Discussions after G Reyes lecture

0:00 You're talking about Hill's talk there, are you? Yeah, yeah, yeah. No, I agree. It's astonishing how deeply he thinks about connections between the geometrical and algebraic ways of looking at stuff. Well, and how able he is to make a relative attempt at nodding at minuses. Well, in that respect, I have to say, speak for yourself. Your acquaintance is a good deal less nodding than mine. I'm afraid I lose track of all but the general. You know, qualitative ideas pretty early on with him. We've still been able to make the sort of, why are we making that? Yeah, that's very true. What I can't... Yeah, I think he is. What I can't understand is why when he's expanding the ideas to philosophers, and when he's actually chosen a title for the relation, or the true relation of logic and set theory, he doesn't actually expand the examples from logic and set theory, which is the one thing which might actually get the philosophers and mathematicians really interested, because instead of expanding mainly stuff about homology... That's right, but the real advantage of category theory is how it is making the connections to all the other parts of mathematics. Yeah, I agree, but I think that pedagogically you probably need to bring that in after you've already grabbed their interest in terms of the over-constructions and not even set them. I mean, what I would have done, if I was not here, would have been to say a lot more about this. The notion of set abstraction as an instance of discrete vibration, which Jim Lambert touched on very briefly, and which I think is the thing which would motivate and orient people in the logic and set theory audience, which is a philosophy of man, much more easily than the stuff he comes out with.

2:30 The trick of this theme, real mathematics, is a lot more solid than it's not basic. I really like to keep that in mind. Sorry, thanks. I'm Ivan. I must not get in the way. You were there when Mayberry was going on, when Mayberry's column was rising. And, gee, I mean, Mayberry really believed that the second article framework is one following. But not the set theoretical framework of the Zermatt and Frankl as a first order or higher order theory. Just the basic informal notion of set as collection and extension. That is what he thinks is more solid than anything else because that's part of the hygiene of mathematics. That's part of the plumbing. That's the thing that you, when you push back to ultimate regress in definition, that's where you have to stop. It's not these things that set theory and axiomatic theory is more secure. That's absolutely not his position. That helps a little bit. Sorry, I mustn't keep you.

5:00 Guess we'd better get going, anyway. Thanks again for a very lovely night at Westbrook, and see you all later. Are you coming back? Yeah, I sold my stuff down there. Okay, thank you. Thanks very much indeed. It's nothing like a spell as it was down in Cleveland. I see actually we're already... They probably won't stop actually on the dot, will they? Is this us here? Oh, sorry, we got in the wrong far. That was my own fault starting there.

7:30 I agree absolutely with your reaction, and obviously want to understand them more deeply. But unfortunately, I do find it extremely difficult to follow once he gets into the depth of the mathematical connection, particularly in areas like combinatorial topology. I'm afraid I lose that quite early on, whereas the guiding ideas that orient me in the way that he thinks about struck in this deeply geometrical way, and obviously also related to his idea that ultimately it'll turn out that physics drives mathematics in a way that hasn't really seemed available as a philosophical position since almost 200 years. That, just speaking myself, I find... I think that he would present more effectively in a way that got not just the real expert, the people who are capable of doing, you know, deep mathematics, but the logicians and the, I'm not saying that there are no logicians who can't do good math, you know, your conventional sort of, you know, working tasky trained logician, or people with interests in, or philosophers. People in philosophy departments who are very interested in math and philosophy of math, but who naturally, because of the history of the subject in this century, tend to approach the subject through logic and set theory. That's most of the math they know is logic and set. I mean, most people doing philosophy of math in philosophy departments... You know, don't know much combinatorial topology, but the thing which I can't understand, really, if he wants to get these ideas talked about more widely, is why, maybe it's just that his own understanding of the interconnection is so deep, he just can't believe that people need to have their hands held, he's forgotten how.

10:00 And yet, that introductory book he wrote on conceptual mathematics, he obviously hasn't done that when he's teaching his own students the stuff. He introduces it in an extraordinarily accessible way. That is what is so extraordinary. He introduces it in such a clean, accessible way. He motivates everything so concretely. He was published by Buffalo Press, but I think it's a kind of private affair. He and Steve Shaniel wrote it together. It's not just Lorvier, it's Lorvier and Steve Shaniel. And it's their first year, it's their freshman course. He took the introduction of the categories, but there are bits of it that are really almost embarrassing. Like, you know, they're real sort of kindergarten stuff, and it's in dialogue form, too, which puts some people off. It doesn't put me off, but I know there's some people, so it's naff. But I think that's unfair. But, as I say with a detail, the way that he introduces mappings, sections, and retracts of maps, pullbacks in categories, in this very intuitive way, motivating them every time by examples, most of the time with concrete, physical examples. These are examples taken from the structure in a very concrete guide, I think they are, that I can't understand why he doesn't, shall we blow the horn and let them know that? Oh I see. And particularly I don't understand why he doesn't say more about set theory and about this view of set abstraction just as an instance of discrete vibration. And the way that one should think of five rings of maths from, in the context of homosophy, from space to space, so that the notion of set just falls out, it's the case, the limiting case of, yeah, because if you really want to motivate this idea that geometry is underneath arithmetic and set theory and analysis, I would have thought that was the right starting point, particularly talking to people whose, you know,

12:30 People whose conception of how you approach philosophy of math is very largely driven by logic and set theory. I would have thought he would have said much more about areas like discrete vibrations and how they fit into the general scheme of category theory. Part of this is that the foundational interests have been somewhat misplaced. Yes, sure, sure. I mean, it's true that he was often using relatively fancy parts of mathematics to explain. The idea, yeah, I think it's a big deal. When he was talking about figures, when he was talking about, you know, figures and incidence relations as the, you know, as the, you know, in the context of ideas about unity and identity of opposites. Standing here on this side, as contrasting with algebraic structure, algebraic ways of thinking about structure on the other, I just think it would have helped an awful lot if he'd given some, you know, some really concrete motivating examples early on. Particularly considering the way that he thinks that the notion of set actually comes out of this way of thinking about figures in geometry. He really should have given more... Concrete motivating examples early on. If he really wanted to hammer these ideas in the way that I would like to see them hammered to philosophers of math, he would have rammed concrete examples down their throat until the real mathematicians in the audience were almost sick and tired of it, because he doesn't need to get to them. They already know how interesting and important these ideas are. Well, many of them do. But, well, that's because most mathematicians just got interest in foundations. It's not because they don't know, yeah, and it's not that they, and okay, as far as something like topos theory is concerned, okay, there are now lots of people practicing it, this is fairly...

