Symplectic topology, Wigner-Moyal distribution / Ray Brummelhuis talk (contd.)

0:00 This was, of course, was why I was angry to become chairman of it. And, you know, he's chairman of it, and we've, well, the other trustees are Mark Lashie, Ray from France, Pierre Cartier, so we think now, but we don't want it to go off in half-cock. I want to wait until we've completed the catalogue and done a few other things. But anyway, once again, certainly see you next year, but I hope before that. Cheers. That's terrific. Actually, I'm sorry, I must just pop back in there and wash my hands. I'm so sorry. Sorry, I'm sorry. Sorry, Shani. Is Ray going off as well? Sorry, Tim. Is Ray going off as well with you? I just want to say goodbye to him. No, I don't think so. Oh, he's not? Oh, good. Okay, well, once again, we're feeling. Have a safe journey back and try and bring the same fine weather with you next year, this time around.

2:30 Thank you very much for your time. A contretemps about me being in Italy, didn't we? You detected the tension, though. Yeah. Which one? The one you went? Iacchio and myself. Well, he makes claims which are manifestly wrong. No, I'm not correct. On the other hand, when he was talking, his wife was talking aloud, you know, down there in the bathroom room. It was very strange. Oh, she upset more and more people. Somebody was organizing all the food for everybody in the village. I don't know what her name is, but Mrs. Arnold, certainly. She's like, bloody woman, I've got to get out of here. Oh dear. So I don't know what happened to her. Well, this was your hostess. Yeah, she would do all the hard work in the village. She organised the villagers to feed us, because that was part of the money for keeping the village going. Yes, yes, that was a lovely idea, but just for another one she was... And poor Maureen, I don't know... Well, when Mrs. Harrenhoff suddenly announced she was allergic to mozzarella cheese or something... I don't know what it was, all I heard was this. Oh, well, dear, well... Maureen walking away from... No, but I was very angry for talking through Mark Edwards' film. Mark Edwards was showing a film and it was a film about environment and other types of issues. Crisis in society.

5:00 Crisis in society, essentially. Was that what it was called? I don't know. No, that's the word description of it. They were setting almost an issue which he didn't relate to and he gave up in the middle of it. Well, actually, it was rather interesting, but he did actually say, oh Basil, can you sit down with me? We're sitting in the church, you know, underneath, and he was interested in what I was doing. So I thought, okay, calm down. Shall we start moving? Yeah, absolutely. What's the drill today? I think we're going to open up. Ah, I see. So we're the ones that are holding the thing up. I think so. We probably still need a car, though, don't we? No, that's not a unique voice, I mean. But you see, this is what bothers me about him, is he's picking up that and he hasn't got Yakir's intellect. No, that's true. So he's the one who really ought to get on and read and know what he's talking about and not just pick it up from our, from Yankee. You were telling me that you were very impressed with that research student of Jeff's, who sent me the paper. What was his name? Siphone. Siphone, yes, Siphone. Yes, I read his, he sent me the overheads of his talk in Beijing, which unfortunately I tried to arrange to have recorded, because it looked very interesting, but they screwed up. But he sent me his overheads, and it looked extremely interesting. I would like to get together with him. He's a very... I'm not sure what his math's like. No. Well, he's trained as an analytical philosopher and not as a mathematician, so his math, of course, is mostly coming from logic and set theory. But he is, as far as I can see, very painstaking and careful in the way that he presents the theory. I'm sure he could get up to speed on it. Well, he's coming from, as I say, his background is not as that of a mathematician, it's that of an analytical philosopher, but a very serious analytical philosopher of physics. He's certainly required to know a fair bit of...

