FW Lawvere

0:00 These are the rules that, according to them, have regulated the course of this study. In particular, a reading of the flow and the approach of Cauchy, de Riemann, Cefal, and Baudelaire to the question of the course. So, we are in 1918, of course, but it is certainly not indifferent that the two will have a strong return to the past, but they relate to Cauchy in a very different way. One or the other, to include the reflection of the concept of reflection in this interpretation of the passage of human history, of the contribution of Wegerstrass, etc. It gives rise to a whole set of reflections. I will not be of use to give you a list of sources, but they play a very important role in the memory. The other point that I took from the theoretical reflection on the object is, of course, the different concepts that are formed. If you have a mathematical object, you would be in the category of the well-defined, well-defined object, with the importance of putting the whole thing back in its place, the demonstrable. I'd understood the general point about figures and match faces, but I hadn't understood exactly how it worked to illuminate the point of the Euler. Yeah, well, that I've understood, but I understood how it provides the perfect fitting for explaining what... Incidentally, what was your discussion with Shanshuang, you know, the guy who gave the talk about the grotesque duality? That sounded as if you were having a very interesting conversation.

2:30 I knew I'd met him somewhere before. Yes, yes, he was familiar to me. Which one? That's a fantastic idea. I didn't actually say, let's work on it. Just, just sort it. And he stopped using a very small guy, which is a very interesting thing to say. I wish I understood his talk better, but that was my approach. You may have understood this program well enough, but I always had this problem, you see, that although Wiesel and Rosenbeek were both, in their own way, positively disposed towards the project... And so they were wildly enthusiastic about these things. But they didn't quite understand it because for both of them, it involves certain aspects of logic that they feel insecure with. And on the other hand, I am totally insecure with complex analysis, which is the main mathematical content. And so, you know, there could have been a collaboration, but neither side had quite the preparation needed. It may be that now that sort of general topology and theology has been thrown around, even in a lots of color systematic way, there may be people who have enough of a grasp of this orientation that they, to me, he might have... No, the project was to prove Grauert's theorem and vast generalizations thereof, as Pouzel and other people did, but to prove them as direct corollaries of the classical Cartier-Seer theorem over a point. By the method of proving the latter in an arbitrary topos, instead of just in sets, and then applying that to the case of sheaves over the parameter space, be it analytic, C-infinity, or more generally, as they say, found out, originally we got a proper map of analytic spaces, meaning a family of compact analytic spaces, parametrized by an analytic space, but the key point is that the fibers are compact analytic in a concrete sense, but what the family... The parameterizer of the family chart, to be quite general, well, why not make it a topos? And having made it a topos, you see that the statement of the theorem internally becomes essentially the same as that of Cartan 6. Take the direct image and start with just a single compact space and map to a point.

5:00 Push down a coherent chief for that and prove the result is a finite dimension. If it can be done in a general, arbitrary topos, or even a much more general topos than sets, maybe there might be some restrictions, then you could just immediately say, well, let's take the example of topos on some parameter, sheets on parameters, apply this theory, and then you get the data. Problems which become algebraic when you look at the machine theoretically become merely algebraic. The difference with this is that it requires some functional analysis. Now, there is functional analysis. If we can just relativize that, this is really the point. You find the internal formulation of this. The statement doesn't mention functional analysis, but the proof does, you see. All proofs involve it, including Cartesian theorem. So that has to be internalized. And there are, I already hinted at some of the methods for doing it, but just a question, could it always, I mean, certainly certain parts of it were done pretty quickly, for example, if you consider the notion of a compact topological space, well, this can be internalized now, because of the divergence of different intuitionistic correlations. You could formulate it in various ways, and some of those don't work very well, but others do work very well. So there's a good definition of compact space in a general total. And there's the theorem that if you take the sheaves on some given topological space as the ambience, and say, well, what is the compact space in the internal space? It's just a proper map. It's just a proper map viewed externally. It's just a proper map. Do you understand what I'm saying? I will be honest, I just don't know enough about complex analysis. This is what I said just now. No, no. It's just the quality part. Okay.

