Discussions FW Lawvere & others

0:00 I've looked at some of the early papers, but to be honest, I haven't had time to look at anything in the last... Oh, there's a book, is there now? Ah! Memory, Volatility Systems. Ah, excellent. Right, okay, I shall try and get hold of that in that case. I'm sure with all that she's worked on, it is a very large book by now. Okay, that will bring me up to speed, I hope. That's great. The last part is on the application to cognitive science. That's to construct. Great. Well, I'm actually looking forward to studying that. When did that come out? In July. Oh, so it's very recent. It's this year. That's why I haven't seen it. No, last year. At the end. In fact, in July last year, but really in December because... I'll just look on the Ossetia site and see all the details there. In fact, God knows when we'll ever have enough time because we're going to this tiny private institute. Maybe one day we'll be able to afford it for the library. It's the rest of the... Of course, like always. 100 and... And they have not even said, we have had only four. Four, that's terrible. And not only that. Do you use the index yourself? Not the index. They ask us to have all the text. You proofread it all yourself, of course? Academic publishers don't even proofread your text. They expect you to do all the work. They expect you to proofread everything. They expect you, more often than not... I had to highlight all the words which were in the index. So they were, well okay, but very often these days they actually expect you to do the index. And then, of course, as you say, you're lucky to get three or four, you know, exemplars. And of course you don't see any of them at all.

2:30 No, no, which of course you've also expected to do all yourself. Yes, all of ourselves, and all the text in CompuScript. Exactly, and of course they'll be sending it all to somebody in Bangalore to set, you know. Exactly, exactly. So of course they can't expect any kind of expertise from the printers any longer. And all of this they have to take their lives. I'm sorry, do you think with academic publishers, it's incredible how... I'm afraid it is incredible how they profiteer in maths. There's a small publisher in the independent publishing house in Italy which was published about three or four years ago. You may have come across them. It was run by this guy named Domenico Sica. Polymetrica, which publishes quite a lot of philosophies, maths and mathematics theory. And they produce really very good texts. I mean, true, the authors still have proof of it themselves. But they produce texts which are... Just in terms of the publishing standards, you know, the binding, the proofreading, the paperwork, it's just as good as anything that people like Eusebio or Verlag publish for a fraction of the price, you know, for only about 25 or 30 years. And even though they have a much smaller... All of this is a very interesting topic, and I think it will be a very interesting topic for the next few years, and I think it will be a very interesting topic for the next few years. Ilsepia is only just a very small component. I mean, they're not really doing anything for the academy, it's not going back into scientific research. And they have not done a very good publicity job. No, they never do these things. They have asked for a list of people who could be interesting, who have a complementary copy. No, no, no, this is a story which I hear from many, many, many mathematicians and scientists and authors these days. It's very depressing. But now you can find on the Google Scholar, by Google Scholar, you can have parts of the... You can actually go online? Okay, that's what a lot of people are doing these days, especially with the really price-gouging people. Actually, you said they're not the worst, but the worst are clueless, along with it.

5:00 What a lot of people are doing now is making it a condition of accomplishing all that they've been allowed to accomplish, not a complete version. No, that is not a complete version, but... Because they know that the tax price, the only piece they're ever going to buy is a margin. Yes, it's not a principle, no. There might be one private file or two in a handbook. Yes, yes. Very, very rare. The whole thing is very self-defeating. That's the business model that they have devised for themselves. But I think it is getting to the point, especially where you have online journals, it's very noticeable in physics, the subscription price now for the electronic version of classical and quantum gravity is probably the most prestigious journal in general. The online annual subscription is now something like 2,000 euros. Absolutely incredible. Obviously only affordable by very well-funded, you know, large academic institutions. And it is heavy as well. And as a result, there is at last now beginning to be a movement, as you can see, you can google various discussion sites and blogs, including actions that have been touched on, I think, in Catherine's list, in one or two of her posts. People are now urging... ...establish scientists who are on the editorial board for these journals to actually resign from them and to cease, and to establish independent centers for publication, online publications, which are just not subject to this kind of proper theory. And it is beginning to have some effect, especially in physics. There are now quite a number of physics journals where the people have actually so many... The senior people have now resigned from the boards that they all refuse to read more than any longer, that they no longer have any sort of credibility. Those journals have actually gone off and in some cases been refounded. The American Institute of Physics, well of course the great thing in physics is that almost all paper is now published on the LAR. There isn't really any motivation, except of course that, you know, you could work in rhetoric obviously. But apart from that one consideration that it is refereed to publications, there's really nothing to be gained by publication in the print media at all, because everything is now put in the archives.

7:30 I think the thing is that these big international... In the last five or ten years, before the technology undermines their profits, I don't know what you'll see. I don't know what you'll see. Well, of course, there's a lot of profit theory. You know the ones I see. Yes, you can get that. But the thing is you just have to have access to an academic ISP before you can download stuff. Yes, but only if you have an ISP, only if your computer is linked to an ISP. Thank you for your attention. If it's not, then you don't get access. And of course it's also, whereas 20 years ago, if you were independent of something, you'd just go into your university library anywhere and just pay a subscription fee for the day. If you don't have access and you don't allow it, of course, they cannot get across to you. You need something online, electronic. You could have every access you could have in the classroom. For me, it's more a kind of practical interest. They say, what is object, whatever you want, but...

