Discussions, incl. M Wright, G Khatcherian FW Lawvere, A Kock (contd.)

0:00 So that the strange-looking axioms come about and you describe that in terms of the indicators instead of in terms of the actual sub-objects. Yes, so the non-assessitivity is not really, as I say, is captured quite trivially by... This pushing and pulling is functorial, i.e. associated with whatever. There's only this strange way of re-interpreting it that gives it... The lack of functoriality in the first variable, which you have in the case of Gerritsen. Should we just... I think I know a get-come. Yes, that's probably a good idea, we'll walk and get them. Yeah, it's okay, we'll probably see them. Oh, I see, you actually get them from here, it's self-service, how silly. Oh, I see, no wonder. Yeah, I didn't realize it was self-service, you can probably get them scooped in a can. This is where the service is around here. He's got a huge bottle. Yeah, I have one. That's a good idea. Excellent thing. Yeah, what's the matter? Yeah, I know, we're just going to buy them there and bring them over here. They'll let us do that, won't they? I mean, we'll help you out. What's he saying? He's not saying no to one service, is he? He said it's not self-service, it's a waiter.

2:30 Oh, okay, well, fine. Ah, Billy speaks Italian. Yeah, he speaks well. He speaks very good Italian. I don't think you're dealing with an intranetiquette here. I think you just go and get served. I asked for a big bottle of mineral water because we don't sell it. We just have a beer or something instead. I don't know, just have a Coke or something. Not that they don't have it. They don't sell water that way. Well, we have to sit down. I think, to be honest, the problem with Como is that it is a typical tourist trap, and the attitudes you get are the attitudes of people in a place that has been spoiled by mass tourism. Well, there's always going to be another dumb tourist along, and they don't have to try very hard. And we certainly found that with the hotels. I think in South Europe people are not so busy, so just sit down and wait, the waiter will come. We're not busy, but we're thirsty. Yes, but everything takes its time. I can handle that. Is this your first time in Canada? Or have you been here before? Yes, this is my first time. There's another villa in Verona. I used to be here. That's also owned by the University of Milan, I guess. But anyway, that's, as I say, I now begin to understand that construction, what the indicator maps are doing. I hadn't fully grasped it from reading the letter, but now I understand what there is.

5:00 Categories with endo-map, well, categories with, with every map, another map goes back, a star, and then you add another one, and then it goes back to another category. Well, no doubt. I don't know where they are exactly. So, well, for many, for many reasons, the category of public spaces, I mean, to take even a concrete example, The category of Hilbert spaces is a much more manageable object than the algebra and the mathematics of a single figure space. You can, in fact, encode everything into a ladder. In fact, in fact, the ladder is equivalent to a form. And yet, if you lay it all out, then you can see that part of what's going on is simple things like Kodak and other things that have to do with the star and so on. So, a star algebra. There's a star category with one object. There's a star category with many objects, which we have external, but at the same time, somehow, at least from my point of view, much more understandable. Because we don't get involved in all these things of coding external stuff in an ad hoc way, internally, and then trying to express it in a formula, then we get these strange things. Not associatively, but in a way. There are several examples of this. There are two by two matrices and a star alphabet, another star alphabet.

7:30 And if the thing is infinite dimensional to start with, then the second can be embedded in the first. But the second embedding is an ad hoc choice. So, on the one hand, it's good to know it's there, but on the other hand, it's also good to have the actual direct sum of the few other stations and the other options, the few that you use. What was the point you were making earlier about the suggestion? I mean, it's not directly related to what you've been saying about the... The way the endo maps work in that category and the way that you capture the non-associativity involved to illuminate what the reasons were. But earlier on you were saying something about this suggestion, I think again in the context of endo maps in the appropriate category, that instead of having a fixed truth value object you might have some kind of a sequence. You were even suggesting it might be motivated operationally in terms of... No, no, it was different, I realize. It's a completely different thing, but I wanted to ask you about that as well. Just consider, now forgetting the category, that category, the lattice of all substations is Hilbert space. Alright, so now, as Jerry has pointed out very clearly, that particular lattice has the following kind of structure to take any given element of it. Then you have this Sotaki arrow, and this Sotaki whatever you want to call it, which is an adjoint pair of endomaps of this lattice considered as a process. Now, the class of all adjoint pairs of endomaps is closed under composition, it's a monoid. Sorry, say again. It's closed under composition as a monoid. Yeah, yeah, yeah. Just couldn't hear a monoid.

