FW Lawvere

0:00 ...that I did out of the seminar. That was absolutely fascinating. I got a much clearer idea of how it was. Well, you knew that already as much as I did, but my God, you're in the presence of a barbarianist mind, you're ever likely to meet him. I do honestly think that the mathematics over him really has addressed the issues in a way that even, well, as I say, in some ways, although... Just probably he's going to complete a book on this Florence. Yes, I'm hoping he will do that. We're hoping, of course, to get the talks in Boston printed as a book. This would be absolutely fascinating. Well, this would make a brilliant... Well, I'd better take what you had in mind. Well, seriously, what he was talking about just now was absolutely fascinating. I found that far, because it was more philosophical. Articulated and conceptively motivated and clear than the talk. I understood quite a lot of the talk, more than I thought I would, especially. For me, I still, I didn't understand some examples, but on the other hand, I had very clear ideas.

2:30 I'm really so glad that we got him to come on board. Of course, I'll be lucky because I'll be hearing all this again in two weeks' time, in less than ten days' time. So I will have recorded both versions and then I can transcribe and he can revise them. So I'll see you tomorrow at 10. Yeah. And if you can bring the recording from the other day. Okay, au revoir. Au revoir. It'll be great. Tomorrow. Yeah? Okay. Oh, fresh air. I needed that. Fresh air. Yeah. Okay. See you tomorrow. See you tomorrow. See you tomorrow, Andre. Take care. Cheers. I'm feeling quite... Well, I don't know. I think I've been a bit of a heel towards Andre. He's not such a bad guy after all. Well, he's a bit thin. He doesn't realize sometimes the things he does, I mean, just how irritating they can be. Like, well, I told you last night, I don't want to go back there again. I think his heart's in the right place. Yeah, I know. Well, yes, that's the problem. I mean, he doesn't really want to, no, he doesn't want to hear about the real, he just wants a nice, easy, you know, understanding, light explanation of his own favorite buzzwords. I think you're right, that's the problem. I have to say that paper he wrote on identity and categorification is just awful. Well, of course, he took that from Byers and Dolan, which is itself a giveaway. In fact, it was rejected by Philosophia Mathematica. Colin was the referee, and I actually saw it. Don't ever mention that. I mean, obviously, it would be very unprofessional. But I saw his referee's report. It was absolutely damning. And quite rightly, it was an awful paper. It's all about the category of animals and the category of pairs and all that. You were saying that the recent book contains a review of Saunders's autobiography by Colin, in which he puts the number of the points of correction which are... Yes, I'm sorry I didn't bring it with me, it came about two weeks ago, so it should be in the library when you get back. In fact, I can bring it to Boston, just in case you haven't seen it.

5:00 Well, in fact, you might as well ask him to bring it, or I can bring it to Boston, because you'll be seeing him, you know, sooner than the mail will get to you from him. I don't think so, no, really. I think Colin and I can take an awful lot of punishment before we get out of your company. I learned so much from what you were just saying then, to put Andre on the straight and narrow. I think that's almost as much as I did from your talk. I would suggest that you let me transcribe both the version of the talk we gave here and the version you'll be giving in Moscow, actually, and then from the two you might want to use that as a basis for writing it up. I got a pretty good... extremely well, extremely well. Well, I hope you didn't think... I was a plant, of course. I did feel I had to ask the question about quantity, because otherwise they might have, as it were, missed some of the payoff. But I think you're absolutely amazing. Yeah, but the idea is, the idea is tremendously clear. By the way, we can cut down there for the Luxembourg lecture, back to your hotel.

7:30 I think the, I think the, the, the... That's, so to speak, the point of the talk. Yes, yes, no, I didn't... In that sense, I didn't actually get to it. But nonetheless. Anyway, my wife, my wife... The 2006 talk in the summer is published in TAC already. Not a reprint, but it's coming up. Right. Oh, I certainly want to read that, yes. Yeah. This talk is going to happen after that paper is done. Okay. And with the background of this talk, you should be able to understand exactly what's going on in that paper. Okay. Well, I shall go back and read that. That's on TAC, you say? That's it. Well, I like the point that you've made. You're now centering on the idempotent, or the condition about the central idempotent, as a way of simplifying and subsuming the distinction of the level of the epi-mono-actuarization. And I think it's also quite helpful that you're now... You know, that you're no longer using the Petit Gros terminology.

10:00 Yes, yes, of course, yes. Vertically. Vertically, yes, yes. It's wonderful, it's absolutely clear. I also hope that it does... Get across, I mean, you know, I really like Peter Thompson, but the wrongness of his going around telling people that all popsicles are calyx. Oh, yeah. Well, he never, he never, he's not that proven. Didn't he? Right. Well, people who have certainly used it have based it on his results. Actually, he's proven. He's proven. The whole idea, popularizing the calyx things. Because he did that. I knew it at the time, this is a diversion. He's diverted thousands of people for years. Basically, he doesn't care much for it. Well, Kallik, by the way, is due to paint it better. Ah, yes. The word. And then he's selling it to somebody else to change the word. Well, I much prefer Peter's, you know, the description of them as, you know, QG. It's quite indesignable because it brings out much more clearly what... Yeah, but that's... That's a different issue, of course. Yeah. What are you saying, Greg? Did I miss something? Well, maybe I missed something. I thought that the... Yeah, okay, so... No, no, no. What do you mean with the epic aspect? Ah... The localic is the monic. Is the monic aspect, okay. Yeah. Yeah, but they are, as you say, okay, they are respectively the epi and mono aspects of that, cases of that, you know. Right. What really helps me is that I see now, actually it's quite a nice little paper from way back, you probably remember, about choice and extensionality principles in a topos. It's a very short paper, only about six pages. We've written it a long time ago, about 1987 or a fair amount before I knew him.

