Lunch conversations

2:30 It's yet curiously coming from the side of the very people who was precisely precisely because it was supposed to connect with issues of general ontology at the same time it certainly doesn't connect because it's a general theory of structure in the way that these people conceive of having a general theory of structure i.e. something which allows one just simply not to discuss the issue of what objects are. It does in fact connect with...

5:00 I haven't said in front of Bill because he doesn't like the term ontology, but I would say it does actually connect in a much subtler and more indirect way with the concerns of ontology via an understanding of, a clearer understanding of these notions of 40 times cohesion and variation, that could be made more explicit, but that again is something we could just discuss. I'm trying to squeeze an awful lot in here, I know. But the point about, well, I think at least one PhD thesis was hatched in this room this morning, which would be nice for somebody. Yeah, no, he's got that problem about mathematical thesis, but I couldn't try to do it. That wasn't what you were getting at when you wrote to me about three or four weeks ago at the list of suggestions saying that in Florence he was talking about this general principle. I thought that was an aspect of it. Whatever you talk about, it exists. And that's the sense in which he wasn't inside. DeMello would say, we've talked about all the ordinals, they don't really exist from here, so we'll talk about something else where they exist. I mean, it is still the same attitude in a way. DeMello does want to say, whatever we can talk about, it exists, but he's going to build it up from here. Bill says, well, we've got this category in categories, and everything that we're ever going to talk about is already in there. We just don't know what it is. We just don't know how to carve it up, so let's look at principles. Yeah, yeah, so now that has made the same point to me about, yeah, that was what you had in mind, which is what I thought. And I expect this is not, this cannot really be made up for a form of academic education, prisms, but it's a very, it's a guiding principle which could be made. Perhaps a good deal more explicit, or certainly more explicit, than it has been. Yeah, well I took that to be one of the things you wanted to talk about and I thought that we, well, at least, well, certainly talked about things which are clearly connected with it this morning. Okay, so you're basically happy with this, the way it's been? You're happy with it? It's been a worthwhile experience? Yeah, I'm very happy with it.

7:30 I think it's great. Yeah, I'd like to hear more from him on the stuff that we were talking about at breakfast, which we didn't of course get on tape, about the Burbanki theory of structure and why it was okay. I mean, obviously, there was an awful lot of insight there, and they did know a lot of category theory, and I don't want to go too much into the details, you know, by... Detailed historical nitty-gritty of why, you know, they were so focused on isomorphism is an interesting thing, but I'd like to kind of pull back and see, given the general, as a salmon hang of masses, it appeared at the time that the body started, why they just didn't get a general theory of structure that was powerful and flexible enough to do what they wanted, and could we have it now? In a way, I think I've attributed it a little too directly to André Bay. What I would say now is that that was the proposal Bay wasn't prepared to block. He didn't really like that one either. But he wasn't prepared to block it. He was prepared to block use of categories. Okay, I think that's a compromise. I mean, not a compromise because it started before the idea of category was even available. But I said in one article that Bay championed the structured set. But that's an overstatement. He didn't like that. It's pretty clear from what Carly has said. But Bay is still the one who wouldn't scotch that. He wouldn't stop them from doing that. But the question is, should all there be some underlying general theory? They apparently thought there should be one. Well, they were the general staff, you know, you are supposed to have... In his work on these kind of things that have to do with things related to Grotendieck, did he never use the categories in number theory? No, no, in fact, he doesn't even give a general definition of morphism. He defines projection from a product. He defines a few kinds of morphism. It would be painful. You could do it. I mean, it's mechanical now, but painful, and he doesn't do it. He's not even interested in general morphisms between varieties, let alone some categories of varieties. What kind of varieties is he using? Is it a real variety? No, no, these are varieties over arbitrary fields.

