Leibniz' Archimedean infinitesimals (last part) / discussion / closing ceremony / evening conversations (contd.)

0:00 Which is very loosely speaking to say whether they exhibit co-variant or contra-variant functional reality with respect to their variation relative to the domain over which they vary. And within this construction, the case of points just falls out as a very special particular case. You can represent points just as the case where you have a global top and bottom element of the lattice. I don't know if there's anything conceptually wrong with it, but it's simply less fundamental than the picture in terms of inclusion between parts. In other words, Boole and Schroeder essentially got interrupted. Well, inclusion between parts, essentially inclusion between parts of the domain. ...over which the variable is varying. But this brings back in the intuitive idea that the variable really does deal with the smooth parameterization of continuously varying quantities, and not as the picture of foundations post-arithmetization of analysis, post-Dedekind, post-Candor has taught, to do with things which are... Point-like. And this is the limiting case, which is naturally forced into place within this wider geometric picture. I mean, that's what I find is so amazing about his vision. The way that he engages with these really deep conceptual issues in Foundations of Mathematics is quite extraordinary. Actually, he's not the founder or the inventor of PEE. No, no, no, no, no. Shees were around in the 1960s. Shees would have... Well, in fact, Shees were around even a little earlier in the work of Luray, and Luray, and Sayre, and other people. But Grotendieck certainly did a huge amount of development of the machinery. Grotendieck used sheaths, he didn't actually invent sheaths, there were at least three different approaches to sheaths in the late 40s and early 50s, of which Le Rays was the most influential. But what Grotendieck was... There are two that allow for a lot of different perspectives and views in our axiometry. I have a question about that. Well, he realized, I mean, God won't correct me if I was absolutely incredible, but his use of sheath theory, he enormously deepened and generalized the broadened sheath theory. So that whereas when he came into the subject, sheaths were defined as sheaths over a site with properties which were very, which made them very similar to an algebraic variety.

2:30 There's a lot of dependence on the set theoretic machinery, and particularly on the opens of the underlying space, so they were basically sheaves over sets of open neighbourhoods and made the whole construction vastly more general and powerful, and it almost kind of stood it on its head and broadened it so much that the case of sheaves that could be represented as sheaves of open neighbourhoods just became one very special restricted case. We have a much more general application of the definition of geometry. Which partly rested on his definition of scheme, which were a huge sweeping generalization of algebraic varieties, which in turn, of course, brought him to topos theory, which he, I mean, the original definition of a topos was the, was in the setting of the Grotendieck program in algebraic geometry for providing the machinery to solve the Weil conjecture, the Weil conjecture, André Weil conjectured these, the, this was probably the most influential single program in late 20th century mathematics. Andre Weil conjectured... Andre Weil, Andre Weil, Andre Weil, W-E-I-L, not Herman Weil, W-E-I-L, the brother of Simonides. Andre Weil conjectured... Oh, I'm trying to remember the exact statement of the Weil conjectures. It was to do with the... The number of points and curves. That's right. Basically, it's a kind of huge generalization of the idea of a Riemann surface. And the points that were defined over the complex field in the Griemann Code, there's a huge, huge generalization and deepening of that idea. And he conjectured that this was an incredibly powerful conjecture. In fact, he had to be a mathematician and a genius even just to conjecture it, to see the conjecture, that there was this condition which would connect with these properties in number theory, which might even, although that was looking even further ahead, give one a route to the proof of Griemann hypothesis. In fact, that was at the back of his mind. But on the way, there were these conjectures which had to be proved. They were very powerful. Nobody at the time that he formulated them could begin to see how to set about proving them. Grosendieck, in the course of ten years... He created all the concepts and schemas that were needed proof. He didn't prove them himself, partly because he never was particularly interested in dotting i's and cropping t's and proving things, you know, it was left to Deligny, who was his pupil, to prove them, and in fact by a method which Grotendieck detested because he thought it lacked the kind of generality of his approach. Deligny got in by a back...

