Evening Talk

0:00 All right, that sounds a little more complicated, but actually it's a very simple invention. Yes, yes. You can actually define it. It's not just a message, it's a sign. You know, the next estimate, you add in the next, you get an estimate. Oh, you agree, this is a very good... I agree, yes, because I was taught this truth by Imre Lider, this person, Imre Lider. Imre Lider. You know, he's from Minakari and he's from Cambridge. He's an impressive man. Very good one. He's a specialist around this theory. I don't know many of the other groups, but that is certainly very illuminating. You have to remind me, who's the class leader? Well, Ramsey's. Well, Ramsey's. Well, you've got to guess. Okay, so just the vertices and the margins. In principle, in principle, I think... Okay, suppose you assigned every two elements of a set of mathematical equations, for every set of two mathematical equations, you assigned it a color, red or blue. Okay, so every two elements of a set of mathematical equations. It's got to be a genuine double. It's got to be really cool. Okay, so every double has a color gradient. The Ramsey Stereo says there's an infinite set of general numbers such that any pair of things in it is a color descent. They say it's an infinite monochromatic set. That's, that's Ramsey. You can do it.

2:30 Then, there's an infinite set of images, every pair taken from which is one. And the curious thing is, the finite version is easily proved from the infinite version, as Harris and Harrington figured out a kind of variant such that the proof from the finite infinite version to their version is almost as good as their version, and yet you can't prove the planetary version of Harris-Harrington. Thank you. This is a question whether looking at finite purpose is possible or not, and whether presumably there is some version of a gravity theorem that you see, and whether you might be able to do a characteristic of that too, and whether that does populate an abstract theory of the problem. It really has more structure. It's sort of like, roughly analogous to the total bearing on a canister, yes, because you have B here, and when you have X here, it's more structured, and even though it's a conservative expression of theories, it becomes more than the X that you have. So why that X is not equal to the theory of finite composes might be true. I don't know. I mean, it seems to be almost the same as the thing as we are in the world today, in terms of quantum mathematics. But it's not clear, because it really does have more to do with mathematics. Seems to have more of a history problem. Oh, no. Paris had a history problem. What's stronger than the other way?

5:00 No, no. I know what I'm saying. There's a theory about that. I don't know. I hope you can do that. That's also something. More, you see, more likely the other way. You've got to hear the other instructions. That's probably the only way. If you take any model we have in the Institute, you can construct a category where the objects are just intervals in the descent of all x between 0 and n to each element n of the object. So it must only be objects. Now, a map is some complicated combinatorial configuration on the right side. It's acting on that, those things, and so forth. So it would seem that, it would seem that you would get a glycogen-closed category in that way out of any model. It's important to start with the armature, you know. So that could be an equivalence between the category of models, if you want to look at this, and the category of models of the supermaterials on the total base. Or not. They don't quite know what it's at. I'm not sure exactly what's written. This is about having a thing in the interface. You're thinking about this in kind of a strange way. Theory of finite, uh, theory of finite problems. What I was saying was, do it in the form of what's written, Dick. You can construct a certain category. What kind of a category is it? It might be that this thing is so equivalent, really, to the abstract. And I said, one of the non-standard models for it. Thank you.

7:30 Yes, you wouldn't have math for the mathematics. You wouldn't have enough math to ensure these intervals between the non-standard elements would be enough math. Yes, because they wouldn't see it in some sense. It may be considered to be a finite linear order. There, you can get sort of mini versions of static recursion, because you can define something by recursion along the finite linear order. You just have to stop when you get to the last element. The data has to... Thank you for watching. Quantum mathematics would be a medical, medical science, quantum mathematics would be a quantum physics, and quantum mathematics values would be a quantum physics, and quantum mathematics would be a quantum physics, and G-sets would be a group, instead of like grid sets, or something like that, or, or, or, or, or, or, or, or, or, or, or, or, or, or, or, or,

10:00 You know, it's another type of scientific example. So, all these are social movements that satisfy certain rights, that satisfy strong climatic conditions, which is a very concise and aggressive way of doing things, but they don't, there's no such thing as having one that's going to generate too much opportunity. If you want to, if you want to give them a kind of virgin title, you know, that's a good thing, but that's the problem. Thank you for your attention. One of the biggest things we were testing was whether one of these final tests was going to be a mathematical experiment or not. We found out that there was none, so we found out that the model was out, so we decided to set it free. So again, they should be at least as numerous as possible. I propose that instead of getting coffee here, we can go over to my house and have some coffee. How about that? Sounds a very nice idea. No distance at all. No, no, no. I feel kind of guilty about leaving my brother there. No, sure, but it would be nice to go there anyway. He'd be charmed if we go over there and start talking about climax offices and gravity. It'll send him right off to naughty land. I agree. Remember, you're speaking to an outsider, so it's all new to me. You know what's depressing? I was just saying life. Life? I've used Ramsey's theory of proof. I've taught it. I've spent hours presenting it. I can no longer remember how to do it.

12:30 If you could give me an afternoon, I might be able to do it. It isn't that bad. I mean, the point is you use sort of iterating pigeon-fish. It's not like eating a meal where you've got inside you.

15:00 Maybe ask them, do you want to go ask them? Yeah, he's just a bit tied up. Richard's already asked him, but you can try again. It's got to do with it being the eve of the bank holiday.

17:30 It actually connects, I think, with some of the things Bill was saying this morning about the nature of these... Things whose domains are determined by this notion of etendu, you know, these things, these toposes where the maps between fibers are not necessarily local homomorphisms. I think there's something very interesting going on there, which connects with these... The correct understanding of the way one thinks of the case of sets in a topos, which I want to ask about, came out of some remarks he made to me a long, long time ago. Well, it's to do with this business about there being, in the case where there are not local homomorphisms, you know, defined between the fibers, there... The thing, because there's some kind of non-trivial action of an underlying group or monoid of the space, for instance, as an example, the thing is varying in, I mean, this is obviously to use pictorial language, but this is kind of varying within itself in a way which introduces a kind of double topology on the thing so that it may be locally a topological space but not globally a topological space. There's a kind of...

