Conversation on train: FW Lawvere

0:00 Instead of being white as they are now, they are white. I think it's wonderful. It's really a beautiful analysis. Rigs, inseparable objects in the topos of directed graphs. I would love to have heard that talk. The other thing, again, I didn't quite catch what you said. And the things you presented in the appendix, in the Propagandist appendix to the book, that you can view, that will help the scientists continue to talk about this whole business about looking at particular determinations of constancy, seeing how strongly constant, and all the really interesting phenomena that were, as it were, allocated at the borderline, efficiently and increasingly strong determinations of constancy.

2:30 These are invariable topological instruments and one of the greatest interests of mathematics. And you were saying that in connection with this idea that obviously the set theorists have got, that the power set is in order. You were saying that you might have a different way of thinking about the power set operation altogether, Oh, oh, oh, oh, oh, yeah, yeah. I'm afraid I was going to catch you in the middle. No, this would be some idea. You know, the GCH, the Director of State of the Academy, powers that concept. We know that it increases cardinality. There aren't any cardinalities in between. So, and then on the other hand, E equals L is much stronger, it divides, but it's... At the same time, it's in a way a kind of subjective nature because it's this wild fantasy that's being able to destruct things, sort of taken into tight control, enables one to sort of erect a scaffolding which will produce this category of sense. Oh, yeah, I must tell you, okay. Yeah, okay. Anyway. I still, I still, yes, okay. Get back to the double power theory. Yes, the double, that's just an example of another functor, you see. Yeah, but you might look at it in a different way. In other words, the idea might be that there might be some kind of objective property like GCH that's stronger, but in between, you see. Right, right. So that there's some other functor, you see, whose behavior is, so to speak, as trivial as possible. Right. In the same sense that GCH is saying . Thank you. Thank you.

5:00 Thank you. Thank you. Thank you. Thank you. Thank you. Thank you. Thank you. Thank you. I doubt it's a double power set, but... No, but no, you just chose that as a possible example. It's an example of a functor that... The point is just exploiting the functorial viewpoint of the power set operation, rather than... Yeah, yeah. No, but I was racking my... I had to write this... You see, I wrote this, okay, for the... It was two years ago that we... My talk was going into the Bulletin of Symbolic Logic. Right. It's mainly about education, but I mentioned this toward the end as an example, and I sent a copy actually to this Spanish fellow, Ferreros, who was writing to me, and this is one of the points, he wanted to save me from embarrassment by teaching me, but I actually most satirically believed that. Anyway, leaving him aside, the actual editors also said, well, they understood this. It might be better if I explained it more. So I did. Well, you did make, yes, you did make an allusion, a very frequent allusion to all the intervening determinations. So I wrote a couple of pages actually explaining this. In a way it spoils the unity of the paper because this is not what I intended to make a main point, and now it is going to be a main point, but I did think of an analogy, just an analogy, but I remembered my old days as an experimental physicist. When I worked at the cyclotron, in fact about a month ago, I happened to be in the Midwest, so I drove to Boomington to see if the cyclotron was still there.

7:30 I knew that it had been put out of service and a nice shiny new one had been built somewhere else, but... But for a while it was kind of like a museum where it was just, you know, it's hard to move or anything like that, but unfortunately now it's, I could go in the same door that I used to go in 45 years ago, you see, and the outer workshop portion was more or less the same, and the booth where I used to steer the cyclotron's magnetic field was over there, but the actual space occupied by the cyclotron is now... There are all sorts of people in white coats doing all kinds of things, individual micro things. Ten people with white coats behind closed windows. What were they actually doing in the site then? I don't know. Actually I had to catch a plane. Oh, you didn't get to find out. So it had in fact been dismantled. The cyclotron certainly had been dismantled. Anyway, I remember from those days, you see. I learned about vacuum pumps. Now, I'm sure things have gone further since then, but to reach a high vacuum, you have to use more than one technology in succession. So first there's a pump, something that, like what you would think of as a pump, and it works away, you see, to its maximum, maximum, maximum, you see, and then... Then you have this, what to any person is a vacuum, but of course isn't really a vacuum, and they need a better one, okay? So a completely different technology has been introduced, namely something involving oil droplets. Thank you for watching.

