Evening talks (contd.)

0:00 Yes, yes, of course. So, in fact, one needs a vast variety of thinking models to come up with that. The general theory must somehow come up with all kinds of things and not just some of them. I mean, it's not going to be a practical answer. Yes, yes. It's tough to say there couldn't possibly be something called fundamental physics, because there's so many ways of... ...Europe again, and then after you told me I could... I don't know. Ah, yes, yes, of course, if you like. Great. Then we will arrange a call, yes, definitely. ...various places, computer science and also engineering, like my class. There are engineers in Italy, in addition to Davide, who have an appreciation of how topos theory can be of vital assistance to them.

2:30 Yes, it's good to know that. Well, at a minimum, you know I'm promoting that. There's somebody who's promoting that. Well, that's certainly good. Well, it's funny, from the window, from the balcony, from the parking lot, you can't stand up. Plus, watch that one. That's why, if you sit down, you have to show your back to the camera. Oh, shit. Yes, it's just around the next corner and then that takes you straight down to just another half a block.

5:00 And you're going to be in Firenze as well later in the year, Bill. Did I understand? Yes, you're visiting Alberta, that's great. Bonita Galileo. When is that going to be, in October? Great, oh good. Oh, but you are going to be in Europe before that because of the Grassman thing. Oh, that's right, I was just trying to think. I was thinking you were going to be there in September as well. In Potsdam, just outside Berlin, on September the 16th, actually it's for three days, and one day they are going to go to Gdansk, or Danzig, to use the old German name. Oh sorry, Stettin, of course, not Gdansk. Why did I say Danzig? That's right, to Stettin, I'm so sorry. Yes, it's Danzig, of course, but I don't know why I said Danzig when I meant Stettin. What do you call it? I've got one of those rain hood things if you want to borrow it. Oh, no, you've got one. That's pretty good. No, I've got one for myself, so if you need an extra one. Oh, gosh, yes. Oh, yes. And also, you don't keep losing... I always lose umbrellas usually within about two or three days of buying them. No, I always rely on a rain hood now. Right, it's just around the corner here. Well, we didn't come quite as a direct... We could have come that way, which is the way you walked with Davide this morning, but we'd already sort of peeled off with Anders to...

7:30 But it's okay, it is... You'll see it when you just get to this next corner. The building's just ahead of you. I forgot, of course, that you've got to go to the next one along, haven't you? To the neighbouring place. I'm sorry, I just suddenly remembered you're not in Clifton Hill House, are you? You're in the... What's... You're in the place next door, aren't you? Oh, okay, in that case... Yes, it's just, that's just further on. No, it's this way to the left. That's Clifton Hill House just ahead. I'm so sorry, I was forgetting. I think you'll see it from here. I'm sorry, I'd forgotten that you were in. Yeah. No, if you come here, your place should be, isn't that it there? The place where you're staying? I think so. Well, this is Clifton Hill House. What's it called again the place where you are? Manor Hall. Yes that is, that's Manor Hall, just along there. Straight, straight, straight down the pavement here. It's the next building, it's the very next building after this, after Clifton Hill House. Okay. So we are at 10, right? At 10 there. Yes. We could come here at 9.30. Yeah, we could do that, yeah. Ah, good, yes. So shall we meet? Right in the lobby, right there. Where the lights are. Where those lights are. Ah, the lights. Yeah, yeah. Okay. That's the lobby. Okay. You don't want me to walk you back down there. I'm quite happy to do that if you like. No, you should not. Yes, I think as gentlemen we should do that. Old fashioned. Thanks a lot. That should be your place.

10:00 I certainly hope so. Yeah. Yeah. Well, that's the neighbouring Hall of Presidents. I think that's the name of it. I certainly hope so. Do you recognise it? I'm as I say I'm sorry because of course it's not the one where we're staying but you'll soon see from the well there are only two halls of residence here in this in this resident in this bit of Bristol so I think it's got to be this one cool sorry you okay I'm afraid I'm not sure where the entrance is of this one do you not recognize this maybe you came in the other side I'm so sorry but of course I'm afraid I'd forgotten that you were staying in a different building It can only be this one. Can you get in the back here? You've got a key, haven't you, with an induction loop? Yeah, that you're going to the right place, yeah. Well, they've probably gone to bed by now. They're here until about 8 o'clock. Yeah, it's the building, it's just... It's OK, yes, I'm certain it was. OK, no, not a bit. No, I wanted to make sure you got the right... Fabulous. You take care. Take care. Cheers. Yeah. Oh, that's a relief. Doing well in Bristol. I'll remember tomorrow to take the shorter route, the one that, the one that Matthias took with you this morning. Misleading.

12:30 Completely misleading categorization, yeah. Yes, this is the same trend of losing, look-a-losing, pedals. Oh, I never learned their theory, isn't it? Something to do with logic. See, who gets cited? Now, this is really striking. You just mentioned who you see. Steve Vickers, he's just the sort of person who pulled it in to doing this stuff without even realizing it. Yes, because he's such an innocent. And, of course, John Baez is the person. Oh, no! Baez, Vickers, and conspiracy in a certain direction. And what, in the article of Ellusi they cite, is that ridiculous thing in the LM... Oh my god, that would be... No, I was going to say, that would be serious. No, of course, Ellusi's a brilliant mathematician, it's just he has to be able to account for not having such an ignorant attitude towards... Background witnesses. Oh dear, that is bad.

