Conversations after FW Lawvere lecture / discussions at JL Bell's house

0:00 This is obviously what happened. Dennis Thatcher was secretly exposed to Polish magicians in his youth, and the whole strategy has rested on the conviction of... Well, I hate to say it, but having been in college with Michael Portillo, I'm quite sure you could con him into reading books and not realizing what was wrong. This is what really terrified me. It's all about algebra and the compressibility. So it is natural they would fund him for that kind of stuff. Well, yeah, yeah, but it's a real logical problem. Yeah, but they don't really fund him to do relevance modeling. Let's come clean. But he doesn't get much. He does have to do much. They all get that kind of money. Good luck. They all get so much money. They pool. They pool. They grant. The theory group in computer science, of which she's a member, there's probably almost a dozen of them. They just pool their money and they just take out what they want. They hire a number of graduate students and, you know, if you want to actually use the graduate students, they run to the library and put a copy of papers for you. Go ahead. But there's so much money there, the graduate students can just... Where's the money?

2:30 They can have any, any equipment that they want. And this bugger, I'm so proud of him. He's got, he's got a beautiful, well, what? Yes, if it weren't for the seat, you'd see. That was a pity. Well, having heard it, I mean, what do you think of this amazing character? La Vieira, I mean. La Vieira? Oh, I didn't know that. Yeah. Well, that's right, you know, that's right. Of course you've read his stuff, so you know. Not all of it. You know, mid-1960s trained, a reasonably bright, tasky trained set theorist or logician. It must be a little bit, not for something like John Belk, because he's a cute and aware of the issues, because he knows Bill, but for somebody in a philosophy department at British University who trained in logic in the 1960s, it must be a little bit like a... The late 15th century Aristotelian, suddenly finding themselves transposed to a seminar in Paris in about 1750, I mean it's just, where the hell do I get my lyrics, where the hell am I in this stuff, I mean you hear, there are moments when you kind of glimpse what seems a vaguely familiar bit of the landscape but from a very different angle. Yeah, but then, you know, there are whole parts of it where you don't even begin to see how that angle on your own familiar landscape even begins to fit in his way of thinking. You can see it around the room. Ooh, I know Rome. Yeah, yeah. No, but I really am feeling awful because I made such a rambling point about Groton's... And the point is, it was actually... It was a sensible point. The epsilon in the equation of decidable topos just is what gives you extensionality, and your talk was entitled The True Relation of Logic and Set Theory, and I would have said something about extensionality. End of point.

5:00 But he does think at such an extraordinarily deep level that he could connect with the world. Well, I'm not saying that all his ideas are going to turn out. I'm sure some of them will turn out to be genuinely scary, but... It's quite remarkable that he's sort of been... I don't know what I was expecting him to look like. You hadn't met him before then? No, no. You'd obviously have heard about him. Oh, yeah. I don't know if you expect me to look like a Killeen Tourette, you know? Well, I don't. I think Bill looks more like my... Bill looks like an Old Testament prophet. He actually looks less like an Old Testament prophet now than he used to. He looked like a folk singer, didn't he? Yeah, yeah, okay, well... All right. He looks like something in between. He looks okay. So he looks like something in between a 1960s folk singer and an Old Testament prophet. Yeah. Well, I think the folk singing bit is partly because of the extraordinary... He suddenly decided to adopt the Stetson and Cowboy voice. This is completely new. Yeah, that's new. He wasn't... That's a new departure. There's one particular folk singer, Stan Rogers. Yeah, I don't know about... Well, what I'm saying is that Scott looks like your ultimate buttoned-up CEO, right, and always comes first. Well, I went to his lectures at Oxford. He was professor of logic at Oxford in the 70s. No, I was at Cambridge. No, no, no, no. I was at Cambridge. I did philosophy at Cambridge and then learned logic. Such little bits as I know, they're basically going to undergrad and then grad lectures in London for four years. And then he did an MPhil in London, and then PhD. So were his... Well, I never actually... I went to... Dana Scott had a seminar on modal logic in Oxford in the early 1970s, by 1974, which I went to.

7:30 And gave a paper up, because that was my MPhil. So was he doing the sheath? No, sheaths, he hadn't... He was just... He hadn't... He didn't really start doing the sheath theoretic semantics until the late 70s. That stuff is all in the 1979 Durham conference volume. Yeah, the... This was... I'll be absolutely honest with you, I don't think that in 1973-74 that Scott would have known... Dare I say this? I doubt whether Daniel Scott in 1974 actually knew what a cube was. No, I mean, that's not being dismissive. I mean, algebra and geometry just hadn't come into the subject. Remember, it was only two years since Lorvier had formulated the Thomas axiom. And, you know, people have begun to make the big pitch for the relevance of all this stuff and logic. When he introduced topological models for axioms for modal logic, they were all in terms of lattices of open sets on topological spaces. And it was really just a topological model for intuitionistic theory. It was all very interesting stuff. And some of the ideas... How they converge, how they naturally fitted into the Sheik framework. But it's like Lorvier said, I mean, you know, the moment that Lorvier heard about building value models, which was at the Tarski seminar, he saw, I mean, just like that, how they fitted in as a fragment of his geometrical theory, how they fit into the Sheik framework. It took him about seven or eight years to convince people. A lot of other people contributed too. It was difficult enough, I mean learning forcing and independence from, well of course the Scott Solovey paper was never written, but it was difficult enough for somebody like me who's not that bright, who's coming into the subject from philosophy anyway, it's difficult enough learning them from John Bell's book. I mean that involved an awful lot of very hard learning and going to a lot of lectures and really working at it.