10:00 It's a nice analogy, isn't it? It's nice, you see. In other words, with ordinary geometric morphisms and toposes, we can imagine getting down to the level where the action of choice, which is the main marker of constancy, or even the continuum hypothesis. But to go further, see, Gödel had to introduce oil droplets, a much more sophisticated technology, and you have to work, work very hard in order to achieve this zero. I imagine, I imagine that much the same is true for temperature. You approach absolute zero. First you have an ordinary refrigerator, and then you have something far more sophisticated to bring you down to those things. It would work a lot in order to achieve nothing. I'm sure the people who actually do the engineering for the large accelerators, which of course have to be kept very close to absolute zeros, just a few. Millions of a degree would be able to tell you all about that, about the different technologies that are used in conjunction or in succession rather. There may well be three or four technologies. Yeah, it'd be interesting to find out more about that. Very interesting. And obviously they, you know, they start by using locks or whatever. They use those to cool it now, then it doesn't get anywhere near as cold as you need to get for these. Thank you for your attention. All mathematically meaningful constructions just depend on the category of sets that you might, that you could extract from any set theory worthy of the name. Yes, yes, but all the important things, the categories, all the various.

12:30 It might have all kinds of other stuff, but at least it will give rise to something like a category. Indeed, as Dedekind and Carreros told me, you know, emphasized this, that already Dedekind quite consciously defined all sorts of things in terms of maps. So one has to realize that of course L is incredibly structured because of the way it's done, but then it gives rise to a category. And because, you know, these oil droplets were so very carefully propelled, it turns out that that category that pops out is indeed colder or even more constant, even more of a vacuum than that, which I don't know what you might have said. Even more frozen, in fact, we couldn't even say. Yeah, you couldn't even say, yeah. Yeah, maybe I didn't know more about the temperature, perhaps, and I could have just... That's what I don't know. That would also be a nice analogy, because of course you could actually use the terminology of frozen to make a point about it. Right. We are thinking of the complete freezing of variation. And also the fact that you never can really achieve it. It's also in the physical. Yes, in the physical. And also, of course, a nice analogy in the mathematical case where one has, as it were, the traces of variation in the case of many of the, well, the Boolean-Valley models, on which we've been here at this point. It's all very naturally thought of as filling in a corner of a deeper geometrical figure. You do have that issue. But as I say, all those models really are. But the reason I have this illusion, Norrset should be big for forgetting that they're supposed to be working in the context of constancy. Because I think, I claim that Cantor himself... All of these models, as I thought, on the borderline, you see, they are actually constructed by first having something which is genuinely variable, very slight genuinely variable, and then almost fixing it, you see. In other words, if you have a domain of variation, then you... In the course of the course of the course of the course of the course of the course of the course of

15:00 In fact, you've made the point about the compactification, and just looking at the basic open sets of a big, sort of, candle space and looking at the compactification. So, seeing the whole notion of cardinality in these models in that context becomes topological meaning structures. I'm way back, I remember you mentioned about the compactification. I think it was in the Bristol, or possibly in the island I call it, 25 years now. And at long last, as I say, after a quarter of a century, people are beginning a class to take it on board. It's been a long time getting there. So the hope is that with this few pages of exposition and this physical analogy, the deceptivists will... Finally understand what we're talking about. It's always been my hope that someone with some technical ability in the experience of the field would actually make these more or less equally precise. The degree of constancy, you see, is still just an intuitive thing. Yeah. But I think it could be defined in terms of, as I said, basically morphisms between topos that don't have adjectives. Right, right. There's Gödel's own collusion of elements that he, that the original he, perfectly knows. I mean, that would give one a whole, yeah, see, that would give one a whole different way of thinking, or this would give a different way of thinking from the point of view of the traditional secularity way of, I mean, why it is that they, you know, they've got, that they're fixed up, you know, sorry, it's hard to get, um, why it is that they...

17:30 The assumption that the power set has got huge, and really just taking power sets as a function is really, you know, very restrictive. Very restrictive. Well, the point that you were making, that one might, in the case of these... This is more of an idea for getting still more constant, but without invoking the L construction. There are some properties similar to DCH but stronger. In other words, between x and 2 to the 2 to the x, there probably have to be such and such, but then if we assert that there's no more than such and such, it's like the DCH, but it might be stronger if you take the right puncture. Yes, if you take the right puncture. Well, as you say, I mean, it's a very suggestive, it's a very specific control, isn't it? Yes, it's quite misty still, isn't it? I guess because we're going along the river all this time. We're actually in the bottom of the valley here. Yeah, this is the canal, but it's actually, there is a river just nearby.