12:30 He makes the point there about the connection with the separability. I'm not British or anything, but... You're, uh, okay. I'm sorry, I'm sorry, I've got, I'm sorry. Well, no, he makes a, uh, he, he, he, he makes a point. No, no, I'm talking about the moment right now. Oh, okay. I need a couple of cues. Oh, okay, let's get you one. Go across there. Well, I think you're talking about the moment. Yeah. That's, um... Oh, hang on. Oh, no, they're up close. They just simply put them there. It's all right. We'll get a cup of tea. What would you like? Would you like a tea? A tea? A tea. A tea citron. Great. Are you sure you wouldn't like a grog? You know, something like a hot lemon or something like that? Not too good. Would you like a tea? I'll leave you to go back for some rest, I'll catch up with a load of emails and stuff, I'm four or five days behind with stuff I need to do on the internet, and I'll pick up your scarf and bring it round to the hotel this evening, so you have it stood. I'll just leave it at the desk. So, you're saying we should meet this evening? Up to you. I'm quite... Up to you. Up to you. To be honest, I think I've got so much to catch up on that I'm... It might be an idea to leave it for tomorrow. Not that I don't want to... Exactly. I mean, not that I don't want to get every single thought to your company,

15:00 but I'm thinking, one, you know, you look like you could do this and that. More importantly, there are many people who want to talk to me. Exactly, and I'm also thinking I'm hogging you a bit. I was hoping to meet Marcellinatis, for example. Yeah, I think the only problem, the reason they weren't here, was the strike, the transportation strike. And also there's this girl from Milan, she came to this when I was speaking for Aluni. Giovanni Cipolletti could be a star in this. I didn't know that was right. Well, with any luck, these people might contact a hotel or they might come tomorrow. I'll tell you what I'll do. I'll give you a ring about seven or so, and then if nobody has one, if you feel like coming, I'll give you five. And just so that you can get into training. I hope that will help. There's some honey in there as well. Do you want me to ask him for some honey? No, no, no. Well, it'll, if I say, it'll sterilize the nasty little, the cockeyed. Bugs. Johnson has a saloon here. It's not one for the three of them. The entire is more or less. I have to say I think it's particularly a Cambridge Vice. Of course. It's a Cambridge Vice. Yeah, yeah, very strongly a Cambridge Vice. Again, all of us have been asked the question about, well, anybody know when the terminology started, when the terminology started, because there was a discussion, and they escalated it, and that was assuming it was attacking a whole blast.

17:30 It seemed like it was working. In any case, the Cartesian problem, the kind of morphism that Johnson says, oh, it must be a back formation from Cartesian progress. And if you look at where this Cartesian code comes from, I've explained it to you, because Eisenberg and Kelly, the division of codes, and there was a case where the attention product was actually a categorical product, which was eventually called Cartesian, but in any case, therefore, Cartesian code, so in a way, it came after Cartesian. No, no, it came from it. The French who invented this certainly knew that pullbacks over phase is just a categorical product. Categorical product. It's written that the thing is categorical product. Relativize it. Call it Cartesian. Extend it to the phase. It's not just that it's analogous. A great many vibrations are obtained from that basic vibration by a body in some function. Yeah. To offer these kind of conjectures is... But the whole discussion on the headboard is very degenerate. People are pulling around baseless lectures like that about history. Well, I'm afraid that most mathematicians are really very uninterested in the history of their subject. But when the people who are in the generation that have created the subject are actually allowing the history to be distorted. That's very bad. That's a requirement for having good categories of space and why though particularly did they choose the term Cartesian?

20:00 The product name? Yeah, because the tensor worked rather than, well it is, you know, in the special case. Descartes is alleged to have found out that... Yeah, yeah. ...coordinatized the plane. Coordinatized the plane and narrowed it down, yeah. The product of two copies of one. Yeah, yeah, yeah. It's just as simple as that. It's as simple as that. In fact, if we accept that story, then we can go further and apply Cartesian to general finite limits, not only product spaces, but also solutions and equations within those, the sort of thing that algebraic geometry would be about, these surfaces and so forth, by solving equations, by solving equations with equalizers, product and equalizers with all finite limits. It's very common to attribute the word Cartesian to a category that has finance in it. Yeah, yeah, no, it makes perfect sense. Well, it's just that Steve and I say that it's Galileo. Well, you've just taken the words out of my mouth, because I was just about to say, but couldn't you also call it Galileo? Because it's actually as parameterizers of motion. In understanding the kind of maps-based properties that a good calculator space should have, you could just as well attribute this to Galileo. Well, independently of the mathematics per se, it's just the product, you see, but the fact that the categorical product is defined by a universal property, it means that, you know, we know that a point of the product is an ordered pair of points, but what you need for describing motion is an ordered pair of paths, where the domain is time rather than a single point. Right, yes. Right, so you have... Something that's moving in free space traces out a path in the plane and also a path in the altitude. So this pair of paths gives a single path so that the, you know, it's no longer just a matter of elements in a narrow sense. You have to have figures of time and shape. Exactly, exactly. So this leads immediately to the idea of universal property in a way that a mere point case would not, a membership case would not.

22:30 So, in many cases, Galileo made more clear that it had the universal complex and actually used it in that way. Okay, so my question stands. It would have been a nice idea to have called it the Galilean product, but at any rate, we don't understand it. I understand why it's the Cartesian. Cartesian closed. But now we're talking about Cartesian categories that are not even closed. No, no, I realize that's a different thing. If these finite limits have enough right-edged joints, then locally Cartesian closed categories and right-edged joints and charters actually map spaces organically. So the fact, as you say, that they're not even necessarily Cartesian closed categories makes it very unlikely that it was just a back-formation from Cartesian closed categories. I'm very surprised by that. It's a strange thing. But we were getting back to the point about, you know, the propaganda that all opposites are localic and, you know, the erroneous way that this completely misses the... He definitely did promote that, yes. Yeah, that's my impression. In the form of saying, oh, that literally was, worse was Johnstone's, something like, all toposes are generalized basics. Yes, I think that was the title of the thing. Without fixing on... ...locality, such. Because, of course, you know, there's a theorem that any purpose can be covered by a locality. Basically, you take the cohesion, if you have some cohesion, you spread it out until it's no longer, actually, it's a description like, a little like the internal one I was giving. At the level of the category, you can sort of spread out the site and make it into a post-set. Ah, yes, yes. And then, of course, the cheese on a post-set is... Yes, okay, so they're that sensitive. And the covers, you see. Oh, that reminds me. It doesn't have to be space, though. It could just be with monomorphisms. That's what I've often thought, you see, that these results, because the locale cover, and then there was the QD cover, in other words, the other half of the thing, there's also a cover.