15:00 One thing was two-chatted words. This, at that time, was sold on the two-category side. John Pelican was speaking about it because his work was entitled to it. It was his main instrument of his work, yeah. What was Dana's objection to two categories? I don't know. This was presented as subjective judgments that nonetheless had to be taken seriously by this famous guy. Very strange, Mr. Green. The way his work in domain theory has gone, it doesn't seem to make sense to me at all. What was the reference he, John, in his talk alluded to the fact that when he first come to Cambridge, his experience had been, for a meeting, his experience had been a very unhappy one. So much so that he hadn't... He was taking almost 20 years before he felt comfortable in the place. But he said to Peter, you know what I'm talking about, but of course I don't want to speak ill of the dead. So I have no idea who was being alluded to in that exchange. He's been in Edinburgh for a year now. Yeah, yeah, yeah. But just recently he died. Yes, yes. Yes, very easy as they say. He's almost like a suburb of Bristol. I guess so, but he didn't lecture me. He wasn't in the post-grad course. But you were doing a pure math, these then, role of computer science?

17:30 Really? We had a course on category theory, but it was him. Yes, I am. Oh gosh, that's a smart thinking. Sorry. Yes, hello Richard. I'm fine. Oh, I see. So does that mean he's left his key here, or...? Okay, that's fine. Thanks for letting us know. Yes, in fact, Bill and Matthias and I are in the lobby waiting to meet him because he said to meet us at 9.30. Okay, what was the problem? Had he made a mistake about the time of his flight or something? Okay. Okay, well, I'll explain to Bill and Matthias. Okay, I'll take the key then and we'll come on and meet you in the department. Okay, thanks for letting us know. Thank you, cheers. That's very kind of you. I'm supposed to pick up this key. That's great. Thank you very much. Anders had to leave much earlier than he anticipated. So, sorry? Oh no, okay. Of course not. Thanks, thanks. You don't need me to sign anything or... Okay, right. No, for some reason Richard just, well he was gathered, rang to say that Anders had had to leave much sooner than he had anticipated. So he's not going to be able to join us. I think he might have just sort of made a mistake about his flight time or his train connection or something. Anyway, that's the message. We could on the way there, yes, actually, yes. There's a little place on the triangle that's quite nice. Oh, even better. We went past it, in fact, last night on the way down to the place where we ate.

20:00 Can't you? Oh, it must be another place, because they have... Oh, good. Perfect timing. Ah, nice to see you. Does this work all right? If you can sleep? The speed was a little bit awful. No, because I don't use the tools to sleep with the clothes that they do today directly on me. Oh, oh. Because you need them to sleep on you. No, I need them to sleep with me. British have taken to duvets in a big way in the last 20 years. We never used to have them at all. We all used to use sheets and blankets, that was universal, but now it's a more recent thing. You never used to see duvets at all in British bedrooms, but now in the last 20 years they've become more and more common. No, no, no, no, they used not to exist, but now they're becoming much more common. The Germans have always used, well I don't know always, but they're universal in Germany. I think it's a habit that we got into, we kind of acquired from the Germans. Oh, okay, yes, because French people like things like that. Yeah, yeah. That's exactly how I came to complain a lot about it.

22:30 Do you still want to go and get a bite at this place in the... Oh, you're going? Okay, fine. Actually drew you into topos theory, Olivia, when you're... Your first your first degree was in mathematics I take it yeah and then When I was in Turin, entirely on my own, I studied logic and category theory. But category theory was motivated by homological algebra. I studied a lot of homological algebra, and then, while studying it on old patient books, I was not entirely satisfied because I could see that some theories could be proved in more abstract ways, How to do that, technically, at first sight. And when I discovered just the concept of mathematical duality, I had a kind of illumination because I said, okay, now we can rewrite everything in this more, well, not everything, but...

25:00 At least you can see at which level of generalities it proves to lie. Exactly, and indeed all the key concepts were there. Yes, exactly. And so that was the key illumination, even though I was quite ignorant at the time about those things. You really, you really kind of recapitulated the way that the subject was actually developed by Grotendieck and his school, so you must say this shows remarkable mathematical maturity and judgment. Well, I was in my final year. Somebody's starting out. Yeah, I was in my final year. Well, it's still pretty impressive. Well, I like to form my own ideas. Yeah. I used to have very strong feelings about it. It's that way, I was going to say. Yeah, just go back. Yeah, hang on. At least in some specific area. Yeah. Oh, hang on. And so that is how I can use the category theory. Right. So as machinery within homological algebra, which of course, as I said, is exactly the way that it grew, and exactly the way that topos theory grew too. Because so many people who come into it from logic, of course, learn it, what I think of as completely the wrong way around. They think that topos theory was developed as a kind of gadget. In model theory, this is for them a very, very unfamiliar and weird, obviously inspired by ideas from algebraic geometry, idea of how one should think about a model. And it's true, of course, that Bill's thesis was precisely on the contour of semantics for algebraic theories, but that itself was coming out of the algebra and thermology, topology, rather than out of logic. Rather than being a gadget for logic, it was actually showing us a general framework for subsuming logic, for taking logic back within mathematics, where it really belongs. But the logicians, at least a lot of them, still don't get this. Especially the ones, especially the logicians like, because I trained as a philosopher, not as a mathematician.

