Conversations FW Lawvere — restriction of toposes to be generalised spaces (to be localic / QD)

0:00 I'm so used to this. I'm trying to think what is our best route from getting to the Niport. Shall we be lazy and get ourselves a taxi? Fine. Okay, let's do that then. Oh no, I think he's saying he's taken by company. They're not here. They're about to happen. It's a little bit awkward to... It wasn't raining, I'd say, yeah. We're at the meeting in the PA Prize for the Ocon applications in category theory and... I didn't realise that there was a video on that. That's right. And there was the, yes, you gave a very interesting paper on, well, really on catharsis and graphs.

2:30 No, no, no, that's a different book. No, you're right, no, that is a different book. No, no, no, no, sorry, no, no, you gave up my time. No, no, no, no, sorry, no, no, you gave up my time. No, no, no, no, sorry, no, no, you gave up my time. No, no, no, sorry, no, no, you gave up my time. No, no, no, sorry, no, no, you gave up my time. No, no, no, sorry, no, no, you gave up my time. No, no, no, sorry, no, no, you gave up my time. No, no, no, sorry, no, no, you gave up my time. No, no, no, sorry, no, no, you gave up my time. I think he is, yes, he is. Well, anyway, in the course of that, not in the course actually of your own paper, but in the course of the general introductory remarks that were made by Gonzalo or by... What did you get in the past? I'm afraid my Danish is... All right. Hi, good morning. We'd like to go to Bruneiport, please. Bruneiport? Bruneiport. Bruneiport, sorry, my pronunciation is Danish. Newport. Newport. Newport, yeah. Newport. Actually, do you know where the Romeskade, the workers' museum, is? The Rumeskade, yeah. The Rumeskade, okay. If you could drop us on the term of Rumeskade. No, okay, then it's not the Newport. Oh, I'm sorry. That's because I gave you the wrong direction. The Newport is right there. Oh, what am I thinking of then? What's the name of the... what's the name of the... Norport! Norport! Okay, it's about... Norport! Norport! That's... I'm mispronouncing things. Norport. I'm supposed to be the tour guide. Norport! I knew it sounded like that! Yes, Norport, the... You said Newport and Corby's were home. It's right there. Okay, well thank you for correcting me. Yes, the Norport is what I should have said. Norport. Norport. By the rumours card, I know. Okay. Because in one direction is the pedestrian street, whereas this... If you want to check out, and there may also be a shop in that direction, and on the other direction is the Worker's Museum, which if it is open we could... No, what I was going to say is there's a remark by, I think in fact by, John McNamara book about a discussion which they had with you in the course of the meeting about how one should define the notion of entity.

5:00 And it's a very interesting answer, which I wanted to ask you about. Maybe if you tell me the answer... Well, of course, what I need to do is consult my notes, which is not very easy to do just at this moment. Any hint might help. Well, it was to do with the stable behavior of things under pullback. It was obviously rather more specific than that. I think the case where you have not just preservation of product and coproduct under pullback, but more restrictive pullback which gives you co-equalizers and preservation of co-equalizers, it wasn't that, it was something a good deal more general and it was the additional, what appeared to be the additional generality that I wanted to ask you about. Well, that obviously wasn't a good question to ask right now because I'd have to look up the notes I made. Oh, maybe it had to do with the idea that through all sorts of different... The definition that's referred to in the Mecklenburg, it's something quite different from that. As I said, I'd have to go back and check.

7:30 This is a house. I wasn't aware of that actually quotation. No, no, that's absolutely fascinating. You know, in his little two or three page, in the midst of the philosophical notebook after he's made notes on Hegel. On Hegel and on the Aleutic school and on, yes, all that stuff, yes. And he has his own little essay on mathematics. I didn't know that. I'm ashamed to say I didn't know that. This is one of the, this is the one where he talks about the vital development of knowledge. Thank you for watching. I'm trying to avoid getting my feet wet because, unfortunately, I've got a great big pool in one of these shoes, and if I put it in the water, I need to get myself a new pair of shoes. Thank you very much. How much was that? Forty-seven. Forty-seven, thanks. Can I get rid of my coins? No, no, don't worry about it. That's therapy, isn't it? I might shake you off on the handbrainer. Sorry about this. No, it's okay. Just tell me when I've given you the right amount. No, no, I won't give you the right amount. Hang on. That's it. No, it's OK. Thank you very much for getting us here. I think I will take you up on that. No, it's OK. I'm going to change it. I was going to get myself. Yes, the same thing about this. I will maintain you on this bench. Yes, indeed. In the statement, this is a house. This is a house. A tiger is a dog. Curiously, the example of a dog, I think, was actually cited in the, in the discussion tomorrow and raised.

10:00 Yeah. It's a dog. It's a dog. It's a dog. It's a dog. It's a dog. It's a dog. It's a dog. It's a dog. No, I mean, the point about modeling is that there's the man himself having just as much of a discussion. It looks like it is over. ...fade his voice to Russian stuff. That is something I've never learned. ...work with fuel supply. ...brandless... ...similar to those wanted for confrontation... More than confrontation. Yes, actually more than revolution. The first must have been an early social democratic, possibly at the time of the First International. Yeah, Lithuanian sculpture. The English Siemens Union brought the statue. Yes, it's excellent. Very interesting. This is the first time I've been here. Good things about it. Okay, that's okay. That's perfect. It's just gone. Oh, there is a reason why I shouldn't take it off. It's basically why these places were set up. Good morning. Could we have two entrance tickets? I do apologise for not speaking Danish, I'm afraid. And my friend here is actually a pensioner, he's like, well he's going to take part in that, he said, well it's all right. No, he doesn't, I have to say, he doesn't sound or look or talk like most pensioners. Do you want to get these? Okay, that's square away for the taxi.

12:30 Thank you, thank you. That's quite the museum, isn't it? Right? Right, thank you. Thank you very much. Here's a poster from the Spanish Civil War, the defense of the three. Why don't we go around the museum? Well, now, isn't this extraordinary what you were talking about just this morning? You're talking about the 17th Congress of the party, the last one before Starling Black. This is a famous one. This is the Danish party. Oh, the Danish party. Oh, at the same time. This is the 17th Congress, whereas the Starling Black was the 19th. Oh my God. That's very interesting. Marxists and Marxists? Marxists. Now that's interesting. I wonder what it is. It's spelled. You can see the proper religions and stuff here. Oh, the picture was hanging in the house and then he got out and took it out of the room. Excellent motion. It's obviously the secretary of physics. Well, they probably took it the right way in 1958 because of depicted style and they led it together.

15:00 This place might be that. It'll probably be that. Oh, no. Not the end of the hotel. So, what you've got on is just your long-distance vision? Right, right. Oh, I see. Got it on the check? They'll probably be in the hotel if I don't have it. Right. You do need them though, don't you? Well, I can just read without any glasses. I'm going to write that down in your actual words. I'll remember. Unless you want me to write it down. Yes, I'll wait until you write it down. Well, you don't normally get that. I don't. Yes, I don't believe a postcard is that. That's the courses. Precisely what they've been telling us for the last few years. Well, he's the man depicted in this poster. Oh, it is the last one. 1972. Leader of the Danish party until 1956. Is the text excluded from the excerpt? Yes, presumably. Well, we'll try to find out more, but certainly from the data it sounds very likely that he was a victim of that. Yes, of course. Thank you. That's a little questionnaire they'd like to ask us about where we come from. Are we male or female? It's unfortunate that the painter in that particular painting does make him look horribly like Richard Nixon.