10:00 If you have this idea of structural thinking or something like that, even if these people don't put it explicitly, that object is a structure, it's very influential in mathematics. I might agree with that. I think that our notion of objecthood goes along with, as it were, whatever the salmon head of mathematics happens to be, given the overall shape of the mathematics. Yes, there is also this science approach, more practical and concrete themes. But which also connects with conceptual issues apart from the recursive functions of the kind of categories in which the kind of, the kind of math basis of the categories in which things like algorithms and recursive functions and the notion of a function of the rule actually kind of live in a geometric aspect of that as opposed to a pure procedural aspect of that.

12:30 Yeah, so it's the approach of... The interesting thing is the way that things like SETDE or the membership-based SETDE became the default. ...project for the philosophy of mathematics in the eyes of philosophers without really anybody in the mathematical community being aware that that was what was happening in the department down the road. And there are a lot of them very, well, what we can now see in retrospective... ...except you have this thing like von Weikin which is of course not exactly... This idea of foundational philosophy, but still it's membership-based. It's organized kind of the whole mathematics, you know, vaguely about this thing, around this thing. Well, except it wasn't. It's set. The Polkowiki volume on set theory. ...which is aptly described as the worst book ever written by a whole sector, by serious mathematicians, was not intended to be the foundation volume of the Varga project, and there never was a foundation volume, but insofar as there ever was, it's the one which is all about the mother strength, which was a kind of rather pragmatic variant, if you like, of what you might call the immediate raised structuralism. For me, actually, the story, even you think like, you know, Hilbert... It seems that for this new way, let me begin with the example of the category of the book, and let me give an elementary book about the functionality of relations, so a relation has to be functional if...

15:00 Thank you for your attention. As a syntax, the theory of sketches. Well, it's not an approach to syntax, is it? No, maybe not. I was just wondering about that. Sketches. Syntax is a way of presenting something. Yeah, that's what I was thinking. And here, you've already got it. You don't have to present it. In that sense, that's fine. Because reacting to your terminology is a bit of a distraction. You're not presenting this category by generating simulations or something, do you? I assume you have this category when you do something, but I'm not sure. You're asking about this stuff, I don't really like it. No, no, no, I didn't think you did somehow, no, no. It doesn't really seem to be getting at all at the de-Volta Brake structure of what's going on in terms of the aritids and operations. People did this sort of thing in the 50s, though. Yeah, yeah. Pucca, for example. Many, many, Peter Pucca, I don't know. Many, many people. Well, it starts with the idea, you see, well, we have a billion groups.

17:30 And what about greeks? Why? Why? There's no motivation. There's no reason why we would do any of these things. In a certain sense, because here there's no real motivation other than some vague analogy of a previous slightly less general view. So it's not his people. Yes, he has a lot. Saved by the bell. That really is impressive, isn't it? So, and yet, whether it's a plan or not, it works out. It just seems to be related to some kind of underlying philosophy of formalism, of a very, very bare, old, bare bones, old formalist mind. That syntax of formalism is philosophic. Yes, yes, yes, exactly. Oh, very much so. They're just doing it because other people did something similar, no other reason. Very bad, eh? Well, and also, in a stronger philosophical sense, that they do think that just bare manipulation of symbols and the level of presentations, expressions, that there is, in a sense, all there is to it, but you see, in a way, I mean, they're well-rounded mathematicians. If you have some exercises of that character, it's fine, you see. But when it becomes a school in itself, it does affect nothing else. Sorry to be so critical. I'm really sorry to be going over this.

20:00 Well, it just seems to be one of the strange uses. Very unfortunately, because some of them are very good at calculating, even about that thinking, sort of artificially. Well, thinking of, you know, the conditions for there to be separators of pairs of maps is the right way of thinking, but it's just, in this case, it just seems to be turned in on its own entrails, and I didn't want any real, you know, insights going on in terms of the... The thing is that the real categories, the real categories, at least in my opinion, are not quite so nice as that. There's some action that is a little bit tight, a little bit like a human being. So then you can do some of the calculations. This is the sort of approach. Whereas to apply it, what are you going to apply it to? I'll give you an example. I mean, you remember at Giselle's meeting there was this guy, Chargois, for example. And he's the only one. I mean, also the other one there. All these guys, you see, so they're doing analysis. They're doing partial differential equations in a way that's What is the devorganization of the topos which classifies the fields of the fixed character?

22:30 Well, for a minor characteristic, it's going to be a few of those here that you can browse over. I'm pretty sure it's a plain characteristic here that I haven't actually checked out. It's an ignorant question, but when we look at the organisation and so on, is there any joint-factor theorem behind, two-dimensional one, which can be used, no? I don't think so, no. Because, one is, you know, as I say, it's not the largest model. It's the largest, dense one. And it's still not functorial. It's still not functorial, all right. Better than real innovation, but it's still not functorial. But there is a mask in the program, so the program will aid you. Well, I'm not kind, I can't afford 50 euros for dinner last. I wish, I wish I could count my money. Have you a chance to get some time with your colleagues? Yes, right, that's true. I didn't notice that. You should, you should. I've been feeling very guilty about that already since I got your number. But I must admit, I would like to be on the fly on the wall when you talk to Peter. Do you happen to know if Peter's staying around tomorrow? I have no idea. I hope so.