10:00 However, if you, if completely, you take two, or three or four, let's say two, adjoint endo-pairs, which are determined by single elements in this way, and form their composition, it's definitely another adjoint pair, but it probably is not induced by a single element. So therefore it seems perhaps it would be natural to consider a larger monoid, maybe to start all adjoint pairs. The case where it is induced by a single element is where you've got the omega is. There's no omega in this. Sorry, my misunderstanding in that case. Except in so far as... Yeah, it's a kind of omega. What sort of monoid can have its identity? That's right. It's another particular kind of monoid. So, buona sera. Buona sera. Speakers include geography, mathematics, mathematics, mathematics, mathematics, mathematics, mathematics, mathematics, mathematics, mathematics, mathematics, mathematics, mathematics, mathematics, mathematics, mathematics, mathematics, mathematics, mathematics, mathematics, mathematics, mathematics, mathematics, mathematics, mathematics, mathematics, mathematics, mathematics, mathematics, mathematics, mathematics, mathematics, mathematics, mathematics, mathematics, mathematics, mathematics, Let's have one each. Ten glasses. Four. Oh, four. Two each, I see. Two each. Okay. Eight. Eight. Otto. Otto. Otto. Otto. Otto. Otto. Otto. Otto. Otto. Otto. Otto. Otto. Otto. Otto. Otto. Otto. Otto. I hope I owe all of ice cream. I don't care what kind.

12:30 We'll soon find out. We'll soon find out. If you didn't, you could have some of mine. I like pistachio. I'm wondering what the net result of eight glasses of mineral water will be. In this heat, I don't think very much. I don't think you'll sweat it all out before you... He wants to call it out. Fine. To conclude this story, the composition of Hiddensburg was made after Hiddensburg. Yeah. But still, as I vaguely understand the operational significance of all these things, the composites of the sequence should be considered, not only for the elements, but to do one thing after another. Does that make any sense? And it makes sense to just come to this degree table. What's further from this is monoids. These monoids are going to play a piece. And then you can recreate the lattice zone. Yes, it could be the lattice is just the lattice zone. Basically, remember, it's only... I hope you're not boring. It's tied up with the order, with the product, just as this one, the order relation is tied up with the product, so we have this idea of this talking, this sort of a jointness, but there are others. Specifically, the composition of the theory, it would not be of the form, I've gone through this before, this is not of the form of this, but why restrict each other to these, I mean, here you have associations and everything, the only difference seems to be the long-term institutions, that certainly is not the same as that.

15:00 What sort of monoid? I mean, you can talk of an abstract monoid, which the identities you have to be... Oh, okay. No, I'm okay, I'll have a kiwi frizz, I'm quite happy. I'm not a connoisseur of ice cream, it sounds all right to me. I heard he was coming out to tell us that the mineral water and the ice cream was finished all together. Chocolate and vanilla for me. Good for you. There wasn't. But it is interesting how, naturally, when you realize that you are, as it were, in a category and sandwiched in between these two other categories, you recover what appears to be the deep mysteries of quantum formulas and really reappear financially and intelligently. I don't know, I mean I never saw the quantum logic as anything more than it is in science. I'm afraid I made a mistake, a great deal of a mistake, and I respect that. Something people were doing, I said, from the start, let's do it a little bit. But maybe, I don't know, in a way it's a long list. There's a good look at this. I don't think we're going to get satisfaction in quantum theory out of quantum law. No, there I think we are all solidly agreeing.

