Dinner conversations

0:00 He's a Christian Jew who makes a fetish of making Utrecht really snide and ugly, and also having things that are out of control. You know, they really accompany him. They really make me think that he's actually pretty disgusting. I used to know that she was his ex-wife. I didn't speak about this on other occasions.

2:30 How do we figure out the... Not for your wife. Two dimensional cells.

5:00 Yes. Independent. What's your name? Mike Wright. We did talk yesterday, didn't we? You also talked about some of these ideas about rescuing. No, I wasn't talking about rescuing. No, you were talking about... Well, maybe about...

7:30 That was in a different context. Oh, yes. That was a nice meeting, not least because Angus McIntyre gave a lovely talk. That was the first meeting I met Angus McIntyre. It was great, very enjoyable. We had a long and very interesting talk. I spent a whole day walking up to the doggoneck from the back and all the time Angus Macintyre was telling me about, well, about many things, but especially, actually, about change of policy. It was a very exhilarating meeting. They've had a couple of those meetings since, I think, happened though. Have you, you haven't been back there again? No, no. I thought it was an absolutely beautiful little place. Yeah, it was, it's really nice. Actually, after listening to Steve's talk today, I, I think I had more of an idea of what you meant yesterday. Okay. About using machine theory stuff. I mean, still, still, I don't... Steve certainly explained it much more clearly than I would have done. I don't, I... I still have only the vaguest conception of what that is, but I do think that there's, I mean, it does look as if there would be a way of viewing this in terms of... There are a number of three-valued models assigned to each point. Topological space and using the topological structure to define the conditioning at each point. And so I thought that that might be the kind of thing that we could do. And so what I actually used would be a very special technology, but it would be interesting to... I would like to look at this more generally and see what kinds of conditions you have. Absolutely. I'm very happy with that. I'm glad you remember that ceremony. It was very enjoyable.

10:00 I want to know more about you. More detail on this. It is going to be funny. It doesn't matter that there's a specific object. I'm trying to compute it. It might even be infinite. I don't know. But it's an adjoint program. You know, there's a standard, a very standard culture that goes from the so-called second order differential equations into first order. You've changed the original space to the tangent level, so you get a first order equation on the tangent level. At the level of first order equations, it's easy to see what the time should be. It's a one-dimensional space in which you have an infinitesimal translation of the usual source. But then we want to shift. But we want to represent those motions back in the second order category. So we apply this left-hand... Oh, there's a definite object there. So that the map... It's something that carries the second order equation, but for any infinite dimensional system of the second order, like Maxwell's equation or something like this, any motion is just given by a morphism, a lawful motion from this thing, and that's what I call the higher dimension of time, but it's not, in a certain way, it's nothing exotic, but it's just classical physics. It's just seeing the classical physics more more systematically. Of course, it may have some implications for non-classical physics too, but at the moment I'm just talking about it. Change of the shape. Change of the shape is rather a very big, another kind of meaning for motion. Because we always think that this is the element of the shape. It's some kind of way, somehow to move like one thing.

12:30 So at the time there... The whole of the shape-building chain, the more that we can understand this whole shape, the whole shape, it is very difficult. And I feel that usually the higher we build, the higher the difference in relativity will be. How do we get the shape in the physics itself? So the continuity of the... In addition, some of the core of the English language is connectedness. One connectedness to two connectedness. These are very high levels of thinking. In physics, there is no such a... Sorry, by shape you mean what kind of connectivity of the components and space, or...? No, no, no, it's only a... Homotopic? No, homotopic. No, homotopic. No, homotopic. No, homotopic. No, homotopic. No, homotopic. No, homotopic. No, homotopic. No, homotopic. No, homotopic. No, homotopic. No, homotopic. No, homotopic. No, homotopic. No, homotopic. No, homotopic. No, homotopic. No, homotopic. No, homotopic. We don't know how well this connection to the one-dimensional, the neighborhood of the one-dimensional, is very mysterious in the classical physics. The one-dimensional? One-dimensional. Yes, I have a completely logical definition of what is one-dimensional. Yeah. It is... Oh yeah. Because... There are many kinds of space categories. Well, there are many subcategories, and subcategories of a certain kind, namely reflected, left-exact reflected subcategories. There may be many, but the idea is that these should be thought of as dimensions. They may not be natural numbers, they may not be real numbers, but it's a certain lattice. All of which, for this category, should be the possible. So space has such and such a dimension if it belongs to that level of the subcategory.