20:00 You know, to and fro between the local and global properties, I suppose. And this is what prevents it from being like a set, because for a set, you've got to have the condition that things, well, the fibers are discrete, which is, because it's, this all comes back to the point about whether the notion of a set as a collection of elements and extension is really fundamental, or whether it can be understood in terms of some more... Well, primitive, topological, geometrical conditions on the fibers. Topological, geometrical issues is a naïve notion. Yeah, well, it's already at sea, I know, but this is the discussion, this is why Richard and I got here late, because this is the very discussion we were having on a more general footing about how one should think about the ordering of concepts in foundations. I know, I know you're... Bill's point about the fibers... Yeah, but of course in the case of these non-trivial 8-on-2, you can't think of them just in that way, because the whole, if they were like that, then they could be like sets as domains of variation, but then the whole point is you've got this kind of internal variation, which mixes stuff up in such a way that it can't any longer be separated out into... Things which live on the fibers where they're discrete. Well, okay. Well, anyway, I think we'll have to revisit these issues. I really can't hear what Bill's saying, and I can only barely hear what you're saying. I think it's not so much that I'm going rough as a jerk. The mathematics logic is not involved in all of these problems, and we don't have all the people to teach us what to do, and it's not time for them to teach us what to do, because we don't really have anyone to explain it to us by saying, well, this is a problem, and we don't really have anyone to tell us what to do, and that's why I like to apply.

22:30 Yeah, that's not really true, because that's not really, uh, that's not really, uh, that's not really, uh, the way that this positive, I call it positive, I call it positive logic, I call it positive, some people call it geometric, but other people call it, uh, dynamic, and other people call it coherent. Do we want to make a move? I put in 1370, which should comfortably cover. There are also a number of other fields of study, such as mathematics, geometry, algebra, mathematics, physics, physics, mathematics, physics, mathematics, physics, mathematics, physics, mathematics, physics, mathematics, physics, mathematics, physics, mathematics, physics, mathematics, physics, mathematics, physics, mathematics, physics, mathematics, physics, mathematics, physics, mathematics, physics, mathematics, physics, mathematics, physics, mathematics, physics, mathematics, physics, mathematics, physics, mathematics, physics, mathematics, physics, mathematics, physics, mathematics, physics, mathematics, physics, mathematics, physics, mathematics, physics, mathematics, physics, mathematics, physics, mathematics, physics, mathematics, physics, mathematics, physics, mathematics, physics, mathematics, physics, mathematics, physics, mathematics, physics, mathematics, physics, mathematics, physics, mathematics, physics, mathematics, physics, mathematics, physics, mathematics, physics, mathematics, physics, mathematics, physics, mathematics, physics, mathematics, physics, mathematics, physics, mathematics, physics, mathematics, physics, mathematics, physics, mathematics, physics, mathematics, That was what we spent the last hour on. Yes, yes, yes, it was quite extraordinary. Some kind of complete theory to do with the notions of form and matter worked out on the basis of what he calls this business of intensive and extensive quality types. I've got a very good set of notes and of course the recording rolls so you can get them. Just a minute.

25:00 Well, I hope you'll take up some of the things we were just talking about now, about finite opposites and your programme, and, you know, the stuff that we were just discussing now, because that was fascinating, but I just couldn't hear a damn thing that was being said. It's awful as you get... It's not that you can't hear things, it's just that your hearing becomes so much less discriminating. You can't pick out voices, individual voices, against the background. No, no, we won't do that. Okay, where are you staying whilst, you know, we're keeping you out of your house and home? Oh, so you haven't got to go miles, in the miles out of your way. Oh, that's quite a fair old way. No, it's not. It's not? No, it's just over the road. It's built for a road. Okay. Okay. It's just that white label. Okay. Oh, so not so far. Oh, by the way, whatever you do, don't let me forget. I moved out of your place, and obviously to leave it to Bill, last night. I have got your alarm clock. Yes, you have an alarm clock. You probably pissed Bill's. No, no, it's not Bill's, because Bill has mine. It's a history. You mean that thing that was in the bathroom, do you? Ah, that's a timer for the... Oh, it's just a timer. Oh, is that all it is? Well, the point is, well, the point is, it allows me to tell the time. Anyway, I have it. I'll make sure I return it to you, but just remind me. And also, also, I borrowed your copy of Hobsbawm's autobiography, just for a little bit of light bedside reading. It's quite interesting. Thanks, mate. You are a prince amongst men. Oh, no, he will not appreciate it. He thinks Hobsbawm is a dreadful revisionist. Yeah, take care. I'll see you. Thanks. No, no, no. I'm going in to see him at nine because I want to finish off stuff. I want to do some emails on his computer, which I can't do, obviously, because we're going to have the talk in Richard's office. Actually, I suspect we will have it upstairs if you and he are going to be there. We'll see. But I'd just like to get that out of the way.

27:30 Well, I have done the application. I've still got to photocopy it and mail it, but that I can do. It won't go now until after Easter anyway, I imagine. Is there any post on Easter Friday? Well, in that case, I ought to try and photocopy it and mail it tomorrow. But I've got it all done. How stupid. Why wouldn't they let you do it online? Ridiculous. Well, you know, the British Society for the History of Science, you know, their special grants application forms you cannot submit online. You've got to send them in. No hard copy. Ridiculous. Anyway, I got Alex Burr to write the reference and he's done a nice job of it. They don't realize we've got an equal because it's Italian for tea. It's not quite equal. It's Italian for tea. It can't be as bad as French tea, which is beyond description. Dishwater. This is absolutely disgusting. I've got some time to catch up with you. I'm going to show you an interesting architectural feature of this house. You know, that's how thick it was in the 1780s. And it's also, this part of the house is a pie-shaped slab that's stuck on.

30:00 You could look out and where that roundabout is is where the public execution is. This terrace had a meeting. They had to change their name because everybody's cut off at the same weird addresses. So they wanted to change the name. So I suggested Gallows, but it was turned down. I think it would have been a neat name. It's neat on our character. It's called Coffin Place, the original name. Coffin Place. ... to know what... Cambridge people. You are curious about... The reason for my insulting Italian tea was, do you want milk? No, I know you don't. So you can have Italian style or English style. You'd like milk and tea? How about you? Just the tea. Oh, just a little bit. I like that too.