10:00 I think it's a lot easier to learn it if you can do some algebraic stuff. Yeah, yeah, because then the topos theory wasn't around. Well, it was, but nobody in the philosophy department is doing logic or anything about it. You beat your head against that for a while, but then you have to beat your head against one second, right? Oh, yeah, it is a lot easier, yeah. And actually, I find the second book on local set theory is actually much easier. I think it is an easier book to read, in some ways. No, no, maybe not. I noticed that one thing is that everybody... I think, I tend to think geometrically anyway, so it's not so... Yeah, I find it much more... I mean, all the, you know, the automorphisms, the groups of automorphisms, and the permutation groups, models of, in the earlier book of the, you know, the independence proofs, the one that he did, the Boolean-Berling models, is that, yeah. I found those more difficult to work through than I found the stuff in the local... But of course I'd heard him expound the ideas on local set theory so many times by then that I was familiar with, you know, so it's probably out there. There was a very curious little seminar, like a private little, totally unofficial seminar. The LSE that he ran in the early 80s with Bud Moshe Marthola, which was, oh God, I mean, it would really have got a bundle on Lorvier, which was, it was really held, it was actually held like literally in secret evenings, it wasn't advertised, which was Marxism and the Foundations of Mathematics. There you go, which was Moishe, John Bell, and about half a dozen others, including Cyril Smith. Very well, that's who, I know Cyril Smith very well indeed. He was a member of that seminar, obviously. Oh yeah, in fact it was his idea, well his and John's. It was through that that I got to know about Lorvier's papers. I started reading Lorvier's papers because of going to that seminar. I have a lot of them. It was a good seminar, actually. I had some very good mathematical physicists attended it. People like Felix Barani and... It didn't really have much to do with marketing. Well, except in a very indirect way. They had a broadly...

12:30 They all had good and they would say that reasons which had a political as well as a purely intellectual dimension for not for rejecting kind of received secular mathematics. The model theory. But it was a very interesting seminar to go to. Lots of good ideas got kicked around in it. And that was how I came to learn about it. Because I got all these papers and read them. Well, I, of course, couldn't make head or tail. You can imagine what it's like, absolutely. Yeah, yeah, actually, I think his exposition is much clearer. Now, of course, the thing that is extraordinary about the way that he does exposition is that he starts on what seems to be a very simple, accessible level, to the point that if you were a lazy listener, three minutes into lecture, you might think, God, this is going to be pretty shallow stuff. I mean, maybe a minute and a half in. And by the time you're five minutes in, you realize this guy is going to be such incredibly deep insights into... There are so many fields of mathematics that you couldn't know. If you lived for a thousand years, you'd never learn this stuff, but you'd still die wanting to know the math that he knows, and he creeps up on you really softly in that way. Have you seen his introductory categories? The weird conceptual mathematics thing? Yeah, yeah, yeah, he sent it to me. Sent it to me with a very nice little letter of... Well, it's such a nice job of getting... I think so. Well, yeah, I think so, too. I'm glad to hear you say that, because some people I've heard... I mean, Colin thinks it's embarrassing. It's sort of, well, okay... Colin forgets what it's like to be a total beginner in these subjects.

15:00 Does he teach first-year students anymore? He teaches them. He teaches them very well. I mean, yeah, he does, actually. He does, actually. You know, Colin's a good... But I think he has forgotten, you know, what it's like. I'll put it this way. I'm sure that the brightest students get much more out of Colin than the average ones do. I'm not a critic, because I like Colin enormously, but I would like... I would have liked to have been taught mathematics by Colin, but I know I would have had to work very, very hard to, you know, to keep any interest in me as a student, and I had been. I think, no, I think it's a good, good book. And one of the things that his marvellous work, Bill, is that he really does. People who have that kind of depth of passion to understand, just got to understand or die, which obviously also usually have got a mission. They explain all that and they really don't leave no matter how many times they get busted that you cannot make these ideas but the essentials as in that book that they cannot be made available even to people with who say that you know even to people who are totally Math-phobic. And obviously, you know, that audience, he would probably, you know, once he got beyond the very elements of that, he probably knew. But on the other hand, some of the brightest people who thought of themselves as math-phobic for years might follow him, even as far as these ideas about graphs of, you know, dynamical action and fixed points, which is an There's an extremely interesting application of all this stuff, which ties in with Etan Booth, and also with this way that one, you know, one is, you know, the big difference between the way that he understands domains and the way that a task force should understand them, which is what I'm struggling to get at. Everything LaPierre-Seven does is part of the political program, which is... And that's a particularly effective part, because you count at Western here.

