Afternoon & evening talks (contd.)

0:00 Elephant, okay, okay, okay. Another example about the standard equation is, I think, you see, if you look at, just power set, omega to the x, comma... Those things that deserve countable suits, countable suits, then the ordinary power set is included in that, because you can take an arbitrary suit and commute with countable suits. Yeah, yeah, yeah. If you're looking at a set that has less than Oulam, that would be the equivalence, in other words, just assuming that from 2 to the x to 2 you have something preserving Kalamu suits, should be represented by an actual set, subset, if that were true then To preserve finite intersections, it would be automatically reducing it to the... In other words, the thing that's induced by unioning with an arbitrary subset, if it moreover preserves finite intersections, it must be a cyclical. The prejudice is literally true. In that case, it gives an even simpler way to define measurable cardinals. It literally is just a countable union. Did everyone come back now? No, John isn't here. I think he's about to come back. And I think he's probably the person who most of all needs to hear what you just said. No, no, I'm just here basically to...

2:30 And that gentleman is Richard Pettigrew, who is the guy who put this whole meeting together. So this has been a hassle over the week. This is the second week beginning the day. Right. Peter just came to see Bill's talk on the first of May. Good question. We haven't really had a chance to do a planning meeting, but the plan for today is definitely to allow Anders and Bernardini to talk as much as possible, because there's no time for us to have a review by any chance to do it again. But that's for today. And then Thomas is asking if we have any agenda for today. You're going to be competing, of course, with math if you come again. If you don't go away in disgust, how will you do A? I'm most unlikely to be in disgust. There is no other possibility to work. An external examiner was unhappy with the exam. I said, well, I'm happy with it. He claimed it had no applications. I regard the abstract structures, you know, adjoining products as logic, and use them to characterize familiar constructs in specific categories, such as set on the implicit as applications. I don't know, but on the other hand, it's a conceptual answer. I think it will be worth very quickly running your point around in a measured posture. Oh, no, we mustn't stop there. No, no, let's stop there. Sorry. I'll do that privately.

5:00 Sorry. So we are... We are introduced to what synthetic quantum mechanics should be. Very, very rough summary. We can summarize what I said before, but it's a bit long. Continuum mechanics of a suitable category of cohesion should allow us to do, I hope, a big improvement compared to classical continuum mechanics. One is the theory of space and one is the theory of dimensions. So let me repeat that by continuum mechanics we understand the study. All of these are examples of the motion of matter at a scale where the extension of the matter is greater than it is what it is, in the sense that given an amount of matter, it will be modelled by some given version of value, and then its evolution through time can be studied at its scale, but for any time. For any time, the same amount of matter can be seen at different scales, ranging from a discrete set of points, a particle at one extreme, or perhaps at a single point at the other extreme. We started to say that of course the first requirement is that he has to host the synthetic differential geometry, and to do that he certainly must include an object equal to his genetic line, satisfied with the reduction and so on.

7:30 Besides geometric line, perhaps to do quantum mechanics, another object should play an important role, an object that now we can not buy a different symbol, but perhaps can be called by another name, but in one sense can be called an object of pure quantities. More likely, this should be the object of the taken reals, according to what we were saying. So, on one hand, the geometric line should be, the role of the geometric line in terms of, in physical terms, in one of parametrized motion of matter. Mechanism is the motion of matter three times, and the array should be the object of parametrized motion. Something that describes the results of measurement of cohomology without the measure of cohomology. And then what I wanted to say after this is that in order to say if there are more features that we should ask to the category of space, I think we have two strategies. We examine all the basic construction of the classical continuum mechanics and check what kind of construction we need to do. This is one way to transcribe the classical continuum mechanics. And the other one is to try to develop an alternative way of mechanics. I don't know if you'd agree, but I would be tempted to quote the alternative way to do mechanics, like Hertz or Hertz. Too many guys whose names begin with an H.

10:00 The next point we can discuss now is just to give a big idea. No, no, no. You need it. Two guys from the South. They call this rugged weather, yes. I can give you a very, very big picture of what the basic structure is of the classical continuum mechanics theory is, and can give you some, perhaps, some insight, some suggestion to you. And then we can compare, say, the alternative HL, what we should find within the category of space to do continuum mechanics. First of all, we have to find an object which is the background to do any continuum. In general, the object of space-time is an object whose global elements deserve to be called mechanical events. What are events? This is a nice expression that I found in a note paper that I liked a lot, because events are like atoms of experience. I like to see what are events in physics. And of course in each domain of physics, atoms of experience are different. In mechanics, the mechanical experience is mainly with position and time, so mechanical events are just two.

12:30 Listing classical mechanics are two basic qualifications that are here and now, so the position and time. So the first object we want to, of course if we do thermodynamics, of course the experience is richer. We have to describe events not only by position and time, but also by a measure of how hot is the body. And so on. Different fields of physics have a richer notion of science, because they inspire different domains of experience. What about mechanics of a continuous body, like a ball of jelly or something like that? And we have before the introduction of bodies. This is just the background where position and time of bodies can be described. Of course, we need the basic object in which we can describe position of bodies before even defining what a body is. So this is just the background. And of course, in a very general setting, events are mixed up. Space and time in general are... They are basically the same footing. But in classical mechanics, time plays a different role. In classical mechanics, instead of merely announcing space-time, we deal with a space-time bundle, where there is an immersion of time that places... So, in classical mechanics we have a space-time bundle, in the sense that we have a map from these space-time objects to another object that describes time. So, I don't know if you know how symbology can become a bundle in the sense that it is a map that extracts the time from the event. So, this gives time to different worlds. This is a feature of classical mechanics. So, for example, the general relativity, this should be a trigger manifold, so they can turn at the same place. Classical mechanics, the time change is specific. So, another point of classical mechanics is that this bundle becomes a trigger bundle.

15:00 Trigger bundle, where the object at this time becomes a new object. Just on your point about the Ulam measure, I mean, the thing which I'm taking from this as a kind of philosophical outsider is just how extraordinary the idea of smallness, I mean, ramifies across many, many fields of mathematics. It's not simply confined to these issues of logic in some narrow sense of logic, although obviously one sees it particularly. There's not a particularly clear form there, but in fact, as you say, in statistics, in the theory of measure and distribution, the same idea is showing up, clearly. Yes, that's an extraordinarily deep, unifying idea, which I'd love to understand more. Perhaps that's one of the things we can come on to later in the workshop. Yes, it's really horrific. Thank you for your attention.