15:00 Consolidated field. Let people pile up results. There are lots of people doing it. It's very application-centered. But the way that most algebraic geometrists approach it now is to say, well, where are the deep theorems? Thank you for your attention. Much better. I think it's done. It's starting to do its stuff. I hope so. I hope so. I hope so. Well, what I want to talk to you about today, if it's possible, is then the chance we can come down on Tuesday. Thank you for watching. Would you travel Monday and come Monday evening? I am. Well, we'd like to do that, but we wanted to talk to John first, because I know he wanted us to stay... I think he wanted us to go to Toronto with him tomorrow. But I don't think there's any reason why we shouldn't go to Toronto with John and then travel on from there to Buffalo. It is the right direction, anyway. It takes me only an hour.

17:30 From Toronto? No, but don't do it. Don't do it. It takes two hours. All right. Well, no, we'll put, we'll put, we'll put Alberto at the wheel for once. I'm sure he'll do it in an hour class. No, I'm going to turn around. So that would be ideally, yes, if we could come on Monday evening. Is that all right with you? Yeah, yeah, we'll see what, we'll see what kind of movie we've got. Well, that's what I was thinking. It would be crazy. Yes, that would be ideal. So we could arrive at your place tomorrow evening and then leave on the Wednesday morning. Did he tell you how to get there? No, but he's standing right behind you. I'm sure he will. I think you're right. Okay. Much better, thanks. Yeah, yeah, yeah. It was bacterial and it's already... Now I've taken four of the antibiotics. It's obviously knocking it out. Well, of course you've got the whole bottle, but I couldn't take the whole course. But I already feel much better than I do. I slept, I literally went out like a light last night. I didn't even wake up. I went to the biopark, passed aid. This is the first really good night's sleep I've had. No coughing. I've just had one enormous relief. But about one o'clock in the morning. I think this whole issue, particularly of Frege, and the principle of compositionality, is a very deep source of many of the, in logic, intervening stress.

37:30 Yeah, let's go and do that. I'd love to hear. I know that although it's a very simple elementary point, indeed, it's just very terrifying for me. There's a very simple point about the diagonal and the way that one arrives at, I mean there you see the origin of this drive to the completed abstract kind of time that the places want. The completed abstract, the completed kind of kinds of abstract objects is already there. So that all the structure, all the richness of structure, general and particular, and local and global. All of this material that is captured in a more directable way by the constructions and category theory, for him, is just arranged on this hierarchy of these.

40:00 And I think that the origin of that, and what I was struggling to express, and I very badly contemplated the question to you about how Jonathan's idea, the results about the QDO, the way you think of the epsilon relation, That's all in relation in terms of the intrinsic structure of geometric structure. Yes, sir. Well, I was referring to what you said at the, you know, . He was trying to push logistic without saying so. So he was saying that, well, something about, you know, the logic of an arbitrary topos is already captured in this little thing up to an epsilon difference. Yes, that's my point. It's not the membership relation. It's saying, well, this reality is somehow negligible. Oh, I see. I haven't understood that. I thought this was his way of saying, you know, this is what epsilon amounts to. No, I haven't understood that, I'm afraid. I must go back to the show. I'm sorry, I haven't... The other three... It's a common way of speaking, right? Yeah. In Cauchy-Weierstrass analysis, you have this epsilon delta, right? But these are small quantities. Oh, I guess I... And this is carried over into ordinary physics, of course, by saying, well... The difference is negligible. Something is negligible. So between the arbitrary total and the QD total, he claims this is an epsilon difference. This is a minor difference. The cause is the point of view of pure logic. You can't tell a difference. In other words, if you want to have a completeness theorem for a higher order of intuitionistic logic, All of these formulas that are purely pure logic, it suffices to consider QD topos.

42:30 And so from that point of view, it's an epsilon difference. But to me, all the difference is that all the growth topos, as you see, are precisely living in what makes them different. See, there's an actual QD reflection. Take any topos. It has a sort of QD core. This is one of his connotations. And the logic of the two, in some narrow sense of logic, is the same. That was his thrust. When I turned it upside down on him, I said, well, look, this construction really reveals quite precisely. The space that makes a growth topos have a content is precisely not something that's within narrow logistic. It's within, yes, yes, yes. Now that's the point I wanted to get at. Precisely. I want to get in trouble by being ironic. In my Komo paper, I say, in John Stone's wonderful paper, it's a good paper, he mentions that that's a lot different. So I say it was precisely this epsilon, what he called negligible. I'm saying it's the whole thing. I mean, not the whole thing, but... No, I'm sorry. I'm just on the street because I was relating to the... There is a space for the defeat of, the triumph of geometry over narrow logicism. Yes, I know. And I want to understand that. I really do want to understand that more deeply. Perhaps when we get to Buffalo, we can explain more. But of course I've not read Johnson's paper because I haven't come across the reference to it until I saw your Como paper, which I only just saw when... I'm really trying to promote that people should read his papers, just not accept the conclusions that are in the logic system. As is typical of his philosophical reasonings, he has all these fine mathematical constructions, and he slips in at the last moment the major philosophical conclusion without any real justification at all. He slips it in. So by using this appreciative term, epsilon difference, I've somehow made a major philosophical point, but it's wrong. And, independently of that, one should read the paper, because it's a very simple construction, and it really helps establish the direction of the paper.

