FW Lawvere (contd.)

0:00 Yes, well, I don't think, I'd hate to read Benabou's Nobel address. I think it would make, you know, the Christians and Hitler on a bad day sound like, you know, a sweet reason. And of course it's a conspiracy by the British establishment, particularly in the story, because, you see, he actually has some quite good points. Johnston suggested substitution for Cartesian and pro-Cartesian, that they should be called prone and supine. It's an absolutely incredible idea. It's a pretty silly, in-group, English, Cambridge path. He hasn't stopped to think about how you translate these things into French, or Spanish, or Italian, or German, and that in fact in German and Italian particularly, French too, you couldn't use them. They would just completely, you know, they wouldn't be the sort of neat duality of the map and the inverse that he wants. It's just a silly English pun. And Benefit is right. But nobody's going to listen, because he prefaces it by saying, you know, lying, cheating, thieving, you know, dirty people, you know, tenth-rate mathematicians, intellectuals, terrorists and plagiarists. It is crazy. Peter Johnston is an extremely sweet, natured guy. He claims to be, and he has a huge ego. Well, I mean, God, talk about lots of cattle. And here's a man who, if he isn't given credit for everything, you know, claims that people are in conspiracy to, you know, destroy him and assassinate him. And he's accusing somebody who... And Johnston had to write that book completely alone. It took him 20 years. He tried very hard, I know. In fact, he tried very hard to get a... he approached four or five major collaborators, but he could never write one for the scope of a teleprompter. And he approached several other people as well. He certainly wanted it to be a joint effort, but did it with four or five people, but they all wanted it basically just couldn't find the time to do something about it. So he ended up doing it himself, and he certainly didn't want that burden.

2:30 Now he's being denounced for not having written the perfect book. Yeah, his preface is that he should have written the perfect book. He should have said more about this in the preface. He probably would have let down the line. I know, I know. It can come later. Exactly, it's just crazy. To attack him in the terms of benefit tax, it's not... Well, as I said, the greatest just can think about it. It's evidence in the guise of the previous icon, which is sad, because he has a great mind. He is very important to our town. You can see underneath when you sit with him that he's actually a very glad, emotionally injured person. And now, of course, by publishing this, he'll never be allowed to publish on calculus. Again, you probably won't ever be invited to another meeting, not because there's some powerful conspiracy to keep you down, but because you cannot call people thieves, liars, you know, murderers, you know, plagiarists, you know, in print, and expect them not to be... I'm very glad anyway he's not going to be the chairman of your jury. But I don't know him, so there's no chance I would ever ask him. Yeah, no, but the point is that unless you had cited him, I know he wasn't even around. In every single line, every single paragraph, he would have won. So that's not the way it is. No, no, no, I know exactly. You've got the... Atiyah, obviously, would have been probably would have been probably would have been The chairman of your, you know, of your jury anyway, I guess. I have no idea now whether he'll want to come to Boston. I can perfectly understand if he decides not to. Oh, okay, thanks very much. Mostly business.

