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

5:00 This was a battery of instrumental techniques. This was a battery of techniques and instruments without any interconnection.

7:30 Because, I mean, see, the system in England is such that at least on those first-year courses where everybody's involved in playing, if you tell the kids something that they don't understand. Oh yeah, then we have to teach them to play. You do have to, but you're not allowed to teach that, I'll tell you that. I think the English. But the awful thing is, it's obvious to everybody in all the sciences, the three-year exams, the old three-year undergraduate course of science. All of these are viable because, you know, 10, 15 years ago when the old system was in operation, the kids came in with roughly what an American freshman would know at the end of his first year. So you can go in to what would amount to a second year math course here. And the other thing is, of course, they do nothing but one subject, so you've got people that are doing nothing but math. That meant that they could concentrate, I mean theoretically of course, practicing what it means that people that got stuck doing something they didn't want are forced to drag their way through two more years of some, you know, reluctance.

10:00 I mean I don't forget all of them, but it did occasionally happen. But now we're going over a four year system. And the hypocrisy behind all this is nauseating because what's happening is the government is doing all this interference with education, and they don't want the general public to get the idea that as a result of their reforms, we have to add another year onto the science courses in the university. I mean, actually, you could make a case for this. The key to getting a slightly broader, broader opportunity to do more stuff. I mean, I think this is that. He's, um, well, it's quite a few years since I tried to study his work. And I tried to study it when I was out of the philosophy at Cambridge and before I really had... So, of course, I didn't get very far with it. I know he pursues this nonlinear spinner field there, which claims to be able to derive experimental recurrent GR and quantum mechanics. And I tried for some time, for about a year or so, to... I just started to work, but then I realized I didn't have the necessary mathematical things, I had to go away, and I haven't come back and looked at his stuff since I began to learn math, seriously. I know in his philosophical papers his position seems to me to be rather suspect, extremely suspect. He relates the, he would say, the ontology of this theory to Spinoza's. And to Wheeler, of course, to the pure monistic ontology of geometrodynamics.

12:30 So, of course, that from the point of view of... Philosophy made me a little suspect of the motives. Also the fact that all his papers seem to appear in journals like Nuova Cimento, which I, well, pretty notoriously, I think, not very, well, not noted for the standards of their refereeing, but tell me what you think of this. Well, heaven's sake, keep the appointment. But have you looked at any of his other papers apart from these? I've got about a dozen of his earlier papers, the ones which appeared in Orbis and Mentor in the 60s and 70s. Yes, his view is that general relativity is the more fundamental theory. He thinks he can derive quantum mechanics within general relativity by its non-linear spin of the generalization. I tried to read it, but actually there was another student like that before we actually changed him to... Mathematics is the thing that goes back to it. You spoke of Kramer. Ah, yes, yes. Was he the guy? Was he? Was he the person that you had the very interesting conversation about general relativity with, Kramer? I remember you having a long conversation about GR with one of the people who was there. I remember. But whether that was far or long, I can never say.

15:00 There was a style of writing that didn't... Well, that's what I found reading, trying to read his papers, but of course I assumed it was because I just didn't know him back. Well, no, at the time that I was trying to study him, it was clearly because I just didn't know math, period, going back and looking at him now, but certainly his philosophical papers seem to me to be really from a point of view of an extreme Spinozistic, geometrical man, yes, this whole issue of non-separability and quantum mechanics. Yes, this whole issue of theory. The idea that it raises, that it does raise profound philosophical problems. I think this just rests on a conclusion. You say you've waded right through these two volumes and not found...

17:30 It was a very, it was certainly the sort of book that I would have been very excited by seeing, you know, ten years ago, before I began to, you know, to learn what are the principles and the answers to it, because that's exactly how such books are written. I think I can say, you know, I would say that I have certainly learned more real philosophy from reading introductory, from reading conceptual mathematics than I have. I don't know, I'm not prepared to open my mouth on that, because as I say, it's a long time since I've read it, and I wasn't ready to read it when I did. But certainly all of this... His papers appear, seem to appear in very, not exactly fringe journals, certainly in journals that are well known to have rather lax refereeing.