17:30 You can't be introduced to that kind of thing until grad school, but if you're truly indoctrinated in traditional mathematics... Yeah, you see, he taught calculus using the elementary period of the category sets at Reed College to, you call them sophomore, don't you, to first year. Freshman, sorry, freshman, yeah. Back in 64. Yeah, okay, I'm sorry. Look, you really have to forgive me. I'm sorry. I mean, I've only been one day. I mean, I've been almost exactly 24 hours in Canada. It's my first time. And I've been, you know, I go perhaps about, because of my business, about something like four or five times a year to the States and have done for the last eight years. So I have to adjust. Yeah, I know. Not being disrespectful, it's a much better place, I've said so already, but no, I think he is a born teacher, and the thing is he's going to have, which he does have to wrestle with before they can be made, it's probably good that he doesn't devote too much of his energy to physics stuff, but if that was all that he had to do, he'd do it very, very well, very conscientiously. The more I listened to him, for instance, the stuff he says about philosophy, when I first heard it, I thought, well, this guy's obviously a brilliant mathematician, a genius, but, you know, his philosophical views are very naive and untrue, you know, terminology about...

20:00 You know, I leave that sort of stuff behind, you know, Morris Cornpaw. But after this thinking, I realised how wrong I was. I mean, it really seems more and more to me that that is the right way to think. There is an awful lot in what he says about ontology and metaphysics as pursued in either analysis of philosophy or in... It's quite interesting. Maybe these are all the tools that we need in order to render the exact conception of the world. It's certainly the most beautiful way of presenting a journal to people with a background in computer science or astronomy. But on the other hand, the philosophy, what he said about it, is dumb. That's very superficial. What normalist is going to buy a language which has a proper... where you have to believe in a proper class of expressions? Yeah, well, that's it. I mean, this is... I mean, and it's... I mean, that's... that's just not thought through at all. No, I mean, and it's because... And all Platonists... and of course, you know, Platonists should be happy because they can have all this, provided they accept intuitionistic views. Well, this is, well, and this book, I think, I mean, it's written for the mathematical intelligence. Yeah, sure. That's what a philosopher, what a mathematician's pretend to philosophy. Fair enough, okay? Yeah, fair enough. And then you compare that, though, which, I mean, essentially, he didn't actually, well, I get you, but he mentioned a couple, just in passing, he mentioned some real theorems, you know, but he didn't present any theorems. No, no, no. But that's not his style in doing mathematics. He didn't reformulate any theory. You know, he's proved to us that he didn't do that at all. Well, he's not proved as much as he could have done, but that's not what his mathematics is about. But, you know, the philosophical stuff is quite interesting. I mean, the way that Randolph was, was what you would expect. A mathematician trying to do philosophy would work. It's a complete contrast to the, I mean, the sort of scientists who invented the philosophical field.

22:30 Yes, yes, which is, yeah. You know, it's like Einstein. The first reading seems like a superficial philosopher too, but when you read it again... And then you realize the philosophers are going to live off this guy for the next century. Yeah. And I feel the same about Laubrere. I do actually think the philosophy of maths is going to live off Laubrere, I don't know, for about the next century. But it'll take 20 years to happen. It'll take 20 years to happen. Except that if they'd started where they should have done, it wouldn't be happening now. But I imagine that never happens. But they will. Well, that hardly ever happens. But on the other hand, it has happened now in physics. It has happened in philosophy and physics, but very good work is now done, I think, in philosophy and physics. But did it happen, I mean, it doesn't happen immediately afterwards. No, of course it doesn't, no, no, of course it doesn't, but it isn't a question of, yeah, I mean, these people really have to know the point of their mouths now, and they are good, the best of them, and I would like to see philosophy and maths in the same position, I would like to see philosophy. Philosophers are there talking to mathematicians on the level of Lanvier today, so that you could actually give a much tighter philosophical presentation of Lanvier and at the same time also help... It's just beginning, because people like Alberto, with their work on mass nouns and pert nouns, and it is in philosophy. But that would be interesting for the bulk of the philosophical community, because there's still, I mean, it's just now that the philosophical community is able to talk about set theories. Well, the bulk of the philosophical community, yeah. You know, it's surprising how many people in Italy there are who are interested in those. I'm amazingly impressed. The caliber of philosophy, philosophical thought about logic, I don't want to say philosophy of logic, but the caliber of philosophically aware work in logic. The number of foundations of mathematics that's done in philosophy departments in Italy I think is much higher than anywhere else in the world at the moment, and I'm surprised that the trouble is that an awful lot of it has not been translated. I'm working now on translating some of Alberto's stuff. I'll bet he does no right in English, but it needs a lot of reworking. I just finished reworking a paper of his, a kind of historical paper, on Kant and naturalism, but they're so good by comparison with what's going on, I don't know about Canada or the United States.

25:00 There are obviously some good, there is a lot of good work, but certainly in England now, that's what I think particularly around... Montreal. Yeah, Montreal, especially Montreal, and here in the West Coast area. Well, it seems to me, basically, if you take a radius of 700 miles around Lake Erie, you've got what you've got to call Billy Muffin. You've got Colin in Cleveland. You've got everybody here. You've got the Montreal people. It's actually the area to be. It's where I'd love to move and actually try and get into the subject of going to the right classes and that's what I'd love to do. You know, it's unfortunate for me, you know. John drops out of the blue into this department. Yeah, well, it's unfortunate for John, too, because he was very unhappy in London, as I'm sure you know, because he was very... Yeah, well, you know... Mind you, don't let him, you know, don't let him commit to everything he says about the horrors of England, it's true. He was very, very, you know, he was very depressed the last few years he was in England. But now that he's here, he doesn't talk about it. But he was very, yeah, he was very depressed because, you know, he's not in a good place. It's just not at all a good place to do logical philosophy at the moment, unfortunately. But that can change. It can change. It doesn't have to be. It doesn't have to go on forever. I mean, Canada in 1940 was probably not particularly much of a place to do logical philosophy. Things could change. I think that it also wasn't that great a place to live if you were a professor. In England, maybe. No, no, that's true. But that's because of the general marginalization of academics in the U.S. as well. But also, you just do it generally how I stand as Libby in North America. I like everything I've seen in Canada. The fact that the school...