27:30 And for a long, long time, I don't, of course, have anything like a natural mathematical aptitude or maturity, you guys, but I do have a deep interest in the conceptual development of the subject, and I can see, even now, why for a very, very long time there has been this conceptual obstacle for the logicians, because it does involve a complete dismantling of their, and re- You know, reconstitution, rebuilding of their entire conceptual framework, not just the technical machinery, and one which brings with it, it seems to me, really deep insights into what the real philosophy of real maths should be about, which is something which, unfortunately, people in philosophy departments don't have a clue about in the most part, but it's very exciting, though, when people start to... When the people do want to learn it, like John and Richard, well Richard particularly, much younger, and so I like to do what I can in the way of making a little propaganda at the right moment to the right people for the cause. Well, there's part of it's just that they're scared of having to learn, start over. It's the so-called fallacy of sunken costs, which of course occurs in many departments in life. You've made a huge investment in one, certainly. Oh, sorry. You did say it was well hidden, Matthias. Yes, very well hidden. Ah, smart. Did you see me? It was a little cafe somewhere. Aha, aha. Oh, thanks. You're a born scout, obviously, Matthias. Time spent in reconnaissance, they say. No, I would never have known about this place without... Ah, yes.

30:00 Is it part of the university? It is. Yeah, okay. No, that's why. No, I was thinking you meant the little, there is a little fair trade cafe on the same street where we were yesterday, but further down. No, this is, this is great. No, I think that's part of the problem. As I say, it's just the sheer amount of re-learning they'd have to do. But there's more to it than that. It's a conceptually deeper problem than that. It's to do with a resistance to geometry, I think. It's to do with left-brain, right-brain opposition between people who are naturally analytical. Perhaps you need to combine everything in an harmonious way in order to understand. General resistance to geometrical concepts and methods and insights, and it's very, very interesting to try and understand it more deeply. I'm hoping that's one of the topics that we may be able to explore before the end of the workshop. Obviously not in such a broad brush way, but staying in touch with it. Perhaps Maya Bill's work on, and your work indeed for that matter, on Gros and Petit Topos. And he's told me something quite fascinating about, well, many years ago in Buffalo, when Grotendieck was there, but Bill was not there at that time, he was in Italy, he was in Perugia at the time, he didn't attend this seminar himself, but he did. I'm going to study the notes of it subsequently and speak to Grodendieck about the matter. Grodendieck presented this, he gave this talk to the Buffalo Mathematical Colloquium, I think in 1973, which embodied, it's something of great interest to me because it seems to embody this, really a kind of total programme for this. For subsuming logic into this much more natural frame of the code, and by understanding the right classifying rings for different theories, so that the sub-logic classifier automata, which of course is the natural setting for the expression of logical notions of relations and quantifiers, would just fall into place within this wider scheme of inter-dissecting or interlocking of ring classifiers as the fragment of the overall framework.

32:30 ...which deals with logic. But again, it's really a vision for, I'll tell Chris, for bypassing or perhaps better subsuming logic within geometry. And it's an absolutely fascinating vision. I would really like to hear about it in much more detail. So I'm hoping that's one of the things that we can get around to talking to before the end of the workshop. But it's very important that we hear all about your detailed, topos-theoretic research. Topics as well as they're obviously extremely important very very interesting. I really enjoyed your talk in Cambridge it was one of the nicest, one of the ones I got most out of yeah I'm just wondering if you it's the coffee oh cakes I'm gonna have a sandwich I think. Oh, you're a member of the minority, aren't you? This one, and Léman's phone. You know, I ask him things like nuclear space, relativity, topology. So you met him for the last time 20 years ago? That's right, 20 years. That's almost exactly 20 years ago. It's almost to the month. And he was already out of the world of mathematics at that time, of course.

35:00 Oh, well, he was already out of the world of mathematics in the 1970s. Yeah. But he worked privately and he also consented to come back to give lectures under the sole condition that he would be allowed to lecture on his ideological views also in the same position. Did he do that? Yeah. Oh yes, very often, very often. This has become almost a famous tradition. That is to say that someone who has ideological views will be given a chance to speak about them in a separate meeting, not the same as an mathematics lecture. That's why Wesley Fuller, Kambler... But before I arrived to give a mathematics talk, they put up signs all over town saying I was also going to give a talk on Marxism. I didn't know it myself. Now that was, I don't know if that point came across. No, that didn't come across. I found all these signs and I thought, well, I guess I'll have to live up to this. And so I quickly put something together, which wasn't too shabby, I think. No, I think it's pretty splendid. It's just they had the tradition. For a guy who finished up, for a guy who finished up running a hedge fund, I think, you know, Wesley first started out on a very different path. He's an interesting guy. Well, at least he uses the resources of his capital fund to support things like the meeting for Peter and Martin. The other thing that's very much demanded is lots and lots of money for the mathematical talk. In order to support his talks, he had a commune, a small, settled settlement. Has anyone ever claimed money for giving mathematical doses? Is that unusual? I did a colloquium talk with the department at Leipzig. Excuse me, there is an honorarium. But usually... Usually it doesn't... it barely covers the travel expense. Ah, yes. I'm used to that. I'm not used to something better than that. In the past it was more so.

