2 sets remarks — Map between spaces & noncommutative geometry (contd.)

0:00 How did it all go? Excellent. Presenting...

5:00 I'm asking, did you... I shan't be too upset, but did you actually get all the guidance we can put on points? We had a discussion, not so much with Pierre, because Pierre was...we spoke about... To your point of view. ...anything at all that you need from me in the way of, you know, covering expenses for your petrol. Do you want to get outside here or in front of the castle? Whatever you prefer.

7:30 The castle is probably a little bit more... Well, with the rubbish and everything, I don't mind. I don't be sure of the castle. Thank you very much for your time, and I look forward to seeing you again soon. Thank you for watching this video, I hope you enjoyed it, and I will see you in the next one. I also grasp the significance of that language. That's why I felt so bad about having to interrupt simply because of the mechanics of getting Laird-Corey here, but I'm glad. I suspect that you will be in contact with him a good deal further. Probably, yes. Now, it's really very interesting because the more facts you can get out of that talk to me, the more part of it I want to know. The linearization is still close here, and so there's more things to come out of the market, by the way. What do you think of the Kohn contract, by the way? Well, I think that Kohn is coming closer and closer to this.

10:00 Now, to get rid of that, I tried to explain to Kohn many years ago, but it's hard to grasp the information that he wanted to have. It's hard to explain. Yes, the first step is the... Then there's the linearization, which coarsens the equivalence, and that's what they're doing in the studies. That's a good comment as well. Yeah. Exactly. Yeah. It's just that it's weird. It felt like it was merely implemented, basically, by choosing a basis, and some of these are invertible. It totally connected with my theory of quotient completeness and all that in the metric space, which is how you say it. It was connected also with what I call the . But unfortunately, let me know where it goes from here, because that sounds very exciting. So this evening, if you're giving a couple of rooms in May, let me know what's been going on, and we could touch on some of that. Okay, it certainly sounds to me very productive. Okay, okay.

12:30 We've come to relax. Thank you again for your time. Thank you very much. I hope she's better soon. Okay, I'll see you in the Ecole de Mar. Yes, certainly. When you talk to her. Oh, of course, we need to talk about the episode we're meeting too, but that could be later. Oh, I know. I will speak up on July 30th. Ah, you told me. Yeah, I just wanted to see how much damage it has done to the shower. Has it been completed? Yeah, this shower is up. Well, I think I didn't really tell you. No, there's a heart. Don't worry about that. Thank you for your attention.

15:00 It was a Bristol sort of specialty. Harvey Rose, who's now right from the middle of the United States, wrote a textbook on this subject or something like that. Yes, could we go? Well, since you didn't have a chance to send me your email, because I've only got one T, unfortunately, so it's been locked out. If you want to go online and do your email and stuff, I'll show you a password. Okay, yes. This is the connection point to the world. I'm afraid so, there's only one. We're going to have to solve this with a calculator. No, I'm going to try to connect my computer. Oh, sure, you can connect. Yeah, well, this is what I was wondering. I've got mine, but I haven't dared to connect it. Once that is done, it should be all right. I thought of that, too. With this clarification of what's going on with the Kahn program, they're taking the coarseness of the quibbles and trying to connect them. They're trying to connect them with the configuration. They're asking you to use complex vector spaces, which in those categories are not making real numbers. In which, in the Kahn case, they're dealing with the same thing as the Kahn equation.

17:30 Mathematics is an elliptic mass from one to the other. I mean, distance is decreasing from one to the other. Nisometry is a special case, actually, for learning the distance. But, you see, there's a much closer notion to that. Well, I mean, there's a closer equivalence between Justin's linearity and a bigger category of math. Same object. See, that's why he kept emphasizing that. He didn't quite get it before. Same object, same math. And the more math in the case of Witten space, it's a very natural... You want to get from the point of A to the point of B. And you say, well, how much does that cost? It costs a certain amount. And let's do this by a certain method. Let's say we're going to take the train via Laval as opposed to the train via Hells. There are different ways to get to B. Given any one, there's a minimum cost. So if you test it, so if you're having an assumption that each point A gives you a definite point B, completely, just like, sort of like random lines, they're different, and you're saying each point in A is for every point in B what's the cost of doing it, rather than telling what the choice is, instead of telling what the cost of doing it. So there's a constraint on this, obviously, if you put both A and B on that equation, if you go from one point in A to another point in A. And then there needs to be a triangle of qualities. The sum of those two parts, the sum of the two types of products in the U of M, if they develop equal to the parts of the other state in the base point, that's the only way on the other side, if you go from the point of A to the point of B, and then from the other point of B, that has to cost, that cannot cost less than going straight.