7:30 He made some very interesting... because you know the Bohm Legacy Conference was much wider than just mathematics. Oh, sure, sure, sure, sure, sure. He made some very, very interesting interventions. I wish I'd known about it. I should have not tried to get down to it. Well, I probably wouldn't have been able to. It's David Peat who arranged the whole thing, and I asked if certain people would come, and the going was financing, so I didn't get any of it. So I'm paying €30,000 for this. Don't get me wrong, I realise myself, as you know, I've been in exactly the same position, financing meetings on a rather more smaller scale than €30,000, but even so. You're always having to struggle to get to the bottom of it. Well, good, excellent. I'm sure that's a very worthy course. Good, good, good. Oil money. Good course. Good, absolutely. Plus it brings goodwill from the village, which hopefully stands to respect. ...opportunity to talk about broad conceptual questions or foundations of physics when he's doing his meteorology, given his colleagues, I think, probably not many of them have taken the same sort of interest that he does. ...to the point where I will have an internal language. ...there are other things, like where you come from the whole into something. If you say, okay, I understand this tune and I'm playing it in this key, and now I'll just play it in this key, and never mind the tune, and you've got all kinds of variations that are related to the tune that are just interesting to yourself walking in the domain of the key. With some sort of an idea that has to do with the two, but it's also vague. But what you're actually doing is producing from the whole, with the whole being the restriction of staying within that key. As soon as you leave the key, it's gone. You stay in the key, and then all sorts of things happen. That's very interesting. This is for a human being at a very simple level. I think music does actually provide one of the best ways into trying to explore these ideas about the contrast between holistic and atomistic views of the world.

10:00 Do you have a problem, Bjorn? Because... Do you have a problem parking? Oh, yes. Or even as we speak, you know, I stay in the... No, just general patterns of increasing and decomposition of structure. Music, it seems to me, does provide a very interesting template against which to test some of those ideas, which you might call the polarised spectrum of holistic theses. We all know from introspection and from the accounts of the great composers that, I mean, Beethoven, Mozart, they never composed, they never composed, as it were, crotchet to crotchet or key to key, I mean, they had a very, you know, and this seems to be related to kinesthesia, to... The way that great musicians see music, I mean, just see a symphony in their head, rather than, after all, they have no staff, so... And they have been... Yeah, well, there's a bit more fun exaggerations than that. Yes, but I think, even in modern times, I agree that it's remarkable to which degree music can be created within the modern... Thank you very much for your time. Having a grasp of the total architecture of both the plot and, as it were, the message, and then filling in the details, you have a, it's interesting actually, maybe the novel does, because there are some novelists who clearly work more from a, Dickens would be the obvious case in point, who, as it were, have a vision of the completed characters, have a very, very strong vision of the characters, of their realities, characters, and how they would behave, and then the...

12:30 You know, the plot really derives from the characters to a great extent, and there are others for whom the plot is clearly primary and the characters are, well, if they're a good novice, the characters are never subsidiary to the plot, so they're never merely plot devices, but they may not be, I mean, that's just Agatha Christie, you know, but they may, within the total kind of architectonic of the work, it may be that the... The plot is driving the creation and realization of the characters rather than the other way around. Yes, sure. But somebody like Thomas Mann, for instance, is clearly more of somebody who has a completed vision, I mean a very, after a minute's completed vision of the whole work, and then the characters are really the expression of their very, they're a bit like instruments in the orchestra, they're more like the instruments within an orchestra, whereas for somebody like Dickens, Obviously the characters, they're perhaps a little bit more like a Veronese painting, it's the kind of glorious colours that are primary, and then the way that those give rise to a finished vision of the plot comes after that, I think, you know, this is a... And there are obviously people for whom it's the equivalent of painting by numbers, and they of course tend to be the airport trash novel. Of course not. I do know any more than writing by numbers makes Geoffrey Archer a novelist. I'm not sure, but I'd even dignify it with the term reductionism. It's just absolute dreck, isn't it? Well, I can honestly claim not to have ever read any of his deathless works, but I was once dragged to see a play that he had written. I can't remember, because a friend of mine was acting in it.

15:00 No, no, those were very unlikely the kind of things that Ponder, I mean, was interested in. There may have been other reasons why. Oh, he wasn't actually sent over there, he just simply had sent his same letter, and I'm not even sure of the story, I just heard that it was one of them as part of it. I mean, he was just persuaded to send his ideas to various famous European mathematicians whom he had learnt of, his existence he'd learnt of from. I remember getting a paper from the Royal Society on a completely crazy theory, and it was written by, um, I have to get the initials right, D. E. Littlewood, who is one of my heroes, you know, I don't know exactly what he's doing, and all his representation theory and how he, yeah. And in later life, this was done through crazy physics. Oh, right, right, right. Well, Erwin Siegel is another example. That's what I was just about to say. Siegel is, of course, another case in point. Who happens to have a Siegel device in entire cosmology? Do you have this? Which, um... Yeah. It's really interesting. Look about your forehead. Yeah. And you sometimes get... I mean, you do periodically get at philosophy and physics conferences, and, um... You know, ontology of space-time structure type conferences. You still do get people coming along giving talks on Siegel and cosmology and saying, well, why won't people go and look at this? Because it's all jolly convincing and really mathematically serious and just shows that... Well, the point is it has completely knocked out by what we know. The point is it's completely unphysical. So which Seagull was this? Irving Seagull, Irving Seagull. Oh yes, yes, he did a... I think he wasted about, you know, much of the last 20 years of his life doing all this cosmology and thinking about it. And every now and again I would get a paper from the Proceedings of Mathematical Sciences. He's one of our members and we've had trouble getting him to read his paper. Yes, it's tragic because he did such great work as a mathematician.