17:30 And this is an exhibition on the work of a Danish satirical artist called Klaus Albrecht, which I'm looking at the posters of, which is a fine cartoon. May I pay for this one, please? Oh, thanks. No, I was actually just wanting the card itself rather than a snap, thanks.

20:00 So I got a little bit more of that. Well, that's fine. Well, that's fine. So they are all time for Witten to come up with other things. Yes, I wish they also did. I wish they also had a quality to have explanation. It's probably right there. Yes, yes. I was thinking about that. Right, so the business didn't smash it. That's right. That's how it starts. Interesting, isn't he? Ah, yes. Yeah, I think she said upstairs, didn't she? There's got to be a cafe downstairs. It's an interesting building. Ah, here we are. 19... It's 1960. It's obviously very... Crucial year, isn't it? Yeah. Or is this just a sort of question? Yeah. It's like finding these sort of things and then being set up. I think an exhibition which George Bush was ever likely to be taken around.

22:30 I've only heard about it by chance. And of course that was the key period. Oh, yes, yes, yes. Oh, yes, the invention of, and of course, our own, you know, sad, home-grown, you know, cloned British versions of the same, which came along. And, of course, the technology which had largely been developed by the Germans. This makes me feel my age, because I can remember my father's first ever tape recorder, even though it was like that. Because of course it was also the era where I believed that macroeconomic policies and demand management would impact the exportation of our employment. That's interesting. I didn't know that even in Denmark, which is held up as the model of social democracy, that until 1961, people who are in receipt of unemployment benefits could be deprived of credit.

25:00 This is the sort of thing my father did. He was very, very keen on it. Oh, you were the radio amateur. Oh, yes. He was a very, very keen amateur. Spent a lot of his time. Oh, yes. He was a very, very keen radio amateur. He made his own sets. Because even as a child I remember telling all my friends, or even some, the name of Maxwell, and, you know, my father's so great, he was a great influence in that way. They made them sign on the labour exchange every second day. Since in 1950 only one third of the white column employers were unionised, today in England it would be much, much less than one third, it's probably not even 10%. Kind of de-skilling and... ...operated and very, very much to the benefit of smart physics. Yeah, yeah, yeah, yeah. Given, of course, all possible encouragement by a so-called new lady. The Democrats seem to have this idea that women should work.

27:30 Yes, yes. You see, I've always felt that that's the fundamental idea there, you see, is that... I mean, of course, if women work, they should get equal pay and so on and so forth. But the basic idea is... So basically, they decided at a certain point that they're not going to pay a worker, male or female, enough to support a family. So basically, they're halving the wages. And this, of course, is one of the... Or worse, because the women were paid less. Absolutely. Or, of course, forced in working. ...being forced into the de-skilled and non-unionized sector of the economy, mainly retailing, where of course they could be exploited almost, and of course discarded without any kind of redundancy payments. And this of course has been one of the principal ways in which operations have evolved. So you can see it so effectively. Much far more ready for all the basic attack. They are one of the ones that have developed a demographic problem because of the collapse of the Earth rate, which of course is also the result of this, the force of the wave of the world. They typically are the same place with a common middle class and a common middle class and are inculcating this commodity fetishism so that they think they have to have two incomes in order to live there.

30:00 And then they wonder why 30, you know, 30 or 40 years on, they have a virtual collapse of the death rate. Then of course that becomes the excuse for the dismantler. Who are dismantling the welfare system too. Just saying, well we can't afford to pay pensions because the tax base has just come out. This is because we're going to have to, so again you can delay the operation of the law before we can process it a little bit longer by raising the retirement age to 75. Which is already seriously being spoken about in Germany. It's clearly going to go up to the early 70s, you know, within the near future, and the disappearance of the U.S. Yes, this is obviously the 30s. We seem to actually be going back in time in the sequence of, yes, this is the announcement of the war. It's the 1st December 1939, so it's a war break, an outbreak of war between Germany and Poland. Actually surprising how much teamish one can understand when looking at something that's pretty obvious. I've been comparing the famous radio station there too. Oh, you mean the incident when the Nazis set up the incident of the attack on the... Excuse me, the pretext for the... Yes, yes. ...figure war and what not. Yes, and of course it was actually carried out by the murder of concentration camp prisoners who were dressed up in purgatory uniforms. Right, right. You know, I think it was carried out by a thoroughly religious reactionary religious ideology, but not by any means the enemy of religious ideology.

32:30 These kind of things, like also the Pearl Harbor attack, which is similar in a way, these things happen not by... Thank you for your attention. I think it's important to convince these people who want to do something that it might be a good idea, or it might succeed, and so forth, and in that way encourage them and not act as a ruffian. Because all these revelations that they knew how it was going to happen are too many to... Yes, I agree. And of course the same has been said about Pearl Harbor, although in the case of Pearl Harbor it was... Objectively, obviously, the outcome was to get America into the war, which is funny. Right, and then there's something that's related to the industry, which is what kind of society that you rally people around you with. I'm the only one who agrees with that. Yes, I'm not in London at that point. So yeah, this is clearly the sell-out of the checks on the Chamber of Declaration. I think this is the sort of, the non-aggression tactic that he put back immediately after the meeting, the thing that he actually famously waved from the window down the street, saying that he'd come back, but the second time in our lives that he's come back from the window down the street, he's been on it. He's going back a long time actually, it was the second time, the first time that he was referring was 1870, it was the Congress of Berlin, which was 1878, so it would have only been a few very old people, his audience would actually remember the first time. But, well, the Mattiotti is funny, they used the name of Mattiotti in connection with the...

35:00 In connection, yes, yes. Well, this would presumably have been, yes, of course, because it's the name of the foundation, I think, the Matioti Foundation, which would have been supporting the Spanish people, the Spanish Republic, helps in the Spanish war. This is, of course, one of my most conserved papers because it's saying straight forward the communists, the communists set the German Reichstag up against the classic pretext. Yes, absolutely. Again, probably, actually a classic pretext. Partly engineered, partly permitted to happen. I mean, the view that the Nazis actually found the Reichsbank and wrecked it themselves and then set up Van der Lubbe as a scapegoat is now, I think, discredited. It seems pretty certain that Van der Lubbe didn't collapse out of place, but the fact was that, of course, he was very unfortunate in our life. Isn't that simple? And at that point, it was clearly tactic that they go to play right into the hands of the Nazis. And clearly they allowed it to happen. But somehow you see that as a confusing issue. You see whether the imperialists directly ordered this or whether they simply seized the opportunity.