20:00 Well, I'll look up topos theory and see how profound they are. ... called Realism, Husserl, and Realism in Logic and Mathematics, which Alberto considers to be a good book, a good introductory, a good, very short introductory book on Husserl. But I think he uses topos there just in a very, very basal way, in a very idealistic way, as to say, well, a topos is a universe of... ...in which we can project our goals. That is to say, it's a self-made thought that we take hold of certain aspects of the world by thereby we actually create a universe. It's a very idealist notion in the background. For the metaphysician, wanting to acquire technical knowledge is enough. If he were to excel in the old method, it would be a serious obstacle to acquiring some knowledge instead of learning it on a mathematical basis.

22:30 I think I can guess. Yes. In the case of Goldblatt, I think it would be rather desirable if you were literally embedded in concrete. This is John Bell they're talking about now, a local... Yes, they're talking about his political science. Which is the article on Hegel. Oh, it's simply in alphabetical order, right? I see, hence, category theory. There we are.

25:00 No, no, no, no, no. I just love him very deeply, because he's only here two months out of the year. He's here right now, and I haven't seen him. Phil was just remarking that apparently Tragesa has written an article on topos theory for this encyclopedia. On topology and metaphysics. We're a bit taken aback about what he says. Topos in the sense of classical theology or...? No. Unfortunately, he thinks he's talking about the topos of... He thinks that Goldblatt is a very good introduction to himself. No, I didn't. In Liechtenstein. Oh, no. Is it expanded? In Liechtenstein. I don't know. I mean, it's a press. It's a policy. Oh, well, it's a book. There's nothing to do with it. You've got to resonate with it. This is the thing you were telling me of the other day. This is funded by the... No, no. I don't know what it's about. I just noticed it the other day. The... The... The... The... The... The...

27:30 The... The... The... The... I'm afraid you've probably caught it from out. Yes, because all three, I'm afraid, I've been sleeping in the same room with these guys. Yes, and you got it in London and I got it over here. Alberta had it in Cleveland. I think I must have got it in Cleveland. Well, I've glanced at what you, in this article about Hegel, it seems to me to be very... That would be enjoyable, but I'd rather we didn't have to be smitten with plague to do it. I see logic articulates the patterns of a mind, already given a capital, a higher case letter, which expresses itself in the current reality, telling the design of logic. That, strikingly, is an almost completely unintelligible sentence, and that makes the basis of my mere possibility of a dramatic structure. I don't do it anymore because of my dad, and he doesn't do it. It looks like a, uh, in this, uh, in this kind of way, you can't do the job because you no longer tell the world how to do it, but it's time again. I guess that's a bad attitude because that's sort of what life's all about. If you give up on that sort of thing, you give up on life altogether. Yeah, I should like to do that. Well, exactly, if you get hungry at the end of another day.

30:00 You want to go to the beautiful Indian Ocean? Yeah. You like Indian Ocean? Thank you very much. Yeah, we could like to go there soon. The only subsistent reality is the whole, which is may be grasped by thinking about it, not by struggling to understand how concrete aspects of it actually get its reflection. It's a fairly orthodox presentation of Hegel, the theological aspects or the way the theological use I must admit I would like to see an article about topos theory, just how awful it is. I assume you've already looked it up in the article. Well, it's not that he completely distorts things. You see, his book on Searle I think is not a bad book. I think it's an introductory book. I think it's quite a clear book. And I think the use made of this in illuminating Husserl's ideas about horizons is actually quite revealing of the way that Husserl thought. I didn't say it's revealing of being, but it's definitely revealing of the way that Husserl thought. Do you know that little paper of Alberto, the real phenomenology of logic? Yeah, well, I mean, actually it's sort of connected. But obviously he has understood. It's 4-4.

