??/07/89 Discussions, incl. FW Lawvere, G Khatcherian & M Wright on extensive quantity & other crucial topics (contd.)

0:00 Thank you, thank you. Well, there is, uh, do you know this, uh, man? How did these people like that do? Yeah, I know, Fatima told me, I was surprised. Sorry, I don't know what you're talking about. Back to the bankers. Oh, yes, yes. Actually, that is one of the things, since it's not obviously one of the most important things I wanted to ask you, but it's about the only thing I can think of that there'll be time for you to answer. What is the example of the, we don't yet know what to call it, the pulling forward of intensive onto extensive quantities, Burnside rig, the thing of which all these examples are? Illustrations, whatever we're going to call this construction. What is the example that you tell to the bankers about? Price is intensive. Yeah. It's about a supermarket. You see, you go shopping in a supermarket. There's a certain space of items. Each item has a price, a rate.

2:30 Price is really a rate, so an intensive quantity. There's an intensive quantity known as price. In the space of commodities, on a particular shopping tour, you take a certain amount of each. So many pounds of lettuce, so many bottles of whiskey, so on. So this is obviously an extensive quantity on the space of commodities. This can be viewed as the push forward of a much more concrete, I mean the commodities are very abstract, this can be viewed as the push forward of a much more concrete distribution, namely the actual things in the store are classified into commodities by a classifying map. You push forward the amount that you actually select and you get another extensive quantity on the... The basic commodities and you take the product of that with the price and intensive. Intensive quantity acts on extensive. So the product of those two is now the cost, which is still an extensive quantity over the commodities. Now the total value of that is what you pay at the checkout counter. Yeah, and if you don't, they call it topology. Total of an extensive farm. Topology thing. The product of intensive or extensive, and the particular push forward known as total, are all three essential ingredients in understanding what you do. Yes, exactly. Yes, why not indeed, because I'm always looking for concrete ways of thinking, I guess because of my lack of mathematical credit, I'm always thinking of concrete ways of understanding mathematical constructions that would make them easy to communicate to people. Okay, these particular people, no, they wouldn't need to. This illustration is so wonderfully clear and simple that a child could understand it.

5:00 So they have some sort of momentum. It's precisely because of bookkeeping. I mean, you get totally confused if you don't realize that at one moment you're considering this extensive quantity as being on the space of abstract commodities, and another time as being on the space which is the actual store, because it has different properties. When it's living on one, it's a privacy when you push forward and you see it living on the other, so... Yes, and, uh, no, the people... This is why people don't understand anything, you see. Because these things are not made explicit. The actual calculations... For one thing, how you arrive at them, why you do them, and why you should believe they make sense. You see, this is hand-waving. Uncle. I mean, these courses... Or worse, authority. Probably the courses that you teach. Also, certainly ones with antigen in there affect filters. Because vast numbers of people have aspirations to improve their... Earning power, whatever you might call it, but only a few of them can actually be used by monopoly capitalism. So they're filtered. Those people who manage to survive this hand-waving might get a job. The others, once they're completely baffled by it, won't. Or, in the case of university, you know, get into medical school or get into MBA school or whatever it might be. So, it's part of the pro-need process, you know. It's using the hand-waving to set up some explicit formalism. This is one space, this is another space, there's a map, there's a push-forward process, which would take less time to explain than the hand-waving. Yet it would be definite enough to be, after hours of study, comprehended by a much larger number of people. Yes, yes. That would threaten, because then they would know some mathematics. They wouldn't be as good for the third level. It was found that it was much more efficient if the third level did not know any of that. And of course much more comforting for the elitist doctor-professors who revealed that they were obviously a super-species.

7:30 This is a subjective reflection of the subject at present. Normally, you know, when John Dewey castigates science teaching as being authoritarian... He's appealing to that subjective component, you know, as though it was some evil intent on the part of the professors that the thing is that way. You said yourself, it's not your evil intent. You might develop it. You know, a professor could become an authoritarian reactionary, of course. But I mean, it's not, even if he does, that's not the reason why there's this horrible system that people are forced to pursue. It's only the subjective thinking which is the reflection of being, social being. Yes, I mean, the athletic professor's social being eventually causes him to be an obscurantist. Yes, I understand your point, I'm sorry, I do understand the point, yes, and it's the Trotskyite tendency to reduce everything to personalities, to the motives of individual personalities, or indeed even the capitalist... The tendency to reduce, to explain fascism as a phenomenon to children, entirely in terms of the psychopathology of these wicked, even individual men, horrendous, a one man or a group of men, or... You know, to reduce it all to individual psychology, and besides it pays, it provides another form of dope, a form of pornographic propaganda as well, let's say the masochistic sort of thing, also helps to fill that role in preventing people from analysing anti-consciousness. I mean, isn't it awful what evil thoughts that man has? It might be evil thoughts of no interest, really. Except when you analyze. Well, I mean, the category evil in application of thought is a useless category. It's like the business you were saying the other day, the fact that they always hold this image of death.

10:00 You know, blot out thought, blot out the hunger for analysis, for understanding, by making people emotionally weak, by preying on them. Yes, I agree. There isn't time for anything else, so thank you, Bill, from the bottom of my heart. Thank you, comrades. Thank you so much for everything. All right. Okay. These two weeks have been the changing points of my life. Can you imagine how much of a change they have been for me? So much I still want to ask, but I know I will see you again. We'll be in touch. Thank you above all for... Well, very soon, I mean, because I'll get the tape to you right away, as soon as I get it. And the letter of Mayberry and... Yes, yes, those two. And I'll also try and... I'll also persecute Redhead until he gets the troll suit. Well, be nice now. Yeah, well, I've got to, because I've got to keep my connections there. It's hard for me to tell, you know, these British people, whether he took it seriously or not. I think we might have him not, but just not indicated it in the artist's tone of his voice. Well, as I say so much, but just one last thing. If you had, and if you literally had just three minutes to sum up, because that's all the time that they've given you, how would you... Oh, it's too general a question. How would you analyze the error that the Phrygians and the people like Scott and Thornton have fallen into about the relationship between geometry and logic? Why is it that they cannot see this? No, it's too, it's too diffuse a question. I understand the question, but I don't know... I think that it has to come either at the end or the last paragraph. I think probably it has to come at the last paragraph of the book, of the next.

12:30 But why is there so much anti... well, they would call it anti-naturalism, I suppose. Frege himself was very proud of this, that he was an anti-naturalist and an anti-materialist in logic. My mother-in-law likes to drink, so good for her. Thank you very much for joining us. There's just so much to learn. Thank you for watching. It taught me one thing, one great thing. Well, reading your papers had already convinced me of this, but I hadn't known or understood enough until meeting you and hearing it, and that is that I don't have to buy a software package of anti-materialist metaphysics in order to understand logic, in order to do... Logic is not... You don't have to buy that package of anti-naturalist or anti-materialist metaphysics in order to be... And they very nearly got me to the point where they convinced me that you did have to and that therefore we might as well give up trying to understand the subject altogether or, you know, this is why I wasted all my time with Finkelstein because I thought, you know, well here at least is a... and you don't have to re... and you don't have to throw away that vast body of scientific knowledge, ever deepening scientific knowledge that is mathematics. It's already...