15:00 And in certain examples, these will be natural numbers. They will just be as many as there are other examples of real numbers. But just take any one. You see, what typically happens is that... But there is a kind of jump operator. So one should be the jump of zero. Now what does that mean? It means that, you see when I say it's a reflective subcategory, it means given any space, I can sort of extract the n-dimensional skeleton, where n is not necessarily a natural number, it is just the name of this whole sub-category, but I can extract that skeleton. On the other hand, I can always compute the... The number of components of a space, well you see if you take the zero dimensional skeleton, of course it has a huge number of components because you've exploded all the connectivities. One dimension should be the lowest level such that the skeleton at that level has the same number of components as the space itself, whatever it was. So that's the smallest level, so it's a completely qualitative, depending on which category you're in, tailored to that category, what one dimension is secure. So this is one of the bases of the how do we think about the real, the sum of the practical objects, perhaps, I think, with the same and our operation and the objectives. You say, it is science. It is a very good question. I think that the most impressive thing about, for example, Brownian motion, and in some of the, this is my, I'm not an expert about quantum mechanics, but one of my friends, he is a physicist, he studied that.

17:30 The empty in the sabbatical has two dimensions. This is a very interesting representation of quantum mechanics. Oh, you're talking about some kind of sub-quantum medium analogous to Brian and Moshe, right? And so what was it that you said to generate to... I'm sorry, I didn't quite catch... Because this is the best of the idea of the problem. From the motion, the totality of the error possible, error possible, to the plus, in the final, in the final, do that each, trascurly, we must have a final, a holistic integral, and only a one-part, other than the principle of maximality. But in quantum mechanics, there is some interaction between the different parts of the method. There is some passage. So, he extends this idea to the sub-quantum. In the sub-quantum, more processes represent the patient from the brain mostly, and another to its class. And he says that there has to be a condition to have the, to have the, to have the, to have the, to have the, to have the, to have the, to have the, to have the, to have the, to have the, to have the, to have the, to have the, to have the, to have the, to have the, to have the, to have the, to have the, to have the, to have the, This is very good for Hilbert's thing, how do we fix it? In contrast, I told Michael that there are two notions of time. One is the inner time that we explicitly write down that it has a parameter key.

20:00 We also have the interaction of the experiments. The connection of the experiments is very difficult and it is, let's say, the origin of the problem of observation. It is deeply connected to our understanding of time in the usual way and the quantum field of conduction that, at the time, I've been reading. How do we, how the two systems contact is a very difficult problem. This is, this is a, this is a sort of a commotion of shape. So perhaps the basic type is the 2D model, but as you know that in other cities, it is a 2D model, it is a very 2D model for traditional speakers. But perhaps it can be extended to many different levels. And perhaps all of this is isolated at the same time, like some notions to... The project is to make possible the new undefined system. This is the outside map of my laboratory. So this is the end of the lecture. The end of the lecture. The fiction of the prediction is not determined. It is very important to understand these things. The end must be the goal. This is the end. This is the end. Thank you. Thank you, Albert. Thank you, Albert. I have a paper which will appear soon in the Journal of Spear and Flight Algebra. I apologize. Categorical Algebra for Continuum Mathematical Physics. Well, maybe for me, what can I tell you?

22:30 Well, in the context of the form of the committee, you're right. In the context of the form of the committee, you're right. In the context of the form of the committee, you're right. In the context of the form of the committee, you're right. In the context of the form of the committee, you're right. In the context of the form of the committee, you're right. When is that paper coming out, Bill? It's continuing physics. Well, I mean, it's true. I really like the study of that. Do you have any interpretation of any of those? It's a special issue, and in honor of Max Kelly's 70th birthday. I'm honest right now. I'll have to give you a pen now to see if you can get that back around on the next day. What's the trouble, is that all the university liners are going to take that again before they see you? Yeah, I can send you that. I didn't already. I did not send you one yet. Not with that particular thing. I was just doing the last corrections to the proofs. Thank you for your attention. Just kind of expounding on them. Once you get into non-autonomic systems. Yeah. Yes, this is a very interesting name, because salt is my, my shu name is, shu means salt, and my given name is Ken, it is a Chinese character, it is a meaning, it is some kind of word, but in Japanese, the category, we translate it in the... Same pronunciation, but different Chinese characters, so I tried to use this as a caveat. That's a smart idea.