32:30 Well, Cambridge people. Peter has invited our day, last year, to give a talk about the models of science here. The thing that he talked about, the inflections, the homotopy, the self-preservation. Well, anyway. I think you don't need to say too much. Fortunately, I'm too expressive. No, no, no, perhaps I'm very ignorant, to judge without being actually correct. And I don't have a question, but probably Peter, if he has invited me. But I don't ask Peter about his opinion. He doesn't write most of the time. Well, no, there isn't any point in inviting him. He's got your point. ...to appear fair, to appear, not to appear prejudiced against the guy because he behaved so badly, and how he behaved so badly that Peter had to actually sort of try to take over the whole, let's say, the authorial behavior. Well, how can you say that? So after this sort of thing happens, you see, then you want to show, well, I'm proud of you, you're not prejudiced. I'm just guessing. I didn't even know he had been invited. No, because he was invited before Brussels, before, last year, not this year. No, no, no, no, not that Brussels.

35:00 Ah, the other. Summer school in Hammersmith, which was in the Ardennes, in Old Bordeaux, not in Brussels. Ah, yes. No, no, no, then it was definitely after that. No, actually, regarding... He has a very strange behavior and he is very unfriendly with Dahl and he is unfriendly and when we were at Peter's house something surprised me and so they formed a little group and you know as... You've been just together evening, haven't you? And I wanted to join the group and say a few words to him. You want to go to the group? Thank you for your attention. If he behaves strangely, it doesn't mean that his work is not good, of course. No, of course not. No, of course not. Do you know him? Yeah, I've met him. It depends on who he's with. I have to say, in fairness, I'm telling you this out of school, but I happened to walk with him and his research student, Henrik, back from the CMS to lunch immediately after your talk, and all that he said was very enthusiastic and positive, and he was saying to Henrik, you really should get together with him.

37:30 He clearly had, because he was talking to Henry, he wasn't talking to me at all, I just happened to be walking with him, but it was quite clear that he had a very, very positive assessment of your work and was keen to try and persuade his research students to go to talk to you, because he obviously thought there would be lots and lots of things that he would learn from doing so, so nothing but positive things. He said nothing negative at all. It may be too much milk. I just simply say that for this work. Here you go. Olivia, there's yours, but you might need some sugar. Do you need sugar? Please, yes. But he does have... He does have this personality trait, you're quite right, of kind of rather lacking to try and take charge when we were in Oberwaldbach at this. He did the same thing. He would stand up at dinner and start making announcements about what was to happen the next day without having consulted any of the organisers or the people involved in the other workshops, and it was extraordinary. I have no idea why he felt the need to do this. He would bang on the dinner gong and suddenly start making announcements about what the programme would be when nobody in our workshop, let alone the other two workshops which were going on, had ever even discussed it. Well, but apart from this, he seems a perfect example. Sometimes I think he takes too seriously the idea that he was MacLean's last student. This is kind of a badge of who I am. I'm not MacLean. I can't remember names. Janalisa? Janalisa. Janalisa. Okay. The Georgian... So you're now the sort of Moses that's leading the... Yeah. According to him. Yeah.

40:00 I mean, this was sort of done at Coimbra. Oh, okay. No, that's easy. Well, there's a certain amount of reasonables. I try not to take it too seriously. But... No, but anyway, you know, even though he had a good opinion of my work, he didn't come to speak to me for all of his... He didn't come to speak to you? That is very strange. No, so you know, and when there was the occasion... Did he urge his... he was urging his research student to do so. Did his research student come to speak to you? Yes, the research student. Well, that's what he urged... okay, that's interesting. He was urging him to do so, so at least. But probably he wouldn't have done that anyway, because he told me, I am very, you know, talented. Well, of course. Yes, of course, interested in the same area. But, you know, anyway, it's strange, because I was really interested in the kind of things that I was interested in, to have a few words, and when there was the occasion to. We exchanged a few words and now we both are fruiting. Strange. No, I didn't get offended. I just took it as a strange behavior and nothing more. A lot of that work is based on the theory that I can show you. Which one? You mean the rabbit? It's a rabbit. He's a dragster here. Ah, yes. Yes, he's very keen indeed on algebraic set theory. I can't understand it. I couldn't understand why so many people got interested in it, perhaps because of the fashion thing. Because I was proposed to write my thesis on it. It's exactly the way of delaying people from learning category theory.

42:30 And so he asked Rosalito for suggestions about toposferic because he said, you can do the easiest category theory but I just put a signature and nothing more than that, you know. Did he ever come to you in Cambridge to talk about this recent work of his other research student on the Tarski pie singer? I just mentioned it to you, I didn't, he didn't. Oh, okay. That's interesting, because I, he, he at one point he was saying, oh, I must go and talk to Bill. You can still, I've still got that. I don't know, do you, are you familiar with what they're, and I know what Tarski did. The glass is really old, so it's going to run. Again, before I was there. Ha, ha, ha, yeah, but still, still, yeah. Well, you seem to have been at all the most interesting talks in Buffalo before you were there. It's basically about, you know, groupoids and so if you succeed in general, I try to, I mean, obviously, naturality with respect to general maps is, you know, that's interesting and kind of weird, but basic constructions like function space are not going to be isomorphically mathematical, only with comparison. So, I mean, to make a really intelligent approach to that kind of problem... In the modern point of view, it should involve comparison maps, you know, of this progression of isomorphism. Yes, and not just some of that. We've got the actual one thing, the negative stuff.

45:00 This is a strange place. That's kind of why I like it. Negative things are only isomorphism in general, not isomorphism. I was having a bedroom success. No, I was really very sorry. The moment I saw that, it was at the Skirting Board and Mount Merdike. Oh, yeah, Merdike. The girl who was living here was speaking. I mean, they didn't have to go and make a whole formal... There's a doctrine or a subfield out of it, which is what happens. So it's basically the people who have some, that would be John here, who have some lingering affection for Z.F. Well, you know, the funny thing is, I mean, you're from the natural axioms for Z, you think. You don't get naturally out of the axioms. You take the natural axioms. You don't get global recursion, so you can't define them. You have elements of elements of elements of elements. Yeah. Which is the bed. I've got a finite set of two elements. A lot makes sense once you realize this. For example, when Pythagoras said number 13, you know, something like this. It's crazy. It's kind of like animistic. There's a definite base of an element.

