Discussions

0:00 It's really a pity that John couldn't make it. It is, because the trouble is he's so committed he's looking after his brother. May I make a suggestion? Do you want to bring the changes? I mean, because we've eaten in the Highbury Vaults two days running now. Do you want to go back to the place where we were before, to this little pizza place or one of these other cafes along here? I don't mind. You don't mind? Well, shall we try one of these just so as to have something, you know, eating in the same place every day? Well, here, as in virtue, you can sit outdoors. True, true, true, true. I'm all in favour if you feel like it. No problem. I like sitting outdoors. I just thought you might be getting a bit bored with... No, sure. I just thought you might be a bit bored with going to the same place each time. No, that's fine. OK. Oh, Christ. Careful. Yes, I like pub food. This would have been my choice, but I just didn't want to force it on you if you... I think you will have some very interesting discussions with John. In fact, you could probably still do it while you're here. I was thinking of going around and seeing him this evening, actually. Well, it's a pity you haven't had a chance to talk more with him, because he is very, he's very unusual in a situation like that.

2:30 Oh, it's your, it's his copy, is it? Oh, yes, you'd better give that. I think that might be advisable, actually, without laboring the issue. I think that might not be a bad idea to make sure he gets that back. He's a bit pernickety about things like that, to put these damn things away when they're not working out. You're going to have an awful lot to do this afternoon. There are also many other fields of study, such as mathematics, geometry, algebra, mathematics, physics, physics, physics, physics, physics, physics, physics, physics, physics, physics, physics, physics, physics, physics, physics, physics, physics, physics, physics, physics, physics, physics, physics, physics, physics, physics, physics, physics, physics, physics, physics, physics, physics, physics, I was particularly interested in what he was saying about the various pedagogical strategies that he was proposing to use in Volume 2 of Conceptual Mathematics. There is going to be a Volume 2, and he wants to introduce the adjoint function. He proposes to introduce guarantee topologies, but we were discussing what could be the simplest possible motivating... This is an excellent example of a great leader of politics. In fact, he decided it was something which could be put across remarkably simply in terms of... My colleague, I told Bill, he kind of agreed in the sense that he agreed there's a possibility... No, no, no, it should be simple. He's a poet, you know, from such a long time. It's kind of so poetic in the sense that... Well, actually, that's not strictly true, because he actually starts from the category of directed graphs. Reflexive directed graphs, because those are the ones which allow you to explain about gluing figures and functor categories in a very accessible way.

5:00 Well, oh yes, just in the very early sessions. Well, it is an immensely central... I don't know what that would mean. So, can I just listen to what these guys are saying just for a moment? They're particularly interested in this particular discussion that they were having. Oh, he knows where we are, does he? Okay, just tell him that we're in the vaults. Just say we're in the vaults. He'll know where that is. The vaults? The vaults. V-A-U-L-T-S. It's the Highbury vaults, but he knows it's the pub. It's the university pub. Everybody in Bristol knows. The vaults. V-A-U-L-T-S. V for victory. No, V for virgin. V for Virgin, V for Victory, V for Vladyshev. A, U, L, T, S. Just say we are in the box and he'll know where we are. Sorry Anders, carry on, this is all fascinating. I'm just arranging all this stuff because I've got I've got a copy all this this afternoon and also well these are actually recordings that were made in September last year at a Congress in Mainz in Germany

7:30 ...called Beyond Einstein, which was a big meeting on developments in foundations of physics, particularly in gravity, with, in fact, our friend Luke Craig, who we were just talking about, but also Roger Penrose, Abiy Ashdekar, and various other people, and the German Mathematical Society made me promise to produce a CD of the entire conference, which I did, except that... The last six recordings I had to make on tape instead of on digital recorder because I'd run out of space, which is what I've just done now, in fact. Yeah. And for that reason... Yours matches, aren't you? And for that reason I need to copy those digitally and then burn a new CD of the whole thing. Thank you very much. I should have done it a long time ago, and I have to get it done today, so anyway, that's the end of it. But please don't let me stop you talking. It's extremely interesting. I want to understand what the, uh, specifically... It was, um, atomized. No, but it was very interesting. I wanted to understand a little bit more about the, the minimal... What was, what exactly was the topic? The notion of commutative language is weaker than vocal, but has the property that, geometrically, these vectors are connected. Interesting ideas, huh? It's not my idea. I was going to say, does this have anything to do with extensivity? I mean, what's the motivation? What's it good for?

10:00 If you have coproducts, then you have coproducts. It preserves coproducts. Pullback preserves coproducts, yeah. Extensive. Extensive is the term. It's nothing to do with extensionality. It is related with distribution. The product exists and it is distributed. So it's not called extension. It's extensive. It's got nothing to do with extensionality. That's what I'm saying. That's a quite different notion. Just slightly... No, it's extensive, and as you say, if products exist, then it's distributive. So it's a purely algebraic notion. It's nothing to do with the way you express extensionality or choice in a topos. And that's an interesting subject itself, but no connection with this one. But what's the motivation? Is there anything, particularly in algebraic geometry, that it's good for, in particular? I'm not sure. I don't think it's the best model to do STG. But in general terms, if you have an extensive category of spaces that you want to embed, preserving the products, putting this topology is the first way of doing it. Right. If you don't know the properties, then you can force it over that. But that's the first step into getting good co-limits, good co-limit preservation in everything. Right, yes. So this is the general part. If you apply this to the case of finitely presented rings, you get this. I'm a classifier for that. My motivation is the following. I'm quite convinced of this radically extensive approach to models and geometry. But I'm not quite sure we... He has come up with the right axiom. He has this way of building objects. The object is radically embedded, really, by extracting them.

