Discussions, incl. FW Lawvere C McLarty, A MacIntyre

0:00 Imagine you have this cubic curve. Find a section. We'll just take the absolute section. Apply Newton-Berger's normalization and arrange the thing in that way over a field.

2:30 Notice that like most of the finite fields, as is well known, has a retraction linear over the base. The bigger one is a vector space over the smaller one. So any map of fields, and that's how it should be so in order to turn the locally separable category of species on the fields to Boolean. The Galois topology is what emerges from the bar topology applied to the site of finite field extensions, but that's going to be the natural restriction.

5:00 Non-commutative becomes commutative. Literally, that can only happen for a very tiny object, for one that's got a point and it's so tiny around that point that somehow it can't. So I think that's a real direct expression of tininess, almost a direct expression of tininess in the way that others are round about.

7:30 What that equation means in particular, and we always worried about this, again, I don't know if you had already solved it or whether you didn't even worry about it. Between Anderson, Gonzalo, and Gavin, we see this thing around. D sub infinity, the same thing as the Boolean group generated by D modulo zero equals zero, where D sub infinity is defined to be part of the line itself, the neighborhood of zero, on which functions are exactly formal power series. I mean, it's a spatial thing. Functions are formal power series. It's a realm in which, as Gabriel points out, You can solve some of the differential equations. So, for example, cosine and sine and exponential, those things always exist as functions on this d sub infinity. Only polynomial functions exist on the whole line. On the whole line, yeah. So, d sub infinity is the union. It's the set theoretic, i.e., topos of the union. All of this is a unit. Therefore, it's the quotient of the sum. Of course, it preserves zero, and that preserves addition, unified addition. So this is the image of this e to the d modulo g equals zero. So the question was always, is this an isomorphism, or are there perhaps further relations in the line? That aren't expressed in the formal calculus of each of those. And that of course depends on the gradient of each of the polygons because you're talking about taking an image and so on. So, well, no, it's the same. The basic calculation shows that in that map, from e to the d into the line, the kernel is exactly what's generated by zero equals zero and nothing else. When I say zero equals always zero, it's a unit. I had to use this idea, you see, that the stochastic sections are the criterion for a math with out-of-mind schemes becoming... Of course, you know, when you restrict from all rings down to these ones that are finite dimensional over the base, it could very well be that... Okay. Angus? No, you didn't even have one, did you? No. Okay. Just thinking of times, I don't... I haven't got more time left. Sixty-nine, I guess, but... Okay. That's a bit more...

10:00 Well, we want to be sitting out. We could go out and get a bite to eat, if you like. And I was thinking maybe we should just spend 20 minutes just deciding how we want to divide up the time for discussion. Well, yeah, we'd better do that before John Bell arrives. Well, that was my thought. All right, Bill, you've saved me. I didn't want to embarrass anybody by saying that, but that has been tactically out of the mouths of babes, in this case mine. Well, yeah, okay. Well, it's like the little boy who saw that the king had no clothes. Yeah, it might just be worth that. And then, plenty of selection of pads and things, and we'll get the...

12:30 The flip chart's going to be okay, is it? You guys, I'm so sorry, I haven't got anything more sophisticated. And then, does anybody need the computer right now? Can I? What is the secret of the computer? You've got it like that. Oh, yes, it's very simple. Well, it's on now anyway. Okay, just... Yellow, it's yellow, green, whatever it is. That, yeah, that's... And then just to go to, if I just... No, it's very simple. Let me show you, it'll be... I just want to go there. You just, okay. Well, all you have to do while it's like that is just click on Internet Explorer, yeah, just simply, the Internet Explorer's on the desktop, just tip on that. If it's slow coming up, and for some reason it is horrifically slow at the moment, just click on favorites and then, okay, it's off now. When you started up, okay, that's now completely disconnected. Okay, just let me, I mean... Oh, this is simpler. Okay, that's switching the power on, that's down there on the tower. See, it's such an old one. Okay, that's the... That's it here, that's a go. That's a go, that's it. And now I'll simply give you my password. It's much slower than it normally is because there's something with this... It's clogged with viruses and this antivirus software he's... He's fitted as... Oh, it's a dreadful, it's an absolute nightmare. I don't know why. It's just in the last couple of weeks, two or three weeks, since he fitted this new anti-ORS software. Yeah, yeah. Well, I don't know what he did, but it's a... The problem is worse since he fitted this antivirus software. I had my standard antivirus which is just upgraded by Microsoft automatically, but I've come round to your guy's opinion of Bill Gates. Next time I am going to go for this. Again, Bill's trying to ask you to remind me again, the little $500 job that...

15:00 The Mac Mini. The Mac Mini, right. No, sure. From the description it doesn't sound like it would be hard. Yeah. Anything which allows me to read your stick, and this stick would be great, but, okay, right, that's my, okay, this is the, okay, Lenohen. Len-i. Len-i. Len-i-hen. L-e-n-i-h-e-n. Yeah, mother's maiden name, the usual. But my grandmother on my father's side was Scottish, so you know that, which is where the Bruce comes. A Bruce, a Bruce. All the males in my family are Bruce. And then, for some reason, because it's so clogged with bloody viruses, you've got to get rid of all of those. Just click until, and then, yeah, and then you have to give it a few moments because it comes up, no, the modem's connected. So you should just be able to go straight on to Internet Explorer, yeah, yeah. Okay, problem solved. But in terms of getting rid of the viruses, in my mother's family, what virus software do you have? I've got standard e-tastic anti-virus software. Well, it seems, all I can say is that since it fitted it two weeks ago, it seems to be generating more bloody viruses than ever before. Does it update? It's supposed to update automatically, yeah. Yeah, but I mean, first of all, the printer suddenly doesn't work. Because whenever I tried it says denied access to printer. I keep getting these from the time that I loaded the last lot of antivirus. That's my server, this is my club internet. I don't know why it's asking me to connect when the modem's already connected, but I'll see what happens. And I might just very quickly... Would you know how to download this maker software which would allow me to read Bill's stick? I really would like to read the stuff. Okay, then. Yeah, okay. Don't need you any longer, mate. Ah, wait a minute. What's happened? I failed to connect? Invalid? What the hell is going on here? This is insane. It's really... It's already connected. It shouldn't be a problem. The web page, I haven't requested a web page. Oh, something is going...

