P Cartier / conversations: FW Lawvere, C McLarty, J Mayberry / John L Bell: talk (partial) / lunch conversations (contd.)

0:00 That's why I'm carrying this. That's why I'm holding this tightly to me. I'm sorry it took so long. I need to see you to pay you for your airfare. Oh, yes. Do you have it? No, you have it back at the hotel, I see. Yes, right. Well, sometime tonight, let me have it so I can then... Yes, because I need to reimburse you for that. Oh, yes, okay. Thank you. We're discussing... But the connection between category theory and the more simplistic logic, or let's say, when I say the more simplistic logic, it's the best subject, it's the principal logic, just on the issue of Brouwer, for example. How do you... All the scope is a connection between the elementary and the upper theory and the scientific logic of the grammar and the philosophical and the mathematics of the grammar. There's very little connection. I mean, I call it Heiting. Why do we call it Heiting Algebra? Because Heiting discovered this logic, you see. Now, he discovered it because he was motivated by a particular conception of power, namely this subject who is creating entities and creating things and sort of in time. So it was a very, very special kind of motion. It was a motion of thought. But just that particular example of motion was general enough to precede the hiding logic.

2:30 But then it turns out that the hiding logic applies to all kinds of motion, whether they have anything at all to do with subjective progress or not. Constructed, at least, I mean there is a sense to it that we are just sort of applying definite algebraic operations when we're doing constructions on calculus theory, but actually to call it constructed puts an association with the constructivist philosophy of Brouwer, and this is incorrect, this has nothing to do in general with this solution. You've already illuminated it with some specific examples, perhaps, which were constructed in an attempt to illuminate it. You went out of your way, I recall, in one of the earlier papers, I think it must have been 1973, to call it the logic of a definite portion of variation. Yes, right. The logic of a definite notion of... I mean, motion in thought is, for Brauer, perhaps, mathematics itself, so... That's right, I mean, it's really hiding. Thank you for your attention and see you in the next lecture. Thank you for your attention.

5:00 Thank you for your attention. My impression is that I agree with you that topos theory, the logic, is kind of more general than industrialistic logic. One should not confuse topos theory with industrialistic logic or constructive logic, right? The topos might very well not be constructive, for example. I mean, even, for example, one can consider the topos of classical sets, which is certainly beyond the reach of traditionalistic logic. But, it is true that there is a remarkable convergence. I think we shall seek logic, ideological focus. Maybe you would not call it convergence, but coincidence, right? For example, the fact that Brauer stated explicitly that the economy even has proof by an argument which is... The miracle that the continents cannot be separated is an insight, a true insight, which I think nobody before has seen. And the fact that in topos, in the general topos, it's the same, is something quite extraordinary. If you use the idea...

7:30 Constructive, logic, logic, logic, logic, logic, logic, logic, logic, logic, logic, logic, I thought he was jumping in as usual. That's it, that's it, that's it. Thank you very much. You know, he really didn't like the idea of the continuum actually having any part to say. I mean, he really didn't. It's not surprising given his religious views. He obviously wanted to get outside geometry. He might be tainted with some taint of the kind of unity of structure of the material world. Yeah, but there's an internal motivation in terms of what his natural interests and predilections were as a mathematician. There's also, I think, a more philosophical dimension. Yes, but nobody would come out. He was extremely scathing. You know, to Verrani and to Dubois, he was... Infinitesimals were really one of his vetements. And it was because, it's true partly because... He had this view of numbers, which simply excluded the idea as numbers. But I think there was more to it than just that. I think he felt that that was some kind of residual... How can I put it? Some kind of residual idea that the continuum still has intrinsic properties which just simply couldn't be represented in a system that he was working for.