27:30 Well, the fact that you don't have the extremities, you don't have the kind of... I mean, one thing that... I'm really sorry, but I... I still haven't matched on to the family name. Is it Klusak or Krusek? Kathleen's husband, the guy sitting right opposite us, talking to the husband of your head of department. What's his name? Krulak? She's Kathleen Okulak. Oh, Krulak. I'm sorry, I just couldn't get... That's her name, but his name is something else. Oh, so she keeps her American name. Brown. Oh, what about you, Jamie Brown? That's what I thought, Jamie Brown. I will never know. So he's the guy who, every time that someone says something about abstract metaphysics and, you know, whenever Bill would say that, he'd forget. Well, he'd forget because he'd have taken it by the time I'd taken it. Yeah, you see, now, I don't think it's into place now, because he had a paper in that... Symposium edited by Andy Irving, Physicalism and Mathematics, which was actually called Pie in the Sky, which was a very hard-line defense of Platonism. And I was certain I'd read that paper, because some of the things he was saying to me last night, I thought, wow, this is exactly like the paper. J.R. Byrne, isn't he? I remember reading all about him. And he said, I think it's garbage. I don't think anybody has used before from quantum theory to do with the two wings and the bell experiment to show that the causal theory of knowledge has got to be. I thought, this guy, I remember reading exactly that argument in that symposium by Andy Irving, by this extreme theoretical Platonist nut, but it wasn't by anybody called Krug or Krulik. Oh, now it's all falling apart. Right, now I know. Because, good, that's a relief. It's a very nice guy. God, he's a nice guy. They couldn't have been nicer to me. And his wife, too. Made me feel very welcome. And he's mainly a philosopher of science, and Kathy, is she mainly a philosopher of science, a historian of philosophy of science?

30:00 Yes, it's a huge project. Whoa, it's still really interesting. The way the bill will use these terms which haven't been used in expanding logic since the 18th century, like quantity, without the slightest embarrassment. And you realize he's really got a hell of a strong motive for using that particular. But I kind of think he does cut himself off from having more of an influence on mathematically competent philosophers and mathematics by choosing to expose his stuff that way. On the other hand, I now see after having followed his ideas for a few years that there's no other way he could actually experiment. I mean, that's a psychological remark, not a remark about the ideas. I mean, I just don't think he would be able to do it any other way. It's just such an interesting area. Anyway, I'm really glad you're here and doing stuff. I think it's a very good place to be right at the moment. Yeah, a really good place to be. It's not a great place to get a job. Where is it? Yeah, well, this is it. It's not as bad as England. What you want to do is, you want to be one of the oxymorons, I'd say. You want to get a job. You know, business ethics or... Yeah, yeah. Well, you don't mean oxymoron. You mean contradictory, you know, you mean business ethics. You mean contradictory. Sir, I'm telling you, I thought an oxymoron was something where you were saying the same thing twice, you mean a contradiction in terms? Oh, right. I'm sorry, I'm a military intelligence. Well, I think that's unfair. I have a slightly higher opinion of the military than you have. But certainly business ethics, yeah, having been in business myself for the last ten years, that's quite a good thing.

32:30 Yeah, that's probably true. I think you're a bit unfair to me. I think you're a bit unfair to the military. The only reason I say that is because... It's not family connections. No, no, it's not family connections. No, it's business. It's actually professional connections. I spend my life taking around veterans on these tours. Most of you are World War II veterans. Some of them are actually very impressive guys. No, some of them were Reagan armies. Some of the best of them were actually. I just had this guy who was the, retired as the deputy chief of staff at the Pentagon. They were impressive. They really were. On the whole, I think the thing that struck me about them is that... You know, their professional formation did actually make them, you know, proof against the kind of really grubby, seedy character deformation that you do see in business. I'm sorry if that sounds really fascistic, you know, kind of code of honour and Bushido and so on, but there is something in it. These guys, the code these guys lived by was such that they didn't deteriorate. People who work in brokerage houses or in the travel business. On the other hand, you can say how can you deteriorate them by dropping crates of napalm on people. It depends, you know, what you... I mean, if you believe that that's your cycle path, it's normal.

35:00 I still wonder about what is the most effective way, pedagogically, to introduce that deepened connection. There are two philosophers whose whole way of thinking about these issues, as you say, has been rooted in a tradition whose distortions they find it very difficult to recognize. I think that the sequence is exactly right for mathematicians. Perhaps it's necessary for some, not for today's audience, but for some audiences to... To make an approach through the history of these distortions, through the history of the pathologies that gave rise to the view of subjective logic as the foundation of knowledge. You're obviously also interacting with these other pathologies. It's also because your thought deepens and becomes more many-sided and disconnected all the time.

