Discussions incl FW Lawvere, G Khatcherian & M Wright

0:00 Oh, you mean the stuff about the CCRs. Oh, yeah, I should never be able to do that. Christ, I mean, that whole construction, you mean, of pulling forward in extensive onto intensive quantities, yeah, that all these things are meant to be instances of. Well, if it connected with the commutation relations of the topos, Bill would already have seen that, surely. I don't know. I don't know. Well, let's talk to him about that. I mean, to be honest, before I got onto anything as specific as that, I want to get more of a feel for this whole way of thinking of logical structure in geometrical terms and just how much of a break with The Phrygian ideas, the whole Phrygian basis of treating the variable it is, because it seems to me there's so much deep metaphysical insight built in here that I want to understand that a bit before I think about how to adapt constructions to...anyway. ... about intensive and extensive magnitudes could fit together with commutation relations in a top... Because it seems a very general construction. You know this commutation relations stuff, it's published in the... What, the stuff on the CCRs? No, he hasn't published that. He only saw the application of it to the CCR paper in the 1174, the... Oh, the construction is in the 1174 book.

2:30 But he only saw, we want to pull in here, anywhere along here, he only saw after that that it was an instance of the, or that the CCRs were an instance of it, that's what he told me. The sketch he was doing, I thought, although he meant it informally, it was in the topos. Oh, in the sense that you've got, yeah, I mean it's... I'd like to find out more about that, because I hadn't really understood until now just how geometrically he thinks of everything, particularly how geometrically he thinks of logic. I told you I wasn't that bad, was I? Hang on. I guessed this. I mean, I work that way. Yes, because... And what was it you said last night about... Yeah, I mean, he really does think of logical relations in terms of, you know, very much in terms of relations of regions in a space in a very strong way. I mean, it's kind of algebrao-geometric logic rather than algebraic logic, but it is sort of an algebra-geometric way of thinking of all logical structure. I must understand more about, because I think that's the answer to all of John Mabry's questions about, you know, the notion of flexion and extension being fundamental, you can't do without it. I mean, really, Bill's got a notion of extensive quantity or extensive magnitude, which is really intrinsically geometrical in nature, to which set identity really reduces in a special case where you have no problem. Strictly complementary sub-objects and set intersection and all of the apparatus of the familiar classical case.

5:00 He's thinking of that as coming and that really reconciles with the other way of thinking of set as extension of a property in the way that it needs to be reconciled in order to make sense of quantum logic or at any rate of ortho-logic. Well, I mean, I know that's terribly general, but I'm thinking of John's orthospaces paper, not the one about manifestation algebras, but the earlier one, the slightly more algebraic one, in Foundations of Physics. Think of it that way, but think of it in that way very geometrically. Just think of it that way very geometrically. Well, no, it's not the set-theoretical tradition, that's the whole point. It was the Phrygian tradition, it wasn't the set-theoretical tradition, because the notion of collection came first for Cantor, not that of extension. It precisely wasn't... The Phrygian, yeah, the intrigue. Yeah. Um, but actually I think your way of thinking, well we need, we need to go shut in the basket here. I think your way of, sorry, a superphysicist with one leg. No, a physicist, I don't know. Well, no, it's just that. With one, a semi-physicist with one leg here and there. Yeah. Well, that's not a bad thing to be. Now all you need, yeah, I just need to find the, yeah. So, you know, I really need to understand more about this, as I said, the geometrical way of looking at all logical stuff. The Lavandam book. The Lavandam book, which one is that? The Differential, from the Differential, Synthetic, Naive, which he recommended. Yes. Well, is there quite a bit about logic in that?

7:30 Well, there will be a little bit of geometry, I think, if you want to do quantum theory, that can only work. Well, I don't know that I am particularly interested in doing quantum theory after... A little bit. Oh, a little bit, yeah. You don't want to logisticize that. Well, he's given my... 11 doms, man. Yes, okay, right, I'll make a match for that. All right, thanks. Can you check this slot? All right, cheers. Oh, Christ, come on, Jerry, don't tell me I've got it. Christ. I haven't got any on you, have you? I can pay you back as soon as I get there. I'm sorry. Oh, God, I'm sorry. You can relax. No, I just left it on the... I might possibly get an ice cream if you... Do you feel like an ice cream? Get that one. Get one of those. Yeah. Thank you for watching this video, if you liked it, please subscribe to our channel and give us a thumbs up. What, um, yeah, I just want to go and get something from the, uh, no, no, it's your, your change, silly. No, you get, you get that. Oh, I see, yeah. Nice tip, nice tip. Ten. No, ten is fine. Yeah, okay, yes, that's right, I owe you, I owe you ten, right. Um, okay, we're just going to get a bottle of wine from Thrasher's. A bottle of wine? Well, I think so, don't you? What a nice thing to have. No, what I'm, no, hang on, look, I haven't got much time, I really want to try and concentrate hard. The, again, this whole geometric way of looking at logical structure, do you, and for you, does it tie up with what you now describe, I think it's a much nicer expression than naive, the innocent way of thinking about.

10:00 There are a category of sets of morphisms, I mean, it's... I only had that view. I mean, he obviously knows... The whole gamut. But he came out yesterday, talked to someone, and he said he was against all this, but... You see, I'm very sorry. I know it was only a little exchange. I'm very sorry I missed it when you were talking to him, you know, just before we set off to the restaurant, about your ideas about the category of discrete... no, discrete category. Well, we never... The kind of innocent discrete category. We never read the thing. I mean, that's not the main point. That's only the calculatory aspect. Conceptually you can sort of think in terms of blobs and boundaries and things like that he was going on. He's just very dramatic. From them diagrams you get to him in algebra. Now if you start thinking a little bit more carefully about the boundaries, hey, you can then create an algebra. So you've begun. You think in terms of calculations. Finite sets are now called calc. The category calc will be a bit more incalculated using finite sets. Well, I think Arithmoid, because I want to keep calc for the case where you don't have precisely Arithmoid, because you have superpositioning compatibility. I mean, you see John also has that very geometrical way of thinking about... Superposition in, yeah, let's, only for the next ten minutes, yeah, yeah, what do you think, what makes you think you'll ever get paid back? He likes white wine, doesn't he? Well, exactly, so, well, we've got a fridge, that's the problem. Get a big bottle, yeah, yeah, get a couple of bottles, yeah, yeah, are you going to be able to give him a lift to the airport? That would be great if you could, because otherwise we'll have, yeah, Sunday would be no problem at all, it only takes about 20 minutes from here, 15 actually, and I know the way, it would just save an awful lot of hassle if you could, it would give me more time to look for the tapes and pick them up, right.

12:30 They're £2.99 each. OK, I'll just take those as well. Right, thanks. So we're now £30. Well, that's a very interesting piece of arithmetic. I don't think you were using calc when you worked that out. These were a Wittgensteinian arithmoid. These were ones which kept dividing and multiplying. Subtitles by the Amara.org community No, I just want to think more about this. I just want to think much more about this, you know, very geometrical way of thinking, because again, it's the way you recover set intersection with John's, you don't have to motivate it in terms of compatibility of filters of test in this operational way. You can see it really purely in terms of a really kind of projective geometry. Changes automatically anyway, but Preston, you might as well press it. Yeah? Yeah. Do you get the results? Yeah. Yeah, the whole thing, I haven't pressed John's all that closely. He's too negative in terms of, you know, his motivating moments in subspace. Then that's all we get, yeah? Well, do you get something? Well, we get something completely.