10:00 And I think, you know, philosophically, he was even dedicated. He had a very definite, almost militant view of it. Whereas I don't think Dedekind did. I mean, Dedekind clearly was much more, well, you're right about Oblivion. Kantor is a monist. It's absolute monist. I mean, you know, he's got this. Which is ironic because, of course, in some ways, well, we'll explain monist. I mean, the people who believe in the absolute maximum, he should have been the imposable, you know, all one. Well, there's more than one kind of monist. We do a lot to monists this way. There's more than one kind of monist. No, no, no, I want to show that, no, in fact, the decisions are very, very difficult. I want to, I want to, that's a good joke, I guess so. But I've always wanted to do what I like to call Parmenidean monies, and Platinian monies, and Cantor's definitely a Platinian monie. No, that's true, that's true. He does, but the first principle is one. And that, of course, is, he identifies with the... No, I mean, Cantor actually writes... The true inference is always the absolute inference. But he certainly doesn't believe that any notion of, as it were, the first principle, the one, you know, the all one, which allows it to be tainted by geometry, the idea that one could actually get an all-unifying structure inside, something which is somehow, you know, geometrical, a limiting structure of whatever exists in any way, whatever is. Something on which geometry provides a handle is something that deeply repugnates him, and I think essentially for, well, for theological reasons, really. Well, as I see the whole of this program of Bill's and, you know, Gordon's work... This is Alex Peller. I'm sorry I haven't introduced you. I mean, I assumed it was. I thought you knew one another. I'm sorry. No, we never met. Although you, years ago, you were the editor of... We had some correspondence over a paper that I... which you finally published, actually. I had a letter from you, a couple of letters on this one. You said you're a referee and you wanted to do this and that, which I did, and came back and eventually you took it.

12:30 You won't remember this one. It's the one paper I have in the JPA. When you were the editor-in-chief, there were two managing editors. I retired ten years ago or so. I've retired from everything. Even from trying to understand what's going on. Were you a colleague? I was a family origin from the 1950s. I did the Institute for a couple of years and Harvard for a couple of years. I've been at the City University for eleven years now. I've read your son published in the JPA. I mean, I think that journal has got so far from where... Well, that's something that you, another thing which you and Jerry Kutcherian have in common. You've both had one paper published in the JPA. That's true, yes. He had a paper in the JPA. Did he? Actually, it's a nice paper, too. Yeah, what happened to that one? The only paper, the only mathematical paper that he was ever published. I can honestly say I had some input in there. I actually sent it to Data. Thank you for watching. On the other hand, he did something that he was so bad at that he couldn't do it anymore. Thank you for your attention. Well, that's going too far. There's been no real danger.

15:00 No, no, no. The only person I must admit I've ever seen actually do that was Martin Hyland when he gave a talk. He didn't work. Well, he wasn't actually, but he did. Was he Martin? Actually, another Martin. Not quite, but almost. I knew various people who were contemporaries of his at Lansing who were very close. I've got some friends of mine who are near contemporaries of mine in Cambridge, I got to know him later, and he's always been this kind of professional young old man, or old young man I guess to say, ever since he was about 21. In fact he's actually less stonery now than he was at 21, but he does do this bit, but actually nowadays it's probably as genuine. He used to do it when he was about 35. The proceedings now, because some of us are getting a bit, a bit, a bit, a bit, a bit, a bit, a bit, a bit, a bit, a bit, a bit, a bit, a bit, a bit, a bit, a bit, a bit, a bit, a bit, a bit, a bit, a bit, a bit, a bit, a bit, a bit, But that was a very interesting... I take it that that very last point that was made was a trailer for your book. Yeah. Well I think that's also going to generate extremely good discussion. I hope Bill actually summarises a little bit more about what he had in mind by saying that he thought that number systems of different lengths might... It might be a reflection of the existence of a further right-anchor to exponentiation, or some sort of factual explanation. It would be interesting to see whether that comes out. He may hate this guy. He hasn't quite heard the stuff before. I mean, it's obviously expressed in a completely different idiom from the way that he would express it. It's a different language, but I mean, you could say you can get at the same thing in Italian as you can in Swahili.

17:30 I think actually the difference here will be more between Cockney and Sanskrit, actually. I hate to say it, but I hate to say toposphere is the Sanskrit A, and yours is the broad Cockney, but it still gets the, at least in your case, you don't have to speak it with the accent of a Dick Van Dyke, it's just a very odd thing. No, I think it's going to be a very good talk, I'm looking forward to it. I mean, that's why I'm looking for this number. Yeah, yeah, that's, um, well. I read the question, it's been really fun. Yeah, we've had a good time. We've had a good time. We've had a good time. So, I'm not afraid to, uh, understand, um, this, uh, result of, uh, I think this is the way I'll start with this. That, uh, a brauer, that the, uh, section cannot be separated. It's to say that the result is saying that any move from the real life to zero-one is continuous. That's right. That's what Viles says independently of Brauer. He also had that. He says it. It's very interesting. He anticipates this idea of a world in which everything is continuous. Brauer never counted as the possibility of infinitesimals, did he? No, no. They're not part of his system. They are in Viles because of his physics. Now, I think this is equivalent, that any map from the Cantor space, from the Cantor space, from the Cantor space, from the Cantor space, from the Cantor space, from the Cantor space, from the Cantor space, from the Cantor space, from the Cantor space, from the Cantor space, The product topology, that if you, by assuming some hypothesis, you can prove that one is equivalent to the other. And therefore, this question of non-separability of the real line is essentially equivalent to the continuity principle for a choice sequence or something like this.