17:30 No, something is seriously going haywire. Yeah, I think you're right. This is... Well, I'll just quickly look in my... I'll just very quickly look in my emails before I do that. And then reboot. I'm just going to my second email package, the one I use, you know, when I'm away, the Yahoo one. My old emails will be on, you know, the Melbourne Express. You know, you have no conception of just how primitive I am, to say the least. Well, right after you do that, I'll just see if I can... Yeah, see, and if there's any way you can... Let's see if we're going to... OK, it's suddenly connected. OK, I don't know what was going wrong before. No, it's suddenly not connected. Oh, OK, so it's not... You can't pick up anything right now. Why? Invalid username, but I've just... Well, very often just rebooting the machine. Yeah, yeah, I think that's what I'm going to have to do. OK. You're probably right, it's probably supposed to show that altogether. Yeah, you're right, it just gives you just a little bit more. Okay, sorry about this. Only if there is any way of loading the stuff on little sticks, especially the stuff about naturality. I don't know why I'm getting this cramp, I really do not. I think the guy who did this just doesn't know what he's doing. That's the problem. Well, the guy he works for certainly doesn't know what he's doing. Yeah, I think that's what's happened here. And obviously it's more of a cottage industry in France than in the UK. I went start, that's the problem. It's my undone fault for not having taught myself what I need to know. Let's just see what happens. The modem's illuminated, so it should be... Ah, no, it's suddenly... No, hang on.

20:00 The modem's flashing on and off. I wonder why that is. Actually, it could be a problem with the modem. No, to me the modem's actually pretty normal for what you're studying. Yeah, that's true. Let's see what happens. Well, at the time that Eilenberg was a member, it would have been... It's pretty natural to invite Clayton along pretty frequently. Well, Cartier and Corey, they were there before Cartier was there. The internal newsletter was very generous. But it's not for public access. The surviving Borbachistas are insanely concerned that people see their articles. Yeah, they're very, very connected. Well, yeah. Oh, because she's done the best work on early Gorbachev. There's a mathematical intelligence arrival by her. And she knows what a differential is. Well, what a differential is. We've already narrowed it down a lot, I guess. But she's done Gorbachev and published some. And also she's willing to relocate.

22:30 But now she can't even tell us anything. She wants something into it. I think that she learned in the archive since then. I think, no, it's just a currently rule. I don't think there's any rule. Well, for one thing, when she's organized the archive, that will certainly be in the moment. For the moment, they don't really know what's in there. For another, that's more harder. And Lillian is capable of very slow work. And since nobody is allowed to see the archive, nobody knows. Is she organizing at a tremendous pace? Is she organizing at nothing? We just don't know. I say it's a mystery after all. I say it's a mystery. So we've not been able to fix it for 30 years. Well, Groton deco, obviously, but then he's out. Yeah. But he doesn't want people seeing stuff he wrote that wasn't for publication. Yeah. I'm not sure. I don't remember what Sarah thought about having to put it on. There have been some people who objected to that. It seemed like a good idea before they were 50. Well, yeah. How did Cartier feel about it? I think Cartier says, yes, you should put what you're going to be, and we should now, I know he feels that we should now declare Fort Buckley dead, the people who have called themselves Fort Buckley. Yeah, the old joke about knee flowers, knee wreath products. Yeah, yeah, yeah.

25:00 Yeah, actually, you said that in print, hasn't he? It was that interview in the Game of Thrones Live. They should have packed it in that the, you know, the whole shape of mathematics has passed and that grand codification is a product for another generation now. Well, and he's one of the people who will say that it's a city not to use category theory. Oh, well, it was... He said that very explicitly in print, too. Grothnig says they abandoned their project of a comprehensive treatment of mathematics. So when did Cartier actually leave? So when did Cartier turn 50? Obviously a little bit later than Sayre, I guess. He's only about three years younger than Sayre, isn't he? He's about 70, 75. I think he's about 75, isn't he? He's in amazing shape. I mean, he's incredibly very vigorous 60. You just wouldn't believe it, I mean, they're so full of balance. I was hoping he was going to ring sometime in the next 24 hours just to let us know what time he's arriving on Friday. All he said was that he'd come down by car from Basse Yvette, he'd drive himself down. He just asked me for directions how to get here, which I gave him as, you know, I hope to God I, you know, see reasonably competent directions, because he's coming via Shasta, of course, which is a way I've never come. I always come down the A84 if I'm coming, which is... But I think it's pretty easy. He basically comes to a vowel and then goes up on the same way that the bus came. I'm sure he can get Fougere. But I asked him to give me a ring, and he also said to let me know whether he is ringing his wife or not, because he still hasn't told me. He said, do you, may I bring Madame Connes here? Well, of course he said yes, of course. But if you are, do please let me know, because I will need to, I'm going to have a rather lot of people in the house by then, and I might need... The wake arrangements are, you know, for you to stay in a small hotel, especially if Madame Cartier is, like I said, creature comforts. So, he said, yeah, sure, and I've heard nothing further. He also said to me, actually my wife doesn't like coming very much, so I suspect he'll just turn up on his own, which could be okay, but I just hope he's going to let me know. And also what time he's going to arrive.

