Breakfast conversations A Bauer & S Awodey / J Mayberry / lunch conversations FW Lawvere, D Bernardini & Silvana Lawvere

0:00 There are a number of different ways to learn about mathematics and physics. There are a number of different ways to learn about mathematics and physics. There are a number of different ways to learn about mathematics and physics. There are a number of different ways to learn about mathematics and physics. There are a number of different ways to learn about mathematics and physics. You will come to the audience, you won't have farce at the end of the day. You don't really care about farce, you don't really care about math. You don't really care about math. You don't really care about math.

2:30 You don't really care about math. You don't really care about math. You don't really care about math. You don't really care about math. You don't really care about math. You don't really care about math. There's a commissioning center that's going to take the data back and then you can work with all of these people and you can work with all of these people and you can work with all of these people and you can work with all of these people and you can work with all of these people and you can work with all of these people We don't even have that here. We don't even have that here. We don't even have that here. We don't even have that here. We don't even have that here. We don't even have that here. We don't even have that here. We don't even have that here. We don't even have that here. The small power of the computer contains at least all the regulars of our program of physics. So it's more difficult to keep up with the number of people who train you. Now, this is a new subject for me. I don't know if you've ever heard of it, but this is a new subject for me. I don't know if you've ever heard of it, but this is a new subject for me. I don't know if you've ever heard of it, Thank you very much.

5:00 There's one thing I want to know about this. There are a number of different fields of study in the field of mathematics, such as physics, geometry, algebra, mathematics, physics, and geometry. There are a number of different fields of study in the field of mathematics, such as physics, geometry, algebra, mathematics, physics, and geometry. Thank you for your attention. If you want to do math, okay? Yeah, I'll take care of that. Okay, first two questions. Where did you need the last one? The real numbers are the answers. No, false. Oh, okay. The first one, I thought I needed. Right? So you have to identify the quotients. Then we can say we are the sequences. They will fraction around. Well, I'm going to define the time of the lecture.

7:30 Your sets are basically equipped with some topology that makes everything work. You don't just throw away anything. But even if you want to, you can find one. Can you just say a little bit about how that works? Is that to do with this business of the V8? Is that to do with this business of using continuous languages, G0 spaces? Yeah, so the underlying thing is already, you know, you have some continuity built in, so then you... So, for example, in Dana's talk, he said, well, what is amorphism? It's something that is cracked by continuous math. How can you have a continuous math that doesn't? So, secretly, everything is continuous. You still may want to, actually, it's still very important that you know how to define an amorphous map. So, what can you do with this? So you just... Oh, sorry. The map says for every open... I'm on my way. Thank you very much for your time. The set of all open subsets, although you don't need a power set for that. That's just sigma to the x. So again, you don't need a power set. You just need for every element of sigma to the x and the topology of x in this case. So, in general, systematically you use power sets by certain exponentials of certain sets. That's a good way of putting it. That makes perfect sense, yeah. Okay, thanks very much.

10:00 Thank you for your attention. I'm just going to go to John at the station to see him. See you at one. That was really interesting actually, isn't it? Oh yes, it was as well. Oh it's okay, you still have plenty of time. You've got your ticket haven't you? And you say all the things that induce anxiety. We've got plenty of things that induce anxiety. It's all right. It's years of practice. Well, that's very interesting, actually, what Andrew Brown was saying just then. I'd really like to understand more about this whole business about these TO spaces and continuous lattices and how they allow you to do, actually do math without parsecs. Very interesting. When you're talking about your intellectual frustration, I'm absolutely convinced you can completely understand what I'm about. No, it is with respect. Yes, I think if I'd started, you know, with the right proof. Oh, yes! Yes, yes, yes, I think I can. I think I can. And that's not because it's all difficult and hard stuff. It is. But it's difficult in a different way. But it is. Yes, it is different. It's more difficult for mathematicians because it's like saying, it's like telling a bunch of jockeys that they're really... They don't know one end of a horse from another. Not really to be riding facing the tail. No, more than that, you should really get on a donkey. Yes, that's it. I think that's probably what... Probably where Peter Johnston was coming from in his reaction. No, but Angus got the point immediately. He said, that's very interesting. He said, I've had thick thoughts along those lines myself.

12:30 And we talked a lot about this. Thank you very much for your time, and I hope to see you again soon. They found it hard because they weren't taking up machinery with them. I mean, I've got to think about that. I mean, that's just the technical thing. That is very interesting. And of course it's precisely through that work on exponential fields at Angus's that the whole thing ties up with this. There's definitely a kind of convergence on some kind of... I hope we get more about that, but I guess you can't have everything in one meeting. You all seem to have taken a step backwards in his understanding of your stuff, but maybe that's because he was concentrating so much on having to think through his own thoughts. I mean, my stuff is the thing that became clear to me. And the connection seems to be through the way that category theorists, or Bill in particular, think about exponentiation. No, it's not? Okay. The connection is that the category theorists have a way of thinking about incunitary stuff that's completely constructive. Yeah, except they would never use that word. Well, it is. No, they do use it. Well, except Bill hates it. Bill doesn't like it. Talk about constructive definitions. Because he sees it as tainted with this braille and subjective epistemology, he wants to say... But the constructive part of his stuff is dealing with explicit algebraic constructions. Yes, yes. They're not interested in bare existential propositions. In fact, that's the whole genius of it. The thing about algebra is that it can... Wherever you can stick algebra in something and turn it into calculations, what you're actually doing is simplifying the logic to universal logic.

15:00 And the existential propositions, all... What revolves around constructing various maps, various functions? Precisely. So it's certainly constructive. The key insight. No, in fact, I completely agree. Okay, constructive in that sense, yes. I completely agree with him that this Brauerian tradition is dead. Because if he thinks that's what I'm doing... That's a philosophical point. No, I don't think it is what he thinks, but it's what tends to make him bridle when people start talking about constructive logic, rather than... No, I know it's not. I'm quite sure he appreciates that. I'm sure he got that at first glimpse. I don't know. Why should he? The motivation for this stuff is not, you know, constructivist, it's, um, for this business of the semi-algebraic sets, the, no, hang on, I think it's, has it come up yet? Is that? No, still not. Still not up there. It's okay, they're going to. That's Milano, that's not yours. No, it's definitely not come up yet. Well, it's okay, it's not like in Britain, they do leave you with enough time. I really do, I really would like to get this train. That's your carriage? Carriage, seat 42. That's it? Yeah. Mind you, it probably won't be particularly cool. Oh no, Sunday, I doubt whether it will be particularly cool. I think the train's probably more crowded on the Fridays than the coming back tonight. It's funny, as we were coming over here, I was saying, why is Santa Maria Novella ringing the bells? It's Monday. Yeah, well, that happens. Yes, I thought that yesterday felt like a Sunday. That's, they have that very moving little plaque, inscription here, it's to the people who were deported. It's on one of the columns here.