15:00 This is a reflection of material reality and it's the best. We have to go through all this? It's a busy day. Yeah, a busy day by the look of it. It's not too bad, it's just possible control here. Did we actually see which one it is? Is it directly to Milano? Which one? Is it 38? Yeah, 38. Okay. So for all those things, Phil, go. Thank you, thank you, thank you. See you, see you, see you. Thank you. Cheers. Thank you for watching. I would hope I've been on them for some time. Uh, shut up. Uh, where do we go from here? Yes, he is, very genuine. Thank you for coming over. Well, I mean, because you came, I was able to hear far more. The ideas I needed to hear because I'm not able to formulate the questions myself.

17:30 He is a good audience, one maybe. Though he was perfectly happy to talk, he is immersed in his subject. I think next time... Did you have any really helpful insights on the... on the... solve any particular problems? No, he did jot down some... something. Oh, hang on, did I... did I take the stuff? Oh, I left it in your car. I don't know. You should have. Well, I wasn't carrying it around with me then, was I? I didn't see you carrying it. No, no, it must be back in the car. I mean, if you open the... I asked him a question and he couldn't get out of those. He asked me, I don't want the answer. He jotted down a paragraph. Well, I'll get it. I'll make sure you've got it. I have that one. Oh, you have that one? Yeah, but I've left all the main stuff. Well let me let me know what's in there because I wanted to see there's a lot about Clippidoc which I want to know particularly too because you see it's one of them oh yeah Well, absolutely, because this is half-stacked with functorial semantics. So whether you can find... Sorry, which is... Oh, you mean it's early. That's the right thing. Yeah, that was one of the very earliest pieces of work he did, right? You see, there is certain trends in his thinking, like get away from numbers. Well, as I was saying, it's the deeply geometrical way that he thinks about all unifying structures. Instead of known spaces, what he was trying to say, let's consider symmetric algebra and take values in the algebra, rather than... Thank you for watching.

20:00 No, it's down there on the right. It's underneath your Atlas. Well, I'm not sure. Maybe there was a thing up there saying, you know, where you pay, kiosk. Take the kiosk, pay before returning to the prize. Yeah, that's right. That's where they're all standing in line. So, anyway, say that again. Instead of taking values in the reals over... Say you have a vector space, so you define it in a product based on the vector space. It's a bi-linear and symmetric function only on a vector space. That's the real product. What he's saying is don't take it to R, but take it itself to another symmetric space. Yeah, but isn't that very much Basil's idea in those papers about the algebra of differential forms, and you see this is why I was asking him about that construction, which I know we never got time to talk about, because it seems to me that was influenced by very much the same sort of Grasmanian. Uh, let me pay for this because you gave it to me. How much is it? How much is it? Fifty pence. Two pounds. Two pounds, okay. Sorry, I couldn't hear a word there. I can't hear a word, I'm going deaf here, I think I really am. I seriously am beginning to go deaf, I'm sure of it. That's why I keep asking you to keep things up, because I haven't understood them, because I haven't heard them, clearly. I'm going to go see a doctor and see if I'm getting my ears stringed out. It's so much trouble. There's about six, six and seven pedestrians. Yeah, I know, that's the problem.

22:30 Well, let's wait till we get in the car. You see, you define exterior algebra. Let's wait till we get in the car, because I simply won't be able to hear otherwise. I heard you and I didn't register. Thanks very much. Yes, thank you. Yeah, yeah, you queue up by the car, yes, which actually is, it is in fact a much more sensible system, I can see that. Just let me make sure I've got those. Yes, I have got them, it's okay. Now, sorry, say, say again. You were just starting to say, so we can... Come on, man. I know you want to get in there. Yeah. Yeah, I will. There is a paper. To the right, you can go out that way. Yeah, alright. Yeah, yeah. Right. The third-fourth page of the law of theory marks that he makes even Clifford Algebras would be defining the inner product of some other symmetric algebra.

25:00 So I asked him, I got a small diagram which is, I don't know, I said I'm going to chase this. He's got an awful lot of maths behind him, blue tickets. Go on. I think you have to insert it, insert it there where it says number one. Insert exit ticket, yes, insert it. It's a mad rush. You know, every Sunday it's unbelievable. Look at that. Yesterday I was murder coming to you to get out of sentiment. But it was worth it. We had a good chat. Oh, yes. Yeah, next time he's over here, I'd really like to get him with John Mabry just to talk, you know, the philosophy of math for the same period. If you do photocopy the Mabry paper. I'm going to. I'm going to do that right now. Will you give me a copy? Of course I will. Yeah, of course I will. I mean, the man is not the Bolshevik. No, he's not at all a bullshitter. I'm sure you can see that now after having read some of his stuff. It's a way, I'm not, as you gathered, either very versed in these areas. No, I agree you're not interested in the sort of things people in philosophy and mathematics do. But, you know, I have an interest in the subject, so it's a good sort of empty point to see what exactly they talk about. Usually, from my point of view, they talk about things that I was not terribly worried anyway. I mean, you made the point.

27:30 The young study took these things so seriously that I never got beyond them. Yes, I know, that's the problem. But it also influenced, and now you say that I see people. Well, I can't say I wasted much time. No, and you were very right not to. But nevertheless, I am interested in this whole area, as you are. But you didn't entirely, you know, not think about it, because you actually came to a rather definite conclusion about how one should think about sets now, and you produced, as you say, this sort of about the spatial origins of the category of sets. I read Loewy's paper in a particular way. Now, this is not against Loewy's ideas, so that I don't feel reassured, in fact. Well, that's what you see to me. That was almost the most important single thing that you were talking about. But I also got the feeling that he has got much more sort of inspiration. Well, I think he's got a much more worked out position. But I, of course, but at the same time he's busy producing so much. Great mathematics that he hasn't really had time to work out the integrated, how to integrate the, I mean, the position is itself very, very fully integrated, but he hasn't worked out how to present it, I think systematically. That's, for me, been the whole object of this exercise. But tell me, I mean, again, I mean, just how did you think of it? I mean, when you... I tell you in a very personalized fashion. I studied this in undergraduate probability, calculating simple probabilities, and then from time to time in the example they say, and what would be the result if you had indistinguishable dice, you throw two dice and this comes out and that comes out. Do the same exercise considering the dice indistinguishably. Now you have completely altered your probability space, something very fundamental. Normally in probability theory you come out with this experimental set of, for what it's worth, in this theoretical fashion.

