Exchanges

0:00 Thank you for your attention. Before she took the position of librarian, she was a professor in Adelaide with a commission to look in the archives. She did a very good job. And so she discovered many, many interesting things. And she put all of them in the archives of IHS. You were speaking of Aurélie, yes, who I've already been in contact with. Yes, I spoke to Boogie. Oh, that's very kind. I mean we did not enter into the details but he said he is willing to do something that's very good that's very good that's very good news he's willing to do something and he has a more general plan I mean the young ladies that you have seen who was with a monitor in the back of the room I mean already yes a new library yes and she's very yes she seems very impressive I talked to her That was my impression too. I got a very good rapport with her when we spoke on the last day. As I explained, I mean, Bourguignon hired, well, Bourguignon knew when Francoise would retire. And he hired already a year in advance. And so she had the opportunity to look in the fight and to put all the fights in order. And there are, of course, treasures of mathematics. There are treasures that no one looks at these treasures. No, I know. Look at these treasures. Bouguillon told me explicitly that he is very positive towards any kind of program of that kind. That's very good news. I told him just in one sentence where we were, I promise you not to diverge any more, so I met him just last Saturday or last Sunday and I told him that you had his attention. I confess I do apologize, Bill. I did tell Pierre. I hope you won't be, and on condition it would be kept completely within our circle. I'm now convinced, Duskin, that this is... Because we have an idea how to do the mp3 and not to transcribe it and so on. No, no, no, no. The mp3 thing, as I say, is very easy. I've now seen the equipment. I'm doing it myself. You're doing it yourself? Yes, I have it on my computer. Okay, okay. It's not rocket science, but it's an extremely mundane and banal thing for you to have to be doing.

2:30 It's good to have a professional to do the whole thing, but I've started myself so that I have some familiarity with what the problems are. So we have this dream now that students will have their iPods and they will be listening to Grotendieck as they lie on the beach. Almost a good idea. I think of course Grotendieck would particularly like that idea of them doing it lying on the beach I imagine. I think it would. It would appeal to his slightly hippie instincts. One of the things, one of the reasons that I'm listening to, you see, in terms of material, I hope to myself, well, basically, it's the idea of blogs. These things will be available, so there will be a place where anyone can send a review, blog, a comment on anyone or anything. Yes, the whole idea is to make it interactive, but also the same type of preserve can happen as well. Basically, the main thing is to have a summary of each one, because transcription is very boring. Yes, very boring, and especially since, as you explained, this material... I have the impression from talking to the people who were talking a bit about what Grundig was doing at that period in IHS last week that... You know, when he was in, of course, nobody knew he was in Buffalo, but he was in the U.S. from 70 onwards, you know, so it was on a number of occasions, mainly, of course, to promote his CIVIVA program, and he was giving quite a number of public lectures because he wanted to get publicity, you know, the quid pro quo being he would give, frankly, by his standards, probably not particularly inspired because it wasn't new material, it was just, you know, things that he... He was quite prepared, as it were, to use the clout and renown of his name and reputation to give... Relatively mundane, in a presentation on relatively well-known material on algebraic groups or whatever, if in return he could get either financial support or an audience for his survival. Particularly financial support. See, I knew about the propaganda aspect, I even attended myself some of these things, but the financial aspect was almost the main aspect.

5:00 But that he gave so many of these lectures, I mean that they gave as many as, what did you say, there were 96 recordings there, so these are obviously three quite substantial courses, very substantial, and the fact that one of them was on topos theory, there must be material in there that is not just expository. I must say the first one is a little bit disappointing because he starts off by saying that topos is a generalization of topological space. Look, excuse me, he doesn't start off with that, he starts off with saying that topos are the proper setting for general topology, with this I agree, but then he interprets this to mean that they're a generalization, which is false, but you see, the first... Which he himself, of course, at that very period in 73 was actually kind of breaking with that point of view, because he was long before that, when he was talking... No, long before that, 1960, he already did, basically. So the first slogan, you see, in other words, has much more to it than the... Generalization of topological space, but he doesn't make that clear in the first lecture, because he goes on to review sheaves on topological space and things like this, but I'm sure this was a momentary lapse or just a mode of introducing the subject. Yes, because after all he is introducing it to people who have not heard of Herthoff's theory before, so one has to make allowances for that. There were classical actions of a group on sets of a given group for metopos. So that's equally a classical, you know, the generalization of the space is equally a generalization of that. And this you could have said but didn't. Well, maybe he got around to saying it in one of the later lectures. I mean, you haven't listened to all 30 or 40 of them. I mean, there's probably about 30 of them just on topos theory alone. There was nothing, I gather from what you told me, there was unfortunately nothing recorded of any of the talks that you actually gave at the Maths Colloquium. It doesn't seem so. What a tragedy. There are three cassettes which are of a different color from all the others, you know, actually pink instead of, and they are, the dates are, I don't remember the exact date of this colloquium, but sometime in May.

7:30 I thought, this must be it. But it wasn't. It was part of the more general course. That would have been an exciting, that would have been a fantastic discovery. That would have been a fantastic discovery. Because one of the things I really would like to do is, you know, maybe, while we're in Bristol in the coming April, It's to do an interview with you about the... Your assessment and amplification of the ideas that were in that Buffalo talk of his about the role of classifying rings in, as you put it, bypassing, allowing us to bypass logic, because those seem to be some of the most fantastic ideas he had, which is obviously setting the bar very high indeed. It's a great tragedy that that one wasn't recorded. But we still we have the list that you say that he compiled and that Jack still has and it would be well I think it would be absolutely fascinating as I say for you to give a presentation on kind of assessment and amplification of some of those ideas so far as they can be reconstructed. I think from what you've told me they can in fact be reconstructed pretty pretty thoroughly So this is a fragment, in a way. Yes, I'd rather gather that the whole sort of Verndi-Tuchtenmüller program is a fragment of that Classifying Rings program. Excellent. He also made it excellent. I mean, making a connection is good, I think. Well, yes, he's working, that's the thing he's working on. But he made it very clear. Oh, yeah, yeah, very good talk. Very good talk, yes. When we decided to speak about the program, we were a little embarrassed because we had to fit two things together, the program of this afternoon and the rest, and I wasn't sure that it would really fit. No, it fits, it fits. I mean, we had various engagements and we had to fit the two things together, but it really does, it really does. Of course we have all these people last week, but that's not what was the intention.