45:00 Well, I certainly need to understand that. It's the rule for reading a Johnstone paper. I see, now I see. I hadn't appreciated the subtle irony. All those little subordinate clauses with some major philosophic assertion, just leave those out. What's left over is the paper. Please, thanks very much. But certainly this presentation of Gonzalo's this morning does seem to me to relate to the way that that thrust towards the framework, the ontological framework of a completed kind of kinds of abstract objects. ...got started and why the work of Frege was very important in contributing to that distortion, the framework in which I should think about mathematics or structure. Also that profound distortion of the relationship between mathematics and physics that is part of... Contemporary objective idealism. You see in people like Wigner and Penrose, the Platonism, the idea that our conception of physics is driven by mathematics, which already has an ontology in advance of any consideration of what there is in the world. I think that's the dialectic that is reflected in this drive to a completed kind of abstract objects which is already there in that at the origin of which is simply ignoring the dialectic of global, local, local, global equivalence relations and assuming everything is given once and for all in the platonic heaven and so that... And therefore displacing these problems, exactly as Gonzalo said, displacing these real world problems of global, local, local, global, analysis of aspects of reality onto the level of pragmatics in the case of our understanding of the workings of language, in this particularly restricted instance, the logic of... The logical structure of these operates in natural language. That whole problem just gets displaced onto pragmatics, something that you do on Sunday afternoons when your brain isn't in gear, which is the implication of the way it works.

47:30 I'd really like to understand how that idea got started and why it's been so powerful in mathematics and philosophy. This conclusion that we can never ever draw a judgment about Spiro Agnew, because we haven't yet taken into account all those aspects that are written down, you see, in the great judgment book of heaven and so on. Yes, no, exactly. No, it is not for us to judge. We're waiting until St. Peter judges Spiro. We're waiting until he gets to the throne and then Spiro, yes. Based on that, you know, instead of accepting this as we will, we have to look at the important aspects and analyze them. Because of course we can judge part of that new class, as Aristotle would say, qua politician. You can avoid forever actually doing that, by speculating on the lack of knowledge about the world. It might have been that when he was six years old, he was very honest in the way that he dealt with his kid brother when he borrowed his... something like that. But the idea that, yes, I agree entirely. So of course it is a marvelous recipe for postponing judgment and therefore postponing action. And making sure that being and doing are kept entirely separate. Yes, I absolutely agree. It's very important to see that. That's why the question I put to you was a very simple-minded one when we were together on the first evening, about understanding the way that one thinks about distinguishability and indistinguishability of context. There is this whole literature on identity in philosophical, in so-called philosophical logic, i.e. what passes for logic and is done in departments of philosophy by so-called analytical philosophers. Who are really our philosophers who were permanently postponed the task of analysis, engaging in real analysis. The huge literature on identity. Is identity a relation? Or is it a problem? Which is full of them, isn't it?

50:00 And, uh, yeah? Reminds me of something that we want to ask you. Sure. We know that there is indeed something... The book, a pair of books, which are the index to Lenin's collected works. What we'd like to do is obtain some of these for our own private library, and now the booksellers in the U.S. say they're out of print, but we believe that actually one of them was, at least one was published in Germany. I'm a communist poet, a publisher from the 1920s and 30s. I've published the complete edition of Lenin's works, of which I have got, I haven't got the complete collection of them, but I've got about 17 of them from New Orleans. The complete collection, which is 45 volumes, you've had that for some time. Yeah, but there's an index. I'm not sure what happened to Lawrence and Wishart since the collapse of the Communist Party, I mean obviously they were, they remained the official publishing house of the British Communist Party after the revisionist term, after 1950, after 1950. But they were obviously very much out of favor in that they were very resistant to revisionist tendencies within the party, and they, I think, certainly by the time the extreme kind of lunatic pseudo-Marxists had taken over at the British Park in the 1970s, and the people who weren't that more revisionist than ...to the point that they ended up actually glorifying Margie Thatcher and saying that she was the, I think you should, the interview with the then secretary or the publisher of Marxism Today, which was supposedly the Journal of the British Communist Party, but which by the time it was finally suppressed and the party officially sold itself a year or two ago, had become a...

52:30 I mean, an outstanding offense against the Trade Descriptions Act is that it had an absolute no connection whatever with Marxism, it became a joke, even amongst the right-wing press that Marxism was the most misdescribed journal title in the world. They certainly had long since ceased to have... I think they had established their own publishing house in the 70s after they were suppressing Lawrence and Wishart. But what happened to the Lawrence and Wisharts? I'm sure that would have been retired at the index for many years. It probably would have been one of the very last things that Florence and Wishart published before they were published. What would the bookstores in London look like? Well, they used to have their own bookstores, but the trouble is there are a couple of bookstores in London. There are a couple of bookstores. There's one in... The area near Lincoln's Inn, I'm trying to think exactly, which used to carry a lot of good Marxist-Leninist publications until about two years ago, was then suppressed and liquidated by the people who ran it, who were connected with the... There's a lot that left the British Army in about 1976 in protest against what was officially described or disguised as the Euro-communist term. But, yeah, I'd certainly do some research on this and try and find out where it's taking place. There are about three or four places that I would start looking. I've still got all the old Martin Wishart catalogs up until about 1979, which I think was the last year that they were an independent publishing house. But that simple point about the distinguishability, indistinguishability, I know it's a very simple point but I want to keep hammering it because I think it's the right way into introducing the ideas.

55:00 The dialectical confluence of purpose theory to philosophers who are preoccupied with problems about identity, which is precisely the kind of problem that people who do what they call metaphysics and ontology are preoccupied with, but which they think they can produce scientific statements about the world, the nature of the world, by thinking about identity as if it were... A relation stored in this platonic heaven and completely prior to all questions about being and doing as actually given to us by our concrete experience and theoretically formulated in a quite exceptional manner. Successive generations of scientists, I think that, but certainly I think that, and there's some things you said also in that paper about how to tackle problems of equality and difference, which I'd like to ask you more about. I'm trying to get... I'll try and get hold of that for you if it can be found. The Lawrence and Wishart. Lawrence and Wishart had a shop in Russell Street, near the British Museum, where they carried all of the Marxist classics. That shop still exists, but unfortunately it's no longer a shop that sells Marxist classics. In fact, there's a very... Telling commentary on what has happened, the British Communist Party, by the end, was a total liquidationist. The official leadership of the British Communist Party sold it in about 1979 to people who now use it, who now sell books on mysticism and the occult, of course, you know, Ouspensky and the art of theosophy and this sort of thing. And the people who used to run it, who were all veterans of, veteran communists from the 1930s and 40s, were of course dismissed and made compulsory.