10:00 But his first step when he wants to talk about a variety over a field is the algebraic closure of an infinite transcendence degree extension. So you're working on an arbitrary base field. So this might not be algebrically complete, it might be a finite field, but his device is, first, extend by an infinite number of variables, now take the algebraic closure of that. As a scheme, you take your ring of polynomials, say, you look at all the ideals, and include ideals of irreducible polynomials, and in that sense, it's like an algebraic closure. All of that has registered in the scheme, but it was all constructed in terms of your base rank, which might be polynomials over some field. No, no, no, we're actually going to look at all polynomials in every countable, in every finite number of variables, and we're actually going to look at the algebraic closure of that field, so we're going to look at all the polynomials, we're actually going to take the fields, so it's all the rational functions, and we're going to take all the algebraic functions. So, in one way, we're doing geometry on an arbitrary ground field. Where we're really living, though, is in setting all algebraic functions on any finite number of variables in this field. Well, you have... More abstract, yeah, yeah, and so it makes it hard to define morphism. He doesn't even define a general morphism between varieties. That foundation... No, no, 40-something. There is also the Van der Velden work on the foundations of... Yeah, I'd really like to hear more from you perhaps later this afternoon, Leo, on the... What describes, you know, the two lines of approach to a general theory of structure of that period, the van der Waarden line and the Babacki, and then looking at it back from now, as I've heard from the point that we've heard from you, I mean, the community of algebra, which was carried out far beyond van der Waarden, so, I mean... Their principal purpose was essentially a local thing to get a good definition of intersections and some of the Italian's had done highly daunting tasks dramatically and it didn't, that in itself at that time did not seem to need to run total conservation so of course if you look at modern treatments say like in Fulton's book

12:30 Even for a variety, even for affine varieties and so on, it requires everything, you use the schemes, you use the vector bundles, and it certainly uses factoriality from the very, very beginning. But that changed. He wants a theory of the whole intersection of two, of two. Yeah. Variety for schemes, not just Fulton was doing that. You require schemes even to get the definition even for varieties. You ultimately pass through schemes in this case. Yeah. The only way to make it work is factorial. This is Fulton's book on intersection. But if you go, like, yeah, projective varieties, but you want to be able to talk about the whole intersection of two projective varieties, not just points of intersection. No, no, but I mean, yeah, but I mean, even to get the local definitions, up to the general case with singularities and stuff, is a very daunting task. I mean, it had not really been realized that homological algebra was all relevant, certainly with the taller definitions. I mean, these are very, very subtle, and it's very important before that. It's not that they taught us we're really ever wrong about a matter, but there's certainly a discrepancy. And they, I mean, they, what they're setting up for theory, the theory of the kind of varieties you do in characteristic theory, and they're in the section theory to get the, I mean, the fact, one of the very few is the intersection theory. It's a product, and the lesson gets you to the, and he says in foundation, there's a sense in its being, you know, machinery to attack the conjectures. The highest level of general...

15:00 One of the things that we should, I think, try and make time for, because we're running down out of the wire, one of the things we should try and make time for is just, even though Cartier's not here any longer, just finishing off Grotendieck, because we never really did get beyond, I think we probably got to about 1965, more or less. But we didn't get to one of the things that Bill's touched on, which is Grotendieck's later. Much later ideas, after he had gone out of the official practice of mathematics, but I'm talking about 1973, the time of the Buffalo Lectures, and this Duskin memoir, which we were talking about last night in the bar. First of all, the complete revision of his earlier definition of scheme, going right back to... Taking a completely different approach there and the reason why he did that. And secondly, this huge program for the use of kind of classifying topos. But the actual redefinition from the one to the main... No, no, no, possibly not. But the conceptual motivation is clearly very different. It's certainly a very important thing. The redefined form is the one that they justify in the introduction to E.G.A. Chapter 1. It's not. I mean, the ideas are all along. And the redefined form turns out not to be as independent as you'd like. If you could talk about the locally affluent... Bill and Sir John. And John? Okay Colin, sorry to interrupt, Leo's chomping at the bit, I need to go and eat, and what you've just been saying, yeah, sure, we're going to go to the place where Angus and I had a drink last night, it's a little bar on the corner, it's nice and warm, raw Arthur, he does very nice steaks and salads and good stuff, I've checked he's got a good selection of fruit and fruit stuff, oh here's Sir Mimi, good. Thank you very much for your time, and I hope to see you again soon.