30:00 But in the absence of other, you know, further knowledge or further reason, each and every one is an inquiry problem. In practice you may say, ah, but the dice is a bit like this. Yes, yes, that's irrelevant to that. Now, where is this notion of identical dice? And what the hell does it mean that the dice are identical? And they don't mean that the dice are similar, because where does the idea come from? It is the Bose-Einstein idea that they had to calculate differently than what Bose... In fact, do you have Pius V? He makes something very similar to my statement. That's why I don't have to be claiming that this is a new idea. In fact, it's an innocent idea which was offered many years ago. It's not now in any way fashionable to make this kind of lecture. I mean, it's too innocent to lecture on. People lecture on, you know, the cosmos and I mean, yeah. Okay? I'm being a bit cool here. So, if you want to study that it arose in black body radiation and so on, you can, in fact, make a whole study of this. It's a very good study of the notion of identity, quality, and the notion that it arose. If you want to get historic there, why physicists are so reluctant. It was a surprise to them, so they weren't thinking of this. Now, when I went to perform in Bath, it was after a search to various departments. I had just done two hours of gambling, and to some extent, I mean, disliking of maths and physics and philosophy, yeah, like you want to go back.

32:30 Yeah, absolutely. Just go to central London. You go straight back to your place. Am I not dropping you a feeling? No, no, no. I'll give Peter a ring. I'll get him to bring the tape over to me. He's got to come in. You can manage, can't you? Sure. Manage what? Manage to get back home from your jobless job? It's a long, long drive. No, well, that's the thing. I think you've already done so much driving. Let's get back to what we were talking about. Sure. I went there to have an open mind and I'm going to do research. I used to study, but this time I've also gone to Alan Peirce. He's a good mathematician. A bit dry, but good. And I used to look in the Universal Algebra, in Grazer. Remember, I was doing a bit of insurance stuff. I've made the money, I've brought the money out, I'm satisfied, but from time to time I went to boards and bought myself a book, I've been mad, so I'm now looking into more general mathematics, I'm not interested to learn about the study of groups of this sort, which are technicalities, which are all of my professional expectations. It may be a crude or harsh assessment, but there is a lot of it going about. It sounds very technical and deep, and it is very technical. It's not necessarily very deep. It's simply that they have a leader, who's a professor, who's got all these little problems. The students first they pick up the definitions and the lectures and they understand what the subject is about and then they churn out the PhDs and these PhDs some of them come good later on their own again and others go to some other department where the umbilical is severed and the 2-3 improvements that have been given have been solved.

35:00 They are stuck because they don't have the initiative, they're not searching minds, they're just problem solvers. Yeah. This is an aside. Back to Alan Fierce. I've been reading a little bit about logic, you know. Centia, you know, this traditional centia, whatever, axiom, this amen, amen, amen. I don't understand all this. I have vague understandings, but I've not been bothered. So I expressed this opinion and he said, well, have a look at category theory. There is a category of categories. I looked at this paper. I couldn't read it. I knew not one functor from another. So what do I have to do? I said, well, read MacLean's book. So I took MacLean's book and I went on and read it. I spent about six months, three months or six months, I understood it, nine months, I was able to talk about it, but I was not. This is when I tried the lecture that Berkley and Lindfield then went. It was a bomb heck of a quality. You want it in theory. Why is it in theory? It was simply a statement. All I wanted to do at this stage is let's get a bit of the syntax across. I'm not at the moment offering any, all I'm saying here is a good way to talk about things and you get intuitions, not necessarily the end product, but you say this seems like they were reluctant to learn the basic definitions again, so they're all false. Perhaps I thought about it as a sort of series of lectures or what it in fact was they wanted a one-hour sort of general talk about it, which is what it was in that case.

37:30 But I normally see what had happened by the time. I thought I'm going to talk to you born and highly and working so forth. And when I walked in there were about 50 people in that sense of category. And I gave them all this spiel about definitions and so on. I should have talked more generally about it. I wasn't all that confident, but I could have made it up. Back to the thing, I have back in my mind these identical particles. Here's a guy who's come out with a new maximization set. Immediately, if you haven't called for the traditional maximization, especially I don't understand sets being members of sets. Yes, exactly. This is why you had to start over. Here was very funny things going. X is in A, and you put two brackets, and this all is in B. The idea of a collection. On the other hand, Lobie, here is a set, one, okay, and one thing is an element, but the basic idea, products, this, that, the other, I need no... At that stage, I'm looking for something a bit more mysterious. I don't know, you may think, I mean, Lovier has come up with this and the elements he's developing, so from a completely different angle, he is also open to generalizations more like, I don't know, I thought this was going to be something really needing a big insight.

40:00 The more you think and think about it, Lovier is saying that if you do, in fact, only the card you have, the very fact is Mabern touches on this, but in rather a very lengthy way, they seem to be coming to see. What are you doing when you see a collection of things and you declare you units? And then you're down to doing a bit of maths on paper, whatever. In your mind you have an intuitive idea of these units, which are just the units. It's as if, in a primitive fashion, someone is on the beach playing with the pebbles. Whatever your formalization, there is some intuitive ideas in your mind that guide you, jot down in you. I've jotted them down in certain particular ways. It's not a conspiracy, it's just a true experience. You have found that this is the way to think about this sort of thing. Is this for your scheme of things? So, what the hell are these identical particles? Okay, I'll cut it short. As I read him, and to a large extent correcting that particular... The elements of the cardinal, these points or pebbles or particles or whatever, have been blended completely uniformly. You have said, I'm not going to worry about this card being a bit fatter than the other cards, which is what you would worry about. You've just covered all of that. All you're saying is, I'm going to consider that as a whole.

42:30 Things that are gathered together as an ensemble is, you know, better. I consider them together. I mean, they don't even need to be, you know, in your mind. They don't need to be here together. One may be there, one may be there. As you said, the glass may be all over the search point. But in your mind, you brought them together. Sure. And when you manipulate them, you don't think of them being all over the world. You keep thinking, I mean, intuitively. These things are now the focus of my attention, so they have been gathered there, so where does this idea of focusing come? It's through our visual experience, maybe through evolution, so we've got this brain also that we use. Yes, absolutely. You probably expressed this much better. I'm talking vernacular. Essentially the way that Cantor described it. It was exactly what Frege attacked, of course, furiously. Well, a little bit about Cantor, because I used to have a... Review of Cantor. I used to have... It's the abstractionist account of set formation, which was the subject of a withering criticism, in fact, by Greger, which... I used to call it to be... Let's come a bit clean about this. I used to think that some of these guys... I don't know, I haven't read them in detail, but I had my bogey, bogey school, let's say, bogey school maybe, not exactly one person. My bogey people were those people who should have gone in and done, they should have developed a career as barristers rather than mathematics. I called it the legalistic. So as long as, you know, the legalism, there was some sort of... The search for the ultimate legalistic, which will make everything secure, this is not anything that corresponds to the way I learn the subject, like maths, when you do it, it's not all that trivial, it's only by learning, refinement, going back, it's in a way, also reflects the way we learn about physics and how our ideas change in the face of new phenomena and the new phenomena...

