Discussions

0:00 No, no, no, I'm completely aware of that. To you. Yes. No, no, no, no, I can, no, I also agree, but, you know, I was just used to it. Implementation could be described in that way. We have these subtypes, subtypes. We want to describe it in the language of finite time-consumption. We'd like to say everything about it that we can say in that language. Language understood, not in terms of defining it, but we have to. We can't say quite everything about it, naturally. Going back to my stupid example of inserting an inverse, if we looked at all possible words involving that, suppose we had the actual inverse, we'd have very complicated things that we can now, which are statements in this general language about this subtopos, and yet we're not able to quite manage to say all those things. You can say a lot of them indirectly just by making it more complicated. The exact formulas and so forth and so forth. I think it's probably why we need to, there's no compromise with the category theoretical issues. One or the other. But there's a problem there too. Yeah. Right? When you come to presentations, there is this maroon-y solution to the problem. Presenting of X categories, you know, beginning of that, right? The judge, just getting quantifiers all together, if he doesn't want to spot my limits, then there's exactly the same sort of problem. It's in the theory now, this is the world. I want to compute the equalizer in math, or I want to describe it.

2:30 So that means if I have another round, once again, if this is term-to-term formation again, because I have to verify that they are equal before I'm allowed to use this term, and so I can't quite understand what the terms are unless I solve this RE problem first, the friend and principal RE problem. However, Moroni had a solution, over and over and over and over again. It's described in the first pages of the book by Lombeck and Scott, for example, and the idea is that you just sort of make a more detailed theory of term whereby to any diagram, like any diagram like this, you always assign it out there, but you don't conclude, you don't say it's equal to that. In other words, it's until you have more and not more. So you just sort of build things up and you impose. And so on and so forth. But then you show that from the machine that you have constructed, if it happened to be the case that those two maps were equal, then this triangle would commute. I mean, you have a third thing, which is a map that measures how much it commutes and so forth. But you don't have to, you don't have an infinite regress, it's just sort of three steps of that sort. And then you satisfy Tarski. Which I always thought must be incredibly important for computer science, but nobody seems to hurt the pulpit now. So, in some sense, the real problem of all this is, is there some kind of analogous trick, which is a little bit more than a trick, but it will work for things like inversion. That's very interesting. I've read that part of the comments. Yeah, now you know. You have some idea more than I do. Yeah, so, you see, you should be able to...

5:00 There are a lot of things that are known to be uniquely joinable, and then go on to say things about them, but things that you couldn't say if you didn't have them. So sometimes you haven't really at all enlarged the theory when you join those terms, but you have to make your ability to reason with them much less recursive and makes you less flexible. Oh, yeah, flexible, yeah, sure. And how things like the, say, monoidal categories are flexible and strict monoidal categories are not. Yeah. But trying the presentation version of that, the thing is that you can add new objects and you can make your, well, in principle it's something, I think, where you couldn't add equations between objects. You could add equations between maps, but not between objects, so you know, it's okay to restrict them or not. The thing is that you would, syntactically, you would add new objects, and you would put those objects equal to old objects. And so you were allowed to do that sort of thing, you know, so we will introduce... An arrow, which is an associativity arrow, and its domain must equal this, and its codomain must equal that, that was all right, but one needed some flexibility, one needed to figure out how to do that precisely, which you could do, because you just said that if you had a new diagram, you could put its object. Equal to objects that you have previously described, so long as you can do it in that order, whereas you can't do it in the same order. Then maths, or mathematics, or someone else, or the guy right at the finish, they define the presentation to be a co-limit in the category of monarchs.

7:30 This is not the way anyone else on the planet presents action. I had quite a lot of trouble with Steve convincing him that the defining of visitation would be a combative bias. I would think that the action calculated in the combative category would be more or less equivalent to the original problem. Pretty much. Well, that was what I pointed out to him. It sounds like you understand. Yes. No, no, no. I understand what it was about. I think the book is correct. In fact, it's really the geometric thing that I'm interested about here. Because, for example, you want to calculate the full realisation of the clear and secure. But it's not clear and secure. So, as you are able to achieve such a simplified speech by using the time for that. Actually, you can make all these deductions, because in a regular category you can always test all these things. If you are a geometric then you can also consider it, you know, this which is in the asymptotic class. So, you know, it's really fundamental, the notion of that. The word you say, geometry, you emphasize, you investigate it. You do. And the fact that this junction is allowed is... But she gave it precisely. But I didn't give the definition myself. But it's the definition that was given in the end. The definition of geometry theory, so I didn't mention it myself. Because of all possible combinations of conditions, we can consider some practical names attached to the end of the song. No, no, yes, you can't relate to that. It's a very strange concept. Also, when you watch your math eyes, then you consider just that, you know, here you have a problem with mathematics, more or less, but you select the one representative for each of the two, and you take the two together.

10:00 And then, at a logical level, of course, it's the same. So, the things that are isomorphic and redundant, but yes, so you can rotate it and, but, you know, it's something that works for it, you know. Actually, this is the call for the presentation. I received just a collection of articles which need not to be closed like that. This is because the platform that sends the received platform to Rihanna is the sub-topos. Thank you for your attention and see you in the next lecture. Because all the covenants, the arch covers, are straight covenants. Can I ask what was the answer to Matthias' original question, that if you ask for the map just to be empty and monotonous, the claim is that you get the question.

12:30 The question is, do you get the exact answer? And if I understood it at all, the answer to that is... I think we can frame this in a more abstract way, that you've got a well-defined two categories of logarithmic probabilities and geometric probabilities. Can you give a syntactic two category that is bi-equivalent to that? The spirit of the Syntactic Academy being the Academy of Geometric Theory. Exactly, which is obviously the motivation for this whole thing, for getting into this programme, for getting into everything we want to say about logic inside Geometric Theory. If you're going to do that, you need a definition of Geometric Theory. You need a construction for each Geometric Theory of a classified topic. Yes, yes. For each classifying top, for each topos, you need to get back a geometric theory. Yes, yes, that's the method. To start with a geometric theory, build your topos, go back to your geometric theory, you should wind up back where you started. Now, you're not going to do that if you have, like, you know, you've got a map where there'd be a model. No, no. When you come back, the thing you get back... So, what he's looking for is a notion of math and geometric theory that's inherently syntactic and then is bi-equivalents of two categories. Yes, it gives you bi-equivalents of two categories, inherently syntactic, and at the same time, as it were, satisfies these geometrically motivated conditions of B. Yeah, now... That you express, you know, what you say qualifies in terms of just, in terms of all that. So it looks like Olivia's claim, right, is... This is a true claim, but it's a limited claim. It doesn't really say very much in that you can pretty much define your notion of sub-objects when you've got a well-defined notion of sub-objects.