10:00 Last week was very, very predominantly algebraic geometry, of course, which, of course, it was intended to be, but that talk that you were commenting to me about on the last day by Gizia, is it? Pizia. Pizia, Pizia Bigupon, was utterly fascinating, the one on functional analysis, that was very, very interesting. I told you that I had to fight for it. I know, which is all amazing, because I think in some ways that was the talk of the conference for me. Yes, you know, when I began to organize the conference, I wasn't... I mean, the title says algebra and geometry. I wanted not to be so restrictive, but I was, well, after the final discussion, I had to agree on the title, Aspects of Geometric Algebra. I wanted to have a more general title, I mean, work of course, something like that. And when I insisted that there should be something about the first part of his work, about functional analysis, well... Since I was a boss, I was allowed to talk about it, and this year I was allowed to talk about it. But that was an utterly fascinating talk. Has anybody noticed before that this... The Bell's theorem, it's just these eight lines of algebra that the physicists make so much fuss of, it's just a corollary of the proof, proof that, you know, the Grotendieck inequality or norms of a normed, uh, it's not, it's very natural. It is, it is. Once you see it, you see it's not. It's entirely natural. Well, it's just the case for two matrices, isn't it, really, for the Grotendieck? It's very natural, it's very natural. Once you see it, it's very natural. There's a certain combinatorial structure behind the Grotendieck inequality. Yes, yes. Yes, but it does appear to be showing up in physics, in this environmental, this contextual norm. And I said I've been, I studied, well I was reading, well not the whole thesis, but I was familiar with the thesis of Gortendieck, it's my, I mean, when Schwarz was appointed to Paris, his first seminar was about the thesis of Gortendieck, he was appointed in the fall of 53 in Paris, as a professor in Paris, he moved from Nancy to Paris, and I was one of his collaborators for his first seminar, that's my real first seminar.

12:30 Before I was attending seminars, I listened, but this time I was a non-real participant, and I remember, for instance, the last chapter in this seminar contained the proof of the topological vector space version of QNET. And once you have realized that, I mean, the nuclear space, I mean, the idea of Goethe and Dicke is that, well, it's discovered, that is that for, let's say, even for manner spaces, it's already relevant for manner spaces. You have a natural tensor product, which is the one which makes the adjunction with the bilinear form. Yes, yes. The classical definition of the tensor, the classical adjunction of the tensor product. Adjunction of harm and tensor. Yeah, harm and tensor. Give you the first tensor product. By duality, you get another one. Combining this with duality you get another one. The first is right-exact because it's an adjoint factor, so by duality the other one is left-exact. And what Goethe did discover is that the interesting class is when the two coincide. For Banach spaces you have only finite dimensional spaces for which both coincide. So you have to go to, not to stick with Banach spaces, but to go to Frechet spaces, which is projective, poor objects of the Banach spaces. Yes, but very special among those. Yes, poor objects. But then, the poor objects which make the two contents of product coincide are really the analogs of flat modules. And if you prove, you can prove, I mean, if you work with flat modules, it's clear that if they are too complex, which are flat, then the Kuhn et Firm is always too difficult, you can't find any way to do it. But it's so interesting that when I discussed with Sayer a few weeks ago, And he said, what do you have in mind to explain? Well, he did not intend to attend my talk, but he did not want to attend my talk, but of course, he wanted to know what I was saying. And I say, nuclear spaces are flat modules. And he's just thinking, but flat modules are not invented. No! I agree, but it's the same idea. Nor, of course, were adjoint factors in 1953, but he still saw them. Yes, and I said, no, and that was surprising, and even shocking. There was no Flandreau. It did not exist, but it's really a fair we invented Flandreau.

15:00 But now it's a platitude. A few years later. That's one of the better puns I've heard. Oh, in title. Oh, it is. Oh, I see. I didn't know that he made a pun on it. I didn't know it was his pun. But if you read in retrospect, I mean, all this work. It's really, the functional ideas are there. The language is not there, but the ideas are there. Projective limit, inductive limit, direct limit, inverse limit, and exactness. All these things are there. All these things are there. And even, I mean, even is very vague. It is very clear in the mind of Goethe and Dix that L1 plays the role of projective, L infinity plays the role of injective. I remember a discussion with him at that time. He knew that very well. But, but, but, I don't understand why people don't want... When we published Lefebvre, I insisted I wanted to have a section on functional analysis. Finally, the journal of K-theory published a few papers, not of the same caliber, but a few papers connected to the work of work-making functionalities. And I had some projects, but finally I gave up. Another project I gave up because I already had quite a lot of work and, well, illusivity. Most of the work, but that was his assistant, and that was important. But what should be, I mean, it seems to be a third idea, the tensor product, the algebra of functions on the product of two manifolds, should be, in principle, a third one, or can it be reduced to the second one? No, it's the same, just the second one. What you can see is the following. L1 tensor L1 in the natural product is L1. In the dual is L-infinity or continuous function tensor L-infinity, which is L-infinity by duality. L1 is a dual of L-infinity. And of course you have in between L2 times L2 is L2 for the Hilbert tensor product. And the last paper of Grothendieck about functionalities is just... Explore all the connections between L1 and L2 and infinity and all the various temporal products.

17:30 And since you have injective, well, every banner space can be embedded into an injective one, is a quotient of a projective one, derived factors, derived factors. And he has derived factors. And I think he was more or less, I think in his mind it was another key. But the point which I mentioned in my talk is that Grotendieck bought this idea from... from Doudanais. Yes, so I was going to ask you, you see, you mentioned about Doudanais, I hadn't realized this. Who is it who introduced to Grotendieck the work of Sebastiao e Silva? Because he cites it in the very early papers. Maybe it was Doudanais who bought him about this? Or Schwarz? Schwarz. I suppose it's Schwarz. Well, I didn't mind to look at this work of Silva, but I had no time to prepare. But I will visit Portugal in the summer, so I will have occasion to discuss with the people there. So you're aware of this, I mean, the thesis of Silva, which is not published. I know the published version. No, no, you see, this is very exciting. From a historical point of view, he worked with Enriquez in Rome during the Nazi occupation. He had one thesis. Enriquez said, it's too advanced, the world's not ready for this, publish only a fragment. Okay, and I think the Pontifical Academy published the fragment. But now the original has been found. I don't read Portuguese very well, but it's very exciting because... It is a general theory of homomorphism. It's as if Silva almost independently discovered category. At the same time, 1945... Okay, when I go to Lisbon in the summer, I will discuss it because... You will find this... No, no, I will discuss it because... Just goes to show how much good archival research can do in the writing process. This categorical thinking really from the 30s and earlier just became more and more explicit.