5:00 A back way, it was an incredibly, he got killed medals for it, of course, but Grosendieck wanted to do it in much greater generality, because he thought he would actually get the Fermat theorem as well as the corollary, which in fact, ha, still, well, it may yet happen, because there is a much more general way of doing it, which people do, through the Thevich cohomology. And then along the way he created scheme theories, he created the Thevich cohomology, he created almost all of the machinery of modern... He did his greatest work in algebraic geometry. But for the first eight years of his career, he was working in functional analysis. He did incredible work in functional analysis. He created a whole series of nuclear spaces which was a huge banner. ...algebra, completely subsumed the theory of daughters, but physicists haven't even begun to appreciate the depth and richness of this theory. All of them are chain arrays. You find me a physicist who has heard of the theory of Newton's phasers, but it just subsumes the whole of mathematical mathematics. I'd like to say, although I've learned this from you, I'd like to say I'm still what Bill's view on the foundations of physics is, but it certainly ties up with his view. The theory of Hilbert space is just a tiny fragment of this much more general and deeper theory of nuclear presence, which is what we should be using to understand the structure of quantum physics. Was there any interaction between L'Ovile and Rotten? Certainly there was. They knew and they only met for the first time, actually Bill was telling me this last week, they met earlier than I thought, they actually met for the first time in 1964. 1964? When Rotten-Deak was at the... He went to Paris for the day on the train with a friend of his. He didn't really communicate or interact with him much at the time, but then later, in 1968, he politically very much made a common cause.

7:30 The other great thing was that he was probably extreme, radical and panicking. But they were very much involved in kind of common political trust and they met again in 1968 and that was when they first really thought mathematics and then in 1971 at the Oregon meeting and then from 1973 onwards which was actually the point at which Grotendie could have already begun to detach himself more and more from mathematics and more and more retreat into his... The first time he went out was after he resigned from the IHES in 1973, but he didn't go out on mathematics day because he went to the United States, he went to Buffalo for six months when Fred Rothstein talked to Louvier. And Louvier just told me this the other day, he left this incredible memoir. Thank you for watching. More or less what this memoir contains, it contains this incredible program, visionary program, for showing how all of the, all theories of mathematics, I say all theories, but anyway, a huge list of theories, which are arranged in this document by going through, in the form of circular chart, what kind of intersecting circular chart with kind of spokes running across it and, and it just consists of this list of about 200. Thank you very much. Each of these theories has a classifying ring, and the idea is that there should just be one single classifying topple for all of these ground rings in the whole theory, so that everything that can be said at the level of conditions on the sub-objects in any theory can actually be said directly in terms of limits and colons, in terms of the relationship of the ground rings of the different theories within this single classifying topple. For example, you have abolished or at any rate completely bypassed logic. You are actually doing everything that you can do because all logical notions, relations, quantifiers, predicates are expressed in terms of conditions on sub-objects, either in a topos in terms of the structure of the sub-objects that represent them, but more generally in any category in terms of conditions on sub-objects. All of this, if Grotendieck is right, is this kind of visionary idea.

10:00 The complexity laid out in this memoir can be completely bypassed if you have the right approach to classifying rings. Yet there's huge speculation which is that there is one kind of, one single ring classifier that will unite all of these stuff. No, no, no, and Bill is trying to persuade Duncan that he has a moral right to break his promise to Grotenbeek and to publish it at least on the Grotenbeek Circle website. And Bill has said that he would be willing to write a 30-40 page introductory commentary on this, because it's very unlikely that Grotenbeek will ever now come back into the process. Yes, I agree, I agree, I have great respect for him. Having said that, Cartier told us, while we were in Cartier, told us, while we were in Hawking, I hate you. Why? Well, I'm sorry. Well, I'm telling you everything that happened as well as I can. Yeah, okay, I agree, and I like your comment. And I did record it. I did record almost everything. I can't do more. And see Cartier's report. Well, Cartier was embarrassed because Cartier just lost his wife, as you know, so he was not very proud. But he told us this incredible news that after a complete silence... In the course of eight years, Grotendieck has communicated with nobody at all for the last eight years. Two months ago, he got in touch with the IHS, and since then he has sent them twelve letters. The answer to this is that the two, you know, which he of course was the head of, he was the first, he wrote. He wrote to them and he has since written 12 letters in the last three months. He wrote two months ago to the librarian asking them to send him a copy of Newton's Principia and then he has written about 10 or 11 further letters and Cartier is hoping that he will be, well immediately they

