P Cartier / conversations: FW Lawvere, C McLarty, J Mayberry / John L Bell: talk (partial) / lunch conversations

0:00 Did you, yeah, we had that long chat last night, actually, in Melbourne. Yes, yes, and I was very interested in what he had to say, and he was telling me some more of the work of this student of his, his doing this work on the, and how this whole sort of quantum sets construction is applied, and obviously it gives one an extremely sort of messy definition of the sub-object classifier, which is not a sub-object classifier at all, but he's aware of those. We'll see where he's trying to make the thing more funtorial. Well, I mean, I'm not going to be incredibly stupid. That's something you said to me last night. I wasn't sure whether to tell you that, but I think that he actually said, you know, I've really seen that Phil was writing what he said to me and came here. Oh, yes, yes. He's a very nice man and a fine mathematician, I would judge. No, it was very good to have him here. I think we made absolutely the right decision after all in getting him involved, and especially in the earlier, he never used the term, no, no, no, yes, yes. I'd certainly like to understand that more clearly. I'm very sorry I missed that. I really am. That's, as I say, one so frustrating.

2:30 Yes, I was very disappointed in... I didn't, but I heard afterwards that I was very disappointed. Apparently what he made was by the fact that he tore up all of his notes for the talk he was going to give and so rewrote it the night before. I think because of bad nerves and it's... But, you know, he's a young man and far to go. Which, as I say, I hadn't heard, except for the Entaglio-Pelikov, and obviously any category is a graph, but what is the connection with Bayer's way of treating these N categories? Well, that's the, you know, multiplications, that is, compositions, all directions. There is a sort of algebraic theory that you have to have definitely. Which his construction does seem to be trying to respect. ...unlike the Mackay thing, which seems to evolve. Right. It does have it, but then, you see, my point was that the organization of that is exactly these graphic monoids. In the case of two categories, or in the case of one category, the basic idea is an arrow. Now the arrow is a generic element of an arbitrary, very complicated category, or graph.

5:00 The arrow itself is a monoid, in the following sense. The arrow, when you apply it to the identity, you get this. I certainly see the connection with what Baez was talking about, although as I say I only got a glimpse of it from the last 15 minutes or so of his talk, at least.

7:30 ... and Taco, are left of all aspects. It looks as though you're getting ready to talk, so I don't want to distract your attention onto other things, but what was the conversation with Angus about ramification? And I guess I'm sorry to ask you to keep repeating yourself, but unfortunately that's the only way I ever learn anything. Oh, it's that one. Yes, of course, because I was forgetting it was down the side of the corner. Can you again remind me of the definition of separated, decidable, and unramified math? Unramified... Well, perhaps... The definition of a map with a question in every category. An object to x equals b slash x. So an object in that is really a math to x in the original category. So a decidable or separable object in e over x is called an unramified math to x, or an unramified problem to x.

10:00 So it's the same concept but applied in a different... So the fiber-wise, yes, okay, I mean, in the case of the ramified objects, one has got this branching, obviously, and the... Well, it's really a map. When you speak, think of it, you think of a given map, and then the fibers are separable, so that a section... You know, it doesn't ever have to, I mean, it's uniquely determined. You have sections over a connected part of the base. Which overlap at all are equal. Yes, obviously. So that's the determinism. Right. So that's really just decidability, but applied in the category of objects over the base. Right, right, okay. Now I think I've understood it. I really have. Excellent. Excellent. Thanks very much. And, uh... Mmm, ginger. Subtitles by the Amara.org community I think I'll have two of these. The other, the other thing I was hoping we could do is to get together with Cartier for long enough to, yeah, yeah, I was going to say, to get together with Cartier for long enough to, to get, to get, to get, to get, to get, to get, to get, to get, to get, to get, to get, to get, to get, to get, to get, to get, to get, to get, to get, A little more background on this.

12:30 It's an Austrian name, I think it begins with a K or a C, no, no, no, no, no, it's a much more German-serving name, he's, come on, thank you, well. He has collaborated with Murdoch. I'm not sure whether he was actually his student, but he was speaking about a conference in Vienna last year on non-commentative geometry in both things. ...legitimately in physics and in mathematics, and how he thought that the work of this man had filled in a very important gap in the understanding of the people working on Kohl's program. I just wanted to understand more of what it was that he was playing with, was that it was to do with topological group points, but it was to do with the right way to do general cohomology in this setting of non-common, non-common messages. I mean, parties are very smart. Personally, I just like to get a feel for what there is that, you know, is happening. I can't see this claim that, you know, Kohn's work is entirely in the rodent spirit, which is what we're hearing from Colling yesterday. I can't see it at all. It seems to me to be a step back.

15:00 A big step back, don't you think? Indeed, with much respect. Mathematics. That's right. That's just right. Well, the same. Yes, the same. Yes, as I say that. It certainly wasn't justified in the article, but I'm glad that wasn't just my reaction on reading it. It was a strange, juddering gear change in the middle of the article and so on. I mean, a lot of what he had to say turned on this business of points having internal symmetries, which of course is the problem here. And how one should treat the loop space and all the space of actions on this symmetry. Well that doesn't seem a particularly, it may not be a deep idea but I'm just going to make sure there aren't any technical subtleties involved. Have you? Speakers will talk about his revision of his entire understanding of schemes, his later field of study. So really, it could be made much more functorial by just treating it as a bundle from rings to sets, to whither appropriate properties, and how it subsumes the earlier. I don't know why he doesn't. I feel like these others have taken this word schemes in the absolute. That's the word. Algebraic geometry means the set of schemes. That's formulated in the earth. Someone said that in 1960. In the old times. It was Sarah Bladendieck. In what sense is the older definition?

17:30 The one that Necrodecus has now repudiated, or at any rate sees now as being subsumed and recovered inside this toil setting, in what sense is that sort of lacking, what is it that, what do you see as being the great improvement in this new construction that you're thinking about? The schemes are sort of analogous to manifolds, that is, locally non-singular. I mean, from one point of view, many affine schemes could be thought of as changing, when it passes to another level where they are all non-singular. And so the schemes are made up, or have an opening. Each of those is affine. So, just as in analytic geometry, again, Erdendieck already invented 360, but a couple of them before, there are analytic spaces that are more general than analytic dimensions that you take, you know, even the, you know, if you just solve the equation, take all the square of all points, there's a part where half the vaccine goes to you. This will normally not be a mathematical theory, you know, unless F and G are sort of independent, infinitesimally independent, and various conditions can be put which will make it a mathematical theory. So it's more convenient to have a larger category where you can freely form these. Sometimes you're actually interested in the objects, but in any case, you're only interested in the manifolds, and you can construct these things and then try to decide whether they are manifolds, instead of worrying about whether they both exist or not. You see, if you're in the... whether there exists a Hilbert speed, for example, to worry about a small relay, the idea is just a Feynman or a Omega, and then so forth.

20:00 So, there should be no question whether it exists or not. That's not the question. The question is whether it is, you know, has some further special properties that are not all on the topos and so does that sort of general method of logic. The other is functional analysis, you see. People don't actually talk about it, but algebraic geometry has its own innate functional analysis, intensive and extensive quantities and so forth. And those objects are, in fact, algebraic spaces, not just, you know, analytic or continuous, but they're actually algebraic spaces in some sort of intracranial sense. And again, in algebraic geometry, they speak about fragments of this, like the Cremona group, the group of all automorphisms in some particular space. It will typically be an interdimensional space itself, so it's not a scheme in the official sense, but it can be entirely understood in terms of the puncture of things, but again, it's both bigger and less, possibly less, manifold-like, or a scheme, again... One should not make it a question of ontology, though we have this nice big world, and then that. In other words, once again, it's a matter of universal consciousness and also of classification. Yes, this, of course, would allow the whole thing to be the same, but this way of looking at schemes would bring the whole thing much more into contact with the whole theory. Yes, well, it would bring various things that even today algebraic allowances would think of as being on the fringe, but they're not schemes, but they do broaden them.

