Categorical quantum gravity

0:00 You know, it's easier than going and worrying about getting it. It's the current version of what I'm doing now. So I get to talk about it in Portugal. So if I submit it to a conference proceedings, I don't have to worry about sending it off to a journal. And you know, nowadays a lot of the conference proceedings are refereed and so there's a pretense at least that, you know, it's good enough for Kansas. Oh, come on, don't do yourself down. It's extremely... Who's bringing it out? Who's bringing out the conference proceedings? Which conference is it? It's a conference in Lisbon. Wow, that must have been very interesting. I spent two years at Lisbon. Yeah, you mentioned it. I have three Portuguese graduates. You mentioned that. I was very interested in knowing more about that. One of the things which cropped up in this Ober-Bolfack meeting on the history of category theory, which I found very interesting, was a bit more information about this extraordinary PhD thesis, which Bill regards as the kind of The biggest single missed part of the entire history of category theory by this Portuguese mathematician da Silva. Which he wrote in Rome in 1943 and which was called the general theory of homomorphism. Did he go to prehistory? Well, yes, except that he's got, he's got, he's, I mean, I haven't seen it because it's never even translated. It's before, it's before, well, it's not actually before the 1942 paper on the general theory of natural, well, no, the general theory of natural equivalence is 1945, but the earlier paper, which already contains, was 42. Was that McLean? The major paper was the 1945 General Theory of Natural Equivalences, but there was an earlier paper in 1942, which came out of their work on the solenoid. Originally category theory was conceived as machinery for algebraic topology, and then of course they gradually came to see that it had far more general applications across much wider fields of mathematics.

2:30 But this guy, De Silva, who was working quite independently because he was in Rome during the war, so obviously he didn't have access to American academic journals, he would not have known of the 1942 Eilenberg-McLean paper. He produced this extraordinary thesis called The General Theory of Homomorphisms, about which his supervisor was. No, no, no, this is the thing. Bill is actually arranging now for it to be translated into English. No, no, it was written in Portuguese, although he was in Rome at the time. He couldn't get back to Portugal because of the war, you know, he couldn't. Well, the thesis presumably was never published. It's sitting in the library of the University of Sapienza in Rome, I imagine. But curiously, Cartier has also looked at it and thinks that it is, in fact, quite extraordinary, 25 years ahead of its time. Well, no, there are people now actually engaged in producing an English translation of it, and it's going to appear on Categories' list on TACMAC. Yeah, it's going to be produced electronically in Categories. The people who ran the Categories meeting in Coimbra. In 2004 or 2005, whenever it was, are translating it. So that would be an extremely interesting contribution to the history. Should we go to... You tell me, whatever you... Are we waiting for anything? We're not waiting for anything. No, no, no, we're absolutely, you know, whatever you feel, whenever you feel like it. Should we make a reservation or anything? No, I just came from there, I just looked in there, because it's not too full, there's still plenty of places. It's getting busy, but, you know, we'd still be able to find a table. No, it was... the... That was a very interesting, very interesting talk. So we're going to meet now on the 17th, is that right? Two weeks from today, yeah. From my point of view, that couldn't be better because, yeah, well, as I say, from my selfish point of view, that couldn't be better because that's, that's the, not error.

5:00 Well, from my point of view, it couldn't be better, because that means I'll be able to come, which I wouldn't have been able to do on the test. Well, it's okay. Don't worry. I don't think anybody's going to be particularly hung up on it. And René Guittard said he would have preferred it to be in two weeks anyway. So, I don't think anybody was going to be happy about it at all. I'm thinking of making a submission to the Gravity Research Foundation. Even though the last two times they did that, they seem to have not even so much as looked at it. But still, I don't know. Which research foundation? Gravity Research. Oh, Gravity Research, yeah. I have a second place award for many, many years ago, but the stuff I'm doing now, they just... You see, they just can't get their heads around it because it's too categorical. Yeah, too much math. Well, I think it's absolutely beautiful that spinfoam actually turns out to be... Not spinfoam, but something as it were, much categorically, no, much more. Well, what I really think is that the space-time is a category, the geometry is a kind of category, and the theory... It's a measure on the category of maths, you know, the HUM category. Yes, that's how I think of it, certainly, that it's a measure on the maths and the HUM category. I'm not so sure about the idea... Well, it's only literally two minutes to the restaurant, so... It depends what you want to do, if you want to go straight off from there. Yeah, it's not... It's a nice mild evening, too. The only thing, I'm a little bit worried about the notion of categorification in itself, because I think it is difficult to make formal sense of it. I think the problem is because one has to specify, one has to do more than simply specify the category on which one's... I think the way that you're using it is shorthand for what you're doing with this projection of structure down from the higher dimensional situation into this nice algebraic and geometrical machinery, which gives you essentially a kind of cobordism theory, is fine, there's no problem with that.

7:30 The definition of the notion of categorification is very problematic, and I'm not sure I'd buy into it. I didn't even want to touch it. No, no, no. You didn't, and that's not a criticism at all. No, I know. But it is a criticism of some of John Byers's statements about it, because I think, you know, this categorifying the kitchen sink, it's just a formula which allows you to categorify anything. I'm very, very dubious about that, because it seems to me to involve a certain amount of sloppiness in the choice of categories. And some quite deep confusions about concrete categories too. I've always been a little bit suspicious of the idea. Partly because it has become a bit of a buzzword. Well that really surprised me. When I invented it... Did you invent it? Yes I did. Oh! Well I didn't mean to... Don't take anything I've just said as in any way personal. I thought it was John who'd invented it. Yes, John did a really good job. No, no, he actually, John makes reference. Okay, well, I should read the footnotes more carefully. I mean, the original proposal was that Donaldson's floor theory could be constructed by categorifying Simon's Witten here. Sorry, you'll have to say that again. The Donaldson-Vore theory could be constructed by categorification in terms of Simon's-Witten theory, that you could take the category of representations of the quantum group and categorify it into a two-category. Well, turn it into a two-category, I'm just, okay. No, categorify it. Well, how do you mean categorify it? The representations of the quantum group have a special basis called the canonical basis, and they can be constructed out of the Grotendieck rings of categories of perverse sheaves over flag varieties. Sorry, over flag varieties? Flag varieties, yes. Gosh, I had not noticed that. There they are. Hi. See you later. He's booked there. No, no, no.

10:00 We've got a booking in there. Hang on. You've got to go in. He's there. No, he's calling you, though. You've got to go. Come on. Come on. Yes, because your father-in-law is going out of his way. And this guy on a different table, he's a very interesting writer, actually. Really? Well, this is a marvelous place, Paris, that kind of thing, running into interesting writers. He's writing this kind of travel book, which in French is not that common, but I think it's a really good one. No, I'm sorry you're going through an extremely rough time. Well, I'm not, she is. Well, she's your wife, so obviously it does impact on you. Well, it's sort of difficult having her gone for a long, long time. But I don't know, I was about, I mean, I was, I've been with women. There is also a chapter on quantum rules. Kirilov is very interesting. His method of orbits is very interesting. You're supposed to try it. I don't know. Talking about people's sons, I mean, what's the reggie that you were talking about? Is that, is that Giulio Regi of Regiculculus? Gosh, he must be getting on by now. I was going to say, I thought he might be his son, yeah. Well, except you just said he's just produced this new work with... I mean, isn't this relatively recent, the work that you were citing? No, no, no, that was like from the 80s. Oh, okay, sorry, that was much earlier than that. That was the prehistory of my work. Well, okay. Oh, right. I wasn't aware of that. You didn't give the date, so... Okay, in that case, obviously it was. I was just thinking that Reggie was still producing papers now. He must be kind of... People told me, just don't bother to talk to this seminar. They can't understand anything anymore. It sounded very sad. I think it's a very good introduction for... ...the paper, that book you've written. Yeah, I probably will take it. It's a good type. The math library is closed because of the strike. What bloody good does that do to anybody? I mean, how is that bringing pressure on the Ministry of Research? It's pathetic.