17:30 I used to be quite familiar with this area, and I've moved away from it now, but I always thought, I've got to think about it. I mean, it's a lot of different directions, you see, in this. You can think of something as something big in the universe. You can think of it as some kind of sheet playing. But it's some freaky stuff. Uh-huh. I don't know if that's helpful or not. I'm going to put this in a wash. And there's also many mysterious things that you can come up with. You start with the chemical relativity and then you've got to look at the least part. That's kind of what I'm trying to identify you with. Perfect. Merci, Marcel. Pardon me for the inconvenience. It's certainly an aside, but the guy who had been a set theorist and then saw some connection with having the iteration of these elementary embeddings, saw a connection between this and brain. It's really quite fascinating. I mean, he's turned it into an industry. He used to be in Brittany, actually. He spoke quite well about mathematics. I mean, he spotted in labor studies some of the algebras, his intuition around mathematics, and this guy, someone who is a very, very good scientist, he spotted that there was a strange kind of...

20:00 Generalised grades, yeah, which of course is a big area of interest for quite other extraneous reasons. I think it's been reported that there's more wiki sellers at the moment. Well, grade theory is a big industry at the moment, isn't it, because of things like quantum gravity. Well, I could probably find a bit out by googling on grade theory. Thank you for watching. Hmm, yes, you'll get an awful lot of entries if you Google on grade theory and you'll get, I think, a lot of entries. Yes, you would get, probably in my case, yeah, yeah, yeah. I know, I know. In fact, if you get grades, you're not very much related to the practice of mathematical physics. You've got a function where you have to have a transformation to start iterating. Uh-huh. I'm not sure about that. Uh-huh. Yes, you're good. That's the stuff that these people doing so-called quantum groups get excited about, isn't it? There's some nice recognition of unifying patterns there, but I don't know whether there's really any physics, but it's interesting stuff. But you're right, it wouldn't necessarily tie into the iterative conception at all. And there is another thing, the theory of principle, the theory of constructive theory, but also the theory of non-principalism, the theory of the association.

22:30 Just start with a few simple ones, and then you'll reach the company. I really don't know what you're talking about. I've got to think of the kind of iterative aspect that's coming from the Brady model the other way around. And they don't want to say no, even though it's not that important. This chap wants to go. Yeah, hang on, I'll wait to tell him. John? Sorry, I don't know. Do you remember? Sorry, Mimi. A set theorist who was working in Britain, who spotted this thing about the connection between generalizations of grades and iterations of elementary values in set theory. He's made an industry of it and it's actually quite beautiful. Name one or two of the French set theorists from the previous generation. This guy will be in his mid to late fifties now. It's just to the tip of my tongue and I can't get it. Well, it'll come back to me. I don't know the names of any French set theorists on the prize. I don't know the names of any French set theorists on the prize. Do you know anything about what this student of John's, who's become his son-in-law, Dickard Rush, has done? He was a pupil of Jensen's, but he didn't for some reason, but I think it was purely for personal reasons. He didn't finish his PhD. No, no, it wasn't that at all. It was nothing to do with his relationship with Jensen. No, no, no, I don't think it was anything to do with his personal relations with Jensen. It was to do with certain frankness in his life. At the time that he was in Berlin, he was studying with Jensen, but he apparently, um, you know... No, alas, not at all. No, I quit. I'm out of my connection with Mr. Deuce. So I don't work as a real estate agent today. I have contacts with some of them, with some of them, but you know, I don't have any contact with them.

