Discussions, incl. FW Lawvere & M Wright

0:00 But now I think I should keep it all secret, because otherwise these versions will use it to create a new religion. What's this you're going to keep secret Bill? Everything. Oh no don't do that, don't become a mutant, don't become a mutant. I'm asking you to do the only way to solve all my actual problems. Of course. I know you do. You tell them everything. I know you do, yeah. In the hopes that at least one of them will get it out. Oh, no. The guy with the mustache, he wanted to look a little bit like Stalin. Well, he was actually the one that I got talking to that I got the impression that he was the only one that was a little bit skeptical about all of this. He was the one who seemed a little more detached from some of these wild... The guy, the nobleman, who wanted to know exactly in what order basic ideas must be introduced in category theory in order to, mainly for pedagogic purposes, but also clearly for our purposes of analysing fundamental notions, he... English was not as good as that guy, the guy at the start, because English was probably the best in the group, although he said less than any of the others, and he would frequently ask him to translate phrases, and when I was arguing with him, the famous Euler-Diderot instance, I was asking him did he really believe that he could use mathematics to buttress this. I think it's really important to think about what it was, what was the position. And he asked this guy to translate particularly some of the expressions. He even had to ask him to remind him what the Eulerian logarithm was. Surprised me. And the guy, you know, replied, but he didn't make any comment himself, and I looked into his eyes as he actually translated some of this stuff, and I couldn't help but see he kind of, he kind of gave a very, very sort of slight upward glance of his eyebrows, sort of...

2:30 As if, you know, include me out. Maybe I was reading a bit too much into it, but I think he might have been the one that was a little more... Through some instinct, it's to him that I gave four of my papers. Yeah, well, I think your instinct is pretty good. Maybe he is the one, he was the one. Anyway. Because when he discussed mathematics with me, there was nothing there except I knew I had an interest in mathematics, you see. Whereas when I talk to the other guy, there's always something behind him, something very, very, very ominous, really. Oh, yes, particularly when we're talking on the train. You know, you can always see him in the wild. Well, this was only the last couple of days that we realized... Well, actually, literally the last day. We heard about it a lot. You probably learned it earlier. ...Christian, you see, but at least I, one of the top confidants, told her that he had just converted, you see. One of the smartest ones, actually. But I thought, well, you know, I know in the States people are also Christian, you know, it's just a normal thing. I didn't take it seriously. Only in the last day did we start to hear some of these... Really hear of it. Listen. Fascist theories. When they declare it, then it all seems very clear. They are, and it's all right. They are, yeah. Not only am I a Christian, nobody else is a scientist. Only Christians can be scientists. The top is quite extraordinary. Lavendome. Lavendome, yes. Lavendome, recursive, geometry. Yes, yes. I mean, Bill's most recent paper is dedicated. Yes, yes. Oh, but Bill's most recent paper is dedicated to Lavendome on his 60th birthday for a first trip to Lavendome. Yes, and you know that, I mean, I will, on the side, tell you a bit of nitty-gritty physics, because that's the way I get to it. Do you know anything about quantals? Vertex. Quantals. About quantals. Quantals and linear logic. Well, no, it's just that we were hearing a lot about it at the... Quantals. Quantals. Well, I hadn't heard of them until Colin, yeah, I, Bill is very much, he thinks the whole thing is a confidence trick, it's based on a, well it's based on really turning a bi, trying to turn the structure, which is intrinsically the structure of a bi category into a single category, and messing up domain and codomain and doing a lot of really rather trivial algebra that has no, yeah, but on the other hand some of the people there are great believers and I wanted to understand more about them.

5:00 Yeah, I thought so, since the pubs are closed, since we won't have another drink. Yeah, this is the way they open. They use the same cups, don't they? Yeah, they use the same cups. That's okay, that's no big deal. And here's your... I don't know, in the... Oh, quantum ideas need a little explanation of how to study them. Yeah, well, we... Yeah, and quantiles probably, well, not that they have much to do with. Okay. Okay. That's right, yeah. That's right. It's an area that we can study, but I don't know. I think it's a genetic device, but I don't see it. Still I don't know enough physics, but maybe it's because I don't know... John Darling's expectations of cosmology, such as NOAA, can raise the likelihood of electromagnetic waves. A whole sort of theory of cosmology being thrown down, and not even one mention of electromagnetic waves. To include, I thought, to include... To include... To include all of them. To include... To include all of them. To include all of them. To include all of them. To include all of them. To include all of them. To include all of them. To include all of them. To include all of them. To include all of them. To include all of them. To include all of them. To include all of them. To include all of them. To include all of them. To include all of them. To include all of them. To include all of them. To include all of them. To include all of them. To include all of them. To include all of them. To include all of them. To include all of them. To include all of them. To include all of them. To include all of them. To include all of them. To include all of them. To include all of them. To include all of them. To include all of them. To include all of them. To include all of them. To include all of

7:30 Well, one thing I am quite clear about, even from my very limited basis of knowledge, is that there is a lot of drunken speculation in so-called cosmology, particularly in so-called quantum cosmology. Speculation means reflection, reflecting, but inside the head, inside the head becomes drunken, it's just going back and forth, reflecting one side of the head to the other side. Yeah, well I think that's certainly what John's, you know, zero. Okay. Zero sitting in the middle and knowing everything, you know. Something quite else. Yeah. Something quite else. Yeah. Did you, were you there when I, when I made my intervention after John Darling? Oh, you bet I was, yeah. Yeah. Because Fatima came up to me afterwards and asked if I'd taped it. And if so, would I please destroy it? Thank you for watching. No, nothing. I thought you said it had been on an American TV program. Yes, but it was a TV program, one of the most rotten programs that exists in general. But there was this one week when it was directed by a British director, who had this particular skit. I saw the skit two times because it was in reruns too. I really admired this, that's why I told it. The one who played Cochise was Bill Murray, and Ghostbusters, and he's a very awful thing, but he's a good actor, so in the hands of the director he was... Co-chief, you know, a very, you know, sort of how you imagine a real honest Indian stance, very silent and yet sort of all observing, ready to do, not doing anything until necessary, and it was very good, very well done, but the Oxford professor, which played by the director himself, you know, dancing about and telling the students, empty out the left side of your brain, fill out the right side of your brain, and move the other...