7:30 A proper map is one whose fibers are compact. Right, okay. This is basic content. Fibers are compact. So you see what I'm saying. So that part is true, and that's an ingredient to it. We're making the internal formulation that would be required, but the main ingredient will be some functional analysis, precisely to internalize more logical spaces. Ah, more logical nuclear spaces. Which, of course, comes back to Grotendieck's work in functional analysis. That's right. Which nuclear spaces were central. It was a great pleasure for me to tell him, you know, in his front lawn at his little hut in the southern trench. Well, topos is great, you know. Nuclear space is great, but nuclear space in a topos, this is what we really need for this purpose. If only you brought him back into science and mathematics. But again, many people, something you told me, if you try to internalize the notion of topological vector spaces, It doesn't work as well as top. Topological spaces, vector spaces don't work well at all. Anyway, compact is different. But bornology internalizes very trivially, very easily, and I had done that also 25 years ago. To internalize the bornology, internalize the compactness, bornology is merely a tool and the proof of that particular theorem. Also the notion of a complex space is pretty well known and you can internalize it. It's just a matter of putting those ingredients together based on some knowledge of what can be expected from the Heiting set theory, what can be expected from the real, it's getting at what the real concept of the theorem is, it has a formal statement, but the physical meaning of it is something of this kind. If we imagine not a static world, but one that's varying over a compact time interval, the theory of a steel chair, we should be able to apply it in the appropriate sense.

10:00 Which, of course, is why it would generalize beyond the case of a set. Yeah, that's right. Yeah, the base, the base. The base would be much richer. The base itself doesn't have to be for a godly reason, really. It's just a good thing. Yeah, but it would, I mean, but you would still have to come shoot the location. Yeah. So, John has been in your ear for several hours? Well, the main reason I let him do that was to keep him away from you, because I could see, one, you wanted to talk to him as well. And I knew perfectly well that if I didn't, if you didn't have somebody sandwich him at the end of the day and told that, he would. I totally appreciate it. If you've never ever done anything else, give it a try. Yeah, and I have to say, you do owe me a drink for that, because believe me or not, it's much, much, much worse than I'd rather. He's just a total psychotic. I'm just a bit of a psychotic. Well, you know, I've taken in what you've said. I knew it was fair. But, you know, I mean, remember, we had this meeting here a few years ago, which you actually spoke at. And then, of course, just before that, there was this incident when Paul Taylor attacked him in Amiens. And I realized that, you know, he was a... I've said that I knew something about persecution many years ago. And, you know, he felt it was very badly done by Amadou Diploma, sort of that sad figure, but I had no idea until today that Peter Johnstone was basically the president. Sorry, you mentioned in your preamble that Peter Johnstone did talk about him in his first book. I'm not going to talk about him. Talk about him? No. Why did Johnstone mention him in his first book? An attempt to avoid... This kind of thing, but we already are known to do it if he doesn't mention it. Oh, incidentally, he is the true inventor of tunnels. I didn't know that. No, no, no. You see, you guys in Halifax in 1970, you know, you and Ralph Kearney were just so particularly confused, and you had a lot of kind of half-finished proofs and half-worked-out ideas, but he sent his students over there and, you know, she came back and reported and he wrote this wonderful memoir which completely cleaned up the entire subject, which is really the first paper published in topos theory, which everybody ought to cite. It's the canonical first introductory paper on topos theory. But they gave all the credit to... Oh, and likewise, having just put this stuff about Sammy Eilenberg in here, who writes a letter to Sammy in heaven, you know, here, Sammy, wherever you are, and see what I want you to know, but then sitting next to me, he says, Eilenberg, look, Kelly, I invented the right categories, everything about the right categories is in my 12-page note in the La Jolla volume, and much more than they have in their 138-page paper, they stole all my ideas, they stole all the credits. I mean, I'm sorry, the guy is completely...