20:00 I didn't mind to explore this. I remember very well the work of Silva, you know, the published version, and I didn't mind, but I had no time to study. But when I am in Portugal this summer, I will do it. So it's, you know, it is Galois theory, it's very, you know, it may be interesting, but the fact of considering general homomorphisms and a general theory of that. And also these homomorphisms were not onto, I have verified this, that they are definitely, you know, there is a difference between image and the co-domain. So it's really a very general... But you know, I mean, what I've, well, first of all, it's very interesting. I studied, I find, about ten show projects. In 47... Giordani must show people, discover what are the flat modules over a dedicated universe. And he probably correctly says the Gaussian-free module, the flat one, while it doesn't speak of flat mathematics, but of a tensor product. 47. So, the first real example after the feedbacks. In the Carton seminar of 48, there is a long lecture by Carton and Sayre explaining what is a tensor product of two modules over a commutative thing. Bobacki was published just after that. Ten years later, we speak of tensor product in a category. And Grothendieck, what is interesting is that the thesis of Grothendieck about tensor project of function spaces was very advanced. I mean, if you don't know the way, what are the groups and categories? His name is included by Bobaki. Bobaki had a strong insistence on the... First of all, in Bobaki, you can pursue distinctions. That's very good.

22:30 It's not like me. You know, my wife died... Oh shit! I thought I understood that when you mentioned the system. And... Which simply further underlines the point about how hugely important is that early work in functional analysis was. It contained the germs of so many of the great ideas that he... So in this whole series of IGS and here there is one lecture which... In fairness, there was Cartier's lecture, which was a general survey of the importance of his work, and then there was this explicit lecture by, I keep forgetting his name, Peziere. That was a very good lecture. That was, yes, we were just discussing an extraordinary lecture, an amazing lecture. And Pierre was just saying that he really had to fight like hell to get him onto the program. Well, that was a very nice lecture. Yes, it was. And Pierre was saying to me a little bit afterwards, I'd love to hear more about how he thinks this might impinge on the whole problem of entanglement relations. I guess you were, actually. Very nice to talk to you. Sorry, this is Bill Lorvier, by the way. We've met already. Yes, I saw you. I'm afraid I missed your talk at IHS, but of course I shall catch it here tomorrow. It's a somewhat different talk. I cannot always do the same thing. No, no, that's good to know. If I were just speaking, then I always would give different talks, but now no one uses this PowerPoint representation to change something. You want to show a lot of pictures, it's more convenient. But then it's some work to change things. You know the cliche, all power corrupts, PowerPoint corrupts absolutely. Coming because I could have bought that. One of the things I've done in the last. My first impulse was, of course, to come in any case. But then when I got the impression you wouldn't be able to come. Expecting to be. In fact, I literally overtook the decision last night. In fact, I wasn't even sure even when I got up this morning whether I'd be able to come or not. Oh, my God. Well. You're in the right country, aren't you? I am, yeah, I am, but the finances are a little... Well, I managed to find this very good deal on SNCF, whereby I could get a return ticket,

25:00 provided I came back overnight for only 63 euros, which was, you know... So I came down, not having a hotel or anything, but I'm sure I can find something. I'm told the Hotel L'Angleterre is quite reasonable, and that... So, we'll see. But actually, now that he's... You know, just merely advertising how much of it Grotendieck did, you know. Yeah, my description of it is, you see, that... You know, he has this example, the bad taste example of how Grotendieck tried to sell his own shit, right? Yes, which you told me about, yes. Well, the dreams, you see, are waste products of an individual, so he's promoting these waste products. I'm sorry to say, I didn't want to say it in front of him, but as far as I'm concerned, he's a public menace. I know that Cartier didn't want him on the program, but he was kind of over-porned by the other people on the committee, because they thought they should have something biographical. I see. And he, of course, is his biographer. I see. No, I haven't discussed it with Cartier. He also has this... Well, I don't... I'm not sure. I think he's very suspicious of the... Ah, yes. Okay. And the other thing which I thought, although it was a very interesting week, and of course everything was videoed, and in fact we've just heard Cartier say he's going to arrange for this excellent new woman they've got here to make sure the archive has got copies of all the DVDs, but I was very disappointed that there was not one single talk with Topos theory in the title, and in all the talks that I listened to, I wasn't able to listen to all of them, there was only one talk which in fact really used topos theory directly, which was actually used in the context of tubular products. This was in, I'm afraid I'm not qualified to follow it at all, but all this work in motivic cohomology proving these things about Hodge duality. They seem to use quite a lot of topos-theoretic machinery there, it's in the background. And Lucy gave a very general talk. I actually texted him with this afterwards, I actually had a go at it, but I actually said, well, do you not think you ought to say something about the development of topos theory from 1969, 1979 onwards, considering that Grotendieck himself said, one, that it was the...

27:30 Le plus vast, the vastest and most important thing he had conceived, and two, that the biggest thing he missed in his entire mathematical life was the sub-object represented, the sub-object of the classifier, and the connection with logic and set theory, which of course doesn't exhaust Topper's theory, but is still a very, very central, important result. Well, I know it, but I just don't know that theory. I just don't know that theory. Oh, you're talking about Lovier and Turney. I said, yes, of course, it's just a huge, just, you know, it's huge. The biggest, you know, the standard reference work, in fact, two, both the standard reference works in the field by Peter, and how many other works, and he's, oh, well, I just don't know that theory. I just don't know their theory. And of course, the AMS asked me to write this thing. It was only a two-page note. You can't say everything in a two-page note. He was a little bit shamefaced, I think, about it. He was a little bit shamefaced. But he attributes it fundamentally to ignorance. He attributes it to his own failure to study the theory, which at least is odd. I don't know. But you see, for me, it is algebraic geometry. There's a kind of French ignorance, you see. Yeah, it's willful. Which is willful, exactly. It may be ignorance, but it's willful. Very much so. And of course, algebraic geometry is the centre of the mathematical universe for them, which is not necessarily a kind of mistaken first approximation. But to limit it to the algebraic geometry that they do is pushing that point much too far. That's where it stops being a useful first approximation and becomes a dangerous distortion. Exactly, exactly, exactly. Well, as usual, singing to the choir. I'm not going to mention this, but I think Cartier is also quite ignorant of it. Yes, on the other hand, I have so much admiration for his mathematical work and his personality, but I agree, but I wish he could shift his center of gravity a little bit, but just that whenever he speaks, well, like Luzi mentioned logic, you see, it's like...