10:00 Thank you for watching. Well, he's a little, a very little like the Baccarinite, Gillies, in appearance, this guy. Very slight, same sort of manner. Would that be anything like this guy? No, maybe not. Probably wasn't the same guy. It was 12 years ago that I saw him. Oh, in that case it wouldn't have been Stephen Price. Stephen Price had only been a big comedian in the last five, six years or so. This guy, you see, he was the director. Ah, I didn't mean to. Well, maybe he's also... No, no. It was that long ago. I have no idea. It could have been anybody. You see the method in my madness. Yeah. Mathematics. Perfect. I studied it. He was presented next week. He walked on to the academy and announced that he would present a paper the next week about it. He was hung up for years because he was trying to get triadic. He imagined that it would somehow come up in the Freedom Book. Yeah, mainly because he was obsessed with this Kantian arrangement of the categories and triplets. He had this metaphysical conviction about space and time. But I maintain even if his philosophy was defective, i.e. Kantian, he was Hamilton, Steiner, and Groszman, the same class and the same leaders. He was some of the last of the great mathematicians. Great universal mathematicians. Oh yes, he's a very great mathematician. He consistently believed, you see, that mathematics and philosophy should have a real relationship and not just a... Yes, yes, not just the kind of relationship that's... ...fragmented. Yeah, yeah. Yes. Raymond, Raymond was the success of Hamilton. He wrote a book about the history of this observatory, which he, well, actually sold it to us. And we were, I was looking at it the other day, looking through the index, and I saw a mention of Don Pedro.

12:30 Oh, yes, you told a bit of a story, the Teller Jerry, yes, very interesting story indeed. Very interesting. In the 1860s, 1870s, a very progressive, Fatima here. Fatima, when she studied Portuguese, had written a long paper about him. He was sort of her hero, you see, because he was a very progressive monarch, very scientifically oriented. He tried to learn every kind of science to help Brazil. Well, actually, one of the things that Jerry is interested in is identity and the concept of identity for elementary particles, particularly whether Leibniz's law is falsified. A certain elementary particle, and he also did a lot of work on power statistics, but I'm not so sure that that was of real value, but he knows a lot of physics, he's a good philosopher, anyway he's been teaching there in Brazil, but he said it was very disconcerting at the institute that they had the people like him on the one hand from Britain and America and some other countries, including one or two from the Soviet Union. Who knew quite a lot of good hard-nosed physics and that, and who taught analytic style of philosophy. And then there were all these Jesuits and Opus Dei people too, particularly Opus Dei, very big in Brazil, very, very big in Brazil. Particularly amongst the engineering faculty. I think I'll tend to be more proof than most. Disciplines against mysticism. And they would get together in their little groups and they would be very worried if people were being taught anything which might be anti-christian or communication anti-christian. And the way that the quantum theory course was taught, they were very careful to ensure that it was the Opus Dei or Dominicans who taught the introductory courses in quantum theory.

15:00 Have I met him? Good heavens, I've never met him, no. What happens now? I've read his popular book, Stability and Morphogenesis, that he wrote a few years ago. And I've read a couple of translations. Yeah, I must admit, the book, I thought, was very confused. On my first sabbatical in 1980, hoping to live with Cartier, who did something about Lavergne and the local sport. Cartier, like all the rest, did not do it. But I gave lectures at the seminar at the IATS in the first year. But he was very... I'm very dismissive of the whole thing. He had to do that because I was a guest and he was the most socially assertive. He's an incredible megalomaniac. He believes, he doesn't believe that he's the mathematician of the century. He believes that he is the natural scientist of the millennium. No, he causes this to be written. He can appear on French TV anytime he wants, and he pontificates in an incredible manner, in an incredible gallant arrogance, even though he's Alsace. We were invited to his house for dinner. He started telling Fadiman, I don't know why all these mathematicians work so much. I don't have to work, I don't have to study, I don't, there's no, when people say I should know functional analysis, who needs to know functional analysis? People say I should know about Lie groups, you know, he works in differential geometry. People say I should know about Lie groups, I don't need to know about Lie groups, why should I study that? I don't need to know, why do they work so much?

17:30 So Fatima said, well, you mean that your ideas just dropped from heaven into your head? Yes, that's right. I think Fatima is a very polite and very restrained woman. I think I would have been a little bit more blunt than that. He said, you know, you are full of... Yes, so I discussed this with Grotendieck. I discussed this with Grotendieck then, you see. I saw Grotendieck. I went... We traveled to this isolated part of the mountains where Gauthier lived and talked about many things, but I said, well, what about Rene Tom? And he said, well, Rene Tom is somebody who literally was subverted by Doudanet and Serre, that Tom was a good mathematician in the 50s, which seems to be true. But then in the 60s, he just became crazy. He said that Serre kept telling him he was a genius, and he believed it. As they did to Grotendieck as well. He eloquently rejected this, basically, but that Tom believed it, just completely became just a pure propaganda tool, and really, so this brings our university, Buffalo, there are certain literary people who... They also invited physics teachers. Yeah, they would. I mean, we have this sort of thing in Cambridge. I mean, George Steiner, you know who is George Steiner. He invented, invested a lot of propaganda in René Thorn. They had something called particularism. A conference on this whole classical thing called particularism. Which means that everything is particular. No general has any validity. No generality has any validity, no moment of particularity. What is the kind of extreme called nominalism? They read at great length from Henry Adams. Sorry, Henry James, I mean. Henry James, the brother of William James. He has some novel in which the chief character is a socialist. But he has this defect, you see, he believes in certain generalities like working class and... So he turns out to be a beautiful character in every particular and also in a particular way. This was the key example in particular. So they invited René Comte as a kind of scientific background.