22:30 Part of science, algebraic objects in some sense, although they do partake of the algebraicity, as opposed to analytic, continuous, smooth, and common, algebraic spaces, in the sense that every object in the category of topological spaces partakes of the idea of probability, even if it's a code of this type of space. So all the objects partake in algebraic systems. Chris was saying last night, actually, that it was extremely important for understanding the development of Grothenburg's thought and, indeed, for his work in algebraic geometry, that he did start in functional analysis. And even before he'd encountered category theory, he didn't actually happen to make that second point. I didn't make that point, but I think it's important because... But no, but it's... Well, tell me more, because I want to try and understand about nuclear space. In functional analysis, there's a natural impression having to do with magnetic functions and rather than working on them. Representing the dual space and the space of analytic functions gets you into the geometry of it, or the analytic geometry of the parts in the CN. Then there's the traditional thing that conflicts in analytic spaces, if they happen to be convent, they're actual algebras. That's a powerful theorem. That's sort of the reason why 19th century... Mathematicians were interested in algebraic geometry, not because of some abstract desire to study polynomials, but because they wanted to study analytic problems. In the compact case, there was a theorem that they actually had algebraic coordinates. So there's a sort of continuity there.

25:00 I had a terrible time getting Colin to even look at some of those papers that don't fit into his scheme about mathematics. This is interesting, that's actually exactly what Chris was saying last night, that people should pay more attention to the functional analysis background and the fact that, as a young man when he was just starting out, he does have a lot of thinking about tensoring product spaces and all of the structural factors that arise in some of the conscious spaces. And that was a very important influence later on. Thinking about algebraic varieties and the development of the schemes there. Can you tell me also about this other aspect of his work in function of analysis? I think that's why I have a different flavor than many of the other categories. I also studied algebraic analysis. I was planning a thesis in function of analysis. Really? Rather than algebraic topology, which most people... All right, John, you go ahead, another bit. Dinner is going to be at 7.30 tonight, and we've got all the details, the coordinates of the restaurants. I think there'll be an announcement before Phil speaks this morning as to exactly where we meet, and it's going to be at 7.30. Unfortunately, I'm not sure if the, yeah, um... The address you gave before is quite near here. The place where the restaurant is? Yeah, it is nearby, so it might well make its way, but there may be some... I think we'll probably tell everybody to meet at the hotel and go down there together. That might be the easiest, rather than leaving people to find their own way there. But I think what we'll do is we'll just tell people where the restaurant is, and then if people... You know, want to make their own way there. Otherwise, and I've actually come to think of it, it'd be a lot easier if we do all arrive together because otherwise people arrive in different drafts and they're kept waiting and they have to sit. Now, I think we'll tell everybody to meet at the hotel. Till I speak to Alberto and Francesco, I'm not actually going to give a time, I should think 7.15, I think 7.15 probably would be okay, but do tell me more about the Grotenbeek's work on nuclear spaces.

27:30 I really would like to try and understand more about what the ramifications for the understanding of the position of Hilbert space within the overall setting is. What do you think about what topological space is, and why you regard it as a kind of a distorting oversimplification of the construction of Hilbert space, or the way that it's... Well, I mean, Hilbert space, on the face of it, means that you identify intensive and extensive quantities that you can't really do. They're always different. So what you do is you... You can artificially generalize the intensive ones by introducing these new L2 images, and of course you specialize the extensive ones to be absolutely continuous. The two things are basically, like this, you know, like this, and that's the, by equal I mean actually, self-deal, sorry, self-deal, yeah. So that's what it is. But the simple left space is, sub-left space. Yes, yes, I was going to ask you, I was just going to ask you about some of those things. Well, these are, you know, maybe they're sometimes confined, but you can consider them as Hilbert spaces. But the point is that you have a whole, a whole system of them. So, I'd like to compare them, first you sequence them, and then you run them. Formalists and intuitionists identify that Cauchy sequences are real, rather than equivalence classes. A specific Cauchy sequence takes the place of the subjective process of coming to know.

30:00 To which you add this fact that mysteriously two different subjects might somehow communicate and come to the same idea, which could only be because of God. Yes, because the old ideas are there already in the mind of God. But there do exist these sequences of Hilbert space, so the basic is that you consider any degree of differentiation. You know what I'm saying, a symbol for, you're going to take the third derivative and the first derivative and the second derivative and the tenth derivative and some kind of finite combination like that, and you consider, and of course you have the basic background, the big measure, so you consider functions which are important. So that many derivatives are square interval with respect to the magnitude. I think somebody, I think I'm getting signals from Marta that she needs me for some information. I'm sorry. I really want to hear this. I'm sorry. Sorry, Martha, I thought you needed me. I'm sorry, I thought you were signalling some information. Where? No, I was misunderstood. I thought you were signalling to me that there was some information that you... No, I'm Italian after all. When I talk, I'm like... Yes, that's another reason why... In fact, we can do that now, if you like. Yeah. Okay, let me see. Yeah, we kind of kick off at half past eight for the same time as we did from here. I mean leave here. Get people together. We're only well enough by now. You have many words to do this. We must have a rather nervous disposition. I think we've got to tear off right now. Well, actually, I believe this sort of kind of cockney criminal flag, it also means sort of something's kicking off. It means a bit of aggro, a bit of real aggro is starting. A bit of serious aggro is starting.

32:30 I'm sorry about that. I'm doing my chairman my duties. Sorry, can you continue what you were saying about the vague measure? I'm sorry, I didn't want to distract you. You can say the functions F whose alpha derivatives, where alpha is a complex thing, take all derivatives of any lower order as well. Our square integral with respect to the ordinary measuring, these are functions of invariance in an n-dimensional space, so they're functions which are, whose derivatives up to a certain point are square integral, and we use the square, well, I guess the sum of the squares of the various levels, including the function itself. So we get this quadratic sort of metric on those functions and we complete it by the Hover space, with the fictitious things in there. But that's the typical Sobolev space. Oh, and I should have said we take a region in the international space, so now obviously there are a lot of spaces, if you take a larger region, you can take a larger, more derivatives, so there are, and then, so there will be... Conclusions between these Hilbert spaces are induced by factors. You've got here, you've got all the higher derivatives, and you've got here, you've got a larger domain, and you've got there, and you've got here. There will be induced bounded linear operators between them due to relations between alpha and beta and u and v. Now, there's a crucial theorem which says that if you, for example, if you... If you have two open sets, they would be, one's included in the other, but you'd much better, the closure on this contained in the other one.

35:00 That sort of well contains. So the thing is that restricting the functions from a large open set to a small open set, and a small open set is an intermediate compact set. This will have a qualitative... There will be a qualitative thing, it's a nuclear mass, what's called a nuclear mass between Hilbert spaces. Okay. So actually, yeah, actually for Hilbert spaces it can be expressed... It's a composite of two Hilbert-Schmidt maps. Okay, I won't... Hilbert-Schmidt maps are those that are given by actual matrices. Not every minute you're out there... No, no. Some are numbers, yeah. A square, a matrix, which is... Where are some of them? You have a square array, but you could take the square... Of all of them, add them up and that might exist. So that's a Hilbert handout represented by a sort of L2 matrix, you know, in terms of Fourier coefficients and sequence spaces. That's the idea. Somehow it's a little bit more concrete than a general linear transformation, but it's also thinner. It's sort of much thinner. It's a compact operator in particular, but it's much better even than a compact operator. And the composite of two Hilbert-Schmidt operators, by some weird twist of fate, is always a nuclear operator. So compact, Hilbert-Schmidt, nuclear. As a condition on the value of many of the maps between topology and vector space, even if I say Hilbert space, they all have this idea that some of the big open disks are being squeezed into very thin, compact, or even better. So anyway, so that fact means that if you take the limit of all this system, so do I, over various open sets and various, there's a similar thing about the derivatives. One, if alpha is really genuinely bigger than beta, in some sense, then the restriction map or the inclusion map between the two phases of functions is again a neutral map.