20:00 I think that river is the Meuse, it's close to the Meuse, there's something like that down there. Yeah, it actually says, we passed a sign going back that way, it says it's the canal of the Meuse. The canal something, I couldn't see the words, something something, the word Meuse was there. I think that, yeah, well, a canal that is made by the Meuse. Incredibly rich agriculture. It is around here, yes. All of Lorraine is very, very rich agriculture. Yeah, that is it. Before the Dukes of Lorraine got their gold, they must have screwed the peasants, yes. They suddenly extracted the surplus product from the farmers, I'm sure. Oh, well, actually, the pleasant part of my hike this morning was I found the marketplace. Every kind of mushroom and the rich vegetables. No, that's the great thing about French provincial cities, they have all that, these wonderful open-air markets. This is actually under a hall, next to an open-air plant. The point is that people still go, that's where they go to get their food. They don't get any processed food off supermarket shelves, it's a fantastic thing. People standing in line to get stuff to cook lunch and dinner with. Right! That's why it tastes so good. It's all fresh and they know how to cook it. But, you know, it just looks so fresh and rich and brilliant colors. Not artificial. No, exactly. Not additives, not... No, no, that's what's so great about it. Well, I'm going to do food in France, Italy. And the cheeses. Oh, I know. It's something absolutely extraordinary about the variety of French cuisine and the fruit, too, of course. Oh, by the way, if you feel peckish before we get into Paris, I've got a couple of bananas I liberated from breakfast, so I thought we might need those before 2 o'clock.

22:30 I don't think there's actually a restaurant car on the train. No, that clarifies things very much. I see now what you meant by an example of the double. I'm just an example of a different function and we're probing something even more permanent than anything else. Oh, the other thing I was going to ask, when you and Steve were talking, I think it was a couple of nights ago, two or three nights ago, I think when we were in the restaurant in the house, he was asking a lot of questions about measure theory, I know that's his big question. Bonjour, Monsieur. Bonjour, Monsieur. Bonjour, Monsieur. Bonjour, Monsieur. Bonjour, Monsieur. Bonjour, Monsieur. Bonjour, Monsieur. Bonjour, Monsieur. Bonjour, Monsieur. Bonjour, Monsieur. Bonjour, Monsieur. Bonjour, Monsieur. Bonjour, Monsieur. Bonjour, Monsieur. Bonjour, Monsieur. Thank you for your attention. I was just about to try my Frenchman. Excuse me, I can vouch for my colleague. He's a very distinguished scientist, a very honest man, and I saw the ticket in his hand.

25:00 We actually get from the British inspectors if you try that. Oh, yes, sir. Well, do it, do it, do it, do it, do it, do it, do it, do it, do it, do it, do it, do it, do it, do it, do it, do it, do it, do it, do it, do it, do it, do it, do it, do it, do it, do it, do it, do it, do it, do it, do it, do it, do it, do it, do it, do it, do it, do it, do it, do it, do it, do it, do it, do it, do it, do it, do it, do it, do it, do it, do it, do it, do it, do it, do it, do it, do it, do it, do it, do it, do it, do it, do it, do it, do it, do it, do it, do it, do it, do it, do it, do it, do it, do it Thank you for watching this video, see you in the next one! Yeah, it's not only Mrs. Thatcher's privatized trains dirty and unreliable and unsafe, but they're also about three or four times more expensive than the perfection damaged ones now. Well, I guess at least it's safe to having to listen to a conversation about that for the rest of the year, at least for now.