20:00 As problematic as the other, they are essentially equivalent. I don't know. Oh, I see what you mean. That was the point you were making. Yes, I now understand the point that you made in your question, which I'm afraid I didn't get at the time. Now I do. Because one is disconnected, right? But somehow the problem of the continuum maybe is related to continuity. Sure. But it can be detached from the problem of continuity because... The sequence of zero and one, that's totally disconnected. Yes, yes. I mean, in the sense, continuity in the sense of connectedness. Yes, yes, yes. But continuity in the sense of continuous function, that's... Well, right. But again, not to present it in a... I mean, if you work in a smooth topos, I mean, we're everything. I mean, we're all objects. Even the ones that you would look sort of classically would be, you know, would be disconnected. The only maps there are continuous anyway, so you don't really have them. On the line, the continuity is determined by the finite initial segment of that. On the counter space, it's fixing a finite set of... Okay, so that makes a difference between the connectives and the continuities. So, on... Or you can't... You have to fix the whole initial segment, right? Well, on the line there is only the constant function equal to zero or one is continuous. There's no initial segment. All of these points can only go to one point because, you know, you've got, as it were, maximal cohesion in the sense that one... No, it still consists of points, but one point can become any other in a completely trivial way because something gets moved around and lived without any attention to the actual structure of the space in terms of its, you know... Because points just are connected components in that space. Is that the idea? I confess, I have a lot of problems with mathematics. I'm going to recover my comprehension of mathematics.

22:30 It gets too technical category, theoretically, I think. Well, that's not a technical category. No, the COVID has asked for topology. Even this outsider is concerned about math. No, I mean that the math jobbing in the classroom is more important. The sponsor is not definitely aware unless I mess something up. You can exhibit numbers. Whichever first cannot be computed. No, I don't think so. I'm sorry? And decomposability, of course, wasn't actually mentioned. Well, except in response to the question. No, no, I mean in Bill's talk. No, no, no, I'm sure he was a little bit surprised about that. There are a lot of things that can't be considered zero unless they're in consecutive sevens in a decimal expansion. Yes, exactly. But what usually happens in these cases, at least if you look at some model of universal continuity, I mean, it boils down to the question of the domain of definition. Yeah. Just take the simple flip part, you know, f is zero at zero, it's zero apart from zero, and it's one at zero. Now, that's it. But, the main, the main is not the whole of one, it's actually that. You can kiss and do, you can do very strange things. Sorry, can I ask you if it's, the exchange that you were having, What's happening with Bill just after the last, his talk, I'd like to try and, unfortunately I almost had to get out of the room, but it's very interesting. What I said was wrong. Well, that's what I wanted to understand, but it isn't respected in the free speech office. Yes, I mean... Oh, okay. Can you fill me in? Well, I... It's simply a technical point. I thought that he was demanding two conditions that actually contradicted each other, but in fact they don't. I mean it's still not clear what the second condition actually means. I'm sorry, just remind me what the two conditions are.

25:00 The first one was what he called the Nostelmos Act, meaning two conditions which are in fact equivalent, and I now prove that. The second one was this condition that co-discrete spaces are connected. The composite you get by going around the two outside functions simply reduces the support function that it sends an object to. And that, you know, I still don't know what that means in general, but it's not the case that it contradicts the other one. Okay, well that's very clarifying. So there's more work to be done there to understand that same condition. Does this also connect with a condition for things to be quantifiable? I hope you felt that the standard of the... Thank you very much for your time, and I look forward to hearing from you again. I think that moving, as it were, down from the mathematical key to the philosophical key in the course of the meeting has actually been the right forum in which to do things. We're having this little interlude of the, well, at least tangentially connected with physics there yesterday. That was the theme. I'm sorry that this party didn't come to pass. I think it would have been interesting to get an interaction, particularly to this point which was raised about the complexity and the historical development of the notion of point and the point having internal symmetries connected with the operation of the orbit.