15:00 You can pretty much define your notion of sub-objects if you don't have it. If you don't have an existing notion of math, and you don't have this going on in a general level, you can pretty much define your notion of a sub-object and then you can characterise it, and she claims she can characterise it by not needing to enter the signature. Which I think seems to be true, but also seems to be reasonably obvious. And that gives you what she wants, which is the kind of stability and the pullbacks of this flat J continuous. Mathematics is not giving you really anything that you weren't getting. I believe this was sort of done in the 1970s, when people took this kind of definition, but she evidently doesn't know it. Well, she said to me this morning, she wants to do this kind of thing for herself, but you do land yourself with re-inviting the wheel a lot of the time when you do that, and also going in via this detail, this kind of mixture of both syntactically motivated construction using a common sense. Yeah, yeah, I mean, I can see it. The understanding is that the outside of all this is a clueless model of mathematics. Just let me get my foot in. What I'm changing from this is that there's just a little bit too much of a notion actually involved in the way that we do it. Notions are motivated by the need to patch on this construction, this existential structure. But we are not yet clearly fitting it into the electrical field. And then she's going to do this whole infinite thing, possibly like that, which is... Which turns out to be a red herring, yeah. But I can see now, because given what she's focusing on this, on the caverns in construction, she probably won't see the infinitrix as the red herring, will she? And it's also the spirit of the syntax-semantics relationship which he got started and which you've got a whole string of results and categories that have been expressed in those forms, the most obvious, the most prominent one being...

17:30 There's a thing between finite limit theories and one different variance of that. It constantly strikes me that even after, what is it, 50 years now, even 50 years after the discovery of count extensions, you can say in terms of quantifiers, in terms of count extensions, in terms of the last substitutes, the logicians still don't... If we're going to go for lunch. Yeah, I was just going to say, I don't want to be a lunch. No, no, no, I don't think I am. I don't think I'm at all affected by the German. I think I'll be destroyed. Let's let, let, let, let, let, let them go. They'll just come to a natural. This could go forever. Do you know where we're going to go from here? Yeah. Ah, yes, yes, yes, yes, starting from the function symbols and variables, yes, you obtain, yes, yes, it's the same, what changes is just, the terms, no, no, I think, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no.

20:00 It was just what you termed as a category or a product. The context corresponds to conjunction, so you have to consider an either or. But I was suggesting that you have the... The term is form of category. You don't have the structure of the terms. You have to consider the form of category. Well, the lattice of the requirements that are living around this could be pretty fresh. I think it would be a good point to break, don't you? Thank you very much, Olivia, for certainly generating a huge and, I think, very, very insightful discussion of the best we've had. So, you could limit yourself to... You said the compositional theories. You don't have anything. So, no, you are talking about those which do not have... The ones which do appear with relations, no matter.

22:30 Curly, so I'm purely relational. Yes, yes. Except toad. No, it is toad if you look at it. But certainly there are projections and there are permutations. Toad-free presentations, that's all. The point is that single-sworded theory of that kind always has exactly the same category of terms. Okay, shall we, well obviously we'll meet again at 2.30, which is, of course, that's why I wouldn't tell John to stop. What is the time now, John? It's about... In person. Oh, okay, that's fine then. We can leave stuff in... Oh, hang on, I don't have... Is it safe to leave stuff? I don't have anything valuable. No, well I have... I think we should... Yes, that's very interesting, and also, of course, I'm hoping it will give more occasion for Bill to... To revisit precisely the topic that we were starting to talk about on the way here, which is that we have to press 50-50. Ah, 50-50. 50-50-50. No, 5-0-5-0. Ah, okay. No, because unfortunately you did 5-5-5-0. There we are. That's okay now. Which is precisely, which of course you did touch on, naturally in the course of the discussion. This whole program, which Grotendieck laid out in that Boston, yeah, I'm sorry, Bill's in the little handrail, he's out here with you, just wait for him, he might, he probably does by now, because we've been there, but let's play it safe, it's only literally two minutes up the road, it's just the corner there, turn right, well, yes, he did, this happened, in fact, in, this happened in Cambridge to my great distress on the last day. Well, I didn't lose him, well he lost himself, but he said to me at lunchtime that he was going to come back to the CMS early to sit down with this, the guy who's going to sit down with him to tackle this issue of straightening out all the...

25:00 Rubbish entries on Wikipedia that give net-dark information about topos theory and so when I didn't see him at the afternoon sessions I thought well that's okay that's obviously what he's doing he's got his you know he's got the bull but he's he's he's got his head down bull between mixing metaphors you know he's got the bull by the horns doing that. No he had in fact wandered off in the direction of the Cavendish laboratory and Yeah, no, no, no, that's somewhere else. That's somewhere else. But it's beautiful there. Yeah, well, of course, in English, I mean, come on, you've been in England long enough not to feel. Well, actually, my best friends from Sydney, she's a lovely girl, Jenny, Jenny Bacigalupi. Well. I'm not sure exactly where they, but Guido's been there for about four years in the philosophy department. He's a philosopher of physics. Sydney University. Yes, Sydney University. But his wife, Jenny, is Australian. She's actually a musician. She's in fact a conductor, professional conductor. Very rare. There are not many women professional conductors. But they've just moved to Aberdeen. Yeah. They've got a little... Baby girl, well she's middle of the baby now, she's a toddler, she's about 18 months, and they've had, well because he got, because he was given the chair at Aberdeen. No, Aberdeen's actually a pretty good university for philosophy. Well, that's the thing, they're obviously hitting the wall now, coming from sunny Sydney in the middle of the summer to Aberdeen in the middle of the winter. I don't think Jenny is exactly too ecstatic about it. I'm glad you explained to us that it's a granite city. What does it mean when there's a cloud in the winter most of the time? You can't see any buildings. Exactly. Although in fairness, in the summer, when you get days like this, which you do occasionally even in Aberdeen, it is absolutely beautiful because of all the silica particles in the ground. Of course, the whole thing just sparkles. It's like the buildings have all got crusty with diamonds.

27:30 Absolutely, but the problem is you don't get days like that. You don't get many days like that. Yes, they're pretty few and far between. That's fair enough. It's very interesting in Edinburgh to see, I guess, professors would be appointed. When you get a professor, I suppose, they come with an entourage. It's like a barber room, I think, somewhere, where there's a whole bunch of students, fellows, students and friends. Right, yes, especially somebody like Atiyah when he's going down there. The idea is clearly for them to be a permanent host, but two years later they leave. Well, it depends a lot on which part, climatically speaking, which part of the world they grew up in, I mean. Well, yes, yes. If they're already inured to... What clearly happens is that two years later they leave, it takes one year to arrange leaving, which means that with the first winter... The first winter finishes, either that, yes or there, well, how do I say it, they... They have to toughen up if they're going to survive that it was the climate that bred the Scottish Enlightenment. But, of course, one thing I think people do, many people, especially Americans, completely forget just how far north the British Isles is. I mean, that it is, because of the Gulf Stream, they completely forget it's way north. Even the really sunny bits, like, you know, the Siliars, are well north of any part of the continental USA. They forget that by the time you get up to Aberdeen, you know, you're... I mean, Edinburgh is the same way, but it's in the same latitude as Hudson Bay. Yeah, yeah, exactly. It's the same latitude as Hudson Bay. Or, you know, or Vladivostok. They completely forget how far north Britain is. Do you think we're going to the same places as yesterday? Well, it depends entirely what people want to do. I'm actually all for having a little bit of a change and maybe going to one of the little restaurants or cafes along here. Yeah. How do you feel, Mateusz? Shall we have a change from the vaults just for one day? Go to one of these places? Well, whichever one you fancy. They are a little bit cheaper than the vaults, I have to say. I like the vaults, but their pub food is quite expensive.