12:30 The next axiom he mentions is for this object to be connected in a very specific sense. Yes, yes. For pi o to be one. Exactly. The first model of SDGs, far from the best, the thing you start with is the ring classic number. If you check the axioms there, it goes perfect until the calculation of the object of real numbers. If you try to check that it is connected, it is not. For general reasons, you can force it to be connected because you have the best topology that inverts. Well, you're having to introduce the best topologies and additional structure. You'd like an intrinsic characterization. If I had a good description of topology, I wouldn't be very happy, but I'm happy. If you insert zero, pi zero apart, that's a math. You want to invert them, and that's the best way to do it. I was expecting it to be this joint coverage of all of this, but I have no good reasons at all. I'm not sure if it is true. So there are either two possibilities. I'm too stupid to realize what the right apology is, or the connection is... Bill is proposing us an axiom. It's too strong. No, I see the horns of the november, it's very interesting, very very interesting. Of course, I still don't, I mean I have evidence that Bill is strong. More or less. I'm still calculating with various people. When you talk about the specific model... There's no particular reason why you should say radical is synthetic. It's just a matter of if these are connected or not. But connected, of course, with respect to a specific piecedal, meaning a piecedal that goes out of D.

15:00 But that might not necessarily be pi, that might not necessarily be one. No, that's not what Mathias says. But anyway, that's a completely general question about a standard topology. In fact, a generic ring, a generic commutative ring, because it's more generic. Right. In other words, it's a more basic, it's not just pertinent to characterizing... Now, the reals in SDG, it's a more general, in top theoretics, it's a more general universal property. Pi zero is not anything unique. Sure. Well, I get it, but in this case, pi zero has to be defined in terms of, discrete being defined in terms of d. And the n-imap from d is constant. So R-connected is, that is related to the question of whether primitives are unique. Is that right? Because purely in the ring classifier, primitives are unique. Primitives, of course, from R to R are unique. That's clear because they're not from R to R. It's just a polynomial in one variable. In the case of primitives, they're not from R to R. Well I've certainly got a much better grasp of what the issues are now. Thanks. Actually, I'm broadening this out to something a little bit, a slightly more general question.

17:30 One of the things I'd really like to try and do with Bill while we're here in this workshop, and I don't know if you agree, but when we get back from Cambridge... Because we never did actually have a kind of planning meet about what were the topics we should try and list, you know, try and cover in the time, and it's probably good that we don't have too rigid, but all the same I think it would be an idea just to have people kind of list the topics that they would most like to have discussed, you know, given the time available. One of the things I'd very, very much like to hear more from him about is his take on this... Extraordinary far-reaching program which he tells me that Grotendieck laid out in 1973 when he gave this talk at the Buffalo Math Colloquium, which he didn't himself term but which Bill has labeled his program, his Grotendieck program, for bypassing logic, which essentially was to to work out what the relationship between the classifying rings of all these algebraic theories. He drew this diagram, apparently, actually on a table, on the tablecloth when they were having dinner in a Chinese restaurant. It was a big table, it was like 12 place sections, and he covered the entire tablecloth with this diagram, which was in the form of a kind of circle, but with spokes running through it like a wheel. And the idea, and please recall this is a non-mathematician speaking to you, so I'm afraid I almost certainly get this. If I'm wrong it will be very confused but I hope this gives a glimmering. The idea was that this picture expressed the relationships via their respective classifying rings of virtually every structure in mathematics it expressed as an algebra and the logical, the things which as it were capture logic which are the things essentially which can be encoded in the sub-object classifier, the topos. Just fell into place in the center of this diagram is just one place which is where well where where the classifying rings were actually what they classified was were sub-object classifiers but this of course was just a reflection of the fact that the subject matter of what Bill likes to call narrow sense logic likes to call narrow sense logic or subjective just is you know the theories of properties and relations.