57:30 Oh, how I like it. You know, but at the same time, that's practically everything. Very interesting. Are you actually looking for the citations on any of those things? Oh, um, yeah. I mean, MacLean, Concepts and Categories in Perspective is the obvious one, or at least two of them. I guess you knew that. Yeah. That's certainly Concepts and Categories in Perspective. Yeah, he mentioned it somewhere else, too, but I don't know. Yeah, but I think it's in the fullest account I've seen as Concepts and Categories in Perspective, and I think also for that first reference, too. To talk about groups really, you have to say, for instance, the law of associativity for this, and then you write something like this, you say this, I'm going to, this is an isomorphism here, and if you express the fact that if you take A and here if you land on So actually, let me just add that this repo goes here to A, B, and here to A, B, and C, and similarly here to A, B, and C, and similarly here to A, B, and C, and similarly here to A, B, and C, and similarly here to A, B, and C, and similarly here to A, B.

1:00:00 So this is then the theory of a set with an item called a model for it. It's a graph map from this into the arrows of the category itself. It takes A to an identity arrow and B to a function whose depositors don't exist. So, a model for it is actually a set with... So, this is sort of being a category. We don't ask this to be a whole category in the function. Yeah, but in general it'd be more like a sort of a data for a category. Something that looks like a function on that part of the data is a model. But I don't quite understand how these ideas connect up with Bill's ideas about figures. About what? About figures. You know, the notion of figures. Well, there's a figure in there. Yeah, there's just a figure in this particular... It seems to be a very deep way of understanding composition. Alberto, I should mention, talked a while back about this, because some of his ideas about a pre-category of part, these ideas that you might have an ultimately kind of homotopic, theoretic construction underlying. ...giving you conditions of composition of maps. I mean, you know, there have to be topo-theoretic conditions that are satisfied before you've got compositions, before you actually get a category out of this, a pre-category level. Very good idea. Do you see a link between this stuff and the ideas about a pre-category theoretical...

1:02:30 I mean, the way he gets the... Composition, the conditions for, that's the composite of any pair of arrows out of the condition of having an endo map in the, in what's already a kind of pre-categorical construction. It was difficult for me to follow any of it, but, you know, it's something that I thought about, but, you know, This is an interesting framework, and I'll take them this way, perhaps, in the next morning's session. Yeah, but you haven't before. You can go with Robert, I guess. Yeah, well, where are we all going? The Chinese restaurant. Oh, right, okay. You have to, I'm taking him. Okay, well, who are we, Phil? Okay, I'll go with Robert, then. Alberto, have you got somebody? Have you got somebody to go with? Have you got a car? Have you got a lift? Have you got somebody to go with? Uh, yeah, I do, I think. Okay, you can come with me. Okay, just hang on a second, let me put all this together. Right, well, let Alberto know that, so he doesn't sort of get isolated. Okay, as long as we're not... We're heading for the new restaurant. Okay. Right, so let me stick all this on. I'll meet you guys down at the park. If you guys figure out how to get your own car, what's going on? Oh, right, you have got somebody. Okay, and then I'll, before you go, Fatima, you and Bill and Daniel are going off this afternoon, all right? Yeah. Yeah, well, we've got, I know, but are there not a few questions you need to ask about it? There are? No, it's quite clear. Okay, all right, I'm sure it is, but we know which direction we'll be coming from and everything. We can call them up and tell them where we are.

1:05:00 True. And then get completely lost, yeah. Right. So we've actually seen our agri-falls on that one. No, no, we'll take you the next step. Okay, brilliant. That's much nicer. Don't try to do it. No, no, we won't. No, no, you take the wood then. Okay, we'll put that down. Sorry, yeah, do it right now. It's okay. Take my... Of course, I need another... I mean, I would need another... What are we doing on the back? Just write down the... Just write down the list and do it. If you follow this, I think you can go round. Okay. Okay. And would you tell, when can we... Also seems to me to be, to need some more, a more naturalistic embedding. From a philosophical point of view, it needs a more naturalistic embedding, perhaps in some kind of, well, the construction that you've been working on, a pre-category of paths. Particularly the way that Bill understands figures and incidence relations is much less rationalist, formalistic than this. I think, I really, one of the things I really hope we're going to get a chance to do on Tuesday is for you to take him by the balls. Ask him for his reaction. Do you feel that he's ready for that? Having come all this way, I think it would be a pity not to get his reaction. But it's your work. It's your work. He may not feel it. It just seems to me, with his motivation, the geometrical aspect, that it does deserve.

1:07:30 From the point of view of statistics, I managed to fit five of them. Oh, Jesus! Welcome to Canada! Yee-hee! We were talking to Bill last night. Unfortunately, I was feeling very, very... The sinus was making it very difficult for me to hear what was being said, particularly when he's so soft-spoken, there was background noise. But do you know when you asked him the question about... Of which the discrete and co-discrete, the inclusion as discrete and chaos is in the case of the Boolean topos. It's the lowest level. This is what he calls the zeroth level. You know that he replied, unfortunately I wasn't able to catch everything he said, but he replied by saying that construction also applied in a number of other categories. He mentioned that he went into some lengthy exposition about the category of more logical spaces. Well, I was not able to follow. But also, as it were, a very simple example in the category of partial order sets, partial order discrete sets. In fact, I don't think the partial order was what he meant to say. He was relating it to, well, he was saying that the after Hoban involved a description on the left where the way in which the maps... They could separate out elements or map points into intervals so that, you know, they could no longer separate points into intervals, but in the general idea of the construction was just mapping a potentially relativized notion of points, training on the interval structure of it.

1:10:00 It sounds very like Jerry's ideas about the stars and his use of the simplicial category in his thesis. To analyze the behavior of identical particles, which are just by using the office. Well, I don't know what you already did, since I didn't write. All right. I know you didn't have a chance to read his thesis. I didn't realize he'd never sent you his thesis. I will make a photocopy of his thesis and send it to you, because I think you would, or certainly if I can't photocopy the whole thing, certainly those chapters, because it's a nice and elegant construction. But it seems to me to be almost exactly an instance of what Bill was talking about in this connection. You have made, in any case, the copy of that paper by Henry Cuomo, and there in the introduction, the introduction to the city, something is delivered to Cuomo since there is just a representation, a further presentation of all these ideas. Yeah, no, that was just about the boundaries of, yes. ...negation of the boundary operator and so forth. The introduction to the procedure that I... The thing which Gonzalo was talking about, yeah. ...exactly the expression of... That's what I thought. ...of chronological equations and the example that he made to make clear that there was this balance between the degrees of... Figures and points with respect to quantities. The example we made is an example containing qualitative distinctions, although I didn't realize that that is the reason of my question today.