20:00 So, Rene, Tom gave my colleague, well, I couldn't stand it, but my colleague went off. Detailed notes, what did this guy say, to these particularities, is it full of shit or is it, so he goes on, mathematics is a landscape, actually a slide, mountains, dangerous mountains and paradoxes in the background, and then there's a sunny beach, you see, and then there's a stream flowing down, and each of these is a graph. It's a beach and a mountain at the same time as being a stream. It's a very versatile sounding object. This object in its relation is very versatile, I think a little too versatile to be convincing. I've read the book Structural Stability and Morphogenesis. I was going to ask you as a matter of fact isn't this isn't there something in common with this you see this is a very pervasive theme in anti-science I now see in a way which I hadn't understood until I guess in some ways until meeting Bill how very To deepen influence, this is on Finkelstein. I think it's where most of his ideas come from, this kratalos. Well, we say Heraclitus. There's an argument not particularly relevant here as to whether the actual historical Heraclitus did in fact believe in this. Knowledge is impossible because everything is so completely in flux. There can be no stable, no, there are no determinate elements to serve as a basis for the study of. Subjects and their relationships, so therefore there can be, there is effectively no objective world or no access to the objective world. Whether the real Heraclitus believed that or not, that's what I mean when I say Heraclitus.

22:30 And that, I think, is a very clearly influenced Wittgenstein's ideas. Do you read these last two papers? Which ones? You mean the relative... Well, when you say the last two papers, do you mean the ones that he gave out in Oxford? Yeah, I did read them, actually. I read the relativistic quantum automata one, which was... Well, that wasn't too bad, I thought, the relativistic quantum automata one, because that just stuck to... This semi-group models of, you know, dynamical actions, just comparing, you know, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no But the other one I thought was actually a bit better than his normal, than some of his other papers. Yeah, and I would like... Well, I never heard what the question was. No, he said, I reject authority. He said, oh, yours is the voice of authority. I have spent my life rejecting authority. His idea was that every category is really semi-glued, that when two maps are not composable, then the composition is defined to be zero. Yeah, this is the thing he's been hammering at me for the last year. You see, so I told him this was wrong. I mean, I told him that many different categories here had thought of this, and they tried it, and they thought it was wrong. The definition of category is the correct definition, because... There are thousands of tests that people have tried to do, this kind of thing, to pretend, you know, to ignore the bookkeeping, really, is the idea, to ignore the bookkeeping. And he wants to ignore the bookkeeping because he's hung up on the second, you know, principle of anti-science. It's just not, yeah, it follows, yeah. It's not correct because it doesn't give correct functionality, you see. A functor will not preserve this fact of being zero. A functor between two categories should be a functor in the usual sense, but if you make this monkey with the definition in that way,

25:00 then you get the wrong work with such structures. But basically, that's basically the reason. But in every kind of particular example where you try to do this, you see that you get a mismatch with the real phenomenon if you do this. This is what I explained to him. The point is that all of this, what I was telling him, I mean he can choose not to believe me, or he can choose to disagree, that's okay, but he doesn't do that. He rejects this concrete experience in favor of his speculations, saying that this concrete experience, you're trying to force on me this concrete experience of the science, of the society of science, that's authoritarianism, the same point of view as John Dewey, that science is authoritarian, trying to force on me this experience of the society of science, that's authoritarianism, the same point of view as John Dewey, that science is authoritarian, trying to force on me this experience of the society of science, that's authoritarianism, the same point of view as John Dewey, that science is authoritarian, trying to force on me this experience of the society of science, that's authoritarianism, the same point of view as John Dewey, that science is authoritarian, trying to force on me this experience of the society of science, that's authoritarianism, the same point of view as John Dewey, that science is authoritarian, trying to force on me this experience of the society of science, that's authoritarianism, the same point of view as John Dewe And this, of course, is just crazy subjectivism. It's to reject all human culture. It's to say there is no cumulative knowledge. There's no cumulative knowledge. There is nothing for me to learn from. There is nothing for me to learn from. You know, I am on my own with it. Yeah, that is absolutely crazy. The reason he's so hung up on with this idea that a category is a semi-group... He thinks of domains and codomains as, in terms of a kind of underlying dynamical action, he thinks of the maps, the morphisms, as a dynamical action, as processes, and he thinks that matrices are more fundamental than maps. He thinks that every mapping is really determined by a matrix. Which solves the Maxwell's equations and in that way produces the electromagnetic field. This is a paper... ...that I studied 15 years ago, 12 years, 15 years ago. Oh, yeah, and he thinks that the... ...commented on them by Wigner. Wigner says this is a very interesting, encouraging idea. And he thinks that there's no exponentiation in the category of quantum mappings. There's only product. And this can be explained. He even has this... And I must admit when I, because I knew no mathematics really at all at the time, when I was first exposed to this, I thought this was a really deep idea and I was hung up on it for at least a couple of years, I admit that really, and I'm really glad that I've been shown, you know, where it's wrong and why it's so wrong, but he thought mapping, you know, he's got this paper called Quantum Mappings and you are

27:30 Well yeah this business of linear logic yes so of course right now he's just discovered linear logic and quantiles. He was telling me all about them on the phone the other day. He's really excited. I mean I have to say I really... Well go ahead. Let's go out of this place. Yeah, let's go. That was a nice meal. Thanks. Good meal. He's got this weird idea that every mapping is determined by a matrix and the ones... It doesn't have domain and co-domain because those would be states and there's no such thing as a determinable state, you see. So really off-diagonal elements are objects. I told you this example. I like very simple examples, right? So I was very pleased when I was visiting Greece five years ago waiting for a bus. Did I tell you this story about the bus? Waiting for a bus, you see. The point is, you see, that these people who think that there's only process, no states, you know, that morphisms exist but they don't have any domains or co-domains, you see, they're very much against stasis, you see. Any idea if there's stasis is a bad thing, I see. Dynamics is prior to logic. That's actually a Finkelstein slogan. So waiting for the bus, you see, I suddenly realized that there was a sign there, you see, that says stasis. Because in modern Greek, the word stasis means bus stop. I mean, it might mean other things. It means the stop of the bus. Now, you see, this becomes clear then. The bus system, the main thing about the bus system is it goes from place to place. But, if it didn't stop, it wouldn't be of any social utility.