47:30 Well, the numbers are potentially infinite. And so you've got you've got a definite number of these things well you can take each one of these things is potentially divisible as math matters every any piece of matters so you take one of the units out of the number and now you've got a you've got a but it's only potentially because the potential is I think this is in the Parmenides, the Von Neumann Organs, okay, so now you've got two things. But the dyad itself, so now you've got the set consisting of itself. Yeah. Okay, but for the two-element set, it didn't build them up like you do the Von Neumann Organs.

50:00 Yeah, so that the idea that the set is a set or is a unity itself can be added to it. I have a paper on that. My paper called the Qualitative Distinctions, Toposes of Generalized Graphs, with one family of examples there, which is precisely based on that slogan of Coltrane's. I wanted to illustrate that, and the backing up, I mean, what you felt, I would think of it as mapping up, given that... The set, you could have another set mapping two to it, so the fibers are going to go back and forth. Using the same set? I mean, the point is, the Plato's example gives you the number of, gives you unlimited error. Yeah. You always have a recovering of a given set by a map. Fibers are, you can even limit yourself. They say you can make one of the fibers have two elements and all the rest one element. But there's no reason why you can't make them all larger at the same time. No, but what I was saying was, if you want to understand the set theory from the categorical point of view, to sort of mix in the, you know, the way of looking at it with that is just a complication. It's a huge, unnecessary complication that you don't really see what it's all about. And that, that's why I, that's why I... They wanted to retain the sort of full structure of the category, but they didn't really, I mean, for me, the interesting issue is there, what about the real bare-knife, what's the relation of the real bare-knife with the, whether you think of it in your classical hierarchical way, or in some more categorical way, what is the actual relation? They don't really deal with that.

52:30 Where do they deal with measurable cardinals? This is a very natural question that arises. There's one idea that they have, which Steve Audie took over and a set of other people. It's sort of viewed now as the key ingredient of the great term theory. That you have a functor, which is instead of a genuine power set, some smaller power set. It'd be something like indivisible. No, they don't even do that. It's like classes. The idea is that the power set of B is not a genuine power set. It doesn't satisfy the random transformation rule. So this is really an unnecessary complication because the key construction does allow you to construct something like a model that's really just be made as embedding by the coverage model. We started with small, as soon as the F is consistent, then there's a small category that's basically modeled with this kind of thing on the axiom, but if you take sheaves on it, you can get the category, but you can take the sum of all the representables, so you have a single object and there's a large theory that sort of represents class in all sets. But actually, Glass and Shedrach kind of... There are memoirs in the math society where they do the same thing. That would be a step ahead, considering in that direction. No, I don't know why they stopped, actually, the value of these investment papers, because they went on the right way, I think. I don't know why they stopped it, but they just wanted to... Well, they were trying to categorize set theory, you know. No, but many things were classified, not only, but a lot of modern theory and finite policy and they wrote this important paper on Boolean classifying hypotheses where they made explicit a connection with some of the particular encyclopedic theories and then that was a good paper and then they wrote...

55:00 A couple of other problems were also viewed, but that was not good. Well, it was partly because they were together. Last year, the professor in Michigan, Shadrach, had a postdoc in computer science. Ah, okay. So they could work together because... Because they were together, yeah. And then Shadrach went to Penn and got entirely involved in computer science. Yeah, I know all those things, but he doesn't know them all. Chenderoth was a student at Buffalo, and his old Ph.D. was there, and he was my assistant for the courses, one of the more active. He had this European idea, since he was the graduate assistant, therefore he should actually assist the courses. How eccentric! Well, I appreciate it very much. He would make up exams and give tutors, tutorials, all those kind of things. Whereas a graduate student doesn't normally do that sort of thing unless he's specifically paid to do that. As for example Dolan was, and Dolan didn't do it. He never did anything right, so I'm sorry. Not our fault. So his official advisor was actually John Mott, and he took all my courses. In fact, those are one of the few authors that are in class. Oh, and I found out something only a year or so ago. I've always thought, well, Andreas Blas is a set theorist.

57:30 He's a very good set theorist, but he also has a very good categorical... He thinks more like a mathematician than like a logician. But this is something very exceptional. Then I noticed his professor was Raoul Watt. Raoul? Raoul Watt. I don't know. Yeah, a very famous K-theory and geometry, upper-grade topology and analysis and the same kind of thing. He was actually a student of a very good mathematician. He probably thinks more like a mathematician than a musician. I know, because if you compare that paper on Boolean classifying topos, I think at that time, at the time at which it was written, it was really remarkable. Because still now... There are several ways in which, what are we talking about, mid-70s? Yes, I think it's 70s, 80s, 82. Yes. At that time, you see, there were many powerful things that were coming out. For some reason, most of them were not pursued. It's very unfortunate. We're back there in society. Joël, for example, in the mid-70s, Joël was one very creative participant in all these kinds of questions in all fields, but then somehow he just became very narrow technician. I mean, I wouldn't even say that he could read. He became interested in technique, not only interested in mathematics. I guess so. And he latched on to particular concepts, which is the positive category, which is even actually due to somebody else, I mean to Borgman and Block.

1:00:00 Whereas previously, he had a very creative idea of the categorical version of the calculus theorem, and many things which were not published, but they were part of very active seminars which were done in Montreal in 1973. Because Martin told me that the Jovial actually didn't publish a lot of these ideas, but why did he have such an idea? Were there, were they in a different form like the ones you've learned? Yeah, very different. I mean, he was not a fuzzy thinker like me. I mean, when he would give a lecture, he was very, very precise. The action was very precise and the result was very precise and very striking. He wouldn't give a lecture unless the result was something striking. Now, I think he gives a lecture just because he has to give a lecture. He became very, uh, routine. Well, he didn't burn out. Yeah, he certainly didn't burn out. He kept running, but... Mike, did you... Oh. Sorry, I'm okay. I'm on with him. You're up in, uh, slumberland somewhere. I was. Now your eyes are there. Yeah. I was just saying it was probably hopeless to... I didn't expect to get any kind of reasonable content of discussion to go on. I didn't know if it would change. Suddenly it wasn't the case. Well, at one time, I think. Back in the 70s. In the 70s, even now, what that is, is good. I've been reading some Bobaki's definition of structure. I mean, the bloody thing is unreadable.