25:00 So if you talk to Mr. Deuce, I'll talk to you first. Well, of course, okay, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, Oh, sorry, I thought you just suddenly thought of the name of the set there. No, no, no. His first name may be slightly red, but... It's so annoying. I mean, I should... To me, if I'm talking to philosophers, I don't want to say cohomology in the title. They close immediately. Or they decide, oh, this is some little thing people do. Bloody good work this morning. Bloody good work this morning, comrade. Really good work, really good work. Thank you. I'm going to explain that an arithmetic scheme is exactly what Kronecker thought arithmetic should use. You're given a problem, you're given some diagram of an equation, this isn't an integer polynomial, maybe I'll mention at least that he actually wants them over the integers, but anyway, you've got an integer polynomial, well you look at the polynomial ring, you look at the ideals on it, which are all finitely calculated, you just, this whole subject works with finite lists of final polynomials over the integers they're natural. And I mean, this is what Kronecker thought it should do. But what I want to stress is, this is what arithmetic schemes do. They have this replication being some weird thing out there. Well, they're just what Kronecker thought it was going to be. They're simpler. We're going to look at complex space, we're going to look at complex varieties of defines, we're going to ask which points have rational thought, etc. No, it's simpler than that. You're just looking at lists of polynomials. But you have to look at those. The problem is given to you as a list of polynomials.

27:30 We can do better? Well, we can do better. Oh, I didn't realize this was just the waiting room. Okay. Okay, that's good. Allons-y. Oh, we're going in here? Absolutely. No, no, no, no. Do you need to get back in the house? No, no, no, no. Okay. Thanks. You can sit next to me, I want to ask you. Come on, you sit next to me. That would probably be a good way to set out an expose to philosophers. Oh, I never thought of it. By the way, I live... Are there any polynomials or not? Uh... Are there any polynomials or not? I haven't even thought about that in my mind if there are polynomials or if there aren't. No, but it's a question whether there are. No problem. Yeah, I just hadn't thought of it. If there are over a field, that's... Yeah. What did you get up to this morning? Anything else? Thank you for your attention. I want to go to the right. Yeah. Not all that much, actually, that side of town. It's quite safe. It's just that there aren't all that many interesting bits. I mean, the old center of gravity is really the bit where you walk along the square where the theater is, and then either take the road or go. All the interesting things really to the left and right.

30:00 There are a lot there, you know. By the way, Carl, I was a referee for that book. Oh, yeah? Which one? Well, this one is about mathematical practice. Do you know what that is? It's all very incestuous. Where is it? Well, it'll come up for press. I don't know when, but it looked interesting to me, given the few dozen people. Oh, this reminds me on practicalities. We do actually have to go down and extend the height of the car by one day. But I think I can do that without any of the named drivers. I think I can just do that on my own this afternoon. Now, tell me something, Michael. I want to understand how did you start all these archives? I'll tell you that another time. Well, for Gromley, the scheme sits there and has points. It's not, the covering is a secondary issue. You might ask what it's covered by. Oh, I see. Right. I agree. The notion of the general structure is more simple. It's going to have a lot of schemes, a lot of intermediates. Yeah. So it's very, if you're expanding the word schemes, I would appreciate it, but I'd be happy with that. Yeah. I'll tell you now, you know. Smaller. These are probably smaller. Oh, she's coming. Well, you're getting really hungry. I'm sorry. I'm sorry. Well, the French are not noted for fast service. They do like to enjoy their food. We're getting there. Don't worry. You are in provincial France, they do things slow here.