30:00 Oh, Lucy is much more extreme, though. In his case, yes. But in Lucy's case, you can see the shutters come down and say, well, I just don't know that theory. In Carty's case, he is interested. He wants to know. But anyway, it's marvellous you could get here. If only I'd known you were coming, I would have brought down what I have just completed, which is the catalogue of all of the recordings I've made of you since 1989, so it's 20 years ago this summer, since the Cambridge conference. Well, it's only a catalogue, I haven't actually started to... It's quite good. There are already about 200 recordings in there. I don't mark where the toilets are. I can tell you the toilets are right at the end of the corridor there. Right at the very end. The men's is straight out at the very end and the ladies' is just before it. you're planning to go down to see alberto directly from here after this to go to rome and then right okay so bernardini oh great who is coming to um fantastic that's great oh that's even better i don't know cox exact dates yet that's great because i i after we had that skype conversation i really really pushed richard It's hard to say, look, you've got to find the money for Davide Bernardini and, well, obviously, and Joachim Koch as well, if he can be arranged. Yeah, I haven't heard that he did find money. Well, I think he's got enough at least for one, which is Takes Care of Bernardini. I don't know about, it would be great if we could get this as well. But when, which dates, sorry, which dates would you actually be with Alberta? Yeah. I mean, I'm just wondering what your kind of timetable would be. Well, I'm flying to Rome on Saturday. Right. Staying in Rome Saturday night, going to Florence on Sunday. Right, so you're moving around. Seeing Roberto on Sunday afternoon and Monday after his class. So the 25th and the 26th. Taking train back to Rome. Sleeping at Rome and taking the transatlantic. Gosh. Returning to Washington on Tuesday. Wow. Well, you haven't heard the half of it.

32:30 Before coming here I went to Alea as I had dreamed. Oh no, I was going to say you didn't. I did. Oh you did. I did it. I envy you so much. I did it. I was wondering if you had. This is the sort of thing I always wanted to do and I did it. Oh, I do envy you. You'll have to tell me all about that. Well, it was quite an interesting experience. I must say, the main speaker, which was... What, our friend Barnes, the eccentric guy with the jabot. You know him. I know. I've seen him. I mean, I don't know him personally, but I know a fair bit about him. He actually was in a costume. Yes, he always wears this ridiculous 18th century costume with the jabot. Oh, he wears it. He wears it. He always wears it. With the trousers. With the trousers, yes. He wears it like a kind of 18th century squire. Of course he does. Yes, yes. Then with the nose, he has this nose, he looks incredibly, he makes quite a figure for a theatrical presentation. Yes, he does cultivate this dandified sort of personality. Then he insisted on speaking Italian. Oh no. Which he'd learned from a book and not from living with Italian people. He was just repeating, you know, you take an interval, you divide it in half, you divide this and this. He said that about 18 times. He didn't say anything. Didn't say anything. This was very just, everybody, there were about 75 people there, people, many people from Brazil, many people were there, and Mexico, Brazil, Italy, of course, Poland, and so on. Everybody agreed that he didn't say anything. But it was very good to meet some of these people. Yes, I was going to say, hopefully you had some useful contacts with these fellow... Some of them were very, very nice people. We had very useful contacts with these people. And the ancient city that I always wanted to see was very, very interesting. Well, I never expected... There was this town of Ashiya. Now, a modern town where we stayed, which was formerly ocean, you see, I mean, it's one of those places where the coastline is silty, so there's a railway, which was right on the old coastline where the city of Alea was, between the rail line and the sea is Acheia, which is the town.

35:00 There's a fantastic gate, stone gates, perfect, wonderful engineering to enter the city. It's on top of a hill. You have to go over the hill to get to the city. And I asked, well, where did the slaves come from? And the guide said, well, we don't really know if they were slaves. Maybe the citizens themselves did. Free labor? I don't think so. Not in the 6th century Greek society. There's the same line about Egypt, you know, that they're promoting that the pyramids were all built by voluntary labor. By volunteers who simply loved the pharaohs so much that they laid... This just seems totally implausible. Well, I think that's putting it very monthly indeed. I think I'd have recourse to what Grimdy carried around an office sale in his pale to describe that. I think, you know, a complete load of... The alternative, of course, the fallback position, if you can't persuade them with the obvious lie that it was all built by no volunteer labor, that it really wasn't slave or slave, it was a much more martial institution. I mean, the fallback line is, of course, because the slaves were extremely valuable, so they were always very well treated. Well, yes. The other kind of garbage that you get. Yes, well, I mean, you treat your racehorse as well. No, that's not... I don't care how well they were treated. This is not a main point. Of course it's not. Of course we agree on that. No, but I'm saying these are the tactics that these people use. And of course they completely contradict themselves because they use quite contradictory arguments. Apparently this has become a common procedure with these ancient sites. The whole group, we were escorted through the city by a theatrical group. Oh, they hate those things! Oh, that's awful! This is probably something that Berlusconi has envisaged on them. Because you know that the administration of all the ancient monuments in Italy, the whole ministry which takes care of the cultural patrimony and heritage of the country, was handed over to this pile of Berlusconis who has no...