15:00 Well, yes, but it's concrete and it's geometrical. Only instead of being something that's really completely orthogonal to the way that one normally thinks of logic, which just makes it seem to be an algebra with these peculiar properties but no real logical relevance, you have semantics for it in terms of an existing quantum theory, but that doesn't really get you very much further forward. No, you do, you have a unified, yes, it's just that this, this geometrical way of thinking of, of everything I think provides a much, yes, that's it, yeah, much stronger unifying framework. It is where the logic arose. What do you mean, is this where logic arose, period? I mean, it's of... Oh, no, you're talking just historically about quantum logic. No, it's not, I mean, it's... It's come from a geometric history. Yes, yes, sure. The problem has always been with quantum logic that the lattice theoretic aspects are really far too crude to, you know, to recover. I mean, von Neumann did two things. He created the continuous geometries. Exactly. Research program in algebraization and algebraicization of the quantum mechanical formulas and the C-star algebras and continuous next one after this and functional valued operators on C-star algebras and other more complex algebraic structures, which did pick up a lot of really real hard, pretty mathematical physics. And on the other, the latest theoretic... There's a way of looking at the logic, or looking at the sort of alge... whether it was a logic, whether it was just simply an algebra, a poor man's von Neumann algebra, it was just the... but anyway, the logic, whatever that means in this context of the...

17:30 Subspaces of the separable Hilbert space, linear subspaces, which was just a lattice theoretic structure which had these properties which differed from classical logic but which made no connection at all with any of the other stuff. I mean you just couldn't get any, it was far too primitive to make contact with that level of structure in the Hilbert space that you got the algebraic structure from that helped you create real physics. I've got the references to the papers. Yeah, I mean, there was a lot of Paul Neumann's early work in the 30s. I mean, there's the... but nobody ever could think of a way of connecting the geometrical... Yeah, sure. I mean, nobody ever... yes. I don't know. Probably. He was just getting up when we went out. You know, I just want to get more of a feel for the overall, you know, the overall insight, the overall intuition, the way that, you know, he actually thinks, like, you know, of the metaphysics of logic. Yeah, the trouble is, the questions are so vague and general that they, you know, they're not really sufficiently... No, I don't mention, in fact, I don't mention any... Leon, is it? Leon? Yes, Leon Davidovich. Davidovich, Davidovich, Davidovich, Davidovich, Davidovich, the son of Bronstein, who he was, Davidovich, oh is it Bill, oh it's okay, he's in, it's okay, we just wanted to see Lilo, no problem, sure, okay, Davidovich,

20:00 Christ, the devil is silly for God's sake, and he hasn't, the last thing I want to do is to, yeah, yes, I know, I know, I know. It was amusing and thanks for the information because this was not a concoction of this. They actually did practice this way of thinking. I think with respect you do miss the point. The point may be a wrong-headed one. But it's a completely cogent philosophical point. Yeah. Well, yes, the point of the force of this is that there is no question of... The point is about consensus in agreeing to accept the form of representation, in agreeing to accept the form of representation, which he thinks of some as proceeding. It's a pernicious form of idealism, but you mustn't dismiss it as, you know, containing nothing of, it is a substantial philosophical claim, I mean, it's something which has to be confronted, not just, you know, glossed over, that this attitude of Wittgenstein, I mean, you know, this man was, I think, a great philosopher. He was trying to understand what he called, you know, the hardness of the logical must. Which is, after all, exactly what I want to try and understand also, and what exactly, and exactly what the profound difference, just let me say something, Jerry, please. Is this showing your head? Yeah, but you do, you do, you know, you do, somehow, the profound difference between the way that Bill thinks of variables and of logical structure in this deeply geometrical way, the way that you think of it also in this deeply geometrical way. The difference between that and the way that, therefore, one thinks about logic and being together is so different from the way in which the Phrygians and the people who believe that one cannot say anything about logic and, indeed, about mathematics without having a notion of truth that is detached from and underpins an ontology that is…

22:30 Quite different from that of the natural world, natural structures, that ends with seeing natural structure as just a little blob of structure at the bottom, at the end of the tip of something else, that makes people very dismissive of attempts to think about how the notion is as fundamental as mapping. And object actually arose from experience of the material universe that made Colin McClarty, when I tried to discuss your paper about identical particles and the simplicity of category, not, I know there's no deep mathematics in it. But there is something very important about this matter of the spatial aspect of the category of sets and how it arose, and how, indeed, the calc arose in the course of the evolution of consciousness. I mean, this to me is a profound question because understanding it correctly, getting the objects and relationships in there correct... Conceptual order is the key to understanding, rebutting a lot of very influential, bad metaphysics. I mean, all of the metaphysics that has gone from Plato to Frege and to set theory, yes, into modern set theory. Yeah, okay, so I'm sorry, that's why I ask you to... It's not a fine mind, it's not a deep inside, but it's just one that I want to try and make much more precise. But don't... Yeah, but you see, don't dismiss this interesting stuff about timekeeping as just a lot of card tricks. There was a...

25:00 I landed into a certain page. This is a very... Yeah, yeah. Unless you would tell me what the point was. Well, the point is that consensus on agreeing to accept the form of representation is all that there is to forms. There is no relationship between forms of representation and structure in the world. Except what we impose by adopting conventions. Now this is the most extreme form of subjective idealism, but precisely because of the way that the Platonists have left, yeah, have a thinking of logical structure, it seems to many philosophers the only coherent alternative. And that's why bloody Wittgenstein has been so influential. Now, a cup of tea? Right, we've got some wine and stuff too. Oh, and a cigarette, oh! I didn't sleep. Well, he went and slept out in the, er, in the closure there. He went and slept out here, and no wonder he couldn't get any sleep. Don't know why. Yeah. For various reasons. Anyway, er... About the time I was dying... Oh, no. Oh, Christ, Gerry, you know what we've gone and done? We didn't put the bloody cigarettes in here. Shit. I, I, I didn't, er... No, no, you're right. I mean, well, I got cigarettes and I put them down on the counter. The girl hasn't put them in there. Yeah, she's charged them. Oh, sorry, Bill. Well, look, it'll only take a few seconds to go down. Well, I'll tell you what. Just let me put this lot in the fridge and then...

27:30 Yeah, because I guess, um, cigarette's probably a higher priority for you than a cup of tea right now, isn't it, Bill? Oh, okay, we'll have a cup of tea. We'll just have a quick cup of tea first, then. Well, I was just thinking, my God, I've only got another three hours of Bill Moore beer, and I... What do I most need to know? Everything. So where do I start? If you'll count. Just mention Crosby 2000 years. It seems to have been a very serious scholar, actually. I find it surprising. Why on earth did he get to...? Who was that? Oh, Cyril. I don't know if Cyril is particularly well-versed. He's not particularly well-versed in math. He's a very nice guy. In Marxism, he's pretty well-versed. Particularly in Marx himself, I would say he's well versed. I mean that is to say in the writings of Marx he's well versed, I would say. I mean not to the extent of being a real scholar, but he's better versed than most people who call themselves Marxists in this country, but that would not be difficult. Given the state of the British Communist Party today, which is just an absolute sick joke, designer T-shirt yuppiedom, they have no more... I doubt whether Mrs. Thatcher has a more sincere set of genuine admirers than the present leadership of the British Communist Party. No, I mean that as an absolutely serious statement. I am not making that as a flip, you know, ironic remark. I mean that quite literally. In fact, some of the extraordinary people that they now get to write for that magazine, the most grotesquely mistitled magazine I know in the world, Marxism Today.