25:00 Of course, as you get more and more general notions of generalized space, these can cover it more closely, given topology. Nobody's ever investigated it. Well, I was just going to ask you about that. You could have a closest... It won't be unique, but you could have a closest... This is the strikers marching. Not yet, but I will, don't worry. I think I want to go out and join the demonstration, I shall, without your permission, monsieur, but I think I understand these strikers. On the contrary, I think they, you know, would probably welcome our support. These people are all young people, I feel, these young people. And they look half of them as if they're only high school kids. You say students. Or is this a good place for meeting girls? Could be. Distraction. Could be. No, I was actually going to ask about the Nordsterrensland. About how this... ...connects with the behavior of coverings on the... Is there a connection? I seem to remember a paper of Collins, which started with a presentation of an unusual skeleton. Well, it ended with one. It was the same idea. Yeah, yeah. He was just taking a simple case. Yeah, very good. I mean, it's an example. I mean, if you imagine that every space is actually the sum of its components, which is often true, okay, and then to say that this map is epic is to say that each single component receives an epic in one component, but now if you say that that one splits... Well, then, of course, you've got an actual point, and then you've got points, so you've got the case of, of course, the weakly decidable suboptic condition, I think, is that.

27:30 In other words, in some sense, I'm keeping the treatment here to be such that it will cover algebraic geometry over a non-algebraic, of course. Precise, let's say, in the case of Slew and Wittig, or even algebraic geometry over an algebraically close to it. But anyway, there's just a map from one into every object. Except, first, it's empty. So every object is either empty, either x equals zero, or there exists a graph from one to x. Now that's the form that he used. Yeah, right, that is the, I think that is the... But that implies we, yeah. I mean, he called it some adjective, you know, Johnson, which is quite good. It's a special, you know, it's a special, slightly stronger formulation, which is true. In the algebraic and prose case, it avoids a certain complication in explaining all the recognizable, but that's the form that Hilbert proved it in. You understand, right? I mean, the list is a mouth-shell of thoughts. It's about defining surfaces and so forth by polynomial equations and individual states. Now, if you set up these things in a very natural way, there's a definite object that gives the set of all solutions in the particular column of the equation. But you might not have any points. Right, yes, yes. And Hilbert's idea was that if you overanalyze those fields, there will be a golden point there. They exist. He called most of them because equations would eventually have the form f of x equals zero. Which is both good and bad in the history of math, you see, because that cures the fact that you might as well talk about f equals g, but of course it does lead to certain subjective manipulations, like factoring polynomials, to figure out where the roots are in that, based on factoring. If you have this problem in the equation, it might generate the whole ideal, in which case I wouldn't be able to call you.

30:00 But that's a trivial case where x is equal to 0. To say that that's not true is to say that x is not equal to 0. Yeah. But then the theorem says that if it's not 0, then it has... I think the axiom exactly in that form is in my set theory paper, my 64 set theory paper, because it applies to many more categories. I think in fact Colin even specifically makes that point that is in your ETCS paper and his introduction to that paper, which is a very great introduction to it for the CAT reprint which I got of course. I think that point is in there. I think it is. Either that or he's made it elsewhere. No, the paper of his that I'm thinking of there is this paper about, well there were two versions of it. One was the more technical version that he wrote for the JSL and the other was a more expository conceptual version that he wrote for the... General philosophical logic. And one was called points are only ever points of spaces, I think. There's sets of, there's like, sets are, sets are sets of points of spaces. Yeah, it was later, later. Yeah, that's right. And the... Later than that he actually mentioned this as a novel. Okay, so yes, because of that, I don't think that is in the paper. No, he's, he's, he's... And then there was... He's proving a conjecture of mine, in some cases, to be too ambitious. But there was an interesting point in the paper that I read, which is from about 1987, I think, about the corollary in terms of the coverings, you know, the condition they should always localize. Yeah. Every covering is decidable. Yeah, that's correct. Again, it's just the point about how one thinks of the case of Setts within the general framework of algebraic geometry. Cantorian negation. That's what his title means. That's where Setts comes from. Yeah, exactly. That was my point. And it's obviously his. And that was actually, I was reading that paper back in 1987 that really got me thinking about this.

32:30 The title should have been made a little stronger. What he wants to say is that's where Setts comes from. Well, I think after reading your 1989 paper, yes, it's obviously, well, I think what's happened is that all of the high school, student, university students all over France have come out in support of the strike. And, of course, you know that almost the entire civil service came out yesterday. I mean, this lie that they're peddling, you know, able to expose, but in fact there's a steady drift back. But I am very worried about this provocation last night at the end of the train. That sounds like an absolute classic. Well, there was a news item just literally as I got back to my hotel last night. There was a trade. The last trade had been delayed with several dozen people injured, unfortunately, and they claimed that, of course, they had been sabotaged by the skiers. It's much more likely that they've done the good work than by state agents. There were several anarchist contingents I was looking at the demonstration here. There were plenty of red ones as well. Whose side the anarchists are backing on is always a question, you know. But I just mentioned them as an example. Oh, it is possible that you might get some individual with mania to do something else. People will try to do it on a trend and think, you know, because they think they're living in a revolutionary situation. But I don't think so. I think it's much more likely, much, much more likely to be done by... Such an energy can be instigated to actually do it. My conjecture is that's what happened. 9-11? 9-11. Oh, 9-11. That could very well have been. I think it's, uh... I mean, these guys killed... Supposedly, if Victoria's at all true, if it's at all true these guys actually killed themselves. But on the other hand, it doesn't mean they weren't helpful enough to quit. Well, I'm certainly convinced that, um, there's something far more that we'll know about later,