17:30 Well, it's very close anyway. It's probably pushing it a bit to say the Barbians wouldn't. And it is a city of 150,000 people on its own. Smaller, a lot smaller than Bristol, but still a city. No, I was told Bath was 100, I suppose it depends where you draw the boundaries. You could be right, it might be 100,000, but it's a lot smaller than Bristol, but still a quite substantial city, a very historic city. Bristol's got about a million. Yeah, Bristol's about half a million. Bristol's nowhere near a million. Oh, OK. Bristol's nowhere near a million. Even Edinburgh doesn't have a million. No, Edinburgh has half a million. I thought, oh, OK, well, I've probably looked at all these figures long ago. The figures I've seen are all probably inaccurate. Yeah, I would have, yes, yeah, so it's very... But it's big enough as a city not just to be a dormitory or a suburb of Bristol. It does have its own identity. It does have an identity. It does have an identity, very strongly. No, it doesn't have the cities of Bristol. I want to go to a decent art house cinema in Bristol. Yeah, yeah, exactly. Bath does not have a concert hall. No, no. Bath does not have an opera house. Or anything of the size of, say, the Hippodrome. No, no. On the other hand, it's got probably a better theatre than, in fact, the Bristol Bath Theatre. Yeah, yeah, that's right. It's got a sort of semi-artist cinema. Yeah. And it's got some nice small-scale consequences, you know, like... It's also the reason that Bristol doesn't get a huge number of tourists because Bath is the, you know, the absolute gem of 18th century architecture in Britain. It's the place and it's actually a Roman city. It not only has the Roman remains but it also has this incredible concentration unrivalled in anywhere else in England, I would say, so I'm carefully not saying Britain because you've still got Edinburgh, but certainly unrivalled in England.

20:00 ...of absolutely stunning 18th century architecture, as a result of that it gets almost all the tourists in this part of the country. Although the funny thing is, if you want to choose what is the single best piece of architecture in Bath, the single most striking building in Bath is Medieval. What, the Abbey you mean? Yeah, yeah, it's the only big medieval. It's very, very late, isn't it? Late perpendicular, yeah. Yeah, yeah. Very ornate perpendicular architecture, with a fantastic fan-bolted roof like King's College Shuffle. Yeah. Yes, yes, because by that stage, all the stress was being taken by the buttresses outside, so they were able to put in enormously high ratios. No, that's true, that's true, that's true. It was actually just that they were very good indeed at supporting a huge amount of roof space on very slender, slender vaulting. But I mean, it's only many years ago. Yeah, yeah. An 18th century city, yeah. But the thing is, all the tourist coaches tend to stop and barf. Oh well, Bristol's a great place, it's a fantastic place. But you're right, it's about four times, if I may, maybe even five times the size of Bach. Yes, it's like the length of the population, and you're going to go through the Trinidad. Now, the problem with the Triad here, the Crystal Temple is, it isn't where you want it to be. It's like a three-quarters-an-hour walk. Yes, it's a heck of a nuisance for you. I believe that is course of course of course of course of course of course of course of course of course

22:30 You must surely have the will of my first Australian editor. I was very tired, sitting in a room with an older friend of mine, and you know, that's only 10 pounds for Father John Terry, who was an Australian priest, Catholic priest. Now, it's 200 years ago, that was a lot of money. But then he continued, asked Father Terry to say, a requiem, I suppose. And then it went on, if he refuses, withdraw the 10 pounds. There, that's what it was. He obviously didn't have very much faith in Father Kerry, did he? Well, he was the only Catholic priest available, and I suppose he had no choice. In fact, of course, it is literally just like slavery, except in this respect that, from what I've understood and what I've studied, the conditions on the emigrant ships ...were frequently even more terrible than the conditions on the slavers, because the slavers who had bought these people had a vested interest in getting them to the plantations alive. It was the same with the convicts. Exactly. Well, the convicts too, but a lot of the emigrants, the ones who just went because they were absolute destitute and they were watching their children starve, they were already terribly weak before they even came on the ships. Once they had paid their passage, there was absolutely no economic incentive on the part of the... People, the agents and the shipping companies, to try to keep them alive until they got there. They might, of course, have become indentured laborers once they got there, but the people for whom they would have been working in Australia, effectively, virtually as bond slaves, were not paying the passage. So, in fact, the condition on the emigrant ships was, and I'm not even sure if the survival rates on the emigrant ships went even lower than they were in the slaves. Not many emigrants actually went to North America, rather than Australia, because it was cheaper. Yes, I suppose it was nearer. So that's why the Australian experience is quite different from North America. Because in the USA you've got all this support for, you know, can't we? Republican Champagne, which was the environment in which Australia has never had that, between the convicts and the gold miners. Yes, yes, it was. Whereas the evidence is the famine.

25:00 Yes, yes. Yes, okay, this is interesting. They were free, it was all free, yes, yes. They're people like the 49ers who went out in search of their fortune rather than fleeing, yes, fleeing, fleeing, well, what's effectively almost genocide in terms of... The famine, yeah. No, that makes sense, doesn't it? Very interesting, very interesting. Stefano Berardi from Turin. That was very good. Anders, you wouldn't like another part of the molten water or anything, would you? No, I want some coffee. Yeah, you can get coffee here. I think I might join you in that, actually. I think it's just care, those categories. Sure, probably, yeah. Probably, because people will hate categories. It's because they can't understand them. Probably because it's so close to logic. It's sufficiently close that you get people... Yes, especially things that categories that are in contrast with the theory. Thank you very much indeed, cheers. Who's this? Who's at the end of the philosophy of mathematics at tonight? Maybe you can do it when nobody else can. Yes, yes, maybe.

27:30 Maybe he'll sort of get his head straightened out. Who is this person you're speaking of? Gabriele Lott. Oh, yes, him. Oh, gosh, yes, him, yes. I've heard him talk in Paris on several occasions. He comes to Paris quite often. Ah, really? Yeah, well, there's quite an interchange between the ENS in Pisa and the ENS in Paris. They have quite a lot of... Visitors from Pisa coming to Paris. Yes, but he was elected to this position in Pisa just before. Oh, okay. Before he was in Turin. Well, he's come to Paris on more than one occasion in any case, even if it wasn't in that context. Ah, giving calls on... In philosophy, in the philosophy of mathematics. I think that's an Italian guy who organized that meeting with Kohn and this big meeting in, what was it, the year 2000? Yes, the one where you spoke. That's Pepe Longo. He's at the Economa. He's different altogether. He's a very strong supporter of you. Unfortunately, I don't think he quite understands. Well, that's a different issue. Categories to this keep carrying on about the coherence of the normal categories. It's not simple. This was Lolly speaking. This is Longo. And then the rest of the hour saying why do they keep saying it's so complicated when it's actually simple. Because he has the wrong definition. Because he has the wrong definition, yes. That does sound a bit like Beppi. But his heart is in the right place and he says lots of things, sensible things, about the kind of broad history of mathematics in the 19th century, about Riemann and Dedekind and Frege and the relationship between geometry and arithmetic, which I think are broadly correct. We wanted to make mathematical structures in computer science, the category theory in computer science, and Peter Fry talked him out of it. He said it should be mathematical structures, along the lines of general and pure and applied algebra. He also spends a lot of his time these days working on applications of category theory to biology.