37:30 ...experience in museum curatorship or in archaeology at all, and he was just some kind of media friend of Berlusconi who was appointed. There was a tremendous outcry against it. I mean, all of the museum directors in Italy went on strike for about three months. So it could be a consequence of this, but unfortunately we have the same sort of thing in England now. We had a couple of guys who were part of the museum anyway, but they just stood there. They didn't do any of this. No, no. Presumably, if you'd asked them some questions, they might have had useful information. They might have had. But the thing is that this, at Aliyah... This was completely distracting. I mean, these people there in front of you, they were talking in a very dramatic way. You can't avoid listening, so actually to actually see, I mean, many people said this, but it would have been much better to have silence, you see, silence to contemplate the ancients and so on. You think about what it means, but these were completely... Constant distraction. It's like Muzak in a supermarket, only worse, because, you know, supermarket doesn't really matter so much, but in somewhere like that you don't want to be distracted. This one pretends to give you some content. There is no real motion, yes. Is it true that we... Oh, no, I would never have expected Jonathan Barnes actually to have had anything at all so penetrating or useful to say about Zeno. That doesn't surprise me. I thought from some of the material that I've read that on some of the pre-Socratics, like Anaximander, that he might have had something interesting to say, but I just don't know whether that was so or not. Oh, well, I'm sorry. I'm disappointed to learn that. On the other hand, I'm really glad you got to see Aliyah and to meet a lot of interesting people. People said, well, maybe he doesn't know anything about this, but after all, he wrote books on Aristotle. You know, I'm dreaming to learn more about Aristotle, so I'm thinking, well, should I read his books? And they were saying, well, yeah, he did something. But then, you see, on the last day, on the last day, it was very interesting. You know, he's talking in this strange Italian, very strained in a way. And then he's talking about something, and he says, Posso dire la verità? La vera verità? A different tone of voice than he had before. The benefit is, yes, I love Aristotle, but 95% of his philosophy is wrong. So he loves something 95% of which is wrong, according to him. I'm not sure I want to read his book about Aristotle.

40:00 I got a completely different idea from the whole thing. There was a course about Parmenides just before, so I missed that. I heard some echoes among the 75 or 80 or so people. I have the feeling, you see, that when they say not, they just mean a dialectical notion. But it's no kind of mystery at all. They know that there are electrical passions there, and they call it not. Yes, I find that's my reading. It all makes perfect sense. There's no big mystery, and many things need to be clarified. This always presented as a mystery, as though this knot was some, well, literally an inconsistency, it's not an inconsistency. Now, one thing about Barnes, though, I found, I did agree completely, he hates Graham Priest. Well, I think we can, that's at least sound. That really is complete garbage. We could share an experience on that. Yes, champion of inconsistent logic. Yes, really is the kind of guy who gives logic a bad name. Oh, yes. And gives contradiction a bad name. Yes, yes. But I notice he never uses, well, he does sometimes use the word contradiction, but he actually prefers... Paraconsistency, non-consistency, it's interesting that he rather deliberately avoids, yes, contains contradictions, but he seems, I've just noticed in his writings, the priest steers clear of the word contradiction whenever he can, except when he has to, which I think itself is symptomatic of the problems with this. Well, no, the reason I asked, the reason I was pressing you about Alberta was because I'd just love to try and get down there for a day, if it was Sunday, but if you're literally going to be there just for a Sunday and a day and a half, like I said, it would be lovely to be a fly on the wall.

42:30 I might press you to do me a favor there, but I'll tell you about that later. It's all right, don't worry about it. The other thing which I need, well, no, it's okay, that can keep as well. So you say that Jack is now quite happy, in principle, with the idea of putting a... ...putting this website in the public domain and so there's no, he hasn't got any kind of, you know, reservations about wanting to do a thing or, you know, do a lot more editorial work on it. He's quite happy to put it in the public domain, that's good. Yes, as audio, as MP3. Yeah, as MP3 fast. Well, that's what I was really hoping. And he wouldn't have an objection if, as I say, because there is this money from the German Mathematical Society, which they... I think it would be better if you used that money for your own purpose, insofar as you can, because it still seems that we're going to get money from Buffalo for this. What would be very helpful though is if, obviously the Buffalo site would be the site and it would be put together by Ewan. And obviously, you know, the editorial content would all be yours. But would it be possible for the archive to have a mirror site? I suppose so. That would be fantastic if we could. What's the purpose of a mirror site, really? Well, basically it just means that people can look at the material on your site. Obviously it's credited to you. Or even if it's just a link through to your site. A link, yeah, of course a link. Yeah, we want to give a link to the circle. Yes, sure, sure, okay. Would it be, okay, is it possible, and this is the $64,000 question, not $64,000 question, but it is about a $5,000... Actually, 5,000 euros, which is rather more. Could I say to... Because I haven't actually told them about it yet, but could I say to David Rowe and the German Mathematical Society that they would definitely have permission to have a link to the site once it was established? Oh yeah, a link. Okay, sure.

45:00 It's not a big problem. Okay, so I can simply send them something over. They probably want something in writing to say that, so you can talk to Jack about that when you get back. Just to say that once the site is up... The links are normally free of charge anyway, aren't they? Well, they are, but you usually have to ask permission for them to put in. Simply, they would want permission to be able to put that in. The German Maths Society. Okay. No, it's just that I was talking to them and they're going to come to visit me and Fougere in February, probably after the Opa-Wolfack meeting, and I'm hoping, they're trying to arrange for me to get some kind of affiliation, full-time affiliation to a German institution, possibly the University of Mainz. Because, provided I had, you know, this is the way the rules for funding work in Germany, provided I had some official affiliation to a German institution, then they could give some support to the archive, the project of getting the whole thing. This of course is my own material, we're talking about other stuff like all the recordings that I've made over the years, but it would be very nice, as I say, if we could, within this, embed a link to the material and to the Buffalo site, because obviously I will have a whole historical section on category theory in there, all the recordings I've made over the years. Speaking of that, Cartier tells me that you're going ahead with this meeting next week in Bolpark. It's not actually next week. It's about three weeks' time, yes. Three weeks' time, yes. Yes, yes. Well, it's Faudemur. Colin is coming also for the whole week, and Cartier is speaking, so is Colin. Colin also. Colin is also speaking, and actually they've got several people there who I think are going to be pretty... I asked Colin if he'd encountered this pragmatism, and he said, well, he's just a kid after all. Well, he's actually about 32, so he's quite old as a kid, you know. I think this is a little bit of a silencer. Naive, yeah.