37:30 In many cases, that was an easy opportunity. And in fact, you're absolutely right. That is the main pearl harbour between cars. They're all quite close, aren't they? But I stand by it. Yes, well, that was, of course, I mean, given the nature of the Nazi regime, that was so absolutely not good to represent, but also, of course, I do not believe that many of the people who were in the cell of the Wilbur's business of the Nazi members were so well known back then, particularly in the world, because it was just a piece of fig leaf, a fig leaf that Hitler just obviously needed for internal propaganda consumption, and nobody believed that the Nazi's were not the aggressors. You can say that no individual believed it, but if there is no united force publishing the opposite, it carries the day anyway. One sees that very clearly in this, in this whole consensus of which CNN, I thought it was very much the insurgent consensus that Iraq must be attacked, even if we don't know why, it must be attacked, but exactly, no, CNN and the newest neo-gubernatorial organization. The original project for the Gulf War was also a project because the American government wanted to recover this bit of stolen property and we'll just stand by and enjoy it. There are many other parallels, yes. I mean, of course, there's going to be official lies, I mean, there's going to be violations, there's going to be, I'm thinking, there's going to be, there's going to be, there's going to be, there's going to be, there's going to be, there's going to be, there's going to

40:00 Thank you very much for your attention and we look forward to hearing from you again soon. Thank you for your attention. And giving her the information to say that, and then of course she would say that on the pass and stuff, so it could all be blamed on some very peculiar level of administration. Again, in that case it could be done in such a way that she could never actually truly say that she'd been ordered to do this. No, no, as you said. No, exactly. Very striking. This is interesting. It's a typical Danish kitchen of the, I guess, the forties, thirties. Oh, that's an absolutely classic example, yes. That whole thing which was clearly set up, that was clearly set up from the beginning. And these poor patsies, the guys who sat there shivering literally almost pissing themselves, you could see them the day that they appeared at the press conference. I mean, they had just got enough brains to understand that they were silent. And of course several of them were, you know, committed suicide just before, you know, being arrested for fake commerce. There must be, as it were, an end game at which we formally liquidate what hasn't been long, long since there has been a share of, and that is the pretence that there still exists a such disorder. The bombardment parliament. Yes, yes. There must be a gear shift, a step change in the whole scale of the week, wouldn't there?

42:30 But again, would it involve a formal liquidation? The actual conspirators in general and so forth probably actually thought, you see, they had been convinced somehow that yes, this would probably be a good thing and might even succeed and all sorts of stuff like this, but... If they are sufficiently isolated and given a line for a while, rather than being just directly paid by Washington to do this, they somehow were a force. I suspect that what had happened was that they had been, of course, been given encouragement by Gorbachev himself, through deniable channels. Gorbachev himself probably didn't know the entire picture, but understanding enough... To know that this was intended to engineer a coup which would fail, which would end in a complete elimination of the people setting it up, and that would then provide the pretext for the formal elimination of the party, and because of a formal winding up of the empty shell of the, you know, it's gone since, I don't know, eaten out from the outside, you know, socialist orbit. Yes, and at that point, as it were, the exporters could really move into... Super-profit mode under the pretext of violating the energies of the market or benefits of the market, but in the case of the Soviet Union, it's been more of a to-see-alike expectancy, rather than to-see-alike to-see-alike to-see-alike to the mid-40s. This is the original stairs, you know. Yeah. We'll have to go back to it. It's fascinating how these old 1950s... One thing that would be immensely educational would be the articles on all these cases we've just gone over. Well, there's certainly a very strong pattern. People are aware, as you say, they're different from one another. They're different from one another. Sometimes they may be explicit words, sometimes they may be merely the exploitation of a situation, but... Of course, notice the way that the so-called conspirators, actually the patsies who were set up in the Soviet Union at the time of the so-called anti-Kolchok, who were always referred to in the CNN media as the conservatives, or even as the right wing.

45:00 In fact, that was so good that I think they almost gave it away. I think all of Kenya needs population six to make things look a little bit more spontaneous. But then if you get the cross-bunch of countries, then see if that's how things have been. Yeah, right, that's the declaration of the key. I think I told you, we had to be in Kenya at the time. There was the Pan-African Mathematical Congress. So we were watching this on CNN in the hotel. But it was so crazy. For some reason, there was a CNN camera set up on the roof of the building, which had an excellent view of the train and the tanks. It was so clear. Perfectly scripted. How did that happen to happen? So we had our own production. What I always found amusing was the small fragment which came out later. Yeltsin sent a letter of thank you to Pizza Hut. Yes, I heard about this. Pizza Hut delivered pizzas to all its supporters. Pizza Hut delivered pizzas to all its supporters that surrounded the tank. Oh, God. Well, what's Marxist saying about it? The second time is fast. Well the third time is farce on stilts. The site of the monument which I remember going to, my one visit to the Soviet Union in the 1970s, which marked the spot where, the point at which the Red Army broke the German Marshal's Mosque, which contained the...

47:30 At this place, no, at this hour, the date, the 7th of December, no, actually it was the 6th of December, the day of the Cornwall Marble, and I think what we have had on this rock actually broke. Well, there is now, that monument now stands in the car park of a large drive-through McDonald's. I saw an article about it about six months ago. Yes, it's now. They haven't actually taken it away, but what they've done, of course, is to empty it of all significance by building a McDonald's car park around it. As somebody once said, a truly revolutionary, a truly counter-revolutionary, does not demolish the monuments of its predecessors. It leaves them standing, but empties them of all significance. Ah, so we are in fact kind of going backwards in the time with regard to this. Grain for bread, not for this. During World War I the Germans imposed a submarine blockade on Europe. Well, not quite. That's a slight confusion. It was actually the British who imposed the blockade. The Germans were of course attempting to break the blockade through submarine warfare. Leading to an acute, yes, there must have been acute food shortages in Denmark. A very ban on serving alcohol was issued. Yes, that seems understandable. When the alcohol trade was re-established, taxation had increased tenfold. Yes, but it's never gone down much since, because alcohol is still very highly taxed in all the Scandinavian countries. Less so in Denmark than in Sweden and Norway, where it's astronomically taxed. But of course the consumption of beer has rose steadily. There must have been very high levels of emigration during the 70s and the 80s, 70s and 80s, I think.

50:00 This is all explaining how the, because they had to work on piece rates, how the dockers, the gangs, who loaded and unloaded the ships, who operated the docks, how they were administered. In order to, obviously, to maximise the returns as they're all being paid on piece rates. They were actually paid by the tonne of goods unloaded because the safety standards must have been important. When people died in accidents, toiling accidents, 100 kilos, 3,000 sacks a day. Yes, all these things which of course are completely ignored in the way that the history of the real economy is taught. The point is that all of these components are related, and that's why there is a sense of belonging and belongingness in the havens. Working class family for time, I guess. Under the neutralization in the 1800s, the last part began with the power of the steppe or the power of human beings on the steppes. The newly built ships were in need of the same large deck capacity as the sailing ships.

52:30 But to work in the havens, you had to be a layman. Well, that's how my parents certainly lived. I mean, they were never able to afford to paper the place until the old paper was completely worn out. Apparently this, this flat was actually in this building. Yes, sorry, sorry, it's gathering, yes, so they left, and the lady who lived here was, um, left it, and she never found the entire flat, which is, yes, the one that was built right up against, it must have been a, after Wilde's father, yes, led a group of, there was somehow the first kind of British, I haven't known him, I thought he was a very interesting man, he was, uh, one of the greatest, uh, surgeons, great, great, great, great, great, great, great,

55:00 He was the president of the Royal College of Surgeons, and a very eminent anatomist. He did a lot of very interesting, a lot of very interesting things. He was an eminent surgeon, and a very important man in the West. Yes, and then he also led you all the way, you see, was to go even beyond Ireland and even to these barren islands. This was the way of the Gulf. This is how I imagine it. I have disdain for that view. Well, because this all happened long before the Roman Empire. Oh, yes, the false there are probably contemporary. And you are already seeing pretty well what we've seen if you've gone to see Triad. Mycenae. Yeah. Same period. They're probably getting back to at least 3000 years. Right. More. But why the Bronze Age? Population would have needed some type of qualification, but particularly at the far western edge of the country, we just don't have enough of that, the movement of populations at that time. But the skill with which, I mean, in the early 20th century, we were able to address those issues there, and it's very elaborate, as you came to the point of being involved.