5:00 Another reason why it's a bad idea for him to have been on your jury is that he has a long section in this diatribe after he's finished, you know, kind of accusing Johnston of every single sin on the face, and kind of paedophilia and mass murder. He also says that the distinction local-global is hugely overworked, and that most people who use it have no idea what they're talking about, and, you know, they don't make the necessary decisions. But in fact it's greatly overworked and we should stop discussing it, so it might not have been the best. Maybe he's not the ideal person. Maybe he's not the ideal person. Maybe he's not the ideal person. Maybe he's not the ideal person. Maybe he's not the ideal person. Maybe he's not the ideal person. Maybe he's not the ideal person. Maybe he's not the ideal person. Maybe he's not the ideal person. Maybe he's not the ideal person. Maybe he's not the ideal person. Maybe he's not the ideal person. Maybe he's not the ideal person. Maybe he's not the ideal person. You'll have to put up with an awful lot of this kind of self-pity and conspiracy theory, which we can all do without, but you don't. That you'll have to endure if you're going to have any connections. Hang on, here, I'm like, oh, it's not going to do it. This is exactly what I'm... Run, run, run. Well, no, it's not that I would go and say, Bill, I need to start talking from home, otherwise I wouldn't... Salut, salut. No, I said, ah... Alain, bonne lecture de l'article de... Oh, et d'un livre, ah... I've got, at some point before I go away, I want to give you the recorder for this thing. I'm giving it to you now. No, I can't give it to you now because I haven't got it. Well, one, at the moment its memory is full of everything I've recorded here, so I need to drain it. But I'm going to be around till Sunday. And what do you want me to take? Well, that's the thing. The race is on the third? The race is on the third and the fourth. The race is on the third and the fourth. But there's also, isn't, I hope he was telling me that, Francois Bosseux is coming down to give a talk as well? Bosseux? Bosseux, the sheet theorist, the big, big sheet theorist, apologist, Belgian. No, I'm not sure where it is. I have to find out. Look, I'm going to be here the day before I fly to Boston. That will be next Thursday.

7:30 Thank you for watching. I'll send you an e-mail when you're free to come for a drink or something. C'est possible de répéter, what is the word? Vous êtes très gentil. Did you turn it off? I don't remember. No, no, no, no. No, it's exactly... To learn the story about how we organized the seminar, you remember the shawl money, the same kind of reception. Yeah, 800 euros on our champagne reception. He didn't bother to tell me he was going to spend 800 euros on the champagne reception. We only had 3,000 euros in the budget for the whole thing and I ended up having to pay for the bloody hotel room. Oh well, Charles is Charles, I think you probably know what we're talking about. Well, actually, no, but it's good to know. And also a poster, a poster. Well, I hate to say it, it was childish as well, because he didn't have to, he should never have arranged that without at least telling us what he was going to do. And also, he went and had, I mean, it looks very nice, he had this lovely elaborate poster printed at the meeting and all these leaflets.

10:00 I think they were printed three days before the meeting took place. I mean, you know, how much money use is that going to be? If he did it three months before, so he could have sent them out and replaced his own. No, what I mean is there was no way to use his money somehow outside the school. In his emails, yes. Well, even within the school it could have been used. In his emails, he advocates the use of emails and saved papers. Saved papers, so we circulate electrons, yes. On some occasions. Let's not get into arguments about identity and individuality of electrons. Oh, I don't know, you know, you can say they're fungible, so you can save them just in the same way you save money in a bank account, not if you're saving individual euro coins, I mean, that is to say they can be freely substituted by others without altering the value of your account. Shall we go and rescue Bill? Well, I'm not sure, I think... He'd probably like to talk to you actually. Believe me, he'd much rather talk to you than he would to you. How do you think it could be possible for me to send a CD around the planet? Yeah, absolutely, that's what he wants you to do. Okay. Yeah, that's what you like. So maybe I can do it for you? Well, I could even take it to him because I'm going there, as you know, on Wednesday of next, Thursday of next week. Just if we're not meeting? Well, I'm going to be back in Paris on Wednesday. You can meet there. Okay. Are you going to be here? Yes, I'm free on Wednesday afternoon. Okay, well, why don't you give David a recorder to record the thing at Roussillon. Yes, he is. The following three days. Good. So, yeah, that'd be great. Okay. And I'm going to give you the check, too. So it'll be the three defense. Yes, sure. I mean, you'll understand that. Thank you for your attention.