37:30 You know, often I would give a talk and I would get $100 for travel, that's all it costs, plus $100 honorarium. That's quite normal. For you, yes. For me, no. A long time ago, it's not so much. It's not so now, because of the tight rules of granting agencies and so forth, very frequently. But on the other hand, if some sort of superstar, like Hawking, comes and gives a lecture which will have no scientific content whatever, you know, they'll offer him $10,000, probably more. Well, exceptionally, in what way is it exceptionally? They get lots of people who don't know anything about science. He's more famous for being famous. Exactly. Ah, okay. That's it, then. Did he get money from his community? Well, sometimes he got a lot of money from Buffalo. And this was after his... I think the main motivation was to remind him of that case. And of course his commune plans all failed because really the people he gathered around didn't really share his views. They were just out to exploit him and cost life. So he became quite disillusioned about it. He lived as a hermit, completely privately. Yeah. He lived probably about 35 years. No, no, 30 years. He had a girlfriend for a long time. He lived in a nearby town. More from what Cartier said, he just, when he did his second disappearance in 1991, he just abandoned her completely. She came back to find a letter from him explaining why he was so unhappy and miserable and couldn't bear to stay in that place any longer and giving curiously very

40:00 Detailed practical instructions to her about how she should arrange certain aspects of his affairs and also of his Siska's affairs, but leaving no forwarding address and just disappeared from the face of the earth. I mean, it is known where he is. Noise is a weapon. At the time, I had come from Perugia, Chris had come from England, and there were a couple of other people in Montreal, and we heard... The Drodende was in Buffalo going to give colloquium talk and so we all took, well our wives weren't there anyway, so we took off on the plane and went to Buffalo. Okay, now, I'm sorry, thank you very much for putting me straight about that. André Joyal from Montreal. And so, yeah, we had an interesting discussion over the week. I attended his Evening Videological Talk and I attended his Colloquium Talks. Okay, now I'm... Thank you for correcting me on that. I don't have a written report. No. No, alas, there was not. They didn't record those, which is really bad because that would have been more valuable than anything else, to have recorded those colloquium talks.

42:30 But still. Yeah. But you did say that there was... notes that he yes uh hang on it's that way isn't it i'm completely hang on no we're sorry we're in woodlands road now yeah it's just across the street from us of course it is there it is yeah yeah we're coming up from the other side that's why i was disoriented that's okay There are five things I keep in mind, as well as some of the statements about the indefinition of the scheme, and there's no question that he was involved. Oh, that was when he gave... That's the whole significance of why I continue to cite it. Brodnick himself rejected the indefinition of the scheme in 1973 publicly. Unfortunately, I had already figured this out. From Gabrielle, Gabrielle, his junior test, motion and assimilation ideas were similarly unbound by the fact that the definition of what's been laid down had become ossified or had become, you know, it's the next one, it's the next one. Oh, no, okay, that straightens out the history for me. I'm sorry I was confused. No, I didn't attend courses. Those, of course, went on for weeks and months. Yes, yes, indeed. Well, if there were 130 recordings, it must have been for many months. No, okay, and because I had known that you were in Perugia at that time and only became a member of the Buffalo department some years later, I must have got completely twisted around. Yeah, that's very confusing. And thought that... Happening in Buffalo... Yeah, yeah. It's curious, at first, the first time you told me about this, I did have the impression that you'd been there, and then I obviously mistakenly corrected what was in fact an accurate impression the first time round, probably under the impact of this awareness that you had only joined the department later. Anyway, it is important, actually, because... Hello, hi. Oh, and I met Steve Shenwell at that occasion. He was already in Buffalo more than a year before I was, but we started our collaboration before I was actually there because of that occasion.

45:00 He was a great drawing. Yes, I was going to say, he obviously drew a lot of people together at the right time. I was just commenting on how badly my brain had been scrambled by yesterday's very exciting session that I got up and walked out of the restaurant without paying. That's your excuse. Yes, you can say it was that. Well, those of us who had actually stood the barrage the whole day, I think, did quite well to remember where our wallets were. So we're just pulling John's leg about now. Well, I mean, I just, my brains were so scrambled by the discussions that I forgot I had to pay for my meal. We've actually joined an army of engineers. We work with earthquakes. So they need to go and check all these buildings and see which ones are worthy of the story and which ones should be destroyed. I mean, I have a horrible feeling that this is going to pale into insignificance compared to what happens when that happens. It could happen any time. It could be next year, it could be twenty years, it could be five hundred years. Well, it won't be five hundred years. Well, one can't. It's only statistically likely to be sooner. I mean, the last thing in 1944 was bad, but it just sort of, the lava kind of just flowed down the mountain, and people could stand there and watch the evening of their houses, but they didn't have to. Actually, the ash on that, because I spoke to quite a number of, you know, my veterans when I was doing the American World War II veterans tours, who had been in Naples in 1944 during that earthquake, and there was one guy, I remember, who showed me all of the photos he'd taken at the time in Pompeii, and they actually had a...

47:30 You know, a couple of U.S. Army brigades, engineer brigades, stationed in Pompeii, and the streets did fill up with ash to a depth of about two feet. In Pompeii? In 1944? Oh, yes, yes, in 1944. The streets filled up again with ash to a depth of about, not so much ash, more a kind of clinker. Yes, very fine pumice. Very, very fine. And he had photographs of this. It was astonishing. Well, I mean, that was bad enough, but it didn't take any lives because... No, because... Well, it may have taken more than we think. So many people were dying in Italy at the end of the day. Exactly. They didn't destroy Naples, which the next one could quite conceivably do. It just happens that a week ago, one of the state's hotels... There is a problem with predicting volcanoes. They were bragging, but finally they did successfully predict one, a huge one in the Philippines, but successful in the sense that they warned people in time that most people escaped. That was a real success. Apparently this thing is successfully predicted also. Well, to be fair, they're up against, I mean, if you evacuate a whole region of the country, then it turns out that they don't. Yes, for some reason the seismologists are very resistant to using this, whereas the other geophysicists seem to think it's already quite a reliable indicator of an impending earthquake, because they've observed a sharp rise in the levels of radon for three days before this thing. He noticed the rate of us on Blackbuck.