32:30 I do hope we haven't given yet this one. In what sense is the boundary of an object the derivative of the object? And now, let's think of it in terms of a characteristic function, that if one at all points to the object from zero outside, well, then it's precisely at the boundary that the function changes, right? And making how it's changing that could be derivative. And there's all kinds of formulas that could suggest that this information is correct. It's very hard to...some of the aesthetics, there's a puzzle there that we've never really resolved, you know? Topos theory creates the possibility of reasoning well about, and therefore of admitting as real. Entities incapable of direct set-theoretic representation, not that there are entities or structures incapable of direct set-theoretic representation because set-theoretic representation itself is a reflection of the dialectic of reflection of thinking, of being in thinking, but that topos theory itself has created the possibility of thinking, of reasoning about these things and therefore

35:00 The idea of omitting them is real, but he's now already been saying that it is reasoning about them in this way that makes them real. His opposition is definitely that of an idealist, an objective idealist. Of a fairly cautious variety. And that is what I took from studying his book. He thinks that epistemology and knowledge is more fundamental and that there is an only approach being through learning. Well, of course there is an important sense in which this is correct, but in that sense it has to be understood dialectically. The reflection of being and knowing, whereas he thinks of knowing and, no, knowing and being as set in opposition to dualistic ontology, is horrible. In my opinion, maybe, uh, that, you know, did talk to the guy, as well. Oh, you did? To this man, again? No, to Smith. Oh, to Smith, sorry, to Smith. I think the whole thing is, uh, although it has many authors, it's clearly, uh, organized, it's very... Manifesto is the beginning of the soiree. Medieval scholasticism has made modern science possible. It's like the Georgian French. I thought, all right, we've had this subjective idea for a century, and maybe finally the subjective idea was to give us something interesting. After all, Hegel was a subjective idea. Yeah, he must have been very interesting, yeah. But, speaking as a whole, it's extremely interesting. Well, certainly this seems pretty, it doesn't seem to engage with anything.

37:30 Well, think of the 3 as 3 steps forward, minus 3 as 3 steps backward. If you did that 2 times, you'd go 5 steps backward. But if I do that 2 times, you'd go 5 steps forward, right? But now you notice the 2 variables the next time is why I've repeated it totally differently. Well, all of this... Could have been said in one sentence, has actually been said in one sentence, somewhere in a paper by some guy, I seem to remember reading in 1975, where he said that intuitionistic logic is the logic of the definite portion of variation. Some strange chap, I think it's called, I think was rather more concise. In this way of saying, what gets served... Do you see this business about the structure of a commercial corporation? Let this stand as an indication of the sense in which topos theory, as a theory of set theories, Bell, 1988, Chapter 8, that's put down as, I think, a rather faulty chapter about the significance of topos theory. Good and just involves a move from absolute to local mathematics, not good and just involves a move from absolute to local mathematics, not good and just involves a move from absolute to local mathematics, not good and just involves a move from absolute to local mathematics, not good Well, so now it turns out there is something that plays a role in my misquoting. It's essentially a geometric idea. It's an open interval, because if I take the sort of Aristotelian continuum, you could break something up into finite orderly pieces without intermingling. So take an open interval, take a point in the middle, and you now notice you've broken this interval apart into three pieces, an open interval and a point. So this object x satisfies that x plus one plus x is x again. That's very nearly the equation that minus one would satisfy. If x were minus one, then two x plus one would be x. But the point is you don't have the additive cancellation. This object has the property that twice it plus one is x, but it doesn't have the property that it plus one is zero.

40:00 So then it goes on from there, right? It's a very simple idea, but somehow there is a geometric system following this one, and it's just an open interval. I think it's thin in exactly the same way as that article's by that Turing, almost there in the title. Because honestly the fact that two objects may correspond to the same integer, how do these two objects differ by dimension, geometrical dimension? I have this extra feature of dimension, where the interval is one-dimensional, the point is only zero-dimensional, the square is two-dimensional. You just piecewise in your maps, and make two objects come out as you mark them with them, and they have the same order for each of these integers, right? They have the same dimension. I'm just saying, if you don't have dimension, you explain more and more, and it's not the same thing anymore. Do you understand that the relationship between your marksman and my marksman is better than the relationship between your marksman and my marksman?

42:30 Well, yes, it is true. Do you still remember? Do you watch? Do you like my art? Yes, ask them if they are... Oh, well, there you are. I'm glad to see that I'm not the only guy that wears these. And it seems to me just common sense. Oh, this is my new gear, it's really... It's the wind. You know, if you're out in that wind... I can remember... Yeah, yeah, yeah, yeah. I came from Missouri. What part of Missouri? It doesn't even have to be for all of us. It's very curious that, well, it's such a great thing sometimes to be able to talk and be quite important.