17:30 Well, I can tell you that there's no such thing as completely understanding. No, you see that even with Bill. The illusion of complete understanding is probably the mark of intellectual... Some of the things he's talking about are very simple, and he uses ordinary terminology, but on the other hand, the ordinary terminology is very suggestive, because he's telling you this is how to think about this. That's a really important aspect that I'm having. Well, why don't you just explain to me what the motive for this whole topology business and the semi-algebraic sets is. It's just to do with making sure that functions can only become, as it were, when it gets wavy, it only still has to go through. If I may say so, it's a constructive view of it. Yes, but it's a very geometrically phrased constructive view, I hope, because that's the key thing. It's very, very geometrically motivated. But of course it is. What could be more constructive than wanting to do away with the input-shipping RL, with the natural numbers? It's constructive in this algebraic sense. You want to be able to arrange your concepts so that everything flows by calculation. Well, that's certainly the algebraic aspect of it, but there's also this very important motivating geometrical idea behind it to do with the nature of space, and the set theory ultimately kind of fits inside this picture of space. That's the other aspect that is very important. He doesn't like the arithmetical view of space, but the arithmetical view of space is going to keep its grip on the mathematical imagination until people give up thinking that the natural numbers are just a given. That's a very profound remark. That's precisely, I think, what he's driving at all this time. The geometry, naive geometry, plus that conviction, impose the Weierstrass signal integrity. Because it gives you the idea, for example, that real numbers are infinite decimals. And there's only one way to be an infinite decimal. That's the idea.

20:00 Of course, when you look at thinking geometrically, you don't necessarily think of that arithmetical aspect of the continuum, but that's essential. Because that's how you get numbers out of it. No, you've got to recover all that stuff within the geometrical picture, which is what I take all this business about trying to express everything in terms of these conditions on the pi-o or pi-1. ...components like that, and this is to do with why he sees that as such a crucial aspect of these nested agites, and that's what's weird about sets, they've got all these nested agites, they're much more than almost any other category, but that's because... This seems too simple. The thing is, he's got a... If he wants to capture the imagination of mathematicians, he's got to shake them out of their standard views. Harvey Friedman is an ally in this, and he doesn't realise it. Bill doesn't realise it. No, that's the problem. I think Bill, because he's got such an incredibly original and creative mind, doesn't see just how much breaking up of the previous... Not even so much laying the groundwork, but you've actually got to do a demolition job and break up all the existing foundations of looking at the subject before you can even start laying the groundwork for this alternative viewpoint. You can't as it were just plop the alternative viewpoint down in front of people. I've been certain from the beginning that you were, even back in Cambridge, although it was then much less evident, that the big problem... The other thing is he wants to say so much at the same time. He has the kind of mind that sees connections so easily that it's difficult for him to say linearly what he understands synoptically. The point is mathematical analysis, number theory, these are central disciplines in mathematics. The people that work in these things do hard things as well. They do hard things. They've got an inner conviction that they've got a clear idea of these things, and when Bill says this isn't right, they're going to say, well, okay. But no worries.

22:30 It's these central things. And the beauty of his way of looking at things is that if I don't think, if I were to convince the whole mathematical community that I'm right, almost nothing would change from this point of view. Because he sort of already thinks that. For him, you know, he talks about the natural numbers being subjective, mathematics, and all this kind of stuff. Oh, that's perfectly, you know, I mean, that's exactly what I think. It's just that he, having thought that, he just dismisses it. I'm trying to think, how are you going to get these guys that have got what is after all not a contemptible view, which is the product of... ...generations of standing on the shoulders of giants and of incredible... How are you going to get them to see that there might be something wrong with it? That you've got to look more geometrically, that you've got to be more constructive and build things, right? Yeah, I agree. It's a difficult task. I mean, you're absolutely crucial, typically, in all this, in that you really do get the motivation, the philosophical motivation, the historical context. I'd say a lot of people don't. It's quite clear Peter Johnson doesn't get that at all, because that's not his kind of mind, and his kind of mind is always incredibly powerful, very, very, very powerful technician. Colin and John could see what I'm doing, but they won't, for different reasons. John, because he's just so... It's almost impossible to penetrate this depression and this sort of self-integration. Yeah, I mean, when I look at him, I see myself in the mirror, although I also don't think I'll ever get as bad as... And I'm sad, it really is. I mean, I have that, the book. And in Colin's case, because Colin is a fascinating guy and, you know, the master of... There's a touch of perversity. But there is a touch of perversity. There's a touch of... Well, I think it's actually a touch of cynicism, I have to say. No. It's pride. Thank you for watching.

25:00 Well, his own profession is at least as much maths as philosophy these days, and I don't think he could stay around. But he insists on... I'm curious, I'd like to understand more, but I actually don't know that much. You acknowledge the perversity, because you've seen other aspects of it, like the varied others. Yeah, well that is perverse. Thank you for watching. He is very, very impressive, and he likes to impress people with the fact that he can argue any position, whatever you have to say, he'll immediately see. He's one of the smartest guys I know. Oh, yeah, yeah, he's fantastic. He would have been absolutely fantastic as a lawyer if he'd had the problem. I hate to think of his talent wasted. Oh, I would hate to see him as talent wasted. Yeah, but he'd be a much richer man, which would not be particularly... Not a consideration. Not a consideration for him at all. No, no, no, no. I don't think he's cynical at all. But in terms of mastering a brief... Okay. Well, I mean, the cynicism masks... A kind of disappointment at not being recognized as standing head and shoulders above the people that he's going to conferences with, which he does. Yes? Absolutely. I've seen that. So, so... It must be frustrating. Yeah, it is frustrating. And the only way you can get around it is devotion to the truth. To the truth, yes. Alas, he doesn't believe there is any. That's my problem. There's just the various levels of, you know, beautiful conceptual organization and cross-relative consistency proves that truth is out. I think that's a false reading of MacLean, that MacLean doesn't believe in truth. But MacLean's not a philosopher. No, no. To the extent that he has ever... I don't think that... I think the only reason for his references to Popper is that he's probably the only philosopher that MacLean's read in the last 30 years. I don't think there's anything in significance of Popperian about him. In fact, if there's any philosophy at all behind the synoptic view about the nature of the subject expressed in mathematics form and function, it's actually a very old-fashioned Lockean abstractionism.

27:30 These forms suggest themselves in the structure of the world around us. But that's an accurate description of the psychology of a mathematician. I've been around enough really good mathematicians to know how they think, doing different things. Yeah, yeah, I'm sure that's right. And I've learned not to tread on their toes in these classes. But if you are effectively going to tell them that, they all ought to... Thank you very much for your time, and I look forward to seeing you again in the next lecture. It's all right. I mean, you know, if you forget to punch it, then when the guy comes around, you look for the dumb, you know, or a dumb straniere, then you're all right. You know, I feel for Stalin. I feel for his frustration. But, again, that twist of character that... It makes him unsympathetic to other views, you know, or maybe skepticism, or I don't know what it is, but he's not, you know... Cynicism was completely the wrong term to use. You don't attract clients with vinegar, you use honey. Yeah. I mean, it's been a... I mean, I've made some good friends up there in St. Andrews. But those guys haven't a clue of what's going on. Well, they're just locked into their own world. There's no point. I could get up there and denounce them. But that wouldn't do any good at all. They'd just lose interesting contacts with nice guys. I'd just drop homeopathic doses of strictin in their teeth. No, it's absolutely weird. I can't get what they think they're doing at all. Well, it's partly because they're not looking at these things from the point of view of a mathematician.