12:30 I can crack, definitely. It's absolutely pitiful. It's so sad. I apologize on behalf of category theory, because I haven't encountered the talent of these. Well, you see, he's... No, but seriously, I apologize pretty well for his disruption. And saying that he wasn't in category theory. Well, he was. All these people, they know, they know him. Yes, they know him, but I hadn't noticed, no. And I mean, I think the work on five big categories is really important in the connection with De Senti and not the greatest of them. He says, you know, he's just got enough sanity left to say, I know I'm not great in detail, you know, but, you know, I've done good in math and, but it's just this, unless, unless he's given credit in every single down the line, in every single... I'm sorry, the thing I dislike is that he's not... I used to pity him, because I thought he was really rather put-upon and had been unfairly treated, but now I see he's really terribly two-faced, because there's all this making nice to your bill, you know, this is more of a tackle to your bill, and then he says to me, well he didn't actually say it... I'm the guy who created this cleaned-up topos theory. I think they had a lot of good ideas, but they were only kind of half-formulated. And Eilenberg and Kelly stole all my work on this. I think I was completely out of the game. And this attack on Johnston is in your category. Yeah, I'm sorry, I meant to say close category. No, but you also claim that he got derived category theory as well. You know, the funny thing is that the students at this point were out there. The actual paper was written by Blainaboo and... Somebody else? Yeah, he mentioned there was Sherwood Beans with a C-H who went on to get, oh he's messed up, because in Johnston's first book he credits the guy who was the co-author of the paper with him, I'm sorry I forgot his name, it's Sherwood, I keep wanting to say, obviously not Chevalier, but it's a name like, similar, sounds similar, and he credited it, and now in the second, you know, in sketches of another book.

15:00 The work is still in there, but he doesn't have the credit. Well, if you look at the page in Sketches of an Elephant, it just says, this is a classical result. He's talking about things which have become canonical. I mean, if he thinks that Peter's done such a terrible job in Sketches of an Elephant, well, why is he left alone, you know, to do it all himself? I mean, he desperately sought for collaborators. He went over for ten years trying to find somebody who would... You know, take some of them. I'm not saying it's a perfect book. I mean, no book ever is. And, yeah, maybe it should have had a longer introduction. One is tempted to argue with John. Which I'm not. As you said, you give us an opportunity to keep him off you. I do. I hadn't realized until I had listened to him just how altruistic I was speaking. Oh, yeah. The other thing is that Ted wants to talk about it. I now understand exactly what you meant about wanting to talk to you about scientifically serious things. Let's go and get some refreshment at this reception. No, you're absolutely right. Change subject. It's too painful. Yeah, I was thinking about it. More about the duality of the present and the future. Yeah. French and German have an advantage over English. French and German have an advantage over English. Histoire can mean history or it can mean a story. And Geschichte can also be the same thing. It can be either history or history. And actually, of course, the word history in English at one time, until the 18th century, had also cracked the same.

17:30 Well, of course, I mean, there were all these things that got separated. Yeah. Last scene of all, this sad, eventful history. What is it? Demands and requests of all sorts of things got mixed up, so English, I know, wasn't always that way. I'm just saying that... An ordinary English speaker would sort of never think of the fact that story and history, I mean, I'm sure if you look in the dictionary, they still do. Well, but in many cases, the ordinary uses it. So this is the real, if you take this principle that you start with the present, with a problem in the present and go back in order to find it. Analyze the roots of the problem. Oh, I picked up immediately. Don't worry. You see, this leads to one kind of history. But the other kind is about stories. You'll go back and say, well, gee, Alexander Hamilton was such an interesting figure. I can make a movie about that. I can occupy people's minds for 1.2 hours about that, and then they can have a story about that. So it's a bedtime story. Sorry, a favorite story. Bedtime story, bedtime stories for adults, you see, on TV. And so, infotainment. So those are stories, I mean they are just, absolutely, you do occupy people's time, but also, well, certainly that, and also of course to give them a completely distorted property, to sneak in the reactionary content as well, to sneak in the distortion of the real history, hidden reactionary content, concealed reactionary content, of course, obviously that, but the form is actually... There's a dual automatic extensive picture of these.