37:30 You have to temper the wind to the shore now to talk to your audience. In the last quarter of the 19th century. Oh, you mean now? Yeah, yeah. And Frege, and Russell. What, who speak about ontology and metaphysics? You mean people like Shapirovich? Which, well, I'll show you.

40:00 Yes. We will show you the categories of being as such, but we won't have to win an understanding of them through understanding, you know, how mathematical concepts arose through the struggle to understand variation in the real world. We will just declare them to be what we declare them to be. Well, I agree with you. I agree. I think when people like Putnam are terribly unserious, they won't engage with these problems at all. These people claim that they are, that they stem from, yeah, that we're in an even more political realm, long term, that's still secular because of the black people. Let me tell you, the principle of Liechtenstein has got this foundation. There are several people working on it, all British, and that's what they're doing. They're aligned with Bochenski, the well-known anti-Marxist Jesuit. Yes, the Catholic so-called philosopher of logic. And various others, and they claim to be really good at it. And they claim to stem from...

42:30 That I didn't learn. Yes, I mean, to express the domination of a portion of it. That is interesting, I hadn't heard of that. Of course, one can also find that at the convention. Well, you see the point is, I've never heard of this guy, Slourez, but I know no English book that you would be interested in. I haven't heard of him. I mean, you take The History of Philosophy by Bertrand Joplin, just to take the most in the discussion of examples, And then the other, I didn't want to do it on this one. You know why? Because of this divine right of change there. But I never even understood what that meant. I mean, I was very confused. But that's why I had the permission, I had the hope, to polemicize against the divine right of change. Oh yes, of course, that would have been great. And it was determined in England by James. Thank you very much for your time. When it literally happened that way, where the philosophy is needed, and I said, well, what about the objections, right? He said, well, you're right. I said, philosophers have been slacking on the job for 250 years. They're really slacking on the job. I said, well, that's right. He says, well, no, we haven't been. Look, we've been doing all this stuff. And he brings out, you know, all the stories of what he was talking about, the boundaries, the results, and declaring this was possible, that's not possible. I found out that he is the new editor since last year of the morning. The editorship has changed in one trend or another. So he shuttles back and forth between this guy and the buffalo and the hornet. But this, so, you know, he's sitting right there in the case.

45:00 So he gave me this, they have an encyclopedia, and he's the editor, together with Plotanski. That's a lot, isn't it? This two-volume encyclopedia of ontology and mathematics. And it's some of the most, some of the most irresponsible, I think they say it all the way there, because, you know, some of them are irresponsible, but it's all promoting this particular form, this particular thing, so I'm glad to hear a lot of hope from you guys. But then, before I see this country stand in front of me, then I realize, you know, I was just thinking, why is this an intense fashion? Why is somebody else doing something else? It seems to me more a form of... The objective idea is to respond to the deceit, or at least attempt, and it goes back and forth, you see. One idea was defeated by the people coming up with their two dialectics in the field, and so the response to it was back and forth, you see. So pragmatism at the moment is, as you say, on the way, or is seen as no longer the preferred option. Now, in the most circumstances, could you think anything like that actually is good, not like pragmatism, like you just said, isn't this the objective that we're looking for? I agree. I'd like to find out more. I've come across him. I've read a couple of his papers, which I thought were very thin. This whole idea of presenting that system. Without relating it to the actual scientific investigation of the world or the results one in or the course of, it's un-theoretical and un-practical. He also translated Brentano, which I've been trying to read, but it's hard to source myself because there's so little substance to it. I don't know Brentano, but it's sort of the same sort of spectator, so you can try to figure it out, but it's practical.

47:30 A chap called Barry Smith, who Bill is telling me about, well, telling me about and warning me against, Barry Smith, he's now the editor of The Monarchs. He's not the editor. He is. He has just become. You do know me? Oh. Well, you can tell Bill more, because I don't. Do you know me at all? I'm just going to go wash my hands and then rejoin you. Come, stick it on. Wash your hands. Thanks. As a judge of a matter, and the reason that this is a judge is because we have the width of our, that we can capture everything. Your category can only be delimited within our wider category of being such, who are interested in the subject that they call ontology and necessarily suck into this.

57:30 Anti-idealist is the one that does, which people did actually take seriously. So people who speak about ontology, it isn't necessarily a warning signal. I don't think it's necessarily a signal that here is mystification, here is idealism. It is often the case, but I don't think always. I think one of the ways perhaps of attacking these people is precisely by doing what you said at the beginning of the paper, Here are the tools. We believe this is the best general theory of mathematical knowledge that we have, mathematical knowledge as the most fundamental part of scientific knowledge. Here are these traditional philosophical questions of difference and equality and being and becoming. Here is the way that you people have treated them. Here is the way that one would tackle these problems with the tools of category theory and look at the difference. Now the problem with that approach is that they will see that as, ah, you are just proposing an ontology, you are proposing an ontology. You propose an ontology, you see, from their way of thinking. You don't actually, an ontology doesn't come out of the actual concrete process of understanding the world, of understanding motion in the world. So the danger of such an approach is that they will say, oh, well, here is Aristotle, here is our ontology, and this is Lorbeer's ontology, and that is a danger, but it's a methodological danger.