30:00 Yes. Martin is an editor of that journal, isn't he? Not at all, is he? Is he? Yeah. The Mathematical Structures in Computer Science. Which is a complete journal. To Cambridge. Yeah, well it would make sense that he would be one of the editors then. Yeah. I'm sure, he certainly was one of the editors, the principal editors when it was founded. I'm not quite sure what his career is. He's done quite well in the organisation. Because its focus changed a lot, but now it's originally going to be category theory, so it had a lot of value. Yes, all the early numbers certainly did. It became more focused on mathematical structures. Right, right. Whatever mathematical structures were sensible, so that changed the focus a bit, and then you'd need to change the editors. Right, so he isn't an editor any longer, although that was a question mark. I think he is. He still is. Okay. I'm not sure why they cut down the literature. I assume independently that I had a little controversy. Too often it's something like, well, as the year sets, they don't have any structure, they just have quality. Thank you. Therefore, we ought to consider ZF. I just didn't understand anything. I mean, I have every right to consider ZF, but to present that has been the logic. That's the motivation for doing so. It's crazy. What went wrong? I got pretty tight about that. What are the editors doing? They're not catching you. It was blatant blah blah blah. Well, that might be why they didn't take him to... Well, I wonder if he would have put... Supposedly, we're independent events, right?

32:30 So, as usual, it was your implacable hostility to all's ignorance that got you into trouble. Yeah. So, no change there. Actually, about the right time, Barry Smith wanted me to write something for his centennial... for his... Axiomatic, you know? Millennial issue of The Monist. Oh, The Monist. I've forgotten that Bepi Longo was one of the organizers of that event in 2000 with you and Korn, and that was where I met Cartier for the first time, so quite apart from anything else I regarded it as a positive thing, even though we had to listen to Girard as part of the penalty for getting to hear you. I think he's a raving lunatic. He's literally crazy. At least that, that, that, uh, talk. Well, I talked about that when I saw him in person. But the earlier work, of course, way back, it was, it was always telling me a little bit about linear logic. It's really, uh... How is it that you're always a queen with me? Linear logic. I think it's all of them. Well, what product? What can we ask them? I mean, real life and physics and category theory, computer science, or any other thing. I mean, I don't have any business interest in linear logic, but I certainly think things like symmetric, monodal, closed categories, and Cartesian closed categories are interesting and valuable. Yeah, so why have this whole other chapter whose purpose is to... I don't know. I don't know. I don't know. I don't know.

35:00 I don't know. I don't know. I don't know. I don't know. I don't know. I don't know. I don't know. ...create diversions from those actions. Some people, some mathematicians have interpreted it as an attempt in that direction, but I don't think they're successful. There's a lot of rhetoric behind it. There's a lot of rhetoric about using up resources, but does he have any content corresponding to any moves that they actually make? And I'm disguising this, I think, imposed aspect of one of these things. There's a car, there's a ship, there's a pump, bang, and all this. It's just genius to complete the diversions that people might otherwise be able to do. I haven't noticed an interest in linear logic. What's about it? What would you regard as criteria for success? Well, as I said, something that we could use. I've seen people use it in general language. I can't understand why so many people went into it. How is that possible? I was proposed to do that for my PhD and I discovered it. Somebody was promoting it very heavily. It's done a huge amount of damage in philosophy departments, certainly in France. You can't talk to them about geometry and logic without thinking they immediately jumped to the conclusion that you're a supporter of linear logic. My advice is that they were forced into it, but also you can't... funding is withdrawn, more or less, when people are no more tending to accept it's some private granting of information. Let's talk about privatization.

37:30 Mathematical physics is the privatization of mathematical research, but if the organizations who offer these grants have completely other standards from the scientific ones, and yet people are desperate at some point to fund them, then more and more people do these activities, however they're interested in the faculty. I'm very grateful to the government for having embedded me in the Cartesian forest structure and I'm going to submit to it when I report to the government. I've seen no accidents, no actions as well. So for instance, the main way to get them is to appreciate the category. You see, if you've got a small, you're not submitting to the category. And you can do that. I'm going to give you some advice. I'm going to give you some advice on the motor here. John Reynolds. I think you've heard him. Come on. I'm going to give you some advice on the motor. It sounds as though, at least from their perspective, they claim it's a valuable tool in understanding some well-developed programming language. Now, it's different from linear logic, but the basic way is that you don't have this kind of shrieked thing and you pick different categories for that. I don't know anything about that.

40:00 I think Girard is as mad as a hat, and he's ridiculously self-absorbed, and he implores on the standard, I mean he implores on the standard, and I feel it's too tough, so anything I've seen of him is potentially clever, yeah, nuts, and the Linux thing is probably particularly bad, but it's kind of a thing of that nature. I was in his place one time. I remember the meeting in Boulder. Oh yeah. Two categories of science. Absolutely. We both remember for some reason. It's fine. It was very close. It was very, very close to Damien Scott. It was a compliment. But Gerard, that's a major project with the U.S. Army. He said, well, before he was allowed to speak Polish, it was first in Latin America. So I was on the bus going away from Mongolia. He said, well, if you come to Paris, come to my house. In fact, I was just going a couple of weeks in Canada, where he invited me over, because he wrote for Anticopia Pliny, the southern town in Canada, and we got to talking, you see, and quickly he decided I'm a Stalinist, and then, and then he actually threatens... ...to beat me up, right in his own apartment. He said, come over here in this open part, I'm going to beat you up. He was all ready to move. It's funny because the other time that I first encountered the Trotsky guy, he did the same thing. I was threatened with physical violence. That was... Max Dickmann. Max Dickmann. Also a little bit.

42:30 Also mad as a hat, isn't it? Well, temperamentally, that's rather a low boiling point. I met him in Chile, or actually originally in Chile, but it's quite strange, this sudden urge to violence, just because of fundamental discipline. Max Dickmann was one of the main organizers of these Ulldum conferences. That's right. I met him through John Bell shortly after the meeting in Paris in memory of McLean, the one that we put together in 2005. John Bell was over there. He gave three talks at the École Normale. He didn't come to that meeting, but there were nice talks. He was staying at Max Dickmann's apartment, so I was invited along for dinner and I had to listen to... A lot of reminiscences, which some of which was quite interesting, some of which was about Oldham, some of which was about, you know, what terrible things, you know, Bill O'Hara and the Stalinists did to the revolutionary potential in Halifax in 1937, but I was quite able to read between the lines as to what had really been going on there. No, no, no difficulty there to decode what was really happening. And, well, you've just said it all. I mean, here's a man who... ...who poses, like so many ultra-leftists, as an ultra-leftist, and who is doing contract work for the American army. Oh, yeah. I mean, one doesn't really need to say anything more than that. I mean, comment is superfluous. Well, another person is sort of like Wally, probably in that respect, but I don't know if they actually collaborated on this. He's also very polite when he meets me, but he's well-known to hate category theory with a vengeance, and operates on that. It's found in Pfeffermann. Ah, I don't know. It's found in Pfeffermann. So it behaves politely, but then... Yeah, yeah. But he's been a foe of category theory from the very beginning. Well, I mean, I knew him already when I was with Tarski and he inferred me back in 1961.

45:00 Yeah. Well, anyway, he comes into this because at Stanton's, you know. Oh, yeah. And I annoyed him. I gave a talk once in Boston. I did that talk. I used to be Tarski's secretary. Trotsky's, you mean to say. Trotsky's. Yes, you said Tarski. We all knew who you meant. No, I used to be Trotsky's secretary. This is, this presenting himself, this is the same name, form of substance, I guess. Much, much later I learned that he had published all sorts of things about how dialectical interpretation of mathematics is wrong, but at the same time, he was claiming to be the head of a foreign international, which is a Trotskyite organization. I think he wasn't originally a logician, but the main reason the logician was him that we know about is because they collected together. It's his ontology. It's his ontology. Anyway, this guy, this guy who came to the right wing archive and to give all his secretary and other events. I spent a lot of time with Peppermont and Peppermont's wife. Peppermont's wife had such an interest that she wrote a book about it. There's this book called Law, Language, and Politics, and Logic.