12:30 He was at the IHES in 1981. No, no, I knew that. I hadn't realized that in fact long before that he was also actually visiting him at the IHB in 1964. It was in India, exactly. I had no idea. That was when he passed away first. When I first met you, you told me, I was in exile in Paris. Yes, you were. In 64. Then friends, communists helped me. You obviously know he's a very sincere and convinced communist, which has not made his life in the American academic life exactly easy. In fact, his first major appointment was Canadian in Canada, but then of course he was actually kicked out of that. Oh Bill, this young man was just about to move in, and I was just about to say, you know, you really wanted to talk about mathematical and global chemistry, didn't you? Oh, manly rescuing. No, no. Oh, okay, right. That's all. Good, okay, it did look as if you might be rescuing. Anyway, thank you very much. Thank you. I think that's just great, isn't it? I think we are. No, no, no, there's still, there's a whole bottle still there. Okay, great. You see, we always keep these things steady. Oh, excuse me, I'm sorry, I'm doing your job, I'm doing your job for you. That's really bad of me, I'm sorry. Should leave it to the professionals. I told you that I started off with galactic. Oh, yes, yes. On Kelly's book, from which I learned topology in 1950. The last chapter had an actual introduction to it.

15:00 But without using the word category, he calls it galactic. Well, that was his usage. He says what I'm saying here could be called galactic. The method is due to Eilenberg and MacLean, but he doesn't use the word category. He just calls it galactic, because local and global. So he says, well, there's local, then there should be... And then there's global, and then beyond that we have galactic, so I thought, oh, this is great for terminology, so I was essentially inventing fiber categories at home, in my garage, still in my garage in Indiana, I was inventing fiber categories before 1960, before I went to the Sea Island Barricade, so... I hope you didn't hear that, but even before Godendieck, I realized that there is this thing with categories, but I called it galactic cluster. You see, it's a family of categories by category, so it's even, you know, it involves galaxies, but it involves a bunch of galaxies moving together, you see, so it seems like a reasonable way to have a sort of semi-suggestive terminology. I used it in a couple of other ways. Forming the grammatical permutations on galactic provided the description of several categorical constructions. This is all before I went to see Eilenberg. I was still a student of Truesdell, but of course plotting to... I think Mark really converted to category theory. He's so much impressed. He's a great guy. I'm sorry he's not here actually. Next time he was trying to get some time to talk. No, that's... yeah, so local and global, yes. And about what you said at the end of your talk, the main point was to show how reals can be considered as ratios and integrals. Do you have anything written on this topic? Some cute things, yes. Okay. Maybe from my collection.

17:30 You're on your site anyway, I think, aren't they? Okay. It might be. Read them on your site. It's for John Bell's... No, actually, that's not on your site, but I have a copy of that, and I can send that to... In fact, the conceptions of the continuum are not unique. No, no. So, in honor of his 60th birthday... Yes, that's right. The book still hasn't come out. No, I know. It's taken a long time to come out. My contribution to his 60th birthday is about this. Euler's reels, I'm currently vindicated. Actually, I haven't seen that. I don't think you have sent me that. Bonsoir, ça va? No, you have to send me that. I don't think I have seen it. Unless there's an earlier version for a different time. Oh, okay, right. I ought to know a bit more about yours. I dropped in 2004 in Halifax. Mass society. I guess maybe somebody like Taylor. No, you don't know too much. That raises the question, well, how do you, what's it got to do with the Dedican Reel? No, obviously, this Dedican Reel should be a quotient of the Euler Reel. So you have a homomorphism from the Euler reels to the Dedekind reels, which is probably not onto, I mean, it depends on the topos, it might be an epimorphism, so these are sort of the smooth Dedekind cuts, if you like. Yes, yes. But then the interesting thing is the kernel. This map has a kernel which is all the infinitesimal, so the fact that in Dedekind reels you don't see infinitesimal is a result of the fact that you've collapsed the reels. Yes, you've collapsed everything. You have absolutely no cohesion at all in the Dedekind construction. No, no, you do have some cohesion. No, no, it comes from the truth value objects. Although the rationals have no cohesion as an object, the truth value objects have loads of it. You know, they're typically just a sheaf of germs of real-valued continuous functions, so continuous as opposed to smooth, but of course smooth is embedded, continuous is embedded into smooth and topology, and you can always, you see, if you, continuous functions is a sheaf over the site of smooth functions, because if you plug the smooth ones in from the right side...