20:00 quantifies and things which can be expressed in terms of those concepts. But algebraically, this is only a very small fragment of what one can say in terms of the mappings between different algebraic structures. So that was the sense in which he saw it as Grotendieck's vision of showing why we don't really need logic as a separate discipline at all, but everything can be expressed. Everything we say in terms of logical notions can be expressed as, as it were, a fragment of this bigger geometric and algebraic picture. And that to me seemed an absolutely fascinating vision, and one which I would really like to have outlined and articulated more fully, and Bill certainly has had very interesting things to say about it from time to time over the years. It's one of the topics I hope we might bring up for exposition and discussion. To just pull the headline, bypassing logic, is one way of formulating it, but another one, which I think Bill would prefer, is broadening the scope of logic. That would be a much better way of saying it, I completely agree. What do you mean by that? Well, actually formulating the theory of what Bill has many times called objective logic. Narrow sense logic deals with the roots and supports of intensive quantities, so this naturally falls into place within this schematization of the relationship between intensive and extensive, and the ways that the different types of quantity vary with respect to. There's no different kinds of space. A slightly more key answer is that logic does not just deal with propositions. In other words, it's not that. That is very important. But, you know, value should speak for all the things you can, not just the elements and the lattice. No, the lattice theoretic part of it, of course. I must say, in relation to this, that since Bill seems to be in the mood of discussing technical details, I'd rather press him into that.

22:30 I know, I have to say, I'm afraid we're completely... Well, isn't there room for both? We're going to have another week. Isn't there time for both? I mean, I hate to say, but this is where, because you are cutting off most of your audience. If it's going to just turn into a research seminar on the specific machinery that's needed to solve those problems in SDG, that's great, that's a very worthy enterprise, but it does mean that you're basically cutting the audience down to two people. And the others, I mean, people like myself and John. Andrew, it is going to be completely unoriented. I mean, don't get me wrong, but I think there is room for pursuing both. After all, those are the issues which you get to pursue in international category theory meetings and in, you know... Well, you see, that's the point. We don't pursue these kinds of things in international category theory meetings. Because we don't discuss them. Okay, well, I understand the point. But if the details get too technical and, you know, the broad picture is lost sight of, it does... It does cut out most of the people who are there, and I personally would like to try and get a broader conceptual understanding of issues like this, of how he actually sees logic as falling into place. Of course, of course, which is why I don't see this as a straightforward opposition. I think there's room to accommodate both. I think the discussions were challenging to participate in, and I don't think we have a good balance between... ...technical and very visionary, and gossip. Yeah, well, let's try and keep the gossip to a minimum. I'm all in favor of that. I mean, that's probably been my fault, is that there's been a lot of gossip, so I plead guilty to that. I'm quite prepared to sort of volunteer to sort of... I'm all in favor of cutting out the gossip, but I would like to have a little bit more of the broad vision, if there is time for it. What do you mean, gossip? Oh, well, just, you know, anecdotage and, you know, who said what to who. And then going on to, well, more gossipy things. Yes. Well, those are things which we can do, as it were. I think, in fairness, I don't think we've done that much in the actual sessions of the discussion.

25:00 We may have done it in the evenings over dinner, but that's what pubs are for. That's like having a break. Yes, yes, sure. Everybody needs a break from time to time. You shouldn't go on lecturing for two hours without a break. No, no, I quite take your point. As long as it doesn't run out of control and get away. The kind of things you talk about, Cermelo, Cantor, and so on, they're very important, I mean, both for philosophers and for future mathematicians. People have these kind of long-term projects. So that's what I mean by, you know, having all this kind of discussion we had this morning before Davide began his exposition between Bill and John, in which he was explaining exactly what this whole business of making the notion of small entirely internal to, in terms of universal constructions, just how to make this notion of small quite, something quite internal to the category of theoretic. That for me was immensely illuminating. I mean, that's the kind of stuff I'd like to have more of that. But he doesn't, I'm sorry, I disagree, I'm sorry, I flatly disagree. He doesn't write those things down. He's had 40 years in which to write those things down, and what has come out has just been a series of extremely illuminating, powerful, brilliant, ober to dicta, but things from which the philosophers and historians of mathematics have not yet begun to learn, and they won't begin to learn until these things are explained more explicitly. So I'm afraid I flatly disagree with this. Oh, he can write that down any time. What I want to do is... You know, okay, I mean, you know, is technical calculations. Don't get me wrong, I think there's place for both. But I am going to stick and fight this corner very hard because as far as I'm concerned, you know, that's why, not just myself, but I think probably the majority of people, you know, involved actually came. Sorry, I hope that doesn't make us, you know... Are you trying to press my point of what I came here to get? No, of course, it's really exciting. What you've just said now, explaining what the motivation for this particular point in the structure is, to me, is absolutely fascinating. And that, to me, falls rather nicely in, dare I say, what you've just been saying now to Anders, to me, sort of bridges the technical and the conceptual.

27:30 ...sides of the motivation issues rather nicely. I haven't got any problem with that at all, but I really would feel a bit disappointed if all we had achieved at the end of the ten days, two weeks, however long it's going to be altogether, was an intensive session on... Specific research topics in topos theory. I really would like to have at least some time on these broader issues like the Grotendieck programme. Do you understand anything? Yeah, and remember, John's a mathematician, not a philosopher. I'll try to answer this first. Go ahead, go ahead, go ahead, go ahead, fair enough, parallel. That's an aggressive answer. No, no, I absolutely don't, I absolutely don't. How many times do we actually meet you in a year? How many times do I meet him in a year? If I'm very lucky, maybe two or three times. On average, I would think probably two times. I don't, usually. I meet him at the conferences. Sometimes I can sneak in and talk to him. He's very kind with replying my emails. It is this kind of situation where you can suddenly track and this in the mood of calculating details. For example, we did it the first time. Yeah, that was fascinating. That was absolutely fascinating. But then I understood the conceptual notion. Well, it wasn't stupid calculations. Calculations are extremely interesting. The idea that there are ratios of infinitesimals, exactly what that means, imprecise, yeah, but it's much more detailed technically, and then this led to specific calculations, which of course... I proposed a specific axiom of why r should be calculated, and Bill had an axiom, and I proposed a different one that works in the case of the ring classifier.