37:30 The Sobolev inequality is a very precise inequality about how all these squares at various degrees are really seriously bounded by the others. Yes, yes. So the whole point is that these are all bounded operators. Well, they're bounded anyway. They're much better than bounded. They're much thinner than bounded. They're not at all invariable. All things being squeezed into a very narrow portion of Hilbert space. Well, sort of a uniform bound where you wouldn't have expected a uniform bound to be stored. Actually, it's all a sort of multi-index, multi-dimensional version of the fundamental theorem of calculus. Thank you for your attention. If you look at things in terms of riveting, excessive riveting, it looks like things are blowing up. They're blowing up so much that the process of coming back is really a sort of drastic smoothing. You see, so this drastic smoothing is the idea of momentum. That really does give me a very, very helpful grip on the conceptual motivation. It's really the whole idea of the book. Most books don't mention it. Which book is that? It's the one by Peets. He's an expert on the book. It's a book by Peter Peets. So anyway, it's one of the characterizations of... It is a limit, an inverse limit of Hilbert space. But the bonding maps are nuclear maps. The bonding maps all have this jump, like, for example, from an open set to a genuinely smaller open set where there's an impact between, see, restricting spin functions across something, things that the whole of, you know, Gauss-Greenstone sort of comes into play, something like the generalized fundamental theorem of calculus, that you have really a...

40:00 There is tremendous control and smoothing of the system because this restriction must pass this contract boundary and so any sort of wild behavior that you had has become incredibly It's interesting to see how the connection, I mean, again, this was, it was currently developing this theory of nuclear space, but yes, that's what I'm saying, because it's interesting to see how the motivation there, which you can say is all to do with this, how you smooth things out and make, you know, bring it out of the truck. You know, it plays itself out again, you know, much later in the idea of tamed topology. I know it's completely different, but at the same time there is an underlying conceptual connection there, you know, instead of allowing these things to blow up so that they become so wavy that they become what you are really getting at, most of the analysis has been based on Barnard's basis. Barnard's basis is a new form. A mere technique, a mere tool for approximating things. It isn't the real thing. The real thing is the swing functions, the analytic functions. Monarch space and Hilbert space are somehow just particular ways of defining particular functions. Just as one might say that the nuclear space is a limit of... Bonnock spaces, but you can always make them Hilbert, because again, the fact that the crucial point of the bonding maps are these nuclear maps, which are qualitatively shrinking, qualitatively smoothing and shrinking along the way. But in fact, you can even see a difference between typical spaces of intensive and typical spaces of extensive bonding. There are nuclear topological spaces and nuclear bornological spaces, so you can have direct limits of solar-like spaces, or you can have inverse limits of solar-like spaces, and those give actually two different flavors of nuclear forces, nuclear in the bornological or nuclear in the topological, depending on your category, so this is also...

42:30 In the future, which you don't find in most of the books, they talk about only the topological, but the natural, because it's a business of trying to replace extensive by intensive, by the way. Yes, yes. And certainly that's a very interesting point about the Danak space as being essentially a kind of tool of approximation rather than the fundamental structure inside which all of this lives, which is certainly the way it's presented. And indeed, you see, when I say it's a limit, you can... You could take many different sequences with the same limit. You could take only balls or only squares as domains or only... And certain configurations of derivatives, as long as it's co-final, you get the same set of limits books, either direct or indirect, depending on whether it's extensive or intensive. Thank you very much. Why is it called nuclear? Yes, I was just going to ask you that, actually. I'll have to keep an eye on it. So this really abstracted the essence of... Schwarz's kernel theorem, the kernel of the nuclear system. Yeah, I guess that makes sense, yeah. Well, I mean, this is standard translation. German, they have Kernphysik. And it's a surprising statement in a way that every operator between spaces at distributions is actually representable by a coordinate, so even though the map of Hilbert spaces is not necessarily represented by a matrix, it's something sort of analogous, in the smooth case, for the nuclear spaces, definitely not for von Hoppe-Hilbert spaces. In some ways, the binary and Hilbert spaces are more complicated, even though, again, in most cases, if you start with the subjective approximations, then what you're getting at is some kind of mystery. I mean, there is a parallel with the way that, you know, one thinks of the infinitary piano algebra, again, as a tool for subjective approximation of the process of one in the other.

45:00 Did you actually get a chance to talk to Angus about saying topology? You were saying you were disappointed that he hadn't had it, he didn't speak about it in his talk. Really, I'm sorry about that. Well, I hope another time. Okay, lots more I want to talk, but we need to make a move. Thanks again. Right, and I have to sort Marvin out, and then I wanted to ask, and Steve actually, and also to talk to Andre. I don't worry, I'll take care of that. Well, yes, we are actually. I just need to run and use the room. Sorry, Andre, I need to see you about your air ticket, I think. Don't you, how did you, did you come here from the U.S. or did you come down from... I came to hear it from Paris and Slovenia. Oh, you did! Oh, good, good. I'm immensely relieved to learn that because, okay, because I thought we might actually have to be paying for your air ticket from the U.S. No, good. I'm relieved to hear it. But as one of the speakers, you are entitled to some help on your travel expenses, so if you want to tell me, I mean, if you work out, you just drove all the way down from Sweden, I mean, if you're, I mean, is there any way that your institution will be able to take care of that? Yes, so I have a project, I have a project in mind. Oh, okay, right, okay, that's fine. But, I mean, I wouldn't mind. We'll take care of it. Well, to be honest, we don't know if, well, how do you... Yeah, well, obviously we'd be happy to put something towards it. I mean, we're taking care of your room here, as you know. Oh, you are? Oh, yes. That's what I understand. No, your room is taken care of here, so that's okay. Actually, today my wife is coming. I'll just pay. Yeah, obviously you can pay the difference between... Oh, thank you very much. That's okay. Good, good. Actually, that relieves me, because we thought we had made a boo-boo on this, We looked for the people flying from the States who were the invited speakers, and we thought, and then somebody said to me, he said, oh you've forgotten André Maas, because knowing that you were at Carnelia, I was like, oh god no, and I thought I was going to have to find about another 800 colours, which quite honestly we would have had a difficulty finding.

47:30 I thought that was dangerous, I think, so this is dangerous for me to do. I wish I kept my mouth shut. I'm sorry. You realize that is the way to do it. My wife is coming to the hotel. Those are, you know, I remember the time thinking that Robinson was a nice old guy. And he died at 54. So he was 10 years younger than me. At least, that's what I am now. I don't think of myself as an old guy at all. Everybody else does. No, no, no. Bill doesn't. I'm the youngster, so it's Bill's. No, we're, well, no, we're kind of, well, more or less kind of evil. More or less. In fact, somebody was saying to me the other day that, I think it was from Bill, that you had really not changed at all since he shared a room with you in, I wasn't actually, in 1965 or 1966, I don't know, well, there wasn't, there was a little bit more charcoal, yes, that's right, he said there's probably a little bit more charcoal in the beer now than there was then, but other than that, really, you haven't changed. When was the last time you were back in your hometown in Indiana? I haven't been back for 25 years. My whole family is there. I'm so sorry, I was wondering if I could ask you again, I'm so sorry to ask you to repeat yourself, we have some much more important things to do, but I was so fascinated by what you were telling me yesterday about the way that the Programme and the College of Geometry have recently been taken forward by this work. I really would like, if you can… No, tolerate having to go over it again. I really would like you to give me a further resume at some stage, is that all right? Especially of the, what you see as the, you know, decisive development that Kainish, is it? Kainish has contributed.