30:00 What about this one we went in on the very first day? Let me just ask Olivia and Bill what they would prefer. No, that's true, she only just arrived. Well, I'll just ask Bill, because he may just have a yen for the vaults. I'm easy, but I think it would be nice just to have a change once. We were just wondering, Bill, shall we just have a change for one day from the vaults, so we don't have to come to maybe the little place we went to on the first day? Or even there, yeah, which is also quite nice. Yeah, let's do that. Richard is around, but he's so busy with admin that he can't get... In this today, well, he's the warden of his hall and he's also, you know, he just got back from a brief trip and I understand, oh shit, excuse me, he has been loaded being the, you know, the new bug in the department. He's, as so often happens, he's been loaded with an awful lot of admin. But that discussion... OK, well here we are again. This time, let's hope nobody chases us because we won't forget our wallet. Shall we park ourselves and see what the nice lady can come and... OK. Yeah, I think we can all squeeze in here, can't we? This is Unidas lemma, right? Yes, yes, I'm sure you recognized it, yes. This is my t-shirt. And actually, as they say in France, pour les nôtres, it actually tells you on the back that it is Unidas lemma.

32:30 Well, I had my bar. I don't think that's right. Oh, you made it with Yoshiki Kinoshita's supervisor. I mean, I visited him regularly every year in Japan. Anyway, he was a supervisor. I mean, I almost met Junedo, but he had a stroke while I was on my first visit to Japan. There is a separate French tradition with just oxynog, representative bunkers and so on. For one time, they never used the word Junedo. And then there was this rumor that, well, they made an act of privilege at the IECS for such a crazy time. Well, it is striking, he published, he in fact visited it in 1958, I believe, in the year that he published, and exactly, it was coming up with the representability, but on the other hand, they made quite a big fuss in 2008 for the 50th anniversary of Yanida's Lemma, they had a conference at the Ecole Normale, and there was a There's lots of, well, it was mainly for the philosophers. Yeah, I did, and it was, Cartier spoke, and there was quite a lot about Junida's time in Paris, which was the only time, apparently, he ever visited France. So, I, yeah, I mean, he obviously must have taken things from listening to what was being said in IHES. Well, as Bill certainly pointed out, I think possibly it was the first to point out, other people have taken it up as well. Of course, Cayley had already seen a special case of this back in the 1890s. And Dedekind in the case. And Dedekind in the case. And Hossess. Standard method of dealing with Hossess.

35:00 That was quite good to me. Do you have any recommendations? I have been here before, but not that much more often than, well, we were all here last night, but it's all, yeah, I think I'm going to go for the, I shall go, I think, for the Cumberland sausage with the mashed potatoes and the gravy. It sounds quite good. If you just want something fairly light, I would recommend the jacket potato, or possibly the soup if you want something a little bit more, say the soup and something else, I'll say the soup and the potato if you want something. What is the jacket potato? This is a potato baked in its own jacket, so it's kind of nice and crispy, and then you have a filling that the potato is filled with. Either beans and cheese or tuna or these are the fillings that go with the potato. This is a very typical sort of English midday pub meal. So it's a potato with a filling? It's a baked potato in its jacket so it's with the filling inside thing. You have the whole potato split in two and you just scoop out the potato and put the filling in and mix it up together. So it can be quite good. Yeah, it can be quite good. Certainly very filling and good. If you want a hot meal in midday, it's a pretty good one. And quite economic. No, she doesn't. I think he does. Well, last night they came to the table to take the order home. Do we have to order at the bar? Sorry. Or do we? No, okay, no, it's okay. I just wanted to do the right thing. I wasn't bitching. You do have to order drinks at the bar, that's the same in all pubs, but they do food service at the table. You probably take the drinks at the same time. The prices are very good. The prices are not bad at all, no, they're not bad especially. Cambridge prices are extremely high. Yes, I'm afraid the cost of Cambridge has become more and more virtually a suburb of London now because of the fast train.

37:30 Bristol is a bit... I think... I think with the recession that prices are probably going to start to come down quite widely now. Would you like a drink? Yeah, I'd love a drink. That's what I like to see. Yeah, okay. How about any, well, how about, how about else? Olivia, how about you? I'd like some natural mineral water, that's for me. Well, do you want to have a still or something? A still, please. And why is it still? Still. I'll have the Cumberland sausage, thanks, with the mashed potato and the... Onion gravy, and I'll have a pint of Carling Cold Flour. Yeah, Carling Cold Flour, the Carling Extra Cold. Yes, please. Jacket potato with tuna mayo melt. Very good choice. This is the case of the representative power of this exceptional case, which we never got to ask about. Which part? We've written it in the same paper where it's notation for n's and co-n's. Yeah, we know that. Yeah. That's it. Yeah. Yeah. Yeah. The back goes usually... I love that notation. Yeah, it's really nice. Especially for co-n's. Yes. Yeah, it's a little bit more obvious. So who landed us with this notion of

40:00 ...of a variable that the logicians have been using ever since the idea that this is the correct notion of a variable. I think it was, but I just want to make sure. And many people, yeah, many people have taken that idea. The things you want to write, if it actually is a functional, they will often write it with a variable just to sort of emphasize that it's a functional. How were the functionals written? I don't know how they were written before. How does one write the math? Well, like, there exists x with the p of x, so this x is the only variable. Well, the function goes from variable properties to constant properties. I mean, like functional animals. Yeah, functional animals themselves don't do this so much, but in applied mathematics, a lot of things are essential when people are trying to use the bound variables. They accumulate stress as a result of it. There'd be a history of short strings of things varying over time, which would be some simple quark with a bound variable ranging over the time. Oh, like they'll actually write down like a sub-square or something like that? Yeah, well, instead of writing a sigma or an integral sign, they might write an s, or something like that, under it or beside it or somewhere, this variable. The Vergo, which really has no context, I mean to say, it isn't really part of this. It doesn't name any of the entities that are there. But of course, when you say it in terms of maps, it becomes...

42:30 Absolutely much, much clearer at what's going on. The problem with the logicians' notion of it, that they inherited in Frege, was that it had the most disastrous consequences for philosophy, and this nonsense of quines, this slogan, to be is to be, the value of a variable, which would never have got off the ground if only they had thought in terms of the mapping properties between the things having to have properly defined domains and codomains to begin with. And indeed all this confusion about existence, which you pointed to this morning, and in fact what they're talking about is existence on a covering. Well, I was thinking when we were walking down there, the real essence of this is weak and sovereign. The idea of weak and sovereign comes up in topology. There are simplicial sets, for example. So, to every topological space there is a simplicial set, a singular contract. And this has a left-hand line. It's known as geometric realization. So the common-sale scheme has a geometric realization. Which is an adverb. So when you take the composite adverb, there's a comparison. There's a comparison in math which is typically not at all equivalent because, of course, the sort of continuous thing that you've constructed has lots of stuff in it that really can be emphatic even with the finite. You see, so you come in a serial format, and then what you've presented is, let's say, three two-dimensional triangles moved together into a four-dimensional one, or something like this, so you have this infusion. But the idea is that you go around and come back, and then when you take the geometric minimization, you take the singular context, and now you have two fundamental objects, the finite that we started with, and some very big one. So, this infusion, this adjunction, from the point of view of homotopy theory, should be an equivalence, but it's not, literally, at all.