30:00 No, they're certainly not doing that. I have to say, I think that Michael Dunlick has a lot to answer for. Because by making these bright people in philosophy departments think that they could come to, that they could... Do philosophy of maths as a branch of the theory of meaning, but essentially as a branch of philosophy of language, because what this all boils down to, which is why this canonization of Frege is the source of all wisdom, insight, practice, teaching, mathematics, but just consider Dummett's work on Frege, one absolutely massive, great big thick book, Frege the Philosophy of Language. And then a long way, you know, sort of 15 years after, an afterthought, an essentially really disjointed and scrappy, an inferior book on Craig of the Philosophy of Mathematics. Well, I enjoyed that book because I saw a lot of things. I'm not suggesting there's not a lot of good stuff in it, of course there are deep philosophical issues, but I still stand by my point that by approaching all of this through semantics, through the theory of meaning, through philosophical semantics, you take completely the wrong point of entry into the subject, you don't see the connection, and you're ending up with the mindset that dismisses All that you and Bill see is, oh, well, that's just technical apparatus, that's just mere algebra. Well, even Colin suffers from not having Colin. I thought he could teach. He doesn't teach mathematicians. I thought he did, actually. I thought he was adjoint professor in the math department. He's not teaching undergrads. He's not learning how to teach. Teaching people how to think about real numbers, the conventional real numbers, you have to teach them. That's what I would have benefited from enormously. I realize now I could easily have done it. Well, I could have done it, although I certainly don't naturally have a mathematical mind. These things are rather simple. They're simple. And from some points of view, sort of crude, precisely because they have to be absorbed. You're not asking people to believe things, you're showing them how to do things. Yes, but the one of course does lead into the other, at least precisely to the barrier, which you now have to try and get through. Getting people to see that there might be another way of conceiving how the whole thing fits together.

32:30 Doing really advanced stuff. Close to foundations. Yeah. Skeptical about the whole business. Yes. Yes. That's what all these guys are, except Bill. Yeah. Yeah. Yeah. Yeah. But this is because you have to narrow yourself to, you know, to... Well, and you have to master an awful lot of very demanding machinery. Yeah. And you have to... I mean, I never teach stuff that I already know without learning something new. I've taught an awful lot of analysis of the sort of crazy advanced stuff, you know. Where it has to be done. But that's another thing which I've done. It can't be done. Old fashioned applied mathematics. No. But again, that was another thing which I wish you had been there. When Bill, what the hell is they going to put this bloody thing up? There's another. I think it's going to be one of these. What's the time now? I think we're up soon. Oh, hang on, it's delayed. It's delayed, I think. Oh, no, it's the previous one. Well, the previous train is retardo by 15 minutes, so it could be retardo, but we'll see now. Is that what that means? Yeah, RIT is retardo, means 15 minutes late. But that's not your train, that's the Rome train. No, but you see, the thing is... It's a terrible thing to resist all these temptations like intellectual dandyism, skepticism, professionalism, you know, like Sartre's waiter. All these things are terrible. They are. Intellectual temptations. They are. Especially if your grumps commit them as a substitute for despair. That's, of course, my danger. That's why I really have got to get serious and rigorous and organized. I know from talking to Martin about the problems of curriculum. I know that he knows these things. Yes. But I have to make him listen to it. I wanted to convert him, say. Martin is a villain.

35:00 He's certainly far more sensitive to philosophical questions than Keith Johnston would ever be. It's still not up on the other one either. It's okay, it's not going to go until they put it up. But that's stuff about the... I actually got that for the first time through having Bill explain it to me one-to-one when we were having dinner that first night in Zaza. I really got that. ...completely for the first time, really understood what was going on. It was fascinating to see how deeply the geometrical motivation runs in this town. And somebody needs to go back, I mean they may have done it, but I think there's a book in German which I... ...and look at what actually happened historically in the 19th century when... The Eustachian analysis came to dominate the outlook of mathematicians and the people were forced into... Neither of these are Eurostar platforms. The Eurostar platforms were pure. Well, you can never tell because they... I don't think there are necessarily dedicated Eurostar platforms. Well, no, actually those are, yeah, that's true. I hope that doesn't mean you have to go all the way down in the end. No, I don't think so, because it's just impossible. No, that's right. Well, it's got to be these three. The problem is, if you're standing there, you're too far away to see the bloody platform that comes up. Well, I can sort of see it. Yeah, well, you're a little better than I am. Well, I can see it, but I can't quite... And of course, there's also... I mean, that was fascinating itself. I mean, this business about the paleontology and seeing the motivation there clearly for the whole time. Really, in a very intuitive way, I'm just preventing these functions from becoming so wavy, or resisting this, as it were, redefinition of the notion of waviness, if you like, that one gets from the arithmetization of the continuum. Yes, one can no longer appeal to geometric or kinetic intuition in any form.

37:30 You do get wavy functions like that in ordinary analysis. Sure, sure, sure, sure, sure. And they're standard examples and people have to understand them in order to... Well, people have to understand how they arise, yeah. But they can see how the whole... I can see exactly what's happening now, that this business about the components is going to be... The problem is... The components function of the space being the key to understanding how you see all of the structures fitting together. Let me just point out the following. There are two very simple functions. The function f of x equals x squared, which is the parabola. And sine x. You couldn't do any kind of mathematics without the two of those. And the possibility of dividing. X squared times sine 1 over X, you get infinite oscillations, infinitely small in the neighborhood of zero. Now, there are all kinds of fancy things you can say, well, you can't separate components up, but you're not going to sell that to mathematicians. That's so clear, it's almost visual. If there's something wrong with that, you've got to say what it is. The only way you can say what it is is by using infinitesimals. Yes. Precisely. Okay. So infinitesimals have to come back if you don't want to get... if you're unhappy about those things. Talking to Angus... The existence of the sine function in his systems is really critical, because if you've got the sine function, you've got the natural numbers back. No, it's still not there? No. If you've got the sine function, you've got the natural numbers back. So you've got to understand what that means, otherwise... Otherwise, so the taint apologists are taking artificial means to avoid having to face this problem in the field that they're looking exponentially close to. When you go to the complex exponential, you've got these. You've got the time function.

40:00 Just moving from real to complex means that you've got the natural numbers back as well you've got the natural numbers as well of course i mean so what angus is trying to do is what angus is trying to do is to solve this extended problem where you um so i'm keeping oh hang on it's just come up i think what's that say no it's retardo No, it's not retarded. They just still haven't told us, they still haven't said which platform. But it obviously is retarded. Uh, yeah, what can you do? I wouldn't, I wouldn't worry. I mean, they must, I don't... And it's obviously late, or they would have cut it up. There's no other place but this central bit, is there? No, none at all. This is the whole station. I mean, I know this station like the back of my hand. I thought so, but I won't... No, no, definitely. Well, it's late. Well, it's late, yes. It's late. No, no, it's okay. There's no way you're going to miss it. But I'm glad you're here because otherwise I'd be shitting bricks. Especially after all that breath I've eaten in the last few days. Yes, actually, I did notice that, that when you go to God, I mean, eat some fresh fruit. It's the worst thing they don't give you in Italian restaurants, is fresh fruit. It's still not there. Still not there. It doesn't even give the facts, does it? No, exactly, that's what I'm looking for. But it obviously must be late, otherwise... Right here. Don't you know a car there? Well, that could even... Yes, that's where I'm going. Certainly I'm not going down there. The other thing I'd like to get much more of a feel for, and then at least I'd know... The building blocks that I needed to get a proper understanding of it is all this stuff about Galois theory. There's obviously a close parallel to the construction that he's aiming for and what it ties up with the whole business of the component sphincter and these spaces and the way that it connects with the train topology program. It's clearly connected and they accept that it's to do with this whole business of field extensions and the problem in understanding these things is that you get close enough to it and you see that you've got the grub. Yeah, that's it. You see you can't do it...

42:30 No, just by... There's no bird's eye view. You can only do it by, as you say, literally going down and digging out every bloody worm. The only thing that... Mathematicians are united in thinking that's a beautiful theory of every kind. And the reason is to just take the really hard problem and just make it unfold by conceptual analysis. But there are lots of things you get puzzled about when you're teaching it or when you're learning it. There are lots of subtleties about... We had a marvellous discussion. Are we still allowed to hear them?