30:00 My calculations were a bit silly, but once he digested them and produced them backwards, suddenly they made some sense. Yeah, sure. But again, I would say, with great respect to you both, the part of that session which was actually taken up specifically with the calculation was a relatively small part. It was only perhaps... At the very most, I would say, you know, 20%, but probably less than that. No, for me, that was enough. That was enough. Because the general exposition of his ideas about the infinitesimals, and particularly about how to clarify this claim of Euler's about the infinitesimals being raised, sorry, about the Euler reals being raised here as infinitesimals, and how he sees this connecting with his ideas about the Leibniz in Mona, obviously. And dare I say, to us, you know, the philosophers, I mean, that obviously was the most exciting and interesting part. Of course, the calculation is very interesting, and I wouldn't have a problem at all if the calculation had gone on a bit longer. But I would have had a very serious issue if the calculation had taken up the whole of the session. And I think everybody else would have, you know, with respect. So, and in fact, at the end of the day, the calculation, in fact, didn't actually work out there on the board at the time. It actually, you know, you both went away and thought about it, in fact, when you came back the next day. You got it straightened out, which was great. Well, no, I think you had a lot to do with that, so don't belittle your own contribution. I mean, it's absolutely vital. But I really think... You know, there are five or six people involved, and I'm all in favour of having time for the calculation, but if you get something really going with him strongly, he's going to go off with you on your own to do the calculation anyway, so you're not going to lose out in that sense. I hope this isn't going to introduce a tension or a clash, but I repeat, I would like to see a bit more, I would like to see at least one or two days where we could talk about issues like what we were just saying now about this visionary program for the role of classifying rings. Because you say he can always write that down, but he first started thinking about those ideas in 1973. We're now in 2009. And nothing about that program, in terms of really setting it out in full generality, articulating it properly in a way which would make it possible to convey it to mathematically literate philosophers, has been done, apart from what Grote and Dieck wrote on the tablecloth, which has been lying in Jack Duskin's room for 36 years.

32:30 It exists, yes. It has even been recovered. No, in fact, there is a plan to publish it on the website of the Grotendieck Circle. I don't know how we reproduce it. There is a plan to publish it on the website of the Grotendieck Circle. But, you know, this stuff has lying dormant for 36 years. Grotendieck is 81, Bill is 72, 73. I mean, you know, I think we are entitled to try and get a little bit, to push a little bit, to try and bring this to the fullest possible articulation while we're here. Who was the guy with whom Biddle had a long conversation here at yesterday's session with Irina? It was mainly with Irina, actually, the very pretty young lady. Here? Out on the sidewalk, here. While we were waiting to go to the... Oh, oh, he's a member of the maths department. I'm afraid I don't know his name. He's an American. He's an American. He's... Yes, that's right. He came to your talk at Oxford. I'm afraid I don't know his name. He's an algebrist, but I don't really know anything else about him or what he works on. He keeps saying, he prefaces every remark with saying, I don't know anything about category theory, but, and then he says things which make it pretty evident that he knows a fair bit because then nobody does algebraic geometry without knowing, even if they don't realize that they're using it. I mean, you know, you get this all the time, people who say, I never use category theory, but, and then they produce something which is entirely cast in terms of scheme theory and so on. This has almost become the official ideology now to say, oh, we don't. We don't touch category theory, but, and then we produce something which is totally dependent on the tools of category theory. But no, I don't know anything about him. That's the guy you were talking about. About this technical, when I was a really boy, I went to Gilfant seminar. It was very crazy, because it was people very different, very young people like myself, very prominent. It all started like an hour later, you know, a piece of time. And then there would be some, you know, young person would write something. And then it was Gilfant and Fuchs, you know, and Gilfant. And the guy would make the session very, very quickly, like, you know, 10 minutes.

35:00 Mikhail, did you understand anything? I didn't. And then there was this funny, it's actually in the math department in Moscow, this funny culture of kind of giving no respect to tea, you know, it's a kind of right or profane thing, you know, they say, okay, there's a... There's a guy who doesn't, can't understand, but the kind of performance is not very serious, that's for anybody, it would be Gelfand's seminar, but all the stuff is, so the guy would repeat, and then he would say, because I think there is something, I think it's interesting. I think that's it. Yeah, I can understand that. It was a very funny atmosphere. Gilfan was a good teacher. Yes, yes, yes. It sounds like it. In the same way that Billy is. It's a bit like Peter Johnson, because he particularly somehow... I was actually in school then when I was there. We were encouraged to come. Thank you, that was very nice. He would explain something. Even if they were at the same time very advanced technical talk, very strange. Are you a mathematician or a philosopher? I switched to philosophy without never being really a research mathematician. I'm a Ph.D. in history and mathematics. But I have some background in philosophy. So you went to some of these seminars in St. Petersburg. Andrei is very definitely a Muscovite. You're really trolling it there, I'm afraid. That's like telling a Scotsman that he's an Englishman or vice versa, you know. No, St. Petersburg, Moscow, you understand that's, you know, definitely one of them. But that's true that it's a bit different even in character, because in Moscow we had also this kind of philosophical environment that people like myself as well.