47:30 I was probably thinking of... And the man was a nice girl, I mean, I told you. I mean... Who's? Sorry... Madame Boisdard, can you explain? Yes, absolutely. Professor Lové said that he visited, in 1081, he visited Goldendieck. Here in Montpellier, at the time of Gordon Dicker's teaching here in Montpellier, but he lives in Carpentras, near Marseille. Bill explains, actually he visits twice, in 1981 and 1989, and the first time he looks for the info for the exact coordinates of Grothendieck's house, which he normally keeps, very secret, but it's a charming... I don't know if you know this, but at the time of King Luke the 14th, he created cities where the Jews lived. Although he's obviously a great man and has done very great things, I think he is a little bit lopsided in his view of topics, it's not least. This does seem to be a common problem in France. He certainly suffers from it to a much less acute extent than others. The calculus is a good thing because you can be able to use the calculus as a foundation. Category theory, he just said today. What? You mean he was told this by somebody? He didn't believe it, I'm sure.

50:00 No, no, no. He repeated it. No, he was thinking enthusiastically. That is what surprises me. In other words, he has some understanding of the edges, but he doesn't understand the edges. It's very difficult somehow for French to reach, to overcome this logic barrier. Yes, yes. Which of course is why they... ...and yet give house room to charlatans like Girard. Yeah. This is why they have such a big audience for us. Because they think, ah, here is somebody who's really seen through all the trivialities of logic and who has taken the subject to a completely different plane, right? Yeah. And said, how we should do dynamic logic. Yeah, yeah. No, that's right. You're right. That's why they are able to get so much of a purchase. That's right. How could you have such a charlatan in logic? If logic was being seriously pursued mathematically, which is to say algebraically, and functorially, they would see through people like Girard. Girard has now been given a course of lectures at the Ecole Normale, a whole four semester course, which he's holding forth, and which I'm sorry to say there is people who should know better from the philosophy, history and philosophy of math community in Paris are attending and enthusing about. She obviously didn't really understand at all about five categories. We have no need of category theory because this is not dynamical enough. This is Girard. This was just in the last month. Yes, he just started giving this. I think on the 9th of January he gave the first. And Benabu, as I say, to his credit, great credit, I think, and good things about John Benabu, many good things about John Benabu, just as a pity that he's become so isolated and cranky and, well, he always was a bit. Convinced that he did everything. Convinced that he did everything, I know. He always was a bit that way. But still, when it comes to choosing between a charlatan like Girard and somebody like Benabu, there is no choice. A cranky personality, but nonetheless somebody who's still done jolly good work in math and who knows what he's talking about. So he did an excellent job of demolishing, I think it's the one after this, it's the, hang on, I'm no doubt it was, I'm quite clear that it was, but I'm afraid it was beyond my powers to follow.

52:30 Well, we're certainly able to extract the essential point, which is that there's a fibration, there's a fibre, there's a fibre category here in the background, which, well, here we are, this is our, no, he was clearly a, he was clearly a mensch who had done, you know, very important work. I could detect that without, unfortunately, being able to follow what he was talking about. Cultural analysis and complex analysis. Was this mentioned in the IHS talk? Cartier did say that in his IHS talk. Yes, only Cartier, though. He's the only person who's drawn this link. No, as I said, apart from him, there was only one. I told him about it. Oh, well, okay. You know, the point is, in his experience, of course, it's there, but it needs to be pointed out. He's the only one who's learned this, then. The only one who's learned it. I mean, it doesn't surprise me at all to learn where he learned it from. I think he might not have thought of it that way. Quite probably not. Anyway, he is now putting that point very forcibly. Very good. We're going to hear about that. We must be careful where we... No, as I said, I've been putting together the catalogue of all the material since 1989 that we've, well, you've done, that I simply haven't been your kind of recording angel, and of course listening to some of it, and dare I say that unlike your experience with the... Great big lectures, I was not disappointed. On the contrary, I was very excited, even more excited the second time around than I was the first. Really? Well, certainly yes. Yes, it's all, it's all, unfortunately there were, fortunately none of your talks. There were some of the talks that people gave at Cambridge that have, I don't know what... Red or white? Good question. White please. That for some reason have deteriorated and in a couple of cases including Gonzalo's to pretty well to the point of completing the audibility. They probably can't be rescued even with the Cedar digital audio restoration software. I don't know, they may be, it's amazing what they can do with that these days. I don't know why that would be because most of the stuff I recorded back in the 70s and early 80s is okay but I think what happened was there was a massive...

55:00 ...deterioration in the standards of manufacturing standards to say the thickness and durability of the magnetic coating on the cheaper tapes in the 80s because they were starting to be a flood of imports from places like Taiwan and so they just reduced the standards and obviously I use them. Cheap tapes. I should have re-recorded them. But your talks are fine. They're all okay. And in fact pretty well all the talks in that meeting except unfortunately Gonzalo's and also Anders' unfortunately, his talk. It may be possible to reconstruct it from the fragments that are still there. Otherwise, everything seems to be fine. All the Gonzalo lectures and the discussions we had there. Oh you said very interesting things in Bolzano. Really? Yes, that was where I kind of remember talking about the contrast between inclusion and membership and the mistake which have bedevilled philosophers and logicians ever since of believing that there's a global top element to this lattice of inclusion. Therefore, the membership must be something global and absolute. The only concepts are properties. Exactly, yes, which of course connects with his neglected co-domains and a lot of other things, a lot of other interesting things as well about the conversations and discussions that we've had with Alberto and Colin and others over the years on obviously a whole range of topics. And some of which we could, as I said, very usefully revisit and amplify and develop further. And then of course the days that you were in Fougere three years back. But lots of other material as well. All in all, well over 230, 240 hours of recordings. Well, all gold dust too, as far as I've been, sir. Remember, I gave you the unexpurgated version of the... Yes, you did indeed give me the unexpurgated version of the Bolzano paper. I have that as well as the slightly toned down version.