45:00 As for, uh, interesting for the issue that, you know, of course, for someone who can't do this, but, uh, how, you know, how does, how does for non-Maccadians and non-Canadians discourage it, discourage it, seduce it, actually, to come, to get this stuff, much, much, much, much, much, much, much, much, much, much, much, much, much, much, much, much, much, much, much, much, much, much, much, much, much, much, much, much, much, much, much, much, much, much, much, much, much, much, much, much, much, much, much, much, much, much, much, much, much, much, much, much, much, much, much, much, much, much, much, much, much, much, much, much, much, much, much, much, much, much, much, much, much, much, much You should have a supply of pennies, Bill, like the story when he, when one of his, somebody demanded what use was the knowledge that he had just imparted to him, that give this man a penny since he must make profit on what he knows, what he learns.

47:30 Do you still teach the course on introducing categories that the book with Steve is based on, or is that just a... Well, he taught it last fall. He taught it last fall. So you take turns to take turns. The original special was a question for myself in algebra. Semi-algebraic groups and groups in the category of semi-algebraic spaces. I think probably in a few years this will be the basis of the topic. Something like algebraic, but related to the abandoned grease that I mentioned in my language. It's strikingly strikingly different from abstract rules. Why, sir? Because it has this minimal amount of cohesion, which you can say something out of place, and not into a particular category or compound.

50:00 I couldn't find it. So it seems to be a combination of these clusters. Yeah, these groups, yeah, yeah. But nonetheless, the group structure is somehow black or contaminated, because Euler characteristics and dimensions, which are all, these are the only invariants that the cohesive itself has. The Euler characteristics is kind of a generalized cardinality. It can be negative. There's also dimensions. The theory of finite groups, pseudo-subgroups... And all this turns out to have the sense, for instance, if you put this very forcefully, one of these would have made a much better advance than Cantor. That is a very striking... Well, it was finite hardware. Okay, everybody knows what is finite hardware. Yes, yes. But then, Cantor said, well, okay, we could have this infinite thing. These are, what's much more interesting is the negative numbers attached to actual objects. When you count actual objects, you may get negative numbers. The Euler's characteristic contains a sense which that has. So these semi-algebraic spaces are typically infinite in the mutual sense of Cartesian order and cardinality, but they have finite Euler characteristics, and one can use that to, you know, the fact that the order of a subgroup divides the order of the group. Well, this is true for the Euler characteristic. I don't know if either of these are finite as usual, but they have finite arrays, but the ones that use this calculation are really very tightly classified in one of the same style, or even extending these off of the finite. It includes things like the general meaning of the...

52:30 Isn't that the Milan lectures? Oh, he has? Oh, right. You can look at it. I really would like to. Thank you very much. It's just a certain portion of each of the two. And he is proposing to bring all this out in a special number of philosophia-mathematica, isn't he? Yeah. What is it? I mean, there was stuff in the 20s, wasn't there, where they tried to get classical electrodynamics out of... So a generalized version. Yeah. Yeah. Yeah. Yeah. Yeah. Yeah. Yeah. Yeah.

55:00 Yeah. Yeah. Yeah. Yeah. Yeah. Yeah. Yeah. Yeah. Yeah. Yeah. Yeah. Yeah. Yeah. Yeah. Yeah. Yeah. Yeah. Yeah. Well, I can't quite figure out what he's actually doing in the middle of that. I don't know if he's in that frame or whether... Isn't that a little bit bizarre? Yeah. It's definitely not the mainstream, but this is the... Oh, yeah, he's the fringe. But that doesn't necessarily of itself tell against him. But the trouble is that all his papers seem to be published in journals, seem to be like Norvig and Mento and other places which, as I understand it, I mean pretty notoriously not very well refereed by comparison with them. Well, the thing is, I mean... Everybody, or at least some, I mean, the guy's, he's got an academic position here. Okay, so some of you, I don't buy this stuff, but because the guy, people think the guy's a little bit, I mean, a lot of people who are just shouted off to the sidelines in the past, I mean, I'm not, I mean, somebody ought to be doing the courtesy of trying to make sense of what he's doing. Oh, yes, yes, which is, as I say, as I say, of itself, I don't know. I think his program's a very interesting one, as I think I know. But I just, I suspect some of the philosophical motivations are it. But he publishes quite a number, or used to publish quite a number of articles in philosophy journals, too. conceptual aspects of foundations of general relativity. And in fact, he's got a paper in that collection in the Western Ontario series on conceptual problems in the foundations of quantum theory, edited by Hooker, which I'd be very interested to hear your reaction to his ideas.

