Discussions, incl. FW Lawvere

0:00 This is a kind of treasure. Exactly. And so that he took the same view. Well, he was quite right after having seen it. He was absolutely right. An incredible treasure trove. Talking about treasure troves connected with Grotendieck, you've probably heard that... It turns out that there is this archive of 130 hours of recordings. Bill mentioned it. Bill mentioned it. Good, he mentioned it tonight. We've been keeping it slightly under wraps, but Bill seems to be ready now to go public on it, obviously has done now, which is... Well, you've mentioned it to me. Well, I think quite a few people know about it now. Which he's actually got now, which Jack Duskin made back in 1970 when he gave these small courses in Buffalo. Plus the six talks to the buffalo. Unfortunately, those weren't recorded, but the lectures were. What I liked seeing was that it was very direct. You know, you wrote to him and he replied. He replied, yes. Even if he was only going to say, get stuffed, you would always... He wasn't going to say, I'm the great professor, get stuffed. No, no, no, no, no, no, no, no. No, he's obviously... Well, no, he never had to cite him at all. He may have been a difficult man to deal with, but he certainly wasn't like that, unlike a lot of great French mathematicians. Generally, yeah. Well, there is, I think, a particular tradition in France of the grand savant. Yeah. But anyway, we're in the course of trying to get those digitized so we can put them all on the web, hopefully, so I'm hoping within the next six months or so you'll be able to listen to those. They are, of course, expository lectures rather than... but there is... so it's not new material, it's not the stuff Elastity presented in the seminars, but even so, just the lecture courses, of course, are utterly fascinating, and just to be able to, you know... ...to listen to him talking about, and at such length too, because until these came to light... It was believed, and I checked, of course, carefully with the people on the Grothendieck Circle and other experts, that there was only one single hour of recording of Grothendieck, and that was a plenary lecture that he'd given at the World Congress of Mathematics in Edinburgh, I think in 1960. No, in 58. 58, sorry, you're quite right. Because I was there. Okay. Because one of the things I've said is that... I was a rather raw research student, and there was this guy, seven years older than I was, talking down.

2:30 There is a recording of that. It must have been made on one of those old, you know, big Grundig spool recorders. But apart from that, I met Raoul Bott. He just came to the boarding house where I was staying and he made two remarks about Grundig. It's absolutely amazing to be able to just play around with concepts and make them really something and also he was prepared to work extremely hard to make things tautological and that was the real expository thing. Because he wanted to understand. Yes, yes, yes, that's absolutely, that's spot on of course. That is absolutely spot on. And there's a lot of people who say, well, the proof is obvious, it's easy to do. Well, he said he was never going to say that, even if, I think, it may have come... Well, no, because his whole method was to go for the correct. He said that the Paris mathematician didn't really like the way he was doing mathematics at first, until he proved the Riemann rock. Yes, yes, that was, of course, that was his first, you know, spectacular, wasn't it, that, and then, of course, all that beautiful stuff that he did in functional analysis with, well, that was earlier, that was nuclear spaces, but I guess they didn't, they probably weren't aware of just how, just how significant that was, unless they were functional, well, even the functional analysts didn't really take that up until considerably later, they are taking it up now, it's become a major industry now, but, well, he said it was one of his most referred to papers, yes. Yes, astonishing. But there's an awful lot of people in the algebraic geometry world who hardly know it. And also I think that one is part of the success in algebraic geometry. Because he came from analysis, which is local to global.