57:30 Yes, I know, yes, and I don't expect why we were going backwards. In fact, we might be liking to visit and have a coffee. Are you still... Well, I was thinking since they serve here. Yeah. Well, that's exactly what I was thinking. Okay. We'll have a coffee. Two coffees, please. We'll try that more in a half-hour. I'll get those. I'll get those. I'll get these. I'll get these. Oh, this was one way of doing it. Although I would like to learn a little bit more about the actual political spectrum of Denmark. But I guess you can always learn about that from books. I should see the houses of these new working people. Would you like some cake too? I think I might have a small sponge cake, yes. That's a very interesting building. Thanks a lot. Take out the reference in Machiavelli and Reyes to these discussions that you had with him ten years back about.

1:00:00 Definition. Useful definition. So that's just very briefly. About pullback in the category where it's restricted. Rather than having just general pullback, you have restricted to preserve product and co-product. And then seeing the further case of preservation of co-equalizes. In these categories, which you wrote about in the Ironberg body article, In the last section, the so-called Eikandil, or more specifically the non-trivial Eikandil, where you've got the behavior, you've got the action of a group or a monoid, which means that the space is no longer the way of the quotient and the action as well, so that co-equalizers are not present. What, well two things you mentioned, one the possible connection of this with the understanding of the behavior of ergodic systems, quantitative and mathematical, and secondly you were explaining that there's an important sense in which the behavior in the case of the action which is what prevents the space from being a real quotient also has this geometrical meaning.

1:02:30 It also has the meaning that the domain is, in some sense, varying internally in a way that prevents it being a set, but the distinction is essentially related to the geometrical structure of the maps rather than to any distinction between sets and classes, other things between sets too big to be sets in the sense of a distinction. And you refer to the, what you call, the two topological, I mean two in the numeral two, two topological Now, what do you mean by this? Classes, in general, I mean, things that are said to be classes are never going to be abstract classes. I mean, you might imagine abstract sets. Classes will tend to be categories, actually. You know, you could have a topos of open sets, so to speak, as opposed to a mere poset. If you look at, say you take a Grosvenig topos and you look at the category of its set value points. So this is, in fact, the class of all models of a certain theory that presents the Grosvenig topos. But as you see, it's more than the class because even at the level of points, you have filtered coordinates. Friends, strangers to each other, you can take a look. And of course, it's that kind of topology, quote-unquote, which is the original example for James Scott, were really just about that, you see. There were open sets, but these were merely derived from the fact that... And the Scott topology... Oh, I see. You have a set with filtered soups. A post-set with filtered soups.

1:05:00 You can make that the points of a topological space. Now, the trivial way is by Alexandrov, where you simply take, now, a closed set, so let's throw the set, whichever it is, as open. But that's not going to give you any closed sets. But then, Schott said, well, actually, if it's going to be the points of a topological space, it will always, in fact, have tilted suits as well. I mean, it could be that the ordering is trivial. Even his discrete sense, you see, trivially has filters here, you see, you know, right? So anyway, so Scott said Scott opens, or Scott closed sets, or those which are not only closed in terms of going that way, but closed undertaking. So there's a more refined notion of open sets, but it's a new hiding algebra of open sets. So really the whole business about topology being involved in Scott domains and the like is tacked on later. I think it's all an experience because the real content... Scott's construction was already with these special things, which could indeed be viewed as special topological spaces, in fact, the essence is merely the fact that the points already determine the topology, because the points are not distinct, they have to be filtered too, because there's no deeper notion of cohesion than that. It's a sort of loose notion of cohesion, but nonetheless a non-trivial one. Well, the whole same thing happens at the level of topos. Most of the topos always have, in this time, you know, filtered co-limits in the analogic frequency, the genuine category that you have, and the topos itself is behaving like open sets, except, of course, that the membership relation, if you have a so-called open set and a so-called point, then the value of the membership relation is a set, not a true or false, but it's a set. It's a set which is the degree to which you go like this. If you think of the point as a model, then each of the elements of the topos is a type or property somewhere in the higher order structure of the logic of this thing. So the value of the point at that is a certain set.

1:07:30 Right. That's the value of the membership. Yes, I understand. Anyway, so it's clearly two topologies in the sense that we're talking about categories instead of post-sets. Okay, now I think I... yes, now I understand. So it's easy to understand. It's where I'm... yeah. And the general idea is that even if you deal with theories or structures more complex than those which can be classified by topos, at least this much should still be true. That the points, you know, are never going to be an abstract discrete class. There's categorical structure. And probably even, you see, a notion of open set, which would be TOEFL. You might even have something like a scheme in the sense of... A thing which is covered by several toposes, but is not a single topos, just as projective space is covered by affine strings, but not itself affine. Where it was covered by a single topos, and where you had the decidability for sub-objects in that topos, then you would have just got back to the case of it being a set of points. In that sense, a set of points of space. Well, I mean obviously not every kind of logical structure is classifiable by a topos because, you know, it's the kind of structure or some kind of diagram but then it's conditioned by and and equality and exists and or. Even infinite or, but not infinite and, and not universal, and not an embedded implication, entailments, you see, so sort of positive theory, although you can easily imagine non-positive theories. Logicians have been doing it all along. First order theories are assumed to be closed under negation and contain universal quantification all over the place, even if it isn't involved in the actual axiom. At least stepwise generalizing from the purely positive theory towards some more general notion, of course all intuitionistic and anti-intuitionistic and so with a good vast refinement of the classical boolean, but still in the same spirit that the toposes are an adequate expression of the positive logic in terms of one topos of openness. Maybe it's sort of like algebraic geometry in that you can imagine gluing together

1:10:00 No clear pasting together the spaces are individually determined by single photos, but such a thing itself isn't. But nonetheless, you know, it would classify some notion of structure, and because you have these many overlapping toposes, you could still speak of it as having a, quote, topological structure, but too topological, rather than something with a category such as toposes, as opposed to finding out what a topos is. Okay, now I understand where that expression is coming from. That's very helpful. I wasn't able for a long time to understand what was, you know, what's going on. No, no, no, it's not my fault for not having studied it. It must have been relatively coherent, because having jogged my memory, I came up, once again, to that same story. Well, that's not where you were. There was also, it was just the other thing connected with the other, with the further thing there, you see, there was a conjecture, which isn't quite correct, it turns out, but there's this whole notion of, of course, a locale has only a set of points. And more generally, an etendue, I believe, has only a set of points, even though it's no longer a co-set. So, but, I mean, most topologies that you would think of, well, grow topos, this, then you have a class, I mean, in the sense of large, still a category and all that, but a large category of points. So the conjecture there was that perhaps it's exactly the etendues which have a, I even put it in a stronger way, you see, that there's a, there's a sheaf of... Germs. Yes, I remember that. You see it because if you look at, again, at the classical case, if you look at the notion of continuous y-valued function, where y is a given, say, topological space, well, this notion applies to any topos. You've got, you know, y is a space, you can speak of these things. But, moreover, it's internalizable. There's an object in any given topos, which is the, quote, sheaf of germs and y-valued continuous functions on That topos itself is an internal thing and these actual continuous maps are just the points of that internalizability of the toposmorphisms between the two given topos as an object of the domain.