35:00 what happened, that we, we still don't know. Um, but I'm, I'm, I'm quite convinced that the level of manipulation, of certain errors, uh, was quite high. I had that kind of experience, but it took off way back. I think I... have I ever told you about Prince's Game, about the Siege of the Iranian Embassy last year in 1979? Well, I had a very close friend. In fact, he was a very close friend of mine, one of my closest friends at Cambridge. Yeah, well, you better. A guy called Michael Hashemin, who was an Iranian. Thank you very much. I've just been back to Iran after the revolution, he was living in England, he didn't come, his father was a doctor, he was radical, yeah, that's it. Ask the permit? Okay, he's opening, he was closing up as he thought. Dangerous young people might try to invade his bar with their revolutionary ideas. Pardon, some words among them. Yeah, okay, might decide the stage of provocation. Merci, monsieur. Anyway, the point is, is he had been back to Iran just after the revolution and had, yeah, we actually, just straight down there takes you back to your hotel. Oh, well, at least we can walk across the road. There's no traffic. As long as it's all vectoring in on God. Vectoring in on the vector of, you know, whatever, you see. I mean, this is, he's clearly, he's called the founder of ecumenism. And ecumenism. And of course, well, of course, Moore was another, was a close friend of Miriam Lewis.

37:30 Yeah, yeah. And the other great thing, of course, is the cult of Plato. This veneration for and building up of Plato as the greatest philosopher of all time at the expense of Aristotle. The most reactionary part of ancient Greece, which is the Biosphere. Yes, exactly. And held to be the most basic part of science, or the thing which held the key, above all, of course, held the key to the outstanding of mathematics. And Aristotle, who quite obviously... Inevitable limitations was definitely far more scientific and really was turned into a villain, a great bar for humans. The greatest bar for human progress is intellectual progress, I can't afford it. The longest tyranny to grip the mankind was that in which our ancestors resigned their native wisdom to the Stagirite and made his torts their universal life, which I forget which renaissance, but that's a renaissance poem. So, no, Aristotle was denigrated, Plato was put on a pedestal, and Neoplatonism particularly, which of course had a, and Pythagoreanism and various others, which I'm sure also had a close connection with the revival of astrology. And that's what I was thinking, that this astrology conspiracy actually had its roots, I mean, you see, in other words, if Mirandola was, he's famous, see, again, each of these guys did something which addressed the problems in order to gain support, etc., etc., etc. That's why they were smarter reactionaries. But then, so, Campanella supported Galileo, quote, unquote. But Mirandola, which was like a century earlier, no. He has his 900 theses that he wanted to debate in Rome and they wouldn't let him do it, so he was an oppressed progressive intellectual, blah, blah, blah, blah, blah. But the main thing that's always mentioned is he argues against astrology. But what I was just thinking was, you see, that what this means is somebody else had proposed ancient, you know, proposed... All of these things are related to astrology in one form, and he wouldn't have proposed it in another form. That's what he was really arguing against, was not something old, but some new proposal. Sure, sure. Dare I say it, it was probably the 16th century equivalent of loop quantum gravity versus string theory.

40:00 Listen, you know, these people, that's astrology, that's absolute crap, that's, you know, endless addition of epi... Well, they wouldn't have said addition of epicycles, because that was real science. You know the rhetoric, and this is absolutely the wrong road for astrology to go round. Now here we've got the right road for astrology, along which we should be pursuing as well. No, it's loop quantum gravity and string theory. The really smart progressive reactors, the really smart reactionaries, masquerading as progressives, will be the ones doing the loop quantum gravity. And the really smart reactionaries, masquerading as progressives, then, will be the ones doing, as you say, Campanella, Giordano Bruno version of astrology. Or perhaps John Dee. Well, of course, the very fact that he was from Medici, who I probably knew as God, would indicate my probability of success. Yeah, no, no, this is actually in some sense the center of my study, is the fact that Medici, because the overture of Aristotle, the last little bit by Plato, was argued by this guy from the East who came to get naval support against the Turks. And, of course, they didn't give this report, but they had this big meeting to sit. They even had this alleged feeling of legitimacy. Well, that's what it is, the Council of Florence. The Council, it was called the Council of Florence. Yeah, yeah, yeah. It's actually the subject of one of the doors, you know, the famous doors in the baptistry. That's right, that's right, that's the one, that's the one. And it has the amazing, and they are, obviously one of the greatest works of art in the world, but one of them is actually the marriage of Solomon and Sheba, but it actually represents, because all of them iconographically represent recent events in Florentine history, and the one that is the marriage of Solomon and Sheba is actually the celebration of the healing of the schism in the Council of Florence, which only lasted, I think, about four years. It broke down over some theological niceties. Well, there were two schisms. There was a schism within the Western Church. John XXIII was on the wrong side, but he was the one promoted by the ministry. That's why he's buried there. That's right. Yes, he was deducted. Even though his number didn't count, he was an antipope in the eyes of the official Catholic hierarchy. That's right. Yes, of course. Yes, of course.