45:00 So, that's why the whole industry model category came up with that example, because you want to adjoin to your combinatorial category, the inverse. I see. And so you have to identify it as being here, here, here, here. You have to invert that equivalence in order to get something on the sort of, sort of, sort of departing from... There was a naive combinatorialness in two ways, from finite to infinite, and now you're inverting maps of the categories as even more formal, but it's homotopy category, it's homotopy and a naive kind of homotopy, if applied to that case, is the same as the homotopy that we wanted for the objects. You've got, if you go between geometric theories and topos. There are such a comparison of mathematics as you need to invert. Weak equivalence. Is Michael Mackay doing all the work? I don't think so. Well, Mackay and Ray is, so... No, no, yeah, Mackay, Mackay, and Ray is. Well, I mean, a large part of what we were discussing, you know... She has made some important advances in her story. Now, would you also like to appreciate what was done before, as opposed to... I know, I know, but I studied first. Then I also read a book about my friend, and yes, so I hope I'll return here. No doubt, no doubt. That's better. Well, what? Thirty-five years of perfection. Thank you very much.

47:30 What do you have in mind? Thank you for your attention. That combination seems unlikely. But they worked hard on it for quite a while. Yes, for six years. Six years? Six years, six years. They kind of dropped out of the race. But Andrew Peet was responsible for the practical part of it. While Peter was responsible for part A and 13. A and B. What is D about? D is about quantum technology. So that's why I'm saying that it was written after Marcanio. Yes, it was incorporated. This is a very clever guy. He understood very well how to do that. At least in my opinion. Do we need a lot of German papers with mathematical mathematics? Maybe. Maybe. I loved mid-1980s when I visited Malcolm O'Connor in Montreal. There is one on... some results on local pharmacies with dental protection, I think. I think he wrote it. He wrote it? No, no, he is very good. He was at Buffalo for one year. He was a colleague of mine. He was even planning to stay on there, but then he got some kind of major fellowship from the Queen, and of course he couldn't come back, and went to Sussex for a while, and then finally back to Cambridge. I think the thing is that the Freudian Society, or maybe an excerpt, the Friedens says from an excerpt from the Freudian Society, something like that, for five years, because they used to have it around them, I mean, it's been three and a half years through, he got the lectureship in Cambridge, so he went off to Cambridge, and that's what... So while he was at Buffalo, he also made some advances in the kind of... Yeah.

50:00 Before Bach, we went to McGill. I think it was after. I think it was after. I was at McGill for two years. A year. But she was there the last year I was there. Okay. So I just finished my Ph.D. and was able to research colors. Thanks. And I'm going to go to Michael McCartney. Thank you. Can I ask a very general question, and it may be too general to be worth discussing at this point, but how far do you see all of this, these obviously very rich and detailed technical advances in categorical logic since Mackay and Reyes and in The Elephant, as fleshing out, fulfilling, The programme which you have told me you think was implicit in what Grosvendieck laid out in his Buffalo Colloquium talk, because, I mean, fascinating as rich as it all is, insofar as I'm able to follow it at all, it doesn't seem, perhaps I'm talking absolute nonsense, it doesn't seem to make much use yet of what I've... These are understood to be the central ideas that Grotendieck suggested of using the ring classifiers, of seeing the way that topos, the topologic classifier in a topos as being obviously the natural environment in this scheme for understanding. Logical notions like relations and quantifiers fits into this wider program that he had for unifying every... Okay, so...

52:30 It hasn't even, you think, advanced that program all that much now. My impression, too. Can you say this a bit? I'd love to hear this. I mean a huge number for a sheet of paper and you're writing it this way, this, this, this, this, this, this, this, this, this, this, this, this, this, this, this, this, this, this, this, this, this, this, this, this, this, this, this, this, this, this, this, this, this, this, this, this, this, this, this, this, this, this, this, this, this, this, this, this, this, this, this, this, this, this, this, this, this, this, this, this, this, this, this, this, this, this, this, this, this, this, this, this, this, this, this, this, this, this, this, this, this, this, this, this, this, this, this, this, this, this, this, this, this, this, this, this, this, this, this, this, this, this, this The striking thing about this is that there was not any use of logic. There was no logic in a very narrow sense of reducing every issue to be relational. He was simply directly using his knowledge and experience of what is the finite limit and what is the total limit. So these were notions that were described by conditions on those two things. So what this suggests is, as I was saying, that... Part of the idea of the formalization of the presentation shouldn't be as much as possible opposed to the things we've discussed, so they could involve directly finite limits instead of congenitons, as is somehow the case, and they could involve problems directly rather than just separating it into things that can be described in some way. Of course, it's a very important and powerful technique. To reduce things to sub-objects, as you were saying, the sort of nerd-eye trick that coverings can be replaced by coverings of monomers, by taking images of all the errors involved. And that has the advantage that the covering will be small, even if you thought it wasn't, because there's a lot of ramifications. But the particular way, see the normal, and sometimes the normal... A presentation at the Lodger predetermines how you're going to reduce things to relations from the outset, instead of en passant, as you were actually coming up with a particular problem of what sub-object might be relevant to this.

55:00 So formalizing Lodger's next spirit... This program has not been really carried out. People like Akai and I made it partial, but there are all sorts of subtle things that you have to get through because you've done enough and nobody's quite succeeded yet. And they are very subtle, as we learned this morning. The other thing would be, you know, kind of a symptom of this is that the actual logic doesn't get much application, too. There are a few examples that are known, but they're a lot more subtle than one would hope that a technical machine could be offered to look at. Oh, that would be great. We'd be very interested indeed. Fascinating. And also sounds as if it's exactly the kind of thing that Grone-Deke had in mind. It's the very thing that he looked at. Ah. Very, to me, to me, very exciting. It was an incorrect misinterpretation. I'm sorry. Grotendieck never had anything in mind about life. No, no, no, with respect, Bill, that's why, no, no, no, no, no, absolutely, I've completely taken on board that point, which is why every time I've referred to this in the last year since you made that correction, I've always said Grotendieck's implicit, I've always stressed the word implicit, that this was an implicit byproduct of the program that he laid out. Because it obviously does have these consequences for logic, but I've tried to underline that point by always referring to it as Grendik's implicit or, you know, concealed.

