FW Lawvere — Tame topology program

0:00 Oh, sorry, that's fine, yeah, that's in here. And actually, that's what happens with the contraction and stuff, but it does it inside of this. Actually, we did it already, the point of closure. That's actually what I revealed, the eternal mechanism of the universe. Yeah, yeah, yeah. Actually, he was talking quite a little bit about some of the work that he and Berdian had done this morning, but this was in the context of And that to understand the homotopy categories, you do want to go to the derived categories, and to understand the... mostly the homotopy category is really a mystery, because it's not at all a category with nice properties, it doesn't have well-defined categories from the point of view of classical category theory, and yet obviously it's absolutely essential, so that suggests that there is something about general category theory that leads into the framework. There's this whole program of derived categories and these kind of triangular categories, which I just don't know, that gives you the way of analyzing the notion of, essentially generalizing the notion of weakening it, that seems to connect with what's going on in non-quantitative spaces, because Bill just wants to say, well, I need to know what that's like. But then he has got a story to tell about where he thinks the non-conceptivity is really coming from, where he thinks that whole derived category of programming is wrong, and that there is a way of doing it in the right way, because from his point of view, all that is preserved, you know, we've got these pros and cons. There's actually a bit of a context, like you do those special bi-categories.

2:30 Yeah, which is very much his view of them. He doesn't think of them as things that are in my category. He's always conscious of them, they just naturally live sandwiched inside the levels of my category. No, no, that's the way I think of it. I'm actually going to talk about quantum enrichment. It actually takes the both positions, but the quantum itself is sort of bi-categorical. Well, I'm really looking forward to your talks, it's going to be very interesting. No, I just, I included all the elements which I talk about in the essay, but I do like, a way of, that's not sort of a provocation, I just bring it all as a way of thinking about quantum mechanics. Body enrichment and stuff like that. But it's a hard time to think about this, it took me long, I've been thinking three months about this. It's tough stuff, it's tough stuff, it's something like me. Yeah, and that is going to be a challenge, because we've actually had to restrict the talks now to 50 minutes, although I suspect they will overrun. We guessed they will overrun. This was my fourth flight this week. Where have you been? I was in a radar for PHDs. I was controlling a rich category. I think I just barely remember the definition of a quantaloid. That was, um, yeah, that, I'd like to know a bit more about that. That was originally, um... Which of Bill's students was it who did his PhD on quantilizer? Rosenthal. Yeah, Rosenthal, yeah. But he did his PhD on Etan Du, that's right, but he later did work on quantilizer. Yeah, he has his two little books on quantilizer. Yeah, which I'd like to study. I'd like to study that stuff. But I really wasn't. ... when I was, you know, long, long ago, when I was, you know, I'm seriously still trying to stay in the field, was the earlier work that he did on non-trivial 8-armage, because that really did seem to be very... Very interesting from the point of view of this whole program, which obviously is very much a Guiding idea of books, that Walsh's set theory is essentially just derived within this geometric picture in terms of conditions on the components and seeing the sets as just the co-discrete spaces.

5:00 But then that naturally leads obviously to the question of when you've got actions on the space, and so stuff is being kind of moved around on the action so that it becomes impossible for the space to be in any sense of kind of underlying itself, and that I find is fascinating just from the point of view of seeing how the How all of this emerged from a geometric way, I think, you know, of a structure, and Etan did seem very interesting on that part of the unit. I never understood enough about them, but so... So just in terms of the way that the conditions on the caverings, you know, the localization of the caverings, that the condition broke down in an interesting way that was global. She didn't, she's just come along because she's the boss of the department, she's not involved in this. I know her actually very well. No, she's not really, actually. Well, Alberto felt that he had to ask her to come on the panel because she is the head of his department, so I think it was more a kind of internal... Well, because actually one of the things which is hard to avoid in my talk is sort of slamming her always like this. Yeah, yeah, yeah. Because I'd be careful. That's going to be my... I know it very well. Yeah. Well, I was a bit surprised, actually, that she decided to come at all, because it's absolutely not fair to the whole traditional quantum logic. Yes, because obviously your stuff subsumes the whole traditional quantum logic viewpoint into a category, which is exactly how you should subsume. I don't even know how many of the talks she's going to be coming to, but she was at one point going to give a talk, and then she backed off, but she is still going to be coming on the last day for the panel discussion. I don't think so. I've had one or two runs in Witten in the past. I think it was, I mean obviously that was Albert's side of things he has to keep on, people in his own department, so that was just a bit of that.

7:30 And you know what it's like in the Italian universities. Well, I'm in the, I have a position at the moment now, as I said, living in France, very nice, in Brittany, in Puget, in a lovely old townhouse just across from the Chateau. But I'm completely vegetating and I don't really know what the hell to do. I've got to find, the main thing I need to do is to get a job. There's a lot of writing from some of the academic classes just for this kind of copy-editing work, or possibly just some teaching, some supply teaching. It's a lovely place, but I didn't realise how isolated I was going to feel in there. That's always the problem. I'm happy I'm doing much better than I did yesterday. Yeah, you seem to be really flourishing. Everything I hear about you, Garth, your things are going really well. Yeah, yeah. I'm really pleased. Well, if you ever get to the point where you can attract, you know, sort of loads of funding, try and interest them in having a meeting in Fougere sometime. No, seriously, I mean, I have a beautiful house that sleeps about 12 people. It's got a huge room on the floor. It's got a huge room on the floor, which is literally about the same size as if we were in Galilee, where Bill is talking tomorrow. In fact, it's actually a bit bigger. It's got to be exactly the same stuff, so I mean, it would make a perfect list of things, and it would be, I'm quite serious about this, it would make a perfect place for small meetings of, you know, small reading weeks or just short three, four, five day meetings. I was thinking of trying to get Freddie Van Eyst here, and one or two people there, maybe for a kind of four or five day meeting on non-cognitive performance. Sometimes I don't even have any questions, but I have like a dozen questions because I have no budget for all this. I don't even have a budget. It's very easy to get there. And here you can show me a range for all. I could certainly arrange it, but the only problem is what I can't do is actually sort of back down my own talking any longer. This is probably the last one I'll be able to do now. But on the other hand, I can certainly provide a very nice venue.