22:30 So now these two things are paired, and this is going to induce stuff on the cohomology, differential forms, and so on. But it starts from this idea of the duality between intensive and extensive, then specializes it to a particular pair of quantities, one of each kind, which are related by this. I understood at the end of the talk that there's a kind of a strict setting in which it's equivalent to the Dirham, Perfect, did I get my line? The point is the coefficients, because you're on the just ordinary constant with the value. After you take the derivative of the cohomology, you know, then this pair is no longer merely going into this ground field, but it's going into the ground cohomology of the space you're over, with constant coefficients. Right, all right, gotcha. So it has become... The general bundle took the values back to constant coefficients, but it's really three simpletons, so to speak. What is actually a distribution? I mean, concretely, in the analytic context, that takes on a whole new meaning, different from the smooth context, more restrictive. Well, it's related to Voltaire's Unready Consciousness, of course, but I don't think these people know that.

25:00 Volterra's analytic functionals don't exist because they're not really functional analysis or something, but basically the point is, this was great in the first world, you take the linear dual space, which is Reid's idea of what the extensive quantity should be, you can do that, but you're doing it with analytic functions, not smooth functions, so analytic are very special for power series expansion, because power series expansion converges, I should say. Anyway, so in the analytic context things become sort of much more concrete. Unfortunately, it's complex analysis, so that's, to me, still very abstract, but, you know, I think you even mentioned the real case. It all has a reflection in the version in the real case, but the real case becomes mathematically more complicated than the complex one, even though from the point of view of geometric insight, it, of course, becomes the opposite. This, of course, is one of the reasons why people always build up the complex, like Penrose, always build up the complex numbers as something absolutely wonderful, as opposed to simplify everything. They must be more fundamental than real numbers because, you know, they give us... All that stuff. Machinery and all that stuff. Yeah, yeah, that's right, because the world is made of complex numbers. Oh, yeah, no, it's a pun. It's obviously intended to be a childish pun. Well, it's a puerile pun, yeah, as well. But behind it there is actually a kind of serious ideological position about the complex numbers being... Anyway, anyway, the thing is this... Which I have to say I was... The linear dual... Russell Howard is very good at that, demolishing that, because he showed how, well, okay, this twistor construction is underneath the complex number at all. Linear dual. In other words, a certain version of analytics, of distributions in the analytic sense, has a very specific sort of meaning. Namely, these really are intensive quantities, after all, but with respect to a specific isomorphism, which is not natural, but... Not categorically natural, but geometrically, extremely, it's the core of the whole thing, you see, that you get this representation of the extensive quantities as intensive quantities. But, let's say you have this compact domain inside a bigger world.

27:30 So you're looking at, let's say, intensive quantities on this. Well, intensive quantities are everywhere. All of these will be represented in terms of an intensive quantity on the complement. If you want to integrate a function on this place, you take another function which is defined completely on the complement. Now you view them both as functions on the whole thing and you take their product and then you take the Cauchy integral. You know, he drew a loop around the... Yeah, yeah, now I understand what was going on now. Now I get to the point of what was going on. So the Koshy kernel exists, you see, in the complex world, but not anywhere else. That is the... well, it's a specific integral, a specific distribution in some sense. Therefore, it implements a map from intensive to extensive. That's why the mistake, the thinking that these things are generalized functions comes from. Because if you use Lebesgue measure on Rn, that gives you a canonical way of interpreting intensive as extensive. But in a more general setting, this measure, well, it's not preserved by many important transformations. But it's modulo that, that you have this idea of generalized functions. But in the complex case, they can be conversely. Any extensive quantity on a given domain is... The intensive one on the complex, integrated with respect to this canonical measure, which is not the big measure, but it's the Cauchy integral formula, which I guess is an example of the kind of covariance and possibility that tends to show up in the relationship of intensive and extensive. All of this? Or is that a separate issue? Well, it's the idea of the theory. Oh, yeah. The theory is that you will compete with each other. Yeah. Then, given a particular element, because if you do things with a multiplicative structure, not just a linear, you can multiply functions. Even if you can't multiply distributions. Yeah. But if a distribution is a functional on functions, the functions operate on the distributions.