20:00 That's a bimodule, right? Yeah. That's a bimodule. How do you control a bimodule? You integrate over the middle space. In this case, integration means taking the engegmum of the sum. So you go from here to here to here, add that up, and you say, no, you're trying to choose the intermediate station to make the two-step journey the cheapest. The cheapest, yeah. And, of course, in general, you take an answer that may not actually exist at the station, but at least you can purchase it and find out, basically, if it really exists. So that's the composition of biologicals. It's sometimes called the Belmont convolutions. The nth of the sum, the semim of the sum of two things, which is essentially a two variable, but the middle variable is the same. And you take the sum, and it's basically a team. It's really like the convolution, too. Yep, yep. And then the context will be replaced by integral, and the sum will be replaced by product, but it all exactly fits into the enriched category of context. So you have got a notion of map between spaces after all, but it's more general than the distance decreasing sort of national notion is. The analog of enriched puncture is distance decreasing math, but the analog of bimodule is as I just described. From that, you can discuss two metric spaces are more easily equivalent if they exist by modules in both directions which compose, in that sense, come out to be equivalent to the identity on each side. So if the metric spaces are not socially complete to begin with, they just sort of, you know, they represent, you know, a dense set of rational numbers. Then you may indeed have... Well, in fact, literally, if you take a complete metric space and now consider any subspace that's not complete, which is dense, two dense subspaces, they may well be...

22:30 ...totally not isomorphic with respect to distance, distance and math, but almost by definition they will be isomorphic with respect to the bimodules. Right, right, yes, I understand. Because somehow the bimodules really only make a difference up to completion. Yeah. So now... Yeah, I see the connection with the machine. Yeah, I see the connection with the completion of... I never understood it all until hearing it. Well, what are you doing with philosophy if you can read that? Typical for a politician of ours. Yeah, yeah, yeah. That's perfectly reasonable to me, sir. I was wondering who was going to make that remark for us. When he says mathematics is too hard for philosophy, he doesn't mean it's hard. That's well. What will you say about our comradeship? Have you read this memoir? To paraphrase Hilbert. Well, the hell of mathematics. Well, the hell of physics. Physics is obviously much more difficult for the physicist. Yeah, I have never seen that. It's probably a possible, but it's a good one. No, I didn't read it. He went to Cambridge and studied with... I don't remember who Braithwaite was. Yes, Braithwaite. No, no, no, he never studied with Braithwaite. No, no, no. He did a Ph.D. in chemistry with Braithwaite, but he'd become sort of pauperized. Pauperized. Pauperized. Pauperized. Intellectually pauperized. Pauperized, yeah. Before that. And then he, you know, he sort of built, he fell out with Pauper, of course. Well, he fell out with everyone. It wasn't difficult.

25:00 But, you know, I think he's serious about logic and etc. Thank you for your attention. And I think that's what Michael made. No, I don't know if you can believe me, of course, you can barely answer this question. Well, John told me that. I told you that. It was, um, I think it was John or... What about this? What did he do with this? He really attempted to turn the house down and came in. That's right, we were just talking about that. I mean, and then, of course, when he retired, it was back to the status quo. Well, right, well, huge, overnight. Talk about the transformation of, you know, into quality, in this case disastrously in the reverse direction. Yeah. Who are we talking about? Michael Redford. Yeah. No, he just raised, he raised the whole standards of philosophy, well, of general philosophy, science, and philosophy of physics in particular. He brought it into fruitful, living contact with what, you know, real pathological physicists do. Yeah. Which, when you think of the state of the subject, it was in, say, 1975. And to, you know, to do it in one academic generation is an incredible achievement of the world. Thank you for your time, and I look forward to seeing you again soon. Oh, you know, history of mathematics, et cetera, et cetera. Was the lady a Frances Yates? That's right. That's the kind of thing. And, you know, 12 years ago, Michael Ray said, well, you'll make it. After all, he thought, well, with a little bit of gentle prompting from an eccentric nurse in 1989, he bought a little beer there in 1989 to give these lectures.