1:00:00 But on the other hand, I think that is particularly when you were explaining to me last night, you already made a point in Cambridge about the role of toposterian understanding. Issues of distinguishability and indistinguishability and abstraction. That does directly confront the way those people, people who call themselves platonists in mathematics, who think that mathematics is subordinate to their mathematics, that the mathematical knowledge has to be limited to their methodological categories, the way they think, This way of treating certain categories has given a priori, has not, has not arrived. So I think it's probably a good point of departure in attacking them, particularly when you're speaking to a purely philosophical, an audience which is mainly a philosophical audience. You know, for example, a philosophy that made this work. First, I separated the notion of language. And then we did separate, completely separated the notion of language. Which could be the words in which I can interpret this language? It's incredible, but this is the point of departure of the philosophy of the Martian language. In how many languages you could express this? How could you say that this is the right language, with this notion of disembodied language? It's fantastic once more, in the sense that it's perfectly prepared. And then you asked briefly, how do you think that Europe is actually the United States? You have already come to that.

1:02:30 Absolutely, absolutely. But I do see that approach, the idea that the world must be made to fit inside language, on that there are infinitely many possible worlds, all of which fit in, are pre-adapted to the structure of the language, Arising from what I was struggling, unfortunately not very well, to express what I said to Bill at the end of the meeting, which is the way that the category of object became so large that the world, the whole structure of the world, came to be seen as something which had to be made to fit within the framework already given by this metaphysical category of object. We have a lot of pathologies in logic, and perhaps, although I don't know enough about pathologies in mathematics itself, but understanding the way that the origins of early 20th century logic, the position of what you call the deification of points. When Weyl said about set theory that it had far too much sand in it, you know that well. I think that sand is also the sand that Vial saw in Setsia, really the sand that created the ontology, insofar as they admit to an ontology, insofar as they're not pragmatists or formalists of modern philosophy. And I think it's going back to reconstruct dialectically the historical distortions that crept into science. It's difficult enough to get these people in philosophy departments to understand just what the problems are, but as I said to Alberto, it's perhaps an analytic philosopher, somebody like Quine, listening to you today, would probably have a much more difficult day. He shouldn't have.

1:05:00 As a 15th century Aristotelian physicist would have had if they suddenly found themselves in the late 18th century listening to Lagrange, it isn't a question of the tools being, obviously the tools have been developed because of the advent of category theory, And that's a matter of learning, of teaching. But the shift in their metaphysical sense, the categories in which they think about are transformed so radically or put back in their right place because of the understanding of the relationship, subjective and objective, of the relationship between thinking and being. That it becomes, well, it becomes such, as you said, such a project for them to dismantle their prejudices because they have assumed that the questions of logic and cognition and the structure of the physical world were all entirely separate issues and that the question of the structure of the physical world and how we deepen our understanding of it was an issue which could only be approached if you already had... Question of the nature of logic, which they don't even recognize as subjective logic, because there is even, there is a book by Dubin called The Logical Basis of Metaphysics, which of course he means the basis of metaphysics in subjective logic, which I think when one reads it one sees very clearly how the, of course he is an ultra-Orthodox Catholic. But they see there is this divorce built into their position between logic, cognition, and the physical, and the understanding of the physical world.

1:07:30 The problem is not simply a neutral, a subjective resource that allows to build some form of science, but to explain how these resources have been objectively possible within a physical world. That is the question in fact we should have. You complete distortion of the position. I mean, Alberto, using the tools given to him by Toposfer and Categoristeri, attacks this question. And they say, oh, I'd either say, oh, you are a reductionist, or they say, oh, but this is psychologism. I mean, there's something, psychologism, you understand. And psychology has been completely expelled from logic. This was the achievement of Frege and Russell, that psychological considerations, considerations to do with the theory of knowledge, with the question of correct rules of thought, have been completely expelled from logic. Logic has nothing whatever to do with thought. Well-known positions. In fact, I'm sure that most logicians in philosophy departments trained in this century, in this tradition, would, the moment they hear you speak, they would throw up their hands with horror the moment that you say that logic is about understanding the rules of law, or understanding the... Which, of course, it is. Logic can have nothing to do with thought. It's just something which goes on in the brains of creatures or on the surface of one planet, whereas logic, of course, already has the categories of being as such, all of which have, and all of that stuff, i.e. the real world with motion, matter in motion, that all has to be made to fit within that as just a little...

1:10:00 That is the distortion. But I really want to understand how that distortion crept in, and how it became so entrenched. Which is this? Not FKTN. What, the one that we've been working on? This will not cost your career. No, just in a few constructions, nothing more than that. You're all much too kind. This will not cost you a career. This is the most important paper in recent philosophy. It is. It's an entirely new synthesis. It has many implications, in particular for Welsh-speaking languages, and in particular for the last, presently more shared view of the relationship between the languages of the world. This is a new form of practice. In Italy, most philosophers today who are concerned with cosmology and mathematics are in philosophy, science, or pragmatism. More or less, it's basically the same.

1:12:30 Well, or have been until very recently. I think your point about what's happened in the last three or four years may be pertinent, but certainly until very recently. Well, I hope you're wrong about it destroying your career, because I think it's a great paper. I mean, the promotion of pragmatism was very important in the early years of the century. I think it came in more from the United States and England, mainly the United States, mainly through people like, like I said, Putnam and Rorty, Klein, yes. I would like to ask you something about what you have said today. The only thing that I would like to ask you, very briefly, is the fact that when... You know, you present these stratifications through levels. No, I mean, abstraction operators. No, abstraction operators. In the case of the two situations you have presented today, that is, the phantom which is, the phantom which has three separate figures, and the other one which has two phantoms, do you think that these two dual situations are...