17:30 I think one should try to forget the formalism and go back to the physics. Not because the formalism is wrong, but not the right way. This is very good. So not to... Not to doubt. Your good health. Cheers. We can taste one of them in the middle of the water. Slight lemon touch. That's the main point. But they don't give it out of the... I was rather dismayed to be told by, I'm not sure who I'm talking about. Well, yes, that's right, it was Richard Wood, who'd just come from a logic conference in Scotland, I think it was, or was it, well, whatever, but apparently over half the people, half the people there were still doing large kernels, really, but not sure, and other naughty things, but large kernels, thank you, bye.

20:00 I shall be doing small cardinal theory. By small, I mean something less than equal to one. Ah! That is the way to think of it. Miniscule theory. Yeah. Section of one. In a sense, the engineer knew this all the time. Yeah, that's very nice. Very... very nice, yeah. Very good. What can you do to it? You need one of these, don't you? Yes, you do, though. Sorry, I thought you already had one. It's a good idea to put it in cold water. You mentioned the menthe at the beginning. I must admit, that's one Italian delicacy I've never been able to develop a taste for. The first time I ever tried it was when I was here when I was a kid. It just tasted like a liquid toothpaste to me. I must admit it still does. It's just like an acquired taste. This is kiwi-proof. Oh, have some, please. It's quite pleasant. Yeah, it's pleasant. It's a nice way to finish an evening. Mine is a classic. I lived through this book the other day. Egyptian mathematics. All fractions power of two denominations, I think. Power of two denominations. Numerator always one.

22:30 Something like that. 50 of them. Egyptian of antiquity, I suppose. Yes, yes. But only. They made an exception in the case of one-third. They accept that. What can I say? You haven't read that book by David Fowler, have you? The Mathematics of Plato's Academy. No, that's mathematics. The Mathematics of Plato's Academy. It's actually trying to reconstruct from an apological evidence exactly what the mathematics that plays a part in it is. That's the man who had tried it. He took the wrong position about a year ago. Mabry? John Mabry. Oh, Mabry, yeah. That's right. John Mabry. Nice guy. Very nice. Would certainly want me to pass on anything else. We'd certainly want to pass on his regards. Oh yes, you've spoken to him. Oh yes, I actually went down to skate with him in Bristol about four months ago for about three, four days. He actually helped him move house, which was a very good start. So I was wondering, did he do as he said he would and give a course on topos here? That I can't tell you because this was only about four months after the meeting. He certainly changed his ideas, I think, fairly radically, since he's speaking English. He once described, and I take this in the spirit of what you've been getting at, he once described, listening to you, I think it was when you were expounding some of your ideas on physics,

25:00 particularly some of the ideas that you and Steve Shannon worked on, he said he had the sensation of travelling very, very fast speed indeed, in an opera mea, down. Very occasionally catching glimpses through the railings of this fantastic landscape on the other side. Just catching split-second glimpses of the landscape and just thinking how nice it would be if you could only have moved along at a nice leisurely pace on foot or on horseback but there was an awful lot of very concentrated ideas there. Did you ever get around to reading that paper of his, which he has? No, no, the one on, the one in which he, it's quite an old paper, the one from about 10 years ago when he, his attack on category theory. I just accepted your place there, I just had quite a bit of your place. No, you know, she'd take it, did I, did you not take one away with you? Yeah, I've, um... I've been toying with the idea of Roger replying to that, but, you know, there we go. Well, it would be much nicer if he tried to reply himself. Ah, yes, indeed. Well, I mean, not that you shouldn't, but in terms of... No, I mean, I... If Roger replied, it would be better to reply to himself. What would your reply say? Ah, well... Well, nothing very original. It would make the point about, it would make the point about Mengan and Cardinale. I'd also want to argue that one has to think much more deeply about extensionality and his assumption that the notion of a collection and extension is simply an irreducible notion that there is nothing, that there is no...