45:00 And he tells me we have to think, and it takes a bit of creativity also, which is unpredictable, that we come up with these ideas, but the work is a whole very complex process of development in the way I perceive it. You've read a bit about that, yeah? Sure. This is not against, I mean, Laugier has a different way of putting things, but it's not against Laugier, obviously. That's why. So, okay, the elements are already in a classical sense, alike. So there is only this question of naming. Have you read a little bit of physics? Read Weill and so on. They make a big mystery about identical particles and so on. They propagate this mysterious way where particles may be identical and not, may be similar and not identical and so on. Okay, you've thrown everything, and what is it that makes you say, well, they are different? Ultimately, you're saying this one is this one and this one is this one. So there is some sort of ideal space. It's only the position, either physically or in your mental representation, that keeps these particles as individual entities. And what makes you say after mapping, there are so many maps. Unless you're going to say this one went there, and that one went there, and I'm talking, if you like to make the point, very concretely in terms of assignments expressed individually, say A goes to this, so you play act the damn thing, I mean you might as well sit on the beach and play with me.

47:30 So, how are these other particles, the bosons, all that different? We're playing with words like indistinguishable, indistinguishable, yet distinct, and so on. It's a play on words. So, what is happening, I say, well, that is not the point of the rotating experiment. Put in by us, into the category of finite sets, are the distillation of a lot of experience we have gained, is a formalization, a correct formalization, of playing with these clever finite sets, blah blah blah. I don't want to get involved in continuity and discreteness, which is to keep it concrete, just to focus the aspect we want. So what is the difference? All I'm saying is perhaps, I mean, we've been too eager to extrapolate from our macro sort of everyday life. You refine a little bit the classical ideas of mappings and trajectories. And we took it into a context in the micro domain where it was not inappropriate. For one thing, these are not really particles. As soon as you look for them, all right, you may or may not find them. When you do that, indeed, you may have an indefinite number of them piled up at the very same spacetime point, so that the notion of spatial separation, which is the very intrinsic part of the notion of an element of a set of the way of thinking of identity, is itself on the mind, and in fact that's what makes it so difficult, I think, for people first exposed to physics, to understanding. There is also this idea that the, I don't know how much thermodynamics you know, a little I know, but these little probabilities have fundamental, I mean they've been designed already, because they do have drastically different thermodynamic modifications as well.

50:00 Summling is a play which we were talking at a very rarefied level, a very limited level. If there are many in a given state, then the others prefer to go in that state. They collect different things. They are supposed to be identical. Stars rather than A, B, C, whatever. If you read Dolby 1976, I think it more or less says... The classical sets are the stars. You could then say the mappings are the ones that make use. And that little example is an illustration. And it's a better point to sort of expend your mental energy on them, that four glasses being actually here on four corners of the world. So you're saying they divert attention. This is my opinion. I was getting fed up in the conference. I reckon people would ignore the meaty questions and go into it. Diversions, however, may be longer. Well, it's the space, the space has changed. Because even our primitive calculations are based very primitively or useless or hypothetically useless to a species. Yeah? Yeah. Which has given rise to the category of finite set. This, this, this, this, this, this, this, this, this. The foundations entirely puts it back on the other way around.

52:30 Which may be the more correct. I mean, you may argue this. I'd be happy if I saw... There's historical evidence that that's exactly how the Pythagoreans did, in fact, do arithmetic, in terms of grids of pebbles. They actually did grids of pebbles. No, they didn't. The earliest proofs were discovered with grids of pebbles. I never went and looked up, because I'm a lazy bugger. I didn't have... So it is the space, once you start thinking in terms of the space, and combinatorially, it's clear enough, the combinatorics people haven't realized that, I mean, I asked Bill, because in one lecture he was essentially dealing with all the preserving maps between, I asked him if he knew how many there were, so he's done it by inspection, and the numbers are there, it's the connection. Like all the maps from M to N, everybody knows is NM. Now all the other presented maps from M to N, if you ask this out of 95 mathematicians, out of 100... We will not know the answer. It's an equally basic fact. The answer, the physicists know, but they don't know it in terms of all the presemic maps. They know it in terms of stars and bombs. The connection is not a big one, as you pointed out, but I made it into an excuse to use of how mathematics is a risk. How adventurous we have to be to really come up with some intuitive understanding of quantum logic, by intuitive I mean the mission of a well-versed mathematician rather than a layman, I don't mean by intuitive and naive, I don't mean trivial, I mean, I should use the word convincing, but then I'm all subjective, so I liken this to...

55:00 The Punch. You know the Punch magazine? At the last page, the inside page, you get 19... beginning of century cartoons. Yeah, yes, I know these cartoons, yes. Without the captions. People provide the captions. The modern captions are sometimes much more humorous. Well, they're always much more humorous than Witten, simply because humor is so variable, it's so subject to change. Okay, that's... Yeah, because it's such a cultural... Now, you're also allowed, I suppose, to change the drawing a little bit, to some extent. Perhaps the drawing has to change a little bit, and also the... What? The drawing, in this case, being, as it were, the ambient... But when you speak about the changing of speculation, in this case it's speculation. No, I do think that this is tremendously important insight. The moment I read your paper when John first pointed me in the direction of it, do you remember when I rang you up and asked you if you could send me a copy? I was completely convinced by the argument that there is a spatial origin for the notion of a set precisely as captured by Lorvier in the elementary theory of the category of sets. In changing the underlying category, we are actually changing, changing the underlying category is a reflection of a change that's taken place in our understanding of composition of mappings as actually understood as the expression of what it is that this structure has arisen from, the kind of experience it has arisen from.

57:30 But then, you see, that was precisely why I was so hung up on Finkelstein's ideas because Here was a man who told me that, yes, that we know morphisms in category really are an expression of dynamics, an underlying dynamical conception of how the world works. That's very important. It may be innocent, naive, but it is essentially this naturalistic conception of how mathematical concepts, even the most primitive mathematical concepts have arisen, suggests that... There may already be built into the most primitive mathematical concepts hypotheses, as you say, about the spatial, sorry, about the spatial nature of mappings and composition and the way that elements behave, classical assumptions about the way the world works that are in fact undermined by the discovery that it doesn't work that way, that quantum theory... This is quite a different picture of the way of the head. This is why I was so hung up on the idea, you know, these ideas of Pickles about quantum mappings and quantum arithmetic. Yeah, yeah, I mean, it sounds like I'm against fiction, but I'm not, because the man... Yes, it is a very cool idea, but unfortunately it turns out... Sometimes he talks a bit bigger than what he always does. Yeah, yeah. Whether consciously or subconsciously. Yeah, and I'm not, I mean, having met the man, he's a chap I like. Oh, he is an obviously likeable man. He is a sensitive man and he is not an aggressive sort of, you know, listen to me type of man. No, I impose myself. He is not likeable. Not one bit. But sometimes he is a little bit. Oh, his papers are terrible. They are a little bit. Well, I think that pompous, they just are confused, unfortunately, for the reasons which Bill has explained so very clearly in the last ten days. But I think he's addressing himself absolutely to the fundamental areas, just that his solution is...