1:15:00 So, we can be unified from the point of view of these keywords of quantum science and the string theory, because it's one of one thing, especially when you put a string there, put a string on it. In respect to the figures that are found in the string theory, it's one of one thing. I don't see how the record is going to be as decisive as it is at all in the academic world. I actually just couldn't make it over. The group of particulars seems kind of empty out of consideration. There is also this chaotic dynamics of another theory that I have not really figured out in the generality that I was presenting it for, and why that might serve as a less common argument against the part that's coming out. But unifying the three, yes, this was already by itself, just at the level of... Richard has an instruction that really contains both aspects linked together and called it the double envelope. Well, I mean, he was thinking in terms of what I called A and B, and how you take these to be one category, and so it's that thing which has the envelope, which could be a model of some M, but the point is if you attach to each object X the set of all the things, and the set of all the functions... And how these compose, because such a composition is always sort of epistemological, it's always a concrete, directly knowable, because there's this A that you know this A, you don't know the general X, it's a contrast between the more known and the less known, so even though you go through the X, the result is from A to P, so this is...

1:17:30 It's the kind of thing you know very well. I mean, it's a 3x2 matrix or something like that, typically. It's something very computational. It's a way of recording information. You can think of these figures also as being experimental probes. The function is the measurement of the result and the record of the experiment. The record of the measurement that was made in the presence of that probe is something complete. You can write it down in a book, depending on the nature of A, but it depends on that. Describing the object, the general object, to be any possible such thing. There is a set called set of figures, another set called set of functions, which is compatible with the known algebraic and incidence calculations on the paper, which is called the WMF. It was rediscovered in a very special case, but it was interesting to see how far you can go in the richness of structures. There's extremely, extremely special things. This was done by Peterman and Peter. And then worked out by Berlacher and Siegel. Berlacher, Siegel, and Dr. Siegel did functional analysis. So the super simple version... Starting, of course, from the analyzed side, where we define the end, we just have... You have nothing but two abstract sets, A and B, and a set Q of math from A to B. So this is the only real parameter in a way, how big this set Q is. Some of the math from A to B.

1:20:00 So now, that's the data. We now define a category of structures in the modern world. The only axiom on this structure is that the one thing determines the other. But the paths are all those things that are natural with respect to all the given functions, and conversely that the functions are all those things that are natural with respect to all the given paths. There are about three different ways that you can express this. A morphism between key sub-structures, such as mapping the paths in one way and the functions in the other way in the way that preserves the code, or equivalently such that if you follow this map by a function, a specified function on a code image, and follow it by a path, a specified path in the domain, the result will be an element of Q. There's sort of super simple definitions. Then you find that, well, there is an object in the category corresponding to A. A is originally an algebraic symbol. It depends on the canonical structure. And likewise, the beam becomes in a canonical way... Sorry, it becomes in a canonical way... Structured. Structured, yes. We have two examples of this. Right, I understand, the union. It's all generated, nothing but the Q. Yes, nothing but the Q. The amazing thing is how many, how many structures of actual interest can be established here, by partial order sets. In other words, you take A between 2, and B between 2, and 2 is under the C map, but they're ordered to do it. So that a path is a pair of elements that are in order in some arbitrary place, and the function is what's called an order idea, and the answer, that is the important experimental result of the composites, is an answer, you know, is it really true that this thing belongs to the idea?

1:22:30 But the math defines them in general, or just the way they do the math. But, well, let's take, instead, let's take... The unit integral and the real line are intuitively continuous maps, with continuous maps from integral to the line. The resulting category contains all the possible topological spaces which are locally packed connected on the first caliber, generated by sequence. So all the spaces, which are most of those that ever really arise in... Once again, this definition of math captures exactly the same thing. So, an affine space is a flat space, an ejected space, a flat space. Again, all that is just an example. So then, the interesting issue becomes, when is this category defined when there is a balanced pair relative to two? When is it, in fact, that you can play? Well, it depends on two. It may or may not be that you can play. It turns out that, if necessary and sufficient, the y to the power x that exists for any two structures of the pipe itself is true if and only if Q itself should have a structure of the pipe. Because Q itself should be b to the power a, right? It should play that role. It depends on the shape of the queue and whether or not it could play that role with respect to itself or not. See, this is emphasizing the play between the two. I personally prefer a less balanced view, I think, because there's only a topos based on this ABT, from this ABT you can construct a small category, and then there's the topos that come out of those two, so I'm emphasizing the geometrical part.

1:25:00 And let the algebraic part fall where it may, see whether it's perfectly dual or not. And there's also a dual view when you take the algebraic structure to be dominant and then let the geometric part fall. That seems to correspond more to the usual mathematical system. But this idea of the perfect balance is not a bad one either. And it's surprising how many we get. So this was one of the... we proved with this cue, you see, in the C-infinity case, and that one lemma is the basis for large numbers of very good papers by Pulitzer and his students and collaborators about the functional analysis of C-infinity monopoles. Because if you knew that this would be the memo, once you get that thing all the rest followed, it boils down to showing that the field of theory is going to be good. Yes, the structure in itself. And, in fact, there is a school of mathematics in the U.K., which is a very specific field of mathematical analysis. And yet, in that concrete guise, it drives all this generality. That paradigm of the balanced envelope... I must admit that it rather gets people more quickly to something more universal.