37:30 Well, look at Manin! I mean, you know... Well, that's for somebody who's a very philosophically-minded mathematician, spends a great deal of time in Moscow school. I mean, I think Manin is probably, well, of course, you may say, well, that just shows that he's senile or whatever, but he seems much keener these days to kind of talk to philosophers. Well, that's not strictly true. He gave an absolutely brilliant talk at the thing they had at IHES in January for the Grodenbeek. But he's coming to Sweden this summer to give a series of talks just basically on the architecture of mathematics, but it's not just hand-waving Omnium Gatherer, he actually has very interesting things to say. Coming back to the previous issue. Well, I thought I was going to let that go so that we didn't end up, you know, but I promise I won't fight with you. But couldn't we sort of reach a compromise as to how we divide it up? I mean, I'm not saying we should plan things in exhaustive detail, it's a very bad idea, but we should at least have some modus revendi as to how we're going to fit in, you know. It's both sets of topics. But I must say, so far, there's been more general talk about concrete. Oh, I don't think so. I would have said just the opposite at the moment. Because we had some concrete talk on Monday morning. We had some concrete talk today. The rest of the time was really more general, I would say profound things. Well, it was exactly general and profound. So, we had two mornings, or one morning and one afternoon of concrete talk for the rest of the time. Well, we've only had four days altogether so far. That gives you more than 50% of the game. No, you said two mornings and one afternoon. No, no, no. One morning, sorry, one afternoon and today's morning. Oh, sorry, I misheard you. I thought you said today. Well, I think it was a bit more than that, actually, but we won't say. Anyway, we both agreed that these ideas should be better understood. My main... You obviously want to get it. I completely understand the motivation. There's something I haven't said. I think that both of us would be... Now, regardless of how they said it, in more generality, I think...

40:00 The main ideas about geometry and predictions should be better understood by everybody. I think an important way of doing it is to figure it out in full detail, the simplest examples of that, so that we don't discuss about the general ideas which are obviously irrelevant. I agree with you, but people take a lot of persuading of that. I agree with you, I agree with you, but... Maybe a way because the S, which for most of us always have been discrete abstract sets, is therefore not very interesting. Sometimes you understand things better when you vary the parameters, so the fact that you can vary is more or less discreet, and it's the pictures that we do. You made a very clear motivation for what could that mean from the purely particle viewpoint on the mass of air into a body of air that is governed by differential equations into a particle. I mean, that's what I consider today to be a technical topic. But you see, that's curious, because I consider today to be just about the most conceptual thing we've had. So maybe we're not so opposed. Maybe we can actually reach a good agreement. Because today, I mean, well, obviously the thing you were doing at the end about the 30-dimensional space and working out, that was technical. That was interesting, but that was technical. Apart from that, I saw today as being... Is it we're rather more broadly conceptual than technical so but that's good because that means that we it should be pretty easy for us to reach that means that it should be easy for us to reach a convergence on how we want to use the... I disagree with the notion of convergence. It has to be rather loose because I mean I did not, I was not very interested in the discussion between Mayhill and... And Bill Lang was eager to get the meter going. Okay. Well, it was a relatively short discussion. Well, yeah, but it's an hour discussion. No, it wasn't. No, I was recording it and I looked and I promise you, this always happens when you're not interested in something or when you are interested.

42:30 Time always, time as it were seems to be telescoped or expanded. No, it was actually about 25 minutes. It was about 25 minutes. I could even prove it to you by showing, playing you the recording. It was 25 minutes, and out of two and a half hours, which is... Actually, when John asked this question, it was always from the Galileo dialogue. We started the morning and said, well, yesterday... But I have to say, okay, I know you weren't interested in that, it's probably because you already have a completely clear understanding of how Bill thinks about the notion of small, smallness, but you see, but in fairness, I mean, for people who are concerned with foundations, this is not, this is, this is a, this is both a crucial technical issue and, you know, conceptually the very heart of the problem for logicians. Which ideas? Which ideas? He has so many, he's being serious. No, no, the ones he discussed in the first morning. The ones he... The linear and real numbers, which are connected. We aren't homologous. I don't. So... Well, except the... So you're trying to communicate these ideas and make them public without having an actual goal. I mean, that's... I'm pretty sure the ideas are essentially right. No, we could carry on discussing, and you could go quite far with the axioms, because he does, in his paper, right, but, you know, there should be more. Yes, I quite agree, it's important to discuss the Eulerian reels at that, you know, technical level of detail, of course it is. But we could discuss the Eulerian reels in generality and its consequence and its philosophical significance for hours and hours and hours, but... I don't think we need to discuss it for hours and hours and hours, we just need to discuss it long enough to get the issues clear. As he did, in fact, very successfully with it, because he cleaned, as far as I can see, he cleaned up this whole issue of how one internalizes the notion of smallness to the satisfaction of a perfectly bright mathematician, but who had not, who, and John Mabry is a bright person, although he's not a category theorist, and he hadn't understood it from listening to Bill's talk yesterday. He did understand it by the time. I thought that was a useful exercise, and I think spending 25 minutes on that was no... I was looking for an explanation. Why is that a notion that will, so to speak, do these things that have the formation of functor categories?