50:00 Perhaps. I think we're ready to move. Are we going now? Yeah, it's okay, we'll go. Okay, we're ready to move. I mean, I guess the best thing, the easiest thing for me was... You said that there were really two aspects of the programme, two aspects really, of the definition of scheme, as it was left in Gorky's hands, which have not really been satisfactorily reconciled until this work. You had a student who had worked a great deal on the subject, but I'm so sorry, let me... By Mackey, by Mackey. Mackey was in the functional analysis of the ergodic motion and he had invented the motion. If you have an ergodic action of the group on a space, the book is simple. You have people who have the spaces of space. But the action is so horrible that you have no doubt about the set of orbits. Yes, this is why I was asking you yesterday about the connection with Etan Du. This was resuscitated by Alain Conne in his first work. Alain Conne used that kind of idea when he did the work in the 70s which gave him the field middle, which was classification of obey to algebras. At that time they did not contemplate the things in the viewpoint of non-committal divagas. And obviously I can see how the role of the orbit space, you know, connects with the ideas of the symmetry, internal symmetries of points.

52:30 Yes, yes. Okay, I'm sorry, just trying to get a feel for it. But then, it came also in algebraic geometry because when people try to... To define what they call moduli space, which is a classic universal family of some geometrical objects, it is a universal family of elliptic curves, or curves of some chairs, and the point is that, of course, it has always been recognized, and there is, of course, you can do that by using analytic methods, automorphic functions, and things like that, but there is also the problem that... A given curve having a few curves of exceptional automorphism and elliptic curves, so the problem is that it is an object but we are fortunate in standard analysis, we are fortunate that the field of field number has no automorphism. The reason we can name a number. I mean, it doesn't depend on the model you choose. I mean, the real numbers have many, many models. Sure, of course. We know many models for the real numbers. But when you say pi, when you say pi, or let's say, or 2.3, it doesn't depend on this name. That's what Darcy, Darcy means. I mean, it means that at length, the parts that you have known to work with are the real numbers. Make it possible to name, and you don't have to say, well, it's by in some order. No, no, no, it's quite canonical. Canonical, okay. So, one, two, three, four. Are you going to go this way, John? Okay, fine. Yes, that's what I thought, but I'm just thinking of this bus. Hang on. We don't want you to end like Brouwer. I'm sure they have no intention of doing so.

55:00 Sorry. Yeah, this is a shorter way than we went yesterday. It cuts off a corner. Certainly you can. This is Santa Maria Novella. It's one of the greatest Renaissance churches in France. It has the great Masaccio Trinity on the wall, on the left-hand wall. And one of the great fresco cycles by Gerland Dyer in the Clydesdale. It also has perhaps the most geometrical of all early Renaissance facades. It has this extremely beautiful and lucid facade by Alberti. Yes, it was a large conventional church. It was one of the largest religious fraternities in France. No, but it has this quite magnificent bazaar, which you'll see in a moment. I did not. I did not do it. I was wandering out here. No, but it is. Oh, from this side. Yes, no, from this side it has. I don't know. No, it has one of the very noblest of all the early, very early sort of renaissance and stars. But to continue with your... Just a short comment. I visited the office yesterday with my wife, and I was surprised that religion in Firenze is not rightful. No, no, no. Not at all? No, no, no. Virgin and the child is more... What is more... Careful. The Virgin and the Child, and angels and saints. No frightening things have been seen. And no terrible crucifixion, no horrific, nothing like the Grunewald altarpiece. No scenes depicting, as it were, the horror of the crucifixion on which Northern European art particularly frequently dwells. Of course, there was a reaction against that. Of course, Savonarola, who felt that the Florentines had become far too lax and... Far too much inclined to dwell on sweetness and light, and of course wanted to make a bonfire of the vanities, so it led to a reaction from the, but no, but you're absolutely right, that's a very, a very perceptive point, but to go back, I guess, actually, if you can really just repeat what you said to me yesterday about the, how the group, yeah.

57:30 Sorry, I'm just wondering where... Oh, it's all right, they know where. Some technical points about the homology of books. And so I said, okay, we take the complex to define the common root and so on and so on. And they said, no, we take the classifying space. Yes, yes. And I was more or less surprised by that. To me, homology of books has become a device for taxing. This is what we were saying yesterday. This was the aspect I wanted to get my hands on. Saying there had been this duality in the subject. I mean, in a metaphorical sense, for those with dualities in technical sense, so between, yeah, yeah, and so this is what I wanted to hear. So, what you have is the cohomology of the truth, or cohomology of the truth. So, for people who are topologically kind of minded, and I was more actively kind of minded at the time, now I'm I'm the great Huguenot. One of my masters, I've been chivalry, who was a Huguenot, like any mathematician. Maybe even a Calvinist. But anyway, there was this discussion between yourself and Sir as to whether the classifying space or the group action was the more fundamental starting point. Yeah, okay. And so, the real, I mean, the really noble idea was introduced by Goel. In his thesis, and one around his thesis, which was the idea that, the general idea that, very interesting idea, that if you have a space and a group action, if the group action is not good, right, not good, that means many things, and technically, see, it's a...

1:00:00 If there are poles which are fixed above some element of the globe, the action is not free, then what you can do, if you work with homological invariants, you can always replace the space by another space which has the same homotopy type, which has all the same invariants. But now the point is there is a contraction, there is a general contraction, which was what? ...given in some particular cases by Borel and then in very general form by Milnov, and by Milnov gave a more general formulation, and which is associated to any given space in the group, actually another space, at the same moment of the time, but now the action of the group is fluid and it moves freely, no? No elements of the group exist in any form. And then, after this replace, when you can begin to play a reasonable game, of means you can consider the orbit space and the angle of the orbit space and so on. Strangely enough, well, strangely, not strangely, interestingly enough, the same idea was recited in physics. And, first, without any recognition that there was something in common, but now, of course, people are more aware. Now, of course, people are more sure, so, indeed. It's also gauge theory. Indeed. The point is that gauge theory, I mean, to say it shortly, I mean, the starting point by Heisenberg was the so-called isotopic spin, which says that, of course... If you have a proton and a neutron, a proton or a neutron, the only difference, I mean, it was striking that a proton and a neutron are on both the same path, up to one percent, not exactly, but up to one percent. So the major difference is that...

1:02:30 These are the things that, of course, the autonomous energy is charged in the neutron atom. So, most of us, on the very day that we saw that this multiplexing mass could be accommodated by taking into account some minor electrical effects, which this idea has been pursued. Of course. Well, and of course, gauge theory is a great subject in mathematical physics. I hardly need to speak to you about that. No, but it's more than the same idea. No, it's further on. It's further on, down the other end. Remember, we're coming to it from the other end. When you speak of a nuclear, now, of course, for the nuclear, the main nuclear, I mean, the nuclear doesn't know whether he wants to be a proton or he wants to be a neutron. And there is some ambiguity. Yes, sure. Some ambiguity. And this ambiguity means that there is some integral, so the nucleus has some integral symmetry, which is the isotopic group, which is exactly the same, I mean, a point has integral symmetry, and, I mean, technically, I mean, technically we had to deal with similar problems, and then there was this marvelous invention by Kadeyev and Popov, of the so-called world. And when you came to me about Gotham, about Gotham, it's very close, very close to the idea of Borel and Milneau, of classifying things. Very similar. Not quite, but... Yeah, but that's an extraordinarily subtle analogy which I hadn't seen before between what was happening in physics and algebraic geometry. Perfect. Well, I think they're going to have to stop for us, rather than... Let me give you a simple example. Take a two-dimensional trove, and a motion by parallel straight lines in the trove with an irrational slope.