27:30 I mean, that's how intellectually serious he was. Well, is it nice? It was really quite a nice concert. I did something good in my life, thanks. It was quite a nice compliment, really. Yeah, I thought so. Me, and I was Michael Redhead and stuff, and then the first year... I wrote the film whenever it was that you were saying you wanted to do this. Curious fellows. Oh my God, you've been forewarned, haven't you? Made him sound too curious. You were doing something else. I think you were away that year. I was in Australia for a year. I originally invited you the year before in 1988, didn't I? That's right, now I remember it. You were away in Australia with Max Kelley. Then you were waiting for John Ehrman to come and give the first lectures. I don't regret that. I mean, if anybody could be a Wormack. Thanks for Bill, and John Ehrman did a good job, and I still stay in touch with John Ehrman actually, I mean you just compare what was going on then, thanks to Michael Redhead, and what happens now, it's awful, I've been there a couple of times in the last year and it's sad, it's really sad. Well I remember Moshe telling years ago, when Michael Redhead was, he was a sort of associate to the Department of Chelsea. He had no official status at all. I remember Marshall saying, I am going to attempt to get Michael Redhead his head about it. He said that years and years before Michael was even... Michael had no associates. Michael had no academic position for 30 years. Michael did his PhD in 1953 at Imperial College and then never wrote anything or had any academic position for 24 years. He went off and ran and, well, he actually went off and ran a prophet as well. Yes, and he did very well. Yeah, well, he became quite a wealthy man. Yeah, he did rather, I think. But he had no, and then he got told he wanted to go to Heinz in Boston. Yes, Heinz was actually rather impressed with him too, I remember. Everybody, you know. Well, Heinz. Heinz, yeah. Hampton. But I remember when I went to Heinz, well, when Heinz, when Heinz retired, yeah, when Heinz retired,

30:00 The only thing is that Heinz was disappointed because he kind of entrusted the Chelsea department to Michael, and then Michael was given the chair at Cambridge, and obviously he did the right thing in terms of raising the profile of the subject and making, there is no, it's a bully pulpit, he had to take the chair at Cambridge, it would have been a sentimental And of course, once he landed, that department was a bit more of a dull one, you know, he was now at UCL, I think he retired. Anyway, philosophy of science is not really an identifiable area. They're all either, there's no focus on that. Well, at LSE, yes, because they've got that center, but that's not... It no longer has a real focus. They were very good at raising funds, you know. I know, I was there for 21 years. I mean, I had nothing to do with raising the funds, but they certainly had to. It underlines Bill's point again. I mean, the reason that the philosophy of science is now in England so distorted is in the direction of a certain kind of history of science, which is represented by people like Simon Sharpe. It's not true at Oxford, though. No, no, no, it's true. Oxford has for a long time been an exception. There's a very serious school in Oxford around Jeremy Blackfield and Harvey and Simon. It's very serious indeed. These people know what they're doing. They have the respect of cutting-edge physicists. They write books, you know, joint books on quantum gravity. There's a lot of good stuff there. But certainly in Cambridge. One of the reasons is that everybody's welcome. Who have more money than any, who spend, who give more money to history of philosophy and science than the whole of all other sorts of companies. Foundation interested in, you know, promoting a particular line on the history of pharmacology and of medicine that can make their own commercial activities. And if it's history of medicine, if it's history of, if it's history of medicine, any, any doctor in any place, you know, just go along and write your start-up and finish it now.

32:30 There are so many people doing history of medicine, it's just a joke. I'm sure there are people who can do responsible scholarly work in the history of medicine, but they're not the people who are being funded by the World Health Organization. Well, I actually have to have a cataract. Can you just get with it? It's a long-awaited list. Well, with this alleged proof of the Pfeiffer conjecture, which is all about surgeries, and I guess this speculation isn't a good thing. Thank you for joining us. Thank you very much for your attention, though I meant to treat me for a minute, and mainly now it is, and probably for the following hour, as we talk briefly, you know, you can see that the drive is so boring. It's worse. No, it's my day, you know. I know that. Toronto is a local thing, but Toronto decides to truck on it, and they have 500 trucks. I'll be right, I'll go on homology. It's unbelievable. I think there probably are, I don't think there are, but there are a hundred different.

35:00 When I try, nobody does that, but you know, there's no one. And I could very, very easily, in all that reason, because it's, they do, in fact, I think, I think, I think, I think, I think, I think, I think, I think, I think, I think, I think, I think, I think, There are more toll roads in the U.S. I mean, he testified really, I think, to the skill of these guys. He doesn't know why, you know, he isn't just a bit of a man. Well, thanks.

37:30 But it is. Well, when we drive to Toronto, well, it helps to drive it faster. But Penn, well, Penn is more accurate here. Adjusting a little, you know, locally. Thank you for watching. I've heard the name. I don't know if you've heard it. It's funny, you know. I didn't want to ever delete his studio, but actually, let go. You described it as so dull. And then, of course, there's enormous information. I can't see it. My senses will never, ever come back. Well, there's a novel by—have you ever read Wyndham Lewis? Now, Wyndham Lewis—no, he's not my favorite, but he's a painter who— The Vortices were a couple of good painters in the 1920s. Something of a reactionary, politically. He wrote a lot of these novels, most of which I find unreadable. However, there is one novel that's really interesting. He was actually born in Canada. Well, born on his father's yacht, it seems. In a sense, it didn't surprise me that I'd heard something of the group at these seminars years ago. The main combinatorial decisions were put to this point in view, but he actually left. He went to live in Canada.