45:00 Yes, exactly. What John calls free mobility, I call it just being able to form functor categories when you need them. I realized this, this is really just a generalization of the notion of finite you have by saying I said it's finite if there's no non-spatial ultrafilters, if all ultrafilters are principal. If all ultrafilters are principal, yeah. Because these N to the S... In this sense, it's saying an object is small if there's no non-principal, well, better formulated positively, if all the ultrafilters are principal, now ultrafilters in this more general sense. That's a good and clean... And everybody can, in principle, understand that notion of small. Generalizing the notion of finite, saying a set is finite, or let's call a set an object, finite if it has no principle, if all principle, all principle is object. Sorry. No, I got what you said. I completely agree. But what stability property does this notion have? I mean, why does that solve or approximate a solution to a need? So the problem of, say, the one that Grotendieck analyzed in terms of universes, I mean, it would be nice to see that in technical detail. Yes, but what you've just said absolutely, I think, makes my point. That's exactly the kind of clarification. That's exactly the combination of conceptual insight, you know, full articulation of the conceptual insights and the motivation to combine with a clear technical definition, clear usable technical definition that we want to bring together. And you, you know, and I think you've just done it. I think it's what Bill and John were doing this morning. I have to say, I'm sorry, I don't agree that that was wasted, and I'm sorry you feel it was.

47:30 There are other things that you wanted to get on. Yes, because you wanted to get the synthetic continuum mechanics, obviously, is your baby, because of SDG. And I'm very interested. I think SDG is an absolutely incredible program. I would like to hear far more. It's all eager to get started. Yeah, that's fair enough. Well, I'm eager to hear him continue, sure, okay. Frankly, I told myself it's good it didn't happen with myself. Well, actually, I thought the discussion that came after your talk was an excellent example of where, you know, he was talking about it. ...broad conceptual issues, but in a way which was extremely illuminating for everybody and didn't go on too long. I think you would agree that didn't go on too long. No, I'm sorry, I'm sorry, I don't mean to, don't take that the wrong way, please, please. Brother, like a clean mic, I'm going to get a dessert. What a good idea, okay. Well, unfortunately, I'd love one, but I'm completely skint of them. Well, not skint actually, I think I can probably... Get some money out on that card. Oh, I can... I can... If you want a dessert, I can loan you something. Well, that's very kind of you. I can actually give you euros. That's no problem. Okay, could you get a dessert for us, Mattes? I'll have the same as I had yesterday, the treacle, treacle, oh gosh yes, more than enough, oh gosh yes. The treacle tart with an ice cream would be very nice. Treacle tart, well they have it with either custard or ice cream, I had the custard yesterday, except I do insist on paying you back. I do insist on paying you back. I'll give you five euros. Okay, in fact, you're the winner there because the euro is so, the pound's gone down so far about the euro, it's almost a parner, so I feel more comfortable about that. I'm going to stay in my store for half a year or... Well, if you give it back to me and I get out some British money and repay you, do you want me to do that? I do store, I always have a store of euros because of that. Yeah, after all, you never know when... No, I hope it's not that... You're going to be invaded again. It's very, very sad. I think even Anders Ragnarsson is that name. He's completely selfish with the countdown, eh? Well, still, Mike, it's... Oh, still, he's very bright, and he provokes... No, no, no, I mean, it's rather amazing that it still works, this kind of mixed crowd.