1:05:00 So, you start from a square, you draw lines with a horizontal slope, parallel lines, and then you wind up your square to make a torus. So, you have now put each trajectory in this space, and because irrationality is dense in the torus. But if you want to have a grasp on the set of these lines, what you can do is to take a transversal, that means around the point you have all the flow going on. Locally, I mean locally, you will just see a bunch of straight lines. And if you are not interested, if you are just short-sighted and you just say, well, I look at what I see, I'm not waiting for a long time until things go back, then what you see is just a bunch of lines. And then you say, okay, that's easy to get, to control that. I take a transversal line which cuts every of these trajectories in one point. And of course, but the point is that this is not unique. Yeah, of course not, no. Unique. And what does is that it's not unique. And what happens, I mean, due to the Witten map of, well, that's the beginning of the Witten map of Markari in a dynamic. What happens, I mean, so you have this small... Let's say you have this small line, and then from each point of the line you draw the trajectory, and then they go back, and after maybe a long time, sometimes very, very long time, they come back, but not in a different place, and so it means that this... Small transversal is not quite, a segment is not very much different from a single point in terms of topological imperience, but we'll come back and there is an internal symmetry. There is an internal symmetry operating. Exactly, exactly. So that's exactly. Something a little like, something a little like akin to spin is actually operating as it were within each point. And now, of course, and this does connect up with the whole theory of Etan-Do, doesn't it? It has often been said to have a deep connection with ergodic theory. Now, what is really interesting is that, I mean, for a long time there was one, Hermann Weill knew the connection with physics. Yes, in fact. With the physics of his time, not the physics of the last years of the 20th century, but from the first part of the 20th century.

1:07:30 It was more or less forgotten. And Hermann Weyer envisioned both the logic, the mathematics, the geometry, the physics. He was the only one. And interestingly enough, if you read the obituary of Hermann Weyer, written by André Weil and Claude Chevalet, They described all of his physics and the philosophy at once. You spoke of this in your talk. Yes, and it is very fascinating indeed. Of course, the account of the mathematical work of Hermann Weyl by Chauvelet is beautiful. The plot, except that they completely missed the depth of interconnections with his ideas in physics and indeed even in philosophy. I completely agree with him. And they say they have some comment that Hermann Wally was like a proteus, I mean, like V1. He wrote like a hippopotamus, did you say? Proteus. Proteus, yes, yes, yes, yes, yes, yes, yes, yes, yes, yes, yes, yes, yes, yes, yes, yes, yes, yes, yes, yes, yes, yes, yes, yes, yes, yes, yes, yes, yes, yes, yes, yes, yes, yes, yes, yes, yes, yes, yes, yes, yes, yes, yes, yes, yes, yes, yes, yes, yes, yes, yes, yes, yes, yes, yes, yes, yes, yes, yes, yes, yes, yes, yes, yes, yes, I think, well, mathematical motives as well. And he clearly thought of a good deal of it in private, did he not? I mean, from what you were... No, you hinted that, yes, well, you have to have... Okay, I think we're going to have the last, because Bill is about to speak, but I really do want to hear more about how this all led into the non-combatant. Please, thank you very much.

1:10:00 I think so. I think so. There is going to be a buffet, but I think it's after, it's at, yes, no, this will be for us. Oh, okay. I thought it was a, on Saturdays, I thought it would be. No, I believe it's today. That's what the, that's what Alberto Brunzi announced at the beginning of the session. No, no, no, it's okay, no, no, no, that is for us, I'm sure. But it's for after the next speaker. That's off the next session. No, I believe that's right. I'll just check with a... Well, is that in the buffet? Yes. Is that the buffet that we're going to have today? Yes, it is. No, apparently there has been a change. It is today we're going to have the buffet and not tomorrow. Yes. Oh, that's excellent. Okay, excellent. What about the poster? What was said about the poster? Just that if you hadn't already got one, the people who have already left, who wanted to have one, just leave their addresses, but otherwise you should have one before you leave. Tomorrow, today, when will you leave? I'm afraid, I'm sorry for this, I'm informative. I'll have a check with my own currency. I'm fairly sure that they're available now. I'll check. If anyone is more than another, can I ask you just a question? Do I need anything you can... Anything I can do to help? Other than you can, you can tell me where the V.S. Anigidio is. Yes, that's a very good question. I think we're going to have an announcement about that before the start of the next session. And then we have to do that. Let me just ask. It's very near to here, in fact. I believe it's close to here, the V.S. Anigidio. Now, I don't want to give misleading directions, so I'm going to check with Alberto Parisi, but it was said on the first day that it is close, very close to where we are now, that it is certainly in this area.

1:12:30 I believe it's not completely explicit, but I'll get that information to you. No, no, no, no, Professor Heller wanted to know about the location of the talks tomorrow, which I'm not 100% certain of, but I will make sure we have an announcement. Alex was just wondering about the location of the Ascani Gidio, where we are today. Where is it? Show him where it is, hang on. Hang on, I'm flipping it. Sorry. We actually have most of the cash here now, but I think we probably don't want to carry around the large amount of cash. Shall we wait till we're back at the hotel? Okay, well, I still have to go and get some more out of the distributor. I'll try and avoid that. I doubt that anybody's going to dare to try to rob you. No, but some people don't feel... No, some people don't like the idea of carrying around a lot of cash on the street. Okay, well, just tell me when you... I'll certainly have a few by this afternoon. The only thing is, when I enter, I can't really ask me if I have about a thousand or whatever. Well, you'll be eight hundred and sixty, so you'll be all right. Are you actually asking these lectures? Are you actually asking for more than a thousand? Thank you for watching.

1:15:00 Sorry, I was shamelessly hoping that you might be able to continue the story from where we left off, maybe another time, this afternoon, let's hope that we, no, no, it's just me, no, but, I mean, there is a very instructive story to be told here about how the classifying space and the group cohomology were brought back together into the same framework, and I just would love to hear it again, especially. No, okay. Whenever you feel the inclination. I would like to go to some sites. Of course. Okay. I want to try and provoke him into talking a bit more about the stuff he was saying to me yesterday in the courtyard, which I really, really, really, really, really, really, really, really, really, really, really, really, really, really, really, really, really, really, really, really, really, really, really, really, really, really, really, really, really, really, really, really, really, really, really, really, really, really, really, really, really, really, really, really, really, really, really, really, really, Kishler. Kishler, Kepler, something like that. Anyway, this off-screen guy. He's gone from the spring. And he said, you know, say I made this for an office at Mollegato. Are you thinking in terms of classifying? Are you thinking in terms of group at Mollegato? Are you thinking in terms of the classifying space? I don't know, I don't know, I don't know. And, well, especially since Conlon, remember, you know, he couldn't run as fast as he did, you know, and he had 25 years, he had guys, you know, and now he's quite certain they've got the key ideas. He spends everything together beautifully. It's essentially the achievement of this. I don't think he's actually a student of Modo's, but he's a collaborator of Modo's.

1:17:30 Krinik, Krinik, you know, you do know, okay, doesn't matter, Krinik. God, I'm messing that up. It's Krinik. Yes, that's the guy, Krinik. There was this meeting, a big meeting, again last year, of this brilliant co-attender, and it's quite so hard to see him as staff, but it is. Exactly the right framework to do everything, but above all to kind of unify those two viewpoints. I'd just like to get a glimpse of the scenery, which is, you know, and then with a little luck you might even be able to relay glimpses of it to me. But, um, I would like to do more of that about ramified. She turns it into rectified coverings instead of quite geometric morphisms, and it turns out to be a terribly long story with no particular end to it. The bill itself is a three-line definition. It's just a decidable object in the slice category. Yeah, this may be the difference between ramified coverings and asking whether a geometric morphism is ramified. But my answer could answer the question you're asking. Yeah, that's why I don't ask. Every question related to... That's why I don't ask great questions. To be honest. That's why I prefer to ask. Can't hear. Did, is there somewhere we can go for coffee? No, the buffet is for later, the buffet is for work. Yeah, well, not much. I'm still in it. I can't tell you. I'll let him talk. Yeah, so they're setting that up for after John speaks. This idea of a geometric morphism having a ramification has something to do with you look at the inverse image of the suboptic classifier. We don't know that. What is... Well, I mean, if the base is Boolean, then the sub-object classifier is 2 and the inverse image is still 2, so the usual notion of separable slash decidable and locally such is simply that the characteristic functions of the diagonal map, which in general goes into omega, actually factors through this thing that comes from below. Just take a general thing that comes from below, it doesn't have to be two, so you could say that an object is relatively separable or whatever, if the characteristic map of the diagonal factors through.