20:00 Continuous is still continuous, so smooth operates on continuous, operates in the way that a site should operate on cheap, so they are in there, but the map is trying to follow the motion. They're smooth. They have continuous, at least continuous cohesion, not more, but they have no variation. I should have said they were completely static rather than what they were. They were completely static, you see. So the Euler reels, which are made up out of germs of motion, what you've been then mapping onto it, so you get a kernel. The fact that you have this kernel is a measure of the main difference. So it might not be subjective? Well, it depends. Depends on how you define the data completely. The thing is that in this paper, which I guess is not really written up, the thing is how do you define the map from the, you know, to, so I had a very brilliant idea about this. Now, the idea of the Euler real applies in complex analysis as well. There are things that are going to be very difficult. I'm not talking about that. Then the real is supposed to be a line somehow, right? Where does the ordering come from? I mean that the map into the Dedekind reals are many kind of models of the reals. It's based on the ordering. It's really just a unative betting. To each x you assign hom of x blank, which is the set of all rational features greater than or whatever, starting from any model. But you need the ordering. In order to get this, you need a map and two dedicated reels. So where does the ordering come from in the Euler reel? Imagine geometrically the line. There's the left half and the right half.

22:30 That's the part where the quantities are invertible. This is a ring, mind you. So some elements are invertible, so it's really the group. You have an attitude. Well, you have a multiplicative. In fact, this is just d to the d of the multiplication, which is just composition. So that composition is sometimes invertible. In fact, ought d, automorphisms of d, is a sub-object d to the d. Well, automorphisms of d are the same as the invertible real. The condition of mapping zero into zero is automatically taken care of if you think it's invertible. Multiplication is just the invertible wheels. Invertible wheels, that's the part that excludes zero, intuitively, right? The positive part, the one that contains the unit. So in the Lie group here you have this famous thing, the component of the identity. Take the component of the identity. There are only two components. It's a group. It has to be a group because bis-zero preserves products. But this two-element group, and moreover, you can choose one of the two components in a very natural way, namely it's the one that contains the unit of this modification. So strictly positive, therefore, has to be defined to be nothing else but the component of the identity and the invertible elements. We've got the positive part, and that's the strictly positive part, which is not yet the non-negative part, because it's the non-negative part which has to have, you know, your native cutting. But, well, there are two more algebraic steps where you can deduce the notion of less than or equal to, or non-negative, from a notion of strictly positive, as being the things that are... No, I don't. No. There are two simple algebras. Well, there'd be some kind of intersection condition, or... It's all based on... So did you sequel that? You have this... well, it's actually a rig, never mind.

25:00 Yeah, I'm sorry. So a rig should have... a rig should have... if you want to order a rig, you should have a sub-rig of non-negatives. So let's just consider that question abstractly. This is to be understood as the positive part, because we've taken the component of the identity. So there's a subgroup of the multiplicative monoid of this rig. To have an ordering, we need a rig of non-negative elements. So from a subgroup of the multiplicative part in any rig, we can deduce the subrig playing this role. And a typical rig is actually the endomorphisms of some additive thing. And that additive thing is with respect to the, in other words, you have this group of units thought of as the strictly positive. Well, consider all the elements of the rig such that when added to... In this group, they stay in this group. In other words, the translation to the right stays in this group. So those are, take those elements. Those elements obviously form a sub-monoid, additive monoid. Take the endomorphism rig of that, or more exactly, just concretely take all the elements which map that rig, sub-rig. So that sub-community thing into itself. Well, endomorphisms of an additive thing always form a rig. So that's your new rig, that's the non-negative. So you can go from the group of convertibles to a sub-rig of non-negatives. Once you have the non-negatives, you can define the negative embedding into the dedicates. You've got the same, you've got the dedicates in scratch, exactly. But you say it's kind of quotiented out of the Euler. Well, yeah, the Euler embedding is not faithful. You're going to have the phenomenon that x less than or equal to y and y less than or equal to x does not imply x equals y at all.