30:00 So it's a graph, it's a reflexive graph. There's a stasis in it. It's not the main aspect of the system, but it's a necessary aspect. I find this example very compelling. Yeah, I agree. It's a very compelling example. I'm just a little bit worried, I was saying to Bill earlier on, Gerry, that John has become a bit prone to this way of thinking. In some... John Bell, yeah, John, because when I've talked to him about local set theory and topos and this business in his book about reference, thinking of truth values in relation to a reference frame and the fact that he takes up this work of Davis on... Think of the reals as self-adjoint operators, which was applied in quantum logic in every, you know, the quantization just is a Boolean valuation, and that it all goes with this idea of indefinite, you know, there being indefinite objects, you know, Finkelstein. John has rather fallen into this way of thinking. I think he's, maybe I'm being unfair, maybe I haven't understood clearly enough. But this seems to be the way that his, this seems to be the way that he thinks of topos is, which I'm sure is. The man on your right. Have you read John's book? Only very briefly. I just saw it when I got to Milan. I read briefly that section. I read Colin McLarty's. Well, it's not only the book on toposes, the article on infinitesimals, though a very good, very interesting article, does end with remarks very much in this vein. Thus we see that the stasis of mathematical entities is... Oh, what is it? Yes, you know, all these remarks about the... We've modelled variation in terms of the static infinite, i.e., in set theory, but of course the thing about topos is that it has an inbuilt concept of variation. Well, sure.

32:30 But therefore we've eliminated stasis altogether, which means we've eliminated in more than one place, yes, in the Infinitesimals paper. As I said, this was several years ago. My friends in Milan are very fond of this. This becomes a slogan for them. What it really means is thinking dialectically. You mean the buses shouldn't stop? Well, in the case of Wittgenstein, there are no buses, really. There are no buses to stop, but there are no beings, you see. Most people object when the buses stop. I think it's something quite unfair when the buses stop. Oh, but you know, they do not, of course, they are... The next one, it doesn't stop either. But you see, it's all this anti-consciousness, anti-consciousness rejection of experience. This idea that all is flux is, you know, there are no beings. This is a world without being. There's no category of being. And hence no buses either. Buses are all an illusion, like in Buddhism. I am the batsman and the bat, I am the bowler and the ball, the umpire, the pavilion cat. Oh god, him. Lali, Lali, Lali. Lali, Lani or something like that. You were very impatient with the man. Me? No, Bill was far too patient, if anything. Bill was painstakingly patient with him. Well, what could you say? Well, an overwhelming desire to tell him to shut up. Well, I missed him in the end. I mean, he... No, but there are, there are... He didn't contribute to the scientific discussion, but... No, I mean... From a human point of view, he was... Yeah. No, but I'm the last person in the world to be elitist. Which meant nothing in the end. Yeah, nothing whatever. No, but I mean, obviously, I'm the last person in the world to be in a position to be elitist about anything, but there are certain conventions that should be observed when you've got some of the leading category theorists in Europe, in the room, when you've got Bill McBeer lecturing to them about foundations of their subject. I mean, it really...

35:00 I mean, you know, you don't get somebody, total amateur, absolute outsider, you know, biologist, who starts off the discussion with a five, you know, five-minute question and then follows it up with three supplementaries. Yes, this one, yes, this one here, yes, this one here. Well, maybe just once for light relief, but when it happens at every single lecture that Bill gave and every single paper, it gets well beyond the joke. In fact, I... I just waited until I found anything remotely progressive, and I responded to that part. Yeah, well, you were very patient indeed. Very impressive. A little bit further, Jerry. It's not this, the first one, but the second one. I thought that... No, not this one, the next one, just a bit further on. I thought he came close to asking you about hyperdoctrine. Here we are, just coming here on the right. Did he? I remember very vaguely he could interpret it. He wouldn't have known a hyperdoctrine from a dead man. I mean, he's a biologist. I mean, nothing dishonorable in that. His sentences were beautiful. They meant nothing. They meant absolutely nothing at all. He wasn't at a loss for the right fuzzy words. He wasn't at a loss for bullshit to say nothing whatever words. Yeah, but that's the gift of the gab, isn't it? Fuzzy words. In fact, fuzzy was one of his favorite words. I think he must have asked about three terms about fuzzy categories. I had the gift of the gab. You are now well and truly stationed there. That was a very pleasant meal, really. Thanks. You did receive the topology, so you could report on the list when you came in. And then we picked that... I didn't mention Russell. No, that was it. I certainly didn't mention Wittgenstein.

37:30 I have to say, I don't blame you for saying it either. I have to say you were a little bit unfair because John Dawling's never been at Cambridge, of course. He was at London and then at Amsterdam. I know. Hello, boy. Hello. Hello, boy. Hello. Well, it's a little bit like the British establishment trying to tell the difference between Lenin and Stalin, do you remember that? Didn't I tell you about the times of victory? In all of this, the essential difference could be observed between Lenin and Stalin. Lenin was, in almost every respect, more British than Stalin. How many cigarettes have you got? Eight. Well, that's okay. It's so wonderful, isn't it? The mentality! You know, Michael? Yeah, Jerry? I've got cases and cases of my math books, which I'll jump in, jump out of the case. Yeah? Oh, what's going to happen to them when he sells the house? I'm going to take them away, but I'm wondering where to store them. Well, if you want to store any of them here, you're more than welcome. Particularly since I'm going to be moving them. I've got about two and a half thousand books in the garage at the moment, which has got to be in, because I've just had a whole lot of bookcases made, and they're just going to be moved into the room where I'll be sleeping tonight. And then, so... I mean, you'll use them, but... Well, obviously, I'll take care of them for you. I'll make some tea, shall I? Tea or coffee? I haven't got any more wine. I thought, I'm sure we should have got a bottle, but never mind.

42:30 Let's say a cylinder. What's the size of this cylinder? It's the size of the base, which is the area, square meters, plus the circumference, times the height, plus two, which is the boundary.

45:00 In order to compute the size of the cylinder, you need these lower terms, because the area of the outside is the sum of the circumference of the base times the height. Plus this one times this area, plus this one times the other area. So you need the lower terms are an essential part. It's not just a volume. You need the area, length, and order characteristic in order to compute the area, length, the order characteristic of the product. But then it comes out exactly. So because you get these cross terms, you have zero dimensional times two dimensional. I started looking at books on logic, about my thinking, and explained that the beauty about logic is that it has no content.

