Discussions, incl. M Wright & FW Lawvere / Conversations, incl. A Peruzzi, J Mayberry & M Wright

22:30 You just went the rounds, didn't you? Well, you came last. I thought once I got rid of you, because this is how you were when we started out. No, you were bad then. And then we thought you'd thrown it off, and then Alberto went down, and then me, and then now you.

25:00 I'm pretty good at getting up, but if somebody bangs on me, if I'm not up at 6.30, bang on my door, the upstairs to the right. Do you have an alarm? That's okay, although I think I will get up okay. Has John got one? I suspect I'll hear that go off. Okay, if there is one, thanks very much. I certainly feel I've understood more now of this than I did from the St. Sebastian page. I've clarified quite a number of things, though I haven't. And the other thing I wanted to ask you before I go is if you could let me have a look at what you're talking about today, the one that John Bell has, Cambridge and Milan. Not all of them, just the one. I think it was really just the part dealing, it was really just the Menger and Cardinale and that the...

27:30 Is, I don't know if you had a chance to look at it, is that Guy White saying anything of interest in the audience in that case? I mean he has stuff about Eilenberg and fame and a lot of it which from what I've been able to understand seems to be straight forward it's right what he says. I just don't quite understand the... What the payoff is in terms of meriology. But meriology, I think, might be quite an interesting area of discussion. Well, you know, I don't even know if he's going to answer. Yeah, certainly. I'm really sorry. I'm not going to have time while I'm here. Yes, I will. I certainly know I'm not wasting my time. I think you may be wasting yours. Anyway, just a couple of questions I wanted to ask you. What are some of the neat objects?

30:00 That's one of the things. Really, I can't think of any way that it can be made clear. Very, very good. One is the color television picture. Well, that's a pretty typical lecture as well, isn't it? Yeah, and brings out Collins' limits, the historical limits, which is in common with how the autonomy of physics is met, how sight moves forward, not in countering itself, which is mainly the case. Perhaps I'd have to say something about the points about the attraction connection with the well-pointed topos. Well, just about how it just is, it does give an extensionality, quite frankly. Sorry, any last thing?

32:30 Yes, of years, but that's what Smith said. Yes, he said he was at the University of... I think he said he was at Manchester. Or that he had known him when he was at... Yes, but I think he said that's where he knew White from. But yes, he is, he is Brechtian, he did, he was Brechtian, yeah. Do you have a copy of that, can I see, hey? That's the only copy I will get you. Would you let me know what his name was? We've got time. Yeah. Yeah. I mean, are there any substantial reasons that you think that he is? Brecht. Yes, and that's not what you're actually going to get. Yeah, yeah, yeah, yeah, yeah, yeah, yeah, yeah, yeah, yeah, yeah, yeah, yeah, yeah, yeah, yeah, yeah, yeah, yeah, yeah, yeah, yeah, yeah, yeah, yeah, yeah, yeah, yeah, yeah, yeah, yeah, yeah, yeah, yeah, yeah, yeah, yeah,

35:00 I think it's lovely. I do look forward to seeing the revised lecture. I was going to ask you a couple of questions about the questions of the side of an object again, but I think I'm going to... Thank you very much for your time, and I hope to see you again in the morning. All right. Okay. Yes, I can read those things. And I've got them. I've got the details of the Lenin, if you want. Yes. Can I just have a look at this, you know, this evening? I mean, you're not going to work on it now. Well, if I don't see you in the morning. No, I won't. But if I don't see you in the morning, to all of you, thanks again very much indeed. I do hope you get to London. What is the second situation, where the elementary notions, very simple notions, are supposed to be in a particular area, whereas the biggest ones become obscure and complicated.

37:30 I grasp that quite well. That non-standard analysis. Just take the weight of it, which is very, I think, very... Know what a burn-side rig is, yeah. Well, you can once, I think, once you've learned to think about the whole construction Yes, absolutely. The level of exposition is an absolutely crazy way of approaching it. I guess you have the problem which many people who think at that level of depth of interconnections have. You know, just not realising what it's like, but... One, what he described last night, oh yes, that was really, that was really a moment of irony, John, while you were, when we went out to the Greek restaurant, and Bill was talking about Grotendieck, and it was, you know, he was saying... I mean, the way Grotendieck thinks, you know, the man can internalize a 2,000 page proof and just use it as well. I mean, it's all right for people like Grotendieck, but for us ordinary mortals like you and me, you know, it's... I don't know this at all, I'm afraid.