45:00 Okay. You don't mind, do you? I really would just stay quiet. Oh, if you want to continue in Italian, that's fine. Yeah, I realized I was talking too fast in English. Well, I really would like to get something for this afternoon. Thanks. And you're okay. Oh, I'm good, I'm very good. By the way, last night, Bill's been absolutely fine, his cough was perfect yesterday, no problem at all, but last night, I thought I heard a very poor puffer going on, that wasn't him, was it? Um, I don't know. I think it was John Bell, probably, across there, was he? Thank you for your attention. I haven't put so much effort into it. It was exhilarating stuff. They were all tremendous. The last two particularly, which unfortunately I didn't get to hear, but obviously I'll be listening to the recordings and the webcam. And we managed to sort everything out with the hotel and with the flights in the end, in spite of some horrendous problems which we all didn't know about. Bravo, bravo! Yesterday morning I was going on almost suicidal because I didn't think they'd make it. Oh, no, it's all right. They're engaged. They're deep in conversation. We'll wait for them. This seems to be a really hopeful collaboration. Who's the CV boy? Who's the CV? What do you mean? That's the guy my dad sends me to see what he thinks. I'll check this guy out. Right. I'll look at that.

47:30 Oh, oh, oh, well I'll say no more about that. That's between you and me. He's a really nice guy. He was funny. Well... If you think you could handle having a mathematician for a husband as well as for a father... Ah, sorry! Don't need to... No problem. That's a really interesting guy. Nah, just kidding. Ah, that probably also explains why they wanted the out-of-the-way this morning. I found that he had made his own summary and outline of a great deal of material, in a way which I found completely convenial. I was pleased. That's what you need. Well, please tell me. I'm sorry if it's not... I didn't ask you to rerun everything, but just tell us a little of what's been done. Let's just do something to it. Shall we try to delay ordering until they can get a table outside and we can maybe have a... Thank you for watching. Thank you for watching. This is the capital of the world unfortunately. That's the classic illustration of what happens when a place becomes reliant on mass tourism for its economy and when all the old core businesses are driven out of the center.

50:00 My mother speaks Venetian, so at one point she asked for something in Venetian, and they literally were putting the food like this, and they were like, oh! Yeah, I know, I know. It's sad because you realize how the tourists get treated, you see it all the time actually, and there's a separate menu for the locals, and a separate price. But that's the only way that people... You know, the local people can possibly afford to stay in Venice. As it is, I mean, the people have been driven out of the historic centre so that it's all become just a theme park. The French are finding the same. Huge. Yes, yes, yes, yes, yes, yes, yes, yes, yes, yes. They always take advantage of those things, alas. I was going to leave tomorrow. When are you off to Milano? Can I come with you on the train? I mean, I've got to go through Milano anyway to change trains to go to Parigi, is that okay? I mean, that's how I was going to go back to Paris. Yes, that would be good. That's okay with me. You have to review reserves. Oh yes, I know that kind, yes. Which train are you going on, actually? With Cohn, with Alan Cohn. Trying to pin him down on the definition of... You know, Giuseppe Longo, he was here. He's from Pisa originally, but he's now at the Ecole Normale in Paris. So he had invited me to speak there at a meeting.

52:30 There was the same month, it was just before the Truesdell Memorial, so it was first there. Are you asking the same question? I was not. I was talking about non-community so-called geometry. And we had a discussion about that. Discussion. Discussion. Yeah. I know what that means. It was a discussion into which I felt, well, there were parts of it I heard, but other people had the same impression, into which Kahn did not enter in a true scientific spirit. He simply wanted to evade discussion, it seemed. At least he was not willing to give explicit answers to important questions. I think this is one important thing that I learned from Truesdell at the time, which was more than 40 years ago. I was just a boy from the farm. I was part of the university. I had no idea how to behave in scientific circles. In fact, most scientists are very reserved. They don't like controversy, but Truesdell, he had no hesitation to speak loudly against something he thought was wrong. So this kind of tradition, I thought, well, this is the normal way to behave. Later I was severely reprimanded sometimes, which is not the way to behave, but I still think it is. So if it's not intolerably pushy, I mean, can you tell me a little of the subject of your discussions this morning?

55:00 If that's not... I'm sorry, I shouldn't really. I guess. No, I didn't think so. I'm sorry. If you prefer to move on to other things, I understand. There'll be other opportunities. His summary of the relation between reality and thinking, and the bootstrap process giving rise to, becoming parametrizers, and then the price of reality. He pointed out that there is an analogy between the prolongation of time and the averaging of time. In a given time, four states and nine states are characterized by averaging first Cs and UIs at a given time because of their prolongation. The opposite cases of splitting of a map, splitting of a restriction map, actually occur in some sense in that way. Important examples. Yes, thank you. That was a very good observation. I have this cryptic remark that somehow Galileo discovered the fact that states of becoming are anticipated by Fibonacci, I say.

57:30 Yes, I recall the remark very well. This is too cryptic. You didn't know what I meant by that. Fibonacci numbers come out of the fact that pregnancy is the generic state of becoming. This is a generic case in that, for example, it becomes a metaphor. For example, when Marx says the whole society is pregnant with the new, it's exactly that idea that there is a becoming which is now, and yet it's still becoming. Yes, and the connection with Fibonacci and Marx, Raya, Galileo. In the moment, it seems it's sort of understood. Why don't you explain why you would be interested in such a job as ours? Well, we have a problem with epidemiologists. And so we have often a problem with causality and confounding and how to illustrate a problem. I'm trying to learn better ways of illustrating the problem because most problems in my field, people don't really look at the problem. Thank you for your attention. One fundamental problem that we studied is the relation of nicotine addiction and cancer and so on and so forth.

1:00:00 In terms of this word cause, there's basic agreement that... There are many pathways in which one could attach the epithet totality. Spaghetti, natural, special. Spaghetti with plain and fresh tomato, and they come here with this filling, with gorgonzola cheese, or just simple grigol. Spaghetti. Spaghetti levon. Spaghetti levon. Do you want... Antipasto. Antipasto? Bungalow. Bungalow space. Bungalow. Bungalow. Bungalow. Bungalow. Bungalow. Bungalow. Bungalow. Bungalow. Bungalow. Bungalow. Bungalow. Bungalow. Bungalow. Bungalow. Bungalow. Bungalow. Bungalow. Bungalow. Bungalow. Bungalow. Bungalow. Bungalow. Bungalow. Bungalow. Bungalow. Bungalow. Bungalow. Do you think there's any chance that it still looks very full outside, doesn't it? Yes, we really wanted, if possible, to get a table outside. We were asked if we... They said we might get... Fifteen minutes. Well, maybe we should have an antipasto. Oh, but maybe by the... Or what do you think? And then go outside. Okay. Okay. It's fine by me. We're getting a lavande, then. So, we're going to go through all four of them, and then we're going to go through the rest of them. So, we're going to go through all four of them, and then we're going to go through the rest of them. So, we're going to go through all four of them, and then we're going to go through the rest of them. So, we're going to go through all four of them, and then we're going to go through the rest of them. So, we're going to go through all four of them, and then we're going to go through the rest of them. So, we're going to go through all four of them, and then we're going to go through the rest of them. So, we're going to go through all four of them, and then we're going to go through the rest of them. So, we're going to go through all four of them, and then we're going to go through the rest of them. Thank you for your attention. I'm the only natural man in this table.