50:00 So that will take note, and somebody else next time will use that. But you've got a problem, I mean, if you evacuate Los Angeles, say, and then nothing happens. By the time you get everybody out there, you say, come on, Earthquake. Okay. Well, that's the problem of getting, you know, a mass public to understand the nature of statistical likelihood. Okay. You're not going to come to any of it, then? Not today, I think. Hang in there, mate. Okay. See you soon. I'll definitely be back by the end of the week. Oh, sure. I'm hoping that today we might, well we wanted obviously to give Anders and Davide as much since they were both leaving. As much opportunity as possible, so we spent the morning on theories of integration in SDG, which was fascinating, but which did fast become, well, for me at any rate, very, very technical, beyond my ability to do more than just identify the main features of the landscape. And then in the afternoons of Edie's stuff, which of course is much more accessible, but which turns out to be very subtle, very rich, this whole theory of dimensions in continuum mechanics is utterly fascinating, and I had not realised just what a very mathematically rich subject it is. Well, John is an example. Yeah, so that was fascinating, but also a little bit more accessible too, so all in all, a great day. I'm hoping today we might get onto the slightly broader brush stuff about logic in the light of... I hope you don't mind asking, do you know when I'm going to be able to pick up the hundred and... It doesn't have to be now, but if I can do it in the next day or two, it would be very helpful. That would be brilliant. Oh, pounds would be much better for me. Okay, thanks ever so much. Thanks. Yeah. Avanti. Avanti. Au revoir. Au revoir.

52:30 Despite the fact that you can buy books by the handful of guys telling you what it's all about. So I was just saying, one thing that Tom yesterday really convinced me of was how subtle dimensions are. Dimensions. Dimensional analysis. Dimensional analysis. Yeah, that is really difficult. Conceptually, it's difficult to get a handle on what you're trying to do. Well, for me, anyway. I hadn't realized the mathematical depth of this. It's just part of nothing. It's just an underpinning. No, no. I mean, this business... There's a number of books on how to control it, like the magnetism and other things. Yeah. It's just a matter of convention. Yeah. Well, that's always reassuring. Have you ever read Maxwell's article on this stuff? I know that he did write an article for Encyclopedia Britannica. I thought it was a general article on electromagnetism. I think, in fact, the edition in 1870...

55:00 You have an each one and make it into the kind of thing that it's supposed to be, so it's sort of, you sort of generate, of course algebraic in the sense that you have infinite unions, but still algebraic in the sense that you're putting these things together to get something like that. And in the same way, the Tikhonov theorem, quote, unquote. In this case, the apparently rational correspondence is missing something, because the product of compact locales is compact, whether the axiom of choice is true or not. It's only by... you see, so the existence of making the product into a space is... You didn't have to use choice, particularly. That's right, the choice, absolutely. But more basically, and that's a symptom of it. The points of a locale, you know, you could define the points or the homomorphisms into the locale on one point, you see. In general, the locale may not have any, but in particular, the product, it's a product like that. So the actual set of points of a product locale is not the product of the points of the factors. It's sort of got some more. So the theorem screws around to this, that the Tikhonov theorem is always true for the locales, but if you want it to be true for the corresponding spaces, you have to have choice, because choice is always a kind of existence of extra points thing.

57:30 You don't know really why you have those points, but you're saying that you always have enough of them, and then you have enough of them, that's right, yeah, yeah, yeah, yeah, yeah, yeah, yeah, yeah, yeah, yeah, yeah, yeah, yeah, yeah, yeah, yeah, yeah, yeah, yeah, yeah, yeah, yeah, yeah, yeah, yeah, yeah, yeah, yeah, yeah, yeah, yeah, yeah, yeah, yeah, yeah, yeah, It's about the, you see, you always, if you have a topological space, or in general the points of a locale. It always has an ordering, actually. The Talsor space doesn't, but the other ones it does because you can say one point is less than another one of every open set that contains one also contains the other. So you extract an ordering on the points from the structure. You can say it's because, well, locales are really posets and the single point turns into the poset with two elements and when you hom posets you always get posets. But in this case, the post-set that comes out is, it's a filtered, it has filtered, the filteredness is, well, taking the case where the filtered co-limits exist. Is it easy, is it easy to say, do topology on? I mean, is it, would it be a natural way to explain topological notions in that circumstance? Well, I mean, it's possible. Some people think it is, because I have additional cohesion and variations.