30:00 Have you ever seen this? One time was the theoretical journal of the British Communist Party. And given the deep British aversion to theory in any form, it was... But now it's just simply a, yeah, it's a kind of yuppie, yeah, it's just a yuppie, yeah, it is just yuppie, t-shirt designer yuppie. The Greens are a more coherent bunch of thinkers than the reformists in the British Communist Party. I knew them at Cambridge, they were all my contemporaries, people like Martin Jakes. And so it's no surprise that they end up denying the success, in their terms, of Thatcher. Absolutely. And they do, you know. I do. But whoever is in power and who has performed the trick of persuading the chatterers to believe that they are... Cults and praises. Well, yeah, we're going to have a cup of tea. Yes, exactly. I was going to suggest that. I was planning to pay for your dinner. That's all right. You've done enough. This man owes me 20 quid. Yeah, all right, shut up. All right, wait a minute, I'll give you 20 quid. Hang on. Otherwise we're going to start foundations of the world and the universe. Jerry, shh, okay, there you go. It's all good. I went out without my wallet, of course. Borrowed money from Jerry when we got the cigarettes and everything. Which we then went and left. No, okay. No, I'm not around for doing that yet. What I have got to do, let's just get me organised. Priority number one, and we've only got about two hours to do it, is to get those bloody tapes copied. So, I think what I'm going to have to do is just...

32:30 No, well, Bill wasn't with us then. That was after. So, I call it photocopying. Yes, well, they do send you to jail for photocopying £50 notes in this country, Gerry. Oh, I see, that's what you meant. Or even five pound notes. Particularly if you use a dot matrix colour print here. Some guy got caught printing six pounds notes. Six pounds? Six pounds, apparently. Well, there's no such note as you probably know. So the judge said, look, you've done a perfect job in the printing business. You wouldn't have been able to catch it. Except for the fact that you didn't go for five. He says, there's nothing I can do for you. This is argument number six. Then he says, it happened this way. The initial designer wanted one password. To get the best, obviously, and therefore we had to put one password. Oh, you just had them. Oh, okay. Sorry, I hadn't realized. Well, Jerry, do you want to put one? Right, I'm sorry, I hadn't realized that. What did you say? Let's go straight up to lunch. No, I'm sorry, I hadn't realized you already had them. Is this some... Oh, no, it's Jerry's paper.

35:00 I'm not reading the appendix. I need the appendix. Sorry, Jim, I'm asking, but did you read John Mabry's letter? Yes, once quickly, just before going to bed. It seemed to be quite... Yeah, I just read it this morning. I must say, I think he's had some very good... good grasp of the key ideas. Yeah. This one here is beautiful, you see. Yes, yes, yes, about how not to think about the subject in mathematics. The thinking subject. Yeah, well I think it almost always does, doesn't it? Brouwer, no surprise that Brouwer ended up as a fascist collaborator. Well, okay, fair enough. Oh, yes. I remember Hans Kamp, my tutor in logic at Bedford, telling me that the first... These are the lectures that Brouwer gave after the war in 1946 and 1947, the first time he lectured again at the University of Amsterdam, the whole body, the whole student body, who had just waited until he came into the hall and then rose as a single body and walked out before he started speaking. It's a gesture of contempt for the way he behaved during the occupation. But I really do want to try and get some more feel for the It's a profoundly geometrical way of thinking of logical structure, thinking of variables. For instance, I think it helps perhaps to solve this persistent problem for Mabry between the reason he's so hung up on arithmoid, because these are the only things which provide us with identity criteria for the elements of the structures that we're ultimately dealing with,

37:30 Our collections in extension are completely determined by what their elements are and which elements are elements of those, in other words, the cardinale notion of it. I think he's got an interesting grasp there of what unit means. Yes, which he didn't have before. I must try to... Formalism is a category in the background. Do you have this piece with you? There's a pair of shoes and a shoe inside. Hang on, Jerry. Yeah. Yeah, okay. Well, but it's a functor, you see. Ah, if you maintain the connection, then the whole thing makes sense. It's natural. I mean, the choice would be natural, in a sense. Yeah, that I accept, and you've given an explicit reason why. These considerations, I mean, if you cut these two parts, if you look at it in itself, then there is no motion specifically from the back. These are not shoes but wine glasses, but everything else, everything and nothing. It's just a calculator, you see. The process. But the that in comparison with which each of the entities of the kind under consideration is called one, the notion of Arithmos for him, is not, doesn't have to be something, doesn't have to be the same as the theoretic notion of Singleton, the that in comparison with which, does it?

40:00 But I think before, formally, he thought it did. That's right. Formally, this was the thing which was holding him up. He thought that it had to be the same thing as simply as the notion of singleton, but in fact it's a much more flexible notion. One has to have it there to have functoriality, but the deeply geometrical way of understanding a jointness. Let's try to work out this example. Well, let's go up in the car, because we haven't got that much time. Yes, there's a very nice one just up on the hill behind here, which I'd like to show you, if you like some nice scenery. Yeah, so we're going to go down and get the cigarettes first, of course. You see, he says something very interesting here. I'm afraid I've not yet fully assimilated Bill or his revolutionary ideas sufficiently to grasp their full significance. What seems to be held out before us is a proper understanding of the traditional notion of extensive magnitude. You see, I think that this is the insight that the notion of extensive magnitude, or rather that... The notion of extension as in collection and extension, and that notion of extensionality, which for Mabry was intrinsically bound up with singletons, with atomism, is actually, if you start thinking, if you start from the geometrical point of departure, you can just be viewed as one, as the limiting instant, the constant case of extensive magnitude. Isn't that a fair, yeah, one can think of it simply as the constant case of extensive magnitude. But extensive magnitude itself, the relations of extensive magnitude, always ask for the object and always think of the object and its relations, in the case of extensive magnitude, don't have to be analysed, don't have to be reduced in their relations to the relations of points. It's the old point about points, the deification of points, as you put it, the platonic deification of points, where it came from. Yes, we'd better go.

42:30 In fact, we ought to first stop and get... The cigarettes, that's right, yes, yes. Actually, they probably almost certainly have cigarettes at the pub as well, but maybe we'd better not take a chance on that. They were already paid for these. Yeah, right, so better let's go down and do that. Because I've been guilty. I'll be... You tend to notice these things. Physically. Sort of dramatically, I think, you know. What happens is... Unlike most of the cities on the campus, where what happens, they tighten the distance norm and they're entirely treated in the sense that they're not into each other. I'm not into mathematics. I don't know which one it is. That's good. That has some sense in it because it used to be the college is photocopy. It was a PhD in photocopying. Oh, I think that would be, if they gave PhDs in photocopying, I would have got several doctorates by now. Too true. Be careful. Actually, Jerry, you want to get around facing the other way here, so... That would be an advantage. Yeah, what you want to do... Well, if you can't do a left down here, well, go straight on, there's a little roundabout, okay, go down right here, you just have to get around facing the, we go straight down the bottom there and then do a right back into the, that's right, yes, yeah, yeah, because that's where the, yeah, they're going to do it Cairo fashion.

45:00 Yeah, yeah, I see them, I see them and I expect them to see me. Well, it doesn't always work that way. Oh yes, oh yes, oh yes. Come on, men, be nice. We have a guest here, we have... It may be neither Synecdoche nor Wattransite. ...the progress of maths. In fact, we almost got... Well, that's right, we did, didn't we? About a couple of large... A big log? Three very large long log alouts who looked as if they'd come straight after the Waffen-SS training battalion. They were a bit disappointed we'd been challenged. I thought Bill did. I saw him rolling his sleeves up at the time. I should think you could throw a pretty good haymaker, couldn't you, Bill? I don't know if you've ever been in any fights. No, I don't think so. But that remark of Mabry's, I think, is to cross the nub. To create an understanding of the traditional notion of extensive magnitude and to see the notion of extension as the set theorist has thought of it in terms of a collection in extension as deriving from that. You ought to do a left here, Jerry. Back past my place, but then keep going straight on. You may be the first to say this, right? Absolutely.