5:00 Exactly. Hear, hear. I'm really glad to hear you say that, because that was the line which Cartier was promoting in his general survey talk at the meeting. And I couldn't agree more. I think that's exactly the secret. I think the fact that he did that early work in functional analysis was tremendously important. But also, of course, that he just intuitively was thinking in terms of functor categories even then, even though nobody defined what a functor category was. I guess they had just about, but he was discovering the concepts for himself. I first heard the term local to global from Dick Swann when I was writing the notes of Swann's lectures on cheese in Oxford. I don't know Swann's lectures at all. I ought to know your lecture notes. They were published by the University of Chicago in 1968. But in fact, there was not, I didn't in the original notes have on the front notes by Ronald Brown. So my name appears inside. But not prominently. So that's University of Chicago... One of the earliest University of Chicago lectures. OK, I will talk those up. Yes, well, it's always good to have different takes on the way you should introduce and teach sheet theory. I have these discussions with people all the time. People have told me, but Tony Bax said that he found it very useful when he was young. Of course people thought in terms of local to global and back again even in the 19th century implicitly. I mean Riemann obviously comes to mind but yes I don't think anybody had really made explicit before then before that period just how central it was particularly as you say in functional analysis and well certainly what he carried across to into algebraic geometry Very interesting indeed. Anyway, these 130 hours of... There's one course on algebraic geometry, one course on algebraic groups, one course on topos theory. And I'm afraid I've completely slipped my mind actually what the fourth course was on, but the point is that they were all preserved, they're all in English, which in fact he spoke very well, though with a distinct German accent, which is curious considering he left Germany when he was only seven years old. Not a French accent at all, distinct, not very pronounced, but still quite clearly a German rather than a French accent. You see, I see him basically as a writer.

7:30 Yes, but that perhaps wasn't so obvious at that time that it's become later. I mean, he was a natural poet, you know. Anyway, they will, I hope, all be online with a bit of luck by the end of this year. There you go. Fingers crossed. Keep our fingers crossed. One of the things I need to talk to Bill about while we're here. Well, actually, probably when we get back to Bristol, he'll probably be a bit too busy here. Are you able to come next door for a drink or are you going straight back to any of them? OK. Well, I'll join you there later and see you, obviously, tomorrow at CMS. OK. I think I might just have a quick snifter because I only just got in. Oh, Bill's already left, so... Yes, they all seem to. Well, I saw Peter at Bristol just two days ago, so I don't really need to say good hello to him now. I'll pop in. Yes, let's just pop into the bar. Well, if people are going back now. Is it down here? Do you know? Yes, we go down here. Yeah, I mean, if we want to, if we want to go in the bar. Yes, but I'm thinking I might just go in the bar for five minutes just to have a nightcap before. Strolling on to Newnham, or maybe, so I'm going to the prime drive track, so I have a leather jacket, so sorry, is there anybody? No, it looks like people have already gone. The one on time is the one with the standard steps and construction concepts. So fine, you can either say, good, it's good to make that law explicit. We might be able to use that. Another thing is to say, oh, it's a great mystery. How could it possibly be that one day... So there's always that choice. If you discover you have a dialectical move, you can make it into a mystery. It wasn't really clear to me whether Parmenides and Zeno meant to, because in Zeno's case it was, for example, the contradiction between the motion of the turtle and the subjective analysis, you see, so there's again a key, several key contradictions in that one, and we still use them all the time, you see. But again, we could say, oh, by God, he will never be able to move.

10:00 So you can make it into a mystery, but again it's not clear whether those people intended to mainly use it as a mode of mystification or whether they simply were rejoicing in these discoveries and perhaps as a rhetorical device saying something about impossibility and all that. I brought that away in the particular contribution of the teacher just from talking with the various people. They have something in which they display is the actual racetrack, by the way. You know, Achilles and... Just a tourist attraction. I don't know, what do you think about that? Is that a valid choice? The only thing which makes me hesitate is about Zeno and Parmenides, or I should say the other way around, Parmenides and Zeno, is that Periclytus had already... I think he sort of analysed this dialectic with remarkable clarity. You know, being at variance it yet agrees with itself that one could in fact reconcile being and becoming. There was no need to say how can it both be and not be. I mean, he'd already taken us way beyond that, but Parmenides is. He's portrayed as saying, of course we only have a fragment of the poem, and there are other things he says in that poem, particularly the businessman, the moon, some which are actually quite fascinating, don't seem to make any sense at all in the context of a completely, absolutely completely sort of immobile, notion of immobile being.