57:30 Well, I know you. I think given your attitude towards orthodox quantum mechanics, I think I would expect you to find it quite an interesting idea, and I certainly find it a very interesting idea, but that's how I only feel at all. And then over there, there are many followers of Brouwer's philosophy. Thank you very much. And so the speaker at second point, unfortunately for the speaker, touched on Brouwer. And I just said that his conception of language was such that language was practically something dispensable. And they, the referees, asked me to change some things that didn't seem to be sufficient to clear part of the case. All of these have been described as something objective, so this sounded to me so astounding to me, that I had to rethink about it completely. Perhaps I missed the main point of Braulio, so I went to the paper by Braulio Beretti, and I, on the contrary, I strengthened it much more than I had before, that for him...

1:00:00 The structure, logical structure of language is something that instead of representing formal representation of the structure of thought, it's something that obstacles, obstacles courage, and prevents any understanding of the doctrine of chemistry. From the point of view of the systematics of mathematics, I actually had a discussion with another professor of mathematics, but it's new to me, and I'm happy to be in this discussion with you, actually. Developments in differential topologies are related to a very conservative conception of what mathematics is. Classical mathematics in the sense of preset theory. For example, I said, oh look, there's not just classical mathematics in that sense. The universe of the sport, of ontological meaning, of ontological units divided by categories, we just replied that that's simply too static and or why static? What does it mean to static?

1:02:30 In a sense, the term is understood in geometry. It means that the objects and variables I think that what you introduced on the present Saturday was given completely. We wanted something that, in fact, they are so strictly concerned with building classical logic and classical mathematics that I don't think that, for example, they... What category of theory is represented for an understanding of intuitionistic logic? It's probably that I think that since intuitionistic logic is associated with the operations of the sun, if you provide a semantics for that kind of logic, you have a kind of subjective linguistic semantics, what it's meant to hold true. But of course there is a... This was this misconception that I think crept in in Tregesse's article. Well there are, there are mathematicians. Mathematics is a kind of, not as a, they don't see it architectonically, they see it as a historical. So and I mean very often these are people that work in areas like analysis. These, this kind of mathematics is not to be despised. I mean there's

1:05:00 But they're not really interested. I mean, the main reason they know that it's sensible to study what they're doing is because it's a very conservative point of view. They're not interested in general. But it's, I mean, there's some very good mathematicians in their life, one has to admit. And there's also a certain amount of truth in the fact that it is a continuity, a historical continuity. Yes, but from the practical point of view, I perfectly agree. The problem is when they are trying to present philosophical views of mathematics without considering... Most of these guys don't like to present philosophical views. That's the whole point. That's the point. If you're going to do classical analysis, it's so worked over that you just don't consider the possibility that there's any kind of... New insight at the bottom level to get into. And the real problem is to solve technical problems which require considerable ability. I don't understand Tom's view of mathematics at all, after hearing... I mean, can you enlighten me as to what it is? You've spoken to him and engaged him in arguments. But I just don't understand what his view of mathematics is about. Well, would it be fair to say that Tom's views on mathematics are colored by his own mathematical preoccupation? So he begins to see everything through the blinkers of his own system. I don't know, it's quite bizarre. You are a different person compared to other people. Yes, that's the case. How do you mean? The notion of nouns and things like that.

1:07:30 Oh yeah, those are strange ideas. But I mean, those are more like, I always took that stuff as trying to get applications with his own work, rather than... But he also seems to be looking for a fundamental epistemology underlying all mathematical concepts, and he finds it in these differential topological models. I just find that I can't understand it. I haven't made any serious effort to understand it. I can't understand it. I think that this is a contraposition that remains between the qualitative point of view and the constitutive point of view, And now we have to go to recover the qualitative point of view of the kind of physics. Of course, you would say, yes, we have to recover qualitative aspects, but why, if I recover them, have I to lose the quantitative aspects? He's arguing to the point that his classification theorem is going to have major applications, and what it's going to mainly have is applications that are kind of descriptive. I mean, we're going to say what... It's going to be qualitative. And the stuff that, for example, Zeman, trying to buy stuff for the prison, I mean... They're sort of suggestive metaphors, but they're not, you know, you're not going to get any sort of hard qualitative data out of them. And the point is that guys like Dean and Tom will say, well, that's okay, because if you've got an insight into what's going on, it doesn't matter whether you can get numbers out of your equation.