12:30 They think that closing down this website which gives information about all of the philosophy of math and history of math seminars anywhere in France is somehow contributing to this general strike which is supposed to bring pressure against the government and their plans for the reform of the research. It's completely pathetic. I'm sorry but it's so bloody childish it makes me spit blood. My wife was travelling back from France. It wasn't even with our gestural politics of the most childish sort. And they told them your suitcases might not get loaded. But after they took off they told them that the workers... Let's get back to what you were saying earlier on. So, the notion of categorification, when you originally introduced it, was specifically geared to this construction using ring categories. It's a bit broad, isn't it? Okay, just remind me again, run it past me again, because I'm extremely interested in the history. So to construct a two-dimensional P4F team, you need an algebra, and the axioms of a ring category are exactly like the axioms of an algebra, except they're categorification. The algebra, the axioms should be just right for constructing a topological theory in two dimensions. The categories are just right for constructing a topological theory in two dimensions. And topological theory in three dimensions. And so the Pentagon looks just like a force in the ice.

15:00 And so therefore it corresponds to a partner world. Whereas the associative law looks like a tetrahedron. And I see how this actually lends itself to higher dimensional, to a higher dimensional ladder. If you have a two category, the pentagon becomes amorphous. Sorry, what becomes amorphous? The pentagon. Oh, the pentagon, yeah. It becomes a two morphism called the pentagonizer. It really satisfies the natural coherence solution, which is called the hexagonal, which looks like a fraction, but that would appeal to the French. Actually, what's kind of philosophically interesting about this process is that it should be something very general, something that you, at first step, you think is a part of description, so it's not like part of your idea of how you, what you say, but next into a part of... Concrete in terms of, you know, simplicity, in terms of, as you say, tetrahedra and their higher dimensional generalizations. It may be just analogy, but just recently maybe, you know, talk on force. It's very good in winters. It's a peasant dish. I mean, cassoulet, well, it's exactly what the peasants ate after a hard day, a hard day in the field.

17:30 I saw this one earlier. His view of Glassman, his whole theory of intensive and extensive quantity. And that's in Cambridge? No, that's going to be in Bristol. Cambridge is a two-day meeting in honour of the 60th birthdays of Martin Hyland and Peter Johnston. It's the peripatetic seminar on sheaves and logic. It's just in two days, the weekend, the 4th and the 5th. So I can stay there and... That should be very interesting. I must go online and see, because they've probably published a list of talks now. Maybe I could. It's far from Britain. You know, I suspect that Bill and John may be driving over there. I'm hoping they are, so I can hitch a lift with them. And train is expensive. Trains in Britain are ridiculously expensive. Oh, yes. Excuse me. Electricity. Yeah, it's one of the talks he's giving, I'm not sure which one it is, which one it is in the series. His first talk is something really weird called Pantor, Dedekind and Grotendieck, which will be about, his view about discrete and co-discrete categories seen from the point of view of Galois theory. I'm not sure, I think it's on the second day, something like, possibly the day before your talk. All the day after I'm not sure. Your talk is on the 31st of March, isn't it? He's speaking on the 1st of April, so the day after your talk. No, that's the only talk that we do. No, no, it's on the 1st of April. It's called Kantor, Zermelo and Grokenbeek. Kantor, Zermelo and Grokenbeek, that's right, yeah. Interesting title. The most important thing about the category sets is understanding that union and power set, as unary operations, are adjoints.

20:00 A jointness situation, which is obviously fairly typical of categories, is very peculiar in the category sets because it's connected with the fact that the singleton is a free object. And that ought to be suggesting to us, and Bill's line has always been, that the... The very notion of membership as global and absolute is simply an incoherent notion that there's no reason why, when you think of the thing in terms of a lattice of inclusion relations, why there should be a fixed top element, just be relative and local inclusion relations. Again, it's quite interesting because it connects with these ideas that we take some grasp about in terms of an extensive quantity, which I would love to understand more deeply, which he thinks should be modelled in terms of covariant functors in the case of part of domains, in the case of intensive quantity, and contravariant functors into co-domains in the case of extensive quantity. He thinks that it was a terrible wrong turning in the philosophy of mathematics when people at the turn of the 19th and 20th century gave up on the theory of quantity and decided to base everything on classes and relations instead. An interesting idea. There's a great deal of algebraic structure in the theory of quantity that was never... it's actually a far richer theory of principle than logic. Which just deals, essentially, with relations and quantifiers. Bill would say it's the theory of co-variant maps into and contra-variant maps out of domains and co-domains respectively. In other words, it's basically concealed, it's category theory concealed. Only the people who were developing in the 19th century didn't know what a category was, but implicitly what they were doing was category theory. That's why Grassman's so interesting. Grassman had these extraordinary ideas about intensive and extensive quantity, which at the time people couldn't make sense of, but which Bill claims can actually be clarified very convincingly indeed by category theory.

22:30 But it's a provocative and very interesting thesis. Of course we were hearing lots and lots about Maclean in Oval Wolfpack, particularly from Steve Aldew. I would have been absolutely fascinated to have had you there. Which of his classes did you go to? Well, I took the Connors undergraduate algebra physics sequence and I think topology too. And that was in the 70s. In the 60s. In the 60s? Wow. He was just writing on these books where everything was done categorically. And they were handing out bits and pieces of manuals. Yeah, which of course presumably became categories for the working mathematician. Which came out, what, in 71, didn't it? I think so. Actually, it became the McLean and Birkhoff algebra. Well, a new edition of that, because that had been around for a while, hadn't it? I ought to look at that. Which edition? The one that came out around the end of the 60s or beginning of the 60s. Right, okay. Because it went through so many editions. Sorry, give me a minute. Not Berkhoff and McLean, McLean and Berkhoff. Oh, sorry. Okay. Thank you for your attention. First of all, is it possible we could arrange for them to speak, either at Paris CETIEM or at RESISE, and would it be possible if we must have put together a little one-day meeting with others? Yes, yes. Steve already wrote to Mik, that person. Mik told me, I wrote to Steve, so we arranged...

25:00 I'm going to, like, reserve the room and call Normand. I hope you're not here. I'm in New York now. Ah, yes, you told me. But anyway, it'll be somehow in a, you know, Auschwitz, IT group, whatever. But perhaps Colin might make a part of it as well? I don't know. It depends. I mean, can you ask Nick if he'd agree to that? I'm pretty sure Nick would agree to play like his today's thing, but he should write him. I don't think he needs to. I don't think it's, I don't think it's that. I think he's going to be in Paris anyway. Yeah, yeah, but Steve just asked for like two days' extension. If he wanted, he would make the same, I'm pretty sure. Do you know exactly which... I checked yesterday morning, we have 300,000... Very nice. Do you know exactly which days Steve is going to be here and which day you've got set aside? Actually, yeah. I don't. He's going to do something before, but then he proposed Steve. He proposed 11th and 12th of May. And I'm going to check tomorrow what's going to happen. Colin! Colin told me he wasn't going to be... They're asking me to get in touch with you and with Marco Panzer, John Jack, other people, to see if we can find something. The problem is, it is, they are both in Sweden for this Martin Löf thing, it is the same thing, they're both out. But, it's been asked to stay on for two days afterwards to give some additional talk in Sweden. So he can't get to Paris until the 13th. So, I'd like to be a separate one. I was thinking it might be rather nice if we could ask, because Renaud Chollet and Pierre Acheron would both be interested in coming and speaking. I was thinking we could do a rerun of the little workshop on structuralism. Maybe with Cartier, Cartier would be quite interested, Jean-Jacques would be. We could do a recessed workshop.

27:30 He's like, I wasn't proposing. I just wanted to go in there and tell him his job. I just wanted... I just need... I was going to ask him nicely. Well, like with Steve. Steve, I... Mick, hi! I just would like to give a talk for you. Can you just occasionally pay me a... Yeah. Of course. Yes, well, that's all I... Have you got that? Yes, fine. So I just have to tell Colin... Yeah, just you... you might... So I just tell Colin that he has to get in touch with Mick. Yeah. No big... there's no need to make a great mystery of it. I don't want people to come to this. Well, that's my concern, because that's why it would be good if we could make it into a recital. So why can't I talk to Marco? We will not be around either. Well, okay, in that case, the obvious thing is to talk to Marco Panzer or to Jean-Jacques or to... Because Renaud and Pierre are both willing to speak. About Colin. Yes, Colin. Remember, we were all in Oberwolfach, and Pierre Ageron and Colin... I would say that they'd be very willing to speak if they organized a one-day or even better a two-day workshop on structuralism in either in ReScience or Paris 30M, wherever, but for when Steve and Colin are there, at that time we thought they'd be there together. It may be that they're going to be separate for a couple of days, we can check on that. No, June is no good, they're not going to be here in June, Colin's only going to be here for about three or four days after the Swedish meeting, but they're not going to be here in the fall. I've no idea about Steve's position, but Colin certainly won't be able to get further leave from CWRU just to go swanning off. He's already had more than his lion's share of leave already for this year. The point is that he and Steve are going to be in Paris for those three days after the Ferdinand Martin-Loeb meeting, and that's the time that, if it is possible to arrange a meeting, that's the time we'd have to have it.