27:30 Well, I feel proud about that then. No, I was going to ask about, yeah, the other day when we were in the restaurant, I think it was one on Stanislaus, and Steve, unfortunately, very difficult, I couldn't really hear anything because it was very noisy in there. It was terribly noisy. Yeah, as hot as hell, yeah. Very, very noisy. And I could only hear a few words. In fact, I could hear some, most of what he was saying, but I could hear very little of what he was saying. I know he was asking a whole series of questions about the category of metrics, but it wasn't about the metrics, was it? But at one point you said something, which I could only just catch a very few words of, but about this construction involving a reversible negation. And then you said, but why stop at logic? Why stop at logic? And you've been thinking for the last month. I think you said it was an idea that you had a month or so ago, that you could extend it to a kind of negation in space. It sounded absolutely fascinating. Can you run that fast for me? I do apologize, but I just couldn't hear enough. It's just kind of a formal idea at the moment. Well, please, please try me. I think it's all right with you. Okay. If I talk about something else, that's fine too. Okay. The intuitionistic definition of negation. A is simply not A if A implies false. Right, sure. And now you apply this, of course, to predicates of any herity, although false comes from herity zero. Yeah, yes, yes. The inverse image. Right. So you could... You could somehow enrich the notion of negation by saying that the negation of A is the function which is assigned to every truth value or everything at level zero.

30:00 Yeah. A implies that. Right. In the sense that, of course, you have to bring it back up to where you are, and then A implies that looks at the same level, so the negation of A... All of this is at the same level as A, except it's a function of the truth values or sentences of level zero. Right. Now, the one thing, so if you evaluate this function as zero, you get the usual indication. But, you know, it's a more complete invariant of A. In the usual sense, but we have a better chance here. In fact, this process has an edge on it, you see, that it's going to be sort of like double negation. Right. The first negation was strengthened, it wasn't trivialized, you know. In other words, A implies zero is always a sub-object of one, even if A wasn't, so this is almost alleviating that. You see, it's not trivial that A implies B for all B, because the B's are restricted to being, you know, absolute false, absolute true, and all the things in between, but constant. True doesn't help much, but the things in between might. Now, in fact, this idea was used already. It helps to explain why, likely, in 68 or 67, he was able to get the completeness theorem for intuitionistic logic by doing proof theory, so to speak, in a certain topos, but a Boolean topos, because he interpreted everything more or less in the usual way, that is, implication is a functions-based. Conjunction is a Cartesian product and so on. A pair of proofs, you see, is a proof of a convention. A map that maps proofs into proofs is a proof of the implications.

32:30 Right. So this is the standard idea in a map. But he applied it to his movie in so close. And, of course, it's richer than this sort of post-set reflection of the category of all things over x. All of this is a much richer thing than just the sub-objects of x. Right, I picked up that point. Unless you have the actual choice. Yes, unless you have choice, in which case it goes like how strict are all these proof bundles down to singles and ones. Right, down to singles and ones. So it's richer in that way, but negation in particular you see would still be... Subobject to one, no matter what he started with. So how does he manage to get faithful interpretation of negation? Well, because he doesn't interpret negation in the usual way, but in the way that I said. The indignation of A is a function of all truth values, not just of false. In my Siena paper, I pointed out an analogous thing. For metric spaces, as generalized logic, because the role of predicates A there is simply taken over by Lipschitz's maps from the metric space into the reals, and now the naive negation would simply give you the zero set. Think of a space like this. Thank you for your time, and I look forward to seeing you again soon. So you get the, not the level sets, but the sub-level sets. So in other words, for every value lambda, you know for which point x your function is less than the root of the lambda. Well from that you know the function. Yes. Although just knowing where it's zero does not tell you that the closed set where it's zero is essentially the negation.

35:00 So, again, it's sort of a non-trivial thing, because you're using only constant points in the category, but nonetheless, you're covering the... Well, it's a kind of a... nothing everybody knows, but to call it logic is not quite... No, well, it seems to represent just too narrow a determination. So, you know, if you're taking the same idea now in a different context, take any geometric morphism and topos, you can think, well, this is sort of the variable line, this is the constant line, and so you can, given any object here, and given any space, you can define a, you can define a, you can define a, you can define a, you can define a, A space-valued function of sets only, of mere sets, is simply the function space of the pullback of the set into your given space, for every set. So this might be a useful invariant of the space here. That's the idea. But then I realized that for space cocos... Now we're satisfying the extra action that the Morphism is essential and even Phi-Zero preserves products. Yes, I remember you mentioning that. Well then, if you just write down what this means, you'll see that it implies that even this negation only depends on Phi-Zero of the space. It's a partial, but even so...