47:30 Yes, I think that's it, yeah. I think that's it. Which, of course, is why the slip of the tongue with Tarski, because she's also, of course, now written this biography of Tarski. So it's a very natural elision from Tarski to Trotsky. Thus he did write that utterly pernicious, and certainly in terms of the damage it's done in philosophy, department's history of logic, well, anthology, which has ensured that so many philosophers don't know the first thing about the main mathematical trend in work in logic for the last hundred years, but only about Frege and Piano and... Sorry, there was something else I think you were going to add about him. Did you ever see him again after that occasion? But it is extraordinary that he set himself up specifically as an opponent of any dialectical view of mathematics, quite explicitly. These people always attack dialectics. It's a tactic of all ultra-leftists. I told you this before, but the book that she wrote, Bob Van Eyden, where it cites Girard, murdered by his wife.

50:00 Like his boss, except he wasn't murdered by his wife. He died, happened to be murdered by his wife, and then there was a funeral in Paris, and the remark of Girard about this funeral, here for the first time. Trotskyites and logicians in Paris were meeting each other. It's just absurd because the only Trotskyites or logicians that I didn't know in Paris were in fact both. Yes, we're the same people. What's he trying to prove? Yes, Gravien and all these, Girard himself, Dickmann. Well, Dickmann's not a logician, but I mean... So basically... Yeah, it is. ...not to be a logician. The intersection of being the null set, yeah. Well, I guess if you spend your life spreading disinformation about subjects, it gets to be a habit. Seriously, you do it compulsively and without thinking. Seriously, I think that's part of the explanation. You spend your entire life spreading disinformation on all subjects, and particularly on the one that you're most focused on. You just do it without thinking.

52:30 I think there's an anti-consciousness to the very... I've said it before. Yeah, yeah, yeah. And he may have had, like, some, one specific, two specific individuals in his mind. Well, then, the problem is, like I said, people need to work. Yeah. That's where the plot goes. Yes, yes. Well, this is where we should go. And also, it's the whole phenomenon that you've drawn attention to frequently, of anti-consciousness. Yeah, probably. Uh... They've actually deluded themselves, and if you spend your whole life, as I say, spreading misinformation, it all gets to be a habit. Yeah, yeah, seriously. I think that is part of the explanation. Anyway, very interesting information. You hadn't... I hadn't heard that story before, so thank you for telling me. There you are. Try again. Ah, there we go. It's okay, it's our room anyway. Ah, okay. It's been kept brass, thanks. I met Max Dickmann through John Bell in 2005 because it was the year that we had the meeting in honour of MacLean shortly after he died. Very strange guy. Oh, really? No, I didn't realize he'd ever done any work in logic. These days he spends almost his whole time doing, sort of, it's algebraic geometry over the reals. Real algebraic geometry, yes.

55:00 Yes, he does, he works in, as far as I'm telling you, all of his papers, I think, for the last 20 years have been in that area. Real algebraic geometry, just to say, algebraic geometry over the reals. Which I think is essentially vile, you know, old vile style algebraic geometry. He organised a meeting, I have to say, I think it was a rather good meeting, on history of algebraic geometry in Brittany, in my neck of the woods, but right over in Belle Isle, the island that's right on the far west, yes, well he and a couple of other guys, and so I didn't actually record it myself, but John Bell went and he recorded it all for us for the archive. I listened to the talks, they were very interesting. It was, half of it was history of algebraic geometry and half of it was this, you know, real, recent research results in real algebraic geometry. It was quite interesting getting his take on the Alden meeting and on, which was, I suspect, a very one-sided account. Yes, that was why they were reminiscing about it, because they both, he and Moishe Macher, I think, were both also involved in organising it. Yeah, yeah. And it was, Max was sort of the local, and there was people from London School of Economics. Yes, BOSU, which was Bell, which was John and Moishe, and Grendi came, which was... And, like, Barr. Yes, yes. Was it on that occasion that Grotendieck christened Bill the main contradictor, or the chief contradictor? I'm not sure if that was the occasion. Grotendieck had a label or nickname for Bill. He called him the main contradictor, or the chief contradictor.

57:30 It must have been a different occasion. Bill will tell us. There was some occasion on which Bill... Criticised, I mean purely his political views, his saliva of propaganda in public, and they had something of an exchange. I thought it might have been that occasion. No, okay, it was a different one. Oh, he wasn't in it, it must have been another occasion. I knew it was somewhere around that time, he'll tell us himself. The label that he gave him, the chief contradictor or the main contradictor, is one that I think he wears with pride. Ah, it is absolutely terrible to think what has happened to Grotendieck, isn't it? Well, he's fallen completely into religious psychosis and into anti-hostility towards science and generally other forms of mania. Yeah, and Dejah Raghava has a very great tragedy. I mean, Grotendieck, well, into various forms of religious mania, psychosis, well, completely, yes. Oh, completely. He's, well, after he cut himself off completely from mathematics for the second time, the second and final time in 1991. This is according to Cartier, who is about the only mathematician who has been in touch with him at all, and also Sharlow, who is this guy, who is his biographer, who we met at the meeting in Montpellier in January this year. He spent seven years convinced that he was channeling. A Belgian nun who died in the 1920s or 30s and who received the stigmata, and he was telling people that he would receive the stigmata at any time. Except that side. Okay. Well, that's just one symptom. But he also talks constantly about the devil being at large in the world, and for some time, when Carty crossed the line, well, Carty thought at one point that might be some kind of elaborate metaphor, but it became apparent that it wasn't, it really was, he did literally leave, that people were sent by the devil to try and, no, no, I'm afraid he's fallen into complete psychosis. I mean, you know, like, a lot of people come, which is all right. In his case it's deeper, deeper, quite deeper reasons. There are all sorts of things in his family background that might, you know, lead one to think that's part of the explanation because before his father became an atheist and a revolutionary, all of the...

1:00:00 His father's family for many generations have been very devout Hasidic Jews, but I don't, from what I've understood from talking to Faraday and Shama and other people, I don't think that's a main, no, no, completely different dimensions, quite, quite a different dimension from that, quite a different dimension, you know, we have this discussion about the, in fact, the point that you raised about double dualizations, principles actually explaining, actually being more fundamental to understanding what's going on across a whole. There's a lot of substance in measure theory or in the theory of distributions and what's going on in logic with these sort of small, large categories. Seeing what's common to that from a kind of unifying algebraic standpoint, I mean, okay, most people would say, well, that's technical maths, but in terms of conceptual breadth of the issues, I think what Remdick was doing, Remdick's whole programme for the re-description of structure across the whole of mathematics in terms of concepts which were drawn from his work in cohomology, which of course had this common feature both in the work in functional analysis and in the work in algebraic geometry, that they were really exploiting the power of contoriality. That to me is also... I would be in a minority, not any longer of one, but still in a very small minority amongst people who study the so-called philosophy of mathematics in having that work for things and to have a rich education of mathematics again is only very reasonable.