1:00:00 In general, in addition to the one that you're looking at, many contexts... Well, there are people even so extreme to say that if you look at R3 correctly, it doesn't have any global points, it only has variable points, in which case... I'm not thinking of any new way of that sort of adaptive algebra, but it's always the amount of three to which that would be susceptible at all, so... No, I don't know. It's too complicated. It would be too complicated. Well, probably the people doing the teaching wouldn't know about it. Yeah, but that's a different matter. A lot of people usually didn't know the topology. I know, but that's why I'm saying it's not as simple to say it's too complicated. I mean, if you're going to say that, at least you've got to have a starting point where your lecturers understand what the choices are. But my impression is that it probably is too complicated, at least from the starting point. And so on and so forth. Certain topos themselves think, again, if we worked out the rules of entrance and everything so that they didn't blind us to the content, it could be done. But then, indeed, the localec theory of R3 inside the topos is several levels of complication that probably... They don't know if he actually succeeded, but they think stick figures.

1:02:30 Could that maybe add another element to the formal logic? Yeah, because that's, I mean, my impression, which I'm pretty wrong, but my impression is that all the logic I'm seeking is more of a distraction than a help in terms of understanding geometry that's going on, because you're adding this extra level of stuff that people have to understand in order to know when you really don't need this. I mean, having taught this stuff, I think that would be... Many steps to be one step too far, leaving everybody behind, right? I mean, I'm confused. But, I mean, R3, certainly you can draw pictures of, like, upset computer science students and mathematics students. Well, I have to teach them. Okay, so he knows where I've been speaking. I believe you, yeah, I believe you. There were very nice students who said, I'm fine with everything until you started drawing pictures. Drawing pictures at the side of one of the legs, at the x-side of one of the legs. It couldn't do lots of media technologies, lots of vision and graphics, and they actually do need to do some calculus. And it makes no sense to have without some pictures. So one can easily imagine, you know, you can draw points and you can draw lines and you can draw coordinates and stuff like that, but you don't need the logic. Well the logic, you see, again, as I tried to say before, the need for a formal presentation, per se, is going to come up at a certain point. Why does it have to be done in a notation which is totally different from the notation that we use in geometry? These two languages should be compatible as much as possible so that the actual contradiction between them is evident and not just opposite sides of the thing. One is continuing into the tradition of the formal, formal way of dealing with these questions.

1:05:00 I mean, Frege already, the fact that there are functionals was becoming very strong in consciousness at that time. But Frege, logicians think that Frege had this fantastic invention of quantifiers. In fact, he says himself, in essence, this is merely the truth-value case. The real value counts as the real variables for other people. Yeah, well, I mean, that's not a thought. He actually openly... He knew it, but the notation from that moment, the notation, the idea that this was some kind of discovery independent of mathematics or something, I don't know, became... Well, there was something to be used to lay some foundation underneath all of that. It's sort of like, so, I mean... Just to use a crude analogy, suppose one day that all the Cambridge colleges are atheists. Will they continue to be called St. John's? It's very likely it is. Well, I think we're at a couple of centuries transition period now, you see. So don't be misled by the past. Just call it John's College. There'll be a moment when people think, why are we continuing to do this anyway? At that moment, it's just sort of the thing you have to learn to be part of the group. But some people will be saying, why do we have to keep on doing this? It's a perfectly serious point. At some point, they must have lost the same question about the Panathenaic procession to the past. Why do we keep on with these rituals? Why do we keep on with certain rituals? Because people like rituals. Yeah, sure. Which is why it happens. The logic has to be done in a language which is... But can I ask a question? Just as there was for many years a myth, you see, that there was axiomatic set theory and naive set theory. Naive set theory can't be formalized, so we have to accept the hierarchical underpinning of the formalized set theory that existed up to that time. But, well, the fact is, naïve set theory can be formalized, i.e. treated in a careful way, the theory of it presented, and so forth.

1:07:30 Naïve logic as well. Naïve is a term with views. It was, somehow. But it isn't. It means just straightforward, obvious, or the way anybody would think about it. Okay, so I've distracted you, so... But the point about the role of the notion of the variable and how this, as it were, structured our understanding of things is a very interesting one, which at another point, perhaps when we come back to the whole issue of very least implicit program of the mind, has come up very naturally. The fact that locales also arise more naturally than spaces while working with topologies. For example, when you consider some of the classes, you can study them topologically, and that is an important point, at least in my work. So, for example, it's also interesting to take, for example, some of the classes in the syntactic category. It is important that one has not to care about the fact that those localities. are coming from spaces, because that is not the case, and one can, anyway, argue topologically with them. If there is time, I will explain the notion of Booleanization of theory, which I introduced, which is the fact that you can always, starting from a geometric theory, which has a certain classifying tokens, if you consider that some tokens are built on double initial shifts,

1:10:00 All the initial theory which corresponds to this sub-focus and that is the humanization and it's good that you can achieve a very simple description of that topological purpose and I achieved that for the first time while working with topological non-supportive practices so you know it's important actually to use topological intuition while working with them. Right, so, I was, if I'm not wrong, I was there because I wanted to prove that this is the classified topos of our theory. So the main fact that we use is the following result, which is very important in topos theory. The fact that we have an extra-lectrical category between the geometric morphisms From a topos E to a topos represented in this way, as she shows in the category C with respect to J, we can have an equivalence between these geometric morphisms and flat factors from C to E, which are J-continuous. J-continuous means that they send J-complexes to epimorphic frames. So, actually, this is a deep effect, so it's not obvious, and... But your syntax allegories have finite limits, don't they? Yes, yes. So in this case, flat becomes Cartesian. You don't need to delete the flat. No, no, no, we don't need it. It was just... To give the general result, yes, to generalize, because in general you see, in many cases, you have a finite limit, but there are cases in which it doesn't. And so, you know, in this case, the flatness... So the functor is basically a property which generalizes the notion of being Cartesian on a Cartesian category, so it's something that specializes students in that case, but which makes sense in general.