17:30 So with people like that, you don't even know the problems, so maybe I haven't understood yet. Of course you wouldn't have. Who am I to judge this man? Well, I'd say that at the end of his life, I think Gödel, you know, rather fell... succumbed to the same problems. Well, it certainly has a reputation. Well, he published a couple of papers, one on modal logic, and one which was, I think, was never actually published in his lifetime, which is in that background. Which, of course, is his version of the ontological proof. And he also published, there was one paper, which I forget who it was, I think it was, I don't know, Solovey, who had some... I think the last paper that was published by Goethe in his lifetime was about 1974, and Solovey wrote back, I mean, to, you know, obviously not... Note, not the publication, but if this paper had been submitted by anybody other than Kurt Goebbels, he certainly would have recommended its protection, because it was a very, very strange piece of work. It was his attempt to prove that the cardinality of the continuum must be Aleph 2, from topological arguments. He did not approve it as an argument. No, no. Well, he wasn't offered as a proof, but he didn't actually submit a paper on this. He became very strange. Actually, some people say he always was a bit strange, but speaking of Gödel now. Well, the one that Solovey, who was the referee, actually said afterwards, if this paper had been submitted to the JSL and it was written by anybody other than Kurt Gödel, we would certainly have had to reject it. This was the one where he thought he had a proof that the continuity of the continuum must be Aleph 2. And it was an argument from, actually from topology, from Poinsettian topology. Can there be a reason to believe in him? Yes. Well, I don't know, I cannot recall now if he actually used the word. I mean, he is one of the people who proved it continuously. No, no, okay, let me withdraw the word proof. An argument to the effect that the cardinality continuum is out of tune. But he did, and of course Gödel was normally morbidly, morbidly reluctant to publish anything,

20:00 and was so perfectionist that he would not submit. That's why he published so little in his lifetime. But towards the end he did become rather strange, and in the collected works the editors do say, and Feferman and Solovey do say, that in fact this paper, the one that he submitted on, The argument of the cardinality of Newtonian must be allowed too, which I remember hearing about at the time from Scott in 1974 in Keele, because he had actually talked to Scott about these ideas before he submitted them. He spent quite a lot of time talking to Scott about them. And then he submitted to JSL that he'd been extremely ill, had had a major operation about three months before and had been on various kinds of medication and drugs, and that certainly must have contributed to the... I think that's the failure of judgement that was involved. But he was doing a lot of other very strange things at that time, for instance he published, he didn't publish but he produced, they're in the NACLAS, work in modal logic which is, in modal logic he had a proof from S5 that... Well, we thought he had some kind of re-proof of the consistency of the independence, rather, continuum hypothesis. There was some discussion about the continuum hypothesis involved, and really it was that I left too. And so it was not regarded as crazy. Well, the idea may not be regarded as crazy, but the arguments by which he sought out to defend it then were... All of these were seen by his fellow logicians as pretty seriously flawed. That's the impression I have. So you were at that meeting of the Gödel centenary. Did you hear Angus MacIntyre's talk, which rather set the cat amongst the pigeons, I gather. Okay, this is to be deliberately provocative. I don't think Angus actually said this in terms, but this was, I think, the general tendency of his remarks. Gödel's achievement and importance have been very greatly exaggerated. You know, he is not such an important figure in the history of mathematics as philosophers in particular, and many mathematicians, believe.