1:02:30 Subtitles by the Amara.org community Are you on the campus near to the Statue of Eternity? No, where the physics department is. Actually, I teach at the architecture faculty. You teach in the architecture faculty? You must. Fantastic. It's the Villa Borghese. Oh, it's the Villa Borghese. Magnificent setting. It is. I got to go to a dinner there with Bill Clinton in 1994. Not a tete-a-tete, no, not a tete-a-tete, no, no, no, no, we're about... 4,000 people. 4,000 people. I think that sounds a little pretentious to say that I got to go to a dinner with Clinton. Well, that was at the Villa Borghese. They had a dinner for the veterans, or actually not for all, but for about 800 of the veterans at the Villa Borghese, when they had the 50th anniversary of the liberation. And Clinton had come and pressed the flesh. They had a big ceremony on the Capitoglio, and the day before they had had a ceremony at the American Cemetery. Rubbing shoulders with the power to the unrelenting struggle against ignorance. Thank you for watching.

1:05:00 Yeah, well, of course, I'm sorry, what a silly thing to say, Davide, I was forgetting you were actually in Rome, yes, of course. I feel like you're a professional. No, no, no, no, but you're right, this is the man who should do it, this is the man. No, no, no, more than me. No, no, this is, no, no, no, this is the man who should do it. Well, I have one guy who should do it. I think I prefer Davide to do it, maybe. No, he'd be definitely allowed to do it. As long as you promise to take Silvana down the Via Appia, the door, that's the one. Atiyah, Apia, Antica at dawn, that's the one experience. Or early, very early in the morning, just as it's getting light. Because that's one experience that, I think that is one of the most atmospheric experiences in the world. So how did you do this great work of producing this conspectus of Bill's lectures. This was from intensive study of his writings or had you attended... It was based on the writings. On the writings, yes. Right, that's fantastic. Fantastic work of the steward. I graduated in engineering with a thesis on bridge construction. So I am here now, starting from bridges and from earthquake engineering. Thank you for your attention. I just wanted to let you know which kind of process I had to... And now materials with shape memory. Then shape memory. That's fantastic. And then... I can actually explain. Then if you continue to ask why, why, why things, what do you end up seeing? Then you end up seeing...

1:07:30 Yes, that's true. You end up seeing categories. Can you relate to that? I found this quite another way. Sometimes, the first time, three years, two years ago, I found some categories, and then okay. Then later, another category. Then the third time... Well, maybe I should consider, and then I started to convince myself that I could be able to read these papers. And the best way of doing that was by reading and actually making your own presentation to see if you could... No, it's nothing. I self-convinced myself that you should be able to read them. I don't understand how different way of thinking. Because I think people can find very difficult if you try to understand categories in terms of, not thinking categories, categorically. That's very hard. No, that's a very difficult way of thinking. But I'm trying to understand the different way of seeing things in some way. You've made an amazing work to produce a conspectus of Bill's writings after starting from a computer. I wish I could have been a capable of that, but that's a great thing you've done. Would it be pertinent for Bill, sitting here anyway, to ask in what... In what order did you actually read those papers? Was it you? The first one was the one on unit and identity of all the papers. Good. Exactly right. Excellent. I helped to type them. Yes, you did. That's right. I found that the first time was... I couldn't understand anything. But already from the first time I had the feeling that you have something really unique. The point of entry is always, it's always been fascinating to start at the point of entry to any great and deep body of ideas because the way, the order in which things start to crystallize is very important.

1:10:00 Certainly a very good, very good choice. I found it because I typed on the internet some mole, because you can just save a mole expectancy. Oh, that's a choice. So I found it on the internet. Oh, the mole. Maybe a mole. That's a great question. A mole? I mean, I love you. I love you. I love you. I love you. I love you. I love you. I love you. I love you. I love you. I love you. I love you. I love you. I think I have almost a full collection of these works. I miss three or five. And the original title was Belgian Unity, because I presented the idea at a meeting in Belgium. And at that meeting, I mean, the circumstances surrounding that meeting were a stark illustration of the way in which the Belgian ruling class uses the language division to foster divisions of all kinds. Including class divisions, of course. Workers, Belgian and Flemish workers, especially intellectuals, because, well, I mean, it was quite, so I had given some talks and written some things as well. This whole question about how the linguistic problem is being, because there's analogy with Quebec, for example, and Switzerland, which is not so much used as an instrument, but it is used so much as it is in Belgium, and Belgium is the extreme, it's even worse than Quebec.

1:12:30 So there was this whole complex of ideas, so the title had kind of double meaning, you see, Belgian Unity, not the Belgian Division, which was my Belgian paper on the unity and identity of optics, but it was also having in mind the necessity for forging Belgian unity by combating the false... Linguisticism. Some ideas about linguistics also. So eventually most of that was left out and we were left with the bare bones of the paper as it exists. This is a beautiful paper. I remember studying it also. No, I don't remember which stage, which type it was. I think it was the early stages. Well, like all these things, there are many layers. And of course, as indeed in the subject of the paper, many layers in the unity and identity. I remember reading that paper myself and then going back to re-read the categories of space and quantity paper and having a much clearer understanding after that, having read the second paper and then beginning to get a far. I don't know anybody who's followed up on his idea. I don't know anybody who's followed up on my paper because it was just an attempt to clarify one aspect. I don't talk about forces or anything else. It's just the nature of the space in which things are happening. There's already a radical departure on his part from traditional. In fact, I don't think he explained it very well. When he talks about these filters, it's not really the general filter. For example, Davini. Davini wrote a review. It's very disdainful, very negative, disdainful, but it's honest. He's saying it's just a translation, a useless translation of physics into the language of general topology.

1:15:00 You mean of your paper? No, no, no. No, no. But I mean it's a complete misunderstanding of the importance of the... Yeah, but the meaning. I got to know him better because of that. He was at the Tunisian Memorial of Pisa, of course. I'm a member of the Italian family of the heuristical engineers that suffer from some... I'm trying to keep a friend with him because he invited me to Udine. I wanted to go, but Udine is only a few kilometers from the home of my wife, who is still in Dergo, which is quite near to Udine. So I'm hoping that he will let me speak there, even if I do mention an abstract. Of course, I will meet with him. We have a common enemy. You were talking about topology. I think I told you about the common enemy. Yes, you did indeed. But they don't know anything. Oh, but it depends, right? Yeah, yeah, yeah. So I just took a conversation with one of the Japanese men from the abandoned tobacco company. And they figured out something. And he said to me... Well, at least we have a common enemy. It sounds pretty breathtakingly cynical when it comes from the mouth of another tobacco company, but I suppose there is something, you know. Well, you can see those Japanese guys are so funny. Exactly, yes. Well, exactly. Another kind of enemy. Another different kind of enmity, though. Simple. That's simple, still. Lenin would have understood this. Lenin and I have become an enemy. He thinks all topology is abstract nonsense. General topology is not the right default category. Yes, there's a big difference.