10:00 You wouldn't have to pay anything for the... And of course you can travel extremely cheaply to Brisbane at the moment. Cheaper airline. Have you been to Berlin? I don't know. Oh, easily. I've got, well, by the time I finish doing it up, it has six federates of which. It actually has eight, with two of them in my library. And there you can put all of them in. But the ideal size would probably be about six or eight. I don't care if you look at me, I don't care if you look at me, I don't care if you look at me, I don't care if you look at me. There are 12 people, but because McGill is over there for free, but still you have to do all the airfares. Well, if there's anybody seriously that wants to use my place, I would be. I'm absolutely serious. I'm absolutely serious. I'm absolutely serious. It's something to really give me a sense that I have something, you know, going forward to do. I used to organize workshops every year. Well, that was what I was doing professionally anyway. I mean, conference organization and travel generally. This is my business, but of course, as I was just saying, John, the travel business... As long as you don't pretend it's geometry, it'll probably be... Yes, so it's sailing close to the wind, but definitely as long as we don't... It's biocentric. It is, although it's almost biocentric, and it's a beautiful algebraic structure, but just don't go around saying that there's spaces over there. Actually, I can't believe it. Who are John and Cole? Well, I'm wondering. They've just tried their rooms and they're not in. They said they were going to be back here around six, which is, you know, any time now. I was going to wait for Peter Schuster. No, it's that guy who just went past. No, no, no. Well, that's somebody you just got speaking to. Peter, you'll recognise him, you'll identify him. He's about the age of... Is that this gentleman?

12:30 You don't know him? No, no, I know the name. He's been corresponding with me. Yeah. He's on the list of people who... Who else is... He's not speaking, but I haven't got the list of non-speaking participants. Oh, about 30. Yeah, about 33 people who emailed to say they wanted to register. Well, originally we did, but then we were told... Well, two things. First of all, we'd only get funding from the University of Florence if there was an open meeting. And I mean, I approve of that in principle anyway. I don't believe in closed meetings. We don't get meeting funding from the University of Florence, which we badly needed, because, you know, one's bank balance. So, if it was an open meeting, so, and we obviously checked with Bill and that was fine, and they had the places, you know, they were big enough, it wasn't going to be a problem over the, as long as the actual number of speakers didn't increase uncontrollably, and we got, we actually ended with 17 people on the list of speakers, but a couple of people had to cry off, so we ended with 15, which is what we'd originally planned for. Of course, at that time, we had expected it would be a longer meeting, we expected to go on for at least seven. We had a lot more time for discussion. It would have been a lot more leisurely. Then we had to rejig the timetable to cram everything in. But then we got a bit exhausted. I know. The trouble was that having invited people, we could hardly de-invite them. I remember when you first mentioned it. I forget when it was. It was a long time ago. I had the impression it would be long, but it was a smaller group. Yeah, well, originally we were looking at 12 people, and then it rose to 15, but that was going to... ...projectiles, and you actually saw it much, much more cleanly in terms of limits and co-limits, in terms of real kind of categorical framework that anything that Cochran and these people are doing, much more cleanly. And who saw all the bearings of computer science. And this guy, and on general category theory, and this guy is seriously smart. And he just pissed it all away because he was a hopeless gambling addict. The second guy in my life that I've known who's totally destroyed himself. But you don't mean he was really in other rooms? Totally destroyed himself. Did you know him before?

15:00 No, he's not in my introductions, I think. No, no, I knew him way back in about 1983. That's right. No, I knew Jerry Kutcherian well before I knew you. Well before. I can't thank him. No, I'm not suggesting. No, I knew him well before he knew you. Jerry came to me. It's really sad. And extremely bright guy. And he did some nice work and he published journals and I do have to, I mean I can claim responsibility for the fact that Jerry was able to publish in the Journal of Pure and Applied Algebra because I actually, you know, I actually helped, I did actually help finish with, even with my pathetic limited understanding, was able to clean up some of the stuff in his papers and force him to finish them. And send them off to, you know, to Anders and to... It's pathetic! Here is a guy who's utterly wasted his life. But, but, but, but, and now I'm, sorry, I'm in the position now where I'm literally having to ring him up and say, Jerry, you know, I know you're just working... But you told me... Could you find me a job? Yes, but, but, you know, and when you gave him a job. Yes. I'm in that position now. I'm in that position now. I'm literally in a position of saying to him, can you give me an accountancy job with your sister's bloody clothing factory, or do you know anybody who might be able to give me a few dollars a week? Language teaching. He is a smart guy. He's still alive and kicking. I remember I went there often. I sat and listened to him talking about a billion categories and exactly a billion categories and the whole, and this guy understands the whole development. Of category theory, ever since, you know, the Eilenberg-McLean paper on natural... This guy actually understands conceptually how the whole subject developed up to adjoint functors and beyond. This guy could give a course as... This guy is as knowledgeable as Colin on the general history of category theory. He's got a fanatic for notation. Yeah, he's got various weird obsessions. He's a really good guy. I introduced him to Nick the Greek. No, my old friend.

17:30 My other friend. I have other friends. I'm sure you do. When you say, What is your word? Sure, that's what it is. Otherwise, you do something else, at any rate. You're related to other people. Most of the people at the university were obviously fascinated with their own intellects. You are an extreme case, isn't there nothing else? No, no, no, of course they are. Look, in the case, and then they're so shocked, you know, in university when, oh, I don't know, people become entrepreneurs or business studies. It's true, it's a bit of a war, but that's, oh, surely you're only because, come on, because. You're absolutely right. To a first approximation, I have to be, as always. That's the point of my career. And then, of course, because this isn't terribly interesting anymore, then you attract, you talk to Andre Bauer, for example, because computer science- Who I don't even have time to talk about, but he- No, no, he tells you people that attract to computer science, you use different metaphors and you do all this stuff, I don't know, but I was still stuck with the old- Yes, of course, but the point is, surely, surely you know. It has to produce something other than yourself. You have to be some kind of artist. I don't know, produce something other than yourself. If it's all purely... I don't even think Wittgenstein left a trail. It's true with other people, not with many publications. You certainly put your finger on the atasolicism of Wittgenstein, which not many people notice. That's very well thought. That is what is so sick of us. No, no, no, there's a very important point there. That's exactly why Bill, I presume, Bill, I don't know, he thought... There was a very serious point there. He didn't, he didn't, one didn't just want to be an ideas man.