42:30 Because that also, actually, that had come to an end a little bit earlier, the so-called Babylonian captivity, the exile of the papacy of Babylon. Sorry, in Avignon, I meant to say, in Avignon. But it was around the same period. Around the same time. I mean, within 20 years. I mean, certainly. But I think it was described... They came with the next, the eastern emperor. Yeah, that's right. Paleologos. They stayed in the Parisi Palace, I think. No, wow, I think you have the better of me there, I don't know where they stayed. But they did come to Florence for the council, certainly. They came to Florence and they lived in one of the palaces that the Mencius had stolen from some other bank. Yeah, I'm sure of that. But anyway. Probably what poor old Alberto's ancestors seem to have been on the losing side in all of the struggles amongst the mother barons of Moros. Anyway, this philosopher came along, Domistos Platon, and he gave this speech, where Cosimo Evacuo was listening to this, precisely on the question, down with Aristotle, up with Plato. Cosimo Eurekio was convinced and on that basis he founded the Academy. More details that I found out recently. He founded the Academy and he took into his own household the same procedure that he used with Michelangelo as a young child, Marcelo Ficino, precisely to groom him completely for the role of head of this academy, to study Greek and learn to translate Plato and all. Now this, you see, so this is a clear, see the relevance of this to modern world, direct relevance, is that the Warburg family, Warburg family, biggest banking family, they've been compared to the 20th century compared to the world of the Rothschilds in the 19th, yeah, and they're about to see other banking families. But anyway, they were very important and amazingly important in going to history, but the head of the family... Around 1900 or so, they decided to completely emulate the Medici. They didn't put it in those words, but the idea was that, you know, we're the most powerful, we're going to be the most powerful bank and pen, and we need art and culture to support us, and look how successful the Medici were for 90 years or so. This is really that successful. But it's successful in the sense that we still know about it, we're supposed to still worship that culture that they came up with, even if they...

45:00 So, this guy is the oldest of the four brothers, sort of like the Rothschild brothers in different cities that controlled the whole banking world. He's still the oldest one, and therefore he has final final say on anything, but he hardly ever consulted because his full-time job is studying Florence. Yeah, which of course is why they set up the Wahlberg Institute, which is the most important art history institute in the University of London, even today. Which is where all these guys go. It was in Hamburg, of course. It was in Hamburg, it moved to London. But you know about it. Oh yeah, I know all about the Wahlberg Institute. I was telling the story that you know already. No, no, no. Hans Gombrich, the great friend of Popper, who was the head of the Warburg Institute for many years, and Kassirer, Panofsky, and a lot of the great art historians of that period. They were all working for the Warburg Institute. I see you've been doing your homework as well. I know the other Warburg Institute. I knew about it many years ago when I was doing my post-grad in the University of London, and they do of course have some wonderful people. So, in fact, this woman in Italy, the friend of Aurelio Cavoni, that's where she does most of the work. That's the source where you have to go if you want to learn about Campanella, for example, and everybody that surrounds her. And, of course, they're not exactly their rivals. But they stole all the materials. Of course they stole them all. And, of course, I wouldn't say exactly their rivals, but the other big art institute. Which is in the very same bit of the campus of the University of London, not far from where you went to talk to Dr. Riley about when you stayed there, is the Courtauld Institute, which of course was set up by, also by Courtauld, the, you know, the great, yeah, true, I mean, Courtauld financed the importation of the library at the time period. Yeah, that's right, yeah, that's right, that was all set up by, but again, that was set up by... They were fabulously rich, you know, the Courtaulds. Actually, the Courtaulds, I think they did make money in banking, but I think most of their fortune came from textiles. They used to be the biggest pharmaceuticals and textiles corporation in Britain until they were absorbed into ICI.

47:30 I forget at what date, but not until very late, not until the 50s or 60s. Before that, after ICI, they were the second biggest textiles and pharmaceuticals group in the UK. But you're right, they did make some money in banking too, I think. These initial readings of mine, I discovered things I had no idea about, but it turned out to be well known to other people. The fact that in 1870, as a result of... The events where we are standing right here, that the Bismarck granted full equality to the Jews, so that these bankers in particular, in Hamburg, now have the same rights as any other bankers. And so they were extremely happy about this and became completely, totally pro-Prussian, pro-Bismarck, pro-Empire, and so on and so on. Even though Napoleon, of course, had done exactly the same thing for the German Jews six years before, but of course, you know, they had... About this guy, so it turns out, probably not the sort of thing where I learned your theory from. But at least they had been addressing it. Well, I'm sure the Cathy of Thuringians would be used to clear or clean up a lot of our statistics. She is now, let's see, what did I tell you about programming last? She had a Ph.D. in epidemiology. She moved to Oaxaca. ...opened a huge nightclub that she refurnished herself and so forth and so on. Met a young man for whom she got married, by whom she has now had two children. Oh, just one? Sorry. Oh, no, of course, it's your other one that's happened. That one had three. Yes, because that's why it was supposed to be a kid. I have four grandkids. You have four grandkids all together. So they moved to Calgary, Alberta, and got a job with the Alberta Cancer Board. Cancer prevention through anti-smoking propaganda. Proving that you can use St. John's Wort just as well as expensive prescription drugs. To assist the will. He was working for a family member.

50:00 Exactly the same sort of super exploitation that Canadians make against the Latin American workers. Namely, they delay the period in which they don't have proper papers. Therefore, they work 12 hours a day with a good salary and deliberately delay because there are lawyers that are paid thousands of dollars to process these documents and the employer has to grease this, you see, so the employer happened to be an uncle, deliberately did not grease the lawyers, I think only after many, many, many months, and I think he does have a legitimate, so-called legitimate... So does he actually have a living wage because the French district came from the French because that's exactly how they treat all these families. All these steps took place very quickly. The next step was, well, she has this baby and so since, as Michael Moore pointed out, some countries have much better health care than the U.S. Yes, I'd certainly say that Norway is a lot higher. But in any case, from the US standpoint, Canada is unbelievably high. Nobody even believes the fact that you actually get a year of maternity leave. And your husband gets maternity leave as well, and you get 80% pay and all this. That's what they did. You know that Washington has a higher infant mortality rate than Delhi, Beijing. I believe it. The last step is this, that a better job when they come back to Calvary, if they do, they now live in Spain. He's going to a school where he can get a certificate, you see, as a genuine Spanish chef and not just a random Mexican chef. He's very good in science, isn't he? So that's what they're doing right now. They live in Valencia. Ah, I wish I'd known. I would have given them a ring when I was down in Madrid. They live in Valencia, yes. Yeah, it would be nice. And he's doing the school and she's doing...