1:17:30 And, um, there was very, I'm sorry, don't worry, carry on, no, no, no, no, I was just thinking, see, when we were at breakfast the other day, and you were explaining to me in Outline the significance of Grothendieck's work on nuclear spaces, you know, for functional analysis in connection, obviously, with that. Yeah. It was one of the things we were discussing at lunch, by the way. Okay. All right, all right, all right. Sorry, I'm going to be completely intolerable for the next 24 hours until I actually get at least a quick rerun on this. I'm sorry. I'm sorry. I shouldn't be. It's selfish of me. I just would have liked to have heard it. There'll be other times, I know. Well, we can be cruel and test the students. That might be crueler on me, because after that... Well, I hope very much I'm going to hear more. In Italian you would say, Sparare sulla Croce Rossa. The Red Cross. The Red Cross. Well, no, I wouldn't ask you to do that, but I'm sure I will have another opportunity to hear from you, this wonderful, wonderful... I doubt whether I'll be able to afford to travel to Canada. I need a job at the moment. In fact I was talking to John Mabry about this this morning. That's one reason why I surrendered to Bill's desire. I know I had to, went to the station with John and talked a little bit about it. He's very kind of going to try and build on the goods. The good reactions to this meeting to write a reference for me. I'm probably going to apply for a job with the British Council in the near future. It's a sort of cultural organisation in the UK which is actually government sponsored but it's responsible for the promotion of mostly our... It's not by any means the ideal vehicle, it's very far from it, because they have their own agenda.

1:20:00 It would perhaps give me a framework in which to work, to do some more work. Much more compatible with your skills and experience than translating or... Alas, yes. ... copy-reading, as you were saying. Well, I think I might be able to do some perfectly reasonable work copy-editing, for philosophy, but... You don't need to work, because you learn enough to do it. That's the major consideration, of course. Yes, yes, yes. No, that is, that is, of course, what I meant. But I don't really want to talk about me. I want to hear what you and David have been talking about. And indeed what you've been talking about. Lead the pharmaceutical industry into category theory. Yes, but you wouldn't want the... ...relation between the pharmaceutical industry and category theory to become like the relationship between the computer science, so-called computer theoretical. Look at the terrible distortions that that's introduced. No, I agree. On the other hand, it's important to have the resources to promote the time and space for scientific work. I really am hoping, though, that you may be... David, as well, and one or two others would be able to come to Fougere and stay for a few days. Do you think that might be possible, to come and stay in this house in France that I now have? Well, whenever would suit your convenience. But I mean for a scientific purpose, to maybe continue discussions with... Some of the other, it's not a question of thank you, I mean thank you if you would come. It would make a very good centre for small, focused, intense scientific meetings, but it's not big enough for a summer school because, you know, for anything up to eight or ten people it would be almost ideal. Italians who have the same interest in ramifications of their subject. You must feel very isolated in... Not now, but...

1:22:30 But to have pursued such a single-minded work of study and understanding, oh well, I'm just completely awestruck with admiration. Because you couldn't have had many people to talk to, alas, you know, I don't know, it must have been very hard. I'm stunned, no wonder. Thank you for watching. Thank you for watching. In distance, it's quite a long way, but because the French, as a result of the excellent engineering, which of course they got from the Italians, goes it, well, in large measure. Because the French now have such a superb system of superfast trains, superfast railways, it takes only 1 hour and 50 minutes on the train from Paris, from Gare Montparnasse, although it's over 300 miles. Yes, it's very good. It is about 300 miles, but 400 miles is a little less, about 275 miles. But you can be there in well under 2 hours. What made you choose that place? Well, it was really an accident almost. I was looking in Normandy. I didn't know about this. I don't understand it. It's a general project. We went to Honolulu for a meeting last summer, and we had various ideas about what we were going to talk about, but we went to the beach the day before the meeting, and we came up with a completely different thing, which we actually presented.

1:25:00 And she, you know, remarkable that she is, so good with PowerPoint, that she prepared a beautiful PowerPoint presentation, you know, just overnight, you know, lying on the floor with her tiny little pocket computer. I'm going somehow by hook or by crook. I don't have the ideas, but I can do the PowerPoint. But Steve, Steve, Steve heard about this. That's what we need. Steve heard about this. He constructed a model out of sticks and strings of what we were talking about. He's got two models. So within days after we made this little discovery, he had models of it that we could use to explain and so forth. He's fantastically good in that respect, isn't he, in making a really important mathematical idea. Well, I don't want to say pictorial because it's much more than that, but that paper about Euler and Cantor where he starts from the example of the potato, that is absolutely wonderful. That's absolutely fascinating. In Halifax? Oh, yeah. Yeah, we had this week-long, two weeks-long summer school in Halifax in 1995. This was the, probably just prior to the meeting, which was supposed to be the 50th anniversary of category theory and the 25th anniversary of elementary topos theory, which of course originated in Halifax. Of course, of course. So just for preparation for students, preparation for students of that meeting. I was inspired by my first professor. Finally. Yes. She was born in Halifax. Well, that's pretty important, too. She was born in Halifax. We had a summer school, two weeks. And I taught in the morning and Steve in the afternoon. So, again, they were quite nicely complementary. They were so amazingly successful. What I would like to... They had to be so successful... That grant writers in Canada, for a couple of years after that, emphasized that whatever reading it was, they were organizing it to have a summer school, because summer schools are such a good thing. Here, here. But, of course, they didn't have Steve. No, no. And so I don't think their summer school actually worked nearly as well.

1:27:30 Did anybody record this or put any kind of record together other than just lecture notes? I'm sorry, I only asked that question. Marmoledo is the one who translated our book into Spanish. The conceptual mathematics book. But he just kept notes. He did work on the notes. Ah, grazie! See? Gradually. Like Grosvenzik with the nut immersed in the water, eventually the shell cracks and we get... Thank you for your attention. ...a number of people, and it was a wonderful time, but as you know, my ambition is to try and preserve a record of all of this so that it can continue to be used at least as the beginning of... These grant writers were a perfect example of pragmatism, as they were saying. ...that this summer school concept works, and we proved that the summer school concept works in Halifax, and it's repeated again, but they don't have Steve, and we have to make it work. It's not the summer school concept that works, it's, you know, you and Steve. But I do see what Sylvain is driving at, that you are complementary in an extraordinary way, because... Well, Steve's good because I know you think you understand. Yeah, yeah. I didn't. Sorry. Well, let's say you're looking at something and say, Yeah, that makes sense. Yeah, okay. Okay. And then he'll go on to the next thing. He really doesn't understand at all. And Steve will say, So, why, you know, this is this because... And then you go, oh. So he doesn't realize that you don't have to do all the things. So he'll make you understand the most basic things.

1:30:00 Yes, yes. Asking you to take the next step on your own. Yes, yes. Actually, that's exactly what André Jarre was doing with me the other night. Because I fail to do too much, you know. No, I have to say, your method is, dare I say, your method is to jump right into the center of things and to get in. A way which obviously, I was going to say uncanny way, but that's stupid, it's an anti-scientific remark, in a way which obviously reflects. Tremendous care and depth of thought at correct guiding motions and then to trace the connections backwards and at a crisscross from there. Is the papers in hypertext? Yes, yes, that is, I mean, yes, Bill would absolutely make fantastic, yes, your papers in hypertext would be absolutely fantastic. That's something, actually, there's a project, there's seriously a project. That's a project that I could devote the next years to. You know hypertext and the internet? Because that really is the way to understand it. There's the words and then you double click. So his papers, that would be great. Yes, and that would be... His paper, but then... What you had to do entirely for yourself. There's a domain discourse, but if you need elaboration on any point, you can immediately find this. So it's really one of the beautiful things about the electronic text that you can really do that. You can have both. I mean, you can either have the main stream or you can have the detailed explanation in the traditional presentations. Or you can have many, many footnotes that for some reason people don't like, but this way you can perfectly have very elaborate footnotes because you don't have to see them unless you want to see them. Yes, actually, that is also very interesting. Hostility towards footnotes. Quine's noted for his hostility towards footnotes. He actually has a passage in one of his writings about how, you know, people, footnotes are tough. People who cannot worry their material into one-dimensional prose. Well, yes, people, of course, whose material is so one-dimensional. But of course, of course, you know, when you see these deep connections and think dialectically, it's impossible to make a complete linear exposition of ideas.