20:00 You have to leave a public trace. Yes, but not under some egotistical obsession with leaving something behind after you're dead. In his case, it is something a great deal more than that. Oh, surely it's more than that for any of us. Well, for a lot of people it's just that and they're more. I know, I know. That's sad. It's the law. It's the one thing that makes you feel superior to anybody else. That, also, is so sad. For me, I don't know, it's just, have you been superior at all? In that respect, there's also, I don't know, it's... How can I put it? The best thing to do is play the piano or be a musician, you see, where you just have the moment. Because every performance is just complete to itself. That's what's really going with you, isn't it? Of course. From where I'm coming from and always have been, but we haven't really had this conversation until now. But the reasons cannot lie with the idea that there's something that's going to survive the individual consciousness. I've never believed it. I've ceased to believe that from the age of ten. Yeah, well I've ceased to believe that also. We're all atheists. If you actually ask them what the point of all this is, you'll be given various blah blah and say, no, this is important. But the fact is, it isn't important because we can't do anything about it. Thank God we're having this conversation. No, no, I'm not saying come on. No, no, if you ask, no, what you're really looking for, and what you're looking for is for the moment, I mean, sorry, the immediate, the fact, the contemporaneous. Yes, but there is the possibility that the immediate fact, sufficient to itself, in some sense remains a part of reality forever. Mathematics may be the vector, you see, by some chance of the representation of those things We're talking about the metaphysical, we're talking about, you know, we're talking about how time fits into a metaphysical understanding of the world as a unity And of course it has a bearing on mathematics.

22:30 I very much... Mathematics will probably continue to describe something that is still there after I'm dead and you and the rest of it, but that's... So, so what? So what? I mean, it ceases to be of any great good interest after a while. I mean, in terms of that, in terms of the fact that it does that, then... Well, if you're just a helpless tramp. No, childish, I go, which I am. Hang on, hang on, hang on. Which I am. Which you may be. I certainly am. Which I am. When you were seven, I was six. I think I was probably five. Okay. This is one of the most serious conversations I've had in my life, so we shouldn't be too flippant about it, although we can be. I understand passionately. Passionately because I understand exactly, you know, the night I understood I was going to die. What's the most important thing in my life? Go on. It's probably about five. Yeah, so... No, no, this is an absolutely serious conversation. Yeah. And how you come to terms with that is clearly a laughable thing. In some ways, intellectually, I came to terms with it a very long time ago. And I can kind of fit one... I can fit... I can fit reasons with... I can fit directions of explanation in terms of... I can fit ingredients of definition. I can see... I can come to terms with this at the level at which you apprehend different versions of the way in which the world might be a metaphysical unity and the way in which it might involve... in which you might have an exhaustive, non-redundant... There's a samanhang of categories which just took care of everything that there is, and they might offer quite different conceptions of the world, and mathematics or music or whatever kind of human might fit into them in one way or another. In some ways they might fit into them in a way which gave you some solace.

25:00 Which might give you some solace because of a conception of how temporal categories relate to that, the being of which is so generous and not proud of the being of anything else. And in some ways they might just simply, you might just have... To take it as it flows and accept exactly but to have to take it as it comes and which is of course is how you actually live each day as it goes and knowing it will never come again and you and I are coming from the same place of course we did years ago yeah yeah and you and I are coming from the same place yeah what I meant was that it took it almost takes another human being the irreducibility of the other That's the tragedy, you know. Of course it's the tragedy. There are various rather facile objectivist counters to it, but none of them provides any real solace at all. And I don't think, as much as I love mathematics, mathematics is a very nice escape from all, you know, the universe. I guessed it was from the beginning for you, which is why, of course, you haven't done as much in this year as you could have done. But you've done a lot and it's been good. Although, I don't know, it's what life was some way of... The astonishing thing is none of this has ever crossed Bill's mind. I know. But listen, so Joseph Brody, all I meant was that there are many other things you could do, actually. No, I don't know, I'm stuck. And I, you know, I'm reasonably happy. He's a wonderful creator of mathematics, there's no question. No question whatever. He's looking unrecognized now, but you have to say... Yeah, because he doesn't actually prove things, but... No, no, no, that's the reason. In terms of in a hundred years' time, people looking back on what has been key to the conceptual organization's subject,

27:30 then obviously it's going to be absolutely fundamental. I meant somebody internalizing some kind of feeling that... I don't know what happened. Overcoming the pointlessness of one's own existence locally. You know, the feeling of what the hell, why should one bother, you know all that. Yeah, and that's where you've always come from, it's just, to me, if you'll forgive my saying so, you've come from it so intensely, almost to the point of boredom, but that's only, of course, from the point of you are the guy, so at least we can talk. Because you've actually tried to do something about the pointlessness of your own existence, by supporting these, listen, I wouldn't be here if it wasn't for you, but it's, you know, the whole business of the world. People, Collins... Colin keeps telling me that ought to be a solace to me, and I fall around laughing because it's not, for a second. Sorry, I just mean to bore you. No, no, no, you're not boring me. What makes you say that? Please stop making these... No, no, there's an asymmetry in our positions. Yeah. And... In your favor. Well, no, I mean, there's a lot of them who just die as old professors. Poor as hell. Unless we produce something in which I'm trying, and you have to go on, Bill, you're trying. He may not. I have long trouble. We don't really know why. He said, in your case, I'm sure. No, I know you do. Well, no, I know you do, and that's why I, I, I, I, no, just shush. I, I actually feel it is a compliment because you're completely right. Of course you're right. Of course I don't know why, because I don't think there is any why. Well, the why, the why, the why is so far off. There's a certain forcing in the, it's very hard when you have convictions. Bill will give, I hope he gives wonderful lectures. Yeah, I'm a bit worried by the time. Well, it's too late now to worry, so let's break it down. No, don't be worried. He looks... Yeah, he's not been well. He looks pale. Yeah, believe me, if you'd seen him three days ago... He looks more Alex than ever. You would be a damn sight more worried if you'd seen him two or three days ago. You know, he was in awful shape when he came in. Well, he was told by his doctors not to travel. It all hit in three weeks ago. Yeah, I know. Life is shit. He's a hell of a cypress state than he was, as I say, two days ago.