40:00 Who would buy Waters? Waters made 24 donations. He said he couldn't understand these prodigies competing after the man spent his time stuffing himself on other men's ideas like a consultant taking jellies. Mirroring the situation that probably everybody on the planet still exists in. This is a mathematical web of words. That's not even my experience of trying to do that. The only, I've never been able to, the only good guy I've ever saw that made me think about something like that. He told me all the things he did. Sincerity, optimisticism, he talks about that. That was one of the matters. Life's too short to try to work through something like that. I didn't really believe it at all. Maybe a shrewd observer on sociological matters, and a rather up-to-the-hatch-legate philosopher. At least, perhaps, that's your own personal motivation. Right, yes, he said his one last visit. Now, I was the size of London that I wrote this all with, much younger. Decided after that.

42:30 But obviously the, you know, the unity, I mean, clearly these kind of ideas about the unity and identity of opposites do help Bill when he's thinking through behavior of this, fascinating behavior of this, these two related, actually constructions of points and components. I agree with you. Yeah. There doesn't seem to be anything intrinsically about it that you would need, you know, Hegelian dialectics to. Thank you for your attention. But it also makes you confident in the importance of the principle of contradiction, that reality is a dynamic space, a very tricky fact, a mutual fact, a fact, a fact. It's one thing to say that... The hard way of this dynamical system, highly complicated, highly structured, is the negation of the other, in the same sense in which you say that it doesn't range, it doesn't have range, it's the negation of range, it's misleading.

45:00 I agree, it's more than misleading, it's unintelligible. It's unintelligible. I completely agree. What I can't understand is why Bill, this doesn't deepen. ...just view himself as deeply making more explicit by exact mathematical models the first and explaining how he sees it as related to emergence of ideas like number because suddenly those points about distinguishability and indistinguishability in terms of the behavior of sections of maps or retracts of maps on the composition where you have a mapping from a... And so on and so forth, and so forth, and so on and so forth, and so on and so forth, All this conjury of all the masses on Reagan's logic and its heritage, where they speak about this A, not A, they mix up all this stuff. It's just a mess. Absolutely. On the other hand, he does seem to me to be onto something important about the sub-object classifier in topos. It doesn't have to be a bi-pointed object. It can have a much richer structure, it can have a rich structure, it can have a very rich structure, the structure of a graphical monoid, but is that just simply a rich algebraic structure, which in some sense serves for the parametrization of that variation in structure, or does it in some sense have something to do, I mean, is it possible to imagine the semantics for a language in which you could talk about things having degrees of truth? I mean all attempts to do that have just seemed to me to be very misleading and I must say there are many valid logic of Taurism in a swamp

47:30 I mean, I was certainly inclined to think that... Actually, those are... OK, although the interplay between the geometric and algebraic aspects of topos there particularly are the categories, too, is very rich, I mean, that stuff about the sub-object classifier in toposes of graphical monoids, toposes of graphs, it's not actually algebraically all that... Abstract. In fact, it's really a pretty concrete algebra. It's naive and uninitiated. All of this stuff is algebra. Yeah. I mean, I know he's done this picture two hours ago. All these, all this symbolic work, this border of symbols. John Bell said... Whoa! John Bell's coming by. You've got to be careful that you get near, too near the art. Your eyes, you'll be dazzled. And your eyesight will be all over the place. You don't get to be absorbed in it. You're sort of mesmerized by it. No, I can certainly see that danger. In fact, it's extraordinary given the way he is and the way he thinks and the stuff he does. ...that he hasn't attracted the kind of people who are attracted to gurus. He could have set himself up as a guru horribly easily. I mean, outside mathematics. Well, I mean, he has, actually. He's grown. He's grown. I mean, he's so much more glamorous than people who are of intense intelligence. I mean, he's got... he hasn't got a bunch of adolescent women. He's got people like Gonzov already. Yup, Drew and Carboni and Maloney and...

50:00 And then the judge, who said to me, you know, a lot of what Bill says is nonsense because he says it's a genius. Yeah, but that's not treating somebody as a guru. Come on, it is. He says, I always listen to him because he's a genius. Well, so do I and so do you. You also think that somebody says it's nonsense. That's just... Well, Gonzales obviously understands a lot more of the topos-theoretic work, particularly because that's where he, that's the theory he worked in, particularly in synthetically-referential geometry. I think that's very unfair to describe that as gurush. It's not an intellectual influence. Even if it is, you will never have... Yes, we can't talk too much. And what you say is very touching, John. You say how much more gratifying and satisfying to have Gonzalo Reyes for a groupie than for some new-ball young teenage girl. Well, I have to tell you I'm afraid that not everybody in the world thinks like that. Most of these corrupt Indian mystics, I think the whole point is the Rolls Royces and the Swiss bank accounts of the new-born teenage girl. I don't think they'd be turned on by the idea of having Gonzalo Pérez as one. He's a genuine builder. Yes, all right, a genuine... He's not a... He can't find a... Well, he's got Bell's description of his father. He's a brilliant builder. He's a brilliant builder. He's tapping some source of wisdom for you. You'll believe it. I suspect a lot of people are disliking him because of the conceptual sense of his politics. I must admit that until I met him, I would have attempted to kill him. You can't be with him very long before you feel a certain sort of protective control.