1:02:30 He and a handful of other people, like Colin McClarty, like Alberto Brussi, like a handful of others, are in the philosophy departments and I think are aware of just how Laurie Barton's so-called philosophy of mathematics has become and how it has to be reviewed by the kind of... There's still a very fact that the philosophy department has come to any of the sessions of this workshop. I think it tells you quite a lot about how out of touch they are. He did, but he's also the only person in the philosophy department who's actually come, even though it's taking place in their department. Well, there is, in fact, quite a large philosophy of mathematics group here in Bristol, but to be honest, apart from Richard, I don't think any of them is technically up to being able to take in work at this level. For me, as a complete outsider, that's an old academic point, but it probably sounds extremely hard of a thing to say, but I think you'd find the science of it. And so the payoff is that mathematics in the interior is what is carried out. There's something about how it fits into, you know, metaphysical issues about the nature of relations, whether they're always, you know, dividing, it lacks any kind of real precision and any kind of contact with real mathematics except via the bits of logic and set theory that philosophers learn about in their second and third years in philosophy departments, which of course is a...

1:05:00 It gives, for the most part, an utterly distorted view of the overall shape and growth of mathematics. But, you know, as far as I'm concerned, my mission in life is to try and change the situation, to shift it in the direction that the philosophy of physics has already shifted it. And that's why I decided to try and create this archive back 30 years ago now. Overarching view of the shape and direction of mathematics plus a sense of what And also a few of the foundations which actually connects it with the actual growth of mathematics and also what the correct pedagogy of mathematics is, which are the dimensions between the so-called foundations of mathematics tend to leave all this completely out of the picture. Yes, of course. If they were purely technical, then there wouldn't be any point in saying it was philosophy. We're having... John is asking me if I can provide a demarcation criterion for philosophy of mathematics as against... Well, not just any old technical stuff, but even the kind of technical stuff which gives a very broad conceptual perspective on the organisation of maths. Of the kind that, obviously, we're touching on in many places in this workshop. But to what extent is philosophy of maths, insofar as it's not completely moribund, just a branch of philosophy of language, can it actually be distinguished from maths itself? It depends on where you put the foundations. Yes, yes, exactly. If you put foundations in mathematics, which I'd be inclined to, then there's nothing to expect. Well, but if you don't... In foundations, you do have to touch on philosophical issues. Of course. Yeah, I'm just not sure what philosophy is. I suspect I do a lot of philosophy, but I'm not clear on what philosophy is. It's a love of wisdom. Well, I mean, Aristotle says that he gives you a pretty complete account of things.

1:07:30 And he says, well, at this point, the demarcating mathematician just takes these things. But then he's very careful to say what he does, what they do take is good. Well, yes, we need to decide that, don't we? What do you want? I have a suggestion. I have a suggestion. Short distance. And a reasonable meal for a not too great a price. There's a restaurant right over here called Cafe 155. Yes, I know what you mean. You've eaten there. Yeah, yeah, you took me there. It's a nice little place, right? And you can get a reasonable, and if you're pinched money, you can get a reasonable meal. I don't think they've got music. I think you've sold us, John. I think it's a game, set, and match. Anybody opposed? Carry them on. Right, that's where we'll go. I mean, not every restaurant will turn off just because a bunch of double zones come in on the floor. I thought you said, okay, well, look, we don't want to hold up Davide. Why don't we, why don't I go and very quickly check out at the end of the evening that they do or do not have music, whether they will turn it off, and if they say, if they don't, or if they agree that they will, then that's where we eat. Well, I mean, the music there... Well, it should be open on a Tuesday, shouldn't it? I mean, my idea of music that's impossible is where you have to shout in order to be heard. It won't be soft, hammy, rock and roll. I mean, it could be sort of ABBA-type stuff.

1:10:00 Yes, that's enough. Okay, so do you want to make a time to go there? So that Davide can bring his wife and... You can walk, it's right around the corner. It's literally right around the corner from here. Literally. Because I'm across the street. It'll take you about a minute and a half to walk. It's on St. Michael's Hill, yeah. If you just walk across the roundabout and then down past the grocery store. There's the place just ahead. Yeah, we're there. So if you can see it, that's very easy to find. And it is at number 155, as I conjectured. That's why it's called that. Yes. Let's see if John's there yet. No. But then we're actually a little... We can handle this level of music, I think, can't we? We could always ask them to turn it down a little bit. Yes, this would be perfect, wouldn't it? No, I don't think we even need to do that, actually. OK. There you are, Olivia. Cheers. Ah, here's John. Perfect timing. No burst blood vessel. No burst blood vessel. I've been on the boat. I just literally hung up. I've been on the phone to Hendersonville, North Carolina, trying to sort out, I mean they expect that he's recovering from a stroke, to be able to have his own parents, but the people there are not, I mean the people are in some security office.

1:12:30 Yes, the crazy rules they have to administer. Bill, do you want to go to the head of the table? Next to John, don't go there. Do you want to? Oh, I'll go there, it's okay. Do you want to go next to John? Well, who else? Well, Davidi and his wife may be coming, we're not sure. He's going back to see how she feels. She feels like she's obviously not exactly in the, you know... I think she's the happiest of spirits at the moment for obvious reasons. If she does come then we could just get an extra chair in. No, I just, I, they won't accept my son. The old one's got a problem. They won't accept him. The old one's got a problem. So it's got to go. They said it'll only take two weeks to sort all this out. Well, I mean. In the meantime, my son's being bombarded with middles, 4,000 here, 20,000 there, 58,000, which is the biggest one. My God. This hospital is simply, it's just, I mean, you know, there's somebody that, well, I don't think that we need to build, but everybody else would be amazed at what we've drawn out of there. There, well, many of these are very good. We're behind in the international sweepstakes, we're behind in the Becky standard, as far as standards are concerned. But we're number one in capitalism. Yes. Tells one something about the nature of the capitalist system, doesn't it? Pretty. Well, I'll see if there's any other explanation. This is a capitalist system here. There's one in Denmark. There's one in Finland.

1:15:00 This is very modified in its effects in Denmark. It's moving in that direction. That's what capitalism does. I mean, you're just ahead of us in time. Well, let's hope not. Well, yeah, ahead in the direction of... We were talking about Fibonacci and the rabbits and Galileo and just saying that it struck me that the role of the two-pointed D in that This is possibly one of the ways that one could actually introduce the whole subject to beginners. For instance, one of the things we were talking about on the first day was the... ...possible pedagogical strategies for a second volume of Louvier and Chagnon, you know, conceptual mathematics. But Louvier and Chagnon, they won't know much linear algebra, will they? No, not much, but they'd still, I think, understand the point about the Fibonacci example and the little diagram you drew with the eight... And there are plenty of examples of speed-up that you can take from those. Yes, but as you say, starting with the communist world model, and then coming out with continuous purchases from here. You need the two-point here. Yes, exactly. Which, of course, introduces the whole issue, you know, role of the components for the space, which you have to, yeah, yeah.