57:30 And a lot of other interesting material from that time, including, of course, your lectures on synthetic differential geometry, which I think were about the same, your... You gave some lectures on synthetic differential geometry? 1993? I'm not sure, because I didn't record those, but somebody else recorded them, because I've got them in the... there might have been 92 oh of course 94 oh no it's early 94 you're right it was it was only last month You have that tape, right? I do. You actually recorded that yourself. We recorded it. You recorded that yourself. We transcribed it. No, I have both, but I have the original and also the transcript. And then, of course, mentioning Grassman, the very interesting things that you said almost exactly a year ago now, in January 2008 in Como. In Kerma, about intensive and extensive quantities in the course of those three wonderful lectures that you gave, which of course I wasn't there, but fortunately they were all videoed, and I was extremely interested in that, especially. That's exactly the material which they need to hear about in Bristol. There are situations of cohesive topos in a way, and there are situations of cohomology. Yes, I agree. Philosophy really means mathematics. Yes, it is absolutely astonishing. Well, I think Grassman and Grotenbeck seem to me to be two great examples of that, of philosophy really passing into mathematics. And of course, clearly with dialectics operating very powerfully indeed, it's a guiding principle. That's more explicitly in Grassman. I'm afraid philosophy of mathematics has fallen off to an almost incredible extent since Grassman wrote. That's certainly for sure. It's difficult to believe, but our friend Rodin, Rodin is peddling these lectures of Girard now. He's acting as his...

1:00:00 You know our friend who keeps turning up. He's the irritating long-haired Russian who keeps turning up. He was in Brussels last October when you went out on the Razz with Karim Barelst and him. Possibly it was so much you've forgotten all about it. I don't blame you. We all had quite a lot to drink that evening. The name is familiar. No, he's the chap. He's the rather sort of unkempt chap with the very long, lank hair, except he hasn't cleaned up his underwear, he's rather more... I know. He came to Buffalo even. He came to Buffalo, did he? Long before he had long hair. Oh, I didn't even know that. No, I remember he came and he still has short hair. Well, he has, he's still... And, you know, our friend Smith. Oh, him. Openly, no, but it was funny because Rodin, you see, had some idea that... I mean, maybe topos, you see. You didn't understand very clearly, but he was saying, well, maybe I should consider topos. I think he also talked about full-part relations. Yes, yes, yes. But the thing is that Smith is telling him explicitly, no, no, don't use topos, use rheology instead, you know. He had it right in front of me. He was telling him privately, no, don't do it. Well, I mean, with all his faults, we all know he's not in the same camp as Murray Smith. He stumbles along the way, but unfortunately he does fall from fad after fad after fad. It's only later, I think, in Paris that he developed this long hair, this semi-hippie... Well, he's short. His hair is not quite so long now. In fact, actually, his shape, his appearance is not so unkempt as it used to be. I suspect, yeah. He probably went through a phrase. Yeah, yeah. Well, he also got himself a job. Ah, well, that helps. Yes, it does help. Where is it now? This speech is run by this strange guy, Mikti Tlefsson. He's a historian, come philosopher of maths from Notre Dame, whose son now has got all this money for cosmology and philosophical implications. The philosophical implications of cosmology, because I was having a tremendous row with Simon Saunders, who is actually one of the trustees of our archive, I wouldn't have, frankly, I wouldn't have been asked him to become so if I'd known that he was going to be involved in this. They have taken, I mean, I had to ask Penrose because, to be honest, you know, having Penrose as the chair does open doors and to possibilities of funding, which I'm never going to be able to get otherwise. So one has to be, I'm afraid it's cynical, I know, but I am, and Penrose doesn't really.

1:02:30 He doesn't take any interest, he just chairs a meeting once a year, he doesn't have any sort of active role, but Simon is the secretary, and he does have an active role, and he's been pursuing quite helpfully attempts to get some funding from the British Society for the philosophy of mathematics, British Society for the history of mathematics, and then we're staying with him just after Christmas, because there's a conference about categories and physics in London. Which was absolutely disastrous. Nothing but N categories was just Tom Leinster and Eugenia Chung and all these people going through the list of I think there are now 13 different definitions of a... ...of an N-category. And they still can't agree on what the hell morphism in N-category is. The program now is that it's all connected up with homotopy theory, and they've got a kind of nice account of everything via this Ekman-Hilton mechanism, but I can't see that they've got any clear definition at all. And the only person who gave a useful talk there was Iker Murdyke. Thank you for your time. I have a suspicion, I don't know, but I have a suspicion that this is all based on a conjecture of Grotendieck which might have been wrong. Well, even the greatest people make wrong conjectures. It's not even a conjecture. Well, maybe the homotopy types can be explained by infinity groupoids. Yeah, yes, that's exactly what it seems to stem from. But it wasn't proved and they haven't proved it yet, you see, so it's not, it's probably false.

1:05:00 Because the thing is, also, Shanuel pointed this out long ago. To street. The thing is that in fact if you look at higher homotopies. Okay, they're graded by natural numbers and they're all these maps, but it's not a two category, not an N category. The axioms for an N category are not satisfied by the continuous recurrentization of paths and so on, right? And yet they always, not only the gang we're talking about, the Baez gang, but even Street. Yeah, he's already sort of making a propaganda that this is something like homotopy theory, when it's not actually. Technically it's different. And Chernuel said that he... I don't want to misquote, but he said he pointed this out to Street, and Street said, well, yes, I know, but it doesn't matter. It's something like homotopy. Yes, it's just like Kahn's response to your question about maps between, the general notion of a mapping between spaces in a non-committive setting, non-committive setting. That's right. And it's like, well, they're bimodals, and you pointed out that that's much too, much, much too restrictive. And then he said, well, there's something like that. I don't think I'm exaggerating that much. He really was dodging and weaving, and that seemed to be the same sort of dodging and weaving by, let's face it, otherwise very, very smart people going on all the time in this area. I'm sorry to say that already my friend Street was doing that something years ago. It's really new. Well, you see, some kind of vague analogy. It might guide the development of in-category theories, assuming you want to develop that for some reason, right? But to say that in-category theory applies to homotopy is not justified by this analogy. Anyway, we listened to these, as I say, now 13 and counting definitions of an in-category, and a fairly...