32:30 Bill asked me the same question about a little while ago. I think so, very much so, yes. It's been a wonderful meeting. And thanks to everybody here. But, no, I think it's been absolutely amazing meeting you. You know, you have, you know, a very unique collection. Yes, I do, I do. Well, it's not just a, I hope it's something more than just a collection. I mean, like something, some squirrel hordes out of, yes, exactly, I agree. In fact, Leona's just asking about that. And I really want to try and make sure, one, that it's preserved, because... The older stuff, the tape, the stuff that was on tape and not digital, the medium, doesn't last forever, and two, that the most important parts of it, of which certainly I think these exchanges over the last few days will count as part, is actually put to some really effective scientific use, so it's one of the things that Angus and I were hoping to have time to chat about before then, so yeah, yeah, yeah. The problem is, I think it would take, Angus was asking me a couple of days ago, I think that just simply to finish cataloging everything and getting the 20% or so of the most important stuff transferred onto digital, from the audio to the digital, if I was doing it on my own without any assistance, I think you'd be looking at a good 18 months of work. I'm not talking about transferring the whole thing, but I mean it's now pretty well completely cataloged. I'm talking about transferring maybe 20% of the really important, the most important material into digital. You'd have to have some really serious secretarial for doing that, but just getting it transferred into permanence storage medium, i.e. digital, and then putting that onto sites where you could, putting it onto paper is, I mean, some of the most important stuff here should be put onto paper.

35:00 I mean, you know how long it would take to transcribe five days of conversations. I mean, you're looking at certainly three months' worth. I think the most effective way, this is something I really wanted to talk to you about actually, but we'll come on to that. That would be fantastic. I only wish I could offer her. That would be fantastic. Let's talk further. Yeah, I will, I will think about it, I think I will think about it. Oh, the sand! Then what? Yves Flottant. Oh, Yves Flottant, you know what that is, yeah. That's the formula? Yeah. I know one. I know, I know, I know. Sounds good to me. Oh, is that a set, like a set menu? Oh, yes, I think so. Thank you very much for your attention and I look forward to seeing you again in the future. There's probably no other alien matrix around. You never can tell. Arbitrary diagram. He got one of them. He got one of them. What is it? I don't know. But the Newton Institute is doing the day. Don't remember my English. It's not even over. I've got it. All right. Send it to me. I'll give that to you as soon as we get back.

37:30 I don't know. We're going to have lunch, I suppose. Yes, that's right, yes. You can't simply step out of the room and go on clerks and robbers. No, no, no. I don't want to. I'm sure you'd have to go on more than that. I hope to go on more than that. You're crazy. You're crazy. You're crazy. You're crazy. You're crazy. You're crazy. You're crazy. You're crazy. You're crazy. You're crazy. You're crazy. You're crazy. You're crazy. Well, I still think my suggested terminology of measureless cardinals is quite a good one, because, as it were, it keeps both aspects, but they're not characterized by their measure. Yeah, okay, it doesn't matter. Ineasurable. Yeah, you can say immeasurable, but then that still suggests that they're not characterised by the measures, which of course I understand is not the case. But that's because measurable was already negative in the crucial point, namely that there exists a homomorphism which is not represented by a point, that's what they always mean by it. So the usual meaning of the word is already negated. So if you negate it, you actually get a positive problem, it doesn't sound like it. That's kind of the problem one is in, you know, to get a positive problem.

40:00 Or it's engageable, somebody else says that's the point. Analyzable. Observable. Oh, no, no, no. Well, considering that they so often invoke reflection principles, you just get them more confused if you start calling them observable cardinals. Do you recall what John's son-in-law Dickens' thesis was? It wasn't on measurable cardinals, but it was. No, actually the thesis he completed was to do with this. There's minimal parts geometry that John's been trying to develop in the context of his non-Euclidean set theory, but the thesis he started to write under Jensen, you don't know what that was about, because that was part of some very technical set theory, which was, you know, really very demanding stuff. I mean, obviously people don't get to… Two PhDs under Jensen, lest they know a lot about set theory. He's a smart guy. He's never going to have a job. He's just decided, since he's, as I say, going to be independently wealthy for life, he's decided to come and live in France and devote himself to doing mathematical research. He's a strange guy. I'd certainly like to get to the Church of John's retirement party, which is apparently now going to be on the 4th of July. Which would give me an excuse not to go to Moscow, because I'm more and more thinking that I just don't really want to bother to go back to them. Well, it's nice to be, you know, to have an expenses-paid flight to Moscow, but I, one, I'm not going to get anything out of an executive's relativity meeting, and I certainly don't want to give Pavlov any encouragement. And, you know, I would, I'd hate to miss John's retirement party. Best friend for a long time.