30:00 Right. Distribution mu, operated on by f, as a functional, integrates an arbitrary g to what mu would integrate the product fg to. So there's that formula. That's that formula. And now if we have some kind of canonical mu, then we can translate one partly into the other. But in this case the translation can be... I understand now what was going on with this. Blah, blah, blah. Well, in this case, f of z times g of w divided by z minus w. Yes, yes, z minus w. Where does z minus w live? You can't have them equal, no. But if you take one from here and one from there... And one from the complement, yeah. Yeah, yeah, yeah, yeah, yeah, yeah, yeah, ha, ha, ha, ha. Now I see what's going on. This is really what's going on here. Now I see what's going on. This is really a... Oh, isn't it? It's a wonderful feeling when you understand something at first time, especially... They never say that stuff so openly. Oh, that's a wonderful feeling when you understand something like that. Yeah, yeah, just like you do in, you know, basic calculus. In other words, the canonical measure is not, you know, integral dx as the big, it's rather an integral of blank divided by g minus w. Yeah, yeah, yeah. You know, that is the thing that's independent of any choice of functions or distribution, and therefore you can implement it. It's used to implement this. Fantastic. And now I've understood where, you know, really, you know, deeply and geometrically, where all those strange recipes that you had to learn that didn't seem to make any sense to manipulate the quantities are coming from, which is the difference. There's a little bit there, I said, it reminds me a little bit of a lot of what Saunders says in his book in Mathematics, Form, and Function about the...

32:30 About, um, about, um, oh, my mind's gone blank, um, yes, yes, we are, it's down here, the reception's in there, and by the way, I should take, I might forget, take your scarf, because you could do with it right now, I would think, otherwise I might forget it, um, about the Legendre transform, you know, he, he, when he first learned it, he learned it from, I forget, it's still a Legendre transform, you know, he's, I've gone blank, okay, well, okay, um, Oh, well, I'll dig out the passage in Mathematics, Form and Function and remind myself exactly what he does say. OK, well, there's a passage where he describes how it was only when he had to teach it the first time that he realised that it was an example of an adjoint function. And before, when he was taught it, he was just taught it in terms of the mysterious recipes which he could never get into. I mean, he obviously learnt how to manipulate them, but he never understood. Thank you very much for your time, and I look forward to seeing you again soon. We all kind of just told the other. So to me, you're trying to take me to many of the other things. You have this kind of thing, the domain space, the big domain space and the compact within it. That's really a complement of that. And so, oh, by the way, another conjecture I've never seen was written down, but it must be true. I think that if one, that you can derive the so-called Alexander duality from the, you see, the Alexander duality is about, again, extensive, intensive, complement, duality, blah, blah. I don't know about Alexander duality. Tell me about it.

35:00 But it's about cohomology and homology rather than functions and measures, you see. So the cohomology process and homology, instead of... The quantitative measure of common qualitative quality in space again has the same kind of relationship. And I'm sure that you should immediately fall out because the cohomology and the homology are merely gotten by taking the abstract derived functions of these various distributions of functions. That's what Dirac's theorem is all about. Dirac's theorem says that to start with smooth functions... If you take the Dirac complex, you can create the same cohomology as the ordinary combinatorial approach, and the dual of currents, they're called currents, the multidimensional analog of distribution, but if you take the currents, the Dirac homology should say that if you take the currents, then it will give rise again to the ordinary combinatorial of the mind, the way of representing the combinatorics in terms of the speed. Because, of course, hope technicality washes out all kinds of distinctions anyway, but it does retain the quality of the combinatorics, but forgets about whether it's continuous or smooth function and measures or currents that you're using. So I'm sure as much as it's usually stated in psychology, it's something independently of technology. You look at the homology of the subspace and then you ask about what is the cohomology of the concomitant space to see the specific role of the quotient cardinal. Giving rise to Zanderval would be a beautifully good connection between functional analysis and topology. Now, I didn't actually catch the name of this just down here. This is the École Normale.

37:30 They said it was next door to the Salle des Sables, and that's on the ground floor. I'm sure we'll see them when we get in there. Just through the door there. In fact, it looks like we can go in without having to go through the lodge, but actually, no, we do, because the gates are closed. This is where they said they were able to do very well with the security, but it's okay, they'll know who you are. We are at the reception for the John A. Husserl, large space. They are. They don't normally ask you anything going in here. It's through the main door. In fact, it's that same room where you spoke in 2000 at the, you know, when you gave the foundations as a, yes, in the foundations of the Protagons. No, it's the room next door to it, I understand. I don't think they're actually giving us dinner. I think they're just giving us a kind of wine and cheese, no nibbles reception. Thank you for your attention.