57:30 Okay, so I thought I got that one across. I've been very careful not to ever say Brogan Lee's programme for bioplastic logic for the last couple of years since you made that point to me. He had an implicit rule that suggests that he had it in his mind. Well, who can tell what he had? Well, one just doesn't know. Well, it's terribly difficult in that case unless you're going to waste a lot of words. You've got to say something somehow. What one could say without sort of attaching his name to the... Well, it's... ...implicit. The idea that Grotendieck presented in that book, okay, is that sufficiently neutral? Implicit in this work is something... Yes, okay, well that's what I was trying to say, but you know, one can only... Okay, well I'll try to qualify it even further in that case. Implicit involves the connections that the community is working in. Okay, well I'll try to qualify it even more. Precisely. It's important. These things are not just trivial issues. It's like saying the geometric logic is not clear to me. Whether it has an independent status of both of these opposites. And if it's that, then why leave it as it is, rather than modify it relative to the concerns of both? If it has an independent status, we're going to look about. Well, that's what I was saying. Yeah, I know. This is a visual sense of an independent state. Yes, yes. The main form of the reversal of coherent logic is that the coherent category that you get if you take the category of...

1:00:00 The geometric theories informally invert the well-chosen theory, and thus sort of distilling a clear conjecture out of the discussion. In other words, in those cases, if you add something to the physics layout, then you also have actions that have the byproducts of defining everything unequally. I mean, surely that's a map that's not involved in the theory, but that map can certainly be inverted. So if you invert all those maps, if you find out what they are in the book, you in fact get the coherence. Anyway, I certainly see now more detail about how this extraordinary deep idea, which indeed does seem to promise to bring logic firmly back within, The framework of mathematics, real mathematics, to understand just, you know, what the real subject matter of narrow sense logic really is, has, still remains, a great deal of work has to be done to present that. Perhaps we can discuss that further in the remaining days. Sorry Bill. That's just one of these logics, but again, the traditional logic, something like the traditional logic is going to be better. Yeah. These sort of things that people like to put aside, it's kind of a field of achievement for new jurists. Yes, well, not everybody has the geometric intuition like Grosendieck, or, dare I say, like you, or even any of you.

1:02:30 Sorry, can you say what you just said again about commutative rings, Bill? I'm sorry, I just didn't catch what you said. Grubner bases. Grubner bases. Uh-huh. A whole body of technique for dealing with ideals generated by certain polynomials and so forth. So it's sort of a practical level. Computer scientists actually program computers to do Grubner bases. Yeah. Thank you for your attention. We were at the first one. Yeah, in 1999. That's right. I remember that. Anyway, at that one, people were talking about the government. Thank you for watching. So they have these equations of hundreds and hundreds of variables, and they all satisfy differential equations, you know, roughly the exponential type, linear differential equations. So how do they solve them? Well, they replace all these power series by Taylor series. So it's all about some fantastic complex system of polynomials. Yeah, yeah. And now they want to compute values and then solve theories with that. So there's a technology. This is a sufficiently subtle system that they can apply. And it's obviously led to incredible technical advances. That's sort of one important way in which this thing comes in.

1:05:00 Exactly. Which is what the stuff on Grosvenor Bases is. But ideally that's what formalization of predicate calculus should be doing for other... I certainly like to understand that. I have to be honest, I haven't understood that, but I would like to understand it. Forgive me, but I don't understand the connection with formalization of predicate calculus in this connection. Some of which are sort of purely quantitative, others not, and in general they're mixtures of the two. It's so nice when you're looking at the board this morning, where you're inverting a map, and then it would seem very odd that adding the course of the map as literally a model is a sub-object, whereas actually adding an inverse with the equation as an inverse is not a sub-object. Yeah, that does seem very weird, I agree. There is a deeply missing kind of patch. It's like the gap between the abstract system of triangulation and the triangle in its case. And then the space can be made into an infinite version of the sum of those. It's impossible. The contiguous triangles are huge. Thank you very much. Cheers, Lovie.

1:07:30 Actually, I guess I do see the point about the predicate calculus now. I'm sorry. Cheers. Oh, oh, yeah, yeah, yeah, yeah, yeah, yeah. When you understand the predicates of vibrations, anyway, I don't know. On the point you just made about the redesign of transistors, of course, one of the things which has struck me quite forcefully listening to Davide yesterday and in his previous talk on Friday was that... And one of the obvious applications for these ideas about synthetic continuum mechanics and real material science ought to be in solid state physics. This is clearly one of the areas which is most dependent at the moment on purely analytic methods and all these very complicated gadgetry which would benefit, potentially benefit enormously from... ...being figured out again in the framework of synthetic and continuum mechanics. Yeah, yeah, yeah, exactly, exactly. Yes, precisely those things. The whole theory of doping and the other things that they use when they're designing. Transistors. Micros, like this. But somehow I think it's going to be a while before Bill Gates starts giving... Tens of millions of dollars for research in synthetic and general mechanics.

1:10:00 NASA is more likely. NASA, yeah, could be. I am a rocket fuel. Yes. Yes, of course. There may be wing coating. Ceramics, which is an area of intense research at the moment. I keep it by my bedside so before going to sleep I read it and try to learn it. That, to be honest, is useful. What was the question? Sorry. Whether I like the elephant. Oh, okay. So that's the idea, isn't it? At least. No, it's good that he maintains a very objective view of the world. He's not trying to propose any particular philosophy. He just presents the things with the highest possible clarity, and so you are free to imagine and define your own path in a sense, and so that's why I like his work. It's very new to me, so... I think so my path is a very zig-zag path. The end is shown together as the banks of life, so I set it back down together to get another thing, another thing. Five different points of view, do you represent five different people?

1:12:30 Five different Indian minds, two other minds. It's just a story. Yeah sure, it's just a metaphor, it's just a metaphor. Well, anyway, there are many different viewpoints, so one could say that perhaps part of the problem is that cohesion is lacking. Well you see of course he hasn't finished yet because the third volume is still to appear and that's going to be And Peter says that the second, the last volume will be almost as long as the first two together, so... I think it's too expensive for me to buy it, and the university is not about buying it. The main reason I go to Cambridge is to examine Martin's PhD students. They're examined about five years from now, and Peter is my host. These are very familiar to the cabinet. So he has to, you know, I keep getting stories of updates of the yellow. And that's the main way I've heard about it, that something that's going to be a total of about 2,000 pages on toposphere sounds, shall we say, remarkably long. It's remarkable he showed me a lot of things like this, of elementary nature, so it's very concise, so, no, no, it shouldn't be much longer, and thanks to his ability to keep things concise, it's really hard, I wouldn't move, I would write 10,000 pages instead of... No, no, it's really very tiny, so it's full of them. Sometimes you have to add a few pages for one line. What about themes? I have lots of technical context, but there has to be a theme.

1:15:00 Yeah, it's sort of like, why would anyone pay it? If you're not already interested in compost theory, would you want to listen to this? He says that's not a good place to start. I don't know. He says the elephant is not the right place to start. No, no. I'm sure that's right. So what's the starting point for the elephant? That's a formal, fairly formal starting point in which you know quite a lot of mathematics, and there you're going to learn some more. I think it's recommendable to read some literature on that, and books and other things, and category theory, especially because it makes heavy use of category theory, so you need to be prepared for that, so the fact that he writes the theories. So the readership is really the category theorists who are interested in topology. Mathematicians, we know, are not about two categories. That's it, they're small. So we assume knowledge of the theory. Right. Well. It seems a useful thing to have this, it sounds like a fairly encyclopedic account. Certainly intended to be encyclopedic, I think. Who is going to write a book of 3,000 pages, which it will be by the time the third volume is finished, or nearly, without... Making any mistakes, I mean. Yes, but it's nearly perfect. Well, I was going to say, you know, it wasn't, you know, whose.