1:27:30 I'm disappointed because it is immediately, because the dilemma of having few structures is relatively simple, it just boils back down to the same definition, you can test something by testing it, by testing it by testing it, so whether or not it actually works, you can test that already too, but the test comes out and it has to be in a few, so hopefully it doesn't be so. I mean I'll just go through it quickly, but it's... In form, it's no big deal. It's just a dilemma. Once you have that, then this whole motive, this whole very interesting category springs from one very, very concrete classical... Actually, this is the example I often have in mind when I keep underlines and you have to do some hard work to make something really neat like that. But it doesn't come out of some vague speculator, you see. At that particular point, you have to really see whether this key works or not, because some of the keys don't. In the abstract speculator kind of use, all the keys look more or less alike. Maybe they have part miles or something like that. But in fact, it's both. Well, of course, the same point about having to understand whether things do work in a concrete case. Of course, it comes out implicitly in everything you say about figures, in the way that you stress the position of figures in understanding the geometric force against the algebraic aspect. It's a very concrete way of thinking. You say, in the way you stress incidence, for example, the ontologist would say, well, that's much too specific or concrete a way of thinking about it. Geometrical space. Geometrical space is much more abstract than that. In fact, we've got a really de-abstractified notion of space here, because it is a geospatial space.

1:30:00 In a dual role, functions on the generic figure are the same as figures in the case of quantity. But this thing itself becomes the abstract general and arrives before the concrete general. And yet it's part of the relationship between the abstract general and the concrete general. And so it becomes more and more... The way of saying, characterizing, it's a way of making cryptic questions, say, Aristotle, a man of God, say, of this man, as the way in which we have the, the other lack of joy, the, the, the, the, It's called the sensuality. Sensual, local, connected, components come through, molecular, all four nodes that they're used in. So this is why Godin didn't have to draw this idea to use the exclamation point. Because, and you can put it either up or down, so it's left or right, all the way to the left, but it means, well, that's the surprising thing, that's the special thing you can have in this particular course, where you say, ah, special, no doubt, whereas the ones that you have always are the stars. Thank you for your attention. This is, this is my, my talk in Iran in the summer, this is my talk in Iran in the summer, this is my talk in Iran in the summer, this is my talk in Iran in the summer, this is my talk in Iran in the summer, this is my talk in Iran in the summer, this is my talk in Iran in the summer, this is my talk in Iran in the summer, this is my talk in Iran in the summer, this is my talk in Iran in the summer, this is my talk in Iran in the summer, this is my talk in Iran in the summer, this is my talk in Iran in the summer, this is my talk in Iran in the summer, this is my talk in Iran in the summer, this is my talk in Iran in the summer, this is my talk in Iran in the summer, this is my talk in Iran in the summer, this is my talk in Iran in the summer, this is my talk in Iran in the summer, this is my talk in Iran in the summer, this is my talk in Iran in the

1:32:30 ...generate the whole of one... ...generate the whole of one... ...generate the names of the usual sets of genera only by direct limits... ...so every object in the topos is a kind of union of the cotaspeed mapping toward it... ...and the cotaspeed, the trivial cotaspeed, the cotaspeed is mapping into them as a trivial map... ...and the cotaspeed maps out, and in most of the variables that you would think of... But there are cases where that determines everything, and basically two well-known, more or less well-known cases. One is what's called combinatorial topology. Sorry, combinatorial topology. Combinatorial topology. Oh, right. I'm sorry, my sinuses are very blocked, so it's difficult for me to hear. A typical contextual context is either the screen or a photo screen, but it has photo screen parts of various sizes that need to be dealt with in one way, so it's actually, the way it's used is it's a scheme or a plan for actually constructing a space made out of wheels, but it's only a skeleton scheme in one way or another. But it's determined by the university's findings. For example, you would have three points that any pair would form a three-discrete graph, but not all three. This is the one-dimensional boundary of the problem, which is a blueprint for building that. So that category, which is called an important goal in anthropology, does have this striking feature of being generated by the state, which is proposed over and another proposed to the state. And you said that there were two cases, the most striking instances. Phonological, basically the figure is a bounded sequence, but we have a motion of a bounded sequence in X and Y, and that's the way we transport bounded into a bounded sequence, and this is...

1:35:00 This is very important in analysis because then you can show that the vector spaces in this purpose contain binocular spaces and much more redundant spaces where it's a much more general space, like a full subcategory. In other words, in fact, the traditional terminology in functional analysis is to speak about bounded linear transformations. That's usually explained as just a word for continuous, but that's true, it's equivalent. In terms of that, and I'm using the linear step, you have a vector space step, we can define an agent of convergence as well. So the so-called Machi-convergence. It basically reduces convergence in the arbitrary monological vector space down to convergence in the random numbers themselves. We have a sequence in the monological space and then we add limit points, so it really is the limit point. It means that there exists a sequence of wheels coming to zero that should be multiplied by...