47:30 Absolutely. And I mean, I've said to you already what I think historically is about his character, which is why I said to you earlier on that I think that as far as philosophy of mathematics is concerned, it is, in other words, it's been absolutely inconstant, and indeed about logic and the relations between logic and geometry and... As far as I know, no, but what he did do, what he did do... ...was to participate in the founding of a society for... there was a society for the promotion of... actually it was the society for the promotion of German philosophy, not just idealism, but German idealism in philosophy. Now the reference for the proofs, Slugas, are really a count here, who thinks that... ...when he was telling you before, you watched him, didn't you? Yeah, yes, yes. Do a right little right here, Eddie. Sorry, Cherry, I'm sorry. Do a right here and then a left. So you have Frege being exposed by a cartoon. Well, rather, in fact, the deeply Kantian elements in Frege's own thought. And then the left. And this idea, you know, ontological categories are... The world is abolished in favour of language. The world owes its existence to language.

50:00 It is quite literally what these people believe. Well, Crispin Wright did, yes. But at least, I mean, Frege did think deeply about problems of foundations of mathematics. These people now really just are just carrying out the exercises. Yes, this is the nice thing about living there in Hatch End. Within five minutes you can walk from where I am to this very nice country. It also goes up on a high hill. On a clear day, probably to be too muggy to see it today, you can see almost right the way across London from the top of this hill. Wow! The only thing is it would give me enough money to go and study mathematics for three or four years. And all the other things I need to study. I've been thinking quite seriously in the last year or so of going to live in France. This is the view. It's really quite Tarot Hill in the distance. Yeah, it is a nice view. I must admit I wouldn't... And then if we pull in here, the pub is just where all those cars have parked. No, not here, that's a posh hotel, expensive hotel, where Gilbert, the man in... anywhere along here, yeah. Or you can go into the park here. Yeah, you can. Unless you're six foot nine inches high. I walked up here with Colin McLarty the other day when he was staying here and had a long talk with him about the foundation, which I learnt a lot from. No smoking or more? I hope they don't. It's not a no smoking fight in Milan, is it? I think not. I think they're pretty civilised about these things, the Italian airlines, aren't they?

52:30 Would you agree with that formulation there, that he's put it in the way that he's put there. You've also failed to take account of his critique of it. He's moved on a long way in that last week, John Mabry. I think I've done a good deed in getting him to come to that meeting. I'm very pleased, very pleased. Having read those papers of his, I thought, well, he's, you know, he's taken up such an entrenched position there. Is it worth arguing against him? Yes, it is. Well, as I said, there is, well, not exactly a castle, but a big mansion house on the hill. Well, this is definitely more the bourgeois suburbs than the aristocrats, than the absolute monarchy. That's the only thing I don't like about it. It's a very, well, as you can obviously see, it's a very, very, very bourgeois suburb and I don't have any kind of stimulus here. I don't have any friends who live near here. And there's nobody with any sort of sharing interests. In some ways that's probably a good thing because it means there are pure distractions, but it means that for any intellectual stimulus I have to go to Oxford or Cambridge or London for seminars, to libraries. But I've been thinking seriously about going to live in France. Do you mean you're going to spend the money?

55:00 Well, no, not spend the money. Spend some of it, because property in France is much cheaper than it is in this part of England. All right. Thank you. We can sit outside in the garden and we don't have to listen to all this noise. What exactly are your plans when you get together with Fatima are you going to the food yeah well i think we have to Oh, I suppose we can get around this way. Are you... I mean, you're going to be in Switzerland with her from Scotland for about a week, yes? Then are you going on to Italy for a bit? Yeah. What do you feel like eating? Yeah, you order it here and then come back and... We might as well do it here. Oh, yes, we are going to eat here. I thought that was the point, wasn't it? Yeah, okay. No homemade lasagna, would I see. Can they do this? Yeah, they can't do it as well as they would in... I shouldn't think they can do it as well as Fatima does, but they can do it, yeah. Go for the homemade lasagna. I might have that too. To be honest, I've never tried their steak.

57:30 I've never tried their steak. OK, go for the steak. Well, ask them if they can do it fairly well down. Mind you, I should think that it's all... I'll just put it under a grill anyway, I'll tell you what, let me stick to the risotto. Okay, well, three risottos, okay, that's fine. That's all right, I'll have some brie. I might even have some ploughmans with brie in fact, okay, right. Okay, right, and then what do you want to drink? Bill? Non-alcoholic. Non-alcoholic. I've got a bottle of wine back at the house anyway for this afternoon. Pepsi, okay. Hi. Hi. Can we have some food and some drink? What food? Two homemade lasagna and one fireman's with brie. Okay. The clean-up is done away. Not for drink, we're not. Can we have one pint of Marlboro, large Pepsi. Are you doing that as well, are you? Okay. Okay. Thank you. All right, I wasn't trying to confuse them. I was trying to get served. We get served in the morning. Thank you very much for your attention and I look forward to seeing you again in the future.

1:00:00 A beer of shandy. Do you want a pint, sir, or a half? Well, how much of Pepsi do you want, Bill? A pint or a half? A whole pint or a half? A half, thanks. I gave you a 20, didn't I, Joe? Yeah. Good. I think you'd like some ice with that, wouldn't you? Sure. Oh, that's more than you'll... Well, why don't you let me get the food, okay. I'll get the drinks. Okay. I'll tell you what, I'll get the drinks. You get the drinks, I'll get the food. So put that back, Will. Right. You're the only American I've ever met who didn't like ice. It's impossible. I mean, you know, I'm driven mad all my life as a travel agent. Uh, will we go pay separately for the drinks and the food? Can we do that? Uh, okay, why don't we... Well, okay, no, take it all out of that again. We'll sort out the change. I mean, I'm absolutely driven by how they always want to know, you know, at least at 3 o'clock in the morning, where the ice machine is. You're staying in a little pension in Umbria and they want to know where the ice machine is. But they, you know, usually can't live without ice. Very, very large supply, unlimited supply of ice. Of course, the capitalists who first realized that by filling up all the glasses with ice, you... I've made a lot more progress because, obviously, right, okay, so how do you want to, why don't you take that, Jerry? Okay, fine, thanks.