22:30 In fact, logic and foundations have gone off in a quite different direction from that which he expected that they would, and his work, although very interesting in its own right and still important, is becoming more and more detached, one can see it in retrospect as having been at the time, but even more in retrospect, more and more detached from what was in fact the main line of development of mathematics. Now that went down very badly, I understand. I've been around with a lot of people at the meeting, but I still think it's a very... My problem is I can't remember whether I was there at one of the meetings. I think it was. I think it was. And it was a tap. There was a big... This is celebrating... The third of the centenary, yeah. Well, the point is Angus MacIntyre is one of the world's greatest model theorists, and it's perfectly natural to invite him to give the... But he's also a very honest man, and he's not going to go out of his way to heap... Honest people I've never known. He's not going to heap praise on Gödel if he doesn't, you know, just for the sake of it. Well, maybe he thought that this was a message that ought to be got across. I think I probably did hear his talk and didn't get quite the impression that you did. Well, I didn't hear the talk, although I spoke to him about it afterwards because he came to Foucher the following year for this interview. He was one of the people who interviewed Lorvier. But I did see the email, you know, the blog correspondence afterwards. And there are all these girdle admirers and hero worshipers and groupies who... ...who plunged in and were all attacking him and saying, how dare he say this, this is a dreadful thing to say. It was very derogatory. ...much more at a different level. I mean, he was, everybody accepts he was a great man and maybe a great impact, but there were flaws and there were things when he shrunk down and why he was still missing the edge and maybe... And why he hasn't had as much influence in the direction that he, I mean, the philosophers particularly expected him. And why his impact on the overall development of mathematics has not been at all... This is what one might have expected given the claims that were made for the significance of the incompleteness proofs at the time. I think that was another part of Alec's analysis. Indisputable. I think Indisputable had an influence. Indisputable just got the centenary or whatever from Pauli, you're going to get it.

25:00 And the previous lady, Edwin McKee, of course didn't have the same ambition as Hilbert's program, but still, for many, proceeded with the formalization of mathematics. No, we need Du Bois and Wittgenstein. Sure, we need Wittgenstein. Du Bois, yeah, it was somebody who sent it. It's very kind of you, but I won't. I won't. I'll just stick with the coffee. Thanks very much, Andreas. Roger, I really don't want to give the wrong impression. I don't want to give the impression that Angus launched an attack on Gödel. But he did attract a lot of negative criticism on the blog from people who were Gödel admirers. But the point is, he didn't dance around singing dithyrambic hymns. I've read the text of his talk, it's on the web, and to me, everything he says seems to me to be eminently defensible, and actually right, but then I never thought that Godel was the greatest logician of the century either. No, I didn't say much more. I didn't think anybody had much more. That would be far too strong a claim. Kirtley is a very great... Who was the... Well, away from the attempt to prove... away from the attempt to decide things like the continuum hypothesis. Because people came to see the thing more and more, not just because of the independence results, but because of the kind of completely different perspective on models and satisfaction that came out of the development of factorial semantics and category theory and its impact on logic, they came to see the thing as, at least in the form in which Gödel and, of course, the whole Hawking ontological greatness thought of it, as simply an ill-posed question. It was not going to be decided one way or the other as it stood, because the meaning of the continuum hypothesis had just altered in a way that meant it was really no longer an absolute yes or no matter, and that's always simplifying enormously.

27:30 I think that's one direction that it took that Gödel certainly didn't use. In fact, it's quite opposite from the direction he thought that's stronger. Essentially fulfilling the same role that the traditional logicians always thought of logic as the theory of being as such, a grand logic ontologically interpreted that was effectively also in ontology and that should be therefore thought of as being prior to mathematics, in other words first philosophy, that should be put underneath mathematics to provide its ultimate foundation. But from contemplation on... ...on the kind of ontological issues that were raised by the ontological claims that were implicit in these new axioms, we would arrive at a sufficiently definite conception of the universe of sex for the continuum hypothesis to be decided one way or the other. I think that was essentially Goebbels' vision, and I think that's absolutely the opposite from the way that maths has gone. I don't know if you'd agree, but it's a kind of quick and dirty... I don't want to over-simplify the count, but I think the contrast is right. I mean, so it is explained why it didn't, but what did he do instead? What did he girdle do instead? Went off in the direction, went off in the direction, went off increasingly thanks to, you know, functoriality and understanding of models as functors and more in the direction of algebraic geometry. I like all the best mathematics. Sorry, just B. We're beginning to, and we'll have more and more to, so I think as time goes on, you're a multi-judge, but if you want me to give an opinion, I think, yeah, my bet is that yes, it will, more and more, that the 21st century will be at least for quite, well, it will be part of at least one of the most important themes will be, well, one of the most important themes will be the working out of Grotendieck's ideas. Well, he had many more than what you had far more than one program. Let's say his influence, yeah. But sort of the very general, the sort of very general structure sort of played a little bit after the middle of the 1970s.