1:00:00 This is a programme which on the face of it appears hugely, wildly ambitious, which is develop and develop quantum mathematics, or analyse more carefully how these, as you say, these kind of counter-natural hypotheses that are built into the primitive conceptions are actually just finding the right mathematical constructions that will really illuminate the physics, and that's what this whole programme of so-called quantum set theory was about. This is what I was so interested in. In David Hosworth's ideas, it just turns out that this really is based on a misconception about the significance of the so-called quantum mappings, that they don't undermine the notions of identity and individuality that are kind of built into... They do undermine, but those notions are undermined, nonetheless, by category, but they're undermined precisely by the recognition... In the case of the Cardinala, in terms of the composition of maps, the behaviour of the classical set, you know, mappings are determined by their actual elements. There are spaces, categories of spaces, where this is not the case, that they are determined by their actual elements. I mean, it's still... Sorry, I'm burping. It reflects the... It's because I'm very tired. No, my only regret was, well, we had a very good chance before we ever read people. Although the paper was hard enough, there is a whole thing. I want to say a few things about the notion of mapping as a reason for motion. But that is what I want the man to write the paper on. This is why, to some extent, when we're all... There is a danger in that, I suggest you, because it's precisely what's led somebody like Finkelstein into this idea that dynamics is prior to logic and that we can only understand the mappings correctly from the point of view of the motions, you know, the real motions, but we'd have a different idea of mappings and how they're composed.

1:02:30 Well, by representing them in accordance with the recipes of quantum theory, in terms of, actually in terms of, well, creationism. You can't make the symbols themselves appear, disappear on paper or become... Yeah, but he... Discuss these ideas. He thinks, of course, that... State symbols. Yeah, well, that's precisely one of the things Wittgenstein spends a lot of time talking about, the fact that the actual nature of the stability of mathematical symbols as material for macroscopic objects is an obstacle to our understanding the really deeply quantum nature. Well, in terms of... I would say, you know, especially by understanding the composition of mappings or every category, by understanding categories, that's expressing basically semi-groups of dynamical actions in the classical case where classical case, yeah, where classical Bill calls the bookkeeping axioms of the behavior of domain and codomain really are an expression of assumptions that you've made about the... ...behaviour of the mappings, which in fact express the ultimate convictions about mappings and elements that are physical in origin. That's, I think, how Finkelstein thinks. Now, that turns out just simply to be wrong, but I don't think it was a... And it was a simple idea, but I don't think it was a trivial one. I think it was... You see, there is... Do you want to go to Chisholm? Because of the traffic, yeah.

1:05:00 It was Sunday, very strange. It's worse than a weekday, isn't it? Yeah, never seen it like this on a Sunday. Thank God we didn't come out this way. It's bizarre. Yeah, I will... I will go through the butterfly between the traffic and the foundations. Sure, this connects with... I think with this business of linear logics and the quantiles that we were hearing a lot about in Bangor. What is a quantile? Well, a quantile is a locale. It's like a locale, but it's over a non-commutative ring, instead of a commutative one. That's right. There are two ways that you can get a quantile from a locale. There are two different programs, one of which was developed by Malmy in Sussex, The second program is connected with these so-called linear logics, which involve introducing additional values. Do they connect with physics in any way? Well, when Mulvey... He didn't introduce the construction. It was precisely because he thought it would provide a way of understanding quantum logic. I mean, these people are all good category theorists. Are they also good physicists? No, none of them is a physicist at all. So they have a chance. None of them is a physicist at all, except, I mean, Bill probably comes nearer to being a physicist than any of them. He certainly understands more mathematical physics than any of the rest of them.

1:07:30 He just has a marvelous feel for how deep ideas interconnect in almost every area of mathematical thought. That's kind of wonderful, controlling. ...which is the controlling perspective that I want to understand, and particularly how this idea of logic coming out of geometry, or this geometrical way of thinking of the logical structure, which you yourself obviously have too, because of the... Sorry to say that I'm in Lovie. I haven't studied, but I will study. I come often. Topology has got a very dramatic view. You can even think in terms of a prison camp with searchlights. You have to explain that to me. It comes back to me as a very disappointing chapter. Because I can't sit down and do any work. Obviously the only way I'm going to do topology is... I can sit down for six months and I'd go and decipher Johnston from beginning to end. Yeah, Johnston's tough. I can't make any headway with Johnston. I can do it. Sorry, I don't have the necessary mathematics. Oh, you could do it easily. Not necessarily easily. Well, you could do it with time and patience, yeah. Yeah, with hard work. Well, look, tell me… But I can't sort of… Look, I'll make you a deal. …say I understand myself. If you do do it, will you sort of… Make an arrangement to kind of do it with me and help me go through it as well. Yeah, but I would love to do it, I know, from knowing as much as I know. You can kind of teach it, teach me at the same time as you are learning it. You can help teach me how to learn because I would love to be able to understand Johnston's. I can, I mean, I've just got to the stage where I really now do understand what a sheaf is. But then I keep coming across things that make me think, well, I haven't really understood, I must have, because particularly this idea that it's so, you know, such a big deal, what Grothendieck did to generalize the notion of sheath topologies, to sheath over a site, and that's how the whole notion of topos, the Grothendieck topos, originally arose, and I don't understand why...

1:10:00 That was something that took, I mean, there must be something about the ship construction itself that I missed, that I haven't understood properly, if it in fact took the, if this was a really deep insight in algebraic geometry that it took a kind of 20 years of thinking about the best algebraic geometers in France, which means the best in the world at that time, and well, most of the time since, to win through to. So, whereas in fact it just seems a very... Straightforward, almost trivial generalization too, but so obviously I've missed something about the sheaves over topological space, the kind of preview, the earlier, more concrete notion of the sheaf, the more concrete context of the sheaf construction. I haven't understood this, I thought I had. And it just needs a lot more work, I've just got to find the time, make the time. The main thing is instead of the variation being on a partial order, doesn't it be on a category? Yeah. Instead of it being over a partially ordered set of open spaces of a topology, let it be over a category. Fine. But to me, I mean, apparently, obvious generalization. But I must be missing something because the history of the subject makes it clear that this was not an obvious generalization, but one which it required a great deal of ingenuity on the part of... You know, very powerful minds to win through to. So I've missed out something in my understanding. I've actually been playing with examples and models where distortion was thought of. Otherwise, it wouldn't normally complicate this stuff. All the maneuvers. I mean, this is after the event. You want to learn how the whole thing, the genesis of the idea is a bit different. If you just want to learn what it's saying, then you can complete it quickly enough. So why is this thing so partial? Well, let's go back to what we were talking about earlier on. I was trying to explain to you why...