40:00 Well, they're probably giving a version of Truesdell's opinion, actually, and I'm sure they're quite familiar because he openly published it. It might have been diplomatic just to say that, you know, there are one or two people among historians of philosophy and mathematics who provide exceptions to the general truth. Well, that's what I said. You did say that. That's my whole point. I agree. Typically they happen to be people who have done work in the field themselves, up to the extent that there is an outstanding job. No, you made that point. I don't think you have anything to add. I didn't understand the last guy. Oh, I wasn't supposed to listen to you. I think he was just giving a kind of A-level talk about a very vague interaction. Well, he just said stuff about philosophy and epistemology and teleology, I thought at one point that he might have been talking about teleology, that he might have meant me. No, I mean not me in a sense, but what I was saying about the origin of ideas is collectively become explicit by an individual. Basic actions of the original reactionary school of history and science. There's the objective idealist historians who say they're the platonic ideas that you have to find, or whatever, where the subjective idealists say what they could consider the opposite, maybe if there is no such thing as ideas until you publish them, you see. And of course that's precisely John's point of view, too. In fact, it's spontaneously produced by individual minds and nothing's completely ignoring the fact that those minds exist in our society. In the context of a society with social relations and productive relations, and yes, yes, yes, I agree. They're exactly... I thought when you said genealogy... I heard you say that, but to be honest, it really came off as if he was just going to give a very expository talk about Burrell. ...measure of obeying integration, like your gift to a group of professors, to an audience of philosophers, that didn't know the history of mathematics. I couldn't... I'd be so interested in what the previous guy, in a charge one, had said about the Great Big Galaxy, I'm so anxious to understand it. I have to be honest, I didn't pay as much attention to the last book as I should have done. No, I think this is... No, no. No, it's not. It must be down here. Oh.

42:30 Well, let's just try to take this as where I am, you know, by the medium, it's supposed to be a normalian, but among humans, um, I don't quite understand that. I thought you said that it was next door to the sound of the sound. What's that? Oh, it's easy. Basically. You saw the... I don't know. Great, great, great, good. No, we didn't, which is why we were a little bit worried we might not have come to the right place, but we know we are now, we see you. The thing is that they're sort of late setting up the... Ah, that's why I was going to say they're late setting up, no problem. No problem. Okay. Ah, that's great. Actually, while they're doing that, I'm just going to run and use the loop. Don't go and leave your scarf behind this time. No, I must be wrong. No, don't worry. I didn't realize that. Yeah, he was a bit destroyed. He and his wife both had some summer home and were driving home. Driving in one direction or the other. They had to go over a bridge that had no... And they went into the water and were trapped and drowned. Terrible.

45:00 He was very unmatched, wasn't he? Yeah, he was. His thesis has just been republished. I mean, not electronically actually republished, but I forget which publisher it is that does, you know, reprints of, I think, Karmakar and one of the blue paperbacks. Well, I've learned more categorically. I certainly intend to buy a copy and study it. Well, I don't think they're called, but it's, um... It's usually just, as you say, a referee word. Certainly homology. And, um... I'd like to guess he was an O'Malley under him. FOMO paper. FOMO 90. Yeah, yeah. I remember. Well, I gave a talk there with him later. So I start off by saying, 25 years ago, one session was on the beach, before we set up on the beach, sitting around and bathing. That's absolutely fascinating, because I completely misremembered that story. I thought it was great, and I thought it was telling you about it. That's important, because it's so important to get the history right. I had heard the word popos before, and this was in the gay's apartment. I knew the word, but he didn't tell me what it was. He attributed it to Giroux rather than to Stephen I.