1:02:30 Well, can we go outside, would that be possible? You can give us a shout outside. Fine, okay. Okay, no problem. My son and daughter quite independently realized this fact. I always demand no ice. Yeah, so as I say, I mean, I'm... Let's just sit here. I might as well sit here. I'll get next to a chair for Joe. Somebody's taken that, haven't they? Yeah. It's all right. I'm sure I can grab one. Oh, there's one over there. Hmm. Well, it's nice seeing kids playing. I often live in a bourgeois suburb with little... I was going to ask you about below. Do you find... Nothing like this. No, really. You mean nothing like as bad, nothing like as bourgeois? No, I mean the whole sort of conviviality doesn't exist in America. Well, I guess I do find, I mean I do genuinely find most Americans that I meet, particularly from small towns in the Midwest, very friendly. You know, I'm not being corny, but I do find them very sort of friendly, open people. Yeah, that's true. Very, very open, friendly people. But I think partly because social life seems to be so highly organized, in the sense that it's all done through the church or these various manipulative ideological institutions, sort of rotary. It's sort of mixed with strangers like this in a semi-friendly way. There's very few contexts to meet an individual stranger like yourself, you see, and the friendliness is definitely there, that's true. But not just to be able to sit in a with a crowd and talk or sidewalk cafe practically doesn't exist. Yes that's true that's something which and pubs obviously don't exist as an institution that's right pubs and bars because yes because bars in the states are very different kind of yeah usually

1:05:00 they're very you know there are a few which function as pubs well the Irish ones of course Most of their places you can go to meet your two or three buddies and do some very hard drinking. Yeah, which is a nasty aspect of it. You really don't have them in pubs now. The pubs here are not boozing shops. Nobody ever went into a British pub to get drunk. That's one of the nice things about them. Well, to Scottish pubs maybe, the old very hard working class pubs in the east end of Glasgow. The very fact that there's such a separate concept as singles bars shows the specialization. Yeah, yes, that's right. We have one or two very tense, uptight, very single bars here. It's not an institution. No, there are some, but it's very... Hopefully he's going in a single and coming around as a couple. Yeah, no idea. It's a couple come single girl. I used to frequent the Café Odeon in Zurich. You know, it's a place where artists and intellectuals just go there and drink and discuss with one another. And, by the way, pick up girls. Oh, I'm sorry, Jerry. It wasn't a... Quite a bit specialized. If you don't score, there's something wrong with you. Yeah, yeah. Well, yes. It should just be a natural part of life. Right, exactly. Well, these were important concerns a few years ago in my life. I remember vaguely. You go through this patch for a few years, then you begin to gain a bit of confidence and then you end up in the practice. You think it's got to do with lack of confidence? No, actually I think there's a lot in there. You grow up as a teenager. And there is an opposite sex. In some extent you have to discover it and then you have to consider it.

1:07:30 Thank you for watching. Well, no, these are kind of very, you know, ordinary kids, but this pub is all right because it has a lot of working people from Rice Lake, which is a town over there, which has got more, a lot more sort of working people, but Hatch End is very stuffy and bourgeois, and, well, inevitably, because there's a lot of elderly, retired people living there, who are kind of mostly upper-middle-class, retired couples. Your daddy was a doctor? No, no, no, my dad was a forester. Well, he did various things in his life, actually. He was the son of a judge. He worked in the forestry. He had a rather strange life. His father was a judge. They lived in Penner. It's very near here. My family lived in this area. At least four, well, several generations, as far back as they go. I say my family, actually I was adopted as a baby, so they're not my biological parents, but obviously I loved them as my parents, so I never know who my biological parents were. But their family had lived in this area, Pinnock, at least as far back as 1799, which is the oldest document we've got, which is a marriage certificate for a right and a whiten. ...from 1798-99. And my father's grandfather, my great-grandfather, was the squire. He was a farmer. And he farmed at Halsden, which is now, for a hundred years now, it's been a suburb of London. I mean, not even a suburb, really an inner part of London, completely built over bricks and mortar. And in the 18... In the 1850s, 1860s, he farmed there. He farmed about 500, 600 acres of arable, and of course when the whole area was built over by the speculative builders who built them, the mass slum housing mostly for the Irish immigrant workers who were coming to London at the time to build the railways, his land was all bought up. In fact, he became quite a wealthy man as a result. But he had about eight or nine children.

1:10:00 And one of those was my grandfather, who went into the law and became a judge. And I never knew him. He died in 1941, but by all accounts, my mother, who did know him, he was a very hard man, a very rather harsh man, very, very strict with his children. My father was one of three children. He had an elder brother and an elder sister. He was the youngest of three children. He was born in 1910. He had an elder brother who was very spoilt. He was born in 1910. I think he was an accident. I have an impression when talking to my mother. At any rate, they had two children. His sister, my aunt Pam, who I do remember. Very fond of, she's a very eccentric woman. And his elder brother, who I never knew existed until about three or four weeks before he died at the age of 83. Extraordinary. He was born in 1900. Now he was the talented boy, the apple of his father's eye. He had, he was obviously a very bright student at school, because I've still got all the books that he got, math prizes. The First World War came and he went into the Royal Naval Air Service. This is quite interesting. He came out of the war and he decided to stay in the services, in the Royal Naval Air Service. He flew, this was, this is the first World War, and he was one of the, I think he was possibly the first man to actually fly an aeroplane off a fixed-deck airport, aircraft carrier, as opposed to a seaplane, certainly in the British Navy. He was commissioned in 1917, and then transferred to the Royal Air Force when it was founded in 1918, and then back again to the Royal Naval Air Service.

1:12:30 He stayed in for four years after the end of the First World War and he went with the British fleet in the intervention in the Russian Civil War. And when he died, now I should explain that later he and my father quarreled terribly and they never spoke to one another for the rest of their life. I mean so much so that I didn't know until my father, who's already a very sick man himself, that was only about six months before his own death. He told me that he had this elder brother, Ernest, who had in fact been living all his life very near here at Pinner Hill Golf Club. He had a wasted life. It was really very sad. He was obviously a very bright guy. But he'd gone into the service. He was very good looking and had a lot of success. Thank you very much for your time. In fact, he'd also managed to get hurt. I don't know exactly what happened, but anyway, he made very speculative investments with a notorious crooked financier, a man called Hattrick, and they lost, well, not all their money, but they lost an awful lot of their money. The result that my father, who had just got a scholarship to go out to Cambridge, to St John's College, to read Natural Sciences, but who was then taken very ill because he had rheumatic fever, he contracted rheumatic fever and was laid up, and this is 1927-28, and rheumatic fever was a killer, so he was laid up in bed for about two and a half years, and he actually got the scholarship to go to Cambridge to read Natural Sciences when he was just turned 17. He wasn't actually able to take it up until he was 19 and a half, because he was very, very old at this time. He was paralysed, in fact, in the magic field. And when he was able to take it up, then his parents said that he couldn't because there was no money, although it was only a small scholarship. They couldn't floor the rest of the fees. And the reason was that his brother had lost a lot of the family money in this crooked company he was involved in.

1:15:00 Subtitles by the Amara.org community ... about the, um, pulling forward, uh, no, pulling forward, extending onto intensive quantities, about the Burnside, you know, that whole construction of functorial, well, functorial, functorial jointness is, is, I mean, what, what, what do we want to call that, the whole construction of, in the relationship of intensive, extensive quantities, of which CCRs are an instance of, what, what, what do we want to call it? Functorial jointness? What should this, uh, formula be called, yeah, I think? And functorial adjoinments? Is that not sufficiently specific? Well, I mean... Probably not. No, because functors are adjoinments. No. They say you do. Sorry, covariant functorial adjoinments, I should say. No? Sorry. No, covariant functoriality. Well, but there are some covariant functors. Yeah. No, yeah. Yeah, you're right. I mean, the formula itself is easy. It's extremely important in algebraic topology, for example, but it's simply called the kind of formalistic name, the projection form. And in physics, it's extremely important, too, but it's known under a name which gives rise to so much mystification, the Heisenberg Uncertainty Relation. I'm glad to hear you say it. People believe it has something to do with quantum mechanics, but it hasn't, and it doesn't. Well, they believe it has something to do with the underlying nature of the universe, an insight so deep that it actually overtones mathematics and Finkelstein.