30:00 I mean, since they're going to be here anyway, you don't have to pay any of their travel expenses. The only thing which they obviously would appreciate would be having their accommodation called for another day. I mean, if you're going to have something like more serious in workshops, actually, I think I'll not be around. Nick will not be around, you know. So it might be... Well, I'm sorry, but... Just simply ask me, okay, can I give a talk? I say, okay, of course you can give a talk. Okay, well, what if Colin just wants to give a... Okay, we're just calling them as far as to give a call. Who should I speak to, Marko? We might put them together. Fine. Well, like as I say, the only problem is I don't know if they're going to be here on the same day. It's because you just told me that... Steve is going to be here on the 12th, and Colin is not going to be here until the 13th, that I know, because I just saw his email, because he has to stay on in Sweden for some reason for two days, so if it is possible, unless it's possible for Steve to give his a little bit later, I mean, I could always ask him. I've got to speak to him anyway, because I need to get his extended abstract. Thank you for the proceedings volume for Overvoltaic, which I'm supposed to be writing up. It's no big deal, it's just that since they are going to be here, I just thought it would be a pity to waste the opportunity. It would be nice if both of them... I'll be around then, in the second half of May. Well, in that case, instead of doing it through ideals of proof, could we not do it through resize? I know resize don't have any money, but they're not looking for money, they're just looking for a slot to give a talk. Even we can probably pay for that, but I mean, I think it's the kind of organization who would take care of it. Well, that's why I'm asking you. Yeah, so if you would... ...sabbatical or another grant anytime soon to pay for it, but, you know, if we are... Yeah, so obviously we're all going to be on very short commons in the near future, I think, in the... ...more ideal proof of... Yeah, things are kind of looking...

32:30 Sure, sure, sure. ...something more serious. I must send you, there was a very good article by Richard Brenner, who I think is one of the best economists around, about the prelude to this crisis, about the consequences of the entire kind of parallel banking system whereby the banks have effectively hidden the level of systemic risk over the last ten years. They were doing it, as it were, without even most of the time being aware that that's what they were doing, although there was certainly some level of deliberate conscious crookedness involved as well. But I don't think that was the key. It's an extremely good article. I must send it to you. So it's remarkably clear and very, very well informed. It's the mathematicians, you know. They should just have created as many jobs as they needed. They should never, ever fire a mathematician with a PhD from the academic world. They should just hire them and sweep the streets. You turn them loose, they go to the banks. And they convince the banks that they know what they're doing. I know, it's an absolute catastrophe. The Black-Shoys algorithm. The interesting thing, the really significant thing about the Black-Scholes algorithm is that, I forget which one it was, I think it was actually Scholes, turned down, they offered him some extremely high paid position to head a hedge fund, and he turned it down, because he said that, you know, this is absolutely crazy, I mean, these people have absolutely no idea about the systemic risk to how to control them, getting involved, and this was the man whose work they were mainly relying on. I should have told them something. Yeah the problem was that the whole the whole bubble was so everybody had such a huge stake in happy happy talk and in margin and in leveraging the thing further and further because that's where all their bonuses and profits were. It was like when this all happened I said to myself one day you know I know the way my colleagues run...

35:00 Oh my god, oh dear, oh dear, oh dear, I'm sorry. Oh yes, yes she is, isn't she? Oh dear, this is not a good moment, is it? No, it's not a very good moment when you've got a kid and dogs in the house. Except the Guido's out at the moment as well, unfortunately. Okay, well he's of course the guy I really need to talk to, so give us a time and I'll give you a call back. Probably, what's the time now? Katie, don't! Sorry. It's okay. In your neck of the woods it should be about 625. I would say about... Okay, which is 9 over here. Okay. I'm actually in Paris at the moment in a friend's flat. It's okay. He's got the 3.FR service so I can ring any number in the UK, you know, for free. Oh good, okay. So I'll give you a call about 8. Lucky you. Great big hug. OK, that would be fantastic. Look after yourself. Take care. See you soon. Lots of love to Kate. OK, take care. Speak to you in an hour or two. Cheers. It's Mike. Hi. How are you? Yeah, I'm in one piece. Is this a good time to chat? Good, excellent. I always like to make sure because, you know, I don't know, you might have friends or family staying and you might be very busy, you know, working. Yeah, I was just ringing up to, well, first of all to thank you again very much indeed for all those wonderful conversations in Montpelier from which I really did learn a very great deal.

37:30 It was a great meeting. It was also very, very, it was just great getting so much wonderful teaching from you. Also, to give you a quick report about Oberwolfach, about which you may possibly have heard from Colin or from Steve Aude. Okay, well, it was, I think, a much, much more successful meeting than I had feared it might be. And there were some extremely good talks. Both Colin and Steve gave very nice talks, Steve's particularly. About Saunders-McClain and about some recent work of his on the, I mean, that's to say of he, of him, Steve, and his students in Carnegie Mellon on the Tarski problem of the characterization of logic, the characterization of purely logical expressions, which turns out to have a rather neat solution in terms of the action of... The action of topological groups but anyway there were some nice talks as I say and I didn't I think he quite a nice talk about about Sammy Eilenberg and the theory of group why Saunders and Sammy's ideas in the 1940s which was not so interesting but his talk on Sammy was was and he didn't pet this philosophy at least and we had a couple of Very good German historians there. And he does do that, and there's no getting away from that.

40:00 Yes, well, I'm afraid there was a certain amount of that in some of the philosophical discussions that we had on foundations, which left me rather unhappy. He did much less damage than he might have done, so from that point of view, I considered it a victory. And it was a pretty good meeting. There were one or two talks that were not all that exciting. Shapira gave an interesting talk about hyperfunctions. Shapira, yes, Pierre Shapira. Yeah, yes, he came. He gave quite a nice talk about hyperfunctions and about the role of categories in analysis. Yeah, that was a good talk. And yes, well, in fact, that was also the theme of one of Pierre's two talks about the role of category theory in analysis, which was a very good talk. Well, I don't want to be so rude as to say rambling, but a slightly less focused talk on revisiting the old Burbach 1940s theory of structures and discussing how much they had got out of their discussions with SAMI at that time and a little bit later, and also, which is more a purely historical talk. All in all, there was some very, very good talks. There was also a young guy, youngish guy called Pierre Ageron, who's a mathematician from Cannes, but who gave, I thought, a rather nice talk about... I suppose Erisman's relations with the philosopher, the crystallization of his role in the crystallization of sheaf theory, and how particularly the Kazan problem was a very important, played a very important, and indeed, you know, problems through analysis played a very important role in the development of sheaf theory, and the story of the development of sheaf theory is it's usually told it was all to do with, you know, sheaves on topological space from the beginning. The kind of, obviously it is the main line of development. And the other thing which you would have

42:30 What was the De Silva's name came up speaking to you in Montpelier and called attention to the importance of this, the likelihood of De Silva's on general homomorphisms, but how much it deserves to be translated and publicized. And that was one of the points at which I thought, you know, here I didn't mention it specifically. It came up in discussion. We had several roundtable sessions. I have to say it was a pretty fruitful meeting. And of course you've been there yourself, so you know it is an extraordinary place. It's quite an atmosphere. There were two other workshops going on at the same time as ours, both small ones, and one of them was being run by a guy called Laptev, who is the chairman, the new chairman of the European Mathematical Society, who's just got the chair at Imperial, and he works on dynamical systems. But he's a Russian, but he's, as I say now, got the chair in London. And I got chatting to him and got some quite useful contacts. He's actually asked me if I'd like to join a working party which the European Mathematical Society has set up to advise the European Commission on how web resources and archives ought to be made more use of in the promotion of both the teaching and both mathematical research and teaching. So I'm hoping something may come out of that. My Germans, the Ralph Crowe, sorry not the Ralph Crowe what I'm saying, David Rowe and the Le Starck, cancelled their visit to come and see me at the beginning of last week. But the administrative problems that they've got in mind, they've now had to postpone the whole thing until May, probably June. Rather hoping, you know, they were dangling the process. He still seemed very positive.