40:00 What was interesting was how familiar he was instantly with the relevance of this. Yes, yes, because no, he wrote these early things with support. Yeah, that was very interesting. Early, very early on. He was a little fascist in the Mussolini. Oh, I see. No, no, yes, yes. He was also the right-winger. Anyway, but I'm going to leave this up because... I've known him a little bit for a while, but I don't fully understand how sexual are his gifts. It's probably quite accurate, I mean, at least in terms of the way Toronto was perceived when it was character. It was completely a provincial area, but always a sort of a provincial place. Yeah, I went there a few times in the late 50s because my father lived in Niagara Falls. We'd vacation on West Virginia Island. Thank you for watching. That's as far as it went, frankly. It was a funny story because I was invited for one month to a long row and arrived simultaneously and independently for a month while I was there, and I thought it was too very happy moment for me, so I thought I'd see if I could do it in the interest of time. Well, you can have it and say which thing you want to draw from that, but that's something else. Of course, I have to record it for my class last year. In fact, I do actually have some of them on video as well.

42:30 How many salaries do you need to buy a house? Access to education is very important. You have to buy a home. And I mean, it's always the same thing. Let's take schools that are following each other probably like that. And business-wise, the states always complain. They measure all these irrelevant things, like whether you can clean the air and where you can get it. Right, I'm reminding you, too. So I thought, gee, why is Tsiolkovsky suppressing this? Maybe it's because this actually puts the value of four in the third. Yeah, but it is a kind of composed axiom. It has a lot of information in it. Yeah, but I mean, clearly... What do you mean, basing general technology simply on the notion of boundary? No, border. Oh, all right, okay. Frontier, border, border. Well, they try to, it's true, all the others, you know, neighborhoods and open sets and nets and you name it. You could have made it redefined. Yeah, yeah. Oh, so there's one that contains its border and stuff. What? No, no. No, no, because that's what you just said. Well, there's one that contains its boundary. One that contains its boundary. Right, right, I see. That's more or less obvious, but on the other hand, you see, notice that the general topology really came into existence because of complex analysis, you had to get straight on, you didn't just have balls and spheres, you had other kinds of sets, so I guess you might want to sell boundary value problems, so in some sense it would close those things up. And even today, it has a very good quality, and it's found to be really great by Witten. It's amazing that this paper is quite clear, short, and a very prestigious journal. It's never went anywhere. Well, when has it been? Have you ever seen it lately? I've seen it more than 60 times.

45:00 Nobody has published it in an introduction series so far. It became a kind of... Well, there were exercises in Burbaki. I can't remember now. I mean, they didn't emphasize the boundary for the Leibniz. You know, really, it's a sort of Leibniz. They actually said it's false. Oh, you mean because they're using the usual notion of boundary. You mean the intersection of A with the cut. All right. They're open. Exercise open. You know, in case you were so naive to think that from this picture of ovals you could actually derive a general rule, they would try to counter it then, but they chose it as the object opens and yet they took the boundary as the operator. Well, where boundary is the intersection of the closure of the circle. Yes, I remember those exercises, though, because I must have done them years ago. And it's true, they don't make, they don't mention the, the, the sort of symmetric rule. Now, if, if I could collect cash and then use plastic, that would help, because I don't have cash right now. Well, I've seen it somewhere, you know. Not only in your work, Bill, it reads 14 and a 15. So you can have five. Ah, yes, that's right. It's 15-inch. Including the tip and the top, yes. I have, I have, uh, no problem. Thank you for watching. Thank you for your attention.

47:30 No, but we're going to, I'm going to try, which I mean in the Carl Sagan House anyway, because I haven't, I haven't, and then if we take three quarters of an hour, which of course connects with his view of what he calls. But he also claims that this came out of a kind of fundamental wrong turning, which was taken in reaction to the crisis of geometrical intuition. In the late 19th century, when the piano, the space-filling curve, and the Ostrowski's construction, the other so-called pathological functions were, as it were, legitimated by Narenko. And I've always wanted to understand more clearly exactly what he intends by that claim. And that would be useful to start. And it also connects in with this kind of criticism of Frege and the whole... There's a lot there to chew on. And then we can come back tomorrow to what we loosely called Law of Eternity, which is, yeah, yeah, yes, I absolutely think that would be a very good way of going.