27:30 Does not imply? x equals y. Yeah, yeah, exactly. So you have the infinitesimal. You've got more in there than the decidable, so you've got the infinitesimal. I'm exactly saying that the two are interpenetresibly close, but not equal, and yet that difference is wiped out when you pass through the corresponding identity cut, by definition. Yeah, but of course, it's there you've got the difference, the identity is decidable. Yeah, that's just absolutely incredible, I agree, to think of it that way. Nothing is decidable, you see. So that was part of the story of the... I haven't figured out... of course there are axioms to make sure this all doesn't collapse, to distinguish the real case from the complex case, so I haven't figured out exactly the axioms or how to deal even with the complex case. You want to find the unit disc, you see, and all this. I mean, it's there, but the lofty goal is that it's all synthetic. It all follows dust, particularly this object, and then the fact that it's embedded in the category without postulating any algebraic operations. Without postulating? Postulating any algebraic operations. Yeah, okay, yes. In this other case, we have produced an ordered rig, you order addition and multiplication, from an object that has merely a point, and in some sense that's not even a structure because it's only got one point anyway, it's a property that has only one point. But in any case, everything else is exponentiation, and we've come to an operation. We construct algebra from that. That's tremendous. I was very glad to meet you. I hope to read your last paper sometime. Salut, salut, salut. Salut, Pierre. So, in other words, if you imagine a complex situation, and everything works except the construction of the order, because then, of course, pi zero of the invertible elements, the invertible elements is a punctured disk, a punctured plane, but it has only one component, and it has higher homotons.

30:00 It's a whole other kind of thing, really. Yeah, yeah. So, to say... There's some crucial axiom which will give a similar... The idea of having an additional adjunction for the exponentiation is fascinating because there you could have... I didn't get to mention that. Yeah, that's really interesting. Well, I remember this is the thing which, of course, John Mabry immediately picked up on when we were talking to Bristol, because it actually connects with his idea that there isn't really an isomorphism time for natural numbers. You've just got Dedekind, simple infinite systems of different lengths, but I'm not sure how his ideas about how to, as it were, redo the Dedekind construction inside what he calls Euclidean theory would fit into this framework at all, but I can see that they ought to connect with this idea of having an additional right adjoint of exponentiation, like you've got in the case of the topology of infinite. The main thing for that about analysis topos is that if you have that extra, then the category of second ordinary differential equations is itself a topos again, which is a fantastic richness of construction. When I say second ordinary equations, I'm thinking of time, so it's things like wave equations, heat equations. The space in which it operates can be an infinitum of space. It needs to be. In fact, you have a whole ton of space. The fact that it should have the exactness properties, should have the universal property of some kind of exponentiation, truth value objects, all those ingredients. I mean, there's a fantastic richness of construction. There is such a thing going on there. That depends on the exercise, which is, which is, one must underline, not something that's been added on to the old theory. It's something that's there in the old theory explicitly, that if you make, if you make, go to the topos that I talked about, 1960 topos, I talked about.

32:30 It has such a thing in it that the people didn't recognize it. Algebraic geometry has such a thing in it. It's not smooth, of course. So it's not at all... They're even combinatorial. If you interpret Fibonacci's equation, using time, using co-discrete time instead of the usual idea... For discrete dynamical systems, I usually thought of them as discrete, the time is discrete, but the same natural numbers as time can be construed as co-discrete, and that's kind of a tokens, and then that forces the solutions of differential, i.e. second order difference equations, in this case, to be… You know, they have to be within the components of something. They can't pop around like the typical examples of chaos. But that's, again, a topos, so you have a whole... So I don't have to go outside of the topos generated by the classical superficial complexes. Just within that, which is also known as the Bruggen algebra classifier. Modest, apparently, things, even if it automatically has these other ingredients and automatically has the representability of the notion of integral equations, but that's why, unlike simplex itself. Yeah, and the point about things being, preventing things from moving around so that they're always in one component, of course, don't like the point about the central role of the components. Functor, its relationship between functor, which of course is the basis of the whole Cantor construction, as I remember you explaining in Florence, or the way that one should see the Cantor construction from the point of view that sees that theory as essentially a branch of algebraic geometry. Yes, yes, there's that, yes. It's another ball game. Anyway, but... Sorry, I don't want to embarrass you or anything, but this young man has done fantastic work in the history of mathematics.