50:00 The good thing about logic is you don't have to worry about the content of the proposition, you know, A implies B. Ah, is vacuously true, I think is the great phrase, isn't it? Vacuously true. A and A implies B. Two meanings. I didn't forget, though, they really had some questions. Do you understand this internal sort of language? Sorry, internal? I mean, John, John, John Barrett's book. I think I do, because he's been very good, he's kind of taken me through it. I don't understand, I haven't read all of it yet, I haven't read the chapter on sheaves yet, but the introductory stuff, well the first three chapters are really just sort of introductory category theory, really very introductory stuff that he was known for, and then, you know, he gives an account of the cars, and then builds up toffices as, well, very much as local, as local sect there is. But with a very strongly sort of type theoretic. See what I thought was illuminated logic. Did you? Sure. It's gone back. I didn't think you'd done, I never thought that was part of your problem. I'm sorry, I haven't done the display.

52:30 No, I mean. Display, please. You have these objective things, you have maps into these, you've got probes these and Yeah. But they are geometric things, so no need for logic. Oh, but no need for subtext. But then came Bainabou. Very clever, you see. It's due to Bainabou, actually, although Bill Mitchell and U.A.R. and other people... No, no, it's fair. Go on, go on. They said no. And then Johnstone in his book. So they regretted what? Same thing. They all did it, you see. They all did what, sorry? Well, they falsified the real idea of topos by making this... What's the technical term? It's not reification, it's sort of duplication. You have the topos, but they're not satisfied with that. They have to have the language of the topos alongside there, you see. So every object and every map becomes a primitive in some language, and you build up these imaginary strings of symbols on top of that, and then you interpret that back into the topos, and they say, this is the thing, you see, really it's this. It's this language of the topos, which the topos is about. What is this? I mean, it's total nonsense. It's a way of doing modeling. It's total nonsense. You don't need this. What do you need all this crap for? You've got the topos. You just calculate with the topos directly. Why do you need to duplicate it with all these... So the fact that I don't understand... In any case, it's not a real language. In the typical case, the topos is a big thing, right? This is a language, this is a proper class. The fact that I don't understand it, so it's good. I've been, I am immune to it. Good, thank you. I haven't figured out the counter-attack yet. This has been... It'll come. But all the same, I wouldn't have said... Just leave it. I wouldn't have described that project as setting out to abolish logic. I would have said it's setting out to explain why... What logical structure really is I mean it gives you it gives you the the the the objective basis of what logic is about that's what I mean yeah yeah and therefore but this is a particular in such a way that the only only so to speak meta logic that you need is just left exact logic yeah composition forming pairs and forming pullbacks you see that's that's the only

55:00 Well, logic just is interlocking adjoints. The only syntactical. No, but it's not even, you see, these adjoints are internal. Yeah. Not external. Uh-huh. So all you have to do is substitute. You don't have to, but there's logic of, the logic, the language of the topos. Goldblatt and Bell. See, they make this the first thing, the language of the topos. As soon as you have the topos, then immediately say, well, no, don't look at the topos. Let's construct this language. And now it's clear, because we're logicians, so we have to do this language-wise. Is there a technical advantage of this? I don't see any technical advantage of this. How can unnecessary duplication be a technical advantage? I'm only asking. How can concocting? It isn't even real syntax. I mean, a language which is a proper class is not real syntax. This is some idealist invention to call that syntax even. So how can the duplication by constructing this pseudo-syntax be a technical advantage? I'm waiting for a geometric introduction. Yeah. But I had a geometric introduction. I kicked myself ever since. That was, what did I say, 17? 18, yes, you said. 73, in other words. April 73 in Montreal, we had a kind of one-month free-for-all meeting. It wasn't a scheduled meeting, but I was invited there for a month, and then Benabu was invited for a month, and then somehow other people appeared. So we had a sort of one-month... There was a period of seminars every day, you know, rotating from one university to the other, and this went on and on, and a lot of, Gavin Wraith was there, you know, he did great, great things, you see, really, a lot of people, Gonzalo and Markov, he was very good, but Vaynerbu, you see, he stuck this thing in there, and then I kicked myself that I said, well, this is a good, good thing, Jean, you know, because I thought... Well, this is when he was sticking types into the... I thought, I thought this was a good thing because he was...

57:30 You know, making certain things more explicit or more concrete, there's a serious aspect there besides the topos for no reason at all. Yeah. Well, because they just see the topos as a model in the old, you know, logical model theory sense of model. It's even worse than that. No, the language is derived from the topos. Yeah. So that the actual, I mean, in other words, it's as though making a theory of real numbers, that you, each real number is a constant of the theory, not that it's a real closed field or something, like in traditional model theory, no, every real number is a constant, so in the language of the topos, every single map of the topos is a constant of the theory, so contrary to having eliminated stasis, they've made every single variable map into a constant. Well, yeah, I think by isolating, I think by isolating the, you know, the, the, the, the, the, by trying to get, by trying to get clearer on, on the, the geometrical. Well, I have to say, Bill, I think Jerry's right. Are you exaggerating about what? Well, about this... No, I don't think you are. And I think that what this does actually tie up with... Yeah, I will, I will. Just stay there. I'll bring it. Yeah, of course, I'll have a copy. John gave it to me himself. Just hold there a minute. But doesn't this tie up, Gerry, with what you were saying earlier? Hang on, just let me get my tea.

1:00:00 Yeah. But maybe you could take another view that this is one way people understand topos theory, you know, maybe you could take, well, I don't know, there's a lot of languages, I know how to use the topos theory, to make an issue of this stuff. Well, it just seems to me that the top, you know, that the underlying, you know, the very fundamental notion of topos is the category of burying, burying. It's this notion of the, you know, the inbuilt variation in the strongly- Any topos is obtainable up to a prevalence of categories as the category of sets within some local set theory. So obviously the theory, you see, is the proper class in most cases, thus revealing the precise sense in which toposes are to be regarded as generalizations of the category of sets. But it's not at all the sense that you meant that they were a generalization of the category of sets. You meant that they were... Subject. Subject. This is all subject. Yeah, yeah. Yes, you see, that's not at all the sense that you meant that they were a generalization of the category of sets.