1:12:30 Liking for this... Just take me back to Hadley Grove and I'll leave from there, don't worry. ...way of thinking of morphisms, you point out, in terms of the spatial origin of our notion of the structural academy of sets, geometrical way of thinking of it, to me, and Bill, clearly, in thinking about the role that the intuitionistic logic plays, the complete hiding hole for players. We've also had a very strong geometrical intuition of these things as he says domains of variation, not in the sense that he used the notion of a domain of variation, just a space in which values, the space of values of a variable, this is really just a panel of space, I mean it isn't geometrical, it's actually a very strongly geometrical notion of variation. These really did parameterize domains of variation in the world that were not parameterized by the constant case. They actually picked up more of the underlying structure in which the constant case was a restricted instance, or out of which the constant case arose. Well, this is how Bill, I think, clearly did write. This is what he specifically states. I mean, this is not my interpretation of him. This is all in the 1973 and 76 papers. But then, of course, my natural reaction at that time, when I was still struggling with trying to understand, if you like, some quantum metaphysics, was, well, how can we express as conditions on... How can we express, in terms of conditioned variation, in terms of conditions on morphisms, the way that supports the nature of that kind of conditioned variation in the quantum case, given that the world really is, I think this is how I was thinking at that time, I'm reporting on my confused understanding there, not on what I think now.

1:15:00 You know, I thought this idea of physical science that quantum sets have many possibilities in their basis instead of just one, instead of having just points in their basis, they have superpositions of points. Let us try and see if we can understand superposition in a way that doesn't already presume identity. In a way that doesn't presume the bookkeeping axioms, because these already have the classical notion of identity of an individual built in, which is what the quantum metaphysics of superposition does away with. That was how I was thinking, and that's why I wasted so much time. Well, exactly, but you see, what Wittgenstein was saying to me, but you haven't really thrown away everything, you've still got enough structure there to be able to understand these, well, first of all, the structure you've still got left really reflects the process, what's really going on, which is processes, process metaphysics, structuralist acts of creation and destruction, you know, the Heraclitian view of reality. You know, if you actually think in terms of operations as being more basic than this, or morphisms, then you can understand algebraic structures like equilibria for operations, which are in a sense a kind of algebra of operations that you interpret in terms of, he has this idea of domain and co-domain as both carrying a lattice structure, and instead of identity, what's more fundamental than identity, this is for Wittgenstein.

1:17:30 ...is the Galois connection on the orthogonality relation between those domains, between domain and codomain, where domain and codomain really are understood input and output rather than an object in the Mabry sense, in the Frege and Mabry sense, the arithmoid, the things which provide the... The being of which is the being of the abstract unit relative to which other things are identical or different. For Frege, simply the category of objects, which is defined as the things to which it makes sense to apply the identity predicate, for which the identity is decidable. Those things which are, for Frege... The category without which you can't think at all, and you just have to have that as an ontological category, let it be understood this is a strong claim about the ontology that you need to understand mathematics. It turns out, you know, in fact, this is all true mathematics, a very strong ontological assumption about the world, which is contradicted by quantum theory. I know these levels, this is a level of hopeless floundering around, but then I didn't have any of the necessary mathematical background to be able to say anything about, oh yes, yeah, I'm determined to now, but I hope, you know, you don't think it was completely rudderless, it seemed. No, no, you've done well, I tell you, my brain just is not fine enough, it is adventurous, I mean, I'm not... I'm not saying I'm better than you are. No, no, your brain is much better trained and it's not coarse in the way that mine is. I mean, I don't think you're... you're not hung up in metaphysics. I want to go this way. Right here, surely? I'll avoid that. Well, do you want to?

1:20:00 Yeah, I mean, there's certainly room for metaphysical speculation, but not, as Bill was saying, for drunken speculation and control speculation. And for control speculation, you need to know a great deal about the fundamentals of the coherent body of results. Well, you read me in Wittgenstein. You read John Bell. Yeah. Well, which side read John Bell? Which book are we talking about? Oh yes, those papers, yeah. I haven't read the Topos' local set theory books yet, but I've read some. Do you have my... Yes, certainly do. Do you know how to... Sir, can you put your window up? I'm sorry, I'm having a little bit of hearing. Do you know how to manipulate a jointness? You know, to say this is the right joint there, or the joint there, or... Well, in simple enough cases, I think, yeah. Yeah, well, I mean... Where is it? Yes, all of them. It's simple enough. In fact, a lot of the constants are already known from the orbit. It's all there. In fact, I have each and every statement. It's all in a bracket where, by and large, they're one minus. But what are these joint decisions on a partial order? Just the notion of the fung phase is a bit of a preserving matter. Yeah. So you have, via the joint, left the joint, the only thing is you have functions that are self adjoint on the right, in the sense of freedom, self adjoint on the left.

1:22:30 As soon as you know that, you can write down four or five things which usually are the essential generalities. Yeah. Right. Now with these little tools, you should go and generate whatever Wittgenstein generates, it would be much more easy without... Yes, without all this dialogic... He did have a sleight of hand in one of these quasi-implications, so why don't you... I'd like to go and look at them. If you have the logical... I didn't want to go down in logic, in a formal logic. I mean, I'm not against Mitterstad either. The man is going to do something in the same direction, if you like. Yeah, yeah. So I'm not against. All right. When they start fiddling a little bit, I get annoyed. Yeah. Because he's trying to assault my intelligence. I don't like this. Which particular fiddle are you talking about? The introduction and the pieces. Oh, right. What are your pieces? Yeah. My reading was that this man claims, I mean, he's got the, he has mystery, and he's put down this, one of them is the triviality, the only non-trivial stuff is that he, when you look at the book, he just puts it down. Have a quick glance at this thing. The thing is derived at everything. It's just stated there. If that's what you're going to do. Then undo it. I mean, there is no harm in saying, oh, I stipulate that that is the expression, but it's one of the implications. So he puts down two axioms, one of which is a triviality in the system. It's not an added axiom in any case. And then you pretend you, from these axioms, follow the expression, which is the point I was making. Yes. However, put all that aside. We're not now here trying to argue on...

1:25:00 How much are you versed in this logic, paraphernalia? Well, how much... Certainly, you're more versed in... Well, the first or the second part... Yeah, you know this sort of semantic syntax. Yeah, yeah. So you know what Mitterstadt is trying to do. Oh, sure, sure. Okay, now if that is the point... He has a very, very peculiar attitude. Well, you have a weapon in your hand. You have an adjoinment. Here is the paper I have. Here is the evidence. For someone like you, take it up and say, with this weapon, blank, so when he writes down his form he has some sort of semantic... But you mean to tidy up... ...and take it from there and do better than that. Yeah, but you mean to clean up the quasi-implication, the quasi-implication connective, the connective in quantum logic. There are more interesting things to do. But you use that to join us to clean up the, uh, that, yeah, the implication connection, yeah, to churn, churn this family the way you like, but I'm saying here is a little tool, here is a guy who has already spent a lot of time on this, and the good news also is that he also seems to be hand-waving here and there, getting, I mean, this is a good, uh, A good way to get a few concrete papers, yeah? Yeah, yeah, I think that's... This is, I mean, if this is your intention. Yeah, yeah. If the intention is simply to study, you could study all to our plight, like old ladies. If you don't have any intention of propagating papers and this and that, I mean, that's not the ultimate aim of anybody. Well, it would be nice to have something in Berlin. That is an area, it's not, you know, it's not very... Well, that's, yeah, you see, yeah, exactly, yeah, yeah, I know, I take your point, um, it's not very ambitious, no, it's not, well, let me, let me go back and look at the adjoinments in your thesis again, and, uh, and then I'll look at, look at Littlestrap. I think that, um, well, it's so sticky, isn't it? That's the problem. That's why I was surprised that you, when you said you were curled up high.