35:00 Well, you don't know yet. Oh, no, you're wrong. Sorry. You don't know. I have, actually. Not all of it, but I have, actually. I don't look at a copy. Oh, wow. Running into a spider. Oh, yeah, you're right. Yes. Don't worry. Thank you very much for your attention. This link particularly about the Cauchy model of the reals and what happened when Dedeckind and people imposed the conditions... It's questionable whether there's a Cauchy model of the reals. Well, yeah, of course you're right, it's not. But you can look at Cauchy and try to see what... But more to the point, you can look at what Wierstrass and Dedeckind and these people did in... Period of so-called arithmetization of analysis from a much more geometrical viewpoint. In terms of the, you know, if you reconstruct the, well, the delicate construction, as Bill just explained, much more, with much more grip of what's going on than I have, but you can explain this very much from a geometrical viewpoint in terms of imposing the condition of decidability on the points, or in this case the components in the space. You know, everything has got to be in or out of, you know, just the side of it. Exactly, exactly. So you've got decidability of entities, just simply the existence of a decidable object is the weakly decidable sub-object which has a very inherently topological and geometrical meaning and also involves the way that you can restrict coverings on the topos to be locally decidable. There's an excellent paper by Colin McClart on this in Journal of Symbolic Logic, 1987, and there's also a rather more expository philosophical paper about it which is called Sets Are. I think it sets our only ever set of points and spaces. And that's in Journal of Philosophical Logic about a little bit later, maybe 1988. I'll dig it out. I'll try and bring it in, actually.

37:30 Maybe you should learn about the time when it's totally exposed. Yeah, yeah, yeah. But that's very expository. I mean, it's not like what you've just been exposed to. I mean, you know... Yeah, well, you're obviously bright enough to have got very good feel for the scenery, the distant scenery. But, I mean, I've been listening to it for 20 years and I can... I just... I just get glimpses of the scenery as it goes by. Occasionally I get a more clear glimpse of the scenery. It's a little bit like being in a car, travelling at very high speed along a motorway, and trying to look at a landscape through slats in the fence. And just occasionally you get a flash and see the landscape. Most of the time all you just see is the palings of the fence speeding past in a blur. But then it's a question of, you know, how deep can a mine go? But the paper by McLarty, which is, as I say, about the 19th century, well, about the set theorization of analysis, looked at from the point of view of topos theory, is a nice paper. It's kind of rational reconstruction of history. I'm not saying it gives one much idea of what was really going on in the minds of either, you know, Dedeckind or Wierstrass or Cauchy, but there is an important aspect that does get out, which is that the way that the story is conventionally been told entirely from the point of view of set theories, you know, is just one version. You can look at it from a much more geometrical viewpoint, which of course is the way Bill thinks, and the way that I would argue that almost all the really great creative mathematicians have thought. And that's what's so interesting. Anyway, I'm really glad you were able to get here today. In spite of the fact, unfortunately, that I live here, I get to talk. All in all, it was a good day. Are you saying there was no talk about ancient history? Yes, I agree. I agree. Particularly Zohar, even medieval or Islamic, because he's done so much work on it. Or even the 17th century, because I should mention he's done fantastic amounts. No, I was a little bit disappointed that we didn't have any, well, we just have the very general, you know, observations about his overall work as a historian of maths by, um, the first, Rashed, yeah. It's fine, but I mean, all he did was just say, yes, but we didn't actually have a talk about what he did. It's a pity. I would like to have a two-day meeting and have, you know, one half on talks like Bill's and Chava.