1:12:30 For which wise does that happen? And now think why is there a topos? For which wise are life the locales, the spaces, in that sense that they're still internalizable? I don't think anybody's ever solved this, probably even thought about it. I conjectured it would be the etendue, but then Johnstone gave this very, very simple example of a topos, which I knew well, but I never thought of the fact that it has this problem. Not a topo, not an etendue, and yet it clearly has only a few points. Anyway, the Dirichlet topos, which I think I mentioned in my lecture yesterday. Yes, you did. You did mention Dirichlet. Anyway, just a sassy quirk with a million problems under it. A very, very simple example. Johnstone has this way about it in a sort of pragmatic way. Dismiss something and he doesn't think about it a little bit. Yes, I had noticed that. That's why I wasn't pretending to have died the next year. I mean, you know, a brilliant mathematician he is and I've just been looking forward to studying his great book and his great three-part. This has happened two or three times. You see, he's found an example to a conjecture, a counterexample to a precise conjecture of Moivre, but instead of applying his immense powers to pursue the original question, you see, the underlying question, which the conjecture was only... He's just declared the question closed. Yeah, exactly. He declared it closed because of this particular solution of his own, proposed solution of his own. He's done that two or three times. Well, there's also this whole issue of his conviction that all purposes being quotient to science, and it's a separate issue that we were talking about a couple of years ago when we were speaking in Florence, and I'm sure you're happy about the qualitative distinction between purposes. You know, you might say, well, the point, the thing about a specific, the sheaves on a specific space-like object should be that it has a small amount of mass than any other topos. Now, I don't know if that's really true or not, but it's a sort of natural, from a set theoretic point of view, a natural first thing to ask.

1:15:00 So that's the first question, and then the second one is more specifically, does Aton do the answer? I mean, that's a clear-cut question. I mean, having nothing to do with Hilbert, we need to switch topos, so y of the property for all topos is x, the category of knaps is equivalent to a small category. You'll see it if you want, with the sheets of terms, given the convection, those which, well, if that is small, for all x. There will exist, with any checks, such a sheaf of germs of Y-value functions, but that would be, again, a further sort of exactness property or something, a preservation property of these small sets. I don't know whether that's true. That would be a concrete calculation. But it's been lost to a separate question about... but I'm not sure, you see, whether even this idea of being... Small should be part of the seminar's government, but you know, yeah, and it's not clear at all whether this business, or whether it's the prevalence of Johnston's condition, you know, to have been called some FQD, object of special assignment. I don't think, I mean, please tell me how, you know, but I don't think the distinction between the gross and the physical is still, it's still not really the conceptual character. No, no, that's true. I mean, it's clear that one part of the distinction, one aspect of the distinction is that the gross offices tend to contain parameterizers for the cunning. But, clearly there's an awful lot of, the distinction is also being expressed in terms of the behaviour of the respective spaces, generalised spaces in the case of the Petit-Turquoise space and the Capriza space in the case of the Goldberg space, in terms of the homology and cohomology of all things.

1:17:30 I just wondered if you had any, as it were, overarching intuitions about how the dissension between Grosvenor and Petty is likely to be. The view of these properties is almost no critique of those satisfactions. There are always those topological spaces that have one unique limit point. Except for those, those are the only spaces that pass the test of various axioms. The kind of axioms are very different. They're beautiful. Unifying conceptual ideas is one axiom which is implied in this book. And that's what you said about the... But that seems to be one clear aspect. Yeah. And even containing infinitesimal parametrizes, maybe. Is there anything to be said with this kind of bounding on the nature of the... the connectedness properties? I don't know. Yeah, that's right. So the idea of the five-year-old distance line has problems with that connectedness property. But in a sufficiently abstract way that it doesn't depend on a particular idea of what motion is going to look like. Of course, Rodenby pointed out that essentially there always will be something like motion. If you have those properties, there will always be something like motion.

1:20:00 Connected I have left two very distinct points, yes quite fine bits, and of course as you pointed out in the Cambridge lecture is the case where you've got no vocabulary at all, so you don't get a lot of talk in that case, or... The co-case of that, which is obviously what we've got to keep, the free-becoming, which is not, in a sense, we don't have to pay any attention at all to the space that's parametrized it. Thank you for your attention. Well, once again, constancy of that kind of guy, or constancy in this case, kind of co-constancy, completely random variation, both arising as the way they were made, so once again, it's a little case that's specific to that kind of guy. This fact is a consequence of the idea of preserving products on two sides of the same scenario for the existence of a connected, strictly bi-pointed object. Every object is even contorially embedded into a connected object. For example, at this point, some are two objects. Intuitive ideas here that the general spaces are enough, there are enough of them to, any two island universes can be embedded into a bigger one, you see, or any number of factors, just as a typical example of the issue of two things, which might themselves be connected or not, but in any case, it will be embedded in something bigger which is connected, even in a contour way, which is of all parties. People who think that locality and cohomology are universal are completely ignorant. You can't do anything like that in locality and cohomology. Locality and cohomology will never have a connected right-handed object, except in an extreme general case. Yes, except in an extreme general case. That's just the concept. Yes, and Jekyll Johnson is the one who has a bit of an interplay with all topos.

1:22:30 Or nearly, I mean, it was Freud who actually had to go to acidity to launch that story because he thought it was funny, but all topology are a little bit, you know, imperfectly well, this is right, but Johnson effectively figured it out. All topology are generalized spaces. Yeah, it was he who referred to the epsilon difference. So your general topos has a pretty good surjective map from that. For example, sometimes QD is the closest possible, and then it can be further analyzed, but with less good maps, so the pure locality is really incoherent in some mysterious states, and so on. Essentially, you can cover a given line by this, but by the way, notice that... If you look now at the maps from this one or something else, they're likely from a class, but only a few of them could be composed with this projection they are in the class and could be set at the math. That's the Italian difference. No, no, the Italian difference is that this map is good, in the sense that the inverse of it is faithful and is... It's an open map, so the universe is not only preserved as potential quantification, but universal quantification, so, in other words, the logic, the idea that everything is a space, quote-unquote, i.e. the faster motion of space, but it's based on the idea that everything is logic, really, because... Because the idea is that the internal logic of topology is sort of all there is to it, so it basically represents that, yes, yes, in this, yes, yes, whereas in the case of really cohesive spaces of thought, which do involve cohesion, as in algebraic topology, it's pretty clear that you have to go beyond this type of logic, in some sense.

1:25:00 Inside the topos, but just not logically known, we don't see any differences, just using the word connectedness as a predicate, if you imagine adjoining connectedness as a predicate, your logic is no longer preserved at all, this thing is totally different from what it used to be. In fact, it's very light, you see. It's very much light that even sort of includes, as a special case, the classical idea that an ordinary space can be covered by a zero-dimensional space, which is, by the way, completely counterintuitive. It certainly is, but it's... I mean, it's so counterintuitive, I mean, I'm surprised that people didn't say that the problem was wrong much earlier. ...that led the algebraic topologists to reject the basis of the basis. But of course, the problem is that you can, that any dimension, in fact, the existing dimension theory, in fact, Turevitz himself has a book on this, Turevitz and Wellman Dimension Theory. This is all a very simple, relevant theory, but totally counterintuitive. You take the ordinary notion of topology. You limit yourself to a compact, metrizable basis. The keys may include an arbitrary continuous mass in the potential sense, inverse n is open, is open. Well, you can figure out the dimension of these things, but if you have a surjective continuous mass from one compact space to another, it can raise dimension. You see, whereas every reasonable prejudice says that surely a surjective mass is going to decrease dimension if it does anything. But it's all tied up. The Arnaud's curve is exactly a map from a one-dimensional to a two-dimensional, or as much as you like, dimensional face, which is surjective. Brouwer's theorem was that isomorphism is arbitrary, continuous, but invertible. We preserve dimension. So that's okay. So the theory of dimensionality has not been destroyed. Not been completely destroyed, but took on a totally bizarre character.