52:30 I mean, they don't have to go back for many, many months. Oh, you'd love to see them. You might, but this is really a... The winters are a damn sight warmer in Valencia than in Calgary. Well, there's that. The child is really a marvel. His name is Diego. Yes, I know, you told me all about it. Diego is very, very... I'd love to go and see them. I'd love to see Diego. But she calls me up and says, I'm still worried about causalities. So the question is... There's this question, you see, it depends on the background. Maybe in some context it wouldn't be wrong to say that Hitler was the cause, presuming that there's the multi-capitalist class, that there's the aristocratic class, all this stuff, well, maybe still that doesn't imply that things move. It is the potential for moving. So, something to trigger it could be called the cause, you know? Well, I mean, Galileo wrestled that problem right back at the time of the Devota, didn't he? I mean, the whole point of that thing, how much do you have to idealize away from the background? Yeah, yeah. You know, from the background or something, from the real world. Background in which obviously everything is immersed in order to have an effective theory. I think in the case of history it's much more problematic. In the case of mechanics it works rather well. Topology is in layers, is part of the theory. Absolutely, absolutely. What is responsible? What copy is responsible? Never solely responsible. It depends on the background. Now, I agree, with historical conversation it's of course vastly more complicated, but then of course that was Marx's great achievement, to open up the essential defining relations. I think it's the ontology conspiracy which has been the main cause of most of that nonsense that we've witnessed in the last twelve years in mathematics. Yeah, as you explained to Mark over lunch, that it's a delusional... Well, I've been explaining it to you, and you've been accepting too. Put the story together in a way which is both true and believable.

55:00 But the beauty must come before rigor and proof, and the pseudo-physics, I guess, but very much so is the package. They're trying to carry out, you know, and... So Bruno, you see, was a little bit less reactionary than Casanova because he didn't accept the astrology per se, what we know as astrology. Rather, he said, oh, there are these real bodies out there. There's an infinity of worlds. Totally, they're all... Populated in this paved way for Star Wars, Star Trek, and every form of science fiction, which thanks, caused by the accident that I was a participant in the 1952 conference of world science fiction writers, I know that in fact the purpose of science fiction since then at least, if not before, was sociological manipulation rather than Rather than giving young engineers their jollies because of extrapolating technology, which I had previously naively thought, and as many, many, many science fiction readers still do, I think, they think that there. It's a long time since I read any sci-fi, but they used to be a bit this way. I guess the completely naked fascist agenda of people like Heinlein was so evident that when I came across them, I realized that this was really... In fact he used to have direct preaching against Marxism in his sci-fi stories. I do believe it's the first place. I was probably about seven years old when I read it because I had a cousin who was actually in the Air Force. He was doing his national service. He used to come back home to visit as a kid and he used to write amazing stories. And I remember, I honestly can't remember which I later, but it was my highlight because I had no idea. But it had this strange dialogue. These guys were having an argument about the value. Have you ever come across this story?

57:30 An absolute story? Well, as I say, there is one of the characters producing a sophomoric demolition of the labor theory. And I remember thinking, because an eight-year-old was trying to understand everything about the background, and I had never heard of it before, in our society, but I remember it sticking with me, and then quite a few years later, I was a little bit more aware, going out and reading. Yeah, it must have been about... I used to have his autograph in the U.S. Navy. He definitely steered the fight between John W. Campbell and Robert Heinlein on such-and-such questions. We might agree with Khrushchev to have a certain limitation on the number of missiles and the number of... Then, of course, we'll decide, we'll have another study which figures out how many missiles you can hide on the ocean floor so that Russians won't find them. This is part of it. You might just figure out the Russians would do exactly the same thing, of course. Yes, and we have kill probabilities and all this. I'm saying it's, in a sense, a part of it, a small part of it might conceivably have had some actual relation to production of weapons.

1:00:00 Might have had some small relation, but the main thing was not that. The main thing was not that at all. The main thing was the Pentagon to buy these computers because you can say that it has been theoretically analyzed in such and such a way and that, you know, there are different levels of sophistication. You can do this on some very crude level where for some people much more sophisticated levels than for other people. If you want to get to the man who has to convince McNamara then... Since I had some minimum amount of experience with acting, I kept thinking to myself, what the hell is this going to do for the war effort, I mean, what is it going to do in terms of hard work, it's going to increase the profits of the company, increase the profits of the company, increase the belief of the enemy in our supremacy, increase the belief of the American populace in our fact that we're doing the right thing for freedom, blah blah blah, all that. But this is the key role of physics. So you need people to do that, but they're the geeks. Complete transformation of the ideal scientist. Yes, into the fideist. You say 63. Well, 62 is my experience, cultural, fictional. What was the application of stochastic processes, stochastic theories, stochastic processes for this design of all of these? If there is an agreement, it will say, it will say, everything is hypothetical, it will say that, you know, you're allowed to have a certain number of missiles. Yeah, so you're allowed to have 200 more at each or whatever. Now, of course, we have to verify this, and you have to verify this, we have to have an agreement. To the effect that, well, with satellites, satellites weren't perfect then. Yeah, of course I was going to say that. Now they can photograph, obviously, a fingernail easily, you know. But still, in spite of that, you can still hide things. You really want to hide things. There'll be a certain level of satellites today.

1:02:30 Hell, I mean, Saddam Hussein made a hell of a job of hiding the weapons of mass destruction. They haven't found them yet. I guess I could say that. Yeah, yeah, yeah. That's very good. Very, very good. But you see, so with satellites, we might say, well, it looks like there's something fishy there. Certain probability. Something else fishy there with a certain probability and so forth. So there'll be a certain probability distribution of fishiness. Which will translate, because of precise agreements, it's all hypothetical. Precise agreements will say that, well, if we have... The 63% belief that there's something fishy there will be allowed to fly over with planes. There'll be overflights. You see, obviously both sides want to minimize overflights, but they may have to accept some overflights. This was at the period, obviously, when they were having to rely on high-altitude spy planes like the Rava. Yeah, exactly. Just sort of ordinary altitude fights. Yeah, yeah, sure. We want to avoid this, and they want to avoid it, so we'll have a trade-off. So it's the whole idea of this. So this distribution at the satellite level will be translated, will be mapped by a stochastic map into something about how many planes were allowed to descend per day to such and such area of Kazakhstan and so forth. And then the same with another, there's a second level of the same type, because if we develop really serious suspicions, We have the right to on-site inspection, so this whole concept of on-site inspections, which was used in the aerobics, it will come in, but again, we want to minimize the on-site inspections should be minimized, again, they all want the same thing, so we have to agree, so there was this sort of three-level stochastic You know where they use precisely this kind of multi-level stochastic mapping now. The reason I know about this is because there was a woman who gave a talk about this in the philosophy of math section in Madrid.