1:10:00 You can predict sort of large scale. No one understands precisely what they can't get. The whole theory is about the analysis of how something looks very near a singularity, and then they pretend you can do this on large-scale phenomena, you know, butterflies and cusps and everything. There's a small, euclidesimal neighborhood of a singularity, and they want to apply it to things where things are wandering all over the place. It just doesn't work. I mean, I got the appeal. When was it that Tom came here and gave a lecture in the math department and then a lecture for a general philosophical audience? There was a conference on Particularism organized by a literary group. Is this particularism meant to be a bad thing from this point of view? No, no, no, they want to promote it. Oh, they want to promote it? No, I'm sorry. It means that no general theory is possible. Oh, I see now, I understand. Sorry, I thought it might just mean something like atomism. No, no, I understand. Molecular compositionality. No, no. You need to get started. No general theory is possible. This doesn't have anything particularly to do with Connes, but they invited him because he's a big man, right? Well, one, it lends kudos. In sociological terms, it lends kudos to their enterprise. And two, I suppose they see, well, if they've got any understanding at all of the bearing of his ideas, I suppose they see his way of doing mathematics as helping fragment mathematical enterprise. I mean, here is a completely different style. That's the key word, style. Here is a completely different style of doing mathematics. So that all, so that instead of trying to search for a general theory or trying to see how, you know, you just simply talk about different people's styles and the whole thing becomes relativized to some sociology of knowledge.

1:12:30 What I would like to understand about Tom, which I just haven't made head or tail of this book, do his ideas about number, about counting in the context of his differentials, where he thinks about... Based on maps and differential topology, have anything at all to do with the, which I also would very deeply like to understand, these suggestions of the Gonzales race, London, and that I understand from Colliner, were in some of your work in synthetic differential, or synthetic differential geometry, about the finding other. Accounts of natural numbers, another of those given by the Peano axiom, models of the geometric, yeah, smooth integers, yeah, that's right, I was trying to remember, the smooth integers and the geometric, is there any connection at all with those ideas, or, I mean, can you explain a bit to me about this, about that program of the smooth integers? You saw a ton of ideas to get to another night, to lose another night, to get my life going. One lecturer, I heard he pointed out that David Hockney had obviously observed the light patterns on swimming pools actively, because you can theoretically predict that there are only certain kinds of caustic patterns produced on the surface. And that Hawking had actually, without, you know, knowing Tom Stern or even having heard of Tom, had actually painted these things in on his Los Angeles swimming pool. Correct me if I'm wrong.

1:15:00 It's a good idea. I think that this has not to be, I believe, philosophical by Tom and mathematicians. It's kind of become serious. He's obviously pleased. He's a Platonist. He's hung up on this non-existent bird now. He's not going to give up, is he? There we go. Not even really worth spending a lot of time on. But the classification of singularity, that's all it is. Yeah, but that was essentially a qualitative thing. Yeah. I'm a guru. Something that strikes people, philosophers, musicians, hearing him, I had the occasion of hearing him in Florence some years ago, what struck me is this, how could I call it, dogmatism and...

1:17:30 Overdicta, overdicta. Yes, who is, how many years ago? I heard him talk in London once. I never, I really didn't understand much of what he said, but the audience laughed at his jokes with, you know, it sounded like, you know, I mean, it sounded like they were eager to laugh. They were groupies, you know. Well, Ingrid, you know, he said something in my, he said, you didn't have to remark, I was all talking to you. Ah, everybody's talking to you. I mean, look at the school of an academic, the way they're used to blowing trumpets, you know, every time Nixon walks into a room, they used to blow a fan there. So, you know, after a few years of this sort of thing, you know, you're really not fit for... So that's the academic equivalent of that, you know, when somebody makes a... It's not just one good idea. He had a whole bunch of good ideas over a period of many years. And now he goes wrong, which is disgusting. You can't stand to listen to him. Well, I haven't heard him in 10 years, so he gave a lecture. Yeah, this was what? Was this in the Particularism conference? Oh, yeah. This was in the Particularism. He just gave a lecture with that, though. It was a one extended joke, right? There's an actual picture, an actual diagram of the intellectual landscape, which is just all jokes, right? There was no touch of... Yeah, yeah, I saw that. I saw that. I saw that. I'll send it. It's just blank. It's very...