1:12:30 So the only point is to account, to make computations on that example in order to have the unity of these two situations. I hope you like Chinese food. I don't know, perhaps a journal that my library has. Well, he may well have an offering that he can give when we stay with him in Buffalo. ...instruction with him or just listen to him expound it. I think I get far more from hearing him expound stuff here than I do from reading his papers. It occurred to me listening to him that it's very like the behavior of bars and stars in the presentation of particularly the role of the orbits, or what he calls the orbits. And this is something that I'd really like to understand in the context of his ideas about figures and incidence relations as well as his way of thinking of geometrical structure, but of the action of the orbit there. What would you reckon is the naive metatheory for today's talk? What is the naive metatheory for today's last talk? Ah, what a good question. Do you see that yellow sign when you sign who? Yeah. Oh, that's it. So I think there's a...

1:15:00 Do you know what I mean? It relates to a kind of... Yes, yes. We're talking about the... We're talking about sets of objects and so on. I mean, it's just... You know, well, the idea is that there is this pre-category construction, which he thinks is provided by his theory of sketches, which... Well, the idea is that there is this pre-category construction, which he thinks is provided by his theory of sketches, which... Well, the idea is that there is this pre-category construction, which he thinks is provided by his theory of sketches, which... Well, the idea is that there is this pre-category construction, which he thinks is provided by his theory of sketches, which... I know Alberto here has a very different line that gives an account of why you should explain. What is the naive... Hi. Hello. Hi. Hi, how are you? But John, if somebody said to you, what's the naive meta-theory for your informal set theory? You'd say, well, that's just a conceptual confusion. That is the naive meta-theory for everything. We're all sitting together, we hope, after hearing Mackay, but what is the naive meta-theory for this? See, Carl stopped himself thinking in that classical way. I say, well, yes, this is all very interesting as a structure, but what's the naive meta-theory for this theory? This is it. I didn't want to be rude and say that. It wasn't me. It was what John Mabry said to me after hearing Michael Mackay's talk.

1:17:30 He said, well, yes, that's all very interesting. It's interesting, all the algebraic logic. But what is the metatheory of this construction? What is the naive metatheory? There's no correcting some people's conceptual confusion. He's always a good one to argue with. I would like to have heard Michael say a bit more on how he relates these ideas to your thinking about figures and incidence relations. Alberto, I know, has some very interesting ideas on those lines. He wants to ask you about them and get to Buffalo. ...which needs to be stated at least once between generic figures and particular figures. Well, I mean, in particular, you can conveniently speak of them as figures are of a certain shape. The shape itself is what he was calling the figure. Yes, that's absolutely right. Letters, actually, are completely horrible. I've seen his paper on this. Even having the paper in front of me, I couldn't keep that straight. It's a very, very simple concept. Yes, and the notation seems to make it more muddled, much more muddled than it needs to be. There's two aspects. There's the system of notation itself, with the square brackets and absolute values. There's also just really the choice of letters, trying to make everything, all distinctions, be typon distinctions. Bold g, script g, etc. It just makes it harder to... Keep straight what the simple idea is. I meant to tell him that, but apparently he had to leave too quickly. I think if he just changed the letters, it would reduce the reliance on time.

1:20:00 I like Gonzales' paper very much, indeed, his presentation. There are lots of interesting ideas in that, that shed light on the history of mathematics. I must try and raise that. Thank you for watching. If there's something that goes wrong in the translation, or, I mean, I assume the higher order logic is still going to fail as well. Suppose we're doing a classical higher order logic, where it's not going to fail. Thanks very much, Danilo. Does it just turn out that those strings, when you use them in the picture, you don't know how long they're not going to be fine? Maybe the higher order logic isn't incompatible. Well, I think, as opposed to what you were doing... Yeah, but I saw it appear to me you could do something with it. Well, why not? You could just do that. You can't just do that. There ought to be a specific category of formulation there that you could give you by the higher order. I think that would come out with me. Yeah, that's what I was thinking. It wouldn't be incompatible. There will be a sketch that describes the notion of Cartesian force, but no sketch of the force of the function of the equation of the equation of the equation of the equation of the equation

1:22:30 Right, right, so that's what would be in that section. That's right, wouldn't they incorporate it into the simple dynamics? Right, wouldn't they be able to apply it? It looks like a very, very abstract version of Haytham's Completion, in some respects, does it not? I mean, you're using law, and you're still using law. Here's the problem with the mathematics, it's like a fraction of it. Yeah. It's just a few. It sort of does the work of Hink and Model, sort of just in specifying the syntax. How many are good with chopsticks? Why don't you say it's good enough? This is a case of who are not not here. My daughter told me, you know, she's a Chinese. Who is? She told me that the trick, the trouble with chopsticks in Western restaurants is because they don't make them fancy, but they give them classically. But in China, they always use those curved ones. That's right, yeah. Very interesting guy. Is he still in the therapy? Yeah. And he's trying to have a... Well, that's...

1:25:00 You can see how he came into this stuff from model theory, can't you? Um... Do you still remember? He was a... He was an enemy alien. He was an intern in Britain during the war. Really? Oh, I see. Oh, I thought he was French by origin. I thought his accent was so French-Canadian. No, no, he's German-Jewish. But his name, Landbeck, of course, is the chancellor. Well, yes, exactly. It was a symbolic name. I thought his background was Peruvian. He apparently, so he's telling me, I don't know how he got there. He had been some kind of refugee, I think, during his young, you know... Yeah, right. But because he hadn't taken British nationality, they got, well, that was grotesque when you look back on it, because almost all the people there interned were anti-fascist intellectuals. They're being interned because they're... Anyway, he then came to Canada during the war, you see, so he's been here a long, long time. So he's... Well, anyway, so he's probably about, I guess, certainly 70, finally retiring. You don't have to call it the retirement age, so he wasn't, he was a well-known ring-theorist, and that's what he did originally. He and Banaszewski came, well, Banaszewski came later than he did, but they were working in a very similar sort of field initially, and they both had some, you know, and in my opinion... So that's how he came to know so much about Grotendieck and Watts. Oh, yes. Yes. And exactly what Grotendieck is doing. He's an alphabet. I'd say his main work, he's written books on reading theory.