1:05:00 This is precisely the same mathematical technology that has been used since the In the early 1970s, to design all of these, we call them the derivatives in the financial market, and to decide, you know, to calculate when you should put a put or call option on your theoretical, when you should put a put or call, a higher level option that you have, yes, whether or not to exercise the option to buy a notional. That's the amount of aluminum or whatever you're dealing with, say, with metals futures, but it gets much more sophisticated than that because the whole point is that most of these instruments were themselves designed to calculate the financial value of having the ability within, say, a certain time to exercise a non-sum. Some deal that you could, yeah, I'm sorry I'm explaining this very badly, but I think you get the picture, but it's the same, essentially as I understand it, it's the very same kind of multi-layer stochastic screwing down of options on whether it's, exactly, it costs something even to maintain the option of deciding whether to screw down the thing at the next level of mathematics, and that's more or less the way that derivatives work. From what I've understood from the, and these, oh, well, these guys got Nobel prizes for, a guy, there's a guy called Scholles at the University of Chicago, surprise, surprise, and Steen, a couple of mathematicians who were experts in stochastic analysis. And her talk was actually, Yeah, yeah. A big appendix. The actual activity of this group was never really to design anything, but to plan, to be paid to be planned, and planned to be paid to be planned from the Pentagon, you see.

1:07:30 So the main thing on this subject was a big volume, which was a proposal, a quarter million dollar grant for the activity. So there was some story about this computer that they planned to build and so forth and so on in the main body, but then there were many appendices. The technical details, so there were appendices that talked about hardware, always about the hardware that they already had to sell, or were planning to build, you know, it was all very much like a brochure for buying screws and nuts, right? But there was a major, more than one, there was my appendix and others, which was just about this stochastic. The category of stochastic maps and various things you could do with it, functorial testing of hypotheses and so forth, and so forth, with one or two words about this arms control verification problem. So the whole point of this was, you see, the whole point of this was that the actual text of the thing was actually totally unconvincing. It was just some, the officers in the Pentagon, who struggled to understand this and give us the money, were impressed by it. In fact, they wrote back about how they were impressed, you see. This we could do with command and control is sort of like that, isn't it? Do you have a model of command and control we can actually use, which is based on this? So they wrote to me this way. That was the thing. So from this experience, you see that the role of this theoretical mathematics is not really to design. And certainly not to design the command and control systems, either, really, but just as you say, to convince the guy. Again, to convince so that they can have propaganda. See, to the effect that we have theoretically designed command and control systems, and therefore you can be commanded and controlled. This talk that this girl, her name was Mikaela, she was Lebanese, stochastic. Theory met financial. Stochastic theory entered financial mathematics.

1:10:00 There's a study of all the papers that these guys have written between about 1972 and 1983 for which they got the Nobel Prize. I mean, considering what they were doing and what for, a pretty huge prostitution of the human intellect. But on the other hand, it was interesting. But one thing which emerged very clear, because I asked her about this afterwards, was that at no point had any of these guys applied any... Even though, as you say, you were applying the notion of stochastic map in the category theoretic sense to this stuff back in 20 years before. This phenomenon, I myself, one of my students who wrote a Ph.D. thesis at XREX Red Guard, who wrote her doctoral thesis about stochastic maps and that experience, you see that she was supposed to go give a talk at some statistics department. And the guy who was the chairman found out there was going to be category theory and canceled. He said, I refuse to learn category theory. Now, and then more recently I read about, I guess it turned out to be a student of his, who again said the same thing, you know, he made some advance in statistics. But part of his paper was, we definitely have to avoid category theory at this point, let's do this instead. So why is this creeping in? Well, the first guy, actually, I found out, he had attempted to understand a Russian book, and he actually had managed to prove one of the main results in the book without using category theory. But he hated category theory because the book used category theory. He used the category of stochastic maps. But the problem really, in my view, was not that it used categories, but it introduced the categories, and then it introduced some very mysterious idea about the geometrical structure of the objects in the category. But you see, this geometrical structure was not actually there. I mean, it was just a fiction that sort of floated on top, and it was completely bogus. And so if you seriously try to understand this book...

1:12:30 You will become very mad because it's completely, you see, but on the other hand, there is a serious calculation underneath that. Yeah, yeah. And so the guy did this calculation and published it. Yeah, I can see why he would become hostile to category theory. He should learn to not to base his induction on one single example, of course. He's a statistician. I was going to say, he's meant to be a statistician, for God's sake. But at the same time, I can kind of emotionally sympathize with his reaction. Certainly, this is at least one major source of the resistance in the statistics. So he'll be happy with the view that it's just general abstract nonsense. But now, I got to thinking. This Russian book was published in the mid-60s. I did my classified secret document about the category of stochastic maths in 62. This is a secret document about how they're going to deal with the Russians in that organization. This guy got the idea. You know, going back and looking at his books, I'm also disgusted by his book because of this. I don't know if that's a mystical way that he presents it and turns off everybody, but I realize also that he uses very much, in that non-mystical part, uses very much the same terminology that I use and very much the same thrust, you see. Sounds very much like he was stolen, doesn't it? So it's very possible that your secret paper was stolen. But as far as I know, it's still classified. I was going to ask you, has it ever been published? Is it still classified? No, people keep asking me for that. So the bibliography will never be complete unless it's international. No, no, if I ever get a moment free, I will. Are you allowed to do that? Well, I won't publish the same thing. No, no, no, you just publish a declassified... The basic, the basic, in fact the main part of the paper is not that stuff at all. It's a sort of different foundation of measure theory. Well, this would be really interesting. It was a different foundation of measure theory. Oh, this is absolutely fascinating. I mean, that would be really interesting.