1:02:30 Bobaki had a very... The European Union wanted to be a part of something that is kind of formal reasoning until a point where you see that it could be formalized by any columnist, mathematician, and then just stop. It's just like, you know, you do algebraic calculation, but you leave out gaps, and you leave gaps out there if you rely on your readers to do it. I think it must have been two. I don't think he was. I think he avoided some of them. The technical stuff. I'm talking about the historical organization. Oh, something historical. Yeah. So he says, he says, mathematicians always say that's what they do. They often say that's what they do. But we can assign, we can do no more than assign it as sentimental or metaphysical. You know, if your main object is, if your main thing in life is proving, You see, in the first half of the 20th century, that was the official philosophy. Oh, that's crazy! I know, but he was just swimming around. This was the sort of philosophy laid down by Piano. And Piano said that when in geometry we talk about a line, we must be careful to explain to our students there is no line, it's only a symbol L. And all we do is do things that are similar. Or not even prove things, but just manipulate them. This was the origin of this mock thesis, this organization for teachers. So his idea was this is how high school teachers should teach mathematics.

1:05:00 And it doesn't mean anything, it's just a normal thing. I mean, look at McLean. He couldn't possibly think that and actually do mathematics. Well, it is said that some years after he said this, he kind of softened and realized, well, really, there's a line there. But no, but this was the official, you know, representative view. Or Hawking's book on set theory. Even McLean, if you look at McLean's early writings. Look at my early writings, for God's sake. The lingo, the lingo is taken from that philosophical division. It's assuming that, like Mike says, you know, capitalism is another name for the way the world is, this culture is another name for the way mathematics and physics was, and the philosophical world was, and so many of us thought this country's religion was painless there and those people didn't think that. At least of all vile, who was the strongest influence on Maclean and Goettingen, according to what we were being told by Audi in his talk at the time? No, there wasn't that much time because the Nazis were taking over. When did Maclean leave there? Mid-thirties? 1933. He left there? So Bayou was just the formal head of the plane at the time of his work. The thing about Bayou went off to Zurich. If you look at the immensity, I mean you can see what Leigh-Anne Farrell is, he's immensely proud of something to be gorgeous, formal, complication, that sort of thing. And you know, all of it is syntactic. And when he comes around in the final section...

1:07:30 So he wants to, he wants to, as it were, retrospectively project what he's going to do, but when he's already in fact done, yes, yes. But the point is... One basic recommendation is don't take it seriously at all. More about these set theories, more about these theories. It's just tacked on. It's not their point of view. Well, the point... It's their point of view that maybe they were trying to wrap themselves to the environment or something. Well, the center, of course, is unreadable. Yeah. It's unreadable. And the rest of the actual mathematics is unreadable. It's very readable. No, I agree. Algebra and topology is very readable. When he gets to the general definition of structure, it's so wrapped up in the syntax that this language can't figure out what the hell's going on. But basically, his idea is there should be a kind of theory of time. That's it. Okay, so it's not all that far off what a topos is. Yeah, it's not that all that far off what a topos is. It's a type of structure and an element that you can tell them. So it's not that far off. If anybody could have understood it, nobody... I think there are much, there are later editions. That's right, yes, yes. That's one of the things we were talking about, discussing in, in, um, over Wolfpack. The original volume came out at 45 or something. I think this is even, but they had, they had also a theory of mortars.

1:10:00 Yeah, yeah. Thinking about that. Yeah. That's right. But you see, I mean, you show up with, it's actually not, it's not very practical. Model theory, because it's not clear how to treat systematically the models in the algorithm theory. We deal with cognitive theory and the first order theory, but you're talking about a theory where you have all finite types as variables. I mean, there's things that can be interpreted in variable ways. The problem is variance, you see, because the entities you want to look at are both conscious and program. So, if you define morphism as a model, you have to make some choices about positive and negative and so on. So, they're making a stab at that rather complicated problem of having to deal with them themselves. All the structures they use in their subscript volumes, but it's a lot of nonsensical syntax. To such an extent that you'd have to be crazy to actually, or you would end up being crazy. It was simple enough. The idea was that this type and then morphology was occupying some type. And presumably what's behind would be just to take, you know, just take all the sets of all finite rank over some base Martin Davis.

1:12:30 And then you take sets of type omega, of rank omega over that, the universe of this, but you can't do that because if you allow, as you complained about, if you allow sets with sets and so on at the bottom level, then things get named twice. It's going to turn out, for example, you can be an element of the universe which is also a subset of the universe. Could be a function on a fragment of it. I mean, there's all kinds of... But you stick it in the... If you put it in the types here, and you don't even ask what one type is equal to something of another, or what you type, you don't get it anyway. I remember hearing you say that when you were in Los Angeles in the 60s, Montague and Scott got a bunch of problems with having two names in the spirit. Now, you're interested in models. Models are taking another topography, and the model is a fun curve, it's a so-called methodological curve that actually preserves, you know, preserves products, and that gives you an induced map, the level of exponentials, and requiring that we preserve all the stuff, but precisely, you know, up to isomorphism preserves all the structure. So, fine, you have the category of all these.

1:15:00 Now, the morphisms, you see, the natural transformations of those factors are incredibly restrictive. So even if you had a group theory in mind, there's a higher order theory. You kind of use the higher order theory of groups. These have nothing to do, I mean, they're so incredibly special among group homomorphisms. But the category of groups in group homomorphisms, you kind of primarily want as the average homomorphism. Despite the fact that we're looking at the logical morphisms and the fact that there's a problem of variance, the problem is where are the morphisms? Their problem, Scott and Montague's problem, is by taking the background as BF, as a model of BF, with its own membership that was spread. And the theory itself also has a membership structure. Those are the two things that get mixed up. In any attempt to define the notion of a model that can express what you want, there are difficulties in your eyes because of the collision and the things that are true about the model that don't follow from the axiom, because things that are called epsilon here and there. As a result of the background, another thing is called epsilon, as a result of being the values of the puncture of the model itself, you will have identifications which will lead to two things in the theory being equal in the model, even though there's no axiom that says it's equal. And this is what we choose. I think we are talking about this. You don't get it, you don't get it, you don't get it. There is no victim, please. The complete experiment in 1949 was phrased for higher-order logic, but what it really meant was that you're just looking at the models that are interpreted as functions of preserved products on the field and, of course, the logical operations, but they only preserve exponentiation up to a map.