37:30 What I didn't investigate is that if we take the opposite situation, if the t-toe goes, then pi zero doesn't preserve pi like sin. It might be that we get a better idea, but it's a more polar invariant of the sheet and the sheet over a given space. Yes, I forgot about that. Here's one of these. I understand now what you meant by the negation of the space, which I think is the phrase that you used. Unfortunately, as I said, I was only able to pick up very little of what was said. I also see why Steve's interest in measure theory was kind of made into it. Although I think in that context he was actually discussing relationships in classical and individual systems. How stable is that? That's absolutely fascinating. What was the additional idea that you had in the last month or so? Just that point about looking at the Petit Topos case. Well, no, not the first case. Just the idea of applying this semi-generalized negation concept to the case of just plain Topos. Just plain Topos. So there's no logic at all. No, no. I mean, yeah, I mean, exactly the same form as a definite meaning. Yeah, yeah. Because you say, I mean, the kernel of that idea is that already in the treatment of metric spaces, there's a sense of generalized logic in there. Yeah, yeah, yeah. Well, I don't think I'll bet the same. Yes.

40:00 Well, I mean, the level sets are literally investigation. Yeah, yeah, yeah. That is to say, you've got contour diagrams. The Albatrude as a function of the position of the contours. If you knew all the contours, you would know the exact Albatrude. But on the other hand, given contour line, the function implies that value. I think this is the one intermediate stop. Oh, probably the actual, was probably the territorial, the actual border of Lorraine. Probably the barrier of the Duke, the Duke's barrier, I would guess. Probably a contraction of Barrière, Barrière-le-Duc, probably the place where they had the toll station when you actually entered the Duke's territory. Speculation. But I think this is actually the... I think that contours, contour lines and maps, getting kids to think about, is actually quite a good way of introducing them to get some respect.

42:30 No, it's cool. It's cool being stuck with a duck. No, we didn't mean to be boring you, Richard, asking all these naive questions. Oh no, I'd love to know more about that. It was a very interesting talk I went to about a year ago in Bristol when I was staying with John Mabry on the history of Hamilton's work on the Theoric Tides, which apparently did a great deal of it in the 1840s, and also Airy, who was the Summer Royal. And they calculated the, is it the anchidromy, is the point on the surface of any body of water in tension which is always at a constant altitude above the sea. And they actually identified, the oceanographers actually identified these points in the North Sea and in several other bodies of water.

45:00 Using, first of all, Hamilton's theory, which unfortunately did not give the correct predictions, and then a correction theory which was published by Airy, which did, and it was a very interesting talk. It was given at, I'm trying to think of the, the man that gave it was a very distinguished physicist, he was known as L'Oreal, and his name, he's the professor of Westphal, I can't think of his name, he's a sir somebody or the other. It was so bad, mate, it really am. I'm afraid I sometimes worry about it. They say that, you know, normal affairs are tempting to get on with victory, but I do think it's particularly bad. Sir Michael, yeah, yeah, I'm pretty sure he's an adult. Sir Michael, Michael Berwick, yeah. It was an extremely interesting book because it was a, you know, it was a general, kind of, popular, kind of, historical book. I'm very interested in Hamilton's work in Oxford. Because Russell-Hiley suggests that Hamilton's work on the Eichmann equation actually provides it and the so-called Boyd phase and actually anticipates this discovery of the mesoflexic group of the double covering of the symplectic group and the key to understanding the way in which the H graph arises in the case of a kind of obstruction to the... The fractional booleans are introduced with great spacing, but you have to introduce a correction to the Louisville theory. The suggestion is that this is where Buck's constant is really coming from. It's a very interesting idea. The suggestion is that, at least for the case of quadratic annotations, it's already worked out in optics. And in the 19th century. And I would like to know more about that. But I have not realized that Hamilton and Grassman also worked on the period times. Or on these other areas, like the Kyrgyzstan and the Kyrgyzstan.