1:17:30 It's an irony that Finkelstein says, oh, but I learned from Heisenberg to think of matrices as more fundamental than mappings. I think it's pretty exactly what he said to you, isn't it? Of all people, Finkelstein, you know, good... Well, Heisenberg's matrix was an empirical matrix, a relationship between observed energy levels. Well, see, Finkelstein thinks that this is actually abolishing the relationship between domains and codomains, or being more fundamental than that. You should understand that these things are really a semi-group and this is really a dynamical action, dynamical action of this semi-group. At least Heisenberg was uncertain about it. You could replace the uncertainty with Heisenberg's uncertainty. He too travelled the road from subjective idealism to objective idealism. They go back and forth. You know his autobiographical claims, you know, Heisenberg's, you know, when he was... His book, Physics and Beyond? I forget which... Yeah, I've got this book, it's on the shelf. In 1919, when he was part of the suppression of the Bolsheviks in Munich... Yeah, he was in the Freikorps, wasn't he? He was, yeah. Yeah, the officers that he was with were discussing Niels Bohr's work. He himself had not studied physics yet. So from this discussion he extracted the idea that the difficulties with quantum theory was due to the fact that the world doesn't exist. It's all in your mind, and so forth, and then went to study physics in order to establish it. But it is incredible how... In fact, he was well known for not understanding physics very well, you see. But you see what I mean, the line, that line of ancestry and fascist thought, you know I was talking about Hitler the other night, the evidence that the only work of philosophy he ever actually read and studied seriously was Schopenhauer.

1:20:00 And that intermittently, even at the very end, he would sort of turn on his generals, and they'd try to explain some material fact to him, and at that point, pretty pressing, you know, like, you know, the Russian army is going to surround Berlin within the next 48 hours. And he would just do exactly what these Georgians were doing the other day, he would just suddenly turn, he would just sort of turn on the anti-consciousness. Oh, it's because you just do not understand that the world is will, and the representation is a function of the will, you just haven't willed the Russians away hard enough. This is, okay, I've always liked this all the time, but they were intermittent passengers, you know, recorded by the stenographers. It's incredible. Well, it is sanatorium stuff, but there it is in Heisenberg in 1919. And out of this is created a mystifying philosophy of physics so fundamental that a highly intelligent man, and I promise you he is a very intelligent man, and a good man. Uh, Finkelstein can actually, you know, spend his life, waste his life when he could have been doing really good work. It's a computer virus. Yeah. I actually waste 20 years of his life trying to reconstruct the foundations of mathematics. And quite honestly, you know, he came and gave a paper at Cambridge about three or four weeks ago and... I like that it's very strict of you. Yeah, yeah. It propagates all the time. You've got to stop it. Yeah, yeah, yeah. Well, I think all... Yes, but it does. This is, you know, one sees this. And, of course, there's another virus, which is Frager, which is... It's like science fiction we were saying before, you see. In other words... You could read Schopenhauer, if you read it 20 times, you believe it's true, you know, but it slowly creates the categories in which you don't see the world, you see, so at a crucial moment it could just, you know, well, the Russians are there. Yes, will the Russians be there? Yeah, the sanatorium stuff will. Well, that is sanatorium stuff, but there you see it becomes so obvious that we can all see that even the...

1:22:30 Even the kind of ordinary guy who's never thought about physics and philosophy can see right away, well, you know, the sanatorium stuff, you know, the guy, but... You know, when he's confronted with the authority, somebody who, you know, one of these priests with the authority of mathematics behind him, particularly, the great Dr. Professor Wigner, who says that, ah, you think that you can understand mathematics as a reflection of material reality? Oh, no, you poor naive man. No, no, no. Mathematics is so unreasonably effective, and I will show you how, but it can only be a gift from God. It's a gift we don't deserve, isn't it? Yes, isn't it? That upsets me. This all hangs together, this objective idealism. I don't find gifts, but... No, that is not a gift. By the way, thank you for the lunch. Cold! I like this guy. That's another... It's another attitude. They confuse the internal sort of mood with reality. But he went the road from subjective to objective idealism. Well, it's because he really ended in Plato, in Platonism, didn't he? It's the fast footwork that John Barry talks about. I mean, one area is inconsistent, and the other area is inconsistent. If you keep doing that, you can have always psychologically, you know, psychologically it's very, you see, I mean, you see, in other words, this belief that everything is relative, hence we can't really know anything and so forth, it creates insecurity, and so, ah, you know, there must be an eternal framework within which all this footwork is going on, this vast footwork is going on. Yes, absolutely. Well, there's a saying in what, you know, how, you know, how Kurtalis stroke Heraclitus gave rise to Plato, to the theory of ideas. I wish I could remember that German.

1:25:00 So the guy between Leibniz and Waltz. I think that's Leibniz. The guy between Leibniz and Waltz. No, I don't know about this period. Yeah, but I think we could... Shall we move? Shall we move on? Yeah, well, you couldn't tell them otherwise, so let's stroll back. There was a pub in Bangor where Potomac and I had lunch. Fudo-Marxists who support Mach and so on, right? Is this the guy who used to trade with you? So, Porter told the anecdote. He actually knew Adler, apparently, in New York City. The claim is that Lenin once beat up Fritz Adler, you know, physically. I don't know if it's true, but I can imagine it. No, this generation gap business and the fact that it's been taken up by all the instruments of ideological control in the culture is such enthusiasm from the moment it was propounded. The line of ordens I quite like. Though I know the term is crap, if there is a generation gap, the fault is theirs, both old and young, who never learn their mother tongue. In the case of the mother tongue of physics and science, that's mathematics, but I like this line. Even though it was propounded by an old Anglican reactionary, which is very good. No, the teaching of English grammar and Latin grammar and mathematics is extremely important. Mathematics must be more widely taught, more people, more deeply. Because, as you were saying a moment ago, right, it's the lack of knowledge of mathematics which makes us accept all this crap. I accepted for years the quantum. The quantum metaphysics must be more basic. That way we'll have to find a new foundation for mathematics because it has been shown that...

1:27:30 The notions of object. I bought all this Finkelstein stuff. I didn't know enough mathematics. Forty years ago, at age twelve, I started reading a book by George Gamow called One, Two, Three, Infinity. He was just full of this mathematical mysticism. Certain curious things about mathematics are described. But the way is to make you worship, you know, mathematics. The basic point is, as in many textbooks on high school algebra, number one, mathematics rules the world. Number two, you will never understand it. Yeah. That's where I, that's how I was. Yes, yes, yes. That's exactly how I was taught, you see. That's why I think I've missed out so much and why I need to learn so much more. At least I know now where, how to resist mystification. Oh yes, and there's an awful lot of that in the popular stuff by people like Martin Gardner as well. If mathematics rules the world, or even in the Dirac-Pythagoras hypothesis is the world, then as you say, you will never understand it. Oh, the pentagon, you know, that was my experience with the Witten system, you see, that... We had dinner at the pentagon, didn't we, in Cambridge? That ought to be the subject of a heroic painting on the subject of, yes, just entitled, Irony. Here was my beard, in the pentagon. We didn't have the camera, I think. I think that was one case of a disunity of opposites, Bill. You were saying something a bit more serious. Yeah. No, I was in the Pentagon, in fact, once, on a mission for Lytton Systems. At the time when they were engaged in planning the Vietnam War, they decided they could also make money by planning arms control.