27:30 Then you'll get Hawking, too. Yeah, yeah. He's old in his lifetime. That's right. It's not a joke with me that you really want him to finish what you do. It was on something he thought about. It was about his goal, his mind. Yeah. It was something he had to do. But it was still interesting. That's exactly what we did. That was a special time to participate. Thank you very much. Thank you very much. Thank you. Thank you. Thank you. Thank you. Thank you. Thank you. The important ones to me measure drop by a lot, but Houlan was there, Gravitch was there, basically every of the components. But did you not believe Houlan was there? Your very son was out there for a little while, Tom was not officially there, but he was with Houlan. Houlan might have been a bit too young to have been a reference, but maybe not. Well, Houlan was still in Poland at that time. Houlan went to, Houlan went to the States. No, I think he's a bit earlier. I've actually got his collected papers in the library, but I haven't got to do it so I can put my hands on them.

30:00 He's definitely born well before 1917 because he's publishing stuff in the late 20s. There's one paper there of his from about 1929, something like that, 28. It's only been published at the age of 11, so I think he's probably more likely to be about 19... I think he's actually born about 1907 or possibly 1910. He was already on the track as a large cardinal in the mid-30s. So he's got to, so he's, he's, he's in his mid-twenties then. I think he's born around 1910. No, no, I think it's even slightly earlier. I think it's possibly 19, sometime around 1910 or 1916. I mean, I like Alexander. He's been a tremendous source. That's in your paper too. I forgot that. Was it just a personal thing, or was there some... Yeah, for all anybody can see, I'm the only one who can do this kind of stuff. For all I can see, they have patented, or royal patented, whatever it is, in Amsterdam. And so you see, they're working through that year. And then Alexandrov starts his campaign of saying, Viatoris wrote this paper published in 26, very, really beautiful, the first systematic use of group theory in homology, merely expanded on some remarks of Brauer. And by, what, 15 years later, Hoppe is saying that as far as he knows, he may have been the first person to use the phrase homology group in print. Well, Hoppe has to have known about this Viatoris paper. I don't think Hoff is lying, but I think he's been very influenced by it.

32:30 Economical with the actuality is the phrase that they've censored the language. There was a strange talk at the Lille meeting by a guy who wanted to say, or at least wanted to cast a suspicion that Brouwer had stolen the fixed point theorem from some other guy. I didn't think his evidence stood up, but it was that there was supposed to be this guy who had discovered it independently, didn't publish because he didn't have a satisfactory proof, At some point, the assertion was that it had been available in the Amsterdam University Library at a time when Brouwer had been employed on a kind of summer job rearranging the stacks. I mean, it was really so circumstantial that it was ludicrous, but... So it wasn't sent to Brouwer? No, no, nothing as obvious as that, otherwise he would not have done that. So I thought it was a silly allegation, frankly, but there was some sort of discussion afterwards. Well, in any case, how the hell can you accuse a... Solipsist of being a plagiarist. He can't read that one. Mens rea, as the lawyers call it. Even if he actually saw this guy's paper and copied it, he still can't have thought of himself as having done anything wrong. Well, what interests me about this whole paper, how do we be amazed if it's a good one? I didn't think it was a very good one. I'm trying to think of a good one. Okay, well I seem to have found out how to get in my emails. There isn't anything much there. Yes, in fact, it was exactly what Jeremy Gray was at this meeting, he made exactly that point, that there had been a whole history of fixed point theorems and that this guy, and of course Brau would have known about some of these earlier proofs, sketches, proofs, everything.

35:00 For a general continuous math, it's really a leap beyond. Sure. And also showing it correctly, people have been incorrectly proving that the Jordan curve thing for 20 years. It's actually not hard. I don't doubt that people even stated topological physics point here. I wouldn't really be surprised to tell. As I say, Jeremy Gray was very skeptical of this claim, but he made exactly the point that Bill's just made. Okay, shall we get off and get some lunch? No, actually we said we were going to talk just a little bit first about the running order, didn't we? Okay, let's... Well, obviously we'll continue in formal discussions and certainly after John gets here. As it were, meet formally upstairs with the, I don't know, the recorders on, I think, on, well, whenever Cartier gets here on Friday. Okay. Um, do you have a biro? I haven't got a biro or something. I have got them upstairs, just say to me. Sorry. Sorry. Oh, yeah. Have you got a biro? I've got five biros, but never a four. Okay, sorry, guy. And, um, this may be nice. So today's the 9th and so say we start on Friday. Let's not worry about when Cartier will arrive. We can work around that. So we should say Friday. How long do you feel comfortable with? We'll obviously have a kind of coffee break in the middle. We start around 9. We don't have to be absolutely rigid about these things, but it's probably good to have them in the world.