52:30 He's so exposed. We are casually brought to the side. In ways that he would be deeply offended. It's like Socrates talking about his demon, you know, he's in the power of something, almost something like that side of himself. He would be horrified to hear somebody say that. He would totally just speak that way. But from that answer, grasp on the problem, don't understand where we can how, but later, after days, you'll see that there was something that allowed you to answer the question. All other people I asked, what do you think about it? Why do you do it? I don't think this is right. Oh, no, I don't think this is right. Oh, either. Oh, yes, I see now. He's never... When he was talking about his relations with Dana Scott and with the people at the... Notting's seminar at Berkeley, had puzzled him so much that I told these people in 1963 that quantifiers were left adjoined to substitutions, and they just looked at me and stared, when 20 years later they're still, oh yes, of course they did, Bill. I still don't understand really how to think so.

55:00 I see how he understands how all this algebraic structure gets embedded in the framework of category theory and topos, that I understand technically what he means by saying that the quantifier, the quantifier's adjoints, the substitution and this business that was already there in Neist's paper about quantifiers and sheets, is it just the mapping of the internal relations of one conformal system into another? No, it's opposite! He's found just a relative consistency proof, just an interesting bedding. He obviously does think that sheaves and the geometric aspect do with cohesion of topological space, cohesion of the sense in which these spaces have internal cohesion in a way that sets don't, that this actually explains what the nature of domain is. This is the part I don't understand at all. This is... The words I was reading in Gonzalo's papers, you know, when he talks about the category of sets and the substantial axioms to do with the way that set adds up, and how those break down in the cable sheaths over a topological space richer than the one-point space, because it's kind of an internal piece of the space, where I had some glimpse of what was going on, but... And I guess obviously all of this stuff about Mengan and Cardinala and the relationship between the points of Funke and its inclusion as chaotic and discreet, the next level up where you've got more structure in the topological space, so you've got a kind of, you have to look at the components in a sense of simplicial topology, that's all related but I just wish that you give a...

57:30 You know, motivating presentation that would make it easier for what he terms ordinary, us ordinary mortals, to see the path of connections between these ideas. Well, I feel, at least, I want a small bit of wreckage to tell me about water. I mean, it's so easy, so easy. If taken at face value, I can't make sense of it, but I have a feeling that they mustn't be taken at face value. What I would really like to know is what he means by the sense that you're ready to go through and figure that out. I think I'm getting closer to the law of it. That's the question. Day by day, you are seeing amazing things. Yeah. I see. No, I think it has to do with the way that he sees the identity in set theory, the quality time, the way that one thinks of the domain of variation, whether you have any classical semantics or quantification theories, and their absolute identity and relational domain, as coming out of this way that he has of thinking about...

1:00:00 ...quotients of equivalence relations in talk office in this geometrical way to do with... ...because quotients of equivalence relations... Well, for him it's not. It's just one of the ways it gets expressed. ...different matters of... Yeah, but when you say questions of equivalence relations to talk that way is just already to talk set theory, you do have these domains of variation which are... ...bearing internally with themselves, the way explained by the, for instance, in the case of the action of a group or a monoid, which is sort of mixing stuff up as it gets moved around in a way which doesn't allow you to recover sets from the... What's a monoid? What's an action? I'm asking really simple mindsets. You can describe, for example, a monoid without using the outside. Categorically, you can describe. You can describe the category. You can describe one single category. Oh, okay, yeah. It's a kind of category. Okay. It's a good lesson. A particular category can be described. Not only the category sets can be described. The notion of just one set is a particular degenerate case of pattern, since you can describe it as a category. With as many objects as are the elements of the set, and with no errors unless the identity error on each element. Do you think that's enlightening? Well, we think that a set is given as an entity which fights elements.