27:30 Space, which connects, obviously, with his own interests in quantum mechanics. He obviously does take a great interest in raw philosophical questions, rather than historical ones, and it's quite interesting to have got his reaction back from Carolina. However, his wife wanted to go shopping, so he will be back around this afternoon. No, no, no, this isn't a model of mathematics. It's moving to the test of mathematics, this isn't a model of mathematics. Take x to the fourth of x. There are also many other fields of study, such as mathematics, geometry, physics, physics, and mathematics. There are also many other fields of study, such as mathematics, geometry, mathematics, physics, and mathematics. I saw that what had gone up on the board in the blacklist, I was decisive. He just isn't quite sure what the condition means now.

30:00 But I'm very interested in it. It's very good. I'm very glad that the discussion was generated. The only thing which I slightly missed out on in the discussion of John's talk was that, ah, it's a little bit of a surprise you didn't say anything, but maybe all of these figures, this sets down to a greater or lesser degree, Lyle is perhaps the exception, to a mark. The idea that it all has to be one or the other, the idea that they're completely missing the point that there could be many intervening levels of degrees of cohesiveness and which one might be able to get a precise hold. Thank you for your attention. Thank you for your attention.

32:30 The phenomenon that's been established is that we can both make mistakes, not just one. So it should be much more easygoing. That's the Cambridge mindset, isn't it? Has he opened up at all on the other... I mean, I realize this is not at all related to the issue that you were discussing just now, but on this... the restrictive conditions that are obviously built into his claim that all prophecies are locale. Oh, yes. Well, I mean, he doesn't, in a sense, get the point, but he's much more sophisticated. There are more names than that. It's just, you know, the recognition that there is a kind of semi-independent class of roach opposers and spaces, which again, I think, because it is my lecture, which was really about that, so I mean, you can't help but recognize it, but it's a serious one. Maybe we'll do something with it. It was probably going to be a good plan. To try and arrange, in the future, smaller scale meetings where you could just interact with your three or four people over a more extended period, that's something I'd very much like to do and have a hand in. And I know that in one sense you're doing that all the time anyway, but to have it in a slightly more focused way. Actually, one thing I don't have is a garage, because I don't have a car, but there is a hill behind the house where there's a road where people can park, so there's no problem about parking, so there's no problem about that, but we could do a little bit better than the garage, I'm absolutely serious, we're just talking about the possibility of getting Bill and Peter, Johnstone and maybe one or two other people together in the small meeting. And there may be some philosophers, separately, for math, but it will be good.

35:00 We must carry this forward, personally. I'm a little bit depressed, but I think John Bell, when he stands up tonight at the dinner, please not to say anything embarrassing about my knowledge, and not above all... To get the point across, if he is determined to say something, the whole point is that this is not the last of these meetings, you know, this is not the last, you know, this is, the whole point is that we... This is, there's a program of scientific work here to be carried forward and deepened, and not, John, I'm afraid, is not one of nature's optimists, which that may be why he ended up on, you know, the Tchaikovsky side of it, but I'm sorry, I don't mean that, although he's a great, good guy, but I don't want him to send out the wrong signals, but this is some kind of a swan song, you know, just the opposite. Well, I have a feeling that you may be inclined to say that... But I do want him to understand this is a screen. We would have had some interesting insights about that point, but one of the things I'm particularly pleased about is that Bill and Peter Johnstone are kind of really talking critically, which they haven't been able to do for a long time, and maybe having Peter here for a whole week, and listening to this stuff, and getting the motivation, well not that he doesn't, but getting to listen to the successive talks, and also making a point. It actually turned out to be a mistake. Well, that business about you can only do it in a three-sheet toolbox. Turns out it actually was technically mistaken. They've convinced him now that they spent about 20 minutes on that. Oh, Peter was wrong. Yeah. Oh, good. Yeah, well, give me a sense of that. I mean, Peter was obviously the great technician of the subject, as you know. The great master technician of the subject. The master theorem prism. He has frequently called them out in textual mistakes, for sure, because that's the kind of problem I have. This is about the first one that he actually missed while I let him go. And therefore I think it kind of slightly broke the... well, honestly, break the ice is an odd expression.

37:30 But they are now opening up to one another far more than they were... Well, obviously Bill's always been... Hi, John! Always been ready to open out, but Peter's just... well, because he is... But that's really good, because there's a really good interchange going on between them there. And in fact, I think all around the whole meeting, I'm really pleased. We chose the right people, absolutely the right people, aren't we? We chose the right sort of order in which to have the talks, and we chose the right kind of themes moving from, you know, purely, obviously, Bill's lectures, it's a humifying theme. No, I'm really pleased with the way it's going, and certainly looking forward to you. I'll leave you to it. John?