37:30 Yeah, yes, it's supposed to be enjoyable. So 9 is fine with me. I'm not going to spam there with a whip cracker or anything, but I'm just thinking that it's a good idea to have a mutual understanding of what, you know, when we start and what time it is. Okay, so we have breakfast between 8 and 9, we start at 9. Okay, that's a good idea. And I would say we break around 10.30 for biscuits and go through until about 12.30 or 12.45 and then break for lunch with anybody else. No, look, we don't, I'm all for light lectures. There's a possibility we won't have to. Yeah, okay, so I think we'll probably sort of effectively start again about 2, 2.15, and then 2.15 would take us through, if we give ourselves three hours, with a break, a coffee break in the afternoon, to around 5.30. So do people think that is a kind of, for the full working days, is a schedule? And then after that we start. No, I'm saying the actual discussions we should kind of timetable. We should assume that the actual discussions are going to be taking place within those hours. Go ahead and tell me if you think I'm talking absolute crap. Go ahead. No, I think the most crucial thing is where we're going to start. Yeah, right, okay, okay, okay, I mean, I'm all for keeping it freewheeling, and for you guys, I think you guys are probably going to keep going the whole day anyway. I think 9 o'clock is a good time. Okay, 9 o'clock is a good time to start. I like to get up and move around a bit. Move around a bit and freshen up, and we could even say 9.15 or... Okay, well, that's really the only thing we need to agree on there, just the time that we'll start each day, and roughly when we'll break through. Okay, that's fine. Okay, so does anybody want to, so that's Friday, which is... At McLean's house, we always had sharing at work. Oh, I'll make a note of that. Thank you. Life would appear with the sharing of other work.

40:00 It's all just a memory that we might do that one day. I remember mentioning this later to my psychiatrists. Good idea. I was thinking more of, you know, kind of biscuits and coffee, but no, that's fine. No, I think that's a very good suggestion. Okay, so... So that's the 10th, and you guys want to be in Paris on the evening of the 17th, don't you? You have to be in Paris on the evening of the 17th. Yeah, so you're both going at the same time, and I said John and Mimi, so we'll assume that you're going to be leaving, say, after lunch on the 17th. I'll get the arrangements to get you down there in a hotel near the airport, that sort of problem. Topically, at some point you and Angus, with obviously other people in participating, and Pierre would be one, I do want to spend some time discussing the whole background and the consequences of the technology programming. I think Leo would be less interested in that. Yeah. And so I think that should probably take place in the first four days whilst Carter is here. Ah, okay. Is it good for the first day? There's many aspects. Many, and we might even... We can't give you all the time, which is why we don't... It's okay, it's good stuff. Why don't we... Why don't we provision it for our people? The discussion of the Tamed Equality Programme has consequences and it continues, of course. Yesterday, that's fine. Obviously, that will

42:30 take us, I'm sure, in many directions, as you say. It has many aspects. I should try to persuade Leo that this is, after all, very important. Oh, no, I'm not against persuading him on that. I shouldn't simply assume that he's not interested in the report. Oh, absolutely not. No, no. I don't think so. If he turns around and says, you know, no, no. But I think if we talked about Eilenberg and MacLean on the day that he wasn't here... Yeah, I think that's a very sensible suggestion, and obviously we have to be flexible, and as you say, if the subject of tamed topology comes up, as it's almost inevitably going to do in many possibly other discussions, then, you know, I'm not going to say, excuse me, I'm the chairman, we already dealt with that topic, but I just think it's a good idea to have a very general heading of the general sort of... They're the main heads of discussion that we're going to look at each day, provided it's kept. As I brought up with you the other day, I mean, there's a question of various kinds of mission transfer that we'd like to take place. One of them certainly is based on the fact that we have two accomplished mathematicians who have an appreciation of the parts of history. Three historians in mathematics, which is rare. And two guys, you and me, who have a smattering of knowledge of each, facilitate this type of claim. Well, in respect of both of us, that claim is nonsense on stilts. Yours for the obvious reason and mine for its inverse. But yeah, we'll take that. So that's one sort of thing. But that should certainly guide the audience to know what really is math and physics. In other words, the particularity of historians are supposedly interesting in a general way. And Kant here is very interested in historical topics as well. Yeah, I definitely agree with you. So should we think in terms of having two or three of the days which will be mainly...

45:00 History days means analyzing the development of the theories and two or three days which should be we should certainly have varying degrees of you resolve and I'm going to talk only about history. Can you really do that? No, that's absolutely not my problem. I resolve and I'm going to talk only about functional analysis. Probably not. No, and that wouldn't be fair. People have various degrees in that way. Well, and various degrees to which they can bring some particular expertise to the discussion of specific topics, so, yeah, I mean, obviously it would be wrong if we were to spend the whole week speaking, you know, with just, you know, just talking about functional analysis, I mean, you could do that, but I would feel that the exercise had not been what it should have been. I don't think it even could have been. No, no, no, I don't think it could have been. Yeah, yeah, yeah. Any sufficiently deep and serious branch of mathematics inevitably brings in almost the whole of the rest of mathematics. So, Colin, do you want to make a suggestion? What do you think about... I know in my own histories of both your work and category theory in general, I slight Eilenberg. I'm just not as close to his way of thinking, and I haven't known him as much in a way. I'd like to hear a lot more about him. Do you see those as topics we could save for the day? This topic will just come in around the development of other topics naturally anyway. I mean, starting with that on our minds, of course. I suppose, more generally, we could have one day of surveying, you know, in detail, but at the same time with an eye on the pictures moved ahead.

47:30 The development of category theory as a whole could be divided into two sections. We'll have one pre what did happen, one post-1970. You know, even to have an agenda that has these topics listed for a day, without enforcing it in any way, having them listed on a piece of paper, probably. Yeah, that's the only thing I have in mind. That's simply all I have in mind here. I'm just still reeling from your suggestion. I think we should devote one day to those three topics. McLean, Attenborough, Allenburg, McLean. Same as to me. Devoting one day each to the three seems successful. Yes, I think that will be taking far too much time from other things that we need to be talking about, but one day for the, which of course, we could in fact perhaps devote two days to a general history of gap theory and topos theory, and with the first part of discussion, which naturally would be anyway, a particular thing in the general history scenario that I have expected, I don't know, which I think leads to the... Criticized and resubstantiated, but more, I've studied more. How did Grotendieck get from 1950 to this? Precisely what we were talking about the other evening. The world's leading analytical analyst and the world's leading algebraic geometer. Was this just some, you know, I have a theory about it, but citations of particular papers and so forth are sketchy and I think I've never managed to get you to accept it. The importance of complex spaces. Complex, yeah, in several levels.