1:27:30 Right. So he was in on the development of schemes and data, cohomology and sheet theory. Although he was really more of a score-reading theorist than a sheet theorist. So how did he get into the... You see, I thought, having looked at the book, I thought he must have come into it from combinatory logic. No, no. He developed some interest in category theory in the 50s, and, you know, just an algebraic kind of thing. And I think he'd always had some, you know, kind of modules, if you want to call them modules, some algebraic things. And then he... This interest has developed. He's always had a rather wide range of mathematical interests, obviously, and I think he's just crowned with that, you know, a categorical idea for someone who's really suited for the thing he's trying to do. And, of course, he's worked with his, you know, he's worked a lot with Phil Stott. Yeah. But as I said, he is really a, you know, well-known decision maker. Well, it was a nice paper. Oh, yeah. The philosophical claims were a little bit suspect, but he had some new things to add. It's certainly an extraordinarily compact way of presenting Gödel's truth. Well, I showed that also the stuff about model prophecies, there's a way of... I finally, I ended up having to prove that theorem to the system I use, as well as something like it anyway. And it's somewhat different. You know, there's certain differences in the formal systems. Jim's is, what he uses is different from the one that I use in my book. I mean, they're equivalent in some sense, but there are considerable differences in the details because he, for example, doesn't, well, not only does the rules lie, you know, but you can change the rules. But just in the definition of the language associated with the topic, I mean, you know, he doesn't, there are some formal differences in the ways that he does and in the way that I do in my book.

1:30:00 And we had a correspondence that could have turned on, ultimately, on the question that he made. He made a couple of, stated a couple of theorems that I thought really were interesting but couldn't be right. There was an error in one of them. It was because in his system it's a bit confusing sometimes. You add variables, you see, to get the determinant. In my case, you add constants. Real constants. They are constants, but they're actual closed terms. Now, he had this way. Of course, it's true that you can make them equivalent. It's not that there's any really… It's merely that in Jim's case, it's a bit misleading. You forget sometimes that what you're really introducing is constants, not variables, and so you can make mistakes, which he did initially. So that was the stuff that we were doing in the Epsilon calculus, which Jim had written on in a tangential way, but I've been working on systematically for some time now, too. I got Dave, my student, my student Dave, he's taking that over now. What's he working on? Well, he's working on a system of semantics for term-forming operators, and in particular in the intuitionistic sense, which hasn't been done yet. There is no bottle theory particular for the intuitionistic sense in the first order sense. I made a start on it. I gave you a couple of my papers. Oh, well, I will. I've got some of them still. The only other thing of yours I've got is the book. Oh, yeah, no, I've got some more. I've got some more stuff. Sorry, I won't realize you've got them all. No, no, but there's this excellent paper that's in the Journal of Philosophics of London, which I would have published probably—you know, it is somewhat philosophical. Some are more philosophical than I usually think. I would have probably submitted it with J.F. Bell had I really found a proper semantics, you know, complete semantics for the intuitionistic definition of calculus. And I didn't. It's actually quite difficult. It's a non-trivial problem for the intuitionistic thinking. But Dave says that he, you know, Dave thinks that he's actually solved this problem, including the species. And he's been very interesting to me. He's not a mathematician, in my argument. He's very smart.

1:32:30 No, he's a philosopher. Yeah, he's a physician and a philosopher. Oh, God. He's a very bright guy. I'm going to come into the subject from philosophy and make that kind of progress, yeah. He's a very bright guy. I mean, and very, very, how can I put it, he's determined in a certain way. He has two bright brothers. They're all doing PhDs. Excuse me, his father is from Lourdes or something more. Yeah, yeah, yeah, I know. He needs to get away to the college office. Such a nice guy. I think I said maybe he went to the college office, but he didn't die. But, of course, I couldn't actually say, you see, that it was too bad that I couldn't. When I, you know, he gave me the paper that we had in the college office in which he gave me the description of what happened there. But he's really good. He has, as I said, I think only one of them is actually doing PhD, another brother. They've all gone through this department, you know, three sort of gifted brothers. And Dave is, I'm hoping that Dave's thesis can, you know, I'd like to see it in some form published as a book. Thank you for your time, and I hope to see you again soon. He's been drifting around, you know, he's done various things, I mean, but he wasn't really sure what he wanted to do with the mathematics and physics stuff. And then it turned out that, you know, he attended some of my courses and both set theory and stuff when I first came here, and he just showed me, I think it should do very good. I mean, I was really impressed. And I said, well, have you considered doing, you know, why don't you consider doing something you want to do?

1:35:00 I always suggest it, I like it. But then we decided that he got interested in the depth of that stuff, and, you know, so he can alter it. So he's very good at literature. He's great at things. He has a philosopher's way with literature. He's very thorough. He's very thorough. He's very thorough. He reads, you know, he knows, by this time he knows all the literature. He studies it very, very well. But anyway, he lives in Waterloo, he knows the mathematicians, he flies to the University of Waterloo, and he teaches a logic course. He's telling me about the all the spot-rich black kids that he has to cope with. No, he does seem an awfully attractive guy. I mean, we got talking last night and completely lost track of time. Very interesting to listen to. And with an interest in philosophy of math as well. Oh, yeah. Yeah, he seemed to pick up on all the, you know, what was really going on in Bill's talk amazingly quickly for somebody who hadn't heard him before. He also writes for me. You know, I mean, the stuff that he's written for me is very well read all around. So I think I have one other... Well, there's Darcy, so I don't know if you've talked to him. He's not... He's only half my... He's another very good guy. You know, Darcy Cutler, I don't know... No, I didn't meet him at all, I don't think. And then there's a third one who you didn't really meet. I thought he was... Some proper philosophical analysis. You mean, yes, history and philosophy, yeah, rather than... What was that conversation you and Bill were having about cohesion and the continuum the first night?