1:17:30 I don't know the ghost of the part of quantities, so I couldn't express it that way. Besides, where is it from? That's a lot of choice. So have you chosen? Okay. I'm splitting I already have the chili con carne at lunch because otherwise I would have had it tonight. I've had it twice in a day. It's a little bit... I would like to sell an article I have on it in Archive of Original Mechanics and Analysis from 20 years ago. Well, essentially what I was just trying to say today, but also it includes, for instance, what is the... Mathematics in terms of splitting of a sequence, coherent choice of units, three independent basic quantities, but the choice itself, you don't have to say which are the basic and which are the derived, it's just a whole market. Is it in the web? Can you download it? I think it's prior to being digitalized. You're not trying to take your old papers, as Bill has done with many of his, and put them on your web page with... I'm in the process of doing so. ...late tech versions. It would be great work.

1:20:00 Yeah, it would be great work. Much easier to put on the web. Yeah, sure. A few hours I have scanned. But the process is far from being finished. Are you actually retired? Yeah, yeah. Do you have any connection with the university anymore? I have my office and my email and... I can use all the infrastructure. I can even get help from the secretaries if I want and when I travel I have the support from the department but as a former I fill out. The amount I want to apply for, I fill out one corner for 20 cents, because the computer won't accept it if I write zero, but the point about having it is that I'm covered with the travel insurance for the university, once it's supported by the university, then travel into a specific country, so I always make this application apply for one corner. That makes sense. Well, don't everybody speak at once, because you know what you want. Order. Well, off to you. All right, I'll have the eight ounce rums. Medium rare. Rums. And side salad. Medium rare. And what flavor would you like? Chips. Chips and salad, okay. Do you know what you want? All of the favorites, peas and cheese, sir. Are you okay? Oh, I haven't seen that guy. Oh, thank you. And mineral natural water? What's that? What's that? What's that? What's that? What's that? What's that? What's that? What's that? What's that? What's that? What's that? What's that? What's that? What's that? What's that? What's that? What's that? What's that? What's that? What's that? I'm going to join Olivia. I'm going to have the old favorite fish and chips as well. I'm just wondering, I hope this doesn't sound incredibly cheap. Olivia, would you like to save a pound? If we order that as a two for eight, we're having the same. Could we do that? Then we both save a pound, which is quite sensible, you know.

1:22:30 You don't have to have the same thing. I mean, all those, you know... No. But as it happens, it is what I'd like anyway. So we may as well do that, and then we save a pound. In fact, you guys could do that as well, if you want. Thai curry and two very poor things. Mathematics, have you said what you want? Yes, you can have the belly full. Okay, all done. Okay. Would you like anything to drink at the moment or not really? You just want some water, don't you? I'd certainly like something to drink. What have we got on tap? I've got some water too. Some water? OK. A pint of London Pride. A pint of London Pride? Yep. Very nice beer. Very nice beer. Best beer you could buy in England. Good for you. Three pints of London Pride. Four. Make that four. Make that four. Make that four. OK. Good. Excellent. OK. Five. Ha! You see. No, that's a good deal. When I retired, as I retired, they had recently appointed a scoundrel as head of our math department, and he tried to fire three guys. And since I was retiring, they asked me to, as it were, organize a resistance party. You can organize the petitions and everything because what can he do to you? I mean, you're retiring in two months. So I did this. And then I found out what he could do. He took me off email and library prison. That really is Petty Spite, isn't it, of a higher order. The philosophers came to my rescue, so I was turned into a philosopher overnight. And now, I'm a philosopher. There's only one possible Shakespeare quotation that comes to mind, I'm afraid, John. Bless thee, bottom, thou art translated. But, I mean, I'm not paid for anything, but I do have an email account, free access to, I can get a face-to-face call. Yeah, that's a big advantage. Did you succeed in the... Well, one guy did, and the second guy didn't.

1:25:00 It was outrageous. I don't want to tell you about it because you don't want to see a grown man cry. That's what would happen. Bringing all that stuff back. But it was terrible. They were lying. The one thing I did do, I got the Freedom of Information Act to provide us with all the internal emails these guys had been searching for. And it was clear as crystal. The guy they got rid of was a student of Gelton, and had published several papers. And was invited to join the IHES. So, I mean, what does this tell you about the stand? He was so much better an institution than the head of the department. It was basically a number thing. It was disgraceful. Of course, the university behaved... With impeccable, with impeccable honor, they just backed him up to the hill, and then he's once reappointed at a department, and the latest rumor is that he's leaving. I think it was the old British thing, you know. So we're not going to, we're not going to court-martial Al Travers. We'll just put him in the room by himself with a revolver on his chest, and we're going to do the decent thing. This guy would have picked up the revolver and shot his colleagues in this certain world. Limogède is the word the French use. Not here in Bristol, it should be added. We've got a more or less by falling between, you know, they closed down the applied math department, so they offered him, they gave him a job in the math department, well, anyway, he got into sexual harassment, and he maintained his innocence all the way through to the end, and then, unfortunately, he was the victim.

1:27:30 I followed the example of Monica Winspey and the Spanish evidence on her brain code, and this was her DNA and so on and so forth. So the director of Caltech and his head of department called him in. So all this was unknown, but what a bunch of dopes, I mean, you're thinking, why, you know, you're thinking, why would a guy want to do something like that? And the answer was, that would explain anything. And he became chairman. Well, of course, nobody wants to be chairman. Who wants that job? But he did. And his girlfriend got to boss everybody around as well. That's pretty grim. But I was a bit disappointed with my colleagues. They were scared. It was scary how we could do this. I don't think we could do it. Even our very senior members of staff were scared, and I thought that was a little bit cowardly. More than a little. I might have been scared, too, if it had been two years ago. I don't think so. What a joke. The University of Bristol is celebrating its centenary, and it was a working group of students, of course, who started the university. Those were quite worthy institutions, you know, and my God, what was happening? Yes, a reflection of how serious the 19th century was about learning.

1:30:00 Almost every town in England had a Mechanics Institute. Almost any town of any size. Bristol would have had several. They would have had Knightly Extension Lectures. Humphrey Davy and his pals sniffing nitrous oxide. In honor of this, for the celebration, the vice-chancellor has conceded the one-half of the symbol to the University of Bristol, wherever it is, which is announced today. That's lovely, thank you. Here we go. Here we go. It's grotesque. Thank you for your attention. Gradually, the educational establishment of the community reoccurred, but the studies of our time have not done much to do with it, beyond that, let's just say that it's not really cutting back on the connection between us. You are having something to drink aren't you Olivia? I understand you didn't want alcohol, but I would have felt embarrassed if you were sitting there without any kind of drink at all.