25:00 I really would like to understand more about the position of the side of all this. What? Specifically, what you meant at the beginning about the side of all this. Thank you for your attention. Thank you very much for your attention. Is it relevant to this or is it a different question? Well, I'm not sure if it's relevant to this, but certainly something Peter said at the beginning of this talk made me think it might be relevant. What did he say? There was one specific condition for the existence of the signable that connected with this condition for the organization of the signable object. I'll have to go back and look at Collins' original paper, but it's the... The demorganization is what he's talking about. Yes, yes, exactly, and I'm not sure that this is more something normal. The theory of algebra and whether it can connect with these sociological conditions that Colin was talking about in action with me. Is there some kind of relative uniform separability condition? It's a little bit more... Ah, okay, well, there seems to be a... Nothing. No, probably not, mate, it's not. This is the kind of work that we need to do as part of the program to redo algebraic geometry. Yes, I've realized it within a few minutes, but I think that this connects very much with what you were telling me in Calais about algebraic geometry.

27:30 No, I'm saying, you know, how do you see set theory as a practice of algebraic geometry? That's why I'd love to hear more about this. Thank you for your attention. There is a fragment of algebraic geometry, something I want to ask you more about anyway. I better go and rescue the recorders. I was lost, but... Oh, I was lost, I was going to argue. But in fact, Bill was here just to explain to me that this is exactly the program, part of the program that he was propounding in Calais, you know, when he was talking to me in Calais, that this was seeing set theory as essentially a fragment of algebraic geometry, all of these... These conditions, for instance, on the existence of fields, which, you know, the condition, for instance, of the idempotence of split and, you know, the various other things which characterize the infrastructure of the topos from this kind of purely algebraic field-theoretic viewpoint, have these, turn out to have these powerful... What you might call a logico-geometric corollary is to do with the behavior of coverings, which give you precisely these kind of logical conditions for the topos to have decidable objects, to be Boolean or, in this more general setting, to have this demorganization property, which in fact, as you see, as he explained, reduces to Booleanization again, provided that you have certain restrictions on the kind of field theoretic structures. There is underlying the whole thing algebraically in terms of extensions of, for instance, this is all connected with extensions of what they call algebraic geometry. Yes, the existence, the condition for there to be certain extensions of algebraic varieties with appropriate closure conditions just turn out to be translated in this subtle, but not immediate, but still very, very controlled way into entirely logical and theoretic notions.

30:00 Deriving logic and set theory from algebra, but for an algebra in which the geometry is also an instrument, because there is this fundamental duality between the representations and the classification of the operations, there's this very fundamental duality between the algebraic and the geometric aspects of the underlying structure, but the logic and set theory is seen as just dropping out of that, almost as derived notions. I take from it as a kind of very big picture overall what the heuristics motivating this whole program are about, and this is obviously very detailed, subtle applications of it, but it is interesting that he, you know, even I understood, I think, this condition that he said, this has been staring us on the face for 30 years, but we haven't seen it because there wasn't a frame theoretic, i.e. Grothendieckian original, the way that he thought of the, I mean, I think the point that's been got at here is that... The original Grothendieck example of a topos, which is often the category of pre-sheet sort of space, has to some extent been the kind of tail that has wagged the dog all along. It has exercised, obviously it's one of the most profound, it was the whole source of the notion of topos as kind of generalized space, but it has actually exercised perhaps a little too much influence because it's provided this heuristic so powerful that it's blocked off certain other avenues. of which this the late This is precisely what Johnston is saying in obviously a rather more high level, highfalutin mathematical way. And of course one of the problems with Peter, and I have great respect and liking for him, that he's totally unlike Bill in this respect, that you will never get him to talk about philosophy or foundations at all. If you press him, he's a really nice guy, he's a rather shy guy, but if you really press him and you get him with a few drinks inside him, He says it in a self-mocking way, a very gentle way.

32:30 But, you know, he's very, very leery of, not in a kind of hostile way, not in a kind of, oh yes, he can be, and I think, in fact, he does actually have quite a lot to say philosophically, but he's a little bit like, his official attitude is a bit like... Findman was, you know, in physics, you know, don't talk to me about fillons sans filles, you know, it's Findman, always used to pronounce it. It's easier to grasp the Bill stuff. It's less, I mean, Bill is very exciting. Well, he sets out the store more clearly, apart from everything else. Although, mind you, you have to have done a huge amount of work. I mean, it's been said that the only one mathematician ever led... Actually, I just read his first, his dissertation on topos theory. Which one? First topos theory. That wasn't his dissertation, that was his first big book. OK, it was not his dissertation. No, no, that would have been far too long a book for a dissertation. And I finally enjoyed it. In fact, his dissertation was just the section on the axiom of choice in that book. It was the three chapters or so that are about the axiom of choice. Yeah, it's much too long for a dissertation in Cambridge. It would be about five times over the limit of the number of pages. I finally enjoyed this book. It's a great book. It's a very difficult book. It's even so. It's very difficult. But it's been said. In fact, they all said it to me. The saying is that only one mathematician has ever actually learned topos theory from that book. And that's Iker Murdych. It's not a textbook. It's not. Not unless you're a genius. Not unless you're somebody of kind of, you know, Iker Murdych's abilities. In fact, in the introductory chapter of that, he does actually say things which are quite philosophical, in terms of, you know, what he thinks, they're more in the sort of, in the character of frame, what Albert calls frame remarks, and historical orientation, but in fact there's quite a lot of philosophy contained in them. Did you notice how very carefully he was? ...with Benabu in the audience to get out of his way to say which was discovered independent, because otherwise Benabu is going to say, you stole my proof, you stole my proof, you know, it's paranoid.

35:00 And, you know, and if you're going to stand up and say you stole your, I stole your proof, I've got, I've got the document on which I wrote the bloody thing, and I've got the date, and I can prove it was 1976, you know, do you want to see, do you want to see me in court, you know? I don't think he was terrified, I think he was just trying to make sure that Benabou didn't make a scandal, because Benabou is so paranoid about, well, hang on, let's be careful, because he's within earshot, isn't he? Well, Benabou left, yeah. And Peter also. Yeah, but Peter is staying, yeah, but he is staying tonight because he had his suitcase with him, so I assume he's staying tonight. Bill certainly hopes he is because he wants to talk to him about it. Are you going to this dinner? Yeah. Oh, lucky old you. Probably, I don't know, I just... No way I can afford 50 euros. I would love to come, but there's no way I can afford 50 euros. In fact, the way things are at the moment, I'm going to have to borrow off you to get back to Paris, because the trouble is I'm going to have to find 80 euros for the hotel. I might have to, and of course I did have to pay the registration fee here, which is fair enough. I think I might just about survive. Well, you'll probably want to go straight round to the Grand Place, won't you, because there's no point in going back to the hotel first, not for you. I'll just wander down with you. What is the time now, actually? It's 6.15. Well, so it's an hour and a quarter, so by the time you've wandered down there... It's 7.30. 7.30, I thought they'd say, 7.30, yeah. I'll actually go and probably check your mail. There's lots of email, there's lots of internet cafes around the Grand Place that you can do that, yeah. Do you want to go, or maybe... I haven't got to go anywhere, I'm just going to go and find myself somewhere cheap, you know, to get some sweetened... Some fruit and a mule and then wander back to the hotel. If I might see you in that little... We can go to check email now, no? What, you want to go all the way back to the hotel just to check email? No, just somewhere to check email. Yeah, well, around the Grand Place would probably be the obvious place because there's loads of places near there. But we can probably meet later and just drink. Yes, well, that was such good value, that place last night. What? Oh, God, wasn't it? I'm not going to, you know, I mean, you can't beat that. Well, they don't have this law in Belgium yet, I don't think. They have a lot of places where it is enforced as a private matter. I think it's still left up to the discretion of the publicans. It's certainly not a federal law.

37:30 Well, if you remember, they have so much problem passing any kind of law in this country because they have, you know, this constant... There's a little battle between the Walloons and the Flamards, and, well, plus, do you know that, I mean, a country with a population only about, well, smaller than that of British home counties, but not much smaller than the population of Greater London, not much bigger than the population of Greater London, they have eight regional parliaments. It's absolutely absurd. They must have more administrative overheads than any other country in the world for their science. It's insane. Still, at least it means that they have, and they have two, they have two national academies. You know, one for the Flemish, one for the, it's completely crazy, isn't it? Oh well, I'm delighted that... I just wish they would blow up the statues of filthy, filthy Leopold II, because it's like having statues of Hitler on every corner, you know, at the end of the day he was one of the greatest mass murderers of all time. He in fact quite possibly killed more human beings than Hitler. I'll make a move, I'll stop wandering down there, as I say. Okay, well, we could come along. I'll stay around till the start of the dinner. I'd like to just say hello to Peter, because I haven't seen him for a while, and actually I want to apologize to him about the Oberwolfach business, because in Calais I had invited him and Bill to come to Oberwolfach. Which, well, Colin had already said, yes, we must have Bill, we must have Peter, and Peter was very keen to come, and then to discover that Ralph Kromer had trumped everything, and fucked it all up by inviting these people like Pierre Ageron, I mean, for all I know, maybe a nice, well, actually, everybody says he's a lunatic, but, you know, I really am pissed off that he did that without even bothering to consult. Oh, well, one of those things.

40:00 I'm trying to remember, I don't suppose you can recall it, have you read Colin's paper in the JSL from I think about 1987? The one about, it's called canonical, it's the one about canonical points of topos, and it's the one which has the explanation of the conditions for there to be, well, there's two versions, a stricter and a weaker version. There's one called the decidable sub-object condition, and then there's a thing called the weaker, the weakly decidable sub-object condition, and they translate into different conditions on... On the coverings of the space, as Grotendieck topologies, whether they localize completely and whether the smallest cover of the topos as a space is separated or whether there's a slightly weaker relative uniform separability. Do you remember any of that? I want to ask Bill about it because it seems to me to connect in quite a significant way with something that Peter said quite early in his talk, but I don't want to ask him unless I've got the form. He performed the set theoretically, in the sense that they are all shifts of sets, and so much depends on this fact, so for me it's not surprising that you're doing something also more intrinsic. Of course, you just hope for different. True. His motivation was... Yes, yes, although, of course, well, I think, yes, of course, yes, I mean, that's true. I mean, the free shape construction does. And category sets, Grotendieck never considered kind of really, I would say, category theoretically as stupid as given. And so all category theory just came later, I guess. Well, yes, he took it as given, but in a way that he really expressed itself just in terms of the kind of closure conditions and operations he wanted to use in forming, you know, in forming his categories. I don't... You remember that Grotendieck was thinking very much in terms of categories as machinery for proving things about algebraic geometry, about field extensions.

42:30 He was never a foundationalist, even kind of an unselfconscious one. One of his favourite sayings, according to Cartier, was when people kind of raised questions about foundations, which says, don't look down. My only advice about foundations is not to waste your time looking down. Don't look down. So to that extent he was very opportunistic, I think, in his attitude. He was going to find the right constructions with the right kind of closure conditions and then whatever was needed to clean up the distinction between large and small categories. What would drop out of the right viewpoint which would come along, and I think he was quite, I think he found the whole, this was, I think he actually found the whole theory of Fibonacci risk and descent theory quite appealing because it made the thing much more geometric. The stuff which Benabu is going to talk about a bit tomorrow. It's a tragedy actually that Benabou has become so psychotic because he did very interesting work. Conceptually his stuff that I think derived from the Erisman tradition was very interesting and important. Actually, it would be great, I think, to have some conference, say, on foundations of mathematics. I absolutely agree. I've been saying that for years. And one of the things that we should do is to have a thorough exploration of the, precisely, of the different motivations and the different kind of, you know, conceptual points of convergence and contact, both from the structural standpoint, from the point of view of the actual mathematical development, but also... In terms of the different overall and conceptual perspectives as between as between um builds you know insistence on close cartesian close categories are fundamental because you've got to have good map spaces because in the end categories about geometry you know the the things which are really The core, which are at the bottom of everything else, as he says, algebra, logic, analysis, are ultimately these geometrically motivated categorical constructions. The Venabubu viewpoint, which is to do with vibrations, and then this even more general programme of... And then the, what you might call, the more pragmatic internalised...

45:00 You know, relative, well, enriched category theory and the relativised constructions, the kind of things that Kelly and the Australian school majored on, because they were more interested in, well, in constructions which were more purely algebraic rather than geometrical, and, yeah. And then also the persistent point which the people who are, you know, unrepentant logicians in their attitude, like John Mabry, continue to make, which is that... You can't sweep the distinction between large and small categories under the carpet and treat it just as if it's a technical matter internal to the category through itself. It was never sorted out and as things stand at the moment, this is the claim, I don't agree with it. I think Benabou in fact, that's one of the reasons I'm still so interested in Benabou's work because I think he possibly was on the right lines for solving this problem. But the claim that John and people make is that in fact it's still entirely dependent on set theory. The distinction between large and small categories, which is absolutely indispensable, you cannot get the theory off the ground as consistent theory without clearing up this issue about large and small categories, because there really is no such thing as the category of all groups, in the sense that it's this kind of... Itself not categorically theoretically well-defined, large category, but you know you have to appeal to set theory at the end of the day, in order to, but I think, I think, I... No, no, well, this is the argument. I mean, there are... There is no clear way how to... Well, there's no clear way, but there are at least three or four. There are at least three or four strategies, broad suggestions as to how to do it. One of which, which I think in many ways is the most conceptually worked out of which, is still Benefuse. Although Bill, of course, has his own viewpoint. But Bill's ultimate kind of foundational project, which was the category of categories approach, although I can see much more clearly now with all of this stuff about... Figures, space, and, you know, the revival of the Grassman program for understanding mathematics as the science of space and quantity, this kind of huge overarching program he has, where it's all coming from. As far as the direct, directly confronting that challenge, that technical challenge about the issue of large and small categories, the category of categories didn't go anywhere.

47:30 I mean, hardly any, there's hardly been any papers published on it since Bill's original paper. That was partly because Isabel found this. Category of categories, I don't know if you'll agree, but I read this remark, I don't remember where, that actually it's kind of eschizoretic, what he said. There's the remark that Colin made, that it's an idea of pure reason. It's like a Kantian idea of regulative ideal, an idea of pure reason, which is a bit of a cop-out, because it's to say that... No, I think different thing, no. That what Bill did technically in this paper, category of categories, was very much like he gave a sketch of category of... In the technical sense of sketch theory, actually he was just starting to talk to me about sketch theory just now, well no, just before Peter arrived, after that previous talk by Yana Lidze's son, well he was coming out, he just took me outside and said, you know, this is what's completely wrong with the whole Yana Lidze program, this whole program about, you know, Malchev, Malchev classification, the Malchev approach to... You know, treating everything in terms of this kind of algebra of symbols and operations, this is why it won't work, you know, and he's starting to explain, of course, but just as he was beginning to get in his stride, saying, this is a completely wrong way to approach the syntax of category theory, it's just, it completely misses out on all of the internal tissue connections within category theory itself that allows you to model what he calls representations. These people haven't understood the notion of the representation of a theory. And then at that moment Peter appears, so of course we want to go in and hear him, so I would love to understand more about that. There is a lot of presentation, but it's also a step in the process of presentation. Sorry, that's what he said, the presentation of a theory, not the representation. That was my error. No, those were his very last words before Peter appeared, and we went in to hear him. He said the problem with this kind of Yann Lidzy approach about... In trying to sort out what the conditions for the existence of, you know, separators of pairs of arrows, and therefore for the syntax to work as syntax, not only have they lost sight of all the geometrical precondition machinery that you need for there to be separators of pairs of arrows in the first place, because they're just focused in this formalistic way on the symbols, but... They also fail to understand what the presentation of a theory is really about.

50:00 Well, of course, I wanted to hear more, but then Peter came on the scene, which is fine. Pretty good, I have to say, that he... Yes, well, it does. It goes down the steps there. We can go that way, but it's six of one. To go directly to the Grand Place, this is probably quicker. I came in a different way. Yeah, yeah, well, you can go that way, but it doesn't really make very much difference, it's just, you go down through the glass. The, um, oh, it's a very, very interesting day. Oh, in fact, they are all walking down the hill. He seems to have looked much more positively at sketch theory. He was saying that that's a much more serious way of approaching the issues about the presentation of theories than what these people are trying to do, to go back to a completely formalistic theory of syntax, which is... Yeah, there he is, yeah. God, are we going to rescue him? Well, she wants to talk to him about the pre-Socratics, she wants to talk to him all about Zeno. I should never have mentioned to her that he was interested in this conference with Jonathan Barnes. Apparently, Jonathan Barnes is one of her bêtes noires, so I hadn't realized that. Well, of course, he's completely wrong about everything, you see. The only person who's ever understood the pre-Socratics is Karin, so they have to... What a lovely evening, isn't it? Yeah, absolutely. What's the problem? I just want to go to my hotel. Thank you for your attention. Because everything is really a category, so you want to present a category. Yeah. So you start with a graph.

52:30 That is more than just a category, so you have to have a whole sketch. I don't remember where I read who wrote that. When you define category, let's go, come to you. Unfortunately, I'm not able to come to dinner as much as I'd love to. No, but you're going to share drinks now. Yes, sure, sure, if you can have drinks beforehand, that would be great. That was a very exhilarating day. Yes, and then I missed even one of the three that I wanted to see, namely L'Auvergne. Yeah, but as I say, I will burn a copy. I know it's not the same, but you can hear everything. But I know we will go into more discussion. Well, plus also, the paper on which that talk was based, which in fact Bill actually used as his overheads, I have a copy of that. So I don't only have a copy of the paper, which was what he used for his overheads, but also plus the whole of his talk. and the question session so you can listen to everything so it's really you really haven't missed anything at all yeah except actually like to see well of course you would have liked to have been there to hear it i know i understand it's a lot better than nothing okay it's down here isn't it oh no it's a little bit it's wonderful oh yeah oh i see sure all right I'm staying at the Hotel Aristopoulos, which is near the South Station, I don't know if that's any good. Avenue Stalingrad, does that help at all? Which street is it on? Oh, okay, I see, fine, I thought you, oh, you do, okay, sure, I thought you might be, yeah, it's really okay. No, and Johnston was amazing, what he taught was really interesting, I only Who is this student? Olivia, she's in my year. What is her family name? Caramello. She's Portuguese, isn't she? Yeah, she's Italian. Oh, she's Italian. Beg your pardon, my apologies.

55:00 Well, I guess we're going to hear good things. In your case, what is original geometry? If you take just a very tiny category... What object is an abstract line and an abstract circle, you see, in a relationship. Well, then the pre-sheaves on that would be arbitrary spaces of arbitrary dimensions that have that kind of structure, all the circles and how they intersect and so on. If it were worked out, it would probably be the best way to teach high school geometry. So you have it as a text, do you? You should have this as a text, yeah. I would really love to see that. No, no, that would be the worst thing. It would actually, because it would get back. ...precisely to the core of the Euclidean viewpoint about incidents and relations and figures, but making those notions as were completely rigorous within with all the machinery of modern insight into categories. ...faces of all dimensions... Exactly, allowing it to... ...thus getting away from the idea that... ...at the same time the... ...the effect of the plane... Exactly, yeah. They are not only adequate but also co-adequate, that the spaces of interest can be represented as the opposite of algebras, where algebras consist basically of smooth functions into the plane, equipped with operations from the plane to the plane, like shifting around or squaring and so on, and then homomorphisms like that, but in the opposite direction. So a co-adequacy means that the spaces of interest are fully embedded in the category of abstract algebras of that sort. But it is a theorem. Precisely, with no ulam cardinals, you need no ulam cardinals, then it is co-adequacy.

57:30 The ordinary Euclidean plane, you could even make it sound more Greek, you see, what these operations are. So Euclidean geometry based with conic sections as the algebra is co-adequate, but you need no measurable cardinals. Thus again further reinforcing the point which we were talking about at lunchtime, that this propaganda that higher dimensional geometries of course have rendered any notion of space and figure as completely inadequate as a basis for geometry because these relied on geometric and kinetic intuition, our own simple-minded intuition about free space, is just a complete lie, but of course an immensely influential lie. Then, of course, translated with all the set theoretic in the business about the Kimberley Hierarchy, that you've got to have a theory of cardinality, an abstract, you know, oh, we better... You can't have these discussions of the computer. Yes, teacher. True. Yes, teacher. Very good. Yes, quite true, quite true. Quite true, quite true. I am going to dig out that paper of Collins about the weakly decidable sub-object. ...and to sort out exactly what it's about. I haven't looked at it for about 10 or 12 years, so that's no excuse. I should be able to remember exactly what the definition is. Peter said, why hasn't anybody found out this result unless... One reason is that nobody cared about De Morgan except him and his students. Yes, true. That's true. Nobody talked about that. No, because the only paper I've ever read about De Morgan in topos theory is his paper about the De Morgan monoid in the 1977 Durham Conference proceedings. Well, there are a few papers. Well, plus also the point he made, which was very interesting, which is that he doesn't have a kind of frame-theoretic motivation or natural way of being said in terms of frame theory, which may be...

1:00:00 Which maybe connects with your point that, you know, we lost sight of the other aspect of... Oh, no, no, they're around the corner. They must be on the ground plus. They must have turned left here. The only place they can have turned. Oh, sorry, oh, sorry, we thought... This idea that... One of the features of cohesion, what I called adequate cohesion, was that every space can be embedded in a contractible one. Hello! Yes, okay, we have to... No, of course, contractible means... Well, that it and its endomorphism space are both connected. Where connected means relative to the base topos. Right. So now, Gordon had a criterion in his manuscript there about, for this, when the base was sets. And I wanted to just state his criterion as a theorem, but it turned out it wasn't quite right. You needed a condition on the base topos, much more general than being sets. Namely, it was the condition that the object 2, the discrete 2, is an injective object. Right. This is how I put it. Yes, yes. But then Peter Student came, you know who I mean, Kenney, he's very bright as well. He said, well, that's equivalent to De Morgan. Yes, yes, yes, exactly. The base is De Morgan. Which I hadn't thought of at all because I never think about De Morgan. No, not at all. I always thought it was some kind of curiosity, but it seems to come up from time to time, so... But it clearly carries a bit of... Objective, if you know what I'm talking about. And as you say, this whole point about the existence of connected objects is clearly deeply related to this point about... In thinking of the adequacy and co-adequacy conditions in this, in exactly the way that one does in terms of figures and incidence relations, like the Euclidean setting, one is able to bypass these constructions like the Ulam, measurable cardinals. Well, not bypass them, they just don't arise.

1:02:30 That's why I propose to finally define the notion of a small set, to be a non-hook, a non-hook. Yes, yes. Sounds like a double negation, but isn't it? No, no. This also connects with what you were talking to me about in Calais and on previous occasions, which is, why was it that Dedekind was led, one might say, kind of, corralled? ...into imposing his definition of the continuum as being one which completely washed all the cohesion out of the underlying space so that everything was ground down into completely structuralist points, dust particles so to speak. Well, because he'd been got at, of course, by the discovery of the pathological functions, the idea that one needed n as a complete discrete infinity, because otherwise you couldn't keep track of these things like the Peano space-filling curve, or the diastrafts and the... This was in, well, from our point of view of what we can now understand with present-day machinery, this was, of course, already building in lots of very strong assumptions on the nature of the category in which, you know, in which you are. In fact, I really wanted to ask you about that. It's one of the things you were talking about in Calais, which I'm afraid I didn't... We got sidetracked also because I had an awful problem with my neck. But I really... There was a very interesting point you were making about the... About the Dedekind construction, about the way that Dedekind got sidetracked, as I say, by this conviction that geometry is not enough, that we had to abandon this notion of geometrically reasonable mapping and go to some completely arbitrary notion of mapping.

1:05:00 Which, of course, in terms of the great antique topology that Topos covers, can be expressed now in a much more precise way. I mean what it was that he also as you pointed out in your lecture on the continuum in uh back in um 2002 in Nancy which builds in all of these assumptions about subjectives and about injectivity and subjectivity of maps actually builds in all sorts of well What appeared at the time as an inevitable generalization of the notion of function, but which is in fact an unnecessary restriction, taking it away from any kind of understanding of the conditions for maps to be geometrically reasonable. No, I have to say, beware of Karen's propaganda on the subject that I am the only person who's ever understood the Presocratics. I don't pretend to have any expertise on the Presocratics, but her paper on Zeno, Colin, actually, don't report this to her, but Colin refereed it. He thought it was about the craziest paper that has ever seriously been submitted about Zeno. Now, I'm not qualified to judge if that's fair. We're going to get to my paper when it's finally written. What's her name, by the way? Karine Varelst. Varelst. V-E, Varelst. Karine Varelst. She's Belgian, as you obviously realize. That was over a million years ago, but I forgot. She's completely crazy, but she's very likable. She's the kind of extreme left communist that Lenin, I'm afraid, had in mind when he spoke about the infantile disorder, because she had a go at Colin in Paris three years ago, accusing him of being an apologist for American imperialism and worse, simply because he made the tactical mistake of using a bit of Socratic irony on her.

1:07:30 Which was probably, but I have to say, George Rousseau gets on very well with her. Was that term irosyncratic? No, Socratic irony. Oh, I thought it was something Irish. No, no, Socratic irony. Some kind of Irish irony. Well, it could have been, maybe it could have been that, it would have worked better. No, he used a little bit too much Socratic irony and she rather overreacted. George Rousseau and I literally ended up having to separate them to prevent a bloodbath. It was an interesting, interesting discussion. George was there too. George was there, yes, George was there. George was at that conference all the way through. He was a great man, really enjoyed his company. That's the last time I saw him, unfortunately. I haven't seen him since then. I corresponded with him and spoken to him a few times on the phone, but not actually seen him. That was a few years ago. That was in 2005. That was almost exactly three years ago. In fact, that would be three years ago... This month, almost exactly three years ago to the day. After Fougere. Oh, yes, it was after Fougere. Fougere was in the summer. Fougere was in July. Right. This was in the November, October, November. Yeah, it was about. No, that's right. It was the end. It was the beginning, end of October, beginning of November. So just coming up three years ago. Well, he's living, you know, quite quietly in retirement and still keeps very active indeed in, you know, both in scientific and political terms, but doesn't travel much. I know he'd love to see you again. Where is he living? Leeds. I think he's still living in his house. He's got a bungalow in the country not far from... Because he was in Leicester before. Yeah, I meant to say Leicester, of course. I meant to say Leicester. I'm a bit confused. Yeah. Yeah. No, he's still living where he was. And... No, but Caring has got this weird... view that kind of has got this weird view that what Zeno is saying in the texts that have actually come down to us, who claim to be the authentic texts of Zeno, or the nearest thing we have to the authentic texts in the commentaries, that it can all be cleaned up in terms of domain theory. Well, I find it's very difficult to follow her argument.

1:10:00 Well, exactly, it's just a particular category but it's a kind of lattice. I mean there is an interesting point I think that she's got that Zeno had, with the equipment obviously that was available to him at the time, some kind of anticipation of the idea that the structure of... A domain of variation is better thought of in terms of lattice theoretic homomorphisms between its parts, and quantities varying over them, rather than in terms of some abstract notion of point, a frozen notion of point. And that's essentially the solution to the paradox that he proposes. That's interesting, but I've no idea whether... I'm no contextual scholar. I don't know. She knows Greek. She's basically a classicist by training who became interested in philosophy of math. You go here. I go with him. Ah, okay. Sounds like a good idea. There's a separate subject. The one about which they had that big meeting in Belle Isle about three years ago. The Argentine Trotskyite guy who still goes on about Dalhousie and how the principal contradictor constantly overruled him. I'm sure, of course, he's quite a character actually, I must admit I quite took quite a liking to him.

1:12:30 Thank you very much. Thanks, cheers. Shall we go outside? The guy who does all this propaganda for, oh, Jesus, where? First item on his resume. Yeah, does all this propaganda for bimenoidal, symmetry bimenoidal categories is the solution to all the... First item on his resume. Yeah. He's not, is he? Of course. Of course, well, Jesus Christ, I didn't know. Um, him as well. Well, he's a complete opportunist, then, because he's definitely not a religious fanatic. He's just always willing to take money from them. I think he just stays in that ass position, being an opportunist. He just makes claims. Well, he's an opportunist, yeah. His claim that the Templeton organization's activities are not right-wing and religious is bullshit. I mean, it is completely misunderstanding the nature of right-wing activity and religion in this state. Of course, it's not... It's promoting some particular sect. That would be an intimate project. It's promoting religion, it's promoting Buddhism, Evangelicalism, Catholicism, everything simultaneously. Sure, sure. It's this kind of claptrap that Prince Charles comes out with all the time that his mother gets very annoyed about. That's about the only good thing I can find to say about the Queen. He says when he is crowned as king, I think he has more expectations of the continued survival of the monarchy than some other people, but if and when he is finally crowned as king, he wants to be crowned. He does not want them to use the title defender of the faith, which of course is the thing which was given to Henry VIII by the Pope, as you know, for having written... Henry VIII wrote this refutation of Luther, Lutheranism. This was in the days when he was still a good Catholic, but he wants to be crowned, and every successive British sovereign since then has had this official title of defender of faith, but no, Charles doesn't want this because this is sectarian exclusive. He wants to be crowned as the defender of faith. No, no, seriously. He made this speech. It's perfect. Exactly. Precisely. Precisely. I know. I know it's exactly your point, which is why I'm underlining it.

1:15:00 Exactly. I didn't see any hidden agenda. Yeah, I didn't see any hidden agenda. No, no, no, no, no. I'm in front of you. Yeah. Well, no, maybe not. I'm trying to turn to you in the sense that the essence of the agenda for this particular presentation is to lower the standards of rationality and make irrational physics more feasible.