47:30 It was May of 1963. Anyway, then on the beach of La Jolla, he explained to us. I remember at that time, I liked it, but I had a negative reaction. I was sort of temporarily a logician. Well, what do you mean it's set? It's still based on set theory. How can you say it's set? Because of the formulation. The new topos, though of course typically fresh, he plumbed the chalk into the sand and walked off. These people didn't take this as seriously. But he gave a hint, and I quickly realized that... They were actually doing what I said in 64 should be done and it was therefore, one just had to massage the actions a little bit, they no longer depend on set theory, for example, topos. So I quickly recovered from this mistake. I missed out on having had a much more productive discussion. I wish I had had more. And actually I never had much of it. I saw Verdi a day later in Overwolf Park and Columbia University. I had a few discussions with him. I was at the IETS. He actually invited me to dinner at his house, but I didn't go. My stay at the IETS was largely a disaster. I didn't feel like doing anything. I didn't feel like going to... I may have had an actual doctor's appointment or something, but in any way I had to say, again, typically, oh, okay, that's how you feel about it.

50:00 Didn't bother to be preserved. No more, he didn't say, come another day or something like this. Yeah, yeah, yeah. So I missed out really on that. Which is a great thing, because clearly he was refused to make it 25 years after he was dead. When, um, when did you first have, when did you first have, well, why the meeting or perhaps the serious mathematical discussions with him? You must have seen it all at IHP. At IHP, that was earlier. No, that was in 64. Ah, okay, I didn't know that you were at IHP in 64. I wasn't. I was in Zurich. Ah, but you came up there. But Tony said, there's a Burbaki meeting in Paris. Let's go to Paris. Right, makes sense. So we took the train from Zurich to Paris, and I attended some meetings, and I was already shocked, yeah, that the French treated each other. That's what makes Tarkio so marvellous. I know. It's so unlighterous. I know. I know. So Peter Boo is just another example. It's extreme, but I don't think it's something completely out of the French character. I know exactly what you mean. I'm sorry. Tarkio's the only French mathematician who's got any of that kind. Thank you for your time, and I look forward to seeing you again soon. But the first impression they give, effectively, is that they're pretty. So you met Kroger to get 64 years old? I didn't meet him. It was a huge, huge thing. He was just a pupil from Zurich, and I remember when he entered, he was the object of adulation, and he managed to resist. And he told me later the story,

52:30 what's wrong with your name, Tom? He believed the propaganda that he's a genius, and if he could give that analysis, he knew very well, he had experienced it himself, so this is what I saw, he lectured about the Brouwer group, that's what he lectured about, sorry, and then I met him at, how do you call it, the international... I attended this session and I stood up and I denounced everything he said. You know, Mallet style. I think everything I said was true. And later, on another occasion, he said that at Nice I had been the main contradictor. The principal contradictor? Yeah, the main contradictor. The only one he deposed. The others, they didn't like all this, but, you know, because they all saw him as an individual. I'm not sure who was behaving this way because of some very strange ideas floating around, but I saw it as a Marxist-Leninist that, well, these strange ideas are actually profoundly reactionary, and they're not his ideas, he's getting them from the imperialist culture and so on. I did stand out in that way, and we didn't discuss mathematics. I was naive enough to go up to him after, you know, discuss mathematics with him, technically meet and discuss mathematics, and he said, well, actually I'm busy. It wasn't necessarily a response against what I'd done, but I did meet a student, Giraud, to Halifax for a semester, and Grotendieck did visit Buffalo in 1973.

55:00 But you were in Italy at that time. Well, I was in Italy, but I was actually visiting Montreal, and again, a small fleet visiting Montreal decided, let's meet at Grotendieck. Buffalo, let's go down to Buffalo, yeah. So then I did have some very interesting mathematical discussions with them. Oh, you did? So I wasn't going to be... That occasion was in 1973. Did you actually talk... Now, at Oldham in 1972, I did not overlap with them. I went there because I was living in Denmark. My God, this was a Trotskyite meeting. You know, the literature, the very literature you're passing out is obviously Trotskyite literature. What are you trying to do here? By that time... But when you had the serious mathematical conversations with him in Buffalo in 1973, did you talk about this astonishing program that he outlined in the diagram that you told me about that he led with Jack Kruskin, this bypassing logic, buying rings via ring classifiers? I didn't discuss that. I think it was on that exhibit that he... It must have been fascinating, your first, you know, series of conversations with him. Yeah, so much later. From what I gather, he really opened up to you. I know Cartier said to me that he has an enormous respect for you.