30:00 Yes, a little bit, but then, or perhaps I would say it took a different form. Well, I mean, generality is in the eye of the beholder, correct generality, of course, lies in common reality, but that's just me, my prejudice. In algebraic geometry itself, you could argue that there's actually been a kind of deflection away from that of Grosvenor Deacon. ...regarded as, you know, as very concrete theorems. Yes, I agree. That is the way that algebraic geometry is tended to go. And it's very striking in France. I'm not even sure that I really would think that in Germany. No, no, but it is an observation about what's happening in algebraic geometry. It's perfectly sound. And in fact, there was a very interesting meeting in Paris just last month, the beginning of last month, in which, you know, very eminent algebraic geometry. ...gave a talk about what he does, and it's all incredibly standard, sort of front and front indeed, very, very kind of concrete stuff, although I'm assuming, oh, I can't even begin to explain what's going on there, but it's because it's really hairy, but there is the, what the hell are they called? So it's going to come into my head in a moment what they're called. Hang on, Gorenstein rings, yes. Gorenstein rings and the kind of, and they use Gorenstein rings to derive these results on, for instance, on results about, you know, cyclotonic integrals and fields. They're very, well, relative to what Goethe was doing, very, sorry, to Goethe, but Goethe used to be pretty, very concrete. I mean, things which would live in real number fields. The kind of thing that Dirichlet and Kubler and Kroniker could have understood. Well, put it this way. If they saw, if they heard what these people were doing, they would at least have recognized the landscape. They would have said, oh yes, I mean, you know, they would have, they would have thought, I mean, there would have been a huge amount, obviously, of development on which they would have had to catch up and thought, but they would have recognized the landscape. They would have said, oh yeah, that's a theory about psychotropic integrals. I mean, I don't know what the...

32:30 I don't even know what a ring is, but they've got some kind of high-level algebraic machinery whereby they prove these results about number fields. But if you'd shown them what Grothendieck was doing in scheme theory in the 1960s, they wouldn't have had a clue what you were talking about. They must have said, is this mathematics? So yes, you're right, to that extent it has gone back in a more concrete direction, which I think is very interesting. Yes, I agree. I think a lot, you know, the extent of Grotendieck's influence obviously hinges to some considerable extent on the development, on the direction that the topos theory takes, both the, you know, the version of the original Grotendieck notion of topos and its role in algebraic geometry and the theory of generalized spaces, but also it's very important And also the logician's wing, which of course is what these guys are doing. Yes, yes, yes. No, in fact, this is one point that, well, that was one point that Bill was making in his talk, wasn't it? That there was always, there were always two quite distinct roles for the notion of topos in Rotenbeek's mind. One was the, one which is well mapped and well understood, which is as a generalized... Basically, essentially, the notion of generalized space that you need for results about, you know, spaces as carriers of homology and cohomology and functors that, you know, what kind of categories they land in. And then there was another aspect, which I frankly didn't quite understand because, you know, I never do understand Bill the first or even the third or fourth time around, which was this business about the Grotopos and its role in functional analysis. I'm so far from understanding it that I don't even know if you can point it out. Well, it's not a particular aetopos, it's a particular family of topos. Grotendieck drew a distinction between what are called gros and petitoposes.

35:00 Gros as in big and little. His early work in functional analysis, before he went off into algebraic geometry, is all of his early work was on nucleus spaces. Do you know anything about this work? It's got something to do with the behaviour of the classifying rings in these topos. It's a kind of classifying, isn't it? But certainly there are important links with functional analysis, which I'd love to try to understand a bit about. The topos theory that's going to be influential in the 21st century I'm sure is what you might call the broad and established stream in topos theory, the one that's been thoroughly explored by Peter Johnston and others because it's such an incredibly fertile idea and unifies so much. And then of course there's what you guys are doing to see if it can't be a suggestive framework for unification of physics as well. It's essentially a reformulation of things, which may contain a few very useful heuristic hints of how one might modify space-time concepts or attack quantum gravity, but only... we already have a plethora of heuristic suggestions as it is. It may just give a more controlled way about thinking of the world. This brings really to the forefront this question, Cartesian closed versus Schorn. Because there's no doubt that the modern categories are tailor-made for treating composite systems. I completely agree with Pope about that. Do not follow him in all the details. They say you need this and this and that. It's a structure because this is...