1:17:30 And so, in some sense, it's really just a multi-source of first-order mathematics. You have sorts of all the higher types, and they're treated more or less like the other sort of stuff, because you do have the validation maps, but they're not interpretative. You see, but in some sense, this is similar to actual mathematical practice, that you have, you know, Borel maps instead of an interpret. The maps as being things that are Borel maps, sort of arbitrary abstract set maps. So the function type in the theory is interpreted as a sub-object of the corresponding function space of the interpretations of the Roman and Ferdinand types. It's a subspace usually because in principle it's just a map that preserves universal qualifications at each level. Then you can show that it's a comparison with my work, but it's not, there's no, there's no systematic way of dealing with the case where it follows what I've been working on. Well, there's really nobody except Peter Fry. Peter Fry has a large number of films about logical mathematics. Well, there's, I think that, I think what Burbanki was getting at was something rather, he wanted to say, I mean, he wanted to say how a mathematical struggle will follow. He wanted to get a general way of describing that situation in general. And he had lots of examples. For example, topological space, the morphology, the property on the... Of course, this is what I'm saying. This is what I'm saying. The thing is, if you want to talk about the topological, the logic of the topological space involves sets and subsets and so forth of that space. So that's why you're in terms of what you have.

1:20:00 Right. But you can't, you certainly don't want confusions. You don't want the fact that the underlying set might contain as elements, subsets, so then they would be, you might even have an open subset, which is an element. I mean, that's completely irrelevant. Yeah, and that's part of the structure of the co-domain, the background in which you take it. Morphism, as it were, you'd have these types on your function separately on this structure to some other kind of structure of the same kind. So, it's just a way of saying, of getting all this clutter out of the way. If a set is, if the underlying set of a group is a von Neumann ordinance, most of these things are subsets. All the elements are subsets. Okay, so what's the domain? The theory itself, construed as a category, forming as a reclimate... Okay, I see what you're saying. So you're talking about, like the sort of Tarski... The free tempos, that is the theory that's construed as a category.

1:22:30 It's arrows that are reclimate classes, terms that you can construct and... Yeah, so it's kind of like, it's kind of, of course, Blendenbaum Algebras or these Hinken Algebras or whatever. Yeah. We do love the fact that for topological spaces, the structure maps backwards in the definition of morphism, doesn't it? The structure maps backwards relative to the map, and for groups, say, it would map forward. Whereas, I mean, for monological sets, it also maps forward. Monology is a kind of cohesion, just like topology is. The idea of the arrow is the opposite. And even in dinner you see the type is the same in which sets are set, right? Yeah. Rounded sets are open, or the class of rounded sets is the class of open sets, so the level at which... Except the point is that the power set has two variances. In the power set you have the inverse image, but you also have the eigencentral modification that goes forward. And so you sort of have to make a choice. If you have something as a type 5, you have a large number of choices to make about the direction of the thing. So this concerns the definition of morphism. Yeah, morphism between structures. Of course, it means they realize that that's important. You can hardly have done the Earth thing. It seems that it was probably dedicated and... It's just so complicated. In other words, to have this general theory is... And they're not going to be learning any theorems about it either. Maybe if someone did it properly, depending on the problem you're making, you know, you're making a subset of subsets and you've got two different variances, unless you limit yourself only to isomorphisms, maps that are invertible, so you turn them all around, in group representation theory they do that, you know, in the category of key representations. There's a lot of construction theory where it really takes you going back and...

1:25:00 We just put in a group element to the power of 1, and that's another instance of something that's very positive, so you can sort of turn a lot of things into universal co-variants and counter-co-variants as well. But most structures are not invertible. See, that invertibility would be something taking place in the theory. With a fixed group, you have a theory of theory. You've got two different models. Then you have to figure out which ones are going to have the G inverse and which ones are going to have the G. Right. You'd have to take that into account. But I mean in the case of, say, topology. Well, there you make a difference, as I say, and you have to fundamentalize it backwards. But, in my own opinion, the true definition of isomorphism, even in algebra, you've plotted together. Well, the definition, if you say two things are isomorphic, you're not saying, there's no directionality in it. Equivalence. But you should still, so that way, it's got to go both ways. Morphology conditions, it's got to go both ways. Yeah.

1:27:30 There's a morphism, and there's a morphism, and there's also a morphism. There's really no other rational way to do it. No, I agree. I know, you're saying it. But the theorem that for math to be a nice morphism, it's sufficient that it should be a nice morphism on the underlying set. And that depends on the theory, right? Right. Q for algebra, not Q for topology. Basically because there is no single underlying set. Usually there are several underlying sets that are relevant. Well, in topology, so what are they besides the set of points? My view is, I'd say that we should get rid of the idea that the topological space is a default. And in fact, the typical, the opposite way is geometrical. The geometrical moves always from left to right. The structure of an object is covariant, which usually means that it varies covariantly with morphism. Which usually means that the object itself is divided across some conjugary function. Crochet's idea of topology is different. Crochet's idea was to go forward. The topological space in that subject is two underlying sets. There's a point and there's a set of convergent sequences. And of course that's just an obvious observation, but just thinking of it. It's actually surprising to think of it that way. I mean, it would never occur to me to say, I mean, I know this. Obviously, I know that you can give divine convergence in terms of sequences or points and open sets and so on, but to think of it as two different underlying sets is actually quite a useful way of looking at it.