25:00 Very nice idea. Sorry, sorry. Thank you very much for your attention. Come back, he is the teacher of my teacher at the seminar. He bribed me, he said, why do you do the cataclysmic, the cataclysmic? In the United States, there is no cataclysmic. He started laughing at me. The field was out of the way. But the professor, of course, did not know the cataclysmic, but how to make a great... The progress after the half of the 80s, you didn't know that? Yeah, yeah, yeah. I have some climate reasons, so I changed my language a little bit. Because of the intersection of categories. A couple of people. Very interesting indeed. Very interesting. I'm very sorry, I was only able to follow a little. I could expect that you all had some background in chemistry. He was the teacher of your teacher. So in the future of Mariko Yasugi, for example, she's a very nice lady, by the way. You know, for you it matters. I'm aware of it. Everywhere. Considered scientists. And some of the semantics of the computer language, they're usually written down by other people.

27:30 There is a Japanese mathematician, well, a Japanese mathematical student, who was... His background is in Algebras. He spent the last two years at the learning campus there. He's now a member of this couple of high-leaders in the theoretical physics team at Berkeley. His name is Rio. I see him on the third day. I don't know his first name. His second name, I think, is... But he left after a great amount of coming to London, but I don't know if that's a problem with him, that's a problem with him. Well, I know that a few years ago, he was really good. But I feel that Girard's intuition is more likely to you, I feel. Well, he used some category theory. So, actually, he was good. Oh, he was good. He was good indeed. Giroux also spent some time back in Halifax when he developed the topos theory. And the checkers think that in some of the 1970 publications about proof theory... So Gerard, Mark, and Bert, these people, they refer to the fact that what they did first was wrong, and they had to correct it because of category theory. So they were all influenced by category theory from a very early stage, but then they disguised it by a different notation, a different terminology, and so on. There was rambling talk where his main message was, everything you did before was wrong. All of his hard-working followers should simply give up and start again.

30:00 Oh, I didn't know that he was following string theory for this category. Is that such an avatar? He's very concerned to have his own sect, his own religious doctrine, so that they will all use that language and not be influenced by mathematics, for example, or philosophy, or reality, or anything else. I wanted to define what is a non-autonomous. Given that you know what is an autonomous system, they form a topos that is so amazingly great that they could be so exactly having you back here. In a typical way, you take a slice, so-called, and you consider time as such, a dynamical system, and you consider another dynamical system with an equivariant morphism to that, so this is in effect a non-autonomous thing, because the law of motion depends on time, but now, okay, well, if you can do that... You could take something more complicated as the base and apply it to just a line. It could be any other object. And so I started to think about how this would be.

32:30 Oh, and I proved a little lemma about Hamiltonian systems, that Hamiltonian... You cannot have a Hamiltonian system like that, in other words, non-trivial. If you have a Hamiltonian system, an equal variant morphism to a lower dimensional Hamiltonian system, then in fact the law is independent of both directions. In other words, if one geometric projection is equal variant, so is the other one. You can't really have a system divided into two parts where one depends on one more than the other one depends on the other one. It must be sort of completely interdependent, which of course is not a bad paradigm for many situations. On the other hand... All of technology in ordinary life is based on the idea that, well, we have this system here, this square, this table, this Lorraine, whatever it may be, and the rest of the world, while it exists, is only a minor influence in our day-to-day. So my idea is that that's what time really means. It is the rest of the world that influences us. And I bolster this in my usual style with Indo-European languages, you see, because first of all, tempo in Italian means weather. Indeed, as it also does in French, of course, Tom. Yeah, so both in French and in Italian, you see the same word is used for time and for weather. So the weather is that which surrounds our system. It influences us, but we don't influence it. This is a common saying, you can't do anything about the weather. Well, that's because it's having that, so in other words... I think that what we call the one-dimensional time is really just sort of a dialectical negation of this generality and saying, well, we can somehow sum up all that into one parameter as a second approximation. The first approximation is it doesn't matter at all. We have a closed system, deterministic system.