1:02:30 I mean, you meant that they were a generalization of... Well, sorry, it's late, but they really have to do with... Now, the way that domains of variation are parameterized in the actual world, this is the point that Jerry made earlier this evening, I mean, about the way, you know, the way that one thinks of the mappings, you know. See, this even, the name of the book, you know, topos and local set theories, not topos. Topos is, I don't know, so it gives us definition of a local set theory, which is a class, instead of a class. It's a very syntactical thing. Now when you come up in the pseudo sense, there's no real syntax. But then when it comes to topos, so the category of sets of topos, there's a collection of L-sets. Those are the types. There's a philosophical name for this, right? The Cartesian dualism or no? Every entity is duplicated. Yes, I don't know that there is a name for a particular, I mean, obviously we're not talking about dualization in, you know, just the opposite, you know, everything, yeah, well, unnecessary duplication of entities, yeah. But you see, that's because they've got an effective philosophical conception of logic, because they think that logic is a theory of the functioning of formal languages. It has nothing to do with the objective structure of the world. It can only be a language in God's mind, because its syntax is in the proper place. And it's not that they think that logic has anything to do with the literal ontological structure of the domains of discourse.

1:05:00 I'm sure this book is written in such a way, you see, that even if I ask you... You bring me your razor, and I cut out the certain chapters, and then the rest would be unintelligible because it's constructed in that way. I think logic is somewhat geometrical. Well, I think, Bill, essentially, isn't that how you do things, correct? I think, yeah, I think that is, yeah, it's... I mean, you said to Colin, according to Colin... It's the relation of very broad geometrical... Yeah. I mean, to do it very crudely, you know, we used to draw Venn diagrams. And I mean literally Venn diagrams. You make observations. Yeah. And you don't talk of enemies, but you talk of... Inclusion of areas. I'm just recalibrating what I read. Colin has a very nice... Well, he says when you remarked to him that you saw how the Scott-Solidae models were a fragment of the geometrical theory of the Russian theater. Yeah, no, but I mean, he described this thing. You know how there's this thing about... There's a scene outside the window and the artist draws on a piece of paper and he rolls up the paper and puts it in a tube and mails it to somebody else. In other words, the scene is mapped onto the artist's painting. The painting is mapped into the tube, and then the tube is mailed somewhere else. But all this mapping is taking place without ever being in the elements. You don't need to have points in the scene. You don't need to have points on the paper. It's a clear idea of mapping. It's the same with the video. Yeah, that's interesting. I've never heard him give that example. The idea of logic that I tried to offer there in Cambridge was that you want to have the algebraic structure of the inimpotent quantities, what these inimpotent quantities do is record where other more interesting quantities live. So for example, there is production of wheat in the state where I live, right? So the amount of wheat that's produced is an extensive quantity. But you can say, well, where is wheat produced? Well, it's in a certain region. Those counties, you see, those areas, so there's where there's some wheat, you know, and elsewhere there's no wheat, you see, so you have this region, again, no need of points, really, but there's this region where, so you can say, well, where is wheat produced and where is cows produced, you see, so there's intersection, union, and all this.

1:07:30 But it is, I mean, again, the cows are actual quantity, you know, some are big and some are small, and more cows or less cows, and more milk, less milk, and all this, so those are actual extensive and intensive quantities, but in particular, you have sort of degenerate quantities, which are where are these other quantities zero or non-zero, more or less, and those are, those define regions, if you wish, and calculus of those kind of quantities is logic. So it's derived from the geometry, space, intensive and extensive quantity varying over space, motions in space, by passing to this simplified approximation. Where does it exist? Where is it zero? What's its support? And then among those things there is a whole calculus substitution. Yes, and the assumption that it's the way it supports split. Yes, in the kitchen. Support in the sense of measured theory or distribution. Yeah, yeah, but don't you also mean support as in the sense of a topos, the way that it splits, I mean supports, the condition that supports split. Not split, no, the support is... The support of an object is a sub-object of one, right? Yes, but aren't you thinking of even that in a kind of geometrical sense? Sure. That's right. Those are those kind of quantities which is just the object itself. The measure is the object itself. In particular, you have, well, where does this object live? It's a sub-object of one. So you pass to the sub-objects of one by asking, you know, what's the extent of the support of a more interesting object? Yeah, that's what I meant. I mean, you obviously put it much better than I can. That's what I was driving at, yeah. But it includes that as well. It's not connected to splitting, is it? No, but... But that notion of support of which one says it might split, that one is a special case where you have these rather objective sorts of quantities of Galileo, Steiner...

1:10:00 But all this is connected with conditioning. I mean, how the way that variation, actual variation of conditions... Conditions are ratios of extensives. This is the, well, what Mayberry taught... This says that Newton said in his lectures too. Occasionally lecture to the empty room, you know. Extensive, yeah. If you have two extensive funds, like, say, volume and mass, then they might have a ratio which would be deficit. In fact, shall I find the quotation from Newton? Would that be of interest? Might be, yeah. Yeah, I've got it. It's in Mabry's paper, one of the other things. Is that it? Yeah. Yeah. So, page 105. With every topos we construct a theory, we get to this on page 105. Topos are specialized for terminal objects, oh my god, this is awful, it won't even work. They make the language out of the whole damn topos. I mean, not just the topos, because the topos, the maths are thought of as constants of the theory. There's a lot of duplication on top of these concepts, which are, in fact, the only things you're interested in anyway. It's like representing a group as a quotient of a free group, where the generators of the free group are all the elements of the group, to start with. Well, I mean, it's sometimes useful mathematical construction to give it a fundamental significance, but he does it in a way which won't even work.

1:12:30 You take your group, all the elements, you generate the three groups, and then you come back to the group. Yeah, yeah. Except that instead of something simple like group operations, you have all the operations of logic. And all the quantifiers and all the... It's like the whole divergence. I think so. Is it? Yeah, really. But he even does it in a way that won't work. He says... What's the fascination in all this stuff? They save themselves by chasing diagrams. They save themselves by chasing diagrams by having a hundred and a half pages. I didn't mind the logical thing with the diagrams. No, I don't. I understand the diagrams very well. Yeah. I don't mind all this logical symbolism. These are not diagrams, these are deductions. Okay, let me show you something. Look here, look here. Given the topos, the local language of E has as ground type symbols the objects of E other than one in omega. Say that again, I only had the wrong question. No, given the topos, the local language of E has as ground type symbols, well you put it in the first range way, they match the objects of E other than one in omega. This is not even coherent, you see. Why, why is it other than one of them anyway? Because if you have a function from E to E prime, then it might get confused. Some object which is not omega might get identified with omega. And this is just crazy to try to exclude two objects in some construction. It's completely unnatural. I hadn't realized that he does that. Those are just the ground types. And you have to start over again and build up these formal types on top of those, plus the logical operations on those things, and then it's a total duplication. Just to avoid, just to avoid from book people, just to avoid community diagrams, just to know what they're going to use. And truly, you know, when you get right down to it, I mean, the derivatives are not any more crazy.