1:32:30 There will be a lot of work. If I could do anything to help at all in that respect, I would really feel I've done something worthwhile. I'll need all the help I can get. Okay, well you've got it from here, you know that. I may have a companion in the library. That would be good. Was it Michael that stayed in the Red House for a while? I don't think so. To John at that time about physical dimensions and about some of the things you were talking about in the talks yesterday about affine spaces. Yes, you don't see snow in Rome very often, do you? Occasionally. They know how to operate the computer program to standardize evaluations. It's important to understand why those programs work. This is only a tiny percentage among the whole epistemological group, which really has some understanding of them. It's also this fascinating, if you're a psychologist, if you want to publish in a psychology journal, I can publish in cancer prevention journals or psychology, behavioral journals. Or, you know, generally, if the stuff is good enough, the New England Journal or the British Medical Journal or something like that. But then there's facts. You know, like, the psychologists always do correlations. Everything is correlations, correlations, correlations. And the epidemiologists always do observations. And then these people always do this. And you can do, you know...

1:35:00 It's not that those statistics are very long, you know, if you do four or five statistics, one of them will come out significant, and then you're like, okay, well, this is a significant finding, I'll publish it in this journal, because this is the fashion of that journal. Like, the professor's literally saying, well, where do you want to publish this? And then they say, okay, go to that journal, look at the article, and do the statistics of that journal. How to translate those into correlations and back, what one journal means for the research of another field is not immediately clear, therefore. You know, there's a very simple thing with a regression model where you can put a variable in the model, in the statistics program, and then you get an overall odds ratio, or you can stratify, stratify, say, males and females, and in the old days, they used to just stratify, stratify, stratify, because it was more interesting, they didn't have the power in the program. To actually add all these variables, they couldn't do it with stratified, but actually stratified analysis is so much more informative, it's really, really much more interesting, and now nobody does it anymore, because they don't need to, and it's really just fashion, totally fashion, you know the basic statistics, but it's really a strange thing to see how... Yes, it's the kind of technocratic fallacy. It's the idea that, you know, correct notions must just come from the tools we happen to have. Well, nobody really understands, so what they do is they follow somebody else's path. I think in a subtlest, in a more abstract sense, there's a lot of that in mathematical physics. The correct notions come from just... See where the tools, the particular machinery that you happen to have, will take you. Let us not ask what the theory does for us, let us ask what we can do for the theory. That was a saloon. It was a big noise. Seeing that analysis of correct notions is where you start from in seeing whether you do in fact have the correct abstract machinery to begin with.

1:37:30 Well of course that's precisely the point that you were making amongst many others in your discussions of functional analysis and more generally in the way that math has gone also. It's all sweeping. Sorry, Doug. There's also a nice journal, maybe you know, the Journal of Mathematical Psychology. They publish those people in representational theory of measurement. Supers. Oh, supers. Ah. Supers. Really. Which meeting? The one with... This one. This meeting? Yes. Marisa Dalla Chiara wanted to insert him into the meeting. She was visiting Florence. I didn't know this until this moment. More or less the same period of time. Thank you. I would have had something to say about that. I'm glad you told me. Thank you for keeping me from... I might have had to take a strong... What was the expression you used? A strong... Oh, what did you say? Furnicious man. Furnicious man. I think probably the first issue of Truesdell's journal is astonishing, this famous review of Cupich's book. Yes, Cupich claimed that you see some kind of first-order logical theory about particle mechanics and axiomatization mechanics, a response to Hilbert's famous problem, and so forth. And so there was a bitter controversy where Crisdell, out of the scientific generosity, actually published the paper of Supes and friends, with a big footnote explaining how, of course, it wasn't really a maximization of mechanics, but even this I think he later regretted because in spite of what he said... These people claim, well, we have now published an axiomatization for mechanics, and in philosophical circles, this is still commonly accepted as a definitive axiomatization for mechanics, it's incredible. Well, he's just some sort of enfant terrible, you see, I mean, no, really, so in a way, his generosity was wrong, and I think even he might say that, because instead of...

1:40:00 In a, quote, democratic fashion, unquote, permitting this to be published and attaching his... You should have just told them to publish it in a philosophical journal, by the way, that sort of nonsense is acceptable. Yes, and I, when I was studying philosophy, without having any background in real math, certainly not in conceptual math, was labored for years under the illusion that this whole subject of dimensional analysis, of the analysis of physical quantities and dimensions and constants, It was a very, very arid subject which was all just a branch of the Supers in the first order. It was just applied logic of the kind of Supers that blocked my understanding for a long, long time. Perhaps someday this sort of empiricist analysis will have some meaning, but I'm afraid most of what I've seen is pure subjectivism, really. At the beginning, I was reading, trying to read, but I think it's gone. Well, I have tried from time to time. I mean, in principle, it sounds like it could be an interesting project. Yeah. For a while I tried to understand the name of this book. I bought the book by this... it's called Marron, Marron, Marron, Marron. He's in design at the University of... You wrote a book on theories of meaningfulness. Theories of meaningfulness? Yes. It's in this style of organized representation of ordered structures. Oh, yes, yes. Actually, this is a representation of ordered groups. Yes, yes. They have this thing they call the theory of order structures. It doesn't really connect with serious algebra. It seems to me a basic thing should be that different dimensions should not be isomorphic, otherwise they are essentially the same.

1:42:30 Just the fact that you have different objects in a category does not mean they're not isomorphic. It's not enough to, not enough to, not to mention their tendency to completely confuse intensive and extensive quantities. Oh yeah, it's a separate issue, but yeah. So having read the unity and identity of adjoint opposites, what was... ...the second paper of Bill's that you read. I'm sorry, I hope you don't mind my... You mean my paper? Well, no, I was actually going to ask you just simply the order in which you entered into this incredible work of... Thank you for making this presentation of Bill's ideas to check your own understanding, but by all means tell me what your favorite paper was. I'm sure I remember the first one. Then maybe the second one is the categories and the proceedings of the meeting in Buffalo. Categories including physics and then the introduction. Thank you for your attention. I don't remember any chronology. No. I usually also proceed. Yeah, you like also proceed as a, no, synoptically, let's say. Yeah, synoptically, isn't it? I don't know events. Integration, yeah. That's a bit like Grotendieck. The rising seas. Yes, yes. Small sides. Yes, all sides. Because there are people that take subjects and they don't forget all the... Yes, yes. Forget all of the interconnections with other subjects. That's definitely not Bill or you, I can see.

1:45:00 Of course, one needs such people also to deepen understanding in particular areas and to refine tools, but they don't tend to be good at arriving at correct notions. In fact, you know, if you can mix the metaphor, I think it's not simply a matter of the rising sea, which suggests a single level of the water, but more like... The nuts that you want to penetrate are actually in various rock pools at different levels, and the tide is rising at different rates and in different saline concentrations of salinity in different rock pools. Yeah, I can just imagine. I mean, you look at the index, and if there is no index, I say, well, the book's obviously not good. Somebody asked me, did you read this book? I find it hard to define. It doesn't mention to read a book. What do you mean? Yes, what do you mean? If you mean to start from page one to page... No, maybe I didn't... I read two books. That's fine. I'd say that's the one respect in which the... The kind of Christian tradition of reading the Bible with concordances and with looking at text has something to offer, just methodologically, not obviously in terms of the content. Oh yeah, there was a point, the last thing we were talking about, you know, that, well, I mean, his general outline included one... There was a sort of feature that I was very happy about, namely some kind of recognition of collective consciousness and collective thinking as playing a dialectical role with individual thinking. And I was saying that, well, many or most philosophical discussions completely omit the question of... There is a tradition, for example Hegel, of using the word mechanics in a pejorative sense to mean non-dialectical.