50:00 If he was left to himself, he would kick us out of the room and he would just have a look into himself completely for the whole of the two weeks and do nothing except discuss the issues of what he wants to straighten out for his next paper, which is not what we're here for. I mean, come on, you and I agree about that. Yeah, sure, and I do want the big issues discussed, I do want the big issues explored, because it's taken long enough to get built here, you know, and there are so many topics, there are so many topics going back a long way, which intends an extensive quantity, debates of variation, how much to think about this whole issue of quantity and space and tradition, vis-a-vis this. I mean, actually, he's already said a good deal about that, and implicitly, and... ...explicitly with what he says, but I want to understand much more about this point about the, you know, why he rejects the idea of the fixed top element in the lattice of inclusions. Why you thought the piano, Frege piano kind of instruction was so wrong. Well, of course it's because there is no universal domain. I understand all that, but I'd like to see the details about how it connects with this whole, you know, theorization of quantity. There was one very interesting thing that Davidi said. Actually, this is the one point where I, the one point where I kind of thought I was going to reach some kind of modus vivendi with it. Because I did actually think that Edith's talk this morning, except when they got off this boring calculation, which they could always have fucked off and done on their own, you know, rather than having to take off, except when they got off this bloody boring, long-winded calculation of the, you know, potence at the end, that was actually conceptually very interesting, because the motivation was so clearly put across, and that business about dimensional analysis and how one should think, that was a... I would really like to have had a longer discussion on that because that's extremely interesting. I think it would be good to have him at our seminar. Oh yes, he's a very interesting guy. It would be very good to have you too, but it would be a big problem that you have. With this guy we might easily kind of... Put him together with Louis Cray Yes, yes, very different, obviously very different use of topos theory, but he's a very, very interesting guy and I mean, God, it must be hard for him in an engineering department with people who basically, you know, interested, you know, their main interest is stress concrete, you know, and, you know, calculating the kind of stresses of when they put up football stadiums for somebody who has these very deep conceptual concerns.

52:30 Curiously, I bet you anything you'd like, that if he was here, he'd say, no, I want to discuss the conceptual issues, because that's what I come to talk to Lord Weir about. If I want to talk about the technicalities of calculating the stress tensor for a fucking, you know, area of concrete, I can do that with my colleagues, you know, 24-7, any time. I don't think he'd agree with me, but I'm not afraid of that at all. Anyway, I mustn't go on about it, but I do not want to see him lying. He will, if he's given half a chance, he will. We'll never get around to the really conceptually interesting part of that lecture. No, no, he wasn't aggressive. No, no. I can handle him. I can handle him. And I'm not going to bite his head off. I think I restrained myself very well. I think he probably did as well. But all the same, he knows that I'm pissed off with the way that he is telling him and pushing the direction more and more. John Bell had the same problem in Fougere. Well, I have to say, I think rather unfairly, but he was, he got so pissed off, he almost walked out there after the third or fourth day, because he said, this is just nothing that I'll, you know, I'm just getting, I'm just sitting here and hearing nothing except Cartier and LaRue discussing these, you know, kind of, to me, super technical issues in algebraic geometry. And their constructions over various different groundings. And I actually thought we were coming here to discuss an agenda in philosophy of mathematics, you know, the continuous versus the discrete, the one versus the many people, where there's full issues. Well, he was, I think, you know, there I was actually having to argue the other way. That's all very exciting and you will find, if you're just prepared to give a little bit more effort to it, that those things actually emerge and come into focus, provided you allow the discussions on algebraic geometry and the way that you see set to run. You can't please everybody, no. You can't please everybody. But what I do object to is somebody who clearly regards the whole thing as something which they can hijack just for their own personal agenda.

55:00 It was after all, it is John Richard and the Palazzo di Carmo who invited him. This is their meeting, not, if he was in Buenos Aires and this guy invited him, it would be a different story. Richard found some money. Yes, Richard has got the money for this whole meeting from the British Academy. I don't think they particularly expected to be turned into a, you know... But Richard also told me that actually the Bristol people was not... I don't get it, do they? I mean, I like James Ladin, but to be quite honest, it's completely above their heads. They're just, you know, workaday. I know good people, some of them, but they're just basically workaday, you know, modern-day philosophers. They just don't get it. It's like John Daniel Isaacson last night here at the Parliament, saying, you know, what's all this stuff about? Is it really a fact? It's a bit... I mean, it does require a massive amount of reorientation. Well, the kind of objection that Miki Detlefson made, but actually he spoke about Colin rather than Bill, I don't know. And I think that for Miki to say Colin, but it's not like Bill. No, no, no, quite different. What Miki says is that Colin says he's a preacher, he has some stuff, you know, to sell, but he doesn't provide really philosophical issues. I see what he means. I mean, he is a preacher. He's preaching the right gospel, of course, as far as I'm concerned, which is that the underpinnings of contemporary structural mathematics are... To be found in category theory and that the correct conceptual perspective on mathematics is provided by category theory and that the future of category theory, the future of mathematics lies in the kind of increasing unification of the subject via functorial methods. So to all extent he's right, but the problem is that Colin... Focus is little, Colin doesn't address traditional philosophical issues, even in philosophy of math, in the direct way that he could, and that Bill does, of course, when he talks about the continuously discreet and about-grasping. He just says, here's this wonderful new machine, and I'm going to tell you a little bit about the history of it, and, you know, anybody who does anything else is not doing anything of interest to it.