1:32:30 I wonder if we'll look at the suspension bridge which is quite impressive. It is a bit of a problem because it's extremely beautiful, one of the most beautiful bridges in the world I think. But, I mean, nowadays, the suspension bridges, they have these high-technological cables with, I don't know how they make them, but in many ways, but these cables are really made of small blocks of, I don't know, cast iron. Yes, cast iron. In fact, in fact, you probably know the suspension bridge was temporarily closed last week, yes, four days ago, in fact, the day before we went to Cambridge, because it was found that one of the... One of the cables, and as you say, they're not actually cables, they are actually rods, that sheared through, one of them had just, it was just metal fatigue, I mean, it just sheared through, and it was closed whilst, that's right, yes, so they just closed it. I mean, it was obviously not remotely close to systemic structural failure, but they... Nonetheless, to be on the safe side, they closed it for, I think, for less than 24 hours, whilst the engineers did a complete check on it to make sure. I'm considering it's almost, well, getting on for 200 years old now. What is it? It's certainly well over 150 years old, the bridge. Well, it was finished in 1864. Oh, it's as late as that, I thought. Well, that's right, because although Brunel designed it, it wasn't completed until quite a long time after his death. Of course, he had died about 18... Yeah, that's right. Yes, quite a long time after, because he died, I think, in 1859. He was, of course, quite an extraordinary prodigious genius, in terms both of energy and of the reach of his... He's obviously a fantastically good schmoozer, I mean, in terms of convincing capitalists to support his schemes, which, considering that most of them were really so much at the cutting edge of the technology at the time, that many of them failed because they were far too ambitious. In fact, he probably lost money for most of his patrons in terms of turning a bottom, and Great Eastern was a complete financial disaster. We're talking about Brunel. But on the other hand, it was a generation, maybe two generations beyond any other ship that had ever been built.

1:35:00 His railway had ten foot... Seven foot. Seven foot. Seven foot. The same gauge which was adopted in Russia. And in Spain, the Great Western Railway was the only British railway that had the seven foot brawl gate and unfortunately because the other railways and another reason Bruno failed financially, he could not persuade the other railways and companies to adopt it. He pushed very hard for it to become the national standard because obviously it would have allowed for much more stable and comfortable travel and far larger load factors, but financially it was... There were a lot of variables of course yes yes yes and that was in fact one of the reasons why it wasn't adopted on the other lines but but having one having one railway which was built on a different gauge from the others was not obviously a practical prospect so after so in fact it ended up being closed. It was adopted more or less at random, but the original reason for it is quite interesting. Everybody knows, I think, that it is exactly the same distance as the distance between the wheels of a Roman chariot. Ah, well, four foot eight and a half, which is the standard international rail gauge, is exactly the distance between the wheels of the Roman chariot. But the reason... It came from Britain. Yes, yes. So it's not a European standard that took over the... The other way around. ...on the British side. Although, in fact, by the time it was adopted, I mean, the railway systems in France and Germany, even in Russia, were already quite developed. But they did more or less standardize on four foot eight and a half, except in Spain and Russia. The reason that it was the same distance as the wheels between the wheels of a Roman cart or a chariot was quite an interesting one. The earliest iron rails were actually laid right back in the Middle Ages. In fact, in the 13th century, iron rails were already laid in the slipways of mines, in the slipways particularly of tin mines and copper mines, for instance in Cornwall, and the carts, the wooden carts on which they... Copper and tin and other things which were mined were dragged up these slipways, used the grooves which had been cut into those slipways by the Romans.

1:37:30 Some nine centuries before, because many of these were original Roman mine workings, and there were places where the grooves still ran, because on all Roman roads had these, well not all Roman roads, but many Roman roads and slipways of mines had these grooves, which were the standard separation of the axle length of a Roman car. The unit of length was not foot? No, no. It must have come out of the deep end. It just came out somehow. Yes, it would have done. The point is that the very earliest trucks which were made for railways where the freight was actually hauled by locomotives. Were the same trucks which had been designed to operate on the slipways of these mines, where they were hauled up and down by fixed steam engines, not locomotives, but fixed on windlasses and on winding gear, the steam engines which operated the winding gears which hauled trucks up and down the slipways. The trucks which were first used on the, because there was no other rolling stock, were these trucks and of course the axle. The width of these was designed to fit the rails on those slipways, which were the standard distance apart because they had been put in the old Roman grooves from centuries before. And that was why those, so when the railways, the first railway lines were laid, I mean the first passenger railway lines were laid in 1820s in England, in Stockland-Darlington, because the rolling stock had those axles, of course the rails were laid that distance apart. ...to mimic the distance apart of the railway workings and the mines, and there were already existing stocks of railway ties and of railway ties, sleepers which they called, which of course were designed for that distance, precisely to those dimensions, because there were many railways already operating in mines. They were either hauled by windlasses or by horse power or... Through a series of kind of contingent causal connections, the adoption of that as the standard gauge, in other words, actually goes back to the Romans through this chapter of accidents. Brunel, of course, would have nothing to do with that. He thought that was absolutely stupid, I mean, to just lay down under this purely accidental reason for choosing, so he wanted a seven-foot gauge because, well, it was, I think originally he wanted a broader gauge, he wanted about a seven-foot, eight and a half-inch gauge, but he had to compromise on seven-foot, but when the Great Western Railway was laid, it was laid as a seven-foot.

1:40:00 ...and operated on that for the first 30 years or so. And then of course when it failed it was bought up by, it was emerged with the other railways and of course it had to lay new track with the, for a long time it did actually just lay a third rail, and they still ran some of the old seven foot rolling stock on it until I think about the 1870s or 1880s, then they just simply removed the other rail. It ran from London to Bristol and then later across the estuary to Pembroke Dock to the far west of Wales where there was a boat train to Ireland. Originally it ran to Bristol and it was extended later, only a little later, I think it's still in Brunel's lifetime, to the west of Wales. And also down into the southwest, it had a spur which went all the way down to Penzance. Because there's another famous bridge he designed across the estuary that separates Cornwall from Devon, from the west of England, which is the very last thing he designed, in fact he finished it the year he died, 1859, which is the, what the hell is the Tame? Which is still in use, which is an absolutely superb piece of engineering. I thought that the reason why they have wide gauge in Spain. It's the same in Russia, of course. They adopted a different gauge, and it was very effective, and in fact it actually had a significant effect on the outcome even of the Second World War, because one of the things, because Stalin gave the orders for complete scorched earth policy, so they tore up the tracks anyway, but even in the places where the Germans captured the railway system intact, they couldn't of course run any of their rolling stock on it because they had no seven foot... locomotives or rolling stock so they had to lay the additional third rail in order to run their rolling stock and that took many months and given the condition of the russian roads in winter it it i mean some some of the military historians think that it well because there were many many factors which contributed to the to their defeat but but this was not an insignificant factor

1:42:30 they had a huge logistical problem they could not rely on the road net um the because of the And they were quite unable to get the railways running effectively in order to move their supplies until well into the 19th, second or third year of the war, and even then on a much more limited basis because, and that was the reason that the Russians, well actually the imperial Russian government had chosen a 7-foot gauge, precisely so that there could be no interoperability. It goes back to 19... Oh yes, it goes back to the Zars. It was a decision which the Zars government took, that they would lay, that they would adopt a different gauge. In fact, the change, you can still see in Warsaw, it actually changes in Warsaw, the gauge, because Warsaw was effectively the frontier of the Russian Empire. And they used to have, in the main station of Warsaw, they used to have this incredible system whereby... There were some carriages which were designed to be run on different sets of bogies, one of the standard international gauge, but they could also fit on the wider, and they had these huge cranes which actually moved them physically across. They adopted the same thing again for military reasons. Thank you for your attention and see you in the next lecture. It's extraordinary business. But no, that was an entirely military-based decision we were going to make. Hitler had a kind of insane plan, because Hitler was obsessed with railways. He had an insane plan which was actually developed in considerable detail by the Reichsbahn, the German railway system under the Nazis. Had they actually conquered Russia, they intended to relay the entire railway system on an 11 foot plus 11 or 12 foot gauge. They had these plans with these gigantic locomotives and these huge carriages which were going to be about three times the width of any others that had ever been operated.