1:20:00 You know, gamma, upper star, omega, this is what you're smelling, I guess. Now, I think, to say factors through, you'd want to say that's some map on a function space as an epimorphism and, you know, suitably internalize this idea. But you could arrive at a notion of locally QD or whatever. I think there are some technical difficulties in doing this, but that's the idea. And the further idea is that that's somehow related to the un-ramified case of the complete spread. If that is somehow, in fact, unbranched, as it has to be, then it should be relatively separate. Well, we're going to get our own effects. I just want to know where it was. Well, I guess what I don't understand, because in the program it says 11.30, but Alberto just announced 11. So, yeah, yeah, yeah, yeah, yeah, he's the chair. Okay, did we get that? Sorry, I got distracted. That certainly seems to make sense in terms of... Well, except that it's still kind of programmatic. I mean, the final explanation of when is a geometric morphism ramified is something like this should work in general. I didn't quite pick up on Peter Johnson's points that this is actually... This will only work, this definition of panoply. Well, you'd only have little stone dots in a pre-sheet category if every object of the category you're taking pre-sheets on has a global section. But if every object has a global section, then this other decomposition isn't going to occur. Right, so there's not, in fact, a kind of...

1:22:30 Yeah, so if you want this situation, it's going to have to be in chief categories, and pre-sheet categories will never... I will not give it to me, I think. Which in a way, Bill was happy with what he was saying. The whole point of this argument was to say we need to look at more general categories. Yeah, that's the whole point, isn't it? Well, now it's even more general than in previous categories. Yeah, I mean, it doesn't actually affect his point, you know, that Peter's definition, that Peter's frame of, you know, all... Topo-allocalloc is... he's using all in a very curious sense. But is this connected up with the definition between Petit and Gros? No, it's a remote sense. He's saying everyone has a localloc cover. Now, Bill is saying, okay, that's true, but these localic covers omit some really important behaviors that you won't see if you always look at these. And it is connected with the Petit Gros description? It seemed as if he was saying it was, but I don't know. Okay, do you want a coffee? Yeah. I think I just about understand where they came from, but I don't think that there's a certain theory after all. Yes, that's what I thought, and that's what Alberto was saying. People are very confused now. Well, there's nothing. I'm afraid I'm lagging behind the confusion. Oh, well, you haven't even risen to the level. Well, perhaps, and maybe I don't need to achieve it, maybe it'll be clarified before I even need it. Is everybody for a coffee? Yes, that's what we just came down for, yes. Oh, a proper coffee across the road, yeah, sure, sure. Have we got, yes, we've got time, it is 11.30, of course we have, yeah. Yeah. Well, that allows me to go to the bank, too. Some things that happen with schemes is that, for example... Perhaps students at Harvard are interested in geometry or number theory, except for another way they expect the theory to be in an elementary school. Denninger has this interesting project of trying to come up with a cohomology theory that will be characteristic zero. The scheme of cohomology only achieves characteristic zero now by...

1:25:00 This is where we went the other day. Hi, um, cappuccino, cappuccino, cappuccino, cappuccino, cappuccino, cappuccino, cappuccino, cappuccino, cappuccino, cappuccino, cappuccino, cappuccino, cappuccino, cappuccino, cappuccino, cappuccino, cappuccino, cappuccino, cappuccino, cappuccino, cappuccino, cappuccino, cappuccino, cappuccino, cappuccino, cappuccino, cappuccino, cappuccino, cappuccino, cappuccino, cappuccino, cappuccino, cappuccino, cappuccino, cappuccino, cappuccino, cappuccino, cappuccino, cappuccino, cappuccino, cappuccino, cappuccino, cappuccino, cappuccino Okay, well, what we do have is schemes for non-commutative rings. Well, that's what he's now saying that this Kleinecke work gives us. Okay, I think it's that, because, I mean, Mowry's working on that, too. Yeah, yeah, well, that's why it was very ironic. I missed Mowry's thought to hear Cartier talking about this to Doyle. Yeah, but it's quite clear that what Cartier's people are doing is a completely different line of equations. And he said that he thought that Scott's talk about equilateral spaces also had something interesting to do with it, which I really couldn't tell you. Okay, it is. Yeah, yeah. Well, it's to do with the way that you think about equivalence, quotients of equivalence relations and topological groupings. I didn't get it at all, but I recall it. It's something to do with the right way of thinking about equivalence. Hang on, I need to put in some more notes. No, it's only three. Oh, okay, okay. Somebody gets a tea. Okay, that's yours then. Okay. Hand me a couple of sugars. Sure, sure. Do you want the... I was going to say it's not actually, I was going to say it's Brazilian. They re-refined that and just put coloring on it now. Isn't it ridiculous? Well, maybe they don't in Italy. We'll keep that in mind. Well, I remember visiting the States for the first time in about five years. And going into a, you know, I was thinking, the one thing I missed is rye bread. You can't get rye bread in England. No, yes. So I went into this big, huge supermarket, and there was of course a whole wall.

1:27:30 Brother Greger was studying who combines vanity and ignorance and heroic abortions. Who had written a book on functional analysis for dummies, right? So he condescendingly told Vitale to go read this book. Vitale said, he looked around, because they're always afraid that KGB will be listening in. I tell you, he said, if I had written that book, I would go to every library in the country and make sure I'd taken it off the shelf. Oh dear, such is life in the UK. Oh dear. Shall we mosey on that? Yes, sir. Are you sorting it out yet? Yeah, well, yes, I mean, we're getting there. I've now got enough cash in my pockets to pay Marta in four-way cash. So then this? Is to be able to sort out this bugger here. Okay, so my next piece is sage advice. Older man, very young. Don't get mugged. This is funny. No, it's not funny. It's not. No, 7,000. I saw him get... I saw him... Well you didn't actually see it but I saw the results and I was already absolutely no deepest bloody shit when that happened and it wasn't funny it was very very very it was that was certainly the worst day of my life yes I arrived in Besançon at about 5 a.m. in the morning having had to take this money and cash out of the bank in Nice the previous day Believe it or not, I've actually gone into the manager's office to get the money and to count it out. Of course, one of the cashiers that I had to deal with first of all said I'd have to come back in the afternoon to get it. She took me away. And these two guys had actually followed me, clearly, because I knew I was carrying this very large amount of cash, the whole way from the bank back to the hotel and then waited in front of the train. I got on the train and one, well 99% certain it was these two guys, one was obviously sort of macro and the other was Vietnamese. I went to sleep on the train between these totally crazy back points of going directly without sleep for about two weeks, remember what I'm saying?

1:30:00 Well, except it looks like something is at last beginning to happen, except that you don't have a sub-object classifying it. So in what sense is it a topos? And it turns out, well, there is a sense in which it's a topos. Well, but even so, what happens with that topos then? I mean, what is it? It's a kind of weird kind of one on the side. It's kind of par for the pass. What did you learn from this? Well, how to do orthomodular lattices inside of robots. And still have something that works a bit like the sub-object classifier. I guess is the answer, because these people are motivated by the quantum logic. Well, yeah, in some long run, but it's been a very long run by now, and we're still... Motivated. Well, yeah, but you made the point. That's a good thing. Yeah, yeah. You yourself said it. Well, we can't start without John, is that it? Yeah, yeah, yeah. Good, well, that's... You can't... You can't chop it up into pieces at all, and or... and reassemble those pieces, so to speak, to, you know, the original continuum, so... Look, I mean, yes, in that respect, I guess, maybe your first thought is, yeah, I mean, these various representations we continue to all suffer. Right, and the fact that not only, you know, the items are disconnected, but they're very, very far from the kind of extreme cohesiveness that I don't necessarily view more or more personally, but certainly by the priority. And so, yeah, I mean, there are both the problems, you know, well, the representation of all would be quite inadequate for the task of doing justice to that, but the striking thing is that you can... I mean, now, as you know, of course, if you make some of these constructions on the right kind of tops, you can bring these sort of edible constructions closer, in a way, to intuition.