1:30:00 Yeah. I mean, what I teach in mathematics after this is continuous function and all that's produced is the operation of the points and the limits of season. That's how preshade is done. It's slightly more complicated logically, because the notion of convergent sequence is slightly more complicated. But it's a given data. This is the way you collect, I mean, this is the way, this, you think of a convergent sequence as more like the way you think about how you actually measure things. No, all you have to, all you have to define, technically, for the case, one is a convergent sequence in the space which is a generic convergent sequence. But if there's a space, like some topologists have been told, more overhand than real, if you get the non-trivial, highly disconnected, compact space, all you have to define is really what the continuous end of math is in that space. The continuous end of math is the same thing as the continuous sequence in that space. So that's the fundamental data. In the general space, we've got a set of things that are declared to be convergent sequences, and one of the properties is the invariant that is the action of all the quantum and mathematical methods in space. Okay, so your case against taking topology is the default, is it? And so the observations of the form is part of it, definitely part of it, because, you see, one of the main reasons for inventing topology was to deal with function space. We wanted to know how a set of space of cohesions within the maps of X and Y should have its own intrinsic cohesion. There is no successful effort.

1:32:30 The defining graph, in general, has been based on this statistic of all these functions and all the other kinds of functions. It turns out that you can do that for locally compact exponents. But on the other hand, in terms of the covariate structure, it's just a variation of what is the smooth and what is the function on the space of all possible paths and things like that. But those are just translatable by man-made and math-based mathematics. So, in other words, you might have some various side conditions on the phases that you have to check. Fundamentally, if the true reality is that you hope to form the function space for any two phases, it's very hard to find them in this way. What I'm calling the geometrical way. Figures and incident fields. Because the figures in the function space... I guess the figure is on the product phase, exclamation test. And then, as a byproduct, you have a notion of which function it's going to have, and so facilitating that is fundamental to derive, and that makes a huge difference. I don't think anyone's ever successfully, when Horavius in 49 came up with the case phase, was treated in David Gale's paper and Kelly's. Down to all colors of the book and other variations on the speeder on so-called convenient spaces for all variations.

1:35:00 Well, Gerame has this idea, which was you start with topological phases, but then you redefine the maps to make this truth. By using the compact spaces, let's call them caves, let's call them ponds. You use all the compact spaces as the model for figures, so a figure after its domain is compact. Automatically then you have all these spaces defined in terms of compact figures. The category of figure types is pretty big, but the category of all the spaces that are governed by it is still bigger, but it is even bigger. And these things are still going on. So a big chunk of it can be thought of as ordinary topological space, but if you can modify this, it makes it a very little amount of fiction, but it's not an exception in the sense that it's not a school of even busting the other kind of structures, and yet it succeeds in that in general. So what takes the place of topological space? Well, there are several. There is no one default, but this is one that's very, very often used by algebraic apologists and by personal analysts alike. It's the one in which the figure types are just compacted, so you sort of accept the category as compacted, and then define the democracies in that way. Preservation goes in the right direction, compacted. The compactness, the compact maps, they're still going in their own. You're pulling the, if you think of the open covering, then you're still moving backwards along the function. You're using the inverse function, the inverse image function.

1:37:30 Yeah, you do that sort of thing. It's really a continuous map into the three-point space. I didn't think about that. Sorry, the two-point space, the three open sets. Mainly one point is open and one is not. Now that, that replaces the real numbers, the real value continuous functions, the continuous maps to the real numbers. Real value and sophisticated functions are identical in open subsets. If you take the inverse image of the close point, that's an arbitrary closed subset, and the rest goes to the other point. So the algebraic approach is... I'm saying that traditional topology is actually algebraic in character, not geometrical in that sense. The fundamental structure are these functions from... Into this funny little... Yeah. Sapinski squared is another space with four points. We have a continuous map between the little ones which represents the idea of finite intersection. In other words, it's a distributive lattice object within the lattice system. So if you have that distributive lattice, you see what I mean? The distributive lattice is embedded into any reasonable category, which I never want to see. Then the notion of morphism is defined by these other interesting figures. Continuous math is not a mistake. All of this has to do with the expansion of the figures in your space into the figures in Czerwinski's space. Everything you might want to do in the open space has to do with the expansion.

1:40:00 Except in the Czerwinski's space itself. They don't define, yeah, of course they're also, you know, the idea of a covering, a climax covering with a certain nerve, a very fundamental common form of induction, you could have like three open sets that intersect on the corners of your head. Triangles and interiors. All those kind of things are really just, you know, you're going to have continuous maps into those. And so a covering of the space X, according to a certain pattern in the mirror, is just a single continuous map into a separate finite... And what's one of these? Sympathy-like spaces? Sympathy-like. Well, basically, basically every finite... How do you find that coset? And the coset is probably out of the space in a particular way, which is fine, but it's most important to find that coset, because it describes all these stuff in a very algebraic way. For a general space, its analysis in those terms is the algebra. The algebra is not a real function. You try to get it to your open sets and then you can go... Except that you see that the function spaces... We'll have their open sets, but they won't be determined by them. They are determined by their figures, the convergent sequences or their compact stage. They won't be the shims of open sets. They're just important byproducts and not the complete. Well, are you saying Frechet had a vision kind of like this? Yeah. For some reason, Reitz and other people... Sort of what we traditionally thought of as topology decided that the machine was on the wrong track. It's actually mentioned in Kelly's book. Kelly, the book that I learned topology from, there's a chapter about this approach, but it's sort of... I know one thing, but the standard thing that you hear is that it has some very big functional space and functional analysis. When you find out that it's topology, it's open source.

1:42:30 All of these are not described by sequences. You need longer sequences in order to capture them. So if you want to do something equivalent to the algebra, the ecology, the cohomology, you'd have to consider the Nets. And that's where the Nets come from, that's where the filters and dichotomies come from. It's basically trying to get a larger category of models, figure-shaping, because of the idea that you have to deal with the arbitrary open set. So having kind of decided that the general open set was the main thing, then you had to extend the sequences in order to sort of match that. In spaces where you can't remember what they're called, sequences don't work. But anyway, you have to use an algorithm. The non-first axiom. The first axiom. Yeah, the first axiom. The neighborhood. The neighborhood basis. The second column.