47:30 That is very interesting indeed. And I hadn't known that about Grassman. Founders of modern abstract algebra. Absolutely, both founders of modern abstract algebra. I mean, Grassman, you know, much greater, really, and Hamilton very great as well. Oh, yeah. But Hamilton, I think, was not aware or hardly aware of Grassman's work, sadly. Well, they did, absolutely. They did, did they? I know, obviously, Grassman was aware of Hamilton, but as I understood, it was rather a one-way street. Yeah. You know, in fact, somewhere Appleman said something like, you know, in his competitive spirit, that he wished he could do something as well as... As well as Grassman, really? Oh, no, that's very interesting, because I have been told, and obviously I shouldn't listen to people who just speak, you know, on authority without knowing what their authority is. I was told by some guy who held himself out, which was in a conversation we had at one of the seminars in Berkeley, a physics seminar that I attended. I feel they're not sold out to be something of an authority on Hamilton, but there's no evidence that Hamilton's ever read the Australians there or the Rasmussen's there, all that he knew, all that he certainly knew about Rasmussen's work. No, I mean, I've never had a good opinion of him. No, that's rather easy. Well, this guy was a bit of a bullshitter. Yes, he was well-educated, but this is a serious... But the idea that this is actually the fundamental... No, no, this is absolutely ridiculous. No, unfortunately the person I'm thinking of didn't give the impression of being such a person when it came to his historical views. It was all marvelous at the level of Godson. Okay, they did hit the light jelly school of history, and not... Not very seriously though. I remember looking it up when I was at that same observatory.

50:00 Big volumes of Hamiltons. Well, actually the same ones that are available now, I think. Yeah. I don't know if they exist. There's, yeah, there's several papers about grass plants, right? Oh, well, that's very interesting. I really must study this. In fact, I might be able to pick up a copy of Hamilton. And at this bookshop, L'Enchart, because it's the sort of thing that they do, do these wonderful reprints. And they do reprints of the original text, so they're not scholarly editions like this. They're just having reprints, but they are very, very well worth having and studying. The one at the garage, Mechanic Anna Latifi, is a lovely one to have. As I say, we'll go there on a Wednesday. Which I suspect you may find your way around. There's more than that. Yes, we must be heading pretty well to West Manor. I was wrong, the train doesn't, obviously the train does not go through that, or along it. In fact, if you think about it, it would be, although there may be a different line, it does go through there. The German, the French, must have constructed a line directly from Paris to Nantes before 1940 if they didn't go by a map because they wouldn't have allowed, you know, the key railroad line to Nantes is to go through German territory, but it's not to be dependent on a single railroad line at the end of this military region. Metz is also a very interesting city. Not, I think, quite as beautiful as Nancy, but it does have a great deal of very imposing architecture, particularly the Louis XIV period, where it was a little bit like, although, some magnificent squares, nothing quite as magnificent as the Stanislaus, but lots of squares that are almost as big as that. Of course, it's a bigger city than Nancy.

52:30 And then, of course, all the German stuff that they built in the 50 years or so they were there, including a couple of very good museums, and this huge railway station, which was built in a very ponderous, neo-Romanesque style, and was intended to be one of the main railheads for German Ireland. All I seem to have is just the memory. Ah, well, that's all you need, is that it. If you want to call him now, I've got a mobile. I also like the thing about narrow empiricism, which in turn concocts its own magic, its own variant of magic, just as well as our shades of Skinner and his behaviorism. Speaking about the old-fashioned magic, I mean, the straightforward, not even sure I'd dignify it with the term, the chipped and idealist magic, but total lunatic, mystery-mongering magic, did you happen to notice along the way into the station that there has been another conference going on in Nancy over exactly the same days as our meeting, the 30th of September to the 4th of October at the Palais de Congress? There has been a seance of, what do people call it, clairvoyance? Yes, apparently some sort of spiritualist congress going on. That actually explains what the lady at my hotel asked me when I checked in. Have you come for the congress? And I was most delighted that she knew that there was this congress on the fact that it was philosophical.

55:00 Philosophical Insights into Mathematics and Mathematics at the University. So I said, yes, I'm of course now beginning to worry that she might have had the impression that I was some kind of lunatic table-wrapper. I don't think, oh dear, no, I hadn't, I only noticed it yesterday, I noticed it last night. In fact, when I went over to the railway station to get my own ticket to check on the train signs, and I noticed that there was this poster up for it, and it was the same date as our own meeting. There may well be, but I never heard of it. I don't ring any bells. You could well be right, but I'm afraid I haven't heard of that. Well, I assume Royale, although it was obvious from the pictures of the kind of crystal balls that it was some kind of spiritualist, medium, model, you know, National Enquirer end of religious propaganda, as opposed to the more subtle end. You mentioned various forms of selling off the... Yes, selling off the patrimony, selling off national treasures and archaeological sites. Well, you know, there's a number of you that I'd like to meet tomorrow. It must be Italian himself, I guess. Yeah, I asked a couple of speakers. No, no, it wasn't one of the speakers. Ah, okay, and I was just wondering whether you got his name. I'll tell you when it was. It was when we were in the buffet lunch the day...