1:27:30 Do you want to come up? Yeah, I'll just have a quick coffee. Thanks, Rick. Well, you haven't, you've seen my pigsty after that, anything, have you? Oh, what I was going to say is that there isn't, oh right, to, it seems to me that there's a lot, it makes a lot more sense to take the adjointness, the one that you've already distilled, out of, in the case of, and this is the adjointness in catamels, right? And to define the implication connective in terms of that, rather than to do what Holsworth's been buggering around trying to do, which is to take the implication connective, to take the Sasaki hook, the quasi-implication, and then try and find a category for it that it lives in. I mean, that seems to me to go to completely invert the proper order of understanding. Well, there is another more basic difference, which is that the author is a fucking genius, and David Hallsworth is a nice guy who knows a lot of logic, but by no stretch of the imagination is any more intelligent than he is. Well, there is a backup, partly because the guy's isolated and not working any longer in an institution, because nobody was interested in the stuff.

1:30:00 So when you say what L'Orfeo did on Monday, were you talking about the first lecture? Were you talking about the first lecture? Yeah, of course. It's very simple. So, what do you mean, this was all here? As you notice, the man in the current history, he's not this sort of super-brain, in that sense of the word. No, I know, that's what's so lovely about him. He's not the great doctor-professor. I'll give you all the theorems in a second. It's not a walking encyclopedia, but it's in fact quite a bit of a pond of intensity. But here he comes up with such insights. Now in the absence of these insights, to then say we're going to come up with a quantum, you see some people say we're going to study quantum. Yeah, well, quantiles are a contract. I mean, Dills had convinced me of that. And in fact, having listened to these people talking about them, he's actually showed me exactly what the contract rests on. It really rests on trying, turning a bi-category into a single category and ignoring and having messed up all the bookkeeping. Then saying you've got, you know, all this non-idempotence, non-competitivity, which they think of actually as a generalization. All of the local construction is in fact just a reflection of the fact that their bookkeeping, their domains, co-domains, bookkeeping has just been buggered up because they're trying to do things in a one category which are actually a reflection of the fact that they're working in a two category. Things like this, you see, there is a lot of PR. And the other thing is, in fact, he actually said, when he put this to Boisseau, who's the guy who's done a lot of this work in Louvain, and he, Bill, said to me that, you know, he put this point in a seminar to Boisseau, and Boisseau said, well, yes, you're right, it really isn't a good category, but...

1:32:30 You could look at the construction that way. Well, yes, maybe we haven't really... But, you know, you have to understand this is an industry now. There's a research program. I mean, you know, people do a lot of work in this. I.e. jobs, I.e. there's money in it. As you always said, you don't proceed as efficiently as you could. This quantum logic, in a sense, was... Well, in what way? Well, they didn't give me any bigger insight into physics. Yeah, but wasn't that what you were going for? For physics? Yeah. Not that I overrated myself, but you've got to be able to come through a lot of paperwork and get insights. The insights were not there. There was a lot of PR. Here and there in the thesis I have a typewriter without being written. You imagine, once I type something I'm very lucky to find it. As I type it, I go and work on it. I don't know how it's written. Here and there it says, this is the dump. Spontaneously away the journalistic. Yeah, yeah. And is that because it relies on lots of suffixes in its constructors, like German, I suppose? It's shorter, yeah, we have, you know, you could construct very lengthy, but we have a lot of, basically, I don't know how they created this in terms of the formal language, because the alphabet, certainly, is developed by the Greek.

1:35:00 Yeah. So, D, R, E, all this. And so on. Equivalent and more. So the whole language seems to be a bit further, like, you know, anyway, so much for what I mean. What, um, yeah, just don't ask me again. The, uh, the jointness. Can I just have a look at your thesis? Because, yeah, it's quite clear to me that... The poor old Holsworth is just trying to do things the wrong way back to France. He's got his implication connective and he's... Where is he? Oh, he's left the other copy. Yeah. Oh, God, yes, actually, he left the other one. My place. Don't worry, I'll look after it very carefully. Yes, please. Thanks very much. But to get at the... But what are you trying to think of getting at the combinatorial? Features of the simplicial category also from these adjoinances. Have I understood that? Well, you said you could get the dialogic rules for Mittelstaats, for the connectives, out of looking at these adjoint funders. Were you also thinking of getting the combinatorial? The comment hall features the morphisms in the Simplicio category out of them as well. No? No. I mean, obviously those do come out of adjoint funguses, any category. I am using them. Morphisms of any category.

1:37:30 These are just a few fragments. Well, where, sorry, where is the passage that I should be looking at? I thought we went through this when we met you. We probably did, and I was probably too, but since it's about half past five in the morning, I'm afraid I probably didn't retire. Well, yeah, we talked about a lot of things, remember. No, no, in the beginning, a few weeks, you've had... Yeah, it's been washed out by this, yeah. Is this it? This sort of, yeah, yeah. Yeah, this is it, yeah. Yeah. Well, I might, you know, go on and publish it. Yeah, it's very good, the jointness. Well, the jointness which comes out... Sorry, the jointness which comes out from the subspaces of... Here is an example that very literally, very literally, Baker and von Neumann, while they proposed quantum, they very literally, very literally, you can see in that joint, just as you see the heating algebra, I mean how all the man is doing is instead of the clueless type of geometry is to say here is a blob and here is a blob, the blob that's... Now, whether you can drastically soften it to something bigger, there's no conceptional.

1:40:00 Well, they can be. Look at all that point about domain, you know, Bill's point about bookkeeping. There is no systematic way they wrote it either. So the adjoinments may one day be presented this way, and the next day presented that way. So, you have to sit down and gaze at these things and say, is there adjoinments at all? For someone with a bad memory, it's a hell of a torment. All the diagrams are presented systematically in a certain pattern, so immediately you recognize a joint. Two diagrams, where is it? Have you come to the joint list? Obviously this is not an introduction in itself, but if you read around the subject it will obviously be a hard diagrammatic. If you go to the chapter on geologists, you just seem to find everything. There are three or four ways you can talk about geologists in terms of the nitty-gritty combinations. Well, I've read up on it. I can't say that I mastered it. It's one of the things I want to do. Once the whole momentum is broken, to some extent I have to repair it. Yes, actually this is extremely helpful. It's all there.