1:27:30 Because among the definitions of... ...indimensional. There are many equivalent definitions in this book and other books later, but it can be praised in terms of, you see you have a compact rechargeable space, but then it is the subjective image of a zero dimensional space. But among the possible surjections in a zero-dimensional space, if you can find one all whose fibers have less than or equal to n points, or something of that flavor, maybe it's not exactly, but something about the size of the fibers being measured by this n, well then that shows that x has dimension at least n. So it's completely bizarre. It's all put in terms of the sizes of fibers inside zero-dimensional space. It all has to do with what is such a subjective map. It's a subjective process of searching through the whole space and saying, I'm God, I managed to hit them all. See, so it's completely bizarre in the point of view of what a normal person would think the dimension is like, who was rescued as a rational theory because of the Raoult's theorem that the isomorphisms don't disturb them, but the surjections certainly do. In fact, it's certainly the other way around. The tidiness of the new dimension is not bound to anything. But this kind of cover, the QD cover of a topos, is very much like that. The effect may concretely be that in many cases. These things have a very pulled apart character. I mean, you start, you can start off something like this. Suppose you have a site for your topo. It's a nice, interesting little category. You're just looking at this higher order manifestation. Very nice. But what you do is you take the free category generated by that. They're just totally formal words. So you've destroyed any information on how these maps are composed. Well, you could just already take the sheens on this free category. That would be a QD topos, because it would not work with the other thing, because to get something localic, you just look at the divisibility relation in the free topos, but that's, since it's free, it's cancelative, and so you get monomorphism, which is the crucial thing about the sites or etendus at least.

1:30:00 Thank you for your time, and I look forward to seeing you again soon. That is vocally, that's a morphing too, as a special case, not conversely. So that's why QD is a more natural generalization than just merely locality. But you see, you've pulled these things apart, made them totally meaningless and subjective. You take these mental strains of arrows, you never worry again about how they compose. Yes, what were the domains, the code domains, the original names. You remember the domains and codomains, but how they compose, you just put in formal words every time. So, there's no information at all except the underlying graph of the original thing, so it's been totally exploded in that way. And that's the typical view of spatial space, but it's the same way because, in fact, it's really a special case if you look at a separable topological space. So there is a way to enumerate points and enumerate neighborhoods which are good enough to do the accountable family and so on. So you just take, you know, the point of the three-dimensional covering and then do some kind of filter of these. These little enumerated neighborhoods and so forth. In other words, it's entirely put, once again, into the subjective terms of arbitrary coordinate systems because you enumerate, or even worse than coordinate systems because you bring it back down to the level of rational approximations and so forth. It completely destroyed any information that there might have been, especially in the case of the theory. People didn't bother about that so much if it was just a space, because after all, spaces are just spaces. You think of actual concepts like classifying topos for groups, and you find the covering topos, 3D covering for that, doesn't mean anything anymore.

1:32:30 You've taken all, you know, all homomorphisms that finally represent groups, and you've looked at formal composites of those. You're no longer caring about how they compose, you see, so you've lost almost all traces of group theory. Yeah, I'd actually like to understand the group theory of examples and examples of theories as well, but just staying with the example of spaces... I was just thinking in terms of what you were saying about the separation conditions on the space and the way it affects the counter-arm, because yeah, it was the case in... And again, that is amorphous, but coming from, obviously the case where you've got relative uniform severability is the case, I suppose, where you've got the ethos amorphous actually factorizing, it's like when you've got tris, isn't that right, with the weight of this? No, no, cancellations. Every math, not just that any math can be factored, but every math is already cancellative. It's very unlike in both things, isn't it? No, that's as I said, I'm just thinking as it were, the T thing, for like the group or, not any monoid, a monoid which has cancellation. Yeah. But take a very simple monoid that doesn't have cancellation, calculate the Johnstone and Frye covering. There's a lot of stuff that's going on in the world today, and I think it's a great opportunity for us to get to know each other better, and to get to know each other better, and to get to know each other better, and to get to know each other better, and to get to know each other better, and to This is some monster again, because again, you take all the order-preserving maps between finite, totally ordered sets, fine, there's a lot of information to have those composed, but you forget that information. You just put in formal compositors all over your place. And now you...

1:35:00 What I find fascinating is what appears to be the parallel between what's happening here... ...with this kind of behavior, restricting to the locality case, and particularly thinking of the sort of zero-dimensional spaces in the context of mathematics. And what was happening back in the 19th century was the scatological functions with the space-field curve and the long-term curve. Of course, again, the way that the... I guess actually the way that the piano axioms admitted the problem. We've got essentially what we were talking about at the end of the talk yesterday, this kind of dedicated affinity around these pathological functions. It secures the way that there wasn't a foundational debate, in a really properly foundational debate, in a sense, not the trivial sort of inverted commas foundations debate that followed, no, no, no, it would be also similar, but why there wasn't a proper foundational debate then, at the time of the first drafts, and there may have been, but it was totally suppressed as well, and as I was speculating, it was just so valid in the beginning of this, it was somehow, somehow nice and magic, wasn't it? So they just accepted that we have to extend the original function in precisely the way that, because they had somehow accepted already that the existence of the natural number doesn't lead to infinity, so therefore it's, well, anyway, we also have to accept this piano curve, well, you know, maybe it's, maybe the, if I wanted to say, well, maybe the definition of continuity is too lax or, you know. There's something wrong here, actually, but I don't know. I've never seen it. Because, you know, maybe the geometers, you know, people did come up with these objections from this area, which should really be looking instead at the definition of mathematics, but the way that it just became accepted that the general unrestricted nature of a function is just any arbitrary order, so to settle all the pairs, hence no... There are no obstructions to existence and inverses at all, so no one controlling this.

1:37:30 You see, I think objectifying the subject, you see, is... It's simply a useful way of organizing the study of the actual finite processes in mathematics. But to then turn things around and say the fact that it's been objectified is the main thing, we're always going to talk about, you know, this godlike, well, not even godlike in the good sense of the study, but simply that we can actually... At the number 8, we can follow Piano and the number 8 all across the curve of the plane as we went. No, we were just talking with a young man yesterday. He was quite insistent, yeah, well, we replace continuum by countable. Well, that means it's more constructive. It's something that a computer can think about doing. This is a very deeply ingrained thing. It was ideology. Almost as bad among mathematicians as pragmatism among Americans. You really have to dig. You can see why. There clearly was a turning point at the time of Bjarne and Witt, the time of Delikind and Lierstrass, and the so-called charismatization analysis, where the geometers do, unless there's a whole... Literature perhaps existing in private consulates or in the record conferences that were not, for some reason, ever circulated at the time. But the geometers seem not to have put up any resistance to this. You should hear them indirectly. I mean, why does revisionism exist politically? It exists because the workers have given up aspirations for common good, etc., etc., etc. In other words, if there were no tendency like that, there'd be no need for Kotskianism and revisionism and all these… There is such a disinformation center in mathematics, namely, intuitionism, because, you see, Brouwer was appealing to precisely this feeling that, well, somehow this objectified infinity is a bit wrong.