42:30 Oh, you're talking about meeting along now? Oh, your brother? Wow. World's own guest. It is said that she was very pleasant to her personal staff, and it was only the members of her cabinet that she terrorised and reduced to quivering jelly, but then of course, who does that remind one of? I'm not suggesting for a moment that Thatcher enters into the same category as Hick, because that would be foolish. No. No, just because people are personally charming. Thank you for your attention. I do. I do, because then you can explain it to me. I don't say that I'll get it, but at least I'll have a much better chance than I would otherwise. So that's one simple answer. The only thing I was thinking about is, for example, I thought I should write a first chapter. So I thought, well, let me find out what the weather was like on September 47th in Paris. I'll just go read in a newspaper magazine and find out what the weather was like. This cannot be done in North America, supposedly, because the newspapers and magazines don't exist. You can't get any commercial magazines or newspapers from France in 1947, in Cleveland, or anywhere near Cleveland.

45:00 Well, if you are really seriously determined to find that, I'm quite willing to volunteer to go to the... No, to the grand TGV, you know, to meet around, you know, and look up the, well, there'll be a weather almanac for 1947, won't there? Yeah, okay. Well, actually, it's very difficult to browse in the bibliotech now, you know, because... Oh, sorry, I mean, no, no, yeah, yeah, good idea, I will, yeah, who's having what? No, I was having lasagna, definitely, now, weren't you having the, say, loveless? Oh, no, it's over there, he's confused, but, um, I thought, oh, I'll look into having that, uh, no, not in 47, it didn't exist yet, you didn't, you did? No, no, just straight up. Yeah, it must be, because you can buy very old copy, wouldn't you, when you go along the banks of the Seine, the wonderful, obviously not what I'm talking about, the wonderful kiosk where they sell secondhand books and magazines. You can sell, because I have actually got a complete set of pairing match for... In 1953, including the death of Stalin, and for 1956, and for most of the years of the Algerian War, something is actually very interesting. They, for instance, have a big thing in Budapest. They have an interview with Imre... I'm sorry. They have an interview with Imre Nodra. Imre Nodra, who was briefly the head of the provisional government in Hungary before the Soviet Union intervened later. And then have an interview with him and the chief correspondent of Paris Maxime. Quite interesting. Well, but I wouldn't say that would stop you from writing the book, anyway. No, I wouldn't say that would stop you from writing the book, anyway. No, I wouldn't say that would stop you from writing the book, anyway. No, I wouldn't say that would stop you from writing the book, anyway. No, I wouldn't say that would stop you from writing the book, anyway. No, I wouldn't say that would stop you from writing the book, anyway. No, I wouldn't say that would stop you from writing the book, anyway. No, I wouldn't say that would stop you from writing the book, anyway. No, I wouldn't say that would stop you from writing the book, anyway. No, I wouldn't say that would stop you from writing the book, anyway. I can write a chapter on etioclomology without knowing etioclomology, I don't intend to do it. That's a different kind of can and can't. Knowing what the weather was like is really important. I'm sure I could find out. I could go to the library just now and just find out. What kind of food did he eat? How were the housing shortages that much improved? Well, they wouldn't have been that bad.

47:30 I don't know. The last bit I'm sure you could find out from sources that are publicly available in the U.S. I'm sure that was then. Much later, I would think. Much later. I expect the most is that he was vegetarian by necessity. Well, certainly, France certainly had very strict food rationing in September 1947, and it's still very, very strict.