1:37:30 I was only able to pick up part of it because it was on the other side of the room, but it sounded very interesting. It was a very historically scholarly conversation, Zeno and... Oh, no, it wasn't about that. It was about Aristotle. I can't remember. I know we did talk in some form about the question of how... No, no, no, I was talking about the realization of the actual, you know, idea, which is Sweden. That's right, yes. Because we got talking about the beauty of at what point, you know, it really came into, yeah. But I don't think that was resolved. No, because you got called away by Mimi in the kitchen. But what had been going on? Because I came into the conversation halfway through. Yes, but the thing is, in some sense... I mean, Gonzales had this idea too, didn't he, about... Yes, but this idea... I really want to understand that. No, this is... it's not far away, in some sense, from the Kantian idea, but actually of that kind. I mean, not the idea of potential or Aristotle's idea. Yeah. But it can't be completed, you see, so therefore it must be completed. Of course, on the other hand... Kant doesn't apply that to the reality of the world. That's the singular case. The fact is that when you talk about the universe, say, the physical realization would be, say, Newtonian space mind, the whole thing, and the objective, there is actually a discrete infinity of God. This is a singular case of this. It's not a... This is what I've said several times still. Whatever analysis you make, you can't then in some sense use it to infer, right, as has been done, you know, there's been all the arguments by various people that because something along these you can't complete an actual incident in real time, in actual time, it's constant, it says, and therefore the past must be far away, therefore there must be an origin of time.

1:40:00 Now, I don't really think that you can use, you see, I think that the case of applying the idea of the infinite to the reality as a whole, in a complete sense, is a singular case, I mean, you can't, it's perfectly conceivable that there were, and indeed was believed, you know, it's a kind of empirical, taken as an empirical question. In some senses, I mean, for cosmologists, of course, they think, you know, as soon as you're going to tell you, it's going to be bounded. But Einstein, for instance, wanted an infinite world. Oh, sure. I don't see any, and this would be a discrete infinity. That's why he introduced the cosmological constant. Right, right. It would be an actively, you know, an existing discrete infinity. But, of course, it would be some special case, you see, because it would be applied to the physical world as a whole. Now, I build them, I mean, you know, he said, yeah, there would be no way to, that's a... Sorry, I wanted to know what he did say. What did he say? Well, he had to bring, as far as I remember what I, I mean, the sense that I felt that I had, you know, this point, this point, conviction of that special, you know, special nature, it's the same thing that Kant has to do when you ask him. This is true, you're treating, in this case, the idea of the physical. Right, of the physical world as a whole. But it's a similar kind of thing, isn't it? You never get an object. We don't produce. See, the point is, we don't actually have to. We don't produce this discrepancy. It's just already there. It's already there. The one case where it really could be already there, it seems to me, is precisely... It's the cosmological case. Exactly. Yes, he's got to get around that, I can see, from the point of view, the point of departure of it. Yeah, all the other cases, but then I think it's because of this case of applying these ideas to the idea of a question, you know, some way to reality as a whole, which always is a kind of special, it's both a special case and a universal case, one of the things that shows up, of course, in Kant's study. It's still very interesting, you know, the fact that that lands in there. And why do you feel, what would you have said? He said you couldn't have an actual completed infinite, right? Of course, discrete infinity, but he's thinking, okay, it doesn't exist. It isn't there, according to Kant.

1:42:30 We can't produce, it cannot be produced by successive synthesis. But the point is the infinite numbers is not produced by successive synthesis. It's just there. That's the difference. Structural progression, as given in the Arndt axioms, is a subjective notion, and he thinks the objective notion is some sort of geometrical number which... But nonetheless, it's the case that the physical objects in the material world could just be there. It's easier to imagine the idea of a simple infinity of stars. I mean, you see what I mean? Of course, it's quite true that when we then say, make it as a synthesis, the idea of counting them, no, we never do complete it, but it's still there. You can't then deduce because of this that there are only finitely many stars. That's what I'm saying. This is a particularly interesting case. It's an old philosophical problem, of course, which Bill has gone back into in a very interesting way. But it's an old problem. You can't grapple with it, too. I think it's much more strikingly presented in some sense in the temporal entity, because there you really have to say, that is a puzzling thing in a way, you know, you think, well, you know, if the universe existed infinitely, you know, if there was no origin of time, then there actually has been an elapsed number of, you know, an infinite number of elapsed moments, two miles, but they are not produced by successive synthesis, you see, that's the point, they have no origin, they're just there. But that's a special case. Every other kind of domain, of course, Kant doesn't have this notion, really, of an infinity extending indefinitely, you know, I mean, over the star, whatever, but even so, I think that you'd have to say... You could say what he has is the notion of a real instance of the potential infinite. Right, right. But the point is that we, for us, it would always be a potential instance. Yes, yes. But nonetheless it's an objective in the sense that... Yes, that's what I mean by a real instance of the potential infinite. It's not omega star because he obviously doesn't have the mathematics to do that.

1:45:00 I think the physical world, the cosmos as a whole... This is an important, you know, a very important case. This is what caused a lot of the, you know, analysis of that has been important to quite a few philosophical, and I think it's quite important here, too. But then, of course, it can't, that very tension, the antinomy, is, of course, one of the reasons for resorting to the view that the whole thing is, in fact, a transcendental construction by... Yes, but interestingly, he does, that's right, because that's because we then imposed it. Yes, so in fact we imposed it, but obviously that solution is not open to Lorvier. No, but nonetheless, there's still a transcendental real sense, you see, in which the universe simply can't say if it is infinite, if there are infinitely many stars in it. I mean, it's just a fact. The question is, in what sense is it a fact? Wittgenstein also has to think that you know what the later people say about this. There were some arguments about it. Oh, on the internet, yeah. And it's an interesting, this is actually a certain, you know, an important topic. Anyway, I think I'm going to... I should like to understand that point that Colin was making when we were together in Cleveland there about Bill's attitude towards the natural numbers. I don't understand that he does regard the counting numbers as subjective. But I would like to understand what this program that he and Gonzalo worked on, that Collins says he thinks, well it's not a dead end by implication anyway, it didn't connect as deeply or fruitfully with the rest of their mathematical insights as they might have hoped, just what the numbers are in synthetic differential geometry, what it was that they were aiming.