1:07:30 This was on the 7th of January, this was earlier this month, and Ica gave a very nice talk about operands, about operands and generally a nice talk about universal algebra also, and pointed out some of the flaws with their program, in fact. And there was quite a good discussion afterwards. So I went to that because I thought it was the kind of thing, anything with foundational implications, anything which keeps track of what is happening in the broader categorical world, I try to keep recordings for the archive. So we went there to record that. It was an extremely good conversation, a good exchange between the two of them. Van Eysteyn's students, which was far better than anything that had gone on with the kind of encounter of the people in the meeting. Stuber, Isar, Isar. He's Arstuber. He's a doctoral student. No, he's the one who invited me to Russia. Yes, that's right. Well, he's actually now in Antwerp. He's a doctor. Well, he's completed his PhD. He's a post-doctor under van Oostein, Freddy van Oostein. Right, right. Who, of course, also is in the non-commutative game, but I think at a much more serious level, mathematically. He's a good guy. Yeah, he's done some very nice work, actually. Actually, I had with him. Thank you for your attention. I was talking with Bernardini, you see, and pointing out all these things about the Templeton. He didn't quite believe, so I said, well, let's just look on the internet for FQX. You've just taken the words, because this is just what I was coming around to tell you about. Look at the list, FQX. And he's shocked. Oh, this one too, this one too. Completely shocked, did not realize at all. This, in fact, brings me back to where I had gone off, completely off track. Oh, yes, Kierkegaard. No, wait, Kierkegaard. I knew that Kierkegaard was in this. Therefore, when I saw I asked him, well, why is Kierkegaard doing this? And also, Catherine, what's her name? Both of them say that they are very, he is their mentor. Kierkegaard is their mentor. Therefore, they don't attack him.

1:10:00 Well, Kirker was first PhD supervisor. I didn't know. He is too, I think. He sounds hesitant. Well, some sort of mentor. They may just be a friend. Well, they seem a bit more than a friend. But in any case... I think Isard Stubbe is a far more powerful mathematician than Kirke. But anyway, never mind, that's not the answer. But anyway, I mean, they agree this is a problem. Yeah, it's a problem. Maybe it's just a sense of Belgian loyalty, you know, Belgians being, you know, decided they have to be cohesive in the face of the world. Yeah, or even more specific. No, I'm not being serious. No, you were going to say something about that. No, I was going to say, in fact... That's what I was going to say before I got off beam by telling you about what happened with this N category meeting. Having been to the N category meeting in Imperial, I went to Oxford to visit Simon, just to do some administrative things, to get the bank account for the archive and various other things, and to chew the fat. He told me that they had just been given, the people in the philosophy department at Oxford, had just been given this grant for this project of investigating the philosophical implications of cosmology and he had written part of the funding application and Penrose had written some of it. And it turned out this was all coming from FQX. So, of course, I attacked him over this and said, well, this is really very, very bad because this FQX is simply a front for Templeton. And it's perfectly obvious, and you of all people, because Simon has always held himself out to be a very, very consistent atheist, should realize what these people's game and agenda is. And I got a kind of series of horrified denials, saying, oh, no, no, Michael, you're getting, you know, I don't know where you get this from. FQX is not part of Templeton. They're completely separate. They're completely separate. Well, excuse me, but, you know, just look at their board of directors. Look at who is, and then work out where their money comes from. Oh, well, it's true, they were set up with a kind of foundation. They were set up with a grant from Templeton. But, of course, but they're completely separate administratively, and, you know, there's no kind of...

1:12:30 They're completely editorially independent. It's just ludicrous. I hate to say it, it's kind of all this nonsense about bourgeois liberty taken to the ultimate extreme. It's like all these people who say that, well, of course it doesn't matter that Rupert Murdoch owns the papers, you know, the editors are independent. It's just ludicrous. So he was absolutely denying. He said, I would never take a grant from Templeton. But FQX is a different matter altogether because FQX has a different... All appointed by Templeton, I said, how can you be this naïve? We ended up having quite a, I have to say, quite a flaming run. Yes, I'm afraid it is. And I ended up having quite, well, I mean he is still my friend, but having really quite strong words with him. But his objection, for instance, he said... No, no, no. I mean, I wouldn't have anything to do directly with being funded by Templeton, but I can't see what your problem is with FQX because it's completely separate and it's clear that they don't have this religious agenda. They obviously have a religious agenda, but his objective was apparently at the same time... Look at the proposals. They actually give the proposals that they have funded. $130,000 to buy us for this thing. Yes, exactly. Well, you see, Simon wouldn't see the problem with Baez. I've explained to him what the objections to n-category theory are. I mean, I don't pretend I'm the best qualified person to explain to them, but I have explained to them that, you know, there is no reason at all. We'll be able to encode the quantum mechanics, and I think it's not able to do so. I told Pettigrew rather strongly not to encourage Chris Isom or Jeremy Butterfield to come. Well, he doesn't know either of them, so I don't think that's likely. Oh, did he? Oh, gosh, I would have stopped. I would have stabbed on that if I'd known. Well, he knows them. He knows them by name. These are people who talk about physics and took them. Yeah, yeah. Well, Richard is a very, very nice guy, but he is young, and as I say, I'm not used to him, so he doesn't, you know. So if he falls under this, you see, therefore he thought to remember what I told him. I told him, you see, the point of my point is this. My plan for the Bristol... Yes, I want to understand that so I can be of use to it. Well, I mean, it's more specific than I could ever, but my basic plan is... That this should be a session for progress, actually mathematical progress. Bernardini and Koch and me and whoever, you know, wants to participate, but not to debate any philosophical nonsense, you see.

1:15:00 Not that I'm opposed to debating. I'd like to, in this case, I want to use the time and the opportunity. So if they come... I have no alternative but to attack them and then we get into... Yes, and then we just waste a lot of time. Well, we use up a lot of time. Maybe you can explain this. No, I will explain that to Richard. He's okay. I'll get that point across to Richard. There isn't a problem there. There isn't a problem in my getting that across to him. It may take longer for him to understand it than it did for you. No, no, but he's a very reasonable person and I think I'm in very good, at the moment, I'm in very good sort of... Good. Very good relations with him and with John, and I think they are very key. This is the point I've been pressing very hard, is how much there is to learn from Grassman. In fact, I've been kind of noticing the slogan that really the philosophy of mathematics has to go back to Grassman and start over, dare I say it, like Barack Obama's inaugural, you know, we have to go back, start all over again, you know, go back to the ancestors. Oh, you didn't hear the inaugural? Oh, well, I hope I won't shock you by saying that whatever one, obviously, one knows whose class interests he covens in, he's the president of the major imperialist power. But he is a good speaker, and he did actually give a rather fine, within the limitations, you know, new progressive era speech. At any rate, it was a definitive break with what we've got.