12:30 But it seems, on the face of it, that he's taking a huge step backwards from the insights that Heraclitus has already provided, so to that extent, if he had come first... And if he had just been presenting this antinomy, then it might be more plausible to believe that he was doing it for a strategic reason, to say, well, you know, here is this apparent antinomy, how can things both be and not be, what reason, because they become, because we have to go beyond this fixed opposition between being and not being, but having come after somebody who had already pointed that out so clearly. It seems strategically, it seems a little bit difficult to believe that he was just doing it as a rhetorical, post-modernist, whatever you want to call it, device to get the debate going. I appreciate that they were still part of the same community, that they'd moved there. They even visited that. Yes, yes, yes. So, after that... Parmenides certainly did, because we know he was in Athens. Yeah. Again, I... I was always trying to figure out which book to read. I decided I won't read Jonathan Barnes' book on Aristotle, because he starts off with the conclusion that Aristotle is an anarchist. I said, well, you can't possibly. So I just kept spattering some awful sources like the Internet. Because, to be frank, I didn't admit it, but already before that, I got the idea that maybe Catullus was the real villain. This is so extreme, and it was the basis used by Plato when he explained that, well, this shows, of course, we can't possibly know anything about the material world. All we can know is the subtleties. So somehow this was the link between these.

15:00 Some of the more serious philosophers in Plato was Cratulis, the teacher in Plato, and they got that right. As I say, just piecing together things, I found a book on Cratulis. The main thing discussed there was not being able to describe the derivative, but apparently he actually believed that the meaning of a word is contained in the word. It's sort of a St. John's position, from the word you must be able to discover a meaning that is known exclusively. Well, I read something about, I think it's sort of a narrow interpretation of St. John's. Yes, well... The word, the second reading it could be, it was the word, it means that the logic... Yes, it was there. It was the logic of the things, which is technical. Things are and are things are born. Isn't that being a bit generous for St. John? I mean, he means, obviously, in the Greek it is, in the beginning was the Logos, but the Logos here is carrying, yeah, I mean, it's a very Neoplatonic understanding of the Logos. It is this theological understanding, it's the emanation of, the first emanation of the Demio, which it is, this ape kind of divine agent. I mean, it's not, he's not saying that in the beginning was... The Logos in the abstract sense of the reason for things that in the beginning was the underlying reason for all. Well, he's identifying that underlying reason with a completely theological notion of this spiritual emanation of the Creator,

17:30 which of course he identifies with Christ and naturally since he's writing the first Christian gospel, well, the fourth Christian gospel. I think he was just saying something as... I think he was just saying something as... I think he was just saying something as... I see, yes. If he was just saying, if it was just a pronouncement about what Aristotle called first being, that we should try to approach and understand first being through logic, through reason, rather than through trying to grasp it through some mystic intuition, approach it through reason. Approach something already existing, the underlying, reasonably as it were, the underpinning existence first being, I would like to say, matter in motion, but anyway, approach this, the most fundamental category we have for understanding, like the zusammenhang of matter and matter in motion through reason, through concepts that have been created for the purpose of reasoning, then that would be a... Dialectically, a very progressive viewpoint, but if you're saying that there must be an approach to trying to apprehend this spiritual principle, which it does seem to be pretty clearly what he is saying as a Neoplatonist, then that's something which essentially is clearly a completely idealist viewpoint, and the first would be at least going in the direction, a trend in the direction of a materialist understanding, but the second... It seems to be pretty obvious what St. John did have in mind because he's a kind of a Christian, a Christian propagandist deformed in a Greek way. True, true, true, true, true, true. Sure, but he is essentially trying to reconcile the Christian, the gospel narratives with... The framework of Greek philosophy, which at that point, when he's writing, was about the end of the first, perhaps even well into the second centuries, is these Gnostic and Neoplatonic ideas.