1:47:30 Unfortunately, I just don't know. There was an interesting exchange at one point, too, in the discussion after Gonzalo's paper about what I took to be the connective point about the construction of the number sequence in spaces. Now, in terms of the integral-valued operations of kind of integral-valued circular motions, I would like to understand that. I really want to understand the background of that, because it's quite clear, it was already clear, but it's much clearer than ever to me that the way that the Collins approach, well-pointed topos, and you have the topos, smooth spaces, and they have various other constructions, and those are just irreducible to one another, they're just all part of the framework that we need to do different kinds of mathematics. The epistemological basis of which is in something like McLean. That is so clearly not Otter's program, but he does clearly regard geometry as underpinning, as underlying all of mathematical analysis, and a very physical, and a really very physical notion of geometry. This is what I would like to understand more deeply. The early part of it, as a matter of fact, I thought I had made some progress in understanding the Spock's book and getting to, I mean certainly the, I understand the, I think how the, pretty fully now, how the square zero infinitesimal arises in the context of the topos of smooth spaces, and obviously why it is a non-classical topos, and that's all, how that all fits into the theory.

1:50:00 Topos-theoretic framework and how it's been understood in terms of the intrinsic geometric structure of objects and how then sets can be viewed as the decidable. Bill didn't seem terribly taken with your point about Lambeck's paper. Not Lambeck, Mackay rather. He really saw the whole thing as a kind of very abstract way of doing the same kind of completeness proofs that were done by... Well, it's more like that than anything else. Yeah. Sorry, who am I... Who is it I'm thinking of? No, no, no, the earlier work he was citing. Hencken, yeah. It's basically a very abstract version of Hencken model completeness. He used the language to make the model. I think that's what he doesn't like about it. I think he wants these sketches ready for him. I think he wants it to come out of an even more geometrical... I was just asking Alberto when you arrived about these weird geometrical models for arithmetic that come out of synthetic differential geometry and what the history is. Do you know about these 17th and 18th century ideas? Yes, I mean, did they really try and do arithmetic in this setting?

1:52:30 I said I don't know, but I know Cox claims that the kind of differential geometry is done in the 19th century, but maybe so, but I can't believe it. I just wondered if you know if there's any history. It seems that there was an interesting history to these ideas. And certainly the idea that geometry was underneath arithmetic, which after all, even at the very end, when he threw up the whole logic of this program, Frege did revert to it, didn't he? He thought the complex, you know, he actually thought you should get the complex numbers first out of geometry and then define the natural numbers as integral-valued reals within, you know, inside that framework. I will admit to that, it was in the very end, wasn't it? Within a few months of his death, he didn't do more than publish fragments on it. I mean, it was a revival of an idea that had been around before. I just think that it must be an interesting historical study of that. It's the sort of thing that somebody who, you know, doesn't... is never going to be able to do the real math. It's the sort of thing I would like to look at. Write up some survey paper. Yeah, I was just wondering, you know, who are the people I ought to read? I thought, you know, since you know so much history of math, I thought there might be people that want to... No, it is ancient history. I hardly know anything at all about that. But, I mean, something similar must have been around. Obviously, I wonder, in terms of trying to get complex negative numbers out of it, in the sense that, as you said, it's a complete anachronism to describe your doctors as having anticipated... No, I can't... No, I don't think there's any... He didn't have that concept of a number. Here's an old line segment. Oh, what? I thought for a second we were going to drag out Nietzsche. Oh, John's got a wonderful collection of science fiction. Oh, sorry, I thought it was... Oh, John Darrow, sorry.

1:55:00 I just glanced at him and saw the rocket going up on the front cover. No, I had that picture. Is that, in that edition, is that covered? Yeah. Well, is it due now that it's paper about? Oh, it's just a sarcophagus, actually. Think about it. Well, it's about its whole view of logic, and can it, you know, does it anticipate the relativized notion of element inseparability that you have in topos theory? He suggested it does. It's quite an interesting little paper called What is the Real Phenomenology of Logic? They published it in the Searle Studies journal, what it was called, Anna Lecter, and it was early on about four or five years ago. Because both of them are squarely on his own central interests, but particularly, obviously, Gonzales. Natural languages, in which a lot of these issues are discussed, which contains an extended critique of McNamara and Ray's theory of natural causes, which you were looking at in the play, and which is part of a longer work as Ray Capaci didn't... And indeed, even the Mackay stuff, he's, you know, he's all of that kind of protocategory in which to think about in these sketches, but, you know, more in the way that Bill thinks about them in terms of... He's very interested in geometric interpretation. He's got this interesting homotopy theoretic construction which explains when the composition of any pair of maps is allowed, which I know he was, for a hundred way reasons, he wanted a course, he thought about it as well. He just hasn't been very trained on that stuff. And of course the style also, I think the antibiotics that he's taking are...

1:57:30 No, not throughout, even more than these things. Is it just Perth, Philly? Well, I do hope he's feeling better soon. I feel very sorry for him. I mean, I got lost in the details. I certainly got lost in the details of the presentation. I know a bit about the motivation from talking to Alberto. I was trying to say, what is there in this that is not completely original? Except the categories, he talked about subsets of the object. He talked about functions. He didn't explicitly talk about categories. Fair. But he took the graph of the categories. I mean, the category could have been a graph. Yes, you can approach categories through a graph. I have a feeling what's going on here is there is a new way. But what it is, it's for making certain notions, previously obscure notions. Noticing also, there's also this thing, that there's the further things, making them primitive or interconnecting them in different ways. When you formulate something, when you formulate something, bleed away all the errors. This is also this idea of looking at, you know, this society in which everybody lives by taking in everybody else's watch.

2:00:00 Yes, I mean, listening to him last night with Alberto on how this levels construction, which he surrounds his presentation with all this metaphysics about being and non-being, its inclusion as the... The payoff for this, from your point of view, is that it gives you a beautiful way of handling structures in the category of monological spaces, and there are whole areas of functional analysis where you get... Extremely. I don't follow them because I don't begin to know any of the math. But it's quite obvious that, you know, just at the, I mean, just knowing as much topology as Alberto knows, this gives an incredibly deep insight into the connections between functional analysis and topology. And there are also instances of it in much simpler forms. The thing is, that doesn't just work for topos. It works for other categories as well. Well, yes, there's the idea of the aspects on the right, which are the purely algebraic aspects, and the aspects on the left, which are to do with the idea that all of these things are figures, and to be understood in terms of instance relations, and that's in a really very directly geometrical way. The limiting case in a really strong sense that they could be thought of as, in terms of physical geometry, they're the limiting case where an unrelativized notion of constancy will serve to parametrize whatever variation is going on.

2:02:30 Well, what I was getting out of it was that it gives you the most formative, very, very abstract form of your situation and did sort of stand back.