1:47:30 But it's because Hegel didn't understand mechanics. Actually, the mechanics, even of his day, was far more dialectical than most of the philosophical discourse. And so, for example, as we said already, I'm just rehearsing it, but the idea of collective consciousness is sometimes described as mystical or impossible for that reason. Yes, some sort of Jungian notion. But all this means is, well, the state of the collective is just the ensemble of the individual states, and that doesn't mean that the law of motion... This is a product, in fact, the typical, typical, uh, progettas and mechanics, in other words, the state of a system of particles is the, simply the ensemble of the individual states, but the law of motion depends on everything. But then there is a further, there is a further point, namely... If we say, well, the evolution of thought depends on collective state and the law of thinking, well, where is the real world in all that? Well, it's an external force. So the idea that you have S, you know, contact forces, you have a constitutive relation inside the thing, on one hand, and on the other hand that there are body forces and other kinds of external forces, which of course have a very important influence, and these have to be separated. Part of the framework of analysis of consciousness, development of consciousness. And in fact, the particular material is defined by its constitutive relation, the internal contradiction. So in that way, in that sense, it's the connection between individual and collective thinking per se, which is quite fundamental to understanding this, even though... They said that, you know, the materialist account of the origin of ideas must be external to that.

1:50:00 But, you know, the dialectic is actually quite similar to the dialectic of the so-called mechanicals. I'd like to understand the remark you made yesterday... Shall we just decide what to order? ...about Hamilton... Sorry? You said something about a special here today, which is not... Well, I thought that was the... wasn't that the spaghetti alla vongole that we just had? It was a special. Oh, they had it? Oh, did I? I didn't catch that. Okay, well, I'll go for that then. We'll find out what it is. We'll find out what that is. We'll take it. I was going to say, the remark that you made yesterday about the reason why you thought that a Hamiltonian theory could not be a fundamental theory. Can you... That's in the paper, in my paper. I'm sorry, which categorical algebra and continuum, categorical algebra 4, continuum, okay, which I'm afraid I've not had a chance to study. I think, in my opinion, it's from my point of view, it's the most difficult. This point flows out of these laws of motion that I described, basically, as such, are autonomous systems, and it's very important to be non-autonomous. So, one way of modeling the monotonic systems is to simply incorporate time as an additional state variable. So, you have the total states, which have a deterministic law. You have time itself, maybe it's even one-dimensional, which has its own law. And you have a morphism. This is the reading off the time coordinate from the whole state picture, and that's an equivariant map of an enfo. So that's the typical, that's the usual way of modeling non-autonomous. The fiber of this map would be, in some sense, a...

1:52:30 A possible state space, or a possible autonomous system, except that it's not when you move from one category to the next by law, which might depend on the category itself. That's lazy. There's another example where a comma category is relevant. This is obviously a comma category. Time has a fixed special law of motion. And you have all categories, all possible laws over that, and this is somehow the new category, which is non-autonomous in the specific narrow sense. But then there are other common categories which are involved in the so-called bifurcation theory. One considers sort of varying the law of motion by varying a parameter, and that's often... You know, it's coded in the same way. You adjoin the parameter itself to the dynamic variables, but now the differential equation is different. Instead of saying that theta dot is equal to a constant, like moving in time, it's zero because the parameters themselves don't change in the course of motion. You can vary them, but they don't vary because of the law of motion. So the object below has the trivial axis, as a sort of thing that's often contemplated as more of a bifurcation. So what sort of thing could come about as a more general common category of dynamical systems? So then I got to thinking about the word time itself. I'm sure you've heard this story before. Yeah, but it always helps me to hear it again. That the abstract one-dimensional time is only one extreme idea of time. Is one extreme idea of time, yeah. But in the actual Indo-European languages, at least, one has many constructions which suggest a much more deeper, richer sense. So that in Italian, for example, time is the tempo, which is also the weather. Yes, indeed, which is an incredibly complex non-linear system, yeah, but its significance here, you see, is that the weather is somehow outside our laboratory, we are, it affects, through temperature, humidity, and so on, it affects the system we are studying, but the system we are studying does not affect its significance.

1:55:00 That's the crucial point and so that is again a common category in the sense that if you consider the weather its own law as one object and above that a dynamical system with an equal variant map means one whose states in some degree you know incorporate the fiber and the base yes you see so but but But the fiber doesn't have a law of its own. It's only a joint law, which is compatible with the weather law, so there's very much of a one-sidedness to it. Now, if you think of other ideas related to time, like the New York Times. Which, again, it reports everything that's going on around us, which will affect us, perhaps, but we have very little effect on it in general. Yeah, as you said, the time is the... Or in Danish, you see, the word time is tid, which means tide. Tide, exactly. Tide and all that. So, again, there's this old saying, you know, time and tide wait for no man and so on. If you think about a fishing ship or a Viking ship, this is a system in its own right, in all sorts of senses. It's influenced by the tides, but it certainly has no influence. There are many other examples. In English, we talk about the times they are changing and bad tides. This is the best of times and the worst of times, you know, the French Revolution. But you know abstract one-dimensional time can't do any of that but again it has this feature that it's the environment has an influence on a system but the system has little influence on the environment so all these things could be expressed by an equivariant morphism of a dynamic system to an environment yes it's very one-sided so then the calculation shows that

1:57:30 Such a system can never be Hamiltonian, but such a system, I mean, in other words, When you have these sort of mixed things, you think, well, at least locally, the total space is a Cartesian product space, as a space, but not as a dynamical system. So you have the internal variables and the external variables, and you project down to the external variables. The law involves internal and external variables both transforming into internal and external variables in the future. And you imagine that there's some law on the external variables by themselves which perfectly is autonomous and that the projection deserves that. So you have a Cartesian product state space and a law for which one projection is equivariant. Well, the claim is that if the system is Hamiltonian, then one projection is equivariant and so is the other one. And that is clearly a... And this is clearly, I mean, this is an easy computation. It's simply a consequence of the fact that partial derivatives commute, right? If you write down the p and the p prime and the q and the q prime and write down the Hamiltonian function that depends on all four kinds of variables and you just write down the law of motion, Hamilton's law. And assume, but assume that the law for the p prime and p prime does not depend on p and q, that says a certain partial derivative of the Hamiltonian is zero, but partial derivatives commute, therefore the other thing, and therefore you find that the other projection is some kind of general law of action and reaction. If the system, if the environment can affect you, then you must be able to affect the system. You know, and of course this is compatible with ideas like conservation of energy and so forth and so on. But of course these other models, which are absolutely necessary for engineering purposes, there are all kinds of models. So the typical model is certainly going to be dissipative.

2:00:00 You could argue that, well, if it's going to be dissipative, it can't be Hamiltonian either. I think this is a more, you know, I'm not talking about some quantity called energy, but simply the relative dependence of the internal and external. So the internal variables are in the fiber, you see. In general, it's not even a Cartesian product.