30:00 The bill was the one of greatest conviction, so we'll probably come up with... So tell me about Halifax. Well, you know... Yeah, I know I don't. I've heard about seven different versions of the story. There's only one from Williams here. And I haven't judged any of them because I don't know... Max, I don't know if Max really went there in all these years ago. Ah, he's a serious convict, you know, convince-mouse. And he had this, this, uh... He said that when people, you know... It was 1971, yeah? Yeah. And the story that was told to me by, actually by Bill himself, was that the accusation against him, which caused it to be thrown out, was that he had pulled a gun on the professor of law. Is that essentially true or is there more to it? I have no idea about that. All I know is from Max's line was... Max Kelly? No, no, Max Dickman, my own. He invited lots of people. Well, Bill had some very strong convictions about the way things should go, and you— I noticed. But I don't believe that Bill ever drew it down. It was, oh no. Well, I have to say, good old Bill. He possibly would have done it in the old days. I don't believe it. We were told back in 71, this was before the years, we didn't want to have Bill and Grotendieck together, because Bill had gotten up and already denounced Grotendieck's lecture. Well, that makes sense, actually. Well, you know, Andrew Stark, nice man, but he wrote, we, it's still published. I read the Ulan, I've read the Ulan volume. No, no, the only really... I think it was probably disastrous, you know... Have you read Cartier's article about Grotendieck? And he makes the... No, Grotendieck, in five minutes, Grotendieck destroyed two or three years of careful political... Grotendieck was a historical charm. Well, who does that remind you of?

32:30 His opponents were also children, and you could see... For all the more reasons that he did not ask. We tried. I hate to say it, by inviting them to the same fucking conference, you didn't exactly try hard enough, did you? Listen, no, no, what I meant was we had a practical arrangement. All we thought was... We were actual, I don't know, baboons all those years ago. And we didn't like NATO. We didn't want the military to run. We wanted to do things for ourselves. And how do you achieve this? Well, you run an alternative conference. You irritate Gandy. Say nothing of Adrian, who came up and denounced me. Anyway... Which I think is probably really hard. Anyway, never mind about the reasons why you would... But never mind whether you would have done that to yourself. I wouldn't, but that's not... Well, you might have done it. You might have... No, no, no, we were just saying that we're the old lads. We are the kind of, you know, we're old lads. The real old lads. You, I have to say, you bring the average age of the speakers down a very, very long way indeed, Bob, which is great. I guess he's the oldest. No, you and Steve. Is Steve the youngest? I think you almost certainly are, by a long way. Well, not by a long way, because Steve is probably, uh, about Steve is probably, you know. Thank you for your attention. Subtitles by the Amara.org community You were very well presented. That's right, it's Yanni.

35:00 Well, I thought Yanni was about 32 or 33. I would have guessed he was in his early thirties, but I thought you were at least four or five years younger than Yanni. I hope you bring the average age down. You and Steve. I'm Colin. I've got the money. Colin is sitting in between. Oh, really? Oh, aren't you lucky. See, not only beautiful, but intelligent and highly perceptive. Oh, hey, Cully. I know a really neat little restaurant in here. Hey, Cully, why don't you get rid of this loser? Thanks very much. Very well received. When you get on this stage, all comments like that are very great for the receiver. Colin was just saying to me this morning, he was interviewing this very distinguished French Algebraic Geometer a couple of years ago in Paris, and he was really pleased because he referred, he was the last person who ever referred to this young man, the one who honestly said, this young man wants to know, in the countdown seminar, did people think of this young man? Was there a different view of schemes? Blah blah blah. And he was really pleased that there was a 70-year-old and a mathematician in the film, which was really interesting. Okay, well, in that case, are you up for coming out to this place, Azar? It's really nice. Yeah, okay, well, as soon as we can get these guys to make a move, we'll try and persuade Bill and Steve to come as well. Is the Galileo thing, the Galileo, certainly Galileo's study, in the top floors of the museum, is it right? I thought, I heard from Silvani you already went over there to look at it. We didn't go up. Oh, you didn't. It is the, the actual study is the very top floor. The place where you're speaking is the Tribune where he gave his lectures. And that is, it's, it's about two floors up. I just came from there with Francesca and it's not, I don't think, I'm, I'm, I'm...

37:30 I should have checked if there was an elevator but even if there's not it's not it's not a major you're not talking um yeah normally just bound upstairs but yeah I know it's rather serious Yeah, I didn't tell you, so it hasn't been there. So it's like a chest congestion? It's a really bad congestion. You're pointing to your heart here, don't worry about it. No, not the heart. Just the breath. I noticed that when I climbed up the stairs on one of the railway stations, I get totally out of breath. I knew it would pass out. Yeah, I mean, you've got a lot of fluid congestion. It's been really bad, in fact. Oh, you've got loads. I've got a secret Chinese... Remedy that Lisa, my grandmother, gave me, which I think is actually helping. Right, well, let's make a move, shall we? Yeah. Shall I just ring Angus MacIntyre and see if he wants to come? I tried him earlier, but there's no reply. He might just be sleeping, but I'll ask him if... Thank you for your attention. When he says categorical foundations, he means the category of sense. Colin, sorry, I completely forgot. For some reason you never got that. I don't know why. Didn't you get one? You should have done. Everybody should have got one of those when they came in. Didn't you get a little envelope with your name on it? Oh, sorry. It's still sitting there at reception then. You should have got one of those with a map and with a little letter of welcome. You can have a little letter of welcome if you like. Dear Professor. That's actually what I was expecting, to get something like that. Yeah, yeah, I'm sorry. They had instructions to give those to everybody and everybody had an envelope with their name on, but obviously he didn't realise it. This is the final program? Yeah, yes, that is the final program. But it seems to be important. I like it so much.