1:17:30 You know, there are certain methods that one can say, well, I would have presented them another way, but, you know, we have mistakes. Is there anything, is there anything apart from just kind of ravings and unfortunately evidence of psychosis in Benabu's criticisms of Peter's approach, I mean, is there any mathematical, serious mathematical content, even if one disagreed with them, would you? Even if I thought that they were flawed, is there any really serious mathematical content to his claims about the... All this should be done in a completely different way, you know. Ben-Ben-Abu is notorious for his polemics against the elephant and... The elephant? Yeah. But then he's complaining about, you know, his medical... Yeah, he's so... The elephant as well. What's he saying? Well, the elephant mentions him, but... Yeah. Therefore, it's part of the... Well, he uses it. Yeah, he uses it a great deal, yeah. Personally, I think that that's a... There is a paper by ... and they say that they got the inspiration... In Halifax. In Halifax, yes. And then in the Perugia Nodes. Yes. Ah, yes, the Perugia Nodes. He said, Bill will tell you much more succinctly than I can. I didn't realize it was that bad. I mean, there was a there was a fight on the I went to the IETS I.H.B.

1:20:00 And then before the proceedings started, after about all these people had assembled, we stood up and said, I want everybody to take this position. It's kind of like a re-enunciation of the fact that he was on the show, and Oxford University Press, who published the book, and so he gave a little diatribe there, which was... There's only about five minutes left, but we're going to have to say that there's a lot of time left, but it has all sorts of ideas. Are there any ideas? I mean, I find it a bit worth listening to. Yeah, that's my question. But I find it worth listening to. When he gets off of these subjects, it's not worth listening to. But certainly in mathematics, the mathematical issues about index categories are not bad. Well, that was my question, really. Did you have anything to say about the other part? Well, the Oxford University test. It's a bit of a mathematical ritual. I think it was hugely offended by the fact that Peter Saiten in Southampton and him and Uwe Borg were not there. The one about the connection between this nation and modern, you know? Which one? Dissent. Dissent, yes, yes. Yes, he's a bit obsessed about the neglect of dissent. What he says is the neglect of dissent theory, but I can't agree more. Of course it's not pleasant if you read your results attributed to someone else. So I think he was very angry with that. Of course, it's a trite shade of approach. But I don't think that Peter did it for Paris. It was really a mistake, so... That seemed to be a large part of it as far as his attack on the elephant was concerned. But there's more, but he does have substantial, in this leap that he passed out, which you sort of, I mean there were...

1:22:30 Specific, although very broad, mathematical criticisms, which amount, well, they boil down to a track generally, he doesn't like index categories, he thinks they're a terrible idea. He thinks that these, he thinks that what he calls the... The category landers, which is one of his favorite expressions, category landers have not understood sufficiently the whole of descent theory. Therefore, they haven't properly understood the importance of his approach in terms of fiber categories. Well, the general notion of descent, yeah, exactly, in algebraic geometry. But the criticisms are not at all precise. I'm sure he could make them precise if he was just prepared to stop being paranoid and engage in proper intellectual discussion. I think that paper that everyone attacked in the 1980s, the only next category, said it's quite fascinating. It struck me as interesting, having caught it. I mean, he had precise things to say. People keep saying it's a diatribe and it's paranoid and all that stuff, but I mean, this is mathematics. I think it's perfectly reasonable. I mean, there was an issue at the time, remember, people kept telling me about canonical morphisms, right, when there was nothing canonical about them. I mean, he was right. You know, you could hear him, yes, the canonical growth. And he pointed out the errors that were made in this category of things, relating to Schumacher, which, as I recall, he was right, there were errors in there, but it was necessary for him. And wasn't he also seeking to internalize, to completely internalize the notion of the small-large categories distinction in a way which... Wasn't that part of the objective in his paper on fiber categories, vibrations, which I understand it was the program that you were... ...seeking to advance in your talk last week. I mean, I'm an outsider, so I can't, yeah, yeah, to make it in a way which is really internal, so that the claim that the opponents of category theory like Pfefferman and co. have made that, well, the whole thing is just parasitic onset theory to begin with, can just be countered and just be shown that that's a personal mistake.

1:25:00 Philosophically, that struck me as a reasonable name, too. Philosophically, I would call it very sensible. So that's why I've always felt that it'd be worth listening to, on mathematics. Yeah, yeah, yeah. Well, I'm an outsider, but that's my impression, too. The thing about the descent theory is that, well, he was not able to do it. In the years before John Beck had that result, he presented it publicly. Oh, sure. Not only was I there in Markham then as a fellow student and a lie, but we all remember that John Beck's application is the same. Could they have been independent? Could they have done it independently? Probably. Well, they probably did it independently then, many years later. Well, since Pennebue hardly ever publishes anything. Well, nothing is clear about that. For example, he gave a lecture in Tulane just after Beck gave a lecture in Tulane. Ross was there, but because none of us seem to have kept our moves, we can't talk about it. But so many people do it. And the great thing that ever came out of the effect was that it was very, very, very, very, very, very, very, very, very, very, very, very, very, very, very, very, very, Well, that's what happens with people who become obsessed with priority disputes, I mean, and ignore the real substantive mathematical issues.

1:27:30 Well, that's a personal thing. It's not... Oh, it's utterly destructive. It's terribly destructive. Yeah, yeah. It's going to become completely, yes. Yeah, yeah. Because, you know, he's going to go completely berserk and start attacking them, you know, character assassination like he's ludicrous attacks on Peter. Yeah, underneath, my impression is that underneath these kind of ravings to do with priority disputes and his personal resentments, there is still several, which Bill has just touched on, several perfectly substantive mathematical issues which you wish you could bring out and discuss rationally. It's a very elegant thing to focus on, and I remember him talking about his topology without a terminal object at one point, which I thought was an interesting idea to consider, and I remember once there was this meeting in Belgium, and I was there, and Peter Johnson was there, and actually Peter drove me to Brussels Airport afterwards, and then... They never gave this talk. It was a very elegant, very simple talk. It was interesting because I loved it and Peter hated it. And Peter hated it because it didn't have any kind of hard theory in it. And I loved it because I loved the way it sort of expressed some idea in sort of vibrational terms. I didn't care whether it was nice or not. So, I mean, to some extent, you can see Venom and Venus not getting on, because they're just so... Radically totally different styles of mathematics. He would probably die rather than write a book like this. And then if he writes a new book, Peter would probably hate it on the basis of, you know, like, where's the theorem?