20:00 I mean, I think that when he talks about the logos, he does have something pretty far removed from what we would think of as logic in mind as his subject matter. I could be quite wrong, of course. I don't know what you think. Well, after all, what's the next verse? I mean, if he was just talking about the abstract reason... I'm staking my doubts on that. It might be otherwise, that's what I'm saying. But just ask yourself the question of what things really meant in another epoch. You see, I was taught that ancient Greeks were the slaves of Cyprus. The city was breached by going over a mountain and through a pass. So they built a wall, stone construction with an arch and a board that went past. I asked a tourist guide, I said, well, where did the slaves come from? Just assuming it must have been built by slaves. She said, well, we don't really know if there were any slaves. It could be that the citizens in town just built this on their own energy funds. Well, I didn't quite realize this was the case. I knew that there are professors in the media, but clearly they're not just black citizens who have to love that they are so much more than they really are. Freely gave their labor for all this, when they were living in a subsistence economy. Very dubious, but somehow the same mode of interpretation of the media facts. It was being used by Herb, I think, as a big guy, as Dalton, in fact. Again, the epoch could be quite different from what we standardly imagine. Or not. It's being pulled into question. Just to jump back for a second to St. John, I don't really see how you can reconcile a rationalistic reading of,

22:30 in the beginning was the logos. If I could look at a broadly naturalistic or rationalistic reading, that's how we should approach first being, through reason, with what he says next, which is that it came to dwell amongst us, full of grace and favour, isn't it, or full of grace and... I can't quite remember the exact words of the verse. In the beginning was the Word and the Word was God and the Word was with God and came to dwell amongst us full of grace and whatever the other word is that he uses. I mean, that doesn't make any sense unless it's being personified. The Word's become matter. Yes, it has become. It's, you know, he does believe that it's been made incarnate in this this character, Jesus. So it can't be anything like an abstract principle of, It's got to be some sort of semi-mystical personification, creative principle, the way that the Neoplatonists seem to have thought of it, and therefore I think more likely to be the kind of thing that you rightly identify the idea that he's talking about, that the word itself somehow contains, the name of the thing contains the very essence of the thing itself. This is a wildly idealistic notion which theologians often made use of. Apparently there was a very flourishing heresy in the Russian Orthodox Church in the late 19th and early 20th century called the Name Heresy, which according to Jean-Michel Cantor, a historian of mathematics in Paris, He claims had a considerable influence on the Russian school of set theorists at that time, Suslin and Lushin and Alexandrov and those people. And the heresy essentially was that God is so inconceivable, unapproachable, cannot even be apprehended by mortal beings, that it is a blasphemy even to think that you can worship God. But we could worship the name of God. That would be permitted, because there's enough of the essence of God in his name, so you actually worship a word.

25:00 In order to have something to worship, you have to be less than God himself, but which we can yet worship, and so we worship the name. Weird, absolutely weird. But anyway, this was regarded as a dangerous heresy by the Holy Synod, Denounced, and indeed to the extent that the Holy Synod actually prevailed on the Russian government to send two ships from the Russian Navy, cruisers full of armed marines, to Mount Athos in Greece, where all these Orthodox monks were, and they dragged off half the community, about 300 monks, at Bani Point, and took them back to Odessa to be tried by the Holy Synod for this heresy. This was in about 1911. Extraordinary business. It's very well documented in this book about, which is actually a book about mathematics, about the Russian school of set theory. I suspect the thesis is a little overstated. The influence of this theological theme on the set theory, I suspect, is maybe considerably overstated, but it's still an interesting idea. But certainly this business about names... Yeah, yeah, some kind of an echo there. Have you heard about this heresy? But it was in the Russian Orthodox rather than the Greek Orthodox Church. Nonetheless, they were in Mount Athos. Oh yes, they were, but there are a lot of Russian monasteries on Mount Athos. But yes, about a year ago I read, in fact he's published a book about it now, Jean-Michel Cantor with a K. Cantor with a K. History was out there, history of Russian was out there. Down to earth. The claim was that... ...Malziv had to turn into more algebraic...

27:30 ...things. Not at all. Jolly positive development. Push people in the right direction, yeah. Thank you for watching. Sure, is there a... do you know where the Portals Lodge is? Because I'm supposed to get my keys, I think, from... Ah yes, I see. There you go.