you can resolve, you know, you can resolve into the various, it's all a technology that comes later, and in fact, it's the case that only for qubits and deformations, and trying to conjecture that, in fact, that this technique would not work for chordics and above and so forth, because of the way the sub-times fold. How, what geometry was. You're asking about a general overview of geometry and the way we understand why geometry is. Right, going beyond the question of the equation of geometry of the roots and the power of the roots of the projection. What kind of subject is geometry? Yeah, I think it's in other places very detailed. In fact, he had a dualistic mathematics.

27:30 He saw mathematics as typically called pure mathematics. He writes about nuclear things like this, and he talks about refined intuition. Refined intuition is simply something that doesn't have any empirical basis whatsoever. So it's something you make up and play with and imagine and so forth, but it cannot be measured. But then there are all those parts of mathematics which he would love, of course, to The question of to what degree can we take anything from pure mathematics as a tool and use it in an applied setting where we're dealing with approximative relations. This kind of attitude is reflected in a lot of his mathematics and pure mathematics fantasy, many would say, and about mathematical objects that are very abstract and so forth. So he's very, very interested in this course and he does it. So close to your question. It does really. He tried to make it work in a few places. He really wrote a paper about the notion of mathematical objects in empiricals, and he tried to suggest that it came up, by the way, as a reaction to the wild functions of Weierstrass and so forth, which caused a lot of controversy and so forth, and he admitted or visualized or realized or...

30:00 Whatever functions that are continuous but not very differentiable, and this type of thing. So within the context of those discussions, he comes up with the idea that in fact we need a type of mathematics that deals with impurity. No longer some ideal of a refined set of points, but it's in fact a little in-script, and it has distinct things to it. How much mathematics can you do with that type of thing? He keeps referring to this, I think it was in his mind all along, that these are the two sides to mathematics and they are difficult to bring together, so I don't think you could say it successfully did, but he kept coming back with a third volume of his elementary mathematics from a higher standpoint, and in fact, it feels exactly the same. He has no standard work about it, but he deals with it over and over again, so that Lee is also someone who inspired him early on, and he writes about this as well, as someone who's full of imagination and intuition, so that's the greatest example of that type of mathematician. But what intuition is, and so forth, this is the difficulty. But he was an educated 19th century German, he must have read Kant, I mean, he must have. Well, of course, but that doesn't necessarily take you very far, I mean... No, but it takes one a little further than just... It's really, it's really... Not distinguished from being empirical and shallow. And also, I don't think that Klein, when he talks about uncharacteristic, I don't think it really is any kind.

32:30 It's kind of philosophical, really. He created things. When he does write about these things, he sometimes gets away from the visual aspect. These things are really about mathematicians and they can be completely calculational habits or whatever they are the types of thought processes you engage in and much of which goes on in the unconscious. It's important, but there are no standard things I could say exist on what Klein thinks. Dedeckin, well, he admired a lot of things he worked in, but in mathematics, Dedeckin is a modernist and he did many, many things that... But Klein appreciated different styles, so he even had a high, very high regard for his work, but I would say this is a famous paper of his, you know, Babel, fundamental... But if one of his main objects was, one of his main aims was a kind of complete classification of the transformation properties of quantities appearing in various parts of physics, as you seemed to indicate earlier on, one would have thought he would have been... Naturally attracted, perhaps even against his own habits and instincts as a mathematician, to structuralist perspectives. So this seems to be, innately, a very kind of structuralist program. Well, yeah, I guess I would make this the specific kind of classic of invariant theory. Yeah, yeah, yeah. So that becomes, of course, other things later on, and seems a rather specific example of structuralism. If that comes to seem very kind of domain specific. It's very interesting if you look at correspondence with Hilbert, when Hilbert was working on invariant theory.

35:00 It puts Hilbert onto the idea that his work in invariant theory has a huge, broad impact. And in a letter where he just rattles off, you know, you really ought to think about the development of Kronecker's work. It's amazing because he created it as a very visionary mathematician. He didn't have anywhere near the technical units. So he certainly influenced Hilbert to think about the invariant theory in a very modern way. It seems to me that you were just at the point when you spoke of the last stressful solution that you mentioned. So what aspect was not to be appreciated? Yeah, sure. No, I'd like to just say briefly something about that. I mean, that's certainly a complicated question in a way, but there's something fundamental, I think, that goes on that's important to appreciate for new mathematics. It's a bit of an anomaly at first if you think about someone like Weierstrass who did an analysis and had a very, very strict methodological idea of the right way to do an analysis. And yet his colleague working with him, he's done excellent terms, this guy called Steiner who does synthetic geometry. After Steiner died, Weierstrass keeps thinking about it and I think it's shared by all of the people. So synthetic geometry is perfectly okay. What's wrong with what Kletch thinks their methods in various theories apply to geometry is different from, this is bad mathematics, it's tasteless, and it's not methodologically well-rounded, and I think Kletch himself was very much marginalized from him, and Klonica detested him very much.

37:30 I think that these strong tastes, let's call them, have partly to do with methodological purism. And that's what those people represented and what Klein constantly reeled about in England. He tried to promote a very much more integrated and indicative of his style, and it was true of Hilbert and others too. So they appreciated the fact that the Berliners were very much caught up in mathematics that Klein didn't share. He retired in 18- he died about 10 years after that. Yeah, in a certain sense, sure, although he's a bit more eclectic, but he shares the... Yeah.