1:45:00 Yeah. That's right. The examples of spaces where... In other words, the convergent nets, even longer ones, in the function space are still the same obvious thing, namely the transformed convergent nets in a smaller time frame. In a way, the idea of using all compact spaces as models is just sort of a natural extension of that. A certain space, a big space with a favorite point, right, and generally in a particular space, a map of that is the same thing as a convergent zone. So that, let's say that that's kind of the space, it's a computer, why not just take all the contacts? Does anybody need any more tea? I'd love that coffee. You'd like that coffee? Yeah, I asked for one when he came in. Didn't you hear me? It's okay, it's no big deal. That's one of the reasons I'm struggling to keep awake, fascinating as the topic is. Oh, I'm so sorry.

1:47:30 No, I'm teasing, don't worry. No, don't worry. Don't worry. I have some questions on the subject of geometry, motion, and the design of it. Yes, the idea of relativizing things with respect to physics. The fact that it makes sense to consider classes of geometry and motion as localities. The general system is only really concerned with that. Geometric morphisms such as x and r are just the sheaths in x, objects in x. In other words, the terms of the sheaths are x, and that is really an algebraic function. The total is treated as an algebraic function. And in geometric morphisms... The basic controls we find back here. That's what the star can be considered in terms of the plastic. The object is composing the function. So this is just like affine schemes. We're defining the space not geometrically but algebraically in terms of the algebraic function.

1:50:00 Seeing the category of all two things is what we have. It is like, like that. This is, this is what my product is, which ones are exponential. Only the ones that are compact ones. The ones that are coherent ones. Which one? Which among, which among all the S-based supers can be used as exponential, you know, right and right. So there's a restriction. It's exactly like classical ecology in that respect. Because the classical ecology is not really geometry, it's algebra. That's what leads to this lack of a generalized science. You could do a huge thing by just using all of these spaces as models, but more in terms of, for example, consider all gluing and coherent filters, you know, spectrum lenses, given the general filters, the geometric morphism, which is really a model of the left and the right, you could think of subclasses simply in that way, two categories, which again will automatically... This will be represented with a single, single purpose, a single idea. I think, my feeling is that there are some aspects of model theory that we are not able to explain. To represent, to represent a certain concept that can't be represented by a single diagram.

1:52:30 No, no, I stand to believe that it is, in fact, my aim. For the moment, you know, I have restricted just to understand what happens below a certain level. That is a very restricted thing. Of course, one should take, you know, all the topics that are out there. No, that is certainly something that is in my class, yes. And there are many other classes, of course, when I... If you're not aware of where geography took place when we were made from the steps, this kind of a theory should take itself from the two categories. What two categories? I think the category that has filthy food in it. The category that has filthy food in it and some food in it. It's actually a cartoon showing a new kind of food policy. The basic observation that's in the latest class is that there are filthy food in it. If you have a Cartesian problem, a map, if you come to find the map, preserves the children's children. You each carry one separately, if and only if it preserves them separately. This is a very rare thing. A map preserves the children even though they're separate. And for that reason, I would say that if you didn't choose to take the category of the modernist, then there is in fact a category of all future children of the present-day country, which would be the natural, the natural. And you see, the thing is that both the category of topology, both the top and the cohomology are really indifferent to that.

1:55:00 Geometric morphisms between two cohomons are in the category of cohomonic. And if you compose geometric morphisms between three cohomons, the operational composition will be related to each other as well. So this is top over s is enriched in this. I know, I know. I know, yeah. Yeah, no, no, no, he's in my... Other, other... Yes, thanks, John, yeah. No sugar also? Sugar as well, just for the night. I don't know. Just one. Do you have an interpretation of one algebraic theory and another in this category of analogies? The substitutions that they're finding for each other, they're still continuing to be done, and you have two different, two categories, one for each, and the top, both are enriched in the same, so somehow, somehow, this is the two-categorical version of sex, and therefore this first approximation is larger, and if you look at it, if you look at it, I'm getting there. I'm getting there. There are things like, I'm not sure if it's true, but there's something that should be everywhere in the book of quantum chronology, and the idea is showing that there's a little bit of a swindle there, and Peter said that it can be, that it doesn't mean enumerating the objects he put, but it's supposed to have to be more than a common natural number.

1:57:30 I say, oh no, there exists such an enumerator, but this existential quantifier is not an enumerator. Well, in fact, in fact it's a fucking image. There is a classifier for enumerated things. There's a classifier just for things, and they just are. And there's a map between them. You want to take the image. The image is like taking the existence. The existence is not going to be a topos. It's not a factorization. If you take any of the usual factorization systems and factor that, then the idea will disappear. So to really get the proper kind of image, you need to go to the larger world, just like in algebraic geometry and varieties. Is this, in some way, connected with those ideas of topologies? I think so. That's what people are getting at. Yeah, because that is money. I don't think they describe it as well as I do. A real man would be able to think that that is the general principle. The idea that set should be replaced by categories is going to come up with us. Of course, that two categories use another context, but it should be with mathematics to enrich topos theory and algebra, and hence a base for appreciatives of all sorts. I've never seen that. What would the classifying ring for a theory of this kind of filtered limits of toposes in some...

2:00:00 Two categories, this kind of generalization of topos theory would be like, would it be? Well, I mean, this is one of those things that Andre Joyal pointed out, not what I said, but another statement, that for these, the object classifier and for the map classifier and a whole lot of topos, one could just instead talk about the models. And the arrows are just filtered co-limit-preserving maps into sets are the same thing as the objects. So that's a kind of two-category theoretical generalization of sets. Basically, the two categories of all categories are filtered co-limits. ...as a general base, and there's one object in it which is the object class of our topos. Yeah. Out of that. The same paradigm of algebra is applied to the... ...it's to get a ring, a base ring, K, in the category of sets. It's like that one, that topos. Just as a ring is a much more rich structure than a set. So that topos is richer than the general category of the filtered code in this case. And it has addition and multiplication. Yes, yes. Yes, I can see the analogy with... I'll show you a bit of chart. I'll show you a bit of chart. You've got to get the statement proper. Interesting. Was there already a plan? We're meeting at 10. Well, we haven't actually planned the detailed topics for each day. We've allowed it to flow, and it's worked very well. At some point, I won't...

2:02:30 I mean, it's quite close in spirit. I'm not even sure it's close in spirit. Categorify. Ouch! You've already made us all... Okay, we'll make a move. I think if your system of... I don't think that's such a bad way of describing it. Say it without us.