57:30 Yeah, it was the day that you gave your talk. The lunch break after you gave your talk, and you just had quite a long talk with Steve, and then this guy came up and asked about what Aton did with his talk, which is one of the problems in the main case, which is usual. And, yeah, I just wondered if you got his name, because I talked to him a little bit later. He's a very interesting guy. ...and wanted me to write in with some kind of recommendations for the reading. And actually I remember later, you know, you talked about some general issues of proposterity about being a character. And, you know, where this idea that all puppeteers are the talent and that people like Johnson and Pride and Cluster and Cumberbatch and all these sort of conversations. Yeah. I'd gone off by then. Well, I could be wrong. I thought he was from Malsi. He was certainly one of the French. He was a French guy who was a computer scientist. He said he worked. Well, that's right. No, he was from Malsi, but he said he worked at some computer science institute because they had a lot of problems. All the people at his institute were really interested in the same thing. Thank you for your attention. Yeah, I'd say it sounds right. That's the same guy who told me where the Batuu barrier is.

1:00:00 Oh, was that? He was the guy who told me. Yeah, it must be. Well, that's why the marsh is on the goldmine. Yeah. Quaternary is black and so on. Right. Right. Well, this guy could very easily have been French-Canadian. I mean, come to think of it, his English was extremely, was slightly accented and very, very out of line. Speakers include mathematics, geometry, algebra, mathematics, physics, quantum mechanics, physics, quantum mechanics, physics, quantum mechanics, physics, quantum mechanics, physics, quantum mechanics, physics, quantum mechanics, physics, quantum mechanics, physics, quantum mechanics, physics, quantum mechanics, physics, quantum mechanics, physics, quantum mechanics, physics, quantum mechanics, physics, Oh, that's the guy you were talking to in the institute. Yes, he's a very, very interesting guy with a beard. Yes, he's a very, very interesting guy. Now, this was a much younger guy, and I'm sure is the person you mentioned. Thank you for your attention. That's good as well. It must have been.

1:02:30 It must have been. Yeah, I guess that's right. Or perhaps just a little older. Don't worry, I'll hold off the requesting the seminar on TV properties at all. We've got Steve tonight, we've got a chance for that. I talked with Sarah once recently on the telephone. We were about to be related about this boundary parking and proposed status. And what was his answer? And what was his answer? Yes, that's true. Du Donnet was the guy who acted as their rapporteur. Apparently, he did an absolutely incredible job of writing up detailed records of their discussions and meetings, as I say. And of course, he didn't have digital tech recorders to help him. Actually, in some ways, probably with an advantage for some. We've been able to produce much more of a redactional of the discussions than just simply a raw record, so it does seem to have been a... By digital you just mean that the audio is just like... I just mean this, yeah, I just mean this thing, this thing that I've been using to record your lectures, just to press, well, to record.

1:05:00 There's a trick with this, and I'll show it to you. I think there's a disk in there from yesterday, from the last day of the meeting. Yeah, that's the thing I recorded yesterday afternoon. Well, to record, what you do is you press the little red button, and then it will come up with a display. This is the trick. You then press this. So that on the digital display, the thing which is showing you the recording level, the sound level, a little thing saying mono comes up, so it's recording only in mono, not in stereo. Now if you record just in mono, you'll get twice as much recording time per disc. So for an 80 minute disc, you'll get 160 minutes. There are over two and a half hours, 40 minutes of recording time, which is excellent, you can record quite long discussions. In fact, there's a more sophisticated version they have now, where you can actually record at quarter speed, so that on an 80 minute disc, you can have nearly six hours. And of course it records an absolutely crystal clear quality. It's very, very clear. Much better than an ordinary conventional. Thank you for watching. I'll get that transcribed as soon as I can after I get back.