1:45:00 It's part of this plan to take all the German workers to the Crimea, which was going to be turned into a sort of huge summer. There's a set of plans which you can see in one of the museums in Berlin of all of these designs for the new rolling stock, complete with carriages which were the cinemas, entire cinemas which would have travelled on the train to entertain them whilst they went along the Trans-Siberian railway. You know, a journey of, even today, is a journey of something like eight or nine or ten days. Quite crazy, but, well, typical taste to go along with her gigantism. I don't know how... I read a thing from one of Wallace's... I'm still not reconciled with life. With life? With life. I still want to know what is the good polizer in the Sarisky talk. Well, then you calculate it. I'm not in the Salisbury table, but somehow the point is, for geometry purposes, you want anyway to go further than the Salisbury table, and the sub-tables of this, of that again, where the co-equalizer does become one. So you simply want to... Right, but you can do that, right? But perhaps you lose tea in the process. This is a question of epsilon stability. Yeah, which we were talking about this afternoon, yeah. So forcing art to be connected might not be a stable ratio. In the discrete cases, you simply go wrong. That's very interesting. You cannot force by geometric logic.

1:47:30 You can cross all the statements, for instance, that every... If a Newtonian element is the sum of squares, that you can say, well, that's a geometric. But you can do something else. You have zero into the zero point embedded into the phi zero point. And you can put the largest topology where it inverts that value. Yeah, there is a topology. ...which forces this map to be an eyesore. In the process, you might lose the... Yeah, yeah, but that's... Well, because he was... ...translates into the fact that reversing that map is not his mistake. Yeah, yeah, yeah. Well, it's not about whether the map is a mistake, but whether the notion is a mistake. Yeah, the notion. But... It's quite easy to check, but it used to be quite simple. And actually, one of the benefits of thinking, one of the benefits of having got all of this business about the role of homosets clear is that it does actually allow us to make sense, really, pretty complete sense of what these people in the 17th century like Wallace and Newton were debating about when they said that they didn't see how you could mix quantities of different kinds, using the Greek definition of quantity. It could be Pythagorean, and it should be, you name it. There are many prophecies we would like to learn from somebody who is more scholarly than me. Pythagorean means that one plus any square sum is invertible. In other words... I'm sure you do. Well, there was actually a brilliant talk on this that I heard in Nancy in December at a conference with Jeremy Gray held on the history of the notion of quantity from the 17th to the 19th century, which I did, but I didn't receive an answer. I did not receive a satisfactory answer. Because that's an invertible square sum. Invertible square sum, you have the square roots.

1:50:00 But it was quite clear that... So there are many promises you want to enforce. So you shouldn't... I was almost saying waste time on calculating the reflection. I'd be specific on these. Intermediate stages like local ring classifier. What would you say is a good test case? It is, yes, sure. But what would you say it is? It wouldn't be a waste of time to check what it is? No, no. You could possibly... I don't know. I'd like to understand. I would agree. The numbers are... The plethyserisminon, a bounded plurality. Some salt, some ketchup, some oil. Actually, somebody made a little joke about that to quite good effect when Michael Dummett was given the Lakotosh Prize in the LSE about 10 years ago. I happened to be there. And I think it was Clive Kilmister who said And what Clive Kilmister said when he presented the award was, Michael Dunnett is the author of a number of weighty. And highly praised books about Frege and the philosophy of mathematics. Of course, this statement is true if you understand Aristotle's definition of number. The same statement is true, of course, of everybody in this room on Frege's definition of number, which indicates there may be something wrong with Frege, which I thought was a rather good... There was a ceremony where Dummit was given a prize, the Lakatos prize, which I think you've heard of, and the man who gave the prize to him made this remark, which is apropos to what John has just said, that I need a lot of stuff to mop up after myself.

1:52:30 And he said, speaking as he gave the prize to Dumbledore, well Michael Dumbledore is the author of a number of weighty and well-reviewed books on the philosophy of mathematics. That is to say on Aristotle's definition of number, of course on his favourite philosopher's definition of number, the same statement is true of everybody in this room and indeed of everybody in the world. Well, of course, zero is a number and most people have never written any books on the philosophy of mathematics. It's bloody obvious. It's an extension of a concept, so... The timing characters, you know, it's so good. Yeah, yeah, I mean, man, there you go. I hope that was inspiring, wasn't it? Yeah, no question. Really, if I could stand this business about the absolute co-equalizer, the business of the absolute co-equalizer that you were discussing on the first day, connection with the barge and how they can do it. Well, the absolute co-equalizer has a very impredicative definition. That's a co-equalizer that is preserved by any single quantum, period. I think the last is coming on your page. Who's this? Oh, yes, yes, of course. Yes, he was an arch-conservative. He was also an anti-Semite and a Francophobe and various other distressing tendencies. Oh yes, well, like most Russians of his generation, so...

1:55:00 That looks jolly good. Mmm, wish I had taken it off, don't know about mine. What is it? That's the pork. I'm sure ours will be good as well. After you with the sauce. No, no, it just means the rib. It means cut from underneath the belly. I think it's to go with your. Thank you for watching. Thank you for watching this video, I hope you enjoyed it, and I'll see you in the next one. What was it that started the stability of this business which explains me about the, in the case of the infinitesimal construction, this?

1:57:30 It's a question of the, you know, the stability of things. Yeah. I mean, is that reflected in the way that coverings behave, that the apologists on sites behave? Is it connected at all with the condition for coverings to localize? I think so, yes. Yeah. I don't know what it has to be. Well, I don't know what you think. Well, I think it was connected with that. Because that's something which has long interested me. Thank you for watching. So these sites are usually co-sites, so coppers push out, that's co-coppers push out, but somehow the point is that they also reflect that something pushes out along one of these epsilon extensions. This is where I was first introduced to the topos of smooth spaces by Colin McClarty a long time ago. How did Colin do it? I've spoken to him on the phone once since then. He seems to be in good form. Yes, exactly. He's no more a youngster than I am. He's one of those young guys that's coming up to 60 or something. Well, he and I are exact contemporaries. So, yes, in fact, we'll be 60 at the same time. Not quite the same month. I think, actually, he's born in February and I was born in March, but what are you? I'm 58. I'll be... Yeah, well, so am I, so exactly we're the same age. But has... has anything more emerged about this book that he's telling me that he's submitted to?

2:00:00 Well, I've forgotten whether it's OUP or CUP now about the category of categories. He seems to have been guarding completely because he didn't mention anything about it at all in the course of a whole week of discussions.