40:00 I'm sure it's good. It's good. It's good. It's good. It's good. It's good. It's good. It's good. It's good. It's good. It's good. It's good. It's good. It's good. It's good. But Jeff had taken a real hard look at how a group of foundations means what Saunders means by it in mathematics. Yeah, I saw that. It's his very word. No, it's Vianazzi and Arley. It is on most maps, but for some reason... Oh, you guys are going to... Oh, okay. Okay. Well, maybe I ought to... I don't want to be a bad host. Well, if you want to go inside, but, um, for sure, yeah, good. Just the eight of you outside. Yeah, it's fine, I mean, no. Blue is for season. This is the wildest. It's actually warm, it's good warm. Yeah, it's very warm, I think. Okay, we can probably manage, but... Sorry, I'm being a very bad host. I'd forgotten you hadn't met. In fact, you're being a great host. I must tell you, you're wrong. I get used to hearing that. I remember the last time I talked to you by phone, you made the remark that Craig is not all bad, and I didn't have time to refute it, but fortunately I've learned more since I talked to you about it, I'm even working on it, that he's all bad. No, he's not all bad. No, this myth that he invented quantifiers is all you can really credit him with, and it's a myth. He had a much better, much better propagandist than that, that's what he dedicated to that. Maybe Russell, you don't know him. There is a footnote by Zermelo to Cantor's review of his books. A very negative review. Among other things, Cantor says, what is this idea that you can find a number of a concept? I mean, most concepts have an indeterminate number. The concept of do, power, class, you know.

42:30 But the thing is that Zermelo's footnote says, well, Cantor is wrong. He doesn't understand that Frege is the one who's pointing. He should be adapting that. So it's not just his point. He's making a general declaration of allegiance to Frege. So now we think that Zermelo was the first one to axiomatize. Actually, the Cantor-Dedekin point is that he never got an excellent grade, even though it's the one that's the fact that's huge, the idea that that's what I was trying to do, that's what I was trying to do. It's quite, it's quite, I mean, I had not, the other, the others I may look at gave me quite a lot to think of. But this is a general declaration, I don't know if you noticed it. No, no. What does he even mean by that? It's rather vague, it is a general declaration. No, and I don't say he discovered quantifiers, I say he discovered the syntax of quantifiers. He had a sharp idea of the syntax of, well, what's now first order logic. He doesn't explicitly make it first order. There's a distinction he doesn't make. But he really has, I think, a really nice syntactic idea of how to present a logic. All right, it's good. Quarantine-style trade. In America, quarantine would be a spinach. You can always demand spinach. What I can tell you is that what we had the other night... What I can recommend to everybody here is the ribollita. Subtitles by the Amara.org community

45:00 Even I realised that after eating it, I certainly didn't. But thanks for the, no, thanks for the, yes, ribollita. Ribollita. And it is superb. The one that I had here is the best, but especially here, this restaurant does one of the best ribollita. So that's, as I say, traditional Tuscan. That, if you want a really good and authentic Tuscan starter, is a very good one to begin with. Yeah, it's better than the one we had the other night. Yep, that's actually what I'm going to have. I'm going to have a bit of fun with it, I think. The Synthite. You are? I pissed you off. Thank you for your attention. So you think that if people had paid attention to what Cantor was really doing in his instruction and in dedicating, rather than being sidetracked by this Frege-Russell, this Frege-Piano-Russell, I'm not quite so sure about Piano because he played it, but they might have got to the isolated, the essential work. There's a category theory, a couple of generations earlier.

47:30 So you've heard of all this, Laudatoria. Frege immediately writes to him, they establish solidarity. Frege says, oh yes, you were right about the American Revolution and all this. In other words, they really formed a political link there. So why wasn't Peano just another mere algebraist in Frege's eyes? Simply because he had this doctrinal line on the map? And yet, we're told in the official literature, you know, he hasn't known how to become the ingratiator of himself, and his son's ingratiator, and these are simple things, but if you just look at the actual documents, you'll see that he was reading them, unless you're blind, unless you're so blinded by the standard myth. Well, one of the standard myths, of course, is that Frege's wasn't a membership-based theory at all, and that it was the membership-based theory that originated by Donnella to rescue Frege's system. And the basic idea is that every concept is mirrored in the property of the whole universe. It necessitates this sort of thing if you're going to get any non-triggered structure at all. And hence, of course, this idea of a fixed and global distribution of this. So if anointment actually introduced the idea of building a membership, how does it have a sense to serve as a distinction between large and small?

50:00 It would be important because this is a mere convention. As you made, the point you made in conversation with Andrew this morning, he was quite ready to, or rather good on the rise, quite ready to abandon it and quite, it was, he had a mess on him. Some people imagine that by putting finite types above B, there's a stronger theoretical strength, so it might be inconsistent even if the other thing wasn't necessarily the kind of work, but actually it adds very little to the truth here. Certainly, the consistency of ZF is enough to justify, you know, just that little bit more, or the consistency of the girl here, I think, is enough to justify, you know, the consistency of the girl here, I think, is enough. If I'd only known about that correspondence of Frege and of Rihanna. You mean Frege and Husserl? No, no. Kirtle and Bernice. Yeah. They're 1963 correspondence of the very moment I was writing that. If I'd known about this, I would have done some things differently. Have you had a chance to speak with Angus since he arrived? No, I haven't seen him. No, I was wondering if he... Is he coming here? He should be. I was told he was here. I wasn't able to get a message to him. He wasn't answering his phone, so I assumed he wasn't in the room. He's already out somewhere, or else he was sleeping deeply, so... But I thought you might have seen him. He got in about lunchtime, so... John, it goes with you. Natural or? Gasata, per favore. Due, due. Due? Did you want naturale or gasata? Naturale.