1:15:00 That's a good expert. Not in my own naive way. The Georgians are not that much more crazy than the magicians. You see, I really need to understand more than anything else how you really think about logic and geometrical science. I guess this is probably the best way. There's a summary of all this visualization, isn't there? Yeah, I mean, I... So what's that? The regions of the space where quantities live are to be taken themselves as quantities, and then substituted, added, multiplied, imaged, and so on, so the algebra of these quantities which record... So you can think of, you know, the existential quantifier that takes the, say, this set of, the sheet... That's a push forward of a two-valued, more or less two-valued... I thought you didn't go that extensive on it. Yeah. Yeah. So you'd actually think of a mapping from the, you know, the set of all sheep that are owned by some man to that man, you know, for all, you know, there exists a sheep which is owned by some, there exists a man who's the owner of that sheep. There's quite literally a mapping in the same sense that Jerry's speaking of mappings as coming from, you know, the very notion of mapping is coming out of our experience with the, with our experience of motions. And you project that into all men, and then the image of that is the chief owner, or you take the slave owner and the slave, and you project to get the slave owner. So, but you see, I mean... It's a very, quite literally a projection in a very strong geometrical sense. But you can have it more precise, you see. You can say, well, this man, this man owns a certain number of slaves. This would be more interesting from a more detailed point of view, but the question whether he has any slaves or not, you see, means he's a slave owner or not, so it's an idempotent quantity, which is derived from the actual... Yeah, idempotent means X plus X equals X, so if one quantity is X or another quantity is X, it means that...

1:17:30 Does this quantity live? No. The sum of the quantities itself, in general, but to ask where does the sum live, where does x plus x live, is the same as to know where x lives, living meaning non-zero. Let's say it's a positive quantity. So a positive quantity x, 2x, lives in the same region of space where x does, no more, no less. So x plus x equals x. So that's... That's why it's inappropriate. Whereas the quantity itself, if you add it to itself, it's not the same. But you just take the aspect of whether it is there or not. And what does that indicate to you? If you look for identities in mathematics, what does that... Well, it's a kind of first approximation to the real product. I'm interested to know where are there cigarettes. So I know that, you know, in cigarette shops there are cigarettes. This is derivative of the fact that they have a precise number. But that's not, you know, the first thing of interest is to know where they have some or not, you see. I'm just trying to find that paper of John's with the quotation from Newton, because it decimates points. Is it one of these that you gave me? No, it's not. It's a thing called what are numbers, which is just a reiteration of the same point. I think that's the... let's see how, if we could just go again through...

1:20:00 The fact that the basic, the ground types are all the objects except one and the main. Yeah. I mean, this is indicative of, you know, total lack of mathematical care or consciousness, you see, although, again, Finkelstein would say it isn't authoritarian, but I mean, all experience with every single application of category theory, whether it be to homotopic theory, to group theory, to analysis, or set theory, or any other situation, we know this never works. It never works to just arbitrarily exclude things. It is not punctorial. If you map one topos to another topos, the property of not being omega is not going to be preserved. And so this whole construction describes it this way. I mean, even if you were interested in this covering, as you might be. You see, like in the case of groups, it's sometimes useful to take the free group generated by the group and wrap it back on the other side. So you might be interested in that construction, but you certainly don't take... You can't take the non-zero elements of this book, because that's not functorial, don't you see? If you map one to another, that doesn't preserve non-zero elements. So the whole construction is non-functorial, so it becomes a, you know, you get yourself into, in other words, a much more complicated bookkeeping problem by avoiding to deal with the basic bookkeeping problem. And this idea of arbitrarily excluding certain elements is a way to avoid... I mean that's why Finkelstein's ideas about partially defined mattings are so crazy. That's right. Precisely that point. Non-pontoriality. Non-pontoriality. Domains and co-domains, all the hell. Well, of course, he said, of course they are all the hell because they aren't really there anyway since there aren't any, there isn't any being that's... But it's the same fundamental mistake. Can I clarify in my mind something that bothered me when I first studied maths? If you take a group and you go take the elements and get the free group, you must throw in a new neutral element, not the original one. Right. OK. Thanks. Some books are not clear on this. Do you have Razio and Sikorsky? Razio and Sikorsky, I... I'm trying still to release myself out of A-not-not, B-not-not, because A and B-not-not...

1:22:30 I think it's a good formula, that one. I do, but it's one of the ones in the garage. Well, I may have to... Well, it's... But not tonight. Yeah, I'm still in energy. The nearest I came to finding a proof, because here is a formula that looks like it is, that must be connected with this token model operator. To prove this formula, he says, look at 5.1, which gives you this thing. I was trying to work out from this whether we could prove it in 5.2 being just the translation of this into this. Can you suggest anything in the algebra? Can you reconstruct it, please? This basic calculation is the one which... Whatever angle I start with, I end up with the same inequality. The truth value object is defined to be the universal sub-object, right? All of this is defined to be any internal function, and in the next instance it is defined to be any linear function or non-internal function. So in a generalized map, a homogeneous map, it goes backwards. Say, big F. That is really going backwards, so the big F of Y.

1:25:00 Notice that this generalizes this notation here, because if X were equal to 1, this would say, What we called a mu before. On y now, but, called a mu maybe. So these things, generalized maps, can obviously be composed because if you had two generalized maps, that just means by definition actual maps backwards on the function spaces that are homogeneous. So you just compose those. And that's the right thing there. So, also it's really a generalized map because if you had an actual map, And in particular, you'd have this induced map on the intensive quantities, R to the F, that we already used back here. In other words, this is not only homogeneous, but even multiplicative. So the actual maps, the fact that an actual map is zero to any product. The first big F of lambda is lambda. So this homogeneous condition is a weakening of the... Product preservation. It says that any product where one of the terms is constant, one of the factors is constant, so this is the general nature of the actual maps as opposed to the generalized maps that take more over preserved products. The generalized map can be viewed by a lambda conversion. See, this is actually going from, it's, you're saying, hom sub r, bar to the y, r to the x, hom sub r, bar to the y, r all to the power x.