1:30:00 So for that I went, I spoke with a general and an admiral, quite nice fellows really. Well, they're all from Canada. How shall we, how shall we, this was in 1962, if we reach an agreement with the Russians, how many, how many missiles can we get away with hiding on the ocean floor? If we were to calculate these stars on the moon, how can we... How can we, you know, for verification, if the Russians are holding too many missiles, how can we be sure that it's a case of, you know, how can we, you know, but it was, but what they wanted, you see, was, and on me in particular, was a mathematical theory which was sufficiently abstract and impressive to indicate that, first of all, the Pentagon itself would accept it. So that, you know, basically so that people will believe that there is this powerful theory behind that they don't understand but they must accept and therefore what Lippmann systems and or the Pentagon has been proved and justified and has been considered. They have an answer that you have to solve it. Every week the leader of the group on Monday would come with a secret document saying, he would say exactly these words, this is the Pentagon line for this week, get to work and prove it mathematically justified. Oh, I see, yes, okay, that's quite frankly, yeah. Exactly. I went through an exercise like this in the computing field, where there was some stuff that really smelled, if you took it literally, and we had to prove to Lotus that the whole thing worked. Lotus itself. So I proved through. Lotus is what they call the... Lotus? Lotus is yet again one of these bullshitty stuff that they've come up with. They call it formal methods in computing. And as you pointed out, they have a horror of anything concrete. So it's a bit of maybe set theory or algebra applications with a bit of blah blah blah.

1:32:30 And you have to read this as sort of application of mathematics to computing, whatever that may be. Whatever the mathematics may be. Exactly the stuff they were talking about. Water buffaloes, you see. You know, a man who would report every day how many water buffaloes get passed by, you see, and this would all be, all this integration of information would take place in a giant central computer in order to be cost-effective, you see, because, because... But also the napalm strike could be brought down on the right, you know, targets. To send a jet fighter to kill two guys in a rice paddock is not cost-effective. But there were three guys, didn't we? So we should be able to predict with certain probability that we can... Yeah, yeah, yeah. Well, yeah. So... Sorry, I interrupted because I heard you. No, it's the same thing. I'm learning Vietnam business. Oh, with that? Oh, I see. I wasn't... I didn't pick that one up. It's the same thing. It was again one of those sort of pseudo-comedies. So the actual stuff that was going to be implemented, you could discuss it and point out a few difficulties and so on, which would help them, and then something might come out of it. But we had to sort of keep that on one side, and we had to go to the symbolism. Which, more or less, said, oh, everything is perfect, and Lotus has proved that this thing works. And this, by the way, relates to you as a terminator. It was the communication network for... I was going to say, this wasn't the origin of SABRE, was it? I'll get you all the information. This was the Civil Aviation Airport... Air traffic control system? No, it's not. The whole world. Integration of air traffic control systems. Some guy had sat down and produced a nice scheme. And you could still detect... After eight years of committees, you could still detect the original plan, which would work. And this had gone off, and people were making careers and studying this thing. Well, the various countries and various interested companies in various countries. For further discussion, all they wanted was to start making profits.

1:35:00 We don't want to get the engineers. We know the whole thing needs to be rearranged, but the engineers, because they have to make the thing work. We'll have to face the problem, so let's get on and have so much discussion. So in each country there was enormous pressure to present to their government this thing as a finalized form, not further discussion. And we faced this from the GDC end, who wanted the contact. And somehow we were subcontracted. So I didn't know anything about law, but all I could see was some sort of ethical maneuvering. Just call it, let's say, directly, look at this, I'll be happy, because you have to learn their own notations of something, which would then be called Lotus, which would then be, yeah, so one guy who seemed to know about Lotus sat down and did it. And I was dead against it, because I said, we've got to say this thing here in the small domain, we're looking at it. Shall we go back and check some more? Yeah, I'll check some more. They're not going to come. I... you... Well, I've got a machine which is just... You know, you just take two tape recorders with a wire lead between them and run. I can lend it to you. I'll show you. Yeah. There is one problem, which is... I think that those tapes were made on a half-speed recorder, in which case I probably need to lend you my half-speed recorder. I'll check that now. No, no, the other one has a half-speed recorder. Oh, fine. Very useful. Actually, you meant the words. They're very useful. Oh, I should have got her an English-Wolfe dictionary, shouldn't I? Yeah, no, right and then left. Right and then left. Right and then almost immediately left.

1:37:30 I'm going to go through those notes you wrote last night very carefully. I hope I have understood the CCR construction. No, I'd like to hear it again. I think probably more important, since we've had little time, is to... Yeah, but the notes are very good, and I can go right down to the end and then left, Jerry. I guess the one thing I would like to hear you tell me a little bit more about, there wasn't so little time left, is really a very general question, again it comes back to the left here and then straight upwards, to the relation between geometry and logic, this very geometrical way of thinking as a source of logical concepts, like the spatial origins of our logical concepts, it seems to me that from the From a philosophy of foundations point of view, just coming up to it now, Jay, just here, I think if I was trying to provide a brief historical framework for the introduction of the main ideas of the lectures, talking to the philosophers, obviously not to the mathematicians, and it would be in the context of an analysis of how distorted the logical, the way logicians have had of thinking of the variable has been ever since Frege.

1:40:00 I thought that the idea that a quantity lives somewhere in an existential quantifier, this smoke here, not here, has a kind of degeneration of the actual quantity of smoke, the extensive quantity throughout the whole car. Well, and the way that sheaves... The relationship between sheaves and quantifiers, which I know you've already written about specifically in the 1970 paper, but I think more of the philosophical, even kind of pictorial motivation for those ideas would probably be the starting point for philosophers in understanding the depth of the revolution in the way of thinking of the variable. Yeah, yeah. And also, above all, in saying, you know, we don't have to start from truth, you know, we don't have to start from, we don't have to, above all, we don't have to assume that the sub-object classifier, you know, the truth value, the notion of truth value object is more fundamental than that of a sub-object classifier. I mean, you see, I think the logicians, even people like John Bell, still think of it the other way around. They just think of, they think of the sub-object classifier construction as just Another way of thinking of truth values, which happens to give you this, in model theoretic terms, more flexible and comprehensive framework because of all the sheep machinery. But it hasn't actually altered our concept of what truth and truth value are. The idea that they're actually quantity types, that there is a naturalistic explanation for the concept of truth, that it doesn't have to have a platonic, that it doesn't have to live in...

1:42:30 All this business about the language of the topos completely obscures the, what we thought was the very fundamental discovery, you see, that if you, the existence, the very existence of logical operators, as well as the laws that they satisfy, hiding predicate calculus, these are not a priori, these are things that are derived from the mere existence of the sub-object classifier. I mean, including the operators, and it implies are constructed out of the purely spatial relationship. Yes, exactly, exactly. But they've missed this point. They've all missed this point, even people like John, I think, have missed this point, and that it isn't that this validates, the last thing is, the last way of thinking of this is to think of it as validating, you know, the intuitionistic conception of truth in terms of, you know, provability from the point of view of subjective or objective idealism, rather it's that, you know, the actual spatial relations... In the domains of variation in the world, well, or the way that spatial relations in fact come out of a deeper understanding of domains of variation, but domains of variation in a very literal ontological sense, not as a metaphor, which is the way that people have tended to think of the relationship between the variable and its domain of variation, really just as a metaphor. Something that the undergraduate magician has to be house-trained not to think in terms of these, oh, you know, this is a helpful picture for the stupid students who cannot be trained to think logically, who will never, who have difficulty in learning how to think as true mathematicians. See, whereas Anders Koch and I always promoted that X stands for a map into X, an actual map from some other, what do you call it, generalized element or whatever, but it's an actual map, whereas they want it to be something separate. See, it's just a symbol that doesn't denote anything. Yes, and I think that's the fact that Scott himself, as a great magician, has always...