50:00 I think that should definitely be a discussion. Again, Cartier will know a lot about that. Well, that, I think... The picture has to be substantially modified or rejected or harder to be accepted. You know, it's not just a matter of the past. Well, exactly. I was going to say, the whole point about such a discussion is that it would also very naturally bear on the future development. Because, as I keep saying, and nobody listens, every topos has its own punctual analysis. Remind people that there is this organic historical link, although there might be an encouragement to young people to pursue that general idea. Well, I'm not sure it's even more... His name is more constantly tied to it, but people use it in clear spaces. So it's a kind of very general rough heading. Grotendieck 1950-1960 and part two, the shape of Grotendieck's legacy, or how is Grotendieck's legacy shaping, that's the bigger topic, I agree. In the way Grotendieck 1960-1970, that's a whole problem. That's a lot better than algebra and Grotendieck 2020.

52:30 Which is what I meant by my... I think it's looking to me as if we probably need a Grotendieck day. Yes, so that would probably need to be the Sunday or the Monday. Just preparing questions. Well, this is another reason why... Yes, exactly. Well, that's perhaps something we should... We only have a few days. Unfortunately, we have four days, yeah. He's only going to be here for four days. I think he might, with diplomacy and Captain Charlemagne being, as I say, hopefully feeling happy in Fougere, be persuaded to stay a day or two longer. I'm hoping so. And if we just go into talking to him without an agenda, then we'll be treated to some incredibly interesting stories about the Algerian war. Yes, we might just get trapped in anecdotes. No, no, no. Those are the kind of things we can keep for after dinner in the bar in the evenings. No, I absolutely agree with you. Those are the sort of things which should definitely not be kept. Anecdotage of all kinds should be kept out of the serious discussions during the day. The kind of informal chatter in the evenings, but on the main, when we actually sit down. Evenings are defined as informal. I'm trying to stay from absolutely defining things rigidly and ossifying them, but I think that... But where do the tape recorders go? While we're upstairs in the... I mean, this is a relevant distinction. Yes, yes. While we're upstairs in that room from 9 o'clock in the morning onwards, with your permission and agreement, of course, for that. Yeah, yeah, and I've said that to Carter, and he said he's perfectly happy with that, yes. I was videotaped to Berkeley, I got myself by telling a humorous story about Sarah that I thought, not a video. Nothing wrong with it. Yeah, yeah.

55:00 Kind of distinction I've got in mind, yeah, yeah, exactly. Okay, so we're looking at Eilenberg, MacLean, MacLean, Eilenberg, Grotendieck, 1950 to 60, 1960 to 70, and more generally. The kind of shape and direction of his legacy and the ideas, the many-sided aspects of them. The exomatography theory of France now. Yeah, those are obviously all things we need to talk about. Is that a part of his legacy? Yeah, well, again, something which I know Cartier has views. Yes. Yes. Yes. Yes. Yes. Yes. Yes. Yes. Yes. Yes. Yes. Yes. Yes. Yes. Yes. Yes. Yes. Yes. Yes. Yes. Yes. Yes. Yes. Yes. Yes. For the discussion of term topology and O-minimality and their bearings on the whole kind of Reese and whole issue of bad infinity and its consequences, which I think should also lead us back into a discussion of turning points in history, particularly in the 19th century, we will have easily three, possibly three and a half, four days already. I think those would make, I'd much rather have those topics within real debt, even if it means. And, you know, having to cut your arm off and not have time for the discussion of other topics than to just deal, just when it's getting to the most interesting. So why don't we look provisionally at looking at that for the three, say, for a three and a half day agenda, which already takes us almost half the way through. We may have one day's outing. Did Leo have suggestions? He's not as involved with the project. He said he was going to send me a much more detailed list of suggestions. He never did. He did send me. Yes, one thing he did mention was a discussion of the, his interested historian taking the lead here, a discussion of the anticipation of major category theoretic ideas in earlier 19th, 20th century algebraic.

57:30 Clearly people like Cayley, and particularly the development of the browser. And also in the increasing recognition of the importance of duality principles. To give those examples, if he thinks there is some general linking account of how growing recognition of the importance of duality principles was connected with the Pfizer category, then he has to present that. This is stuff that he just said. Somebody said to me on the phone, I was asking him for a suggestion. Did Cartier also mention that? Yes, Cartier also mentioned that. In reality? Yes, Cartier also mentioned that in passing. Oh yes, whether Alexander duality is related to ser-duality. Yes, yes. The word duality is used for both. Yes, yes. Exactly. It could fantasize a relation. The topic for after Leo gets here, and fortunately he will only overlap the most by one day with Cartier, but may not even overlap by one day. So I'm not quite sure what time he's getting in or what time card he's going to go that day. But okay, so it's 19th and early 20th century algebra. And it's roughly blocked out three and a half or four.