1:42:30 Ah, it's even... You get, you can express this. You soon get fed up after you get it under your belt. That's the proof-theoretic stuff. Yeah, yeah, sure, which I wonder is from, from any introductory book and from Bill's books.

1:45:00 Thank you very much indeed. Well, no, I promise I'll look after it very, very carefully. Yes, of course I will. You're welcome. You're welcomed. I want to remiss myself. As long as you... Stupidity, A, knock, knock. He couldn't remiss me. Sorry, I'm not understanding you. A sort of well-known candidate. Mm-hmm. No, not not. Yes, I'm sorry. I wasn't hearing you. Yeah, you wouldn't hear it unless you'd expect it. Right, okay. The other book is this hooker because for this you really stated closely. I think there is somewhere the statement of adjointness made to the catamaran version of the maybe I should mention that. No, this I don't have. Residuation theory.

1:47:30 I thought I ought to give one or two references. What bores me writing in paper, to be fair to everybody. But there again, I mean, if these people don't highlight something, it's not up to you to read every in between the lines to find out the genesis of it. Is it really? No, I don't think so. Well, obviously not. I mean, that would be like saying that Bill O'Veagh plagiarized Schroeder, I mean, when he discovered the... Yeah, of course you don't. I mean, you know... You say more like this in various forms. Well... You regenerate them in different contexts, which is not to say that they are the same ideas in the sense that, but when a construction is as universal as that, as fundamental as that, then... You know Kalmbach's book, do you have it? No, afraid not. Tell me about this. That's John who told me that. I must get hold of it. Not a very good book, but you must. Yeah, I must get hold of it. You see, this, people just take it as a piece of mathematics, and they say what would be an orthomath, and that is where the elements would be. I mean, you're not this sort of thing. Well, no, explain it to me a bit more. Neither am I. I can't explain it more than... Well, you said it's the old-fashioned way in which they say we take an orthomath.

1:50:00 The old-fashioned mathematics. Now this is the PR, the PR relation. The PR of it. Publicly. Oh. Yes. Yeah. Now, this has become a respectable area where you do some sort of... And that's the whole purpose of the exercise. So that book is written from that angle. So it's, in a way, it's a disappointment. And you can still find that in Georgia. It's a mention. Inequality. Yeah? Things like that. But what I'm saying here is a technical sort of advantage. Okay, you've got John Bell and so on. You've got Middlestadt and so on. They tried to... You can easily write that over here. Here you've got the adjournments for the partial order. And the partial order is a very peculiar time. So the notion of adjournments, natural transformations, because all the damn diagrams just commute.

1:52:30 All you have to do is this Galois connection. A lot of the time, contrapuntalism. And then they have a self-adjoint on the right, by which we mean this. Yeah. Then all this follows. A lot of these, we prove and we prove this. The formula of bargaining is this one, in the other direction. Yeah, which is true in the Heighting Algebra. Yeah, well, which is just the double negation thing, isn't it? It is double negation, but it's the third formula that gives you a J topology. Remember when they talk about J topologies in the topology? Yeah, yeah. Joyal topologies. They use this formula. Yeah, absolutely. It's a double negation topology. That formula is this one.

1:55:00 I just want to go with pure instinct. There's pure instinct beforehand, you know, and one of the things to do in this life is to... I hope so, because... This is what you were asking Bill about. Yeah. Well, and he didn't know what the... He said they've been... they've lost track of the future. Well, why don't they just go and re-prove it, I'm sure it's more than capable of. It didn't work, and I didn't want them to go on, because we don't want to waste time on that. I don't want to be anywhere near them. Once you have the algorithms, the proofs become very difficult. All this is the method of it. We want some logical. If we want to do logic, then, honestly, there may be a case for concentrating on this sort of stuff. Yeah. To be able to prove very easily a formula like that. I mean, is this a logical rule? I think so. What? What is this centralization? Is this implication? It's called a quantum implication.

1:57:30 What? Is it the adjointness between A star... That's the right adjoint. Oh, it's the right A, yeah. The notation is... You have this atonement. You have this atonement. Yeah. It couldn't be... Well, it's... Yes, is it intuitionistic implication? Don't be it. I'll be honest with you. Yeah, tell me, go on. You will try and mix that. They give you a long sort of indicative axis, but then they derive... Now, if you can derive their stuff, you also should cut it down a little bit. And you have another angle. If this is what you're doing, it's not a very deep exercise. And use a jointless to get this. Let me be a bit more positive. If you can come up with a story, like the Crick-Joyal semantics, Bell, or introduce also these other adjoinments, because Bell stops it all. Go on, I want to know. Yeah, so if you want to be more positive, all right, there is a challenge there. If this is any indication, you make the leap without working at all with them. A better story.

2:00:00 Why is it that statements are represented as subspaces? There is a transition which people have been into a mess, in which people's statements are now to be represented by subspaces. Why? I mean, the fact that it's subsets, mod those of measure zero, the fact that this is subspaces of a Hilbert space are the refinements. Yes, sure, sure. Yeah, I absolutely agree. Those are all just refinement. It's the fact that once you can look at it, yeah, do the raw view. We've done a lot of this, reading the refinement, and they said the basics, so the broad outline of the definition is right. They've come up with some ideas. I myself have, and it's a nice tidy way to think of it. So it's in an angle. If you have something, say it. I think you can go to the front page and... Hmm. Yeah, I think maybe you're right. Yeah, yeah, yeah. Well, you know, it's not that complicated. You know the references, I mean, look. You have a wide knowledge of the references. I wish I had that, because usually when I come to write an introduction of a paper, I'm at the lotus to what, you know, what preambles me. This should be easy sailing for you. Yeah, yeah. As long as the reader gets something out of it, a reminder of one or two ideas, one or two new angles, it's a good paper, so don't be ashamed to write a paper which is not the solution to all problems, nor even the statement of your beliefs, nor even your life's grand achievement. It's a paper, that's why we need to look at it that way.

2:02:30 Can you throw this past me again? These are the Mittelstadt conditions on implication. I've seen Mittelstadt come up with a lot of axioms like this. Just imagine you were going to write axioms for a boolean and start to write a whole lot of implications. Thank you very much for your time. Now, there will be, in my estimation, a lot of blah, blah, blah, blah, blah, blah, blah, blah, blah, blah, blah, blah, blah, blah, blah, blah, blah, blah, blah, blah, blah, blah, blah, blah, blah, blah, blah, blah, blah, blah, blah, blah, blah, blah, blah, blah, blah, blah, blah, blah, blah, blah, blah, blah, blah, blah, blah, blah, blah, blah, blah, blah, blah, blah, blah, blah, blah, blah, blah, blah, blah, blah, blah, blah, blah, blah, blah, blah, blah, blah. Well, Jerry, I'm going to go away and think about that, and I'm going to come back to you and talk to you about it again. Now, let me know when you want to buy any of these books, and thanks very much. Coming over and helping me make Bill's visit, get things out of it. I'm sorry I spent so much time talking utterly waffly, waffly, waffly, waffly.