1:02:30 You just do not consider which operations. In defining, not spouting, the course group has an operation at such and such time that its elements are defined. ...by simply separating them from the others, without any... Well, I don't use this as a particular illuminating example, since the various times I used it, teaching, it was misleading. I saw that... Because you're already talking about categories, a set of objects thereof. But when you give the list of examples of categories with only one object, then you say, well, let's give, from the other side of the spectrum, categories with many objects, but no errors, no less identity. Well, it costs a lot to use that as an example, to generate these. But I don't understand. So say it. Well, I do think that, I do suspect that that is the nub of it, that he does think of quotients of equivalence relations, that he, you know, without thinking of sets, without thinking of set theory. Well, it's a kind of, it's a kind of preparing process. Kind of? Because you've got to start in BDS race. You've got to have some mathematics going already.

1:05:00 Yeah, I think he'd say that... It's constancy and variation in the world around us that give us that already, mathematics is. The general level that mathematicians use without presupposing that, together with this bunch of constructions, have canonical descriptions written in set direct facts. I think Colin would agree with that. You know, of course you have to start in medias res. That's what Cantor and Dedekind were doing with the situation as regards, you know, the understanding of functions in the 19th century. They were starting in medias res. They were starting from a situation which they found themselves in after the work of Cauchy and Wierstrass. When they imposed, as Colin sees it, this trichotomy condition, imposed the decidability of identity, that's making sets fundamental, putting them at the bottom of everything, so that the notion of space and mapping were no longer autonomous, or even prior as they had been before. They were also starting in medias res in that sense, from the different ways of organising mathematical structures. ...that were available in there and given the problem situation of mathematics that they confronted. I guess you say that the trouble is that the way that history is viewed is absolutely Whig history and Mabry is the supreme Whig historian of all. You're Macaulay and he's Butterfield. I think it's hardly me to argue that anything in science is capable of proof. Nothing in science which is capable of proof should be assumed, okay. And he starts out with this monograph, starting with

1:07:30 It takes to be sort of simple and obvious principles and slowly building up to make it out of the number. Sure. You know, what happens in the, for people to discuss categorical... And one of these simple and obvious principles which... No, but John, what you say, he does start that way. And one of the simple and obvious principles that he takes for granted is that the identity relation on the domain is decidable. And that already builds in, from the topo-stheoretic point of view, a lot of structure. Certainly, it is a point of view which, first, what are the objects you are talking about? Yeah. Transformations, your object, are subject to... Yes, I see that. That's beginning, that makes sense. It has nothing to do with set theoretical foundations or what ever you do, or the way you go about doing things. It derives a touchy question from foundations, of course. You are trying to be as large as possible. It's usually, it's not something that those who are interested in foundations are not really obliged to do. They're not feet and muscles. They're not required to run.

1:10:00 I'm slow a bit. Yeah, fine. You okay? I just want to say, don't have to slow down below me. No, just... He's not coming out, is he? No. No, they obviously don't stop you unless you're about sixty. I would think they probably don't stop you unless you're doing seventy or so. Angels in the house, kind of thing. This is a metaphor that the spiders are on the level of the high building and they are thinking that it's on their webs. It's a representation of the entire building, right? And they kind of ask other mathematicians, do you believe that? What's your argument that how you're allowed to design things is not allowed to order?

1:12:30 Thanks. There are questions. It comes not simply to describe what a group is and how we have to follow certain patterns or not, but to intervene with it and describe and put it in the first place, not on the background. These ingredients, which are recurrent in mathematical groups, and this is quite what, in my opinion, integral theory is trying to do, not just a theory of formal groups, it's a theory of contentful groups, where the common... The apparent basic gradients of mathematical thoughts in, as is structurally close, is putting in and prompt as the principles by which they're more, they cut deeper into the, there's certain things that you have to teach all beginning mathematics students. Yes. You have to teach them what is structural.

1:15:00 What do you mean by an operational set? What do you mean by a function? What do you mean by a function? I mean, there's a remark in that... You see, I don't think Bill would accept that adjoints are a higher-level construction or involve a higher-level principle. I think he would say that... I'm sure he would. Yeah, the notion of one-to-one bijective correspondence in the sense of... It actually comes out of, the correct way to understand it is by understanding points functor as an adjoint. Well, he doesn't think it is the more abstract. He thinks it actually captures this constancy and variation in the real world that doesn't explain where mere ones came from. I agree with that. I think he is right in that respect. I know it appears to be a more abstract explanation, but in the end, I mean, you have the, when you say that the notion of one-to-one objective correspondence is... That has expanded in set theory. Even the informal set theory that is part of the hygiene or plumbing of mathematics, that does involve commitment to an abstract ontology.