1:45:00 It's thought of a sheaf as really just as in the first place. The only sheaf he was ever really interested in was the sheaf over the complete Heiting algebra because it was simply a way of thinking of generalized truth values, but truth value, just generalized truth values as the starting point. You already have the notion of truth and truth value as given in advance with nothing to do with the structure of domains of variation of material reality. Truth is, again, that's the Fregean starting point. The truth is part of the world. There's always speculation. There's a concrete mathematical problem which has not been well solved yet and well explained. If you take, in other words, for a localic topos, the truth value object in itself somehow governs the whole thing, you see. But by contrast, if you take an example like M-sets, where M is a monoid, It does not. In fact, with them as a group, the truth values are only two, which tells you nothing about the group and how it's acting. That's sort of an opposite extreme. So this gives rise to a property that you can attach to a topos, or really a topos over a base topos. Namely, is it or is it not the case that the sub-objects of the truth value object generate the topos? Generate, meaning that there are enough, a family of objects is said to generate if there are enough maps from these into any other objects to distinguish maps and distinguish sub-objects, maps from, not maps into. So, I mean, the localic topos is sort of the simplest case, the sub-objects of one, which are condensed into two thousand objects, sub-objects of one already generate. But there's a next step would be, what about sub-objects of omega? When do they generate? Or sub-objects of omega to the omega? So there's a kind of an expanding sequence of classes of topology, namely those that are...

1:47:30 To some degree, in some higher order sense, determined by their truth values. But the only trouble is that they would think of that entirely in terms of the type theory. Well, no, I'm saying let's take the concrete problem. What about the first step now? For which topos is it the case that the sub-objects of omega will generate? Definite mathematical problem. To characterize, let's say, in terms of a site, what kind of a site on a base topos S, It gives rise to a topos which is generated by sub-objects of omega. As a very special example, just take any monoid. Which monoids, for which monoids is it the case that the sub-objects of omega generate? Well, it's true for the directed graphs, because for directed graphs... Take a new sheet. Yeah, come on, sir. For directed graphs, it's... They make a sub n. The right ideals of n, and a, times a, times x, and a. Okay, so, in the gym, sub-objects, say t, contain them, generate, i.e., if you have two maps, they're different, and they should exist.

1:50:00 So this is somehow the next level of complication beyond localec, because localec is generated by T's contained in one. Well, I say, for example, Vim is a group, not true, because omega is two, and yet you need G itself to generate. It's known that the generic arrow does generate, if you have two graphs and two morphisms of graphs, these are different, they must differ on some arrow, and an arrow in E is a map from I. What's omega? Omega is this little picture here with the five truth values, and it's clear that I is contained in this, for example here, or here. This is a sub-object. There's a sub-object of omega which is generated. So, to give a useful description in terms of what properties does M have, it turns out it must have, it must consist largely of idempotents, and there's one idempotent that's kind of playing a central role, and so on. It's the properties that a monoid may or may not have. You see the strategy to fly in the face of this philosophical speculation by posing a concrete mathematical problem. Well, yes, I'm well-reputed, I know, Phil, but remember I'm not yet able, and please, God, I will be able to pose a concrete mathematical problem in this form, but unfortunately I have to do it in terms of kind of... I'm trying to get a handhold through the philosophical ex-speculation, but yes, I certainly agree with you quite a bit, Scott. This would shed concrete light on philosophical... So what does Scott say when you put this problem to him?

1:52:30 Well, his method is basically to pretend, you know, that he doesn't understand it, there's nothing to it, and then ten years later, some thesis... Announce it as his own discovery. I mean, all this stuff of Foreman's and his about- He kind of accepts things piecemeal, you know, a little piece at a time. All of this stuff about identity and existence in the Scott Foreman work, that, you know, that that's the way to do many-sorted logic, just in terms of restrictions. I guess we'd say spatial, it's taking the spatial aspect of the topos, but they're not, yeah, but they're not, but they refuse to accept the materialist implication of that, but it really is, we really are actually dealing with the structure of domains of variation in the sense that you and, okay, Jerry, when you say that logic really has to be understood from the point of view of this, this point of departure in geometry, you know, for them, That's just simply a picture, thinking about how, I guess, how, you know, to restrict certain operations and still have well-defined, other operations well-defined. I'm putting that very badly, but you know Pullman's paper on identity and existence in the Durham volume. And he's actually opened that with a short polemic against you. Now, Lorvier has said that we should, that the whole... The whole Topos point of view opens the question of, with regard to many sorted logics, of the analysis of the notion of the variable and is brought to light as something which the way of thinking of the value of the variable in orthodox logic had obscured. He's saying, no, that's... The whole ontology which they give there, while they do talk about domains of variation and so forth, it's only applicable to localic topos. Yes, yes. At that time, they had not come to grips with little examples like directed graphs. Their analysis doesn't apply to that simple case. They got enamored with this effective topos, which certainly is not localic.

1:55:00 So they had to expand. It's like a defensive action of conservatism. I tried to make up a story about that. Some idealist story about that and eventually events force them to accept a little bit more, but the principle is you should accept all, you see, is forbidden to the students. It's been like that, I mean, since 1962 when Scott, I must say, was the forward-looking enough to permit me to speak to the Tarski Seminar. More progressive than anybody else in that group. But his own, you know, acceptance was just very piecemeal. I can remember, from 77 until recently, with this transition, you see, admitting that there are two poses which are non-localic, basically. But I think that his big difficulty in that would have been that he'd always thought of topos as, he'd always thought of sheaves over complete finding algebra as the guiding example of the topos, because he'd always thought of it simply as a space of generalized truth values. Well, we're still thinking of truth and truth values as something which are, as something not as in any sense a material quantity or not in any sense a quantity type. And for him, I think it would just be quite unintelligible. The whole Fregean tradition doesn't allow you to think of truth as a quantity, as a quantity. The basic principle that the, the demure fact that the characteristic maps and the sub-objects in this one-to-one correspondence, you have a category, there's a profound restriction on the category, but it implies, on the one hand, that this object has operations like and and implies.

1:57:30 And on the other hand, that the sub-objects of any object do form a hiding algebra, so the whole, both the existence of the algebraic operations, as well as the identities that they satisfy, already follows from the mere unity and identity of opposites. So the program, you see, is that the same fact should hold about space and quantity generally, not just two-valued quantities, but real quantity. That the, in other words, Descartes' analytic geometry with the resulting sort of bifurcation into analytic geometry and synthetic geometry. Which Steiner was promoting synthetic geometry in opposition to the analysis of analytic and so forth. Oh, Steiner who said of Cantor that this, yes, that, what was that remark that Collin put in his paper about, I don't remember, I think. I didn't know about that. Well, no, it's not particularly, but it's just a very, it's just a passing remark, but again, I think it reflects the, actually it reflects the impact which Fred had already had on these people when he says that... ...that you can add and multiply real quantities to itself be a consequence... Oh, no, I'm sorry, I'm... ...of the actual isomorphism of projection between quantities in space. As well as the properties of space itself could follow from that, so that the precise, in a certain sense, the bare statement of the unity between the two should already imply the operations on each, the midpoint operation and geometry addition and multiple quantities, and the fact that you concordatize that there are variable quantities, that it's extensive and intensive and all that. It should just flow out of the mere statement, flow out in the sense of rigorous deduction.

2:00:00 As in the case of existing topos theory was done for two-value quantum. Yes, it is. I'm afraid I was wrong, actually. It was Schoen, please, quoting Steiner. Even though Cantor, as he himself incidentally stated, borrowed the notational concept of power from Steiner, still the corresponding geometric formulation has nothing to do with the kind of thought which lies at the base of set theory. Yeah. But there you are saying that in fact... I want to read that one. But in fact, geometric formulations do have everything to do with the kind of thought that lies at the base of set theory. They provide the correct framework for seeing it in its true, yes, for seeing it in, yes, yes. This is such a tremendously powerful, I'm sorry.