1:00:00 Which still leaves us with one day for a complete break. Yeah, I was going to say about... Well, I don't know, I wouldn't have a discussion about it. You did this more about... I mean, just as a kick start. Yeah, yeah, yeah, yeah, yeah. I'd be 100% for that, but I'd rather Angus said it, because I know you've made it very clear you don't want to be made into the center of attention as part of an insider discussion, but of course I think we should have at least a day on what you and Steve have been talking about. There's another aspect. We hear about anticipations of category theory. But there's also in foundations, in Contra and Dedica. Oh yes, very definitely. It's very definitely part of that, yeah. The whole business around set theory being still a cantor in Dedica by all the fanatical followers of Dedica. I think you should definitely lead such a discussion. That I think is also something which is possible. We should have Wacartius here, because I know he also has. What do you feel or do you think? I don't know. Well, I know that Leo would be interested. Yeah, okay. Well, we can keep going. But I think we should definitely have such a discussion. And specifically also on the historical side, it would perhaps naturally tend to, that might even form a natural bridge from the discussion of topology and the consequences of infinity from the historical side. This is an agenda I've had later before. It's not categories of set theory. And there's the Ponder's broader sense of set theory.

1:02:30 I don't say I have to philosophize, I don't understand it at all, but this is the wrong opposition. One way I've been putting it, comparing set theory to category theory is like comparing apples to fruits. It's apples and fruits. So what do you mean by set theory? As I said several times, there are at least seven discernible meanings. I agree. Well, perhaps in the context of a discussion of, you know, how Cantor and Dedekind's ideas got, you know, by these guys, by the Pregapiano line of development, the Mello line of development, we could also... Enter into a discussion of precisely those different senses in the theory versus something else and get absolutely explicit on the relationships and how a clearer explicit understanding of those relationships could serve the right direction in which. Spattering of knowledge may be a starting point for a more serious search for knowledge about both mathematics and history. Absolutely. There are some places where a smattering of knowledge is actually a good starting point for discussion and some where it's a very bad one and I think, you know, that discussion is spot on. So that's okay. So that's clearly another topic. This is a big question about set theory versus category. The difference in their minds is set theory is what I must know because I think I had it in grad school and category theory is what I don't know because I don't like it. But they don't in fact know either one. And yet they've got strong opinions. They don't know each other in any sense. And yet they think they know each other in the only sense they could possibly have.

1:05:00 It's terrible. But here's all the seven senses. Well, to the extent that they can even be said to contact any of the seven. The other smattering is the correspondence between Frege and Coussereau. I didn't either. I still don't know. Is it in play? Yeah, there's a collection of Frege's correspondence. I know nothing about Husserl. I said I had three courses on him with Bob McIntyre. And Albert says, well, that's why. It reminds me, I promised that I'd ring Albert whilst we were all assembled here, just so we could all say hello to him. Say, sorry could be as if we could, can you remind me to do that tonight if I forget? Cantor, Dedekind, and how their achievements were, shall we say, annexed, distorted, in the 20th century, and the senses, the distinct senses of which I can speak of set theory versus something else. Well, you've got your emails on there. You've posted this on the Catalyst in discussion, the emails which I've seen and which are easily... I can't get out of it that easily, but I think form a natural part of a, you know, is it worth the retrospect, the kind of historical overview that you'd probably present as part of a, you know, general discussion of your and Steve's work last year, or do you think we should make it a separate one? I think separate. Separate, okay. Suggestions for discussion. Yeah, separate suggestions for discussion. Yeah, okay, well I've got it down here in that case. I think you're right, that probably is a day where we do need Lauric Leo here, so that should be after the 14th, that should be the 14th of, yeah, and I think Carty is probably going to be most interested in piano, well, the thing is, we won't be able to, yeah, yeah, yeah, actually, tell him that.

1:07:30 But wasn't it Frege, didn't Frege explicitly dismiss him in writing as a mere algebraist or? Vatican and Schroeder both did. I mean, technicians were the ones who understood him. They definitely didn't understand him in the way that Frege and Herzl did. He was spot on the ball there, he was better. Definitely hit the nail right on the head there. And perhaps we should have a broader discussion of the... The distorting effects of that conception of logic as something which has to be put in place underneath mathematics vis-a-vis the increasingly powerful recognition and ramifications that really all of these come in also, of course, in an exposition of the Grand Programme legacy. And obviously these themes are bound to weave in and out, but I think just having a rough... A series of half-day headings is going to help, you know. Okay, well I've got notes, I'll write that up. Just type a little, there's a provisional agenda given to you and then you can tell me.

1:10:00 But I'll do that tonight. Shall we go and eat lunch? I imagine we'll rebuild it anyway every couple of days. Yeah, yeah, sure, sure. Once John Bell gets the book. Yeah. I thought it would be good for us to sit down for half an hour and have this discussion now, so at least we've got something there. Otherwise we might end up in getting pushed in the direction of everything-ology. That's everything-ology. There's bound to be a bit of everything-ology. Great, you've made an excellent suggestion. You can take it to my room now. I've got loads more where they can come from. Okay, um, it's because you probably want to have a little siesta before we drive to the jungle. Oh, I know, it's really weird. Oh, yes. No, no, it is. Well, I hate to say it, but he circulated a lot. I don't know how far he circulated. I saw something he'd written about, you know, which actually provoked Adrian Mathias to sending out an email. You may have seen it. It's on the phone. Say, what did MacIntyre say in Sicily that so outraged on it? Did you see it? I didn't see that. Well, I can show it to you. What did MacIntyre say in Sicily? That's a very good talk. I think it was one of the reasons for the names of Christ. Exactly. Yes, it was about Christ. It was just an attempt to say what Christ had actually been saying. Well, you're not quite there. But it had this amazing stuff, like MacIntyre betrays logic.