1:20:00 But you mentioned Durkin. How did... Somebody actually ordered this man? I did. Yeah, yeah, yeah. Want some more? Yeah. Let's have some more. Some more now. We're going to share it around. Are we sharing anything at all? Well, I'm happy to do that. I'm happy to do that. Yes, I think that's right. It's like Passing the Port in Oxbridge College's Michael Redhead, who asked me, by the way, to send his regards. Oh, okay. Thank you. So, I would try to, again, or did he never stop doing mathematics? What is going on? No, Grondy. Grondy, Grondy. Well, I think he continued to make, to work on it, but what is the activity? I think that I have no name to call it. It stopped in work and being a physician. I lost something that most contemporaries have lost, something that again is an impossible to evaluate.

1:22:30 In the meantime, if you didn't continue to participate at the University of York, I'll stop even teaching. I'll be teaching for a while. Oh, oh. I thought I saw you right there. I thought I saw you right there. I thought I saw you right there. I thought I saw you right there. I thought I saw you right there. I thought I saw you right there. I thought I saw you right there. It's not annoying, but I just typed it continuously to correct the mistakes he'd made two days ago. That's the only way to keep the stages. Well, what does he want to do, just publish it like that? Yeah. Is there any precedent for that sort of thing? I mean, now there's a notebook we're going to have to publish after he's gone.

1:25:00 Something typical in Indian food that makes it spicy in a particular way, but you knew of what it is. When the Khans were going to join the Hunters, the Catalans didn't have any conflict with the Huns at all. You know, it was just sort of a language more than anything else. And all the stuff that Key and I had worked on was essentially trivialities, and I guess that's not far from the truth. Well, I mean, if the analyst remembers any of his work in the early 50s, or part of any of his other classes, try to suggest that an algebraic geometer is a member of his team, and they can get going on that. Everybody just go back? And the topostherists remember him? No, come on. Well, just take one step back. You don't have to take the whole thing. Thank you, Jim. Why did you say they are in that paper? I'm sorry I didn't understand the point fully when you were explaining it yesterday, briefly before the second lecture, but Johnston's paper on the decision-decidable, the QDA, the book, that correctly understood this represents this epsilon difference, correctly understood it does represent the victory of geometry over narrow logic systems.

1:27:30 I'm trying to understand this. What is that thing? We need a spoon. We've got one here. I guess what I haven't understood is what it was that Johnston thought had been shown by... Every topo has the same logic as its QD core. Yeah, I don't understand about this QD, the QD core business, what exactly QD... The two D means personal, decidable. Both are decidable. A decidable, or an object is decidable, is diagonal, and is in its core, and has a complement in the value of three and seven. Not that it's a new analogy in general, but it's a particular thing of a complement. An object with a QD, or if there exists an epimorphism, it's going to come from the side of the object. Right. A tokosh is QD of every object.

1:30:00 All the QD objects are in a tokosh. From the point of view of geometric morphism, it's actually a projective. So there's a kind of completeness there for logic, or a kind of logic there. Some kinds of things there. Yeah, yeah, yeah, yeah, yeah. You take those as epsilon. Yeah. They're negligibly different. Yeah, they're very different. They're negligible. Whereas, the properties that I think you should be aware of is that your object is connected to the components that are preserved by that product. It would never be true for a piece. Mm, yeah, yeah. This is a relevant topic from the light of time of you. Yeah, I believe in it. There's objects that are described by the internal logic,

1:32:30 they're meant for the nature. You see, a locality, for example, is always P-D. For example, the Kripke model. P-D, you just have information, but you don't have any basis for it. I think that they are... Kind of unconditioned variation. On other things, there may be conditions, but the conditioning is not internal, it's external.