12:30 When you've got this book in your office, the director tells these librarians to go out and buy, you know, a new copy of the Principia from the academic bookshop and they would courier, they would send it by courier, you know, down to where Greg Dick is in his little kind of hideaway in the Pyrenees, which they did, and then there was kind of silence for about... ...a week or ten days, and then they got another letter, and anyway since then he's written about 10 or 11 letters, the last one of which, or the last but one, was actually written directly to Cartier, and Cartier thinks there's just a chance, but the trouble is it's absolutely borderline because he's still mentally in a very, very strange state, he says in these letters that he expects to die quite soon, well he's nearly 80 next year, and there's a lot of spiritual stuff in them about how I... I want to make my piece with all of you because I'm surely going to leave this world of corruption and filth and go into the world of lie and all that sort of thing. Yeah, well it seems to be, it seems to be from the left. No, of course, I mean, Carter takes it utterly seriously. More than anybody's likely to, considering how close they were. No, it's not because you just have to read it. No, it's not because you just have to read it. Of course not. Well, exactly, because in the last letter for Juan de Cartier, he says, you know, after, he said, I think I can, I think I can explain physics to you if you will only tell me what a meter is. If you will only tell me what a meter is. Now, how you interpret that is anybody's question. I mean, I think that could be a metaphor of some kind. He writes to Cartier, I will tell you what I think is the truth about physics, if you can just tell me what I mean. I don't know what that means, of course. But Cartier may have some idea. And what they're hoping is that he would be prepared, because next year...

15:00 It's the 50th anniversary of the IAGS, of which he was the first head. So, of course, they're having a big, you know, programme. And what they're really hoping is that people might agree to come back to the IAGS. I think it's a long shot. It's a long shot. I don't even think it's a good idea. Well, it's difficult to tell. I think it would be more... Sorry, that's my impression, my impression, because... It would be much better. Well, that's not Cartier's idea. That's Gorgi Gnus. I think it would be much better for that to go down to him. Yes, and not distorting the social environment, but not distorting or even throwing it out of the picture. No, but the only thing is that Cartier has narrowly, I mean for the last 29 years, that it would be, well certainly all that time he has thought that it would be a great, great mistake to turn up and distort him. And he has resisted the temptation to do that, because he's one of only about three people who knew exactly where he was. No, no, I agree, but I rather than trying to bring him... I think that's a very bad idea, I agree with you. But you could organize a small meeting or something like that. Who knows? It's still very early to tell. And it is still very... In any case, if there would ever be a meeting, let me know. The one thing you can't do is put him in front of an audience of 2,000 people. If I understand what you said before, is this in relation to the program called... Well, in that case, you must listen to the recordings I made, one of Jean-Pierre Marcmas' talk in Madrid, which is about the science of this topic, and two, the much longer and fuller... This is the talk that he gave in Boston, which of course was discussed afterwards with Cartier and Kahn and Louvert. So I think this is an amazing program. Yes, very bright guy. But my god, the more that I see the depth of these ideas in Cartier theory, the more I realize the idea of the ultimately geometrical roots of fundamental theological constructions, just how far reaching that idea is, and how it shifts the entire... Yeah, it's the opposite way to stop and handle them. Of course. They think that geometry is an exercise of classical logic. Of course. And they're wrong. Classical logic is the proof.