1:12:30 That's wonderful, you can attack from both sides. That's true. Because, of course, Harvey Friedman, the foamer and the foaming... The foaming, foaming. ...the foaming thing, attacking stuff. But how dare you apply logic to geometry? You're a traitor. Yeah, yeah, yeah. No, no. But tell me about Van Gogh. He has a low flashpoint. Well, yes, notoriously he has, but not sadly. And he was in pretty, yeah, I mean, but he got really, really angry at those. Yeah, he did really. Yeah, is this okay for sometimes? Sure. Hello. Bonjour. Bonjour, mon petit. Ça va? Bonjour. Bonjour, monsieur. Ça va? Bonjour. Bonjour. Ça va? Hello, my name is Mike. I am an Englishman. I am a friend of my father. Hello, my name is Mike. I am an Englishman. I am a friend of my father. Hello, my name is Mike. I am a friend of my father. Hello, my name is Mike. I am a friend of my father. Hello, my name is Mike. I am a friend of my father. Hello, my name is Mike. I am a friend of my father. Hello, my name is Mike. I am a friend of my father. Hello, my name is Mike. I am a friend of my father. Hello, my name is Mike. I am a friend of my father. Hello, my name is Mike. I am a friend of my father. Hello, my name is Mike. I am a friend of my father. Hello, my name is Mike. I am a friend of my father. In fact, we could even go down into the terrace, yes, yes, very nice with the terrace. Parfait, ah, bonjour, bonjour, bonjour, ça va, ça va bien? Mon ami Bill, mon ami Colin, et mon ami Angus. It's a bit in demand at the moment, because he was a very close friend of Crick. That I did know. William Crick? Francis Crick.

1:15:00 Crick gives him a lot of credit in this pursuit. Yes, that's right. Well, they knew one another for a very long time because they were both working in the operations branch of the Admiralty in World War II and doing research for the British naval effort. It's an extraordinary story. I don't know if it's in this book or not, but Craig said he first had Friesel's name. Because apparently there was a file of Friesel's locked away somewhere. What if you devised for mining the Baltic using ideas of Wittgenstein? No, I've read about this. Now the story gets better because apparently this was sent up to Churchill. It was actually read by Churchill, this file. And apparently... No, it may be in there. Actually, Crick has a very nice article in there. It's one of the nicest articles. It's a short article and it's... It's very, you know, nonsense. I found it very, I wasn't paying attention. It's, you know, there are other more dramatic things, of course, like Verena's article. I mean, but this one is quick, writes very nicely. Pretty mature. Yeah, it's interesting. Yeah, yeah. I never saw anything like that. Yeah, yeah. I've already heard. Yeah. Her article, I've never seen anything like that. No, I don't know about this. You'll have to fill me in. This is obviously anecdotage, but we're off duty now, so tell us. Right, we're off duty. A couple of Kreisel anecdotes I could tell myself, but I'll let the... The Verena, I mean, the Verena thing, I mean, he and Dyson had been at Cambridge, maybe even at Mount Poole, I can't remember, but, no, I think it was near Cambridge, and he came, Kreisel eventually came to the Institute of Science for a couple of years, Girdle said, and he eventually became involved in a layer, so Girdle was apparently very annoyed at the same time, but the Kreisel, Girdle said... But this wouldn't, I mean, this wouldn't happen if they had brothels of his appearance. I think that was Kreisel putting his own words and his own feelings into Gödel's mouth, which is certainly not above doing.

1:17:30 Verena wrote in this Kreiseliana, I mean, a very, very explicit, open, I think, account of Gödel's voice. Have you read the Tarski book by Verena? No, not yet. I mean, I volunteered to the Lieutenant Fairbairn Towers. That's in the book, too. What did you think of the book? Well, she was in awe of the writing in this book. I mean, she wanted to tell you what a great man he was, but she seemed to take the free route. So various things show that they are all elementarily equivalent and or show that they are decided by the user. Implicitly, now, one would be looking for some analysis of what kind of definitions you can carry out more frequently. Now, of course, in the free semigroup, I mean, this is just basically girdles. I mean, Quine did it, I guess. The theory is definitely undecided. You've got your syntax already there, basically. And so you can then go from there to get girdles. And actually, just to read your parametrization and recursion theory. Yeah, exactly. But the moment you put inverses in, and you've got the... It's a very different game. Tversky himself didn't, a very little was ever done until the Russian combinatorial group series began attacking in the 1950s and so on, and eventually, for short, with very, very great difficulty, there was an algorithm for designing other equations. Which equations of solutions are there? So you would allow parameters for the generatives of the group, and then you would say... Ah, merci à vous! Ah, c'est très gentil. Ah, merci madame. C'est bien fait. Oh, nice and cold. Ah, merci. Excellent. Thank you.

1:20:00 And there were various other things. I think it eventually got around to showing that all the groups were maybe universal existential equivalents or something. None of them are straightforward. But eventually, in recent years, it's more of a sort of law. Well, I mean the program for getting at it. I think it was an interesting thing also, coming out of computer science, that these semi-groups, they earned quite, they tanked it, they didn't prove it, but there was some kind of notorianity for inside three semi-groups, or three groups, for descending systems of equations of this kind. And it was proved, first of all, by... And then a more conceptual proof is given by Russia. This is how I deal with three groups of generators. Representation by two by two meters over the interface. This comes really from this. Really from this, the notherian, you can get, you can then get, you can translate, you can show this notherianity. But then we're after three groups and get it for three semicolons because they're in bed. It comes from the fact that you're dealing with... Every equation you take translates into an equation about two by two matrices over the integer. The answer is sort of a theory that's an algebraic geometry problem. I'm just clustering the data together. So this was the first hint that there might be some connection between algebraic geometric ideas. But that still was a red nape.