27:30 The geometrical topological ingredient already built into that notion, which seems to be his position, that it is a completely reducible notion, provides the starting point for any analysis of collection mapping. Collectional mapping is a false one, although there's a great deal of geometrical structure already built into the notion. The toposetting, which is one reason why I wanted to talk a little about the various forms that it takes, the various versions of well-pointedness and the other variations that it takes. And particularly the topological ramifications of extensionality, that it reduces to relative uniform separability, which is actually topologically quite a strong condition. So, one has to say something about the space that you're in, before you, and I suppose I also want to say that our naive cognitive representation of space involves, actually does involve, a lot of assumptions about the mappings of space, to assume that all of those things, you know, the direction of fit of those elements follows after you've already got. A notion of collection and extension given and in place is a very strong assumption. It's one that he makes without ever questioning it, and I think it should be questioned. It would be a confused philosopher's reply, because I'm not a trained mathematician. I mean, he seemed to make very heavy weather of the... I think he's, I think really he's, although he denies this, that he's really just a platonist under the skin. He just thinks that the notion of set is absolutely clear enough.

30:00 It doesn't require any further analysis and it doesn't involve any intuitive components. This is Mabry's position. It seems to me that it involves very strong intuitive components. The point that you make in your paper on identical particles in a specific category. The assumption that one can calculate with these mappings in this, you know, you're already making a strong assumption about the category that you're in. Well, I suppose I'd also want to look at the point again. Where does this notion of mathematics come from? No, mine was quite just directed to him in the context of his paper. Where does this notion, the notion that he advises, the only possible notion of mathematics come from? Find a fluid plane, you have a map with you, drive you to get there, and you'll find a fluid plane. A hot plane. Yeah. The potential, yeah.

32:30 No, in fact, I mean, Joey does actually have a position on this. It's quite, I mean, I think it's one that I'm inclined to go along with, but it actually comes from the public representation of motion. The assumption that things retain their identity when they go from here to there. In other words, there is already a strong spatial component, a strong component of... There is a cognitive representation of space already built in for the way we think about the category of sets. Geometry and set theory are connected at that deep level. Is that not your...? Are we very wicked people to think like this, or are we just very confused? I think we're allowed to. Are we allowed to, provided it's very late in the evening and that we're in good shape? I think we're in good shape. He took the bullets out, of course. Sensible man. He slipped the bullets out of the sphere. Make sure no mischief was coming to you or anybody else. He's a far-sighted man. What can it do?

35:00 It can work. Lights. Lights. They're lights. They don't shoot, do they? No, they don't do lights. Oh, they don't light it. No, it's not the... I think it's run out. No, I don't know. I can't use it today. You can't use it. I think I counted the number of times you used it this morning. I think I've run out. I think my solicitors will have to write today. They're asking for damages to the record. My solicitors have been punched. It works every time when it doesn't. Are you thinking of bringing him back to England for a holiday, Santander? Yes, I have thought about it. I wish you would. I wish you could stay with him. He's really cute. Yeah, he sure is. He's a very good student. Oh, it could be. It's a large team. It's very civilised. That's really too civilised. No, I was going to say, not really. I think in fact it's absolutely no useful function, whatever. I think somehow they soft out the mathematicians, that's why they keep it. It's a point. You're actually muted. Can you feel maths? Can you feel maths?

37:30 What is physics? It's not a culture, it's not a thing to involve really, it's not a concept but a reality. There's a four-digitized space, Rn, so there's Rn and then there's Rn star, which seems to be isomorphic to it because it's in the form of Rn, but of course you can multiply that isomorphism by constant h and it would still be isomorphic, but it's really the duality. It's a crazy number. In a way, it's not a number, but a quadratic form, which is needed to express certain relationships and results in theory analysis, you know, of, you know, electronics, of course, of those ways, you know, studied ways to lead a duality. It's lovely thinking because it's one of the ways that we can tell each other what we expect from each other. You can hear them correctly. That's why I said, half jokingly, it was really called for in matter, which is somehow matter, which enables one to put the intent to the next stage, which is direct, which is therapy, which is apparently, yes, opposition, which is therapy, which is faith, as such. There are certain measures that you can push forward and then pull back along the sides of the morpheus and come back to one. That's one of the actual specifications that we have a concept in. It only means that the original duality was not the one that was given by these coordinators. You do need a duality, or otherwise you could use a specialization.