52:30 Vino tinto for the whole table, since it'll work out a lot cheaper if we do it that way. So we start with two bottles? Three? I think two to begin with. A litro? Si, due litro. So does anybody want bianco? No, I didn't think so. Okay, do a liter of it also. Ready to eat? Okay, well, I'll start just to, so, you know, kick off and then go counterclockwise. Omelette con tortufo. Possibly con salata dirty? That's it. Okay, Bill, you want to go? Okay. Next one. Next one. I'd like the, uh, what was it? Oh yeah, beef today with creamy cognac, pepperoni sauce. Okay. Medium. Medium. Medium. And, uh, spinach. Nice choice. I thought that was English.

55:00 Yeah, that's the problem. Yes, it's going to fit. And clearly that whole tradition of viewing logic and how to break the spectrum was sort of dedicated and shared and didn't go away. It was always there and, you know, continued clearly through the work of... Yeah, oh, yes. Well, at least it wasn't until around the time that... Yeah, yeah, sure, but... Because of course exactly as how it was first presented to me when I, you know, that this was the cylindrical algebra and stuff that people like Halmos would have been doing in the 50s and 60s. This was a kind of... Nice, you know, formal game, algebra, but no, no, have no real conceptual significance in organization of the subject. Well, it could be about Frege, because it was, it was an attempt to make an algebraic image of what Frege had done.

57:30 Well, is that fair or aren't people like... I'm not sure that's fair. You could say it was an attempt at algebra as Tarski's model theory, but what Tarski did, but, you know... But just Tarski's convention is what personality logic is. I mean, there are these very infinite things. There's one single infinite list of variables. This is completely contrary to the practice or to the idea. So, how most incorporated is that. That's good to make sense of it, you know. At least, at least, almost have the virtue of recognizing substitutional moments of basic and quantified. You didn't quite realize that you wanted to determine the other way around. Yeah. Because, again, actually... Now, I mean, the actual adjoint is a little bit distorted by the fact that it's using an intimate set of variables, one intimate set of variables, probably, rather than, say, the category of finite sets as a model of themselves. I think it's very unfortunate because it did precisely obscure the fact that the first order of logic is really just a presentation of this kind of object structure. That's not really the intuitive concept, but somehow this whole tradition of saying that a presentation of anything there is, coupled with these arbitrary conventions, somehow forced them into that. So you've got a philosophy of mathematics. Philosophy of my doctorate. And I'm chair of the philosophy department. But unlike most of my fellow students...

1:00:00 The chair of math came into this chair's council meeting the other day and I said something I'd been thinking about, spectral sequences. He's going around for the next couple of weeks saying, how many universities can you go to at a chair's council meeting and someone else talks to you about spectral sequences? I mean, he felt really good. Thanks very much, by the way, for the clarification that you and Andre... Thank you very much for your time, and I hope to see you again soon. At the technical point, it obviously erected an additional barrier. Well, yes, not least because he said, and you should understand that there's this thing called triangular characteristics. I won't burden you with the definition because you wouldn't understand it, but he explains, Harvey. Well, if you won't burden me with the definition, it's not going to explain anything, I'm afraid. Well, you see, if the axiomatic method were a standard method, too, he could tell you that, well, this is an axiomatic method. So you can take the derived category of any triangulated category, not necessarily a category of chain calculus, and you get another triangulated category that's just a little bit more complicated. Yes, yes, that was... because I went away with the idea that a triangular category was something various. In other words, you can explain it, as I just did, without telling you what the actual axioms are. You can learn them if you want. But the idea is, you simply, you always make this kind of abstraction of what's the minimum needed to go further. Clarify things. I don't like it when people say, I'm not going to tell you. No, I don't. You never do. That's one of the things, many things that I, but I'm afraid, not that I, I was very impressed with Naishin Han. I'm sorry, I'm not pronouncing his name right. He's an extremely bright guy. Oh, yes, I noticed his name. In fact, that's why I noticed his name in the development of topos theory survey.

1:02:30 I think not only for italian children like myself. I'm sorry, I didn't understand the last... You don't have to explain that, I'm sorry, I'm being very... Well, above and beyond the facts, I always take any opportunity I can to explain things that really do to Volterra and... Yes. ...Betty and... Yeah, yeah, sure, especially Volterra, but... But I'm sorry, people who don't have that point of view realize that the so-called flunker age theorem was actually 25 years earlier. I've forgotten what it was that de Romm had said to him. Truly, but no common sense, right? No, I tease you, is that okay? So, you know, it's a question of his community friendship. He has Poincaré as the great hero. He undoubtedly was. Thank you for your attention. John, you're having one, aren't you? No, I'm sorry. Bob, do you want to start the wine, perhaps going clockwise? No, I think we'd probably have to mix the wine and the water and ask them. I'm quite okay with that. If you want me to ask for more, I don't mind. Do you particularly want another glass for the wine?

1:05:00 Well, you can always ask for one. I'll tell you what, I'll go... No, you start. I'll start, I'll start. Yeah, exactly, that's the idea. And that's the tea, that's what you... Hang on, have they got, they haven't actually put the tea in there have they? That's a bit dumb. Thanks. They don't seem to have actually put any tea in there. Have you not, sorry, where is the tea, the actual tea? That's just water. That's just water? Is that what you wanted? Is that what you wanted? I thought you said you wanted tea. Oh, that's fine. And then pass that around. Looks good. Does it work? No, it just works fine. Excuse me for driving. Really? Covered out the net. That's my nanny. Number two, nanny. What sort of program is this? I've read about things like this in China, where they're trying to sort of assess more effectively the academic leaders. Well, what I do is, well, it's difficult getting the funding to do it, but ideally what I do, when I have done it, is do very conventional research. So, I'm just about to defend my dissertation. I've studied using St. John's Wort for people that give up smoking. So that's the kind of stuff that I do. Oh, I see. But we do very, you know, traditional clinical trials. So you're not so much looking at the clinical components of it? No. No, I want something like St. John's Wort to be very studied so that we actually do clinical trials. Great. But I mean, I have a very small panelist, so I'm trying to get funding to do a larger study.