1:17:30 Lorvier's way of looking at the thing, I think, does ultimately rest on attempting to make rigorous the ideas of abstractor as arising from equivalence relations that really are there in the structure of the world. It doesn't have to be abstractor. You can put five atoms in a one-to-one correspondence with five one-to-ones. That would be an illustration of a one-to-one projective correspondence, but it wouldn't be a general definition. And then you say, ah, yes, and then you say what? That I am a string. Different elements of the domain go to different elements of the co-domain. And every element of the co-domain is the image of some element of the domain. I mean, people... Well, exactly the way you do... Yes, and you've got the category of sets. Well, you know, there's nothing to stop you from understanding it in a quasi-operationalist way. But I think that Brill would say that the availability of... No, maybe not. Well, let me give you an example. I was reading last night, I was reading McLean's piece on category theory. That encyclopedia of metaphysics and ontology. He says a category consists of objects and arrows and ontology. He goes on to say, he says, this is an axiomatic. It's a purely axiomatic description of categories and doesn't require any set theory at all. Yeah, and you say that you have to have set theories. The informal set theory is the semantics of the axiomatic method.

1:20:00 Now, I understand your position well. I also think that MacLean's way of presenting it is not satisfactory. I think that's the whole point. I mean, what else is a category other than a collection of arrows and objects? You can't say that, for example, group theory simply is... The theory which doesn't require a notion of no-sense. I don't think, you see, I don't think anything turns out. I don't think, I mean, I think, well, I do think something turns out. I think the category of groups, the category of cosmological space. I think there is a problem. So does MacLean. You know, I mean, when he quits, when he's out in the bullshitting phase, when he's talking about stuff and he's actually doing the mathematics, he makes carefully, he'd grope and dig his way out. That's the whole point about grope and dig universes. The point I'll have is that you are not attacking the notion of set, just the mathematical theory, but just the category of set. No, no. That's a quantified driving theory. No. No. But you can talk about the set of cultures among categories that I don't know of. You have to put restrictions, of course, if the categories start to be the ones you can, you can't.

1:22:30 You can't always put these kinds of restrictions. What do they do? The same, and so on. They talk about locally small categories. Of course the category of categories was supposed to be a program for getting around that. Not just size limitations in order to go on your usual work of algebra, but structural properties by which you define the size limitation you were going to do, I think that would be a way of saying that category theory is also a sufficient method theory. But it's not said that we can arrive at that point, and probably we will not arrive. What could be the meaning of this? One could answer also that the question of arriving at the monolithic foundations of mathematics has no problem. And that you have to deal with different sources of mathematical concepts. And to acknowledge that just one told a story. Which is Collins' position, as I take it. And what is Maclean's? I think although he denies it, Collins' position is very close to Maclean's in that. He should completely call it a story, since I don't see where it is to be called a story. But I'm equally clear that this is not Bill's position. Bill does think of all of these structures coming out ultimately from, well, I think ultimately out of dynamics and geometry.

1:25:00 Yeah, but I mean, I'm happy. I'm happy with that, because there's a question about the ins... maybe I can put it candidaciously by talking about the difference between hygienic foundations and inspirational foundations. Sorry? Hygienic foundations. Hygiene. Yeah. And inspirational foundations. Okay. Well, or more conventionally by talking about the context of discovery and the context of justification. Now, do I think if you include the problem of discovery and foundations, or the context of heuristics, well alright, talk about the context of heuristics, the context of which heuristic perspectives on structure arise, and the context in which you actually justify as part of a... You know, codification of the ultimate principles on which any mathematical proof, any mathematical argument rests. The question I'm asking is, does the category theoretic foundations envisage getting rid of axiomatic? Does it envisage getting rid of axiomatic? Does the category theoretic approach to foundations envisage getting rid of the axiomatic method? I think in Bill's case it does. So insofar as Bill is implicitly pursuing a foundational program or has a foundational perspective, which he does, I think it does involve refugiation of axiomatics. It does involve? I think it does involve, in some sense, a resituating of the axiomatic method. I don't think Bill does view the axiomatic method in the way that people like Hilbert or...

1:27:30 Well, I mean, I don't know how... No, I'm dead, dead, dead, dead, dead, dead, dead, dead, dead, dead, dead, dead, dead, dead, dead, dead, dead, dead. Because he didn't describe categories axiomatically, he wasn't using set theory anymore. It just strikes me as nuts. Somehow he thinks set theory is a completely unformalized theory. It's a very particular position that you have. If you say that categorists can do at the same time, you can say that your notion of set responds to what most part of the conditions you take as a set is. No, I don't think it's an axiom. If you say it doesn't capture the set, I believe you need to say, when we talk about sets, it doesn't seem to follow.