37:30 An axiomatization of Hilbert space, basically. Exactly. It's a re-axiomatization of Hilbert space, very much of the instrumentalist viewpoint on the formalism. So this is where, but the first thing I'm saying is that all categories are natural. When you want to seek opposite systems, that's true. I didn't mention this yesterday, but the spectral pre-sheet very probably also is the internal log-carb, not just the spectral carb. Sorry, which the spectral pre-sheet is also? Actually, I remember Petri... was it Petri? No, it was Isar Stubbe. I've heard a little bit about that, and I was saying, well, you know, you're going to go and talk to Andreas about that, because it sounds like a very interesting idea. I'm communicating with Pedro, and I completely asked him, and I said we could do this and that structure, and now I'm getting something wrong, but it looks as if we actually have an internal locale already now without having to find much extra structure there, which would be nice. Okay, they have two systems, and they have both this state object, which is an internal locale, and then you can take a product of this, which still is an ordinary product, but the product of two locales is given by what is called a tensor product of the underlying frames. Yes, but it's a very weird kind of tensor product. No, it's not bilinear. Yeah, it's not bilinear. Well, that is for topological spaces. If you have a measure on the product space, it need not be a product of the measures on the product spaces. So this seems like something we would need. And interestingly, and I also didn't mention this yesterday, every quantum state gives a measure on the sub-objects of the spectral frequency in a very straightforward way. And it has nice properties. That's a very nice result. I'm going to try and look at this product of two locales and look at the measures on this, and if they don't all come from, well, measure on the one times measure on the other, if not all of this form, then at this level I do have more measures, so more states, more the composite system. I don't say this will capture the whole of entanglement, but I have some hope that, despite the fact that I'm in teaching closed category, I can...

40:00 Well, you see, this I think is why Bill was so excited, because in fact, in spite of his boorish attack on you for completely extraneous and non-scientific reasons in the trade, he did in fact say to me after, I talked to him just before leaving, that... Have these guys seen the desionization as an adjoint functor? I think you have, in fact, already seen it, but not from a functor. So, yes, an adjoint functor is really neat. So he suddenly began to set up and take an interest. And then he immediately saw, well, look, this is a Cartesian close-up, which is why I've always said that quantum is a big cult. I've always believed that all these monoidal actions which are in the background... The quantum formulas are really naturally sandwiched between the levels of a bi-category. Of course, I know that a Cartesian bi-category is not a Cartesian closed category, but at any rate, it's a lot nearer to it than a symmetric bi-manual category. So he's immediately excited by that because obviously anything which is going to get the quantum story inside the framework of what he calls categories of space and quantity is going to appeal to his overall strategy and his strategic vision. So he really seemed to be much, I thought after I talked to him, much more sympathetic to what you were doing. And he was also very, he majored immediately on this point about the contravariant, that it's only in the contravariant construction that the factoriality seems to work right. And the one which, sorry, I can't think of his name, Lanzmann and Spitzers and co. are doing, the covariant one, is just not going to work. It's just not going to get it right. The maps are all going to be the wrong way round. He hit on that straight away. So, you know, if only he would, if only he would drop the, you know, the, the, the, the, The, the, the, the, the political blinders and, uh, and, uh...

42:30 ...because he obviously liked to see how people react and see if you're a mensch, but the very fact that he was taking such an interest in the construction shows that he, whereas of course he takes no interest at all in symmetric line andoidal stuff, I mean he obviously knows what Bob and Abramski do, now that may be because he has deliberately kept himself in ignorance. There's a lot of ignorance as to how rapidly the experimental evidence that vindicates entanglement has developed in the last 10, 15 years. I suspect he is seriously ignorant about the physics. That's the problem. Anybody who's looked at particularly the developments in quantum computation, I think, can't simply help themselves to some... A version of hidden variables a la the kind of thing that Mackey essentially was after. But there's still, independently, obviously, a great mathematical interest in seeing how far you can recover all of those aspects of the quantum formulas that you have to recover within the framework of, essentially, a Cartesian closed setting. And that's not the least of the reasons for... You know being very interested in what you and Chris are doing. What specifically were the other criticisms he made apart from the fact that you hadn't, that in fact you had, recognized that the Dasein and Daseinization is an adjoint functor? Pedro said that when we thought about it, what we actually have seen in this, because it's just not the functor between, well it's not within this topology. No, no, it's a functor between pulses. Yes, yes, yes. Thank you for your attention. There is another point where I see a certain confirmation that we are on the right track.