1:22:30 This guy who was involved in the claim that he saw this cool device, brilliant, brilliant mathematician, but also someone who was really wacky when he was a kid. He had been a leading dissident in Russia and set fire to himself. Anyway, he began to give more ideas about, as he put it, algebraic geometry over three groups. He made some hints. Now these were taken out by a man, a truly brilliant student of his, Sella, a real Sella, basically found a technology for translating issues about the fundamental groups and so on of surfaces. I mean there's a technology that the Comintern group is well aware of, but basically these guys got a technique for translating with what appears to be a proof that, I mean, first of all, from this point of view, there's a thing called the GSS decomposition, Jacob, Peter Schellen of Chicago, Johansson, well, I mean, it's geometrically motivated, but you can prove it combinatorially at the end as well. I mean, it's about seeing that some things that he has in the end come up with an alleged proof. There's only 600 pages which parts were published, but the three groups are all of an entirely equivalent, but not claiming yet that they are all decidable, and also claiming that there are, it's now claiming, but it's an example of a stable theory, in the sense that I was mentioning to you as well. That is not, I think, claiming itself. That's, that, there's another pair, the Russian Miasnikov, whom I've known for quite some time, an attractive personality, and, um... But not in his previous work, not in various stages of mathematical addition. He teamed up with Olga, she's a serious combinatorial group theorist from the classical period, and they've also claimed this result.

1:25:00 I mean, their proofs were hopelessly unanimous, but they've managed it. At least the last ones I saw were focusing on that. They managed to get them published in the Journal of Algebra, but Zolmanov is the editor who accepted that study. It feels weird. But he clearly has not looked at these papers properly. Clearly claiming decidability as well, and I don't think anybody knows that can believe it. But it's a very, very unfortunate result. I mean, it's fairly clear that Vercela has made a move. The problem was clearly far too difficult to be solved in the 60s or 70s. This is the kind of thing, she was doing natural things using the wedge functions and so on, but the problem is, with these wedge functions you typically get undecidability. It's very finely bound. You've really got to work in the language of root theory and nothing more. The more you try to put in something else that appears geometrically natural, you get undecidability. Positive measures and positive functions. This is a relative triviality, but you get these kinds of things, which aren't going to be positive, we're not going to get any of that. I've been aware of this. Jane Dana Scott, a molecular buffalo, buried an SL2 of N, which shows that it's exactly 3 million and 42 nanometers. The term at one condition, you just transpose the negative term to the other side, which makes sense for the positive term. So SL2 of M is or is exactly the same. If you could convert your negatives, you're going to get SL2 of Z, which is a marvelously well-structured proof, but certainly it's no longer a faithful representation.

1:27:30 This is interesting. There's a free group in there, but it's got a complicated structure. It's weird. Although this could be a part of the free group, which is comprehensible, even if the whole thing wasn't. I can't remember the exact formulation. I'd have to go back to it now. About four or five months ago, Martin Taylor asked me a question. Which had come to him from Serre. Serre was interested basically in something not completely, it wasn't about, of course Serre has made major contributions to the structure of SL2 by this analysis in terms of trees and so on, so you can show that these groups of amalgamated free products know what they're doing. But he asked a question basically just exactly about the determinant identity. It was such a simple question I'll probably never be able to answer. I'll have to figure it out. But at this moment it's a simple machine, I think. This talk with Dana blew my mind at the time. I was already interested in Riggs, but never in Kerr. Yeah, that's beautiful. That's how two of them even make sense. In Kharkov and Islandburg, there's a homology of monolingues and a homology of grooves. If you look at the groove associated to a monolingue, that's functor, that's left adjoint functor. So, in other words, I guess what they said led this partial student to think, well, how do you understand that what happened in the long run was to apply this function, and of course, that's wide enough to drive whatever it is. I mean, I don't know what was Tarski's motivation, really, but it's been really so hard to prove anything. I mean, in actual fact, it's a move. I mean, nobody knows really whether a corresponding thing in a rhythm theory makes a difference.

1:30:00 I mean, it seems to be taken seriously that the Hilbert-Sent problem is decidable. I mean, it's not proved, and it's unlikely to be proved, but I mean, the method used for the integer simply breaks down. The integer thing depends critically always on solving it at a discretely ordered rate, and the equations, moreover, the equations that are used in... She showed the undecidability of the elementary theory, and it's an interesting point, but you need three-quarter of her changes, I mean, that situation, at least we believe you need three-quarter of her changes, her definition has never been improved. I mean, it's a global principle that ultimately appeals to quadratic forms, things around quadratic reciprocity and so on, infinitude of primes and progressions. It's tricky. No one succeeded in taking out one quantifier. Maser has conjectured, he's not the only one who has conjectured, but he gave a geometrical reformulation that you cannot define the image as existential. I guess there's a general belief that you can, and quite often very serious people suggest that the evidence for undecidability in the case of the rational is not. One has never really proved anything about it, except that Julian Robinson came up with the quantifier statements. Another example we're just adding here, enormous. I'm going to try this. Are we going to go both rosé and... No, no, no. Well, okay, you go for the red. I was pouring the white first because I thought that might be your... No, I'm actually quite fond of rosé. Okay, me too. And Bill, how white are you? Many of my accomplices don't go for rosé. No, there's a silly snobbery about it in England still, and I've never understood why. I've never had a problem with it at all.

1:32:30 Now there's a silly English snobbery about it. Except when you get further south, you can drink it when you get down to the Mediterranean, but anywhere north of Lyon, then it's very naff to be drinking rose. It's a very silly approach to history. In fact, it's a pure piece of nasty class snobbery because the perception is, you know, macchias rose, the sort of thing that, you know, people in... The first wine that the middle class in Britain discovered in the 1950s. So all these dreadful snobs look round their noses. So there's this disgusting kind of, you know...