1:32:30 I mean, for example, if you take a model of synthetic differential geometry, I know a smooth topos. And you, of course, I mean, of course, the thing is the design, the usual, the real line there, the primary object is, of course, a decomposable. I mean, it's built into the, you know, the basic axioms. But, interestingly, you can, you can also construct the real numbers, you know, the usual way. I mean, you can do a dedicated one, right, in that topos, by taking the power set of the natural numbers. It's quite interesting that, of course, if you do that, if you make that construction, the arithmetical construction within the Smoot office, then that arithmetical continuum will share certain features of the, it turns out to be a decomposable, or at least… It does have certain properties of cohesion, which you simply wouldn't possess if you liked it in the usual, say, in the policy sets. So, I mean, I don't know. In that respect, you might well say, of course, it's true, that's not, there, that's not, foundationally, that's not terribly useful in a way, because, of course, in the smooth talk, you've already got this, you know, the smooth, you've got the smooth line. In principle, perhaps, the arithmetic continuum could, in some way, have to be closer to this, you know, 3D composability and these intuitive notions, at least it's possible if you vary the mathematical, you know, so they have to be a mathematical framework. In other words, perhaps making the gap between the vials as unbridgeable can be narrated. I mean, one way to relate, I mean, we all know it, to relate the totally disconnected from the continuous is to just think that motion sequences work by forming a good one.

1:35:00 Or, one example I like is that if you take the downward flow set and the rational number to get it down to the wave class, then you have to take the retract in order to move the two kinds of chunks together to get the continuous set. So, the gluing comes from a separate operation. So, an interesting example of that is the constructive set theory that Michael and Axel have spoken about. You don't have the power set, but you do have the motion sequences, and of course it's related to that fact that, in conclusion, it's a continuum, a bounded sequence of rules doesn't have to have the least of the bounds, so the power sets come to haunt you when you think you're trying to... I would have to go back to advise for that work. I'm pretty out of my own presently. Basically, what you should remember is that Ryle's work in the foundation is a small fragment of the enormous amount of work he was doing and has been doing in mathematics and in physics. I know it. So it's conserved, the PhD thesis, which he made in the 29th year of his program here at the School of Mathematical Sciences. It's a borrowed foundation of concepts in mathematics and physics. So he's very aware of the phenomenon. This is your concern. My only concern is that you're coming out from the physical inside, yes, and on. It's clearly not the one you get from scattered points of the set, except here. But what he's using mathematics for is to organize physical work. So, first comes structure. Then perhaps points may be derived, of course, and they are very useful, there is no doubt, but there is a derived fact after the direct correlation of the phenomenon wave with the phenomenon in the system, where you need to organize things which are connected, where you have action states which requires, requires confusion, and then perhaps there may be points. And this language, again, is Rassel. Rassel thought he was going to write it wrong, because by the axiom he made it archaic.

1:37:30 So, as he said, in mathematics, he did a little thing, he writes down, okay, that's the basis for a very nice proverb, but if you analyze it, you will never work on a frame. I mean, it is really not... No, that's quite right. That book begins, you know, with the concept of the incompleteness of arithmetic, meaning he has a radically and, I guess, a very isolated attitude concerning the idea that mathematics is a representation, because it means that it invests a lot of things in parallel physics, and that is the motivation. And that he probably, as you said, seems to be the case in our analysis. It's an important balance, actually, that we are concerned about. Yes, I do think so. Sorry, sometimes it's sort of hard to, I mean, you're right, Weill's concerns were so, so, as I say, I didn't mention that he was writing space time matter before. You know, he tried to find some way of resolving all of these problems. Sometimes it's quite difficult. I know, I know. And of course that also of course starts with a philosophical, you know, sort of ringing philosophical declaration, you know, about the, he uses these sort of metaphors about relativity unifying the inner and the outer. Do you remember the beginning of it? I just wanted to remark that the use of the word intuition is quite ambiguous. I mean, there are a number of opposites, in fact. I suppose that's a fair way of putting it, I mean, yeah, to the extent that Hilbert, of course, you know, he, Hilbert was not

1:40:00 The Hilbert wasn't very philosophically inclined, and so did Wiley was, who was very much influenced by Wiley, by Gerard Wiley, who wasn't. It wasn't Hilbert who was influenced by Wiley at all, but Hilbert was influenced by Kant. Yes, he was, to the extent that he was, to the extent that he was as far as geometry was concerned. I mean, ten more questions. I think there are other reasons, too. I mean, he's fallen out of it, but, you know, he did have it. There was personal friction, of course, between Hilbert and Hartley-Lewel, so there was perhaps all these things together, but yes, I suppose that's a fair way of summing it up. When he writes his book for the whole public, his geometry and algebra, his intuition is right in there, but it's not Rauher's. Oh, well, yeah. In that respect, it's true, it's probably closer to, as you said, but I don't think people like Arrhenius. Hilbert doesn't seem to have been greatly concerned, if you like, with the whole kind of apparatus that Kant elaborated, if you like. I mean, as far as his, kind of, geometric intuition is concerned, I mean, that particular phase of it, oh yes, I think you can see that Kant, a little bit, even in the early one, I mean, in the Grimlockian or Jahn, you know, he quotes Kant approvingly, you know, on that aspect. But I think the rest of the... You know, the rest of the apparatus didn't really interest me, but it did interest Weiler very greatly. I mean, Weiler, in fact, as you may know, Weiler was a communist. I mean, he studied common philosophy very seriously before he went to, you know, to university. And it was Hilbert who shattered his views on, you know, on the common philosophy as such. Because of the advances that Hilbert had made in geometry. And it was the shattering, according to Weiler himself, it was the shattering of his... Ideas of Kantian philosophy by Hilbert that led him to take up phenomenology, whereas Hilbert himself didn't really have very much interest in this aspect. This was always the difference between them.

1:42:30 Yeah, I think that apart from the problem of distribution of continuum, what is important is also the reduction of discontinuity. The miscibility of points. I mean, we tend to think of points as a way to make state of whatever analysis, which sounds to me, you know, a big exception. I mean, in the light of quantum mechanics, for instance, people think that points may get internal structure, such things. So, how does the world picture it? We may discover a continuum to the points themselves. Well, no, that's, I think that's a very, that's a very interesting question. I mean, and it suggests, well, yes, it's true, you know, I mean, a point is defined in Euclid as those parts. And look at the, it's kind of, you know, it's kind of the times of those lectures. How the notion of a point, and also in Chris Mulvey's, how the notion of a point is actually all elaborated, and it actually turns out that it is a highly sophisticated notion which is actually undergone across a considerable number of dialectical developments. Indeed, indeed. Maybe the idea would have been, and this is sort of implicit, that perhaps the notion of a point could get elaborated. ...sufficiently so that it would be reasonable to concept or to analyze the intuity or the cohesiveness, if you like, in terms of that new notion. Of course, we use metaphors like that. Of course, in the intuitionistic mathematics, you know, you talk about ones sort of smeared out and, you know, you don't, you, you, and of course, this is something my power was trying to do, which you would see. But I think that the notion of point is obvious, as Bill said, so we have it. But at this point, even in the mathematical sense, now, in the last couple of years anyway, it's undergone as much development as any other mathematical concept.

1:45:00 Well, of course, there was the argument also about, I mean, even before these more elaborated notions of points, I mean, Aristotle and Brentano didn't think of them. I mean, in truth, but points just don't exist apart from a continuum. I mean, they can't simply be extracted out of a given, given some kind of independent existence. Yes, exactly. This is what Aristotle maintained, and it's certainly what Brentano maintained. And perhaps, in some sense, it might have been what Dedican might have accepted, you know, and surely he didn't really think, I'm not so sure, but, you know, that what he called points, it's just, after all, these are highly elaborated things, you know, equivalence classes, coaching, whatever, or Dedican cups. And it's everything that came up earlier, isn't it? I forget, as I did yesterday, about Dedican thinking. I think it was, no, I mean, it was Dedican really thinking, well, the continuum is already there. I mean, so to speak, the continuum is already there. What he's doing is kind of elaborating away an analysis of it. And then somebody remarked. Did anybody, it was asked, who remarked? Who said that you could then regard him as human? Who was it? I don't know. Was it anybody? It was the paper. It was the paper. All right. I'm going to do it. The paper was ready for discussion. Did it stop? I hope I've already remembered this. If you do the localec topos over the hiding algebra of open sets of the reals, then first-order arithmetic is classical in the localec topos, but first-order real numbers are intuitionistic, so it is possible to be classic about finite objects and be intuitionistic about interim objects. And then we have the last question. Well, just an observation. It seems to me that the number, the notion of point is problematic.