57:30 So I have to say, Colin Dunstan, you know, he's his own worst enemy in that respect. I completely understand that point. And I think it's a pity, because I think the people, if the whole thing had been done a little bit more deftly and a little bit more... You know, a little bit more tactical finesse, and we would have got much further, you know, sociologically. It's Peter who should leave. It is, actually. I wish you'd stay, because not least because you'd help to support me and John and Richard in terms of trying to keep at least a reasonable proportion of the agenda for the broad conceptual issues. Because if it's just going to be hijacked into a purely technical topos theory seminar, then I frankly, you know, well, I wouldn't say I'm sorry I came because I've got enough out of the last four days to more than justify it just out of your talk, Bill, and the discussion. But the bit that I got the least out of, and I'm sure it was the same for you, and it's certainly the same for Richard, I was just watching Richard's face, poor boy, just sitting there, it was a rift of agony, whilst all this, the first day when you went there, they did this long, long calculation on the board, which was just a kind of, it looked like a Jensen diagram of proof theory, and it was just one endless discharge of hypotheses, and the point is he could have done, they could have... ...had often done that on their own. Exactly, it's a private thing, it shouldn't, it's a private thing, why the hell should it take up the whole of the session? It didn't in fact take up the whole of the session, but it took up a car more of it than it needed to. And what Bill was saying about, he wanted to make clear this, what's behind this idea, which you've already heard him talk about, the Eulerian... And why, of course, this connects with his ideas about axiomatic cohesion, why this is a much more cohesive model of the line than Dedicate. And clarifying all those, and you can clarify a great deal conceptually. Okay, I do take his point. It, of course, is also very important as... The climax of this program is to exhibit a model in the rigorous sense of what a mathematician means by exhibiting a model, but exactly, what he means is some really super technical kind of relative construction in probably involving, I'm not even sure what he does involve, but I mean, unless you're prepared to give the motivation, clearly, for the whole subject, then…

1:00:00 You know, because all he would do is just kind of wrap out a series of... As I said, I didn't want to go there. But we would never get any kind of conceptual clarification or any sense at all of why these topics are of importance to philosophers as well as to mathematicians if we were to take the root fields, if you want us to take, we would just be completely blinded by really very narrow technical discussions of something which is only relevant to one particular topos, which is basically the topos of the old. In which, you know, the topos was a second-order partial differential, well, the natural topos in framework for analyzing second-order partial differential equations using this particular, they did, I mean, Bill was saying very interesting things about DD and about why it has to be connected, why it has to be strongly connected. And he was saying things which connected with these ideas of Leibniz about the infinitesimal and about monads, what he calls intensive quality. That was all really interesting. And he was getting obviously more and more pissed off and hostile with this because he's just not interested in philosophy. Same problem with Peter Johnston actually. Look, I'm a very pretty mathematician. I guess that's what I do. But at least Peter Johnston gets that he's much better than he used to be. He's far, far, far better than he used to be. But this guy, this guy is just, you know, he's getting, I mean, he kind of almost wrapped up, well, yes, yes, this is all very well, but why, and then they started doing this endless calculation, which in fact they ended up, he ended up getting it wrong. Bill had to go off and spend the night sorting it out and bringing it back downstairs. So you can see why I'm not annoyed. I actually quite recently read his first book on topology, and I really... Oh, it's a brilliant book. Yeah. I probably could appreciate it. The set is more interesting than the other. I mean, the other part is just kind of an idea to get everything done. Well, he said to us last night that the third volume is, of course, still nowhere near completion. He's going to have a sabbatical next year. He'll try and get it finished. But it's going to be at least as big as the first two volumes, which, again, has gotten us, well, probably about 400 pounds when it comes out, I should think.

1:02:30 Yes, well, that's good for him. But at least Peter Johnson, in fact it was quite obvious from things he was saying after Bill's talk just in private, Bill and John, I mean, he does absolutely get what these foundational issues are. He's certainly not dismissive about the importance of the program of, you know, really internalizing the notion. Incidentally, the person I would most like to have heard... In response to what Bill said yesterday about this program for internalizing smallness in this, and why it is absolutely so crucial to have the cartilages in both settings, is there a book? Because what I'd really like to understand is how this connects with fiber categories. Of course, I already mentioned to John, and actually I'm going to send him because he never read it. He didn't have copy. John? Oh, no, no, no. Well, yes, I think he should be able to read that. John has actually picked up quite a lot of category theory by now. John has picked up a lot on category theory, although, of course, he does essentially remain a classically trained magician, which is why he asks the kind of questions he asks. He's a lovely guy, though. You've seen enough of him. He's just great fun to be around. But by the way, just for the record, as I'm sure you'd like to know this, Bill did say to me on the way to the restaurant last night how much he, seriously, he did say how much he enjoyed your talk. He said, what do you think of Rodin's talk? I said, yes, he's really come on, he's really come on. I used to think, but now, I'm sorry, I went into a silly voice, but he said, no, no, I used to have my doubts about Andre Rodin, but now I think he's really, mind you, I could be blinded because he was saying such nice things about me all the way through the talk, but it's always nice to hear. But I thought he was really good, and I didn't bow his head off too much about him. Because I thought he was really good. He's absolutely saying all the right things. These are all the things that philosophers and mathematics should be hearing. He actually said that. Those are all the things philosophers and mathematics should be hearing. No, you really liked the talk. You definitely won him round. Well, I think he was never hostile to you, but he was not quite certain what he thought. But now he's definitely on your side. He definitely gained a fan there. No, he was very pleased with that. In fact, he even mentioned it in the course of his public lecture, didn't he, a couple of times. So, that was good.

1:05:00 I had a very...