40:00 I put it very crudely here, but this seems to be the sort of idea that was leading to Sater's concoction. But that's a reflection of it. We actually ordered eight, didn't we? Oh, well, maybe that's it. Maybe this is the single. Ah, that might... Well, no, this was what they brought the spoons in. Well, no, because we said large. We said grosser. We said two each. Thank you for your attention. Thank you very much for your attention.

42:30 I did a lot of bicycling on what you call a tandem, double. Oh yes, tandem, yes. I've ridden one of those too. In fact, long distance bicycling with him and my wife. It's not so difficult when you've actually got them one behind the other, but which you say, when you're actually sitting side by side. And there's no balance. My daughter bought this double bicycle for her and her husband, yes. The use of it requires a lot of practice in the city, because you have to make quick decisions and agree on what decisions to make in relation to this. So it's not the front, but the acceleration. When do we go and get before or after that car? The front one is the only one that can brake, or the two can brake. The back one can also brake. Not as bad as it would be if the people pedaling in the car side by side could both brake. That could leave an awful lot of problems. Then you really would have a... He's been wanting to use the brake while I'm driving. I was thinking about the people in the pedal car, but, yes, you're right, huh? Oh, yes, if the passenger had had a foot brake in our car, we would have skewed off the road on more than one occasion. Thank you for your attention.

45:00 One thing is that the pedals are slightly out of phase, meaning that it's a very even way of moving forward, just 15 or 10 degrees, but it's much smoother sometimes. Yes, oh, Thomas, McLean, and H.O.L. I think, or something about grades and consumptives, some consumptives. Could we have a look? I'm afraid we never did get that. But then we can have it out of registration. I never saw one of these. Paulus MacLennan, coherent hexagonal droids, and quantum field theory. Oh, that's right. Is he? Well, that's why he was so interested when he was asked. I was talking to him about Redhead's intellectual book. Redhead's intellectual book was an interpretative introduction to quantum field theory. Which, he was asking about it. We started at 9.5, after coming to a cross-street, quantum groups and hops, half-boiled eggs and hops.

47:30 And that's why I would say that geometry and algebra are the key components. That's what I would like to hear. Then in the afternoon we have a late-ish lunch. You see we were supposed to meet up, yes, we were supposed to pick him up at one o'clock. When, when, when the, ah, yes, that's Italian style. Oh, yes, yes, four o'clock, four o'clock. It's a pity we should have told him to come at two, we wouldn't have had to miss the last session. Yeah, yeah, and chairman is just yourself. I think I know something of what we're going to say about that. Yes, exactly. Boundaries of logical operation. You have a copy of the thing. Very interesting. One of his films marks five concepts which would be very powerful in terms of categories as five-line concepts.

50:00 In any such category, you have two objects, x and y, which is from every object t, the number of them, x and y, and x is nice and working for us. Just the number is the same, yes, just the number. That belongs to combinatorics, I guess. Here's the resistors around the six foot block, I think. Well, that's okay, we'll get one. Also, that's fine. We got to think about going in some sense. So when is that? Oh yes, Richard Wood, of course. Now, I'm going to circle me out of this thing.

52:30 Talk about a copy-braker. No, sorry. It's the doer of the national theme. Copy. Do and break. It's made of fusion computing. So, can I just see the, uh, what term... It's the first time that's been done. We have had a category meeting on the size of 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 62, 63, 62, 63, 62, 62, 63, 62, 62, 63, 62, 62, 63, 62, 62, 63, 62, 62, 63, 62, 62, 63, 62, 63, 62, 62, 63, 62, 63, 62, 63, 62, 63, 62, 63, 62, 63, 62, 63, 62 You had a bank, didn't you? What's the name of the Indian guy? Sorry? What's the name of the Indian guy? What Indian guy? The guy who asks the... The crank, you mean. Thank you for your attention. Yes, he's a sarcoid. He's actually a biologist who got interested in, uh, in, uh, in, uh, in, uh, in, uh, in, uh, in, uh, in, uh, in, uh, in, uh,