1:30:00 Where's the beef? I heard in the lecture several times that the subject was very, very ultra-simple and obviously a universal significance on fractions. And I can't look at very detailed notes. I think Thomas Schreiber has his notes there, has been a great professor, and I think people are used to that kind of stuff. The funny thing is, it's a bit about the same issue we're talking about today, in a way. But one of it worries me, because, you see, this business about open and closed, seven keys when you define inverting and versus. This special field illustrates that in the category of community games, there has to be both, there has to be both. Whether there's a fielding up and then there's a weekly filming spot, you have to be both of those. But if you want to replace things in the category, or you can even have both the categories. But when you come to, if you consider, instead of the two categories of finite limits, it surprises to observe, equating just goes along for the ride, and you can see why, because you take equalizers, and the equalizers are preserved, and so two things are equal if and only if the equalizer is as fast as veritable, so inverting the equal... Inverting the equalizer and being sure to stay within the world of categories that are not trivial would be equivalent to identifying them. So identifying becomes a special case of inversion. And that's true also for more elaborate things that sort of add everything you want. Like free tokos and even tokoses, you still have that same detail. But to describe the subcocos and subocos, you know, using the full maturing of the cococos, it suffices just to invert certain things anyway, and there's morphology.

1:32:30 Well, she's conceptually very interesting indeed. Which I thought, somehow I should consider, instead of just looking at rings per se, you look at categories of modules instead of going into the ring. In the rest, within the categories of modules, this is an example where you have this lexicon. And so again, it just, it suffices to invert. Yeah. Invert and... Stay in the same world, which is a great thing, in order to account for both of these. I remember when Max Kelly wrote his book on enriched categories, I mean, Don Ray wrote a review of it, and it was scathing, right? Because he said scathing, extremely negative. Very hostile. Yeah, on the basis that it didn't include proof. And it didn't include details. Now, I've read Max's book from cover to cover, and it's still the one and only book on enriched categories, and I think that if I was writing it, I'd write it differently, but it's more formal than I'd go for, but on the other hand, I much prefer it to John Gray's book on formal categories, which is extremely kind of encyclopedic. Definition built on definition built on definition with no real selection about which people write definitions, which were definitive. Whereas maths is full of, well, it's not as much as I'd like it to be, but I mean it's basically full of these. It's succinctness, it's a virtue, but the fact that it's short, you can really get the main ideas and imagine that you could well complete the two of them, which I think is a popular attitude very often, and certainly toward that book and toward that subject. Exactly, exactly. Yeah, so I thought, you know, so you could just see that these are two people with two radically different styles, and I mean, one can sort of agree with one, disagree with the other, and you can sort of still see, okay, you can see these two people are going to take each other for what they think they are. But the work, that has to be, you know, I was in Australia. Yes, I had a lot of coffee, thanks. Just a small white coffee, thanks, yeah.

1:35:00 Anyone else? But this is clearly a disaster. And this has been a disaster for the whole subject, in my view. The fact that he wrote such a statement review in a prominently read place is both a disaster for him and for us. So that means, you know, the whole hordes of people, scientists and students and so forth, over the generations, have not looked at those categories. Because of that, though. And it was totally unjustified, I mean, what was, what was Greg Gray's answer to that? He said, well, whenever you do category theory, there's only one way to do it. You have to cover the blackboard with diagrams, but who? Obviously, it's important to have diagrams in the blackboard, but who? And he said, there's only one way to cover the blackboard. And since Max doesn't do that, therefore, it's not categorical. Irrational. I think so irrational that John Gove himself wouldn't have said it. If it hadn't been Max's book. Well, Max did give us ideas. Yeah. Well, it's terrible. Well, I think that's the problem with Peter and Benabu. They should have profited from this. What we're going to call this category is. Don't have it because of that tradition, because people can cite that review 30 years later. And so, well, of course, both of the AMS showed that this is no good, so don't bother wasting your time with anything. You know, what you're saying is, if you take, if you take the sort of the classifying couples for a map, then the classifying for the invertible map is a sub-couple.

1:37:30 But then the geometric theory, okay, so the geometric theory which actually has that extra symbol, no, has the extra symbol, yes, the signature has an additional symbol, right, with some axioms. No, you can also express them. Consider this case. By enlarging the signature? Enlarge the signature. Yes, and then postulating less of the inverse of it. Yeah, well, that should be a sum. Okay, thank you. I mean, whatever your definition of conservative quotient, whichever way you put your letters... No, because I define a quotient... No, yes, but I'm saying that you should redefine a notion of quotient to make that thing a quotient. I mean, it's enough that that thing is not a quotient. It's a quotient if you... The conjecture is that if you take the category of all geometric theories, then you can find certain maps to invert in the morphisms of the universe, not just the elements of the universe. If you form the category of fractions, that should be equivalent to the set of instrument areas from here, I believe. Right? Right, right. That's two different, two categories, and one's the same volume of earnings. That is incorrect, yes, yes. And whatever is your account of quotients should take that back. There should be an account of that. There is. You just have to go through the original theorem in the enlarged statement that you have done. There's no linear definition of quotients. You can, if you try, you can allow that. You can allow that. The fact is that once you introduce a notion, there is an important feeling about it. Otherwise, it's not... I think that this is the definite theory, the definite category of geometric theory. The morphisms are the right ones for that idea, mainly Tarski, but don't say that. So you have these general morphisms, and among those there are going to be monos and epis. Yeah, in fact it was interesting to see if you could fix the signature.

1:40:00 It is an interesting signature. Well, I think that's the content of the main content of the year, although not formulated as a week of the year, but taking a restrictive format of a series that you can express also. Well, it's actually an actual restrictive form of theory to fix the signature. The signature is not that good. Yes, it depends on the perspective that you have. So it's natural or... Actually, the context here is not actually the result, but it's the fact that you can compare, you have two truth systems, the one of geometrical logic on one hand and the other one of Grothenitz-Horgeson on the other. And you can define transformation. So that is how I see the theory, so and it's interesting and you can also give a direct proof of that without invoking any classifying purpose here, so you can take for example You define the syntactic transformations between the two of those systems, and then you can verify that whenever you have an inference rule, for example a geometric logic, so that you can derive certain conclusions from certain premises, then if you lie to all the inference rules, so I mean both to the premises and the conclusions of these transformations, You obtain that the field that you obtain is the conclusion. All of this belongs to the Grotendieck topology generated by the images of the premises. And conversely, you can verify the same thing going in the other direction. So this proves that you have two syntactic transformations between the two proof systems, which prove that they are in a wave. So, this is an interesting timeline, I think. And can be proved independently or classified to opposites, but just argue inside time. I have done that.

1:42:30 My dying the same signature is interesting to me. If you keep the same category, because the fact is that you want to keep the same syntactic category, so if you change the signature, then you don't have one syntactic category, but you have one for each signature, so the fact is that if you want to analyze it, because that depends on the signature. These objects are, first of all, signatures, plus actions. Before this, what you're talking about would still be given equivalent categories. If you contrast it with another one where you have also this third generation, that would be another category, and it's equivalent. You get an equivalent kind of thing, yes. You get two equivalent geometric functions that correspond to the same thing after you've inverted to the math, like that math. So what you're saying is that you can set up this comparison. What happens when you invert everything? Yes, exactly. It is a purely syntactic world. That is the core. It's a natural scientific exercise because you can look at that and it's good that you see on one hand geography and on the other hand logic and variance. It's not all about specifying opposites. It's also just about the real essence is truth already speaking. You're not going to not think that we can win today.