1:07:30 They're very easy. The ones I used to record in the audio ones, as I said, sometimes they didn't record all that well, and you had to listen to the tape many, many times, and there were individual phrases or exchanges where, because there was a lot of background noise, maybe other people were thoughten, or your voice or the voice of the questioner was soft, just didn't get what was being said, but with this, you get perfect sound reproduction. We're getting very close now. The two of them are still born just ahead of the tower. Exponentiation and non-abelian homology. In my dreams, would I be able to follow that? No, I just noticed that the usual definition of homology is a simple, universal, categorical instruction that has nothing to do with having a linear category, The ritual story, d squared equals z, one category of the ritual story, full subcategory, or the bigger one, so where the inclusion is both left and right edge, then it follows, because it's a full subcategory, there's a natural map from the right edge line to the left edge line, it sounds like the wrong direction, because it's full, because it's full you cannot have it in the, it's really going through the,

1:10:00 The thing that you do is you take the image of that map. So that's giving you a third-pronged homology. The example is you take a chain complex and bring in just sequences. And there is left and right adjoins. And these are known as anticycles. The natural map is those modular boundaries, which is the way it's usually described, although in many books. In many books, the anticycles are mentioned also, and they even remark that this is a more symmetrical way of putting it than they will use it, but it has nothing to do with linearity. For example, source and target structures. Well, among those, there are those where source equals target, always. The information of how many connected components have at least one loop. Right, yes, I think. Oh, that's the question, I see. Yes, that's right.

1:12:30 Right, and the light that this throws on exponentiation is... Do you want to apply this or not? How do you find your colleagues in working in homological algebra and in political politics are responding to these ideas, and particularly to the overall programmatic inside them? It's not the open sets, which are really the science that's not there. They're going out of date. They're going out of date. They're going out of date. They express themselves. I mean, is it really beginning to get into the way that they do? Some of it, some of it. Some of it, some of it. Some of it, some of it. I'd rather see more results. No, but coming at it from all sorts of angles, for example, the homology of space, as opposed to the same thing, with field coefficients, satisfies the Cartesian product as equal to the tensor product, where the tensor product is a greater than or equal to the end term. It's the direct sum of the potential products of the p-th term and the q-th term for all pairs being built up to n. So it turns out that that's the homology of the product. So that means that the homology is actually not just a gradient.

1:15:00 It's actually automatically a co-algebra. Because the diagonal map on the space is going to induce a linear map from it. The homology of space into the tensor product is homology in and of itself, because that's the same as the homology of the Cartesian sphere. Now these co-algebras, all these induced maps are co-algebra morphed. They're not mirror, many of them are. The community of co-algebras, where many are tensor category, is actually the Cartesian category. I mean, the tensor product becomes the Cartesian product. Very often they're even Cartesian closed in the co-algebra category. So it seems not to throw away this wonderful information, it would be better to regard the homology of going from Cartesian closed category of spaces into another Cartesian closed category of a kind of abstract nature in the sense that its objects are the shapes of spaces rather than spaces. That should be a Cartesian closed category as well. Maybe it helps to linearize in order to compute it. Conceptually, it should be thought of as a way of kind of very much abstracting the spaces to get these shapes as a functor from one Cartesian closed category to another that preserves Cartesian products. So co-algebra is just one way to construct such a category. And the usual homology, in fact, would feel too efficient. In fact, it gives such a function. But this is something nobody ever talked about. I think Eilenberg not only knew it, but took it seriously. He had a series of papers on co-algebras, which I feel sure was motivated by trying to see homology. I would prefer to think of it in a still more conceptual way, just so it was co-domained in some kind of certain Cartesian closed category. Maybe it even satisfies a universal property, you see, like, but by the way, homotopy is also a category.

1:17:30 Homology is sort of coarser, it's also stable, crossing up into higher dimensions. Is this the content of that paper in the American Mathematical Monthly which you're talking to me about? Showing how the Jordan curve theorem falls out of this much more general framework. It falls out of the stability of homology of a complement, that's the point. The basic idea is this. Simple circles and any closed curve are both images of the same space just plugged in in two different ways. And now you're talking about the complements. Now the complements of isomorphic copies of the same object are usually not isomorphic. So basically there's two embeddings in another space, which turns out that their homology is the same, and then you can produce, in this higher space, then the homology is invariant under this.

1:20:00 So I've been privileged to be at the, he still didn't want me to be a plate.