17:30 Well, when one takes geometry at the depth of the concepts we're talking about here. When one takes... One's dealing with a structuralized notion of geometry. A sui generis, a structuralized notion of geometry. But not one that rests on set theory. That's the mistake they make. It is. I would say that it will become clearer and clearer as the 21st century goes on. The geometry in this extended and deepened sense actually is a huge set theory. It's a force into place, it's a fragment within that. Is that true in the Zerlitz-Kreisel, the von Neumann-Wendt-Kreisel? I mean, is that actually what I'm trying to do right now? It's a very small discussion, but I think it's a little bit controversial. You think Canthor in terms of value of time is actually a number, but Canthor in space is actually a 15 meter cube of mass-measure. Yes, I agree, I agree. A micro-halide is a micro-halide. Yeah, I know, you know that very well. And actually, I think one of the things that Bill has written about, about Canthor, about Canthor's intuitions, that cohesion, how very geometrical his construction was. Canthor had a very... It's extremely interesting, you've got this idea from Steyer. You actually, now I realize, you say that you could read counter-college theorem without thinking about sets, but that's true. No, you can't. Well, that's the Lester-Thomas theorem, but that's precisely the Lester-Thomas theorem. Precisely. The whole power set construction is essentially a geometrical construction. ...matched in the category. That's a nice idea. Now, that becomes clear. And one can understand it in terms of the behavior of coverings. Well, the covering space is relative to the fundamental group of an underlying space. It doesn't have to be thought of in terms of iteration. In fact, the cumulative hierarchy is, well, it's perfectly useful heuristic for the, you know, for the Zamello or the von Neumann construction, but it's not the only way of thinking about set theory.

20:00 No, and it was not conceived as a heuristic. It was conceived as a solution to the pattern. Yes, yes, yes, which Michael Halley makes very clear. And that is the thing to say about the... Oh, yes, well, that's the disadvantage. It does carry very heavily, you know, heavy weights down to the bottom. Above all, it carries the message that, first of all, that membership chains are absolute and global, and that membership is a more fundamental notion than inclusion, which is exactly the wrong way around, I think. One thing I've learned from thinking about topos theory is just how deep the algebraic tradition in logic went. And really, I think the main tradition in logic actually ran Schroeder and Bloch. The idea of an ontologically interpreted grand logic ala Frege-Russell was the lateral channel, the kind of side channel, but when Frege wrote these people it was their mere algebraists who had what ultimately turned out to be the deepest ideas. Because they were the ones which connected with the geometrical intuition, which we needed functoriality to understand. Would you agree, this is a bit slogan, but would you agree that, like, geometry stands for logic in the analysis that you made now? Absolutely. Inclusion stands for members of the nation? Yes. Absolutely. That's my viewpoint. But I must send you the paper that Morvier wrote about Kant. It's absolutely great. Actually, I should send you what I promised to send to Karine as well. I have this finished, actually. It's finished, the dialogue between Kantor and Leibniz. Oh, yes. I thought I'd really understand it. Everyone I've shown it to has very much liked it, but I wanted to make it into a book to do a day two and so on, but I think I should publish this anyway. I've been sitting on it for years, actually. Did you make a nice setting for the discussion? It's called Leibniz and Cantor's Paradise, and the surfeit therein.

22:30 And the complete is that I just happened to overhear them talking in the afterlife, and I wasted my time. Very good. An excellent choice. I completely agree. Nothing I said contradicts that. I agree with you. I like that. You put it together. Sorry, I missed the... This idea that there is a paradise with a tree and a place. Oh, right, right. And the tree is the only thing it needs. Yeah. This is a very common form of mathematics, but only in the Bible tradition, which means that you said the first case, you take it back and you take it with you. Oh yeah, yeah. There's this whole heresy with... And the crowd priesthood is a big cue for Barrett and McKenzie, it's probably the one I found. And it's interesting because the biblical paradise and the non-paradise are the two things, the tree of life and the tree of knowledge. But when you teach from the pool of knowledge, you lose. So it's part of the culture, we spend a lot of time agonizing over whether we personally believe or don't believe in everyone. Socially constructed Catholic cultural identity is not to be called Catholic social identity. I understand what she means, but the point is, I am not a Catholic, especially in my own opinion. He's a very good friend of mine. And he's actually an ordained priest. But, you know, he knew my beliefs and I'm always very outspoken and so for a while he became an atheist and then he said, no, Rick, I've actually, I've become quite convinced now and it's the ontological argument. He convinced me that atheism is true and false. Well, if you want to call what I just said theism, but I don't see how it might be. Well, except that it has no connection whatever with the connection, the idea that that, the being of which you're saying, has the attributes of personality, agency, or any of the other attributes of the Christian God.