1:40:00 He turned it into a community subjectivist, so I have to say, I think that he must have been, at least the fact that his work was widely propagated, was probably the response. They sound more genuine, but not quite the precisely formulated objection. And that's my indirect proof. Well, it's a persuasive argument. I mean, by indirect proof, I mean the education is going to go for the instructive proof. Well, I certainly would like to understand more about what happened, because Brahma, as an entomologist, seems to have thought very deeply about the relationship between... Continuity and discreteness. And yet in his, as it were, his official philosophy, partly because he wasn't equipped with this crazy sort of ultra-subjectivist idealism, which toppled over into ultramarginalism, ultra-fascism, I was going to say, instead it got written up as a debate about the finite of the infinite. In both cases, as it were, a subjectivist, a subjectivist idealist understanding of the positions respectively, we'll find out in a minute, but in fact it was clearly something quite distinct and potentially greater or more connected with the objective needs of science, which one sees with fractures on it and has to ask workers as apologists. I'll probably make sure the relationship between the two is discreet. It's funny because his actual work as a pathologist doesn't seem to have much trace of his reactionary philosophy. He's got this theorem of the invariance of dimension and the invariance of domain and some other constructions too. In just a brief period of time, 1910 to 1912, something like this. So, I mean, one extreme cynical interpretation is that he deliberately, being, you know, a genius, was able to do so. He set out to establish himself as an actual mathematician, because his reactionary philosophy was already well formulated before that, in his didactical thesis already. Yes, yes. He takes actual, seriously, position in telepsopsis.

1:42:30 No, that's it. So, somehow, a person with that philosophical view... For a period of two or three years, it's suddenly doing very significant and impressive and objectively fruitful and so on. And then he goes back, famously, to promoting his reactionary philosophy. And it's almost as though he consciously set out to establish himself. That's probably not true, but anyway. No, I don't know enough about Grout to know how far it is just a matter of his individual mental pathology and how far it was, as it were. There were connections to other hidden or not-so-hidden ideological agendas, but I think he's pretty clear that the foundation for that relationship between continuous and discrete, which obviously should have taken place, should have been generated by the whole aftermath of the alphabetization analysis... Nobody forwarded these monstrous and paradoxical functions. They didn't take this and what we're told was the foundational debate took place a generation later, somewhere around some 20 years, about 20-25 years later, in reaction to the work of Cantor. Mathematically, the idea is ensured and the algebras, the geometries that we're already having, apparently are bad in the field, and this insistence on the complete distinction between inclusion and membership. And the canonization of membership was something global and absolute, out of which all the other problems in maths would be taken as derived. And of course that further reinforced the idea that geometry was just some. Derived structure, which had to be imposed by a position of structure on Hilbert space, and some other defining structure of the notion of space in general, which of course in fact relies on treating the intensive conferences as the fundamental thing that's completely ignoring the way that they actually arise.

1:45:00 It is astonishing that it's taken so long to kind of go back and excavate those presuppositions and not find them. And even your own point about Schroeder and Frege, Fiorano, Berserk and this three-fold attack on algebra is something which the historians of mathematics I think are just beginning. To catch up with the work of Abel and one of the other people that I've read, beginning to show appreciation. I've just had a call with Schroeder at the end of the school war. I haven't completely spoken to him, not a sign of him. Even though, in fact, in the 20th century, I'm going... It really was the mainstream of logic insofar as logic was actually connected with... It was the mainstream, not the expression of life. It certainly was the mainstream of mathematics, but it was completely ignored by the historians and the philosophers in favour of this artificial Frege-Russell-Klein lie. I think that in fact the canonisation of that had something to do with it. I'm lying at the hair side of my mind as well, but you can see me of course, who promised you the idea of global membership. Yeah, at least that's my impression. Even though it's quite clear that he was certainly smart enough to see the inclusion of membership in both aspects of the seminar. It was. Let me see, are they tethered? But for a long time, I mean, I remember that the University of United might have seen your paper for the very first time. 20 years ago, there was precisely a remark that you made at the beginning of, I think it's 73, or a sort of logic book, about there are these two ways of analyzing variation.

1:47:30 All these two things have to do with having a domain as a consistent point, and that actually it's good to do with magnetonomorphisms from parts of the domain variation to parts of quantity. Of which the case where it's restricted to be a qualitative point, or to the point that it is purely just a specific restriction. And that's why the algebraic approach to mathematics... More fundamental, because that's what I mean. Where logic is about, where that's about. Yeah, logic in what I call logic in the narrow sense. And then, of course, I suddenly thought about inclusion and inclusion maps and membership. Suddenly seeing membership as if it were a special case of belonging to you. Belonging to you, yeah. It's a tremendous revelation. Somebody has only been exposed to the logic of thought. I think I was telling you, I had this email of debate with Jose Guerrero, and I pointed it out to him, and he's adamantly saying, no, no, no, daddy, no, because you are really crazy. Strange, right? Thank you for your attention. I think when he was really the first, but in particular, I said, well, no, that's because he used the same symbol, you see, for belonging, whether it's membership or not, and so he... No, no, actually he was just confused on his notation. Of course he meant the same thing that Piano Man was going to say. It all becomes, after all this careful citation and chapter and verse and so on and so forth, it all becomes just a mishmash of half-justified project. Inconsistent objections here. I wanted to ask a really quick question.

1:50:00 I did enjoy that talk that you had in Bolzano. It was about inclusion. No, this was about, you know, well, about possible solutions, but I think the context of the whole understanding of the real relationship between inclusion and relationship. Did you have to worry about that? No, that's a pity. You gave me an early version of you and I never saw it. There were so many things you had to do in an unexpurgated version. Yeah, an unexpurgated version. That's why there's all the references to the Medici's and their mules full of gold. Their mule shags of gold. Yes, I remember that. You can actually commission my particular journal. Oh, those things, the thing with Algarve, it's a strange job. I think actually Alberto and or John might be members of the party. John Bell is just a very, very, very, very, very, very, very, very, very, very, very, very, very, very, very, very, very, very, very, very, very, very, very, very, very, very, very, very, very, very, very, very, very, very, very, very, very, very, very, very, very. What else do you think about? I mean, what else do you think about? I mean, what else do you think about? I mean, what else do you think about? I mean, what else do you think about? I mean, what else do you think about? I mean, what else do you think about? I mean, what else do you think about? I mean, what else do you think about? I mean, what else do you think about? I mean, what else do you think about? I mean, what else do you think about? I mean, what else do you think about? I mean, what else do you think about? Yeah, it just seems to be, yeah, well, my impression of reading, I have read a couple of them, and my impression of most of it is that they're pretty, yeah, soft. And then once in a while I'll have a paper, schema, which is quite an interesting, actually has some very interesting insights into it. In particular, I actually want to point out sets and closed boundary components of complex connected spaces, heating, obviously, the connection between the components under the quantum slumber, so there's a kind of origin of the set theory, or the concept of the set theory.

1:52:30 Very nice work, but apart from his paper, which was a good expository to do with cosmology, and one by John, which was just a very, very, very generous expository paper about classical theory. I didn't say, I had to say that it doesn't seem to be much of any value. And I'm sure that you're right about that. Because anything to do with numerology, I mean, because it tends to be to do with the survival of our team. Thank you for your attention. He wrote to me several times and I didn't reply, but basically he wrote me several questions asking about, you know, would I be able to get hold of you and persuade you to go down and talk, and I said, why didn't you go to the hospital again, so he's quite capable of telling me whether he wants to. To speak to you or not? That's interesting. You weren't really speaking about what to him? I did just at the meeting. I said, well, why not let you ask him yourself? I don't particularly want to run errands for your organization. We do talk sometimes. Let's say once I actually got the courage to ask him, you know, what is this coat of arms under your office, are you proud to say? Yes, he said it's the half-blade M-I-O-N. This was because there was a Russian guy who teaches in Kaliningrad, I think. Oh, you mentioned him also. Yes, a man who defended himself very ably against Smith. That's right, that's right. He uses conceptual mathematics, of course. Yes, you were saying something about him. Yeah, but when he visited, you see, he was presenting a certain point of view, and he was proposing to use topos theory to get this far besides him.