35:00 The next approximation is that everything that happens outside that is just one-dimensional. There are many others. For example, the word, the Danish word for time is tide. Well, it's TDAG, but it's the same word as Old English for tide, you see. So, again, the tide of the sea comes in and goes out. It's a huge thing going on, a massive thing. It influences us. We can't do anything about it. This is King Canute, you know. Absolutely. Time and tide wait for no man. Absolutely. I have to say, I do think David Boer would have done it. I love this. Then I go on. Why is the newspaper called The Times? It's telling us all this stuff that's going on around us that for the most part we have no control over. But it's just proceeding along. And anyway, the word for times in German is Zeitschrift. So it's as long as stuff, you see. So I have a whole rambling argument of this sort. Grasping straws from Indo-European languages and ordinary experience and so forth. But really that's what time is. Why is time one-dimensional? Well, it's because that's the first approximation. Why is heat one-dimensionally temperature or entropy? Of course, it's much more complicated than that. It's again a matter of the temperature and the entropy are about... ... about the problem of, we have our system, there's the outer system, but instead of considering equal footing, which would be incredibly complicated, we sum up everything that's happening outside as just one function of one variable, entropy as a function of capricorn. And again, it shows why entropy and time are related, because... They both are about this basic problem of giving a minimal recognition of the external world, which may be necessary, but of course we can never hope to account for everything, so we make a one-dimensional account.

37:30 Because they have the same knowledge, the only difference is the output of our people which are processing and it's cut off the different dimensions of the two thinking to concentrate the data at the movement. No point is a partial derivative to you. Hello. Expresso? No, no, no, no. Please keep it out there for a moment, please. Au moins deux cognacs. Sept-cinq ans. Excusez-moi, deux cognacs et aux frappés, s'il vous plaît. Sept-cinq ans. Or, actuellement, cinq-cinq ans ici. Thank you very much. So, sorry, in the case of Hamiltonian, there's only one way to talk about this very thing. So, okay, we have, we imagine a system divided into two parts. So there's a Hamiltonian function of q's and then p's and then more q's and more p's. I can just say q1 and q2, q1 and q2 for sure. So the idea is that we have a product space in front of the space to project onto one factor. We assume that if we project onto the second factor, that that will be equal variance. To say that that's equal variance means that the law of motion below is independent of the values above. So, in other words, the H is independent, which means that the partial derivative with respect to Q1 is zero, but then the law of motion says down here that the rate of change, that Q dot is equal to partial with respect to...

40:00 P2. So you just commute the two parts of the derivative, assuming one is zero then the other one is zero. You just work it out, you'll see that it's the same. Which fits, you see, somehow with the intuition of the Hamiltonian system, based on more or less having total energy between all the parts, that if one, it may be that one part is completely independent of the other, but that's really the only thing, the only way that one can depend on the other. So the theorem, the statement is that if you have a product space, and the Hamiltonian there, and the projection to one side is equal to there, then the projection to the other side is also equal to there. I see how that would connect, which means, you see, I never quite realized that generality before, but none of these theories have determined that I see statistical mechanics with a temperature parameter. None of those can be Hamiltonian. A certain portion of Hamiltonian, but then they are modified in order to model this idea that there is some interaction with the outside world, but it's not completely reciprocated. So basically any one of these, take any object in my second order topos, pass through the size topos, it's again a topos, but now it's describing non-autonomous systems like that. That chosen object is playing the role of the external world. Right, yes, I understand that. You can't do anything about the weather, and time and tide race to nowhere. I was reading some lectures at Shrewdale, which were made 40 years ago, and I thought that he was talking about this emphasis, and I thought he was talking about this emphasis, and I thought he was talking about this emphasis, and I thought he was talking about this emphasis, and I thought he was talking about this emphasis, and I thought he was talking about this emphasis,

42:30 There are lectures and theories where there are several temperatures, but there's still only one answer to them. Yes, yes, tremendous idealization. Tremendously powerful. But since it works so well, there must be some rationale for this kind of evaluation. So I see, I see a direction of growth. Now in this same picture I talked about many things. This is only one, in fact this is a footnote. About the tempo and the tight shift and the tide and so on. It's just a footnote. And I thought, I thought we were going to do the footnote even though it's a back note. But in another part of the paper I'm talking about very general notions of averaging procedures, so maybe there's some way of explaining it. That is how some of the rules of mathematics are modeled and genuinely dissipated. Resistance is against things which you can idealize and model and might connect up with ergodic theory. About eight years ago, this was two pages too long. There's a much more general theorem, which is a particular case, and they use a homology, but they're using a specifically apocryphal homology that I never realized was the consequence of an Ellenberg keynote accident.

45:00 Really? It's a kind of... It's so interesting when you see a beautiful, practical theorem coming out of a special case or something. Thank you for your attention. All the way? Or are you stopping in Paris for a day or so? Yeah, I might still have one in London. Ah, good! Oh, she has! I guess she just opens it with, uh, using the word ratio to mean, yeah. She's a homomorphism. No, it's actually the only rational mean. Okay, well you guys, uh, yeah, you're going down this way.