57:30 I mean, the Jew was almost the only person he would still talk to after he tried to cut himself off from it. Yeah, completely. Ah, well, we're a bit late. The party is almost finished. Let me get you something to... I need a sailor. Ah, good. I don't understand. Yeah. Ah, oui, alas, nous en avons parlé tout à l'heure. Nous parlons en garçon. Thank you very much, thank you very much, thank you very much, thank you very much, thank you very much, thank you very much, thank you very much, thank you very much. Thank you very much for your attention and I hope to see you again soon. Thank you for your attention. I think he's interested in the physical category.

1:00:00 This amazing document I was telling you about, which I hadn't heard of until now. Ah, Manipeet Nersi, we're not. Nersi, we're not. Very interesting, in fact, it's a history. I just talked a bit of a disaster this afternoon because of the microphone and the... I can't... I know, I knew. I'm very sorry about that, it's a great pity. I actually thought that was much clearer and more accessible than the talk we did yesterday. The motivation, yeah, much, much clearer. But it's just so disastrous. And while he's got a heavy head cold, he'll speak somewhere anyway. He needs a proper microphone. Quite pretty. But they were all good thoughts. I thought that the kind of spoke about curricularity, that was extremely interesting. Bill just spent 20 minutes explaining to me what it was all about. I think I may have understood a little bit more afterwards than I did at the time, because of what was going on. It's fascinating. No, it was good. It was a very interesting day. The only one I was a bit disappointed in was the one I got from one of my lovely friends. Thank you for your attention.

1:02:30 There's another guy right there. Well, he's done wonderful work. He developed a long period of my material. I'm going to connect it to sense theory. He had a very strong and rich line about him. ...how one should base his theory on theory of vibration, and in some ways it's really parallel to all of his projects, which you can see him embedding set theory and algebraic geometry. But he was always very extremely thin-skinned, and in fact he was a constant. This is probably the background. There are the parents who did it. Ah, yes, yes. They were the parents who did it. They were the parents who did it. They were the parents who did it. They were the parents who did it. They were the parents who did it. They were the parents who did it. They were the parents who did it. They were very curious. And so on and so forth. There have been many examples like Graf, and in other countries as well, but he wasn't particularly notable at that time. But he is a very, very extreme example. And now he's just saying that John's the best. Well, I actually think Graf is a dangerous character. These are really not things that one says about colleagues in print, but the sad thing is that there are some substantive differences between what he makes of Johnson's preface, of some of his terminology, and some of the things that he says, for instance, about Coates and Scott, he doesn't know exactly what he's doing. There are a number of categories which I think are very well justified and things that he says about, you know, the unjustness of the neglect of a fiber-categorist person, I think a lot of what he says is mathematically justified and if he could just simply make it as a straightforward contribution to the debate, he would be listened to with respect.

1:05:00 He can't do this. He can at last only do it by launching, I mean, not just ad hominem attacks, but just incredible, just incredible personal abuse to the level of raving psychosis. He actually told me at the very end, I sat with him after he came away from the film, he was confident that he could get through the film and spend the whole day reciting his list of paranoid complaints, so I kind of sandwiched him at the end of the bench a bunch of times and poked up all the way through. And one of the more common terms here is algebraic theory, where you have the idea of a variable, where you have a variable, where you have a variable, where you have a variable, where you have a variable, where you have a variable, where you have a variable, where you have a variable. I produced this paper, which is actually cited in Johnston's, first of all, if you go to Johnston's, it's actually the second. In which I completely agree that it's the first really major, you know, it's the first major paper in propaganda, it's the first general survey that made everything clear in the state of our time, and I can give you the credit, Johnston was completely suppressed on knowledge of mathematics, and it's actually the second item of Johnston's philosophy in his first book. But, you know, in the drift of that kind of psychosis, I mean, unless Benjamin Moore was cited in every single paragraph of Johnson's book as the inventor of almost every single major concept of topology theory or derived categories, he would, and even then, he would consider that he was being denied credit, and even then he'd probably still find a way of being in a paragraph saying they were all out to destroy him. Socking up a lot of...