40:00 Oh, it was very, very interesting. And I couldn't understand it as opposed to... Yeah, well, I'm sorry. There's all sorts of reasons why it was difficult to follow Bill's talk. Not least his very soft voice. I know, I have a lot to learn. No, no. Obviously, the other way around, actually, I found Charles' talk much more difficult to follow. That's obviously because of my bad French, but also because I know very little about complex functional analysis. I know a little bit about that stuff, but very little. And I'm just aware that this was the first really major result that Grotendieck proved. Other than that, I can know damn thing about it. And Billy could have explained it to me afterwards in about 20 minutes, and I actually understood it very clearly after that. But hearing his talk for the first time, it wouldn't be hard. But actually, the intuitive kind of orientation you get for what's going on... Which is obviously that these technical notions that Isabel isolated about adequacy and counter-adequacy of sub-categories within a large category in the way that this all very, very naturally captures inherently geometrical motions, it's really fundamental, the fundamental motions are really inherently geometrical in a very intuitive way, even in terms of this notion of incidence of figures, which is really going back to Euclid. I mean that, in other words, the mode order is geometry first. I'm not saying, of course, that it's the way that mathematics will actually go, but I suspect it may be because these ideas are incredibly deep. I absolutely agree. I think that's a tremendous project. If I was a lot brighter, it's what I'd like to do with my life. I agree. I think that would be an absolutely tremendous project. I couldn't agree more. I'd really like to help bring that about. You may well be the man to do it. Yes, yes. Yes, you may well be the man to do it. Anyway, I'll definitely come and see you with a recorder on Wednesday. And I'll send you an email before. Right, I think I'm going to make a move, actually, because we're moving out of time. Oh, do you want me to... Oh, no, because I've got to go back to the hotel. What about it? Do you want to go back? The usual, please, then.

42:30 Okay, I'll give you the check, then. Do you actually have the recording with you? Oh, great. I'm going to have to see you again anyway, though, because I've got to give you the actual recording. I've only got two. Actually, I have two. Yeah. What about me? I don't know how you managed to end up with two of them. Yeah. No, but I'd rather give you... Well, no. Do you actually have one with you? I have one. Oh, well, let me have a look at it. It might be the crappy one. I've got a much better one. I've got a much better one. Okay. What? You and me? Yeah. Yeah. Well, don't lose the fucking thing again. That's the list. No, don't worry. I think I'm going to make a move. Oh, that's great. Thank you for your attention and see you in the next video! Bill certainly buys that, because he consistently makes the point about the Erlanger project, which is too narrow, the specification of the notion of mapping space in general, in categories of space, because he needs something much more general than I do. I think I'm going to make some notes, because I am feeling very, very nervous. Well, I've got to give you... I'm sitting up to you, and I can see some more. What are we going to be talking about? Uh, probably I can just... Okay, fine, let's do that. Something like that. That'd be perfect. That'd be absolutely perfect. Because I'd really like to go back and get some rest after the office. If there's any change, I'll give you a report. Bill, I do apologize for interrupting, but I'm really feeling I need to get some rest. You'll be fine to get back to the... I'll give you a call tomorrow and find out what plans you've got. I'm having lunch with him. Oh, excellent. Good. Professor Zell, thank you. Congratulations. Thank you. Have a wonderful day! I still haven't learned a fantastic amount of mathematics and history of mathematics through it, so I thank you very much.

45:00 But I'll give you a ring anyway. See you there, too. Take care. Have a safe journey. See you there. See you, Renat. I'll give you a ring. I've got your email. Yes I am, I don't know if that's what it is. Yeah, I'll see you at 10. I'll have everything then, okay. I really am absolutely zonked at the moment. Because I went to Havasu last night and I managed to get quite a bit of maths along the way. But it was mainly... Oh, I have to say, it's certainly the most fascinating relationship of my life. ...to have been able to get this close to such a great line. And there's no question, you know, I just see glimpses of the lizard scenery. I'm not saying there's not a lot of other very smart people there, but there's lots of them. And so, for that matter, we have the Shab-Shab-Shab-Shab-Shab-Shab-Shab-Shab-Shab-Shab-Shab-Shab-Shab-Shab-Shab-Shab-Shab-Shab-Shab-Shab-Shab.