1:27:30 In other words, the quantities on y, that's what's called the z of x, then, for short, that's our covariant functor, to the power x. The Dirac delta, if you have a generalized map, it might actually be a real map. It just means that it factors across the Dirac delta. A generalized map assigns to every point of x a distribution on y. But it might happen to assign to every point a Dirac distribution, which is really just a point. Which is just a point. So it would be an actual map. So this explains, in another way, the nature of the generalization. So in case of the omega, then these generalized maps, you might call it e of xy, y to the x, because of this construction here. You get that e of xy are the relations. And this observation here ...is equal to e of y. Previous notation e of y with one variable. Special case where the generalized map of domain one is just a distribution. So that's how the relations come into it, as generalized maps in the case where the quantities are just truth values.

1:30:00 In other words, you assign to a point a subset rather than a point in general for a relation. Canonical commutation relations. The formula says that if I consider an extensive quantity on the domain, an intensive quantity on the codomain, then I can compute, or I can take the composite and the notation is supposed to suggest that this is associative, multiply this first or multiply this first. But actually it's stronger than that because it's talking about the location by a variable, by a density. I'm multiplying at one extensive quantity by a variable density, not just by a constant, and getting another extensive quantity. So the thing is, if I take f of mu, that's just a push forward, like the sojourn, for example, concept, extensive quantity here. Now I can multiply that by g, because this multiplication is by intensive times extensive on the same space gives another extensive on the same space. Substitute f into the c first, which would give me an intensive quantity on x. Now that's eligible to be, that's a density on x that's eligible to be multiplied by f. Between that, take cf and then multiply it by mu. Now that's an extensive quantity on x, which I could however push forward. I can take f of that. Is that true? Well, that just means that for all c1, I have to get the same value, or say all c bar.

1:32:30 I have to get the same value. So C-bar integrated with respect to C times F-mu means by definition of multiplication, I take the ordinary product of C-bar and C, integrate that with respect to F-mu. The definition of F-mu was, I substitute into C times, substitute F into that product and integrate it into F-mu. But now I use the crucial factor that substitution preserves product. So this is, I agree with this question, which is C bar F, which is C bar, oh, C bar, left-hand side and right-hand side give the same values, hence they're equal, since they're functionals, and have the same values on all the, there's a proof of this marvelous formula in this concrete case where extensives are just a linear dual, linear functionals on intensives. It's very beautiful. Before we come on to the CCRs, I just want to use the loo.

1:35:00 Sorry, I'm afraid to sleep there. Last night's catching up to me a bit. Anyway, I followed this up to this proof, which is very satisfying. So the claim is this is just a CCR in the context where the usual CCR is a Q. Let's take a different key to put it in a different notation. General p could be any function derivative operator. In other words, p, let's say p of x times v dx to avoid them. So then p acting on a function c means you multiply p of x times the derivative c with respect to x. On the other hand, q c is just q of x, p of x. What is the bracket of PQ? P minus P of X times D by DX of QC minus Q of Q times P of X times P of X times Q of X of X Q of X times PC DX plus P of X CDX, same as this.

1:37:30 Now this is the operator p applied to q, p applied to q. This is for all c, therefore pq itself is just p of q, I mean in other words this particular function multiplied by, I mean this is another q, it's another function which is acting by multiplying. If you take the special case where p is 1, p of x is 1, and q of x is x, so that dq dx equals 1, so that p applied to q, you get d dx comma x equals 1, which is so-called Heisenberg. Heisenberg is just a special case of Leibniz. Or you choose special P and Q. But this is a useful form because you need the arbitrary P and Q. Now you may say this is not really the Heisenberg relation because it says take any two operators which have commutator one.

1:40:00 But the theorem of stone has an irreducible representation of this acting on some Hilbert space x. Then you can coordinate h as being L2 of some space, the line actually. In such a way that A corresponds to d by dx and B corresponds to multiplication by x. It really says that the most general example of Heisenberg is really of that form. But in that sense, it's really no more general. It's a special case of Leibniz. Now, this is the differential form. If you want to express it in an integrated form, the point is that... E by dx is just the derivative of the translation operator, adding 1, the operator which assigns to every c, the new function, c of x plus 1, so, sorry, derivative with respect to t of translation by t, that's what, fundamental theorem of calculus, so, so, so this is, this is a case of a substitution, you see, there's an f, which is adding t, So, on the other hand, my dot here was a multiplication by q, so this formula will become this formula if you interpret f as the translation whose derivative is p.

1:42:30 So, in some sense, p equals derivative, t derivative of translation. The definition of F-translation applied to any C-bar is the same as F-translation, F-translation, F-translation of QF times U itself. So by differentiating, you get the usual statement of CCRs. In other words, we're applying this in the case where X equals Y and where F is just a translation. And both of these are thought of as acting on intensive qualities. Can you just remind me how it works in the case of the ideal, the ideals? There's a construction you showed us the first time around, in terms of algebraic ideals.

1:45:00 Yeah, it was the CCR. You can save me for tomorrow if you like, because to be quite honest, I'm not going to be able to keep my eyes open for more than a couple of minutes. In fact, I was just dropping off then, and it wasn't because this is not fascinating, it's absolutely fascinating. I've now gone for about 48 hours, so I think I have to... ...to crash out. I have to offer. Gerard, do you want to stay here tonight and go in the morning, or what? The only thing, all I can offer you is a mattress, I'm afraid. I can't offer you much in the way of money. Are you sure? Sure, I'm sure, yes, provided you don't mind slumming it. Just let me put a... Just let me put a... Well, I don't know about that. Yeah, of course you're not, no. If you don't mind popping down to pick up some breakfast from the, but I can, well, nonetheless, I'm going to come into your car to take me down to buy some stuff to give Bill breakfast in the morning, because I do. Hang on a second.

1:47:30 You see, one of the things I've always had difficulty understanding is exactly why the difference between the roles of state space, you know, phase space and function space, as between classical field theory, classical particle mechanics, and between the difference between those roles and the underlying ontology. And I'm thinking that this may help, once I've really understood it, it may help me sort out those differences, get clear on them. Anyway, let me show you where I was sleeping. Well, let me show you where you are sleeping and I am sleeping in the same room. I rephrased that.