1:07:30 You smoke it? No. Me? No. No, no, I mean, like, are there alternatives to smoke which kind of... St. John's Wharf is a perp. You take it as a pill, to stop the cravings. I just stopped a month and a half ago, so I'm having a craving like this. It's also used as an antidepressant. It's simply a natural product. It's pretty widely used. Antidepressants are sometimes used to help people stop smoking, so she found this exactly even better. Thank you. It's the active ingredient in those commercial herbal tablets like Calms and Quiet Life that you buy in chemists in England. They usually combine it in those with things like valerian and gentian. St. John's wort is certainly the principal ingredient. Yeah, there are plenty of those tablets. In fact, I've actually got some upstairs. They're available in all the chemists in England now. Thank you very much. There are some people who are true false elements, because there's solid steel and there's false solid steel, but you can get various things that are not really false solid steel, so this stuff is true false solid steel. It's actually really hard to do research on Chinese herbs because they're so specifically prescribed. So they wouldn't... Chinese. Give everyone a table of things. The whole system of understanding the body and diagnosis is totally different. They would relate this condition back to other aspects of the human being. Well, I wasn't sure if it would be the same or not. Well, even then it would be the same. In the traditional sense of it. One thing in that paper of the development of topos theory that you've touched in general terms about the whole program of topology as it had been taken up and developed in...

1:10:00 And how far has that now been pushed in the direction of... I understand that you have these categories, these toposes rather, which are generated by a subcategory that gives geometrical spaces with nice properties. And within this, you still have equational... Not that I don't think that works. You still have equational equations. It's very hard to apply our methods. ...between truth values that kind of give you the infinite, discreet piano algebra, but it doesn't actually occur, I mean, it kind of occurs within the category, but it's not, sorry, um, well, you, yeah, but you, yeah, but not as often, yeah. In a topos, all sub-objects can be obtained by maps into the truth value, namely, a sub-object has its characteristic map and is also constantly true. Now, so, the interesting feature here is that if you want to put it in a larger context first in a topos, There may be many, many sub-objects of those objects which are not equalizers, which are not in the sub-category. They're in the larger... Of course they're equalizers of truth maps, but they're not equalizers of, say, real value maps. Okay, and in this case they're not equalizers of maps between the geometrical spaces. You might be on the left the Rn and things like this. Right. And this is because the maps are rather tame. But the striking feature is that... In the subcategory, every object has only a finite number of connected components. Yes, that's the... I wanted to ask that. And yet it's closed under equalizing. This is quite a contradiction, by the way, from a classical point of view, because R and Rn are already connected, they just have one component.

1:12:30 And so how do you produce things that have many components? Well, you can take equalizing with maps... Wavy maps, where two wavy maps intersect is only a few points or a few pieces of surface or something, so it has many components, but the traditional categories have so many continuous maps, or so many smooth maps even, that these equalizers may have mentally many components. Yes, as in the case where you do have all these famous counter-examples, if you leave that in the interest. Yes, yes, all the famous pathological so-called... X times sine 1 over X and all these things have to do with... So essentially you see this little category that has equalized various other categories. To me, what finally works out is that it contains these finite features as an object of a very special category. The connected components of any one of these matrices is one of these finite sets, so you have sets in a way that don't be finite well, even though there are infinitely many points typically in these spaces, you cannot collect those points into a discrete space. That's the thing I've been missing in understanding it. Okay, that's the whole code. That's what I couldn't understand until now, how this works. This is precisely Cantor's genial discovery, you see, this idea that you could imagine connecting the mirror points into another phase. Yes, yes. This is one of his three great discoveries. But the connected components in this case are not necessarily... Well, I mean, irrespective of how that works out, it's just that I've been much more sort of in the mainstream way of doing it, though, but you're paying me like this. He decided, Cantor decided, that I really wanted to get the introspection sequence in this part of it. The screen space is integral, finite. Well, finite in the old sense of the... That's the idea about this small category. The screen space is integral, finite. And yet those are enough to take care of all the faces of connected components and all that stuff's in there. That's... now I've understood. Thank you.

1:15:00 In other words, essentially just real part of mathematics. That's a nice category. We didn't have the same interest properly in that at school. We find that number of roots and it goes up from there. So having achieved that, we think, well, let's add in some more functions. Let's add in the exponential function. Can we still maintain this? If we don't go too far, we'll get to the piano example. And of course, they basically want to say, of course, we've gone outside geometry. We've gone right outside geometry, you have to understand. That geometric concept just has to be reconstructed. Even if we look at it, we find that even geometry is infinitely more complicated than we thought. Yes, it's infinitely more complicated, really. In other words, they essentially collapse like an accordion. Yes, I wish he'd collapse his bloody accordion. It's actually subjective to the object, you see. The fact that you have this subjective infinity outside the geometry is probably not so great. Yes, I think I'm probably going to have to wait for this guy to finish. But this is tremendously clarifying to me. Because I understood how it was connected with this point about the components of the space and the way that Cantor saw this is implicitly, although obviously you brought out the adjoints involved, but it's implicitly there.

1:17:30 So the fact that you have topological spaces obtained by, you know, equalizing two maps between the unit and interval, which have infinite number of components, and the fact that you have space-filling curves, interval mapped onto them, is almost totally against any intuition. Thank you for your attention. You just have to accept it. You have your starting point, you have to come out and you have to accept it. If you accept the piano, as it's usually presented, it's based on open sets. Open sets are really the traces of characteristic functions of Lepenski's space. It's taken on this locality that has always been essentially beautiful in this so-called beautiful cataclysm from around the world. But it's a copyrighted structure that's all so very close. Well, that was part of my... Yeah, yeah, yeah, that's right. And it's meant well. Yeah. There are so many different equivalents when they're all set up together. Yes, that is really... Yes, I was going to say that's...