25:00 I can make that up. And I think I have a completely structuralist reading of that, to do with levels of... Anyway, let's carry on. Oh, sorry, sorry, carry on. We've cut her out now. So I'm not just a... No, no! I don't know. I accept a structuralist version of mathematics. No, but I cannot make clearer because of this discussion what I meant when I said I cannot be an atheist because I'm not a believer. For me atheism is just the dual of atheism, of believing in God. And the mental, I am not able to make the mental active. Why not? I mean, I do not believe that I exist. I exist. You know why? Sorry, say that again. I do not believe I exist. I know I exist. You know why? I, me. Wait, stop, stop, stop. Because if I say I do not exist, simply by making the utterance, utterance, I refute it already. Of course. If God's existence is more doubtful than my own existence, if it asks the mental act to believe in this or that of its existence, then I actually say I doubt this or that of its existence, and it doesn't interest me. It has no relevance for me. Well, there must be things, the existence of which you doubt, the issue of the existence of which interests you. That can't be a general statement about... I understand the concept of God. If I should have this... I don't. And as I say, he certainly reads carefully. I'm not sure how far he thinks he reads. I'm certainly convinced by the general viewpoint that the girdle was clearly very influenced in his use of reflection principles by his reading of the monadology.

37:30 And I thought, well, maybe I'm just naming the head of the story. I sent in the first one.

40:00 Yeah, well, that's the board. If we go up there, I can get us back from there to the tree that leads to the Grand Platte. So I think we want to go one block further and then turn towards the Grand Platte. We might be going two sides of a triangle, but we'll get there anyway. How are you getting back from... Oh no, you're staying here, aren't you, for a day or two? Tomorrow I'm going to Delft. To Delft? I had intended to try and grab Michel Serfati before he disappeared and crudely... Oh, he's a very good guy. He's a very fine scholar. It's slightly too fast for me. My French comes along. Yeah, he's very insecure outside French. Unusual in that respect, the multiple philosophies of mathematics, but he is. But he's very, very smart. And he's written this lovely book about the... The rise of, well, basically about the centrality of symbolism and the development of symbolism in... I couldn't quite put it together into a whole. I couldn't understand all the bits and pieces. No, no. I didn't quite understand the general thesis about... I didn't understand the general thesis. No, he is a little bit like that. He dodges and weaves. You only get to the punchline in the final paragraph. You're never quite sure whether he has a completely, you know, symbol-generated meaning. Rather, so I can never get what the term is, you know, Percy and Kind. What's the, I don't know, you as well. I'm always, you know, finding designs. Oh, whether he, I mean, not that I dismiss that, because there are some very interesting insights coming from that. Well, yes, I think he would do himself a favor if he was more correct. On the other hand, he gave a beautiful talk about the role of Marshall Stone, you know, Marshall Stone's work on

42:30 Algebra, and particularly on duality principles in algebra, in the rise of, well, in the crystallization of several ideas in category theory, including the idea of the algebraic function theorem, which I think was a beautiful piece of work. And he's very, very, well, I know, I'm an Englishman, therefore I'm an anarchist, so he's paying attention to his bloody red or green light. I remember the first time we went to a conference in Germany when I was 19. I can still see the face of the, you know, the, I hate to say it, but the Himmler's granny and her sister-in-law that were standing, this was in, you know, in 1974, they were standing about seven o'clock in the morning when I started a completely empty street and there was no, there was a tram like, you know, about a mile down the street, no other traffic at all, and I walked straight across and I can still see their faces with their jaws dropping so... And I understood Lenin's remark about in Germany there will never be revolutions because in Germany the police do not commit revolutions. No, sorry, better that. Because in Germany the police do not issue permits for revolutions. No, no, no, that was it. Well, just to say that Michel is quite a very interesting guy. And he's written a very interesting book on the, what do I call it, the general algorithmic. Access, in terms of the development of symbolism and symbolic manipulation, particularly to do with understanding of exponentiation and other operations that took place between the Eta and Newton and Leibniz, centred very much on Descartes. How symbolism was free to fly and generate meanings. In the mathematics of the 17th century. I mean, it's not the whole story, but it's a very important part of the story, which was largely neglected in the British tradition. He was a smart and interesting guy, with a very great range of reference. I invited him to talk about... There was a conference on the impact of categories in Paris in 2005. He gave a very good talk indeed. But he's very... Because he's never been at home in English, he's always a little bit clenched up when it comes to international meetings.