45:00 I could think about this in some detail because I'm writing a paper comparing the two things. It's a little delicate because it turns out basically the dust has not the physical interpretation. At least as far as I can see it, I won't write it quite that way. They say, can we approximate like we also do, which are operators in each context by, well, that's right, operators in the context, and they use the linear way. It turns out, somewhat surprisingly, in their scheme, which is slightly different, it doesn't matter, because I've got to look at this thing for the spectral order, so I look at this thing for the spectral order, and I really write down in detail what we get. In the end, you can't do this, but you can do this when you do quantum algebras. Everything is much more explicit. I can actually calculate what you get. It doesn't matter at all. You can think about the question, okay, if I use this new order, which is... Do I get something different? And the answer is, no, you don't get the same thing. This is one thing where you feel that you're on the right track. Well, you don't just need a post-set, you need that arbitrary joins and arbitrary meets that he finds. You need an actual complete lattice, or at least a bounded complete lattice, called self-explanatory. Which is really a kind of naturality condition. Yes, and you only get this if you use the spectral order, because then the self-explanatory is becoming bounded by a complete lattice. But you don't get it if you use the linear order. No, exactly. Which I think was the thing that Lorbier immediately hit on, because he said, well, this spectral... The spectral, the spectral is giving you some kind of a natural condition. Okay, okay, good that you picked that up because that's the same conclusion I arrived at when I'm thinking about this question. And that's why your, the contravariant approach is much likely to be on the right track than their covariant version. This actually is not dependent on contravariant and covariant but again in the covariant, the covariant approach many, many things just work. The basic point is they don't have coarse gratings. And this just means that basically many things become trivial. Cross-grounding is really an important concept in what we're doing.

47:30 Really for the physical implementation I think it's important. Which must be related to the behavior of coverings in the topos. It must be related to the behavior of the coverings in the exoskeletons. That's where I'd like to see, that's where I'd like to understand the details. This integration theory, which again becomes much simpler when you think about phenomenal algebras, because after all, if you have projections, it's like cutting out a piece of Hilbert space and it should have some size, but if you restrict yourself to having no projections, or not knowing if there are projections, then you have to do much, much more, and it's just not that nice. As a result, every quantum state gives you what I would call a measure on these sub-objects, in particular on these sub-objects which I call clopen, where you can pick a clopen subset of the quantum spectrum at each stage, which is just a response to picking a projection, because clopen subsets of the quantum spectrum are this projection in the abelian algebra. Anyway, so you get such a measure, but interestingly, you can yield to temptation, Basil. Thank you very much indeed, Roger. Go on, you have it. All right, I will have one spoon full. Well I wish I had this moral strength but unfortunately I have to say you look extremely trim to me, very trim. I bet your waistline is exactly the same as it was when you made that recording of you. I do. I was wondering whether you wanted to listen to it. No, I can't actually detect your waistline from the sound recording, I'm afraid. Technology may advance to that point, but right now... Putting it through SIDA. Oh yes, after that we'll even know what your waist measurement was after we put it through SIDA. There's this amazing software that we now use called SIDA.

50:00 ... computer-enhanced digital audio restoration of tape recordings. It's absolutely extraordinary what it does. What do you mean, it makes you look fatter? Yes, it does. It can't actually correct any of the mistakes in the math. But it does clean up the sound so that you can hear... I mean, for instance, if there are several people talking over each other, I'm going to do it within limits, but unless the background is about... Four times the amplitude of the voice that you're picking out. It will successfully take out, digitize that voice, and then you can listen to it, and it's just as if it was just that one person talking. It's probably worse than correcting your mistakes. It means those mistakes, you know, when you're not quite sure, you mumble a bit. Yes, yes, you mumble a bit. It removes all the muffler, just as the dumb idea that you're actually coming out with under your breath. Oh dear, I'd better not say any more, otherwise I think you're going to back out on this archive. I don't know, I think it's interesting. Well, so do I, otherwise I wouldn't be doing it all these years, but let's listen to... No, hang on, have I got the right... I was part of a film in which he was... Thank you for watching this video. So, virtual, virtual Bohr. Yeah, virtual Bohr, that's right. They haven't actually succeeded in turning him into a hologram. No, no, no. It was done by a Danish people. Right, I hope this is going to work. If not, I've got another one. Well, that's too slow. Wait a minute. It probably was a Danish book. I mean, alright, we'll try this one.