1:12:30 But I think we don't need to go into any technical details. When you say Cartesian, you mean Cartesian closed, or do you mean something else? No, no, I mean that it's just finite limit, it has a finite limit. Finite products and finite limits. Cartesian closed simply means closed in the way that the tensor product is Cartesian. That was the... Okay, so it's kind of confusing to be told something's Cartesian and think they're telling you it needs Cartesian clothes. Usually they always are the clothes. Well, that's what she was saying. The flatness is superfluous because her categories are already with finite limits. Yes, in our case it's superfluous. It's just we don't know where it comes from. You're asking whether the name Diakonescu, in my recollection, should be used precisely for that, the fact of flatness. In other words, to find a similar description. What are the good fronters, even if C just isn't that good? There are cases of interest that don't have... Oh no, you often want to take pre-sheets of an arbitrary small category. Yes, yes, yes. So, no, no, that is really useful. Because, for example, there are theories of pre-sheet type, theories which are classified by pre-sheet purposes, which are not algebraic. And in those cases, one uses the same expression.

1:15:00 Okay, but to come back to our case, what do we get from that? We get this example. Let, in this case, becomes Cartesian. Cartesian Jt continues from Ct and L. Then we want to get here. Is that possible? Yes. Basically, it's instructed to describe the part of these words. If we start from the model, then the corresponding factor sends the given formula in the domain of the syntax order. In the model, the model is the dot. And one can check that this correspondence is actually . So it's something very very natural. And this is also a very important point because to have a simple description like that enables you to... Also to study what happens, for example, if instead of J-T continuous, you take something like J continuous for a topology which contains this, which is what I have done to establish my duality theory. You know, they are simple as correspondences. One can reasonably expect to find that things which are natural here correspond to things which are natural here. So, I would like just to remark, you remember, unfortunately I had to cancel that, but I can just rewrite it. Do you remember that sequence in the definition of syntactic ecology?

1:17:30 Well, basically, let's interpret that one in these terms. Let's call it the sequence like this. A model M satisfies this sequence if and only if the corresponding flat counter here sends that, so of course I don't want to rewrite all that stuff, but you remember that seed that corresponded to this, sends that seed to an economic framework. So that is the crucial point, because the sequence is natural logically and it corresponds to this notion of J-continuity of a factor, something that sends a secret to an epimorphic frame. So we can rephrase logical properties in a factorial way. That is the key fact. So we have proved, well, we have proved, I will just give a sketch of the curve. So that is, of course, an important fact because we have that, but we have also a very explicit and natural description of it in terms entirely, in syntactic terms, involving the theory. So this is... and also, you know, it's natural to see...

1:20:00 Once we enter that, one can wonder what happens if I consider extensions of the theory deep. Do I get some topos and everything works perfectly as you need? And this is also interesting because we have a lot of topos theory that can be done involving some toposis of these topos and we can lift all these topos extensions. So we can introduce logical notions, a lot of things entirely coming from geometrical motivations of the elemental forces, and also we can obtain, by using the transformation there, we can obtain syntactic representations for the classifying purpose. or many extensions of the theory in a similar way. The only thing is that whenever you have a quotient, so if you have T, then if you consider a sub-topos, sub-topos correspond to topologies which contain this one, this well-known fact called toposphere, so you can write it with sub-topos in this form, where this is the canonical geometric inclusion. So what one has is that it's very natural to wonder if there is an extension to cry of T which goes here, so such that we can compute it. And it's really possible.

1:22:30 It's like axiomatizing. Exactly. But now this is the thing we were discussing before. I don't believe it's the case that every subject... You may need to change the language key by drawing more numbers so it's no longer a theory in the same language as it started with. Of course, and again, like every such category, it has such a presentation, but it's a different one. It's stronger, not only in respect of having more actions, but having more structures to say the actions about. Add to the signature. Because the simplest example is, say your first signature has one name for a map, and you want to say, and so the classifying topos is just the classifying topos for that, which I think is all pre-teams on the category of maps and one-access programming. One last subtopos is one where this map is invertible, or is an isomorph, and that is even expressible as an axiom. The name of the statement, again, for all y there exists a unique x, which is on the graph, but you don't have any name for the reversed graph. You have to add it to the signature as well to catch the most definite case. So even from the point of view of phonological presentations, it seems to be an actual problem to know which subtopos we are describable without imagining it. It's very much like, I mean, there's a simple case in algebraic geometry. See, the topos is like the, in fact, there's a special case. That's the case that we have here.

1:25:00 The topos of a certain object are like topos for plain objects. In other words, it's actually all the functions from finite sets to sets. This is a topos. I call it R to remind us of real numbers. In fact, with everything you've got, in other words, the continuous functions on a general S space topos, these are exactly the objects of the topos. There are all the, well, you like the she's on space, like the same thing as an organism, but the she's are, we're talking about the whole organism, just the same thing as objects, she's, means objects, so we capture the general, the general, the one special example, the generic object in any, in any, well, now suppose we want to say that this, this now is. Now, a slightly more complicated one is where we have the math classifier. It's like that, except that instead of the category of finite sets, it's the category of finite sets to the power of gold, too. And that contains a generic map. In this toposem, there are two objects and one map between them that have no properties whatsoever. The most general thing is generated. So then I'm saying there's a sub-object, a sub-topos of the math classifier where that generic math is inverted. And like every topos, it has a presentation in the style that she's describing, but it's not a sub-presentation of the presentation of this. It's just a syntactic pattern.