45:00 It certainly didn't show, I can assure you. There was absolutely no evidence of it. But I was hoping to bum a lift with him back to Paris tomorrow, and now I have to try and track to the... Yeah, yeah, he did. But he left early. I suspect I shall be able to find him back. They don't have to buy the bulletin. Well, I can, but in terms of it, it will clean me up again. The train from Brussels to Paris is horribly expensive, the TALIS. It's much more expensive than the trains within France. If it was the same distance on the TGV within France, it would have been about 39 euros, but in fact it's about 78 more expensive. Anywhere else you would, because it is high. It's okay, we're just up here on the left. We're one block up here. We've slightly gone round the houses, but even so, we get there. I think that's absolutely right. I wish I'd possibly be able to tell you what I think the foundation is tonight, but I have a very clear intuition that that's correct and that, you know, if I was stone-cold sober and talking to the right people, I could... Let's begin to articulate, draw up an account of what the connection is. Hang on, this is wrong, wait a minute, we're on the other side, yes, yes, yes, we're on the wrong side of the square. You're absolutely right. Well actually, I think it's actually down that way. No, it is down that way. Well, you can go either side, it doesn't make any difference if you come around to it. Yeah, yeah, exactly. That's right, you just go straight down there and there's a second on the right.

47:30 There's hardly anything in it. Well, I'm glad I'm still on my feet, because that was a fairly serious pounding. And I have actually had three beers in the bath before arriving, so... Yeah, yeah, so I think I shall try and... I could go cold turkey for at least a few days when I get back from here. Well, it does sound a little bit like Sokol, doesn't it? It does sound a little bit like intellectual imposture. Oh man, if anybody can do it and still come out smelling roses, I'm sure it will be you. Quantum gravity has become a horrible buzzword. On the other hand, it's a very serious subject with, although it's still very speculative, there are a lot of very, very smart people who, not the only smart fellow in the field, although he is a very smart fellow, I agree, but there are people like Crane and Byers and others who are also very smart. But the fact that they're all... Of course, yes, you can be I. Well, I've recorded a hell of a lot of stuff with Smollett in the last few years. He's very much known what he wants to know. He's very selective in that way, fair enough. He's an interesting guy. I've known him for about 20 years and have recorded an enormous amount of stuff of his, both in conversation and in discussion, which I think is really quite useful. Thank you for your attention. He's a pretty aggressive old bugger, isn't he? Yeah, he is. My God, don't get the wrong side of him in a minute. No, that's the other thing. Oh my God, yes, I've been, well, I haven't personally, but I've been there with other people, including Penrose, who's been on the receiving end of this tongue-lashing column.

50:00 He can be a pretty ornery old bugger. Well, I have to say, I'm halfway through the woods. I suspect we're not anything like halfway through the woods. I suspect we're about 10 or 15 centimeters away in the woods. It's not entirely clear that we even need a theory of modern... I mean, it's still... I think it's still on the cards, although this would be an outside shot. The whole thing might be turned around, and the first principles of a completely re-articulated understanding of modern theory might actually turn... The general perception of quantum gravity is that it must come from, this is our hotel, I think we have to ring the bell or something, is that, is it, is it where that the... No, that there will be an underlying, you know, purely linear quantum theory from which the... Now, bonjour, you need to come in. Ah, there you are, one glance. Press the right door. Bonjour, sir. I didn't realize that I wasn't supposed to speak to that man in French. You were supposed to speak in French? No, no, the man in the restaurant that I... No, no, I don't think that's the question, but he was just in a big rush. He was just in a big rush. No, I shouldn't have been so sensitive.