1:20:00 Yeah, yeah. Sip some of that excellent tea and wait until this guy... ...buggers off. I'm sorry, it reminds me of the old Tom Lehrer song, you know, the merry archers would serenade and they would not go away till they were paid. I mustn't be unkind, poor guy, he's a nice boy. Yeah, you probably do, yes. No, I'm sorry, it's being on my phone. My father used to play, a lot actually, he used to play the accordion. He was a good accordion player, he enjoyed playing the accordion. He used to play for me when I was a child, so I haven't said the mentality about accordion players, it's just I don't want them to be... I'm slowly realising the key to understanding a child's cohomology is also to understand that almost nothing can be done. It's the water torture principle. I'll give it to you. It's a risk I have to... It's going to say put it in the... I'm sorry for the poor guy because clearly he has to make a scratch living there. No, but that is very, very clarifying because I... Yes. As you were saying, it asks Fox and Steenrod and people already implicitly recognized in the 1940s when they began to question that whole basis for the definition of the point. No, that's the other sense. That's about the structure. Yeah. They contain the usual category of compact morphological spaces.

1:22:30 The usual continuous maps, just the compact spaces, the changed spaces, the spaces that weren't compact, and the maps that didn't change, the compact spaces that were already there, that you have too many maps. And the basic reason for that is that it's not a sufficiently kind of path-centred, you know, they were ignoring the role of, well, they were ignoring the role of figures, you know, more complicated than points, you know, figures other than points. Oh, yes, please, thank you very much. How is cohomology working? Really, I'm... Yes, please, could you... Likewise. Did you order some white wine? No, it's in the meat. Oh, it's in the meat. Can we pass the wine back around and refill? Okay, thanks. Just to have a refill. Won't do your voice any harm at all. Rest it. You give this data function. Francesca said she was coming to the hotel tonight, but she didn't. I give her a ring tonight as soon as we get back just to check that you will have a microphone tomorrow. I'm sure you will, because when I went out there they had a lot of electronic backup, so I will make sure you've got them out, well depending on how you write it, they're very impressive. The coefficients of these polynomials, depending exactly on what you write down, you can get them to have the same absolute value. Well, I understand now what you meant when you said that somebody could have written a headline around 1940. Topologists turn their back on topologists, topologists reject topology. The whole strategy of a conch homology figure is confirmed to be false. There's this propaganda going on on the phone list at the moment. You have certainly better things to do with your time, I know.

1:25:00 But in the context of the denunciation of Caulfield's book, because Caulfield's book contains a couple of chapters on ultimate topology, Of course, general topology was only made possible by the fact that people came to understand i.e. foundations of maths, as we understand it, i.e. Zanello and Brigham. And without that, there would never have been General Apologies, so it remains a natural opportunity. Are there sense that, and we did partway through this work write SGA 4.5, so we should have already been thinking about it. So, yeah, it's absolutely crazy. I mean, it's just historic analysis, yeah. You can get by talking about less than that, although Hansen was also explicit in the book. But your house talkers told me they're confused, they'll need it. No, Sir Mallow's theory made it to clarify what he wasn't just trying to... Simpson has said this explicitly in previous lectures. Well, there's two senses in what you're doing here, though. There's two senses in what you're doing here. The other is that for that I have no idea but I was very surprised standard conjecture because he does what has come to be called naive much more detailed construction because I tried to explain legally in my appendix As a sort of stalemate between actualization and work further on. You mean of the book with Paul? Yeah. Yeah, that's right, yeah. In other words, the actual century that people used is very far from the fake piano paradigm. Also, the only thing that had been axiomatized over was that paradigm. So, therefore, it was thought that most students don't need to know this axiomatization. They'll just be confused by it. Nowadays, they try to do it the other way. So we'll have this thing called naive, which we will refuse to axiomatize.

1:27:30 Yes, yes, yes. What I'm curious about is whether they, I don't think Alster called it that. It would be interesting to know who first coined the label naive. I would be interested to know who coined the word. I would be interested to know. I would be interested. They will probably not, for example, give a name to that category and write an adjunction for it, because they don't want to do those things. Is that as good as it looks? No, I only know this one. Oh, wow. But who was it who first pointed out that... In the context of sheath theory, you could use sets just as sheaths over the one point space. I mean, it's obviously implicit, but who was it who first explicitly pointed it out? The best people really are thinking in terms of the categories of sheaths. But when they write it out, they try to say it all in terms of the next level down. So in that sense, you're not going to find anything quite that way. Thank you for your attention. Combination of things in that way and also things really spartacus about other things. That's what you see. Mm-hmm. Mm-hmm. Mm-hmm. Mm-hmm. Mm-hmm. Mm-hmm. Mm-hmm. Mm-hmm. Mm-hmm. Mm-hmm. Mm-hmm. Mm-hmm. Mm-hmm. Mm-hmm. Mm-hmm. Mm-hmm. Mm-hmm. Well, you're extremely good health.

1:30:00 Thank you. Thanks for that. To all of our health. To all of our health. Yes, very good health and a very successful meeting. Which is that he can. Sorry Colin. Cheap shot. We have a great advocate in our department. She's one of our quite the most important professors. Uh-huh, uh-huh. Sounds well, that's what she meant. She's extremely put out that I claim to know something about mathematics. She has a degree in that. So she just doesn't know why I have to horn in. As soon as she comes on campus, I start claiming I know something about mathematics. This isn't... Martin Lussbaum, is it? No, no, no. Who, who, who? Sorry, who is it? Karl Lussbaum. Karl Lussbaum, doesn't mean anything. You have a second. Do you need some more wine or is it your time? Well, shall we have one more carafe? Would that be okay? One more, I think, would be enough. They're almost empty, so I don't think we're sozzled or anything. It's a nice wine. One more. Do we need any more water? How about the tea? That's cool, that's cool. Thank you for watching.