1:55:00 So, Barry Smith simply, you know, told him without any argumentation, oh no, don't use topology, you're using variology instead, as you do, it's just, oh yes, use the really highly developed science, use something that, you know. Not just misguided, you know, obviously, I don't think it's always misguided, you know, 21st century machinery, as long as we're in geometry, well, just go back to the 11th century when they really understood, so, so, anyway, that occasion, that occasion, we were the three of us, actually, we were in David Barry Smith's office, you see, because a resident was visiting there, and his name is You've told me it's in the email. I can't recall now. Anyway, Barry Smith's office wall. Well, what is a map of the Habsburg Empire? Once again, his view of Europe is still... He's going to get it all back one day. They believe it, too. This woman, in fact, it's somebody... I believe it's actually the... Remember the Danish author? I was hearing about this just casually in conversation at the hotel I was staying at a couple of nights ago. I'm not sure it's the... His family apparently had estates in Hungary. Or in southern Poland, which were part of the Habsburg territory, which were expropriated. And he's been bringing a court case in Hungary, you know, trying to get back, I guess, the title to the state. And this is already one of the richest men in, actually, France. It's Prince Heinrich. It's the husband of the queen. But apparently that family had a place in the front of the world before the First World War, before 1918. Which even after this length of time, and having lived very nicely indeed with several chateaux in France, and the second or third largest wine shoppers in France, and indeed now being married to Quentin Marx, one of the world's best women in Europe, still wants to get back to the States in the East, giving a lot of money to lawyers to pursue these claims, which eventually were ruled out of time.

1:57:30 Surprise, surprise, even the Hungarian capitalists are not going to open that particular can of worms. If on the other hand it was somebody with the kind of clout of the Hapsburg family, they probably would choose to reopen it, because there they would see there was some useful political gain to them. But that was very, very interesting. I mean obviously the subject matter of Mariology, not part of the relations, may be abstract in some ways. There's an absolute empty metaphysical sense, but the powerful relations in a specific concrete case of the structure of systems, particularly systems where you've got dynamical actions going on, the way in which the parts and wholes do connect and provide templates of stability in relation to other structures is, you know... Potentially, the subject matter of serious science. Namely, what is this? Well, just what you pointed to yourself, for instance, in the way you were talking about science. Getting clear on the actual relationship between... Oh, yeah, yeah, yeah, right, right. But also the... So they start by erecting a series of barriers to clear it up, like denying that there is a set, and then neurological sums, and it's a mess, and it isn't developed mathematically, as you said, that's the obvious reason why you can't apply it to any particular topic, it isn't developed mathematically. Even if it could be at all, they haven't developed it, they just talk about it, especially brandishing the prohibitions that it contains, such as an empty set and the last second set. By the way, all such formulations violate a very, very basic tenet of logic, namely Tarski's dictum about well-formed formulas.

2:00:00 Right? Because Tarski said that whatever kind of system you have, whether it's logic or algebra or whatever, you're presenting things by means of strings or symbols and all that, but what expressions are the well-formed formulas should be a recursive set, you shouldn't be able to decide if a string is actually an acceptable expression or not. Generate by rules of inference from axioms or things like axioms and conclusions, but you can't be sure of, you can't enumerate the non-pyramids in these situations. But, you see, if you claim there's no empty set, you immediately violated that because in building up expressions... You may, in some, whatever, you know, language is, in some translation of taking an equalizer, the mass, you see, the equalizer might be empty or not. But the symbol for that equalizer is not legitimate unless you first prove that it's not empty. And that's a RE thing. So the notion of Bell form formative becomes RE and not at all clearly recursive. This is one reason why, you know, just from the point of view of recursive machining and all that, that it hasn't been developed. That's why numerology is not to get off the ground. You can't develop something that... No, that's the problem. Mathematics has never been... I know that Alberto has tried to take some of the ideas in numerology and develop them in, you know, that's his little paper on what he termed action of structure and the structure of actions. And this really in terms of the behavior of spaces. He had some interesting things to say in that paper. If one worked more in retrocapacities, one might be able to develop some of these principles in so-called numerology in a way that, for instance, gave one an understanding of the case where you have a breakdown of exceptionality.

2:02:30 But I don't know, even then it didn't seem to be mathematically particularly well developed. It's just much better developed than anything that the people doing so-called classical mariology are. Because there have been various people doing so-called systems theory, and logical systems, who've got a total bog down. You know where to get going, don't you? You won't be a train in a couple of hours. Yeah, and do you want to go and get some books? Is this all your stuff? I have some stuff back in my hotel, but that's okay. There's just two doors away. Oh, okay. But you want to go and get the book for the Catalina, don't you? Well, let's go and do that. Thanks very much, Bill. Do you let me know about the dates for the seminar? You don't have to. I would like to. No, I wouldn't do that. Well, I have a shot, but I don't. Um, I don't even have one back in my hotel room yet. So, I'm doing my homework, and I'll see you in the evening. So, myself. No, that's mine. Oh, did you have yours? Did you leave yours? Hang on, put it around me and get it. Did you have your carry bag? Did you have your carry bag? Yeah. Oh, look, this is upstairs. Who are you guys? Thank you! I think most people are still sitting. Intellectual. Don't knock yourself. We'll step in a manual mode. Oh, she said not to.

2:05:00 I like the nice old fashioned demo. Okay. I'm trying to do this without breaking the umbrella. There's a little thing on the side. Lenny? Yes, thanks for reminding me about the climb. I hope I haven't taken up too much of yours. It's this way to the... Yes, don't do it from here. My hotel is just on the corner there. In fact, what I think I'll do, if it's okay with you, is pick up my bag, and then I can go with you to the bookshop, and then I'll just hop the... the subway from there to the main station.

2:07:30 Oh, I'm closing... No, no, we'll do that suddenly. This is a... A Calvinist church here on the right. That's all very superstitious. Another, it's all material. Oh yes, a garden side, a super garden side. As I say, one bunch of, you know, actually, superstitious, superstitious people of the world. Oh yes, that's exactly what it is. It's all, you know, psyching up the day for the support. Trying to turn the so-called war into the so-called war. I'm just trying to think. It's OK. I know where I'm at. It's just the other side of the square. We'll be there in a moment. It's going to be OK. Can I just go in behind and get it? Thanks.

2:10:00 Thanks very much. Thanks for an excellent stay. It's going to be a bit of a problem, isn't it? Is there any way you can take... Well, no, just take that one, I said. It's got papers and books in it. Including a copy of Path, Curse of Space and Quantity by one F. W. Law here, which I was studying just before coming away. I finally got around to doing what you recommended in the paper itself, which was actually writing a diagram, trying to open a diagram all day. I've still got a few things I wanted to ask you about, but they'll keep until I've done some more homework. Yes, well, I think I managed most of it, as there's a variety of things I wasn't quite so aware of, particularly about, just as at the end, about some things to do with the homology of physics. The bookshop is just the other side of the cycle department, and its work is hard to still be, for the young academic research mathematician.