1:45:00 What was the second proof system that you said could be translated as the American proof? What was it? It's the one that you take... It's difficult to state that as a real proof system independent of... But anyway, it's... This is the one that you start with all the JT collocations. So these are the axioms, and the... Yes, the axioms of the system, and the rules are, the inference rules are those which allow you to derive from a sieve, the pullback sieve, and in the same way, from the transitivity axiom, that you can derive a certain sieve, starting from a j-covering sieve, provided that that condition in the transitivity axiom of the homology holds. So, you're thinking of axioms as... So, the interesting point is that these... You're just describing two different styles of... Two different styles. ...the rules of inference for geometric categories. Exactly. Exactly. Not for a general. Not at all. But I'm seeing that for the particular case of syntactic metabolism, there is the notion of logarithmic topology as a intrinsically logical notion. Because the axioms can be viewed as inferences and they happen more and more, and inversely also. So there isn't an actual equivalence, I can't really say, well, these are not the ones, but in substance it's like that.

1:47:30 I think it would be a lot more interesting if you had a fire for the government toposystem or geometric motors and stuff in syntax because then you'd have equivalent government toposystems necessarily. There are a number of different things that you'll set up, and the whole sub-optic thing would, you know, the moduli, you know, the moduli could do it. It would come out naturally. Thank you. Theoretical theories, geometric theories, that is what I am, it's not my work, it's not, because when you have a gothic topos that represents the sheets of C with respect to J, then that axiomatizes the flat functions which are J continuous, and you can write the axiom. So there is actually a duality, you know, in geometry. Yes, great remark. So that necessarily characterizes the subcorpuses? No, no, no, that is... Yes, it does, necessarily. The fact is that you know that they exist always, but you don't know how atmospheric construction influences the signature, because there are many different ways of presenting the same classifying tools, so over different signatures, so it's a question of seeing how signatures relate to opposites. Are you saying specific to the signature? I mean, it's not about, not at the level of theories, the signature, but then you sort of wind up with a very strong emphasis on maintaining the same signature. The question is why would anyone care? I mean, because the natural thing to do is if it follows from your axiom that something is invertible, it would seem naturally harmless to add an inverse.

1:50:00 I guess the theory has proved to be extremely useful for a lot of applications already. What theory? The duality theory. For example, I was able to prove a deduction theory for geometric logic by using entirely the point of view on quantum mechanics. Second, another result that I will cite this afternoon, about characterizing the geometric theories such that whenever you... If you consider a sequence over the signature of the theory, either the union of the theory with that sequence is contradictory or is the theory itself. So, it's an interesting question to ask, can you characterize these theories? Yes, the result is they are designed with the Boolean completed theories. Entirely logical and you can deduce by characterizing the atoms of the lattice of subatomases. Then you have a lot of other applications regarding axiomatizations of theories and which can be... Topology can be obtained by changing the representation of the classifying topos and observing that the lattice structure that you have is independent of the representation... For representation I mean a way of writing topos as shifts on a particular side. So a representation is a side of the topos. That's why I just want to be precise about topos. Of course in mathematics there is another meaning for it. I like that. And so you can, by changing the possibility of representing topos in alternative ways, you can change the axiomatization of theories. And so that the lattice operations are preserved, because they correspond to geometric inclusions of student forms, and so if you change the representation, basically the order remains the same, so you can establish an order between subtoposis of one representation and subtoposis of the other one, which is order-preserving.

1:52:30 So sometimes it turns out to be convenient to calculate operations on theories by using an alternative representation. And that is how I achieve that explicit description of the myth of local things and integral things, by changing the representation, because otherwise it would be impossible if I used a syntactic one. But I introduced a semantic one, which is linked to the syntax in an alternative way. And by doing that, I achieved... So, there are many, many implications that we can flow out of that. Of course, I just reported some of them in the paper because it's a very recent work, but probably other people will find others, and so... And also, for example, there is all the connection with J-homogeneous models, which is the syntactic approach that I used in my first paper, which links very naturally to this syntactic thing, so that, for example, you have certain models, which are defined entirely categorically, and you ask if it is possible to assimilate them in the same signature of the theory, so that you can... It's important to know that. And I use that, for example, for my process interpretation of the process of construction of the field. That actually lets me design the model field. I think we'd better make a move because we said to John that we'd be back around 2.30. We need also, of course, to get our bill. It's very interesting. That is very interesting indeed, isn't it? It is, isn't it? This study only applies to one of those isomorphic things. No, no, no. It's more like an expressive result. Say PCF. PCF we know. Okay, so that may be interesting, but it's not.

1:55:00 Category theoretic constructs typically are isomorphism invariant. That's the nature of the subject. Now, when you move to syntax, that's not true, but I mean, if you're making things that are specific upon your choice of syntax, well then that requires justification. Yes, because in general you're going to lose the isomorphism variance. Yes, and it makes it, in a sense, inherently not category theoretical. That's exactly the kind of, dare I say, conceptual remark which is what philosophy of mathematics ought to be about. Good point. Yes, exactly. No, I see the point, which is why these syntactic categories are tricky, fairly tricky, or finding the correct geometrical... I've still got one diet coke and one coffee. Ah, you have a diet coke. I have a diet coke, but I just ate it. Okay, it's easy to forget those things. Hmm, they're still very interesting. I have a much, an utterly naive question, I mean, by comparison with all these very interesting technical issues. You mentioned Steve Vickers' book, Logic from... Oh, I forget who it was, but somebody in the course of the discussion mentioned Steep, because topology from logic, yeah. And I'm quite interested in, as we're looking at things the other way around, logic from topology, which I take it also to be part of the motivation for what Olivia was presenting today. Topologically, in terms of behavior of coverings, the topos is in which the axiom of choice holds, but extensionality, I mean, I'm very, very interested as a philosopher in trying to understand more deeply what is the topological and geometrical meaning of the various versions of extensionality that you have in a topos. To refine the notion of something being extensional, I mean, there are several different versions of, and they're not of equivalent strength, you know, the condition of supports that should split, the so-called weak extensionality, which is, it's just the, you know, there are a whole number of ways in which you can say, a whole bunch of ways in which you can say in a topos, that the axiom of extensionality holds. There isn't just...

1:57:30 You know, there are several versions of extensionality, which I think is itself very interesting, because it seems to me the whole point about the set theoretic way of thinking of structure is that it really starts from the idea that extensionality is just one absolutely globally fixed notion. It's just the idea that a set is determined by its members. They are just collections in extension. That's really conceptually the most basic notion. It seems to be in set theory. And one of the interesting insights of topos theory is that in topos theory, it's still obviously a very basic notion, but it's one which splits naturally into several non-equivalent and more or less, you know, ranking from stronger to weaker versions, each of which has... A distinctive topological and geometrical corollary in terms of the topological and geometrical structure of objects in the topos. And I'm really interested in trying to understand what the topological and geometrical content of the different versions of extensionality is in a topos. And how it relates, for instance, to separability conditions in topology. Weak extensionality, which seems to be equivalent to relative uniform separability conditions in topology. But I