1:15:00 The large part of the paper is about that. And then that's used... So you could publish that on CAT, on, you know... I could publish that, or I could omit it and just go directly to Cass's version of it. Oh, the connection with measure theory is really interesting. Well, you could do both. I mean, measure theory is a system around Borelli and Copeland. This is very important. Real numbers are tall. I mean, the mystical probability of such and such, why should it be a number? Maybe it's on some kind of integral from zero to one, but it need not be the real integral, it could be some other one. The whole sort of mystique of going from data, which is a map from actual things, to classification, turning into Shannon information. It actually works in any cohesive tempo, so the stochastic version is something that I want to write, and the main conceptual content of the thing that I can do in the general way is what my students did actually, from the stochastic view of something, you can, you can, which makes it convex, the convexity idea that somehow between, because... I think it amounts to meaning a lot of instances, stochastic, any two points in the space of possibilities are joined by a line in that way. But from any stochastic, from any convex, you can metrize it. There is a metric based on the measurement of information.

1:17:30 And this assigning of a metric to the convexity is a closed function. Let's go over into tensor products and metrics. A function that does preserve tensor enables you to transform enrichment. So a category enriched into a category enriched in metric space. The typical categories convexly enriched are stochastic processes. Categories are complex. A convex set. The random math between any of these things, even if they have this complicated dynamical structure, it will still be a convex set. So stochastic processes is enriched in some way. That's a therefore a metric basis for this basic construction. And so if you, typical statistical thing is if you want to complete commutative diagrams within the category of stochastic things. But not necessarily exactly. So you have two arrows and you want a dotted arrow which, as close as possible, makes it to me. So that's the method, as close as possible in that method. Yeah. I remember you explaining this when you were staying in Fougere because we had a discussion about stochastic maps. No, no, don't worry. I don't recall every detail. No, on the occasion, I'm saying I'm really glad because I do remember that discussion. You see, all that is true for some very general notion of convexity. So you actually implement or model the convexity of the working process for corresponding interventions that can both be seen in a more combinatorial way or in a more inter-dimensional way or all the information approximately is in that paper in a particular measure theory, say it's in a particular case. Yes, you can just say that again.

1:20:00 By the way, looking at the time, if the statisticians are reluctant to take on the category, yes, well, they're not going to understand it, are they? Maybe do one of these CIA people, not just the KGB person. Fox us on our... because no doubt the agents actually believe this crap about arms control. This is how they're going to fox us, and so we need to develop this, and so this guy gets his PhD at Moscow University about stochastic maps. In the process, unwittingly, or maybe wittingly, no, that's too much of a Machiavellian, turns off the statistical world in the West. See, the West and East are different in the sense that the Soviets have no objection to categories. I'm going back to general mathematicians now. No, that's my point. You know what I'm saying? I mean, here, even in France, but certainly in the U.S., Britain, and Germany, and so forth. Maths is a mathematician, not just a statistician. Whereas in the Soviet Union, they never had this problem that I know of. You get Blavatsky and these people are normal kids in Moscow high school. I mean, it's as natural for them to use category theory as anything else when it's needed. Of course, the book itself was immediately translated into English because of the opposite sort of intelligence by somebody in Israel, actually, didn't it? But this record book, oh, this is very important for us, we've got another record book. Yes, there was quite an industry over there translating Russian textbooks and Ph.D.s and things like that. But in fact it was the reason... He classified it. It's absolutely... No, no, that's exaggerated. I mean, obviously this guy did more than... Yeah, I'm sure he did more than just poach your... but at the same time it sounds very much as if he did base his own kind of access to that classified document.

1:22:30 But what's... We could make a good story. We could actually write a novel. We could have a good story. And the category of stochastic maths would be what? Hitchcock used to call the MacGuffin, you know what he, the MacGuffin, you know, oh Hitchcock, well there was usually, and it was very often a technical thing like a scientific blueprint, or actually at least in one film, Tall and Curtain, it was actually a mathematical proof, and that was the plot device, which was almost always something to do with science. Hitchcock always used to refer to in his film as the MacGuffin, the public weren't supposed to. I don't, of course, understand what it was about, but it was the absolutely crucial plot device on which the whole thing turned, you know. This was the formula that they had to get the photocopy of. But tell me, what was the book? I'm sorry, what was the book? I've heard most of the category theory books published around that time. You said it was translated back into English by somebody in Israel. I believe that's right. But you can't think of the authors, something like Chernoff, C.E.R. Chernoff, yes, I think I've possibly sort of seen it in bibliographies. You should certainly think if you've got the time and energy, and if you do write up these ideas on the Riemannian metric, or these ideas about the synthetic differential to the metric, to me earlier, to the point being that we really don't have it, we don't get it. ... in either the analytical or the synthetic setting. But anyway, if you do press ahead with these ideas in synthetic differential geometry, vis-a-vis Riemann, then... Is that for next week, boss? No, not for next week, but if you look ahead a year, if you'd be interested in coming to Moscow for this meeting that they have every year on foundations of differential geometry, I think you might find it quite interesting.

1:25:00 And this is this guy who has this private foundation. He maintains about 20 guys who work mainly on Finnsler geometry. A couple of topologists. He himself did a PhD in Finnsler geometry about 20 years ago. To be honest, I don't think he's a very good mathematician, but he very seriously wants to support differential geometry. Not at all. Well, I must go and do some work. But aren't you hungry? Well, it's just after half past seven. Would you like to join me for dinner? Yeah, do you want to just go right next door? What a good idea. Just let me go and run you some food quickly.