1:47:30 The notion of subsets, of arbitrary subsets is also problematic, as well as the notion of function. No, that's quite true, but I didn't get to that. I have a perception of it for a while, but I can't. Yes, no, that's quite true. I think that's a very refute fact. It is a coherent idea to have functions without subsets. No, I agree. I agree. There are two sequences of functions. I can't imagine. Yeah. Thank you. Alberto, time to reassemble. Can you just remind people of the time to reassemble? Oh, okay. That's basically it. Do we have the... Do you have to stop the program? Ah, cool. Okay. Well, I'm assuming everybody has a copy of the program, didn't need to be told. Tempos theory. Tempos theory, fine, okay. There was some confusion this morning as well when we were supposed to start. Sorry, may I just? Sorry, okay. Yes, thanks, sorry. Thank you for your attention. There is this restriction to three-sheet talk. No, it's not. Oh, well, I'd like to understand that, because I was a bit thrown by that. No, it's a pity that we couldn't have discredited any of that discussion in public, actually.

1:50:00 I'll ask you later when we're done. But I noticed there was quite a lot of additional stuff that had gone up in the last interval. That's what we'd like. Well, that's why I was too busy running around sorting it out. I'll sort out André Joël now and take care of that. That was an excellent discussion. Very good. You did indeed, and I am of great interest. I haven't had a chance to really think about it, seriously, but I noticed you actually had a reference to Collins' paper on Poincaré that matched your references. I hope I'll have been able to prove to the general trust in conclusions that this was one instance where the... One carries some methodological fixation with not having his premises contaminated by any admixture of ideas concerning the structure of the natural world was actually turned out to be... I have another one there. This is... Oh, no, this is the same, yes. No, this is today's version. Ah, you've been realizing it. Yeah, because it's apropos of what you just said. Here I've included a random algorithm for breaking a stick. At random. So now we have an experimental procedure which will give you... Uniform distribution and independence of the breakpoint. So that's relevant to the point that you raised.

1:52:30 Yes, I'd be interested. May I take a look at that? There's an interesting study to be written of the origins of the barycentric calculus of the whole of that chapter of 19th century. Well, I suppose one would call it. I suppose it's on the borderline, in the borderlands between geometry and analysis. Can there be a book on geometry? Yes, there were, of course, plenty of surveys written at the time or shortly after its creation, but I mean, a scholarly style of... Thank you. Yes. Thank you very much indeed. And you're right, considering the way that it finds its way into so many of the puzzle-boxes, it's, uh... You see, this was written... I've suppressed Ian Durand, though, but I'm happy to deal with it. This is a reaction to the fact that I sent him this paper, the statistical science, because, uh... In the past they've had this very similar thing. But the new editor apparently doesn't want to find this probability, has concluded, and... A rather difficult thing, if you want to keep probability out of statistics, I would have thought. This is what's so incredible about it. And it was reviewed by a member of the editorial board. It's a little like being asked to keep proofs out of logic, isn't it? Yeah, crazy. I think you invited me to give it. I did, I did. That was my title, actually. I'm glad it was absolutely magnificent. Much more one could say. I'm surprised Bill didn't make the point vis-a-vis this business about, well, the turtle. ...in decomposability of the kind of ultra of this whole business of the adjoint cylinders and exercises this dialectical control over the degree of confusion that one has at different levels of construction. A space with construction with maximum cohesion between sub-parts, and yet the thing itself is still disconnected, if not federated.

1:55:00 I thought he was just going to say, well, of course, the problem is that none of these people are dialecticians, which in fact I think is an aspect of the... That's my mind set. You mean, you mean, you mean, you mean, you mean, you mean, you mean, you mean, you mean, you mean, you mean, you mean, you mean, you mean, you mean, you mean, you mean, you mean, you mean, you mean, you mean, you mean, you mean, you mean, you mean, you mean, you mean, you mean, you mean, you mean, you mean, you mean, you mean, you mean, you mean, you mean, Okay, now, my understanding of this is as follows. But first of all, what the concept is not is that you've got all these intervals, and the point is something sitting inside that's nonsense. So, the thing that you're looking for, that's the concept itself. Yes, well that's the obvious. It's a very structuralist work. Well, he wasn't the only one who maintained that view. I mean, Whitehead also had a similar kind of view. It's a view that, an account that Lyle had in common with some other, you know, with somewhat different elaborations, with several, with several... Well, that in a sense seems to run counter to his conceptual motivation, because how's one going to define a sequence in this case, other than there's a, in structural terms, which themselves, as it were, rely on arithmetic notions as the ultimate ingredients.

1:57:30 Yeah. This seems to be very much in line with the notion of profit as a term, not getting rid of it, because either of these is going to shrink under the attempt, right, and cut it. Right. I think this is intuitively satisfying. No, I don't quite see what you mean. Of course, yeah. So that comes from the Aristotelian aspect of the notion. Well, you know, what happens at the point, the boundary point when you chop, you see... It doesn't actually consist of pre-existence. Right. The thing is, but... Yes, you see, what happens is that both of the... Well, you know, the Aristotelian... When you chop a continuum, of course, they're both introduced to... Where there was only one boundary, you've introduced two, you see, and so what you don't have, you can't then, you can't then recombine them without changing them because then one of the boundary points has to disappear. I mean, if that's the source of the idea of indecomposability, I think it's weak. The Greek conception, I mean, which Aristotle inherits. Both you, Cliff, and in what today is the guy's bill, both, I think, want to say, well, that is, of course, it's because the notion of generating figure is the most fundamental concept in understanding the structure of space. Well, of course, then... Well, then there's the other, I mean, yes, you mentioned... There's a special learning instance of that. Yes, but actually, of course, there is that other view I didn't mention. Well, I mean, it's sort of implicit, but, you know, the idea that continua are, you know, very common methods generated by motion. I mean, the other continuum or continuous curve is... And Greeks had this conception too, because they thought, for example, curves like the Archimedean spiral, which the only way you could really generate it was by motion. And in that case, if you actually have a dynamic account, or a kinematic account of what a continuum is, then of course... Surely, you're going to end up with... In decomposability, you know, you can't just sort of arbitrarily chop the thing up. The thing is generated by motion.

2:00:00 So, again, I think in decomposability is going to rise in the dynamic conception, too. Oh, yes, absolutely. Although that view about what continue a word had sort of fallen away. It was a very common view up to the end of the 18th century. Absolutely. And of course it completely disappeared by the time. But it was still recognised that it was what they were refugiating. You know, when you see Russell's. Oh, I know. Russell, sure. Russell's actually got his entire ontology around him, just a period immediately after his turn away from Kant and Hegel. This is a very interesting book on the origins of analytical philosophy by Peter Hilton, it's a very nice book. He discusses this business of what he calls Russell's Platonic Atomism, essentially his take on mathematics and ontology from the 1900s. It's called the Origins of Analytic Corrosion. He makes that point about the consequences of the reception of... I'm sorry, people can't hear me. Go ahead. You don't really need to get those graduates out of the university. The only logistic students we have are PhD candidates. At the undergraduate level, what would you call him? You've got to look at his head now. He's an ordinary, but of course he's an invasive object. Uh, Bianco. No, Bianco. They're a change. I've had nothing but Rosso all week. Just try them all. Thank you. See you later, Alex. Oh, yes. Thank you very much for your time, and I hope to see you again soon.