1:27:30 You mean because you have to put in a name for one of them? You have to put a name into it, but that makes a big difference. That the syntactic of a point of view has to already vary in a genuine and categorical way, not just in terms of sub-theories, in order to account for what it's trying to do. I've got a really, really naive question here. But, I mean, I'm thinking in terms of conventional logic. I mean, it seems to me like there's something somewhere in here, Lindenbaum algebras and Heide algebras are, that we're looking at the phenomena of Lindenbaum and Heide algebras and the syntax, these syntactic categories intended somehow to take in the, take in the, the… Apparatus of defining satisfaction, basically. Am I completely wrong on this? No, it's totally related to that. And, see, those limit-bound algebras... Turn out to be the truth-value objects in the... Well, yeah, of a non-trivial... So the relation between a non-trivial theory and sort of giving no atoms or something, it is that you're getting the algebra of truth-value functions. Topos, by contrast, is algebraic set value functions, which becomes a category, not a poset, because ses is a category, not a poset, and so on. So that satisfaction gets mixed up with the truth in that way. They're not two different aspects of the same sort of thing. In fact, in topos, there's also the truth value thing, which is small. You take the sheaves on that Sierpinski space, so there's a space that has two points and two open sets. That's a topological space. It's a sober space. And so if you can look at sheaves on it, it's the classifying topos for what? Precisely classifying topos for a proposition.

1:30:00 And so you can have on any topo Z, you can have the continuous mass into that, and that then is the omega of the topo. The general truth values of the topos turn out to be the Sierpinski values of the continuous functions. So what's story me here is that I thought the whole thing was this. The idea that in the topos you've got a kind of generalized truth of things. You're saying that's not what's really... They're just variable truth of things. The world is variable. At the same time, the locale topos are not at all typical. So there's stuff going on in there that you can't see in the coin. In fact, the most classified topos... The place where the syntactical category plugs into it is sort of far away from the truth value part. It's another part. I just wanted to come to you, I wanted to make a strict analogy with this thing. When you try to say, look at the line, the line is having... The algebraically continuous maps in the field have this many functions, and so if you want to pass to a sub-object of the line and describe it allogically or algebraically, you see right away that there are two kinds of sub-objects, because you could have, you know, x squared equals 2, and that's describing the closed set that consists of two points. And that is a genuine, I haven't changed the syntax, I mean the algebra functions on the two points are all named by the same old polynomial, it's just that lots of them are equal though, I'm careful, I'm taking approaches.

1:32:30 On the other hand, if I want to, if I want to say that an expert is not equal to two, in other words it's an open set, it's an open set instead of that finite. The algebra functions on that I have to get by inverting. So I need a new name I have to localize. It's the whole thing about localizing names so that you can describe the algebra in certain terms and functions. The open subsets. That will almost always involve introducing new variables, i.e. new elements of the signature that went into presenting. So my idea was if you take all the small precepts from Jay and if you apply that transformation about what do you think? The point is that if you see if you wanted to have a categorical interpretation of these positive logic The topos is too strong or too rich, not too, but I mean it's qualitatively more rich than what's appropriate. So you have ways of constructing things in the topos. In the topos, it's a theorem that if... I have a map, and for a while there's a unique accent in the internal language, and it's true, provable. Then there really is a map there. There really is a map. No, there is no such thing. I'm just talking about the topos, no matter how I got it. So the fact that the map is really there intuitively makes you think, well, okay, fine. But if you want to go back to presenting the things in terms of theories, you have to add a name for that map. The theory has no name for that map.

1:35:00 It has names, complicated names for sub-objects, you see. The formulas are always names for sub-objects. So you can describe pretty complicated sub-objects all over the place. And the maps that you were given as part of your syntax from the start, but you can't describe all the maps. That's right, so you don't need logic in the sense of quantifiers and so forth to do that, because that's sort of purely algebraic, that's for any category you can imagine, you can invert elements in the world of categories, but the point is we want to describe as much as possible in terms of logic, and we bring in these existential quantifiers and equality. And entailments and so forth and so on. We can describe a lot of stuff. We have names for lots of sub-objects. We can make lots of statements about when these sub-objects are equal or when they're not. We have no way of producing new elements. You can sort of define the graph, but that doesn't give you the object. Right. That's what's going on. Right, that's what I'm saying. Ah, okay. As an example. Whereas the topos, per se, does give you the math. There's a contrast between Penrose and... I mean, there was this notion of geometric category that you proposed. It doesn't satisfy that theorem that topos do, and so you will have everything that will exist in the generic geometry. All of these categories will exist already without changing the language. So the traditional logical thing of adding Skolem functions or witnessing constants or something like that... Well, the problem is that all that stuff is... That's expanding the language, right? Yeah, that's expanding the language. So you can add Skolem functions... You can obviously, I mean, no problem to adding... Enlarging signatures, there's a category of theories, not just a subset of sub-theories, everybody can imagine that, but the point is that some of the morphisms in that category are monomorphisms, they count as, they should count as sub-objects, but they don't, they're not in the same language as the thing that they're supposed to be, so they have to change it.