45:00 In the meantime, I've also had some hopeful noises from the British Society. For the philosophy of science, we're also having a meeting on the 8th of June at which they're going to discuss whether they could give some funding support for their project of starting to get an archive online. So I'll just have to kind of hang on until then, but I've been so used to hanging on by my fingernails for years now. A few more months won't make any difference. But I must admit I had rather hoped that the German thing might come to a head. I'm feeling very, very embarrassed at not having yet been able to. Thank you very much for your time, and I look forward to hearing from you in the future. In the meantime, as I say, I'm hanging in there. And I'm also looking forward, of course, enormously to seeing you at the end of this month in Bristol. I certainly hope so. Richard, as I say, impresses me as a very serious person. And I was a little bit taken aback, though, to discover, not nothing to do with him or John, but somebody in the philosophy department had invited... Well, I don't want to be too rude and say a little wretched, but I mean, he has been a pain in the neck as far as I'm concerned on all sorts of fronts, Monsieur Rodin, to give a talk there on the, in fact, on the 30th, the 30th of March, so don't be surprised if he sort of turns up at the first of your, first talk that you give, actually that was impossible to discourage him from doing so, but short of actually locking him in, yes, well, as I say, yes, as I say.

47:30 I do not know, it is a mystery to me, a hundred years, actually, you know, completely, completely, completely took back all of the insights that Dedekind had had and even, was even, they're even worse than Frege's, yes, and of course a lot of, a lot of gnomic pronouncements about, borrowed from people like Schopenhauer and Tolstoy. Yes, I can't understand the, John thinks it's all to do with the fact that he numbered his. It's just like in the original tractators, and for some reason, the philosophers were so impressed by this. Now, I have no good account for author of Wittgenstein. I would even say that again. Well, look at the fact that they still haven't discovered Grassmann after nearly 200 years. Well, 200 years this year, well, there couldn't be more, but certainly 150 years. It's astonishing. The other thing I wanted to ask you, apart from making sure everything is on course, you're actually coming over, I think, on the 29th, aren't you? Is that right? You should probably be there a day or so later. But you're talking, I think, the first day of talking is the 1st of April? I'll certainly be there at least the day before that, or probably two days before. So looking forward enormously to seeing you. How is the Growth&Deep project coming along? British Academy, yeah. Who was that?

50:00 Oh gosh, yes, I'm sorry. Yes, well, it is that kind of a place. It's very much an arm of the establishment, the British Academy. They have very grand headquarters in Burlington House in Piccadilly, you know, next to the Royal Society. It's the kind of humanities version of the Royal Society. Yes, it is. It is to the humanities what the Royal Society is to the sciences. You know, a lot of establishment figures have always been represented on it, and in those days it would certainly have had a lot of idealist philosophers. It has had some good people. I mean, Michael Redhead was, I don't think he was ever actually the president of it, but he was certainly, you know, a member of their council for a good many years. And so, you know, they're not all menace. But how did this connect with the Grundig project? I promise to myself that I make too many promises. No, I was going to say that's the one resolution you never get to keep. Well, don't let it fall off the radar screen. I was just wondering whether there isn't any, because the German mathematical society, as I mentioned to you, had said that they... They were posing to offer this Seacorn grant to myself in the archive, and their intention was that at least part of that should go towards the support of this Groton, Jiggin, Buffalo project, in return for which, of course, they hope that you'd at least put an acknowledgement of their support on the website when it comes out. But if we could get, and if I could just get some funding out of them or one of these other... There are a number of different types of bodies, like the London Mathematical Society or even the British Society for the Philosophy of Science. I mean, provided it was absolutely firm and was there, what would be there to stop?

52:30 Are just paying outright for the whole thing to be digitalized by a recording studio rather than you having to... There are a number of ways to fit in so much additional work when you've got so many other commitments on your plate. They'd probably come around and do it on the spot. I mean, you wouldn't even have to part. If you wanted to make absolutely, obviously to reassure Jack that everything was going to be okay, they could even come around with the equipment and do it on the spot. You've taken on such a huge amount. Well, that's why I was just wondering whether there isn't any way that we could just simply take the burden off you and obviously, you know, while you continue to have total editorial control of it, just get the fact of it, the tedious... Well, we can talk about it while you're over here, but as I say, do be aware, I'm sure that there is not going to be some funding from that. And, I mean, obviously on the understanding that you control the whole thing 100% editorially, but just from the practical side of, I mean, right now I'm actually sitting in the flat of my technical assistant, Harris, and I'm looking at this magnificent machine that he has got here, which is a thing called the Luxman Auto Reverse Triple Digital Cassette Deck, which allows you to record Audio cassettes to transfer them into digital and unfortunately it can only be done in real time which is a nuisance but this thing has got a kind of link I mean you know this thing can be put in a relay with about 16 others so it should actually be possible to do the whole thing I would have said in about three days maybe even two which would at least be a weight off your mind.

55:00 And then just have the whole thing stored in a hard drive for you to keep. I just feel a lot safer knowing that it was preserved in a hard drive and not... It's like my own recording sitting around. I kind of have nightmares about the Templeton Foundation coming around and firebombing the place. Blow against science. Well, they do, somewhere down the line. Well, they probably don't hire them directly themselves. They probably have four or five, which is the usual thing. Just like the chairman, you know, the chairman of... Well, I don't know, the chairman of the United Steel or the chairman of the British Petroleum never actually has the kind of contract killers himself. That's all handled by somebody a long, long way down the line. And what they call opticians, I think, rather like the banner, they're deniable. Pierre Cartier was making very good conversations we had in Oberwolfach. He seems to be fully alerted. He seems now to be very what a menace they are. Yes, that's right, yes, he did. He's very, very fully alerted to just what a menace they are. There was somebody there, I can't remember who it was now, one of the Germans, I think it was, no, was it? One of the German historians, maybe it was Schapke, who was saying, oh, I can't see, yes, exactly. Yes, absolutely, well, he certainly is spreading the word. How are you and Fatima, how is everything at home? I haven't actually seen any of your grandchildren yet. I've seen pictures of them, there's a couple of them. Well, as I said, they'll probably be producing their PhD thesis, but...

57:30 I do see them. It's always great to hear your voice Bill. I look forward to seeing you at the end of the month. We can make more plans about, yeah I will, I will, fingers crossed and I have everything, I've already got my ticket to come to England and the ticket to Bristol, it's all sorted out and somewhere to stay. So I'll probably, yeah as I say with a bit of luck I'll see you at the end of the month. We can talk about the Grotendieck project in more detail then, we'll have plenty of time. Take good care. Give my love to Fatima. Take care. Cheers. Oh, yes, yes. This is extremely well. I expected it would be very interesting. He's always very interested. I'm not sure I absolutely buy the reading, but it's absolutely brilliant to see this material subject. It's such a masterly exposition. The quadrature stuff is very, very interesting. There's something very interesting going on there. I mean, to make it really, you would have to solve the problem about the use, as it were, the higher order use of the infinitesimals in lightness to completely vindicate the position. I think it's just that, I think the truth may be, although one accepts he was a universal genius and arguably the most powerful intellect of all time, certainly pretty well and clearly in contention. Nonetheless, for a man who believes so passionately in the principle of sufficient reason and non-contradiction, he was quite prone, he certainly skirted, he certainly didn't always offer sufficient reason. And he certainly skirted contradiction, to put it mildly, more... Yes, yes, exactly. Sure, sure, we didn't have to... One thing though that I did like particularly is this nice little knock-down argument that he's got against the programme which he believed in in his very early phase of tracing everything back to nominal essences, that one could give a complete account of...

1:00:00 The reality in terms of nominal essences, his reasons, the reasons he later brings against that are amongst other things jolly good knockdown arguments against certainly the ontic version of structural realism, the idea, certainly any version of ontic structural realism of a kind which of course is now depressingly popular amongst some of the philosophers of physics. That rests on set theory, that rests on a kind of extensional foundation, because that essentially seems to me to be open to exactly the same objection that Leibniz is bringing, that it does press to the conclusion, it does do away with, well it's certainly reconcilable with any kind of correspondence theory of truth. Anyway, I think there's a very interesting shadow or parallel of the correct argument against nautic structural realism in there, which is one reason I like that bit of line there. Well, this is fascinating stuff. It's really great. Yeah, sure. Mick, when are you off to the States for this Saturday? Oh, this coming Saturday? Oh, correct. Well, much sooner than I thought. Oh, I thought something that, um... Sorry, something Andre said made me think it. You weren't going until May. Oh, okay. I'll be back there twice before. I'm going back now until the first week of April. Oh, okay. I'll go back the first week of May. Because I wanted to come and talk to you a bit, if possible. I'm going to be going to England at the end of the month because Bill Lorvier is going to be in Harvey Friedman's old... A sparring partner is going to be in Bristol for two weeks with John Mabry and his students for a kind of series of seminars come running workshop on foundations and there's also some going on in Cambridge too over two or three days for the topos theories people there for Martin Hyland and Peter Johnson. So I want to go over to record all of that. So I'll be back around I think about 15, possibly a little bit later. The workshop ends on the 14th of April, but I probably won't be back around until about the 18th or 19th. Are you going to be around then? Great, okay.

1:02:30 Why don't I make a date to come and see you sometime around the week of the 20th of April? Yeah, that's cool. Around time. About a month after St. Patrick's Day, give me a month to sleep it off. You should give me a time to recover. Yeah, yeah, you should give me a time to recover. I'm very surprised to learn that you're going off the week of the... Oh, no, of course, you'll be, you're presumably spending St. Patrick's Day in the States. At a Catholic university. Yes, at an Irish Catholic university, of course, yes, yes, yes. That should be fun. Oh, actually... That should be fun, that should be fun. This is not today. Curiously, St. Pax never used to be such a great event in Ireland itself, as you probably know. It was always, it is now, but it always used to be, until about 20 or 30 years ago, it was always much more celebrated in the diaspora than it was known. I would say in the last 30, 25, 30 years. There's a little song about that, isn't there, about something, something, let's forget St. Patrick and get the snakes back, have the snakes again, I forget how it goes, but, yeah, there's a little song about the man who wakes up, no, it's a song about the guy who wakes up after St. Patrick's Day seeing snakes, and I've forgotten how it goes, but the last line is something, forgot the good St. Patrick and saw the snakes again. So anyway, that should be fun. That should be fun. Objects are points in category of categories, it's at least what Bill says in this kind of thing. What object? How do you mean? What are the categories of the points? No, no, no. If you have category of categories, okay? Sure, of course I know. I know what the axiom is of the category of categories. And then you're trying, okay, you have your terminal object there, this kind of axiom, but then you want to define what is an object in some category, and the answer would be point, you know, it's what Bill says. But if you say that there is not, you know, it's not that it's more pointed, so somehow you have less objects than functors, you know, or you've got a functor which wouldn't be definable in the usual way in something like this.

1:05:00 Well, the functors have got to have domains and codomains. No, they wouldn't have. And there are plenty of cases of examples of domains and codomains that are not defined in terms of their points, I mean we have plenty of examples of that. Categories are a bit different because normally you would say what is fun, something which sends objects to objects. Yeah, and morphisms to morphisms. But in that case if you just define them through category... Or strictly speaking, of course, objects to co-objects. Well, typically, which reverses arrows. I mean, yeah. So it has to preserve, although it can also reverse arrows. But, yeah, I think I see what you're getting at. It's interesting. There is a paper about pointedness in the category of categories. I'm trying to remember where I've seen it. I think it's in GSL somewhere sometime in the night. It wasn't by a builder. It was by somebody. I think it was by a student of... The guy who was Bill's colleague in Buffalo for many years, who died about three or four years ago, the guy, Isbell, John Isbell, not, no, no, it wasn't Isbell himself who wrote the paper, it was a student of Isbell, a PhD student of Isbell's. I'll try and dig out the reference for you. Even you might have category without confidence, this kind of thing. No, that's a very curious notion, of course, you could, I mean, There have always been people who wanted to kind of relax. There's been this program, I mean, Fried and Shedroff's theory of allegories is one aspect of it, of people who wanted to relax the condition for maps to have rigorously defined domains and codomains. They wanted to make a laxer notion of partial, you know, a laxer notion of… I don't think it's ever been made to work satisfactorily.

1:07:30 Because one can rephrase it in terms of other constructions which do behave perfectly well in terms of domains, it's an option, but the conceptual motivation for it has never been, I think, really well made. In fact, curiously, the best example of somebody's, I think, who's actually attempted to clear conceptual motivation for the idea of abandoning the idea that maps must have well-defined domains and codomains is actually Alberto Peruzzi in a very little-known paper. On the construction which he calls pre-categories, which is on his website, which I don't, I'm not sure where it was published, but I've heard that. That, in turn, even his pre-Catholic construction, I remember, in fact, this was a conversation we had while we were here in Paris when he was here in 2005, you know, for our meeting between him and Colin and Jean-Pierre Marquis, and I forget who else, possibly Cartier, about three or four people around the, you remember when we had dinner in Pirishmani? I remember having a conversation with him, with Roberto then, about... This is a rather convincing criticism that in fact what Alberti was trying to do there in the pre-categories paper could be done in a much cleaner, more general way using the homotopy properties, using the homotopies of maps. It's like with Leibniz, I mean, you could say you can do anything without a plasma, but it doesn't mean that... Which came first, graph theory, category theory, homotopy, the homotopy of spaces, or some, and is there some absolutely primitive notion of space, of map space, that has to be present to convert some kind of geometrical structure before you can have algebraic operations at all, which of course is Alberto's claim, because Alberto conceptually is one of these, you know, geometry first people.

1:10:00 I mean, conceptually, that's his position. He really does believe, a little bit like René Thorn, that there is some absolutely primordial notion of space, of spatial structuration, without which one cannot think, which is an interesting claim, which he... Whereas Colleen is much more of a pragmatist. I mean, he just thinks in terms of foundations as whatever point you decide to enter the web of interrelated mathematical concepts, where there is no absolute notion of priority. Conceptual clarity, one way or the other. But that was an interesting debate. I'm very good at that debate. Can we also review both volumes of Hilbert Bernays? Yeah, yeah, yeah. Fantastic review. Absolutely. This again was something Steve mentioned. The reviews of Hilbert Bernays, the reviewer and really... He's actually very generous to Carnap and he, as it were, tries to soften the blow in the end by saying, unfortunately it appears that this is not going to work. And also, which is very interesting, he was a close personal friend of Quine's and they used to go on sailing holidays together. And, although this is only circumstantial evidence, they were 1948, 1949, 1950. Quine was spending a lot of time with Maclean, talking to him, and we know from It is religious that MacLean said to other people that one of the things that he was talking to him a good deal about was the analytic-synthetic distinction, which MacLean thought had certainly not been clarified, and his clarification was going to be the major task of the 20th century's philosophy, and there's certainly some quite good circumstantial evidence that some of the ideas in Two Dogmas were taken from those conversations. You know, Quine was plagiarizing MacLean, but he got a lot of the lines of thought in the discussion with MacLean, which is very interesting. And then later, the other thing which he called attention to was that there was a seminar in, again in, I guess it would have been at Columbia, because Bertrand Russell was part of it, and that must have been in 1940 when he was president of the University of New York, wasn't it, where he got sacked.

1:12:30 I forget who the fourth person was. It wasn't MacLean. MacLean wasn't part of it. There was a seminar on logic and particularly on the analytic-synthetic distinction in, yes, I think it was in Colombia, 1940-41. And we've got very, very detailed notes because there was some Polish guy who was a research student there, not one of Tarski. He did complete a PhD later and went off and did it in a different field. Loosely here being labelled arithmetism. He could have a pragmatic motivation for it and he could also as it were on Sundays have a more conceptual and more even ontologically. Right. Oriented motivation for it. The two things are not necessarily, I mean, there may be some contexts in which they are pulling in opposite directions, or at least in transversal directions, but there'll be other times when they could be very, very naturally operating in support of each other. It wasn't proposed to use for this. Of course not. But I think what you didn't credit was an argument from usefulness. You cannot justify methods of proof. Yes, yes, yes. That's right, yes. Because this is not a justificationist issue. That's precisely the point. I absolutely agree. I think that's right. It's not a justificationist issue. I think that's actually what Klein was getting onto. I think there's far more in common between Klein and this, you know. And this also, of course, plays into some of your concerns about how one should characterize purity of method claims. Thank you very much for your time.

1:15:00 The characterization of quadratic forms and invariance of quadratic forms in this purely geometrical way, which obviously did anticipate a lot of the main line of development of the connections between geometry and number theory. The 20th century seems much more naturally to belong with a kind of Dedekindian perspective on structure to that extent it is there's one thing also I disagree with what Marco said although Cantor is important here it seems to me that Cantor and Dedekind are Okay, they're both concerned with the foundations of analysis, and also they're both concerned with the foundations of analysis in a justificationist way. This is a different agenda from Konica's, but they're pulling in very, it seems to me, in markedly different directions. I mean, Dedekind is concerned with generality of concepts and methods. He's concerned with what is the correct level of generality at which to base the requisite. Concepts from which one should develop the methods of analysis and of course he sees it in the theory of complete ordered fields and it's set theory only really only the incidental sense that you know set theory in this very extended and general sense is just past the intellectual scaffolding of mathematics from that point onwards it's the method of that's what John Mowbray sometimes calls extensional structuralism it's just the method of definition of modern mathematics but Cantor's clearly majored on it Cantor is clearly much more concerned with The nature of these infinities and with the ontological meaning of the totalities with which one's dealing, both are clearly different concerns from Kroniker's, but there are three, it seems to be pointing in three pretty, pretty well opposed, well, pretty, certainly clearly separate directions. Are you going to go off? Yeah, okay. Well, to be honest, I think by the time this finishes, I'm just going to be too exhausted. So what I'm thinking is, the only thing I really wanted to try and do tomorrow before leaving, going back to prepare, is coming and getting the tape deck and getting you to give me the instructions on how to install Audacity, because I won't have time to do anything more to convert these. I won't have time to actually digitalize any more of those tapes while I'm here. But if you can just give me, I'm so sorry, if you can just, is there any way I can make time to see you just for long enough for you to give me the, you know, the kind of instruction manual on installing the Audacity program?

1:17:30 Well, you tell me when's the best time for you. Well, you know, you just tell me, if you want me to come round tonight, I'll just keep going and come round tonight, but I'd rather do it tomorrow if possible. But what's your plan for tomorrow? For tomorrow, I'm mainly free, but I want to have some time for myself. Sure. OK, obviously we can spend a few... Well, it doesn't even have to be a few hours. I mean, how long do you think it'll tell you? If we just focus purely on, you know, how to install Audacity, nothing else, I mean, how long do you think that would take? I'm sorry. Thanks ever so much for the birthday gift. I'm very, very sorry. It's very sweet of you. Sorry. Do you want me to give you a ring later just to fix it up? I will leave my last conference at around 8, I think. You mean this evening? Okay. I'll give you a call this evening about 9 or so. Okay. I'll give you a call this evening about 9 or a little bit before. If I feel good, perhaps you could come for two or three hours. But if you want to just wait. Honestly, I don't think I'm going to be able to keep going until midnight tonight. But I could try. Let me see how I feel at 8.30. I'll give you a call tonight. All right. If we can't do it tonight, well then I'm either going to try and make time to do it tomorrow or else I'm just going to have to do it next time I come back to Paris, which is not going to be until the 17th, that's the problem. That's your speech. You do, well, if you want to stay in the... You might just go online and check the registration, but I think you can probably attend, provided you're not going to take part in the dinner. And you're not staying on the... I can't see any reason why I couldn't just turn up because of the noon and weather. The conference is taking place, it's kind of an open building. No, actually, the talks are actually taking place in the Newton Institute, so that's a project.

1:20:00 You might just check on that. I'll check myself, actually, as to whether you do have... I'd be very surprised... It's the weekend, Saturday and Sunday. I'd be very surprised if you actually had to register, if they made it a requirement. I mean, especially in England, Martin, Hyland and Peter, they're not the sort of people who... Listen, I'm sorry to press you, but I have to... No, because I've got to, the problem is, I haven't had a chance, I need to check my card now in the hole in the wall. I'm not being funny, but Zeki, you know, the lady in the foundation in Sweden, sent the money to me a week ago last Wednesday, that's now a week ago. But when I checked yesterday, it still hadn't been credited to the account, even though she's actually sent me the documentation, and I've now used all the remaining cash that I have to pay the hotel last night. Is there anywhere I can crash at your place tonight? Just for one night, it would only be for one night. No, it's only for one night. Look, I actually wanted to go back to Fougere tonight. The only reason I've got to stay is because Benoit's still got this bloody tape deck, and he still hasn't had time to give me the instruction on how to install this Audacity software. Also, I must speak to Nicolas about this thing that he's doing for me for the, what do you call it, the LaTeX thing for the speakers. And it'll be too late for me to go back to Fougere now anyway. But, okay, just let me put these away. I promise it's just for tonight. I'll need to leave quite early in the morning, whatever happens. Let's put these away. It is my birthday, by the way, in case you were gone, so maybe we could go out for a drink and maybe I'd be able to... That would be nice. Okay. Just let me get my coat and put this away. Okay. I need to sort out some... Thank you. He's got these very interesting ideas about identity, which Jean-Pierre Marquis, in fact, was talking about a little in the Oboe-Voltaic workshop, and, you know, which are sometimes flagged up as marking a kind of turn away from an extensional account of identity towards some kind of grand intentional account.

1:22:30 Obviously, I would be this role that homotopy theory, you know, study of mapping, mapping maps as part. I've never actually understood exactly what, you know, I've listened to Mike McKay. He's very difficult conceptually to pin down what the claims are. He's a colossally bright guy, but not one of the clearest expositors, I've heard, by a long, long way. But there are smart people like Jean-Pierre who do stuff. Often guys like that, they're so far inside the room. Exactly, exactly. He's a classic illustration of that. Dare I say, Bill's another example. Perhaps not as extreme as Mike McKay. But on the other hand, there are lots of people now, like Jean-Pierre Marks and others, in the philosophy and math community, who are beginning to get interested in that stuff. who are certainly very much very very much clear expositors so i think it's an interesting area have a very safe trip to and have a great time i'll see you when you get back but enjoy enjoy yourself in some not today i envy you it's the pope of catholic yeah Oh gosh, I just need to inspect one last thing before... Thank you for your attention. Thank you very much, man. Good for you.

1:25:00 Since you mentioned Steve, I'm so sorry, I didn't mean to jump in. One thing I must ask, but I've still got you here. Has he given you a title for his talk? Oh, he hasn't? Okay. Thank you for your time, and I hope to see you again soon. Yeah, and I'll sit Colin on Steve. Oh, yeah, he's good. Yeah, I saw him doing that. Yes, that's setting a real attack dog. Yes, Colin, when he goes into attack dog mode, tell me about it. Anyway, I can call you Michael. Yeah, don't worry. Stay, stay. Have a good time. So, is there going to be a He-Man seminar? Yes, I heard, but unfortunately it's on the 13th, isn't it? Next Friday, the 13th. Yes, which unfortunately I will not be able to get to, alas, because I... But thank you for letting me know, Joel, it's very kind of you. Especially after what happened last time. It's you and... who are you speaking? Christophe Soulet from the... Ah, yes, yes, I know Christophe Soulet. He's given quite a nice course on mathematics. He will talk next. What a pity, I went, do you know what time? What? Do you know what time? What time? It's 16. Oh, it's 16, so a little later than usual. And in the sous-sol? In the sous-sol. If I can get to Paris for next Friday, I'll certainly come, but I think it's very unlikely I'll be able to, but thanks for letting me know. But that's something else that we're going to start with. Yes, of course, of course. Part of the model and the rest of the model. I guess we're done now, aren't we? There's a list of... Kantor is obviously far more concerned with the nature of the raw material. Obviously, with the justification for, I mean, his is much more a justification as product, and it's to do obviously with the ontological legitimation of the actual infinite, as everybody understands, whereas Dedekind is, I think, far more concerned with, in some ways, with more internal and methodological questions, which is what is conceptually the correct...

1:27:30 The correct level of generality and the correct concepts from which to depart in founding the subject. So he's much more algebraically focused than Cantor ever was. After all, the whole point is he thinks one should start from the recognition of the field, you know, the reals as forming a complete ordered field in the way that... I see that, you know, modern calculus books that teach calculus in, you know, a kind of rigorous, extensional, structuralist way, Spivak obviously being the classic example, as being very squarely in the descent from Dedekind methodology, whereas that's not there in Cantor at all. And Cantor's motivation, I'm sure he's... You can say they both want to found the subject in set theory, yes, but for Cantor and Dedekind, set theory means two, it seems to me, conceptually quite distinct things. I mean, for Dedekind, set theory is basically the raw material of definition. It's what you have to have as the raw ingredients for the method of definition of modern structural mathematics. The cantle is much more focused on, you know, justification. Ultimately, one has to say, it's on metaphysical concerns with the nature of the entities. We were going to have a drink, weren't we? Here, here. OK. Ah, yes, and where's André gone? Because I'm supposed to be going back with him. Is he behind us? He was just behind us a moment ago. Um, who are they? I have no idea whether they went on ahead or whether they... No, they were ahead or... Yes, okay. Well, I need to wait for answers. I don't know why actually, because S is summit and S is sketch, and it's a mix that makes no sense. You're calling two different things P? Yeah, yeah. Wait, if it was about understanding... You're pissing me off! Do I understand the theory of elliptic functions? It's good that we made an effort. You're right, you tried to make me understand, I didn't really... No, no, it's true that it's not very good.

1:30:00 What does it mean? Does it mean that it transforms the initial network into another network? Because it's in the same. It's in the same, of course. Anyway, it will always be a network. It will always be a network, of course. Okay. Because whatever the substitutions you make, you end up on another network. On an elliptic integral where you will draw a network. Yes, it's the same network. So we're looking for... For the isosceles, in modern terms, we are looking for multiplications in the ordinary sense that apply the network in the same way. Exactly. Yes, that's it. Wait, yes, except that it's not multiplication. The matrix of multiplications is B-BA, whereas here you have the coefficient line which is K times A. No, no, wait, no. For A, for the diagonal, you apply them both equally, and for B, you apply minus B and then minus KB. This is to adjust the matrix on the quadratic form. It's because I say but... It's a theory of arithmetic and social ethics. Ah, ok, ok, ok. Yes, yes, yes. Let's move on. I must admit I'm a bit pissed off. I didn't know that quotation from Konica where he gets really nasty with Klein and says basically, you know, you're just a geometry, you don't understand these things, you're not good enough to do the math, you know. I mean, that really is... That's nasty. That's nasty, and I think... I don't even know how justified it was even in the terms in which Kronica was choosing to pose the accusation. I don't think it was justified at all even in those terms. Conceptually it wasn't justified anyway because what Klein did was provide us with an incredibly deep insight. That whole program for the classification of quadratic forms is one of the keys to understanding the whole subsequent development of the relationships between arithmetic and geometry in the 20th century. He spent most of his life working on a subject. There comes up somebody else who says, well, no matter all this. Yes, yes, I mean, I've subsumed the whole thing into a... Yes, of course, it's exactly the way that people who've spent their lives working in invariant theory must have felt about Hilbert. Of course. And there were some very, very smart people who'd worked in invariant theory and who had just kind of seen the whole subject taken away from them. It no longer existed. It'd just been subsumed. Well, it existed, but it's subsumed into... Calling someone a theologian is more polite than...

1:32:30 Yes, they're calling him a con merchant, a faiseur. What would be the literal translation of faiseur? Actually, I don't know, a faiseur is... A poser? No, no, it's a poseur. Someone who takes poses. Yeah, a poseur, a semi-poseur, yes. Here we are. Hi, André, we're here. Hi, Karine. Merci à tous. Merci à vous. Ciao. À bientôt. Ciao. Subtitles by the Amara.org community I'm a bit kicking myself now to have missed the last talk because I went out to do all this email but in fact I thought he was going to give just frankly a rather dry talk about Thank you very much for your time, and I look forward to seeing you again soon. To be honest, I've always thought of the Robinsonian version of non-standard analysis as essentially, well, I don't want to go as far as to say, basically, cheers, cheers, a bag of cheap, cheap, a bag of cheap logician's tricks. I'm just basically using the compactness theorem. But the whole thing is completely parasitic on the existence of the standard model. So, I mean, there's no, in the Robinsonian version, it's totally parasitic on the existence of the standard model. It's not a really, In any sense of conceptually exciting, it doesn't really pose any interesting conceptual questions, as I say, in the Robinsonian version. Now, I think synthetic infinitesimal analysis, the topos of smooth spaces, those do pose really interesting conceptual issues. And there are some versions like the way that Mabry wants to do it, via Nelson arithmetic, We've got a way of making infinitesimals without going via the extensions, the model extensions and the compactness theorem, which are much more interesting.

1:35:00 Really? Where? Do you notice? Okay, have a look when you get back if you can go online. I'd be interested to learn about that. There was one last year, there was one in fact just around this time last year, in March last year. That was a year ago. That was very interesting. It would have been even more interesting my first time in Greece if it hadn't been the bloody general strike and all the streets full of dead rats and piles of stinking rubbish. Oh my gosh, I'm not a very friendly host, I think, but it might look... It's about the structuralist position. It's this harping on the idea that, as it were, structures are just positions in a pattern. The whole point is that there are no interiors, there is no inside of mathematical objects, which is one of the... ...slogans that structure is particularly fond of, and Bill is particularly irked by that claim because it flies completely in the face of his entire program, The Little Foundations, which is all about penetrating inside categories of space in order to reveal what their generating figures are. And even this, I can assure you, just means that he kind of changes his mind continually. Which papers, when you say, I mean when you say his early papers, you mean the kind of preparatory remarks to the category of categories, the things he says, yes he does say things then in the early 60s which do sound a bit glib and which I think he would now distance, well I know he would distance himself from them, he would actually say, he would distance himself quite a long way. The turning point, I think, conceptually for him in understanding his position is probably his dialectical, his 1969 dialectical paper, the one on a joint... Thank you for your attention. And it was only around that time he really began to understand what the full ramifications, the scope of the insight that almost all fundamental structures of mathematics were given in terms of pairs of adjoins, what it was.

1:37:30 No, but I'm saying that that was a very major shift of orientation for Bill, and it did involve a distance from the early... As you put it, it's a slightly structuralist flavor of some of his earliest papers, but apart from the prefatory remarks in the category of categories in one or two other places, I don't think one can really read much of what I call naive structuralism in his position. The whole program for axiomatic characterization of cohesion, this whole notion of cohesion which is so central to him in understanding what it is, why algebraic topology is... I don't say I've understood it completely, but to the extent I do understand it, I recognize that it's profoundly opposed to any kind of structuralism. It's profoundly opposed to any kind of structuralism. Bill really does believe in mathematical substance. He might not put it in those terms, but he really does believe in the substance of mathematical structure. If he's a structuralist of any kind, he's a very, very substantialist kind of structuralist, I think. In New York, I live in Lower East Side, but it has been a very long time since I've been there. Not far from here. Not far from here. Not very far. It's a bit like Bellevue. It's pretty cool. And it's like Cartesian.

1:40:00 Yes, hi, Colin, it's Mike calling. It's Sunday evening over here, mid-afternoon your time. I should have rung your home number. I'm sorry, I made a mistake. I'll call your home number now. If for any reason you're not there, I'll call back and leave a message on your office number. Okay, cheers. Hi, is that Colin? Yeah, I said in my email I'd give you a call this weekend just to confirm everything. I spoke to David Rebois and he's very very keen indeed to put together this workshop and he's got at least four people who would very much like to speak in it. I got a message from Steve which he probably copied to you to say that he's quite happy with talking on the 12th if you talk the following day. And it should work fine. The only concern which Steve had, which was that it might cut across the invitation from Mick de Terversen's I.D. thing, the whole point is that the people in resize and the people in I.D. fellows are in fact the very same people. So it's not really two invitations, it's just one of the same invitation. This would all be quite straightforward if our friend Rodon hadn't managed to muddy the waters, as he usually succeeds in doing, but you probably already guessed that. Fortunately, he's going to be away when you and Steve are here, yes, so he isn't going to be able to stick his oar in and, well, you know, I won't go there, but... I no longer know. In the days when I was involved in the travel business, as I could have told you, it should take a fairly quick Google on airfare site to find out who's paying for your airfare.

1:42:30 Well, if it's mixed people paying, they should, mixed got quite a lot of, I mean, don't, this is unofficial, you don't need me saying this. But Mick's fellows program is fairly flush and could almost certainly, if there's going to be a big supplement on top because of your coming to Paris, then Mick, I'm sure, would cover that, provided you come to, provided, this is the absolute, you know, $64, provided that you're in direct touch with Mick. The problem is that Mick is also not going to be there in May. He's going to be in Notre Dame, I think. In fact, he's over there. He left, in fact, yesterday. I had a chat with him on Thursday when there was this I.D. Well, I had a chat with him on Thursday because I was in Paris anyway, because there was this I.D. Fellows meeting, which had some quite interesting talks on ideal elements. It had three or four good talks, including a very nice talk, actually, about Leibniz, the infinitesimals from David Rubin, who's a very smart guy. But Mick was saying to me that he was going to be over in not just this next three weeks, but also at the time that you're going to be there. So fortunately, he's given me carte blanche to organize everything with David Ribon and this girl, Paola. I'm sorry, I'm afraid I've forgotten her family name because I only met her for the first time a couple of days ago, but she's going to be the person who's managing the shop for the ID Fellows program when Mick is away, and she's pretty level-headed and sensible, and if there's any problems, she's got Mick's mobile and she can give him a ring. If you find out that your airfare is going to be a lot more, get back in touch with me, I'll contact Mick and I'll strike it out with him. The great thing is to keep Rodan out of the loop, because if he gets involved it'll just be a nightmare. But as long as I can deal directly with Mick and Paola there shouldn't be any problem. Okay, good. Okay, I can hear the dogs yelping in the background. Well that's it for now anyway. I'm still trying to struggle with LaTeX.

1:45:00 To put together this, you know, the report for Ober-Wolfach, but I'm getting there, hopefully. No, because I haven't figured, I know this sounds absolutely pathetic, but it's such a nightmare I haven't even figured out how to send it out yet. I'm supposed to get somebody to help me do that tomorrow. Hopefully that will be with you within the next 24, 48 hours. Okay. Right. Take care. It was a great meeting. Lots of love to Pat, Patricia, and I'll see you in May. Okay. See you soon. Okay. Take care. Cheers. It's Mike. Hi. Is this a good moment to call? I don't know. I thought you might be busy with Ben or something. No, I'm okay. I'm okay. I'm good. Is that the thing that, that meeting that John Bell was at as well, about two years ago? Yeah, yeah, which I wish I'd known about, it sounded fascinating. There's going to be a big... Wade Nelson was there too, wasn't he? Oh, he wasn't, it was, oh yes, he was going to be, but he cried off or something, I remember. Yeah, because it was, it was about non-standard analysis, wasn't it, mainly? Oh no, it was more general, let's go on, remind me. Peter Axel I was getting confused with, yes, of course. Yes. And Bob Hale, he's a neo-Fregean, isn't he? Yeah, he's a weird, weird character. I just went at the beginning of this month to something, there was a workshop in Paris for two days, he spoke in, but I have to say I'm very, very underwhelmed by it. I couldn't agree more.

1:47:30 However, so you've got a second shot of it at the time, but you're obviously going to have a chance to... Well, when you've let it fester and it's nicely drawn like a proper well-hung pheasant, you can send it to me as an email, send it to me as an attachment. I'd like to read it before. But I had a couple of reasons for ringing you, but before I come on to those, which are actually to do with... You know, practicalities. And I probably really need to talk to Richard, but unfortunately I've lost his phone number anyway, so I had to ring him in any case. But it's about practicalities to do with Bill's day. But before I come on to that, since you mentioned John Bell in passing, have you heard anything further from him about how mean he is? You haven't? Hmm. I was wondering if I should take the ball by the horns and give him a ring, just to see how things are going. No, I will do that. I'll try doing that in the next day or two. Just, you know, so he feels that he's not being forgotten by his friends, which obviously he hasn't been, but I can imagine, you know, he must be a pretty low ebb. Anyway, okay, on the practical side, first of all, has Bill sent you abstracts of any of the talks he's going to give? Oh, he hasn't yet, okay. Probably needs a little bit of tweaking. I spoke to him about, when was it, about three or four days ago now, just to get his news and actually to talk to him largely about the Cambridge meeting and about André Jaillal, because there's a possibility that we may be putting together something in the way of a debrief, a series of intensive panel discussions with André Jaillal on the same lines that I organized with Cartier.

1:50:00 And Angus McIntyre and Colin and others with Bill two years ago. We'd like very much to do the same thing with Andrew Giles sometime in the next year. He really is flying on all cylinders at the moment. It's quite extraordinary, I mean, how productive he is. But yes, I'm just absolutely bursting with ideas. It's quite phenomenal. And also, in retrospect, just how important his contribution was in the early... I think he's been, partly because he's such an extraordinarily self-effacing guy, not because there's been any kind of conscious conspiracy or anything to deny him credit, but the more I study the history, the more I listen to people like Steve Audi, who really do understand the history in depth, the more I realize that he's been rather unjustly written out of the script, that he had a role to play. I mean almost as as big as Bill and you know quite really more substantial than quite a number of other people like Benabu and Tierney and he's he's really an extraordinary man. I certainly like to get a you know really good intensive week or so of panel discussions with him and if we can do it in the same way that we did with Bill it should be very very productive. I just had a couple of days with him actually almost this time last year in Patras when there was this thing for Anders Cox's 70th birthday which was itself very interesting but of course he was you know he was busy with lots of colleagues at that meeting it was only two days but even just the very little bit of time that I got with him actually with him and Gonzalo Reyes when they were talking More philosophy of math than the technical topos theory. I was just absolutely stunned at what a powerful mind André Joël has, and it's because he... Yes, in fact, he's too nice for his own good. He's so quiet and so self-effacing and so unassertive that I think he, except to the people who are really connoisseurs of this kind of stuff, they don't appreciate just how great his contribution has been.

1:52:30 Anyway, I'll see if I can't squeeze Bill for a... At least a couple of short abstracts of especially this fascinating sounding thing that he's going to be giving about Zermelo and Dedekind. I suspect it's going to have something to do with the stuff that he was talking to me about in Montpellier five or six weeks back, all too briefly, when he was down there for the Grotendieck meeting. About these, you know, the way that he thinks that include that the whole understanding of membership has been got topsy-turvy, the relationship between membership and inclusion has basically been misunderstood ever since Frege and Piano, and how it connects with these ideas of his about extensive and intensive quantity, which are themselves fascinating, but obviously in need of considerable elaboration. Clarification for us poor mortals. And particularly this point that he was stressing about the, you know, in the category of sets, that the union and power set formation as unary operations are adjoint functors. And therefore he thinks there's this kind of lattice of inclusions of which there shouldn't be a globally fixed top element. It's, you know, the ideas are very, very interesting and certainly... One hopes he's going to pursue them and make them a bit clearer. You think it's crap, is music to your ears, because it fits with the Euclidean and Aristotelian notion of arithmosis, being what the subject of arithmetic is about.

1:55:00 Aristotle's amanuensis was having great difficulty in following the material. It didn't just mean the copyists, I meant the guy who was actually sitting there in the Lyceum trying to take down... Because, after all, these are lecture notes, as we... these are... these are... no, exactly. You read... I must go back and reread Bill and Roseborough. Something to do with the way that he sees all of this as fitting into, you know, the view of topos,

1:57:30 after all, of which, you know, well-pointed topos, which are models of set theory, are just. One kind as being sites for homology and cohomology. I suspect all that business about insertion is to do with the machinery that kicks in later when you're doing algebraic topology. I suspect, I don't know, as I say, it's one of the things I hope to find out. We've got three weeks to pin him down on his motivations and that should be a very productive exercise.

2:00:00 Oh, yes, I guess, because he's going to be away for the middle in Cambridge, isn't he? Well, still, two weeks is a long time to have Bill to yourself, so it should be a very useful exercise. Yes, that's absolutely clear. And not only that, but they have a different idea from each other. I don't think that Bill's idea was the same as Saunders's at all. One thing Bill is very insistent on is he absolutely repudiates any talk about category theory being a variant of or a clarification of so-called structuralism. I mean, he regards that as a complete, you know, as a completely wrong idea, and particularly the slogan of the structure is that... Well, no, not structuralism in the sense of, you know, the working method of definition of mathematics. I mean structuralism as propounded by, you know, the people in philosophy departments who call themselves structuralists. People who say that mathematical structures, mathematical objects don't have any insides, as it were. You're only dealing with the external relations. It's a complete wreck. No, but Bill's particularly insistent, not only they not know what they're talking about, but that what they're saying about, you know, the ones only dealing with, as it were, with external relations is the exact opposite of the truth. The whole point is that category theory is precisely the mathematical instrument that allows one to see, as it were, inside the structures to compare. That's what all this business about cohesion and variation are to do with. I mean, far from having, as it were, no internal structure, these things have far richer and more layers of internal structure than categories do. Yes, as categories. It's an issue, I know, because...

2:02:30 Considering that all the issues were already there at the beginning, well, maybe not all of them, but some of the deepest issues were already there at the beginning, even in the general theory of natural equivalences. I mean, one of the things which came out of these discussions in Oberwolfach, which I must send you the CD of, of course, now I've finished burning it. There was one very interesting discussion precisely on abstraction vis-à-vis generalization, vis-à-vis simplification. What work were category theoretic inverted commas foundations doing in those respective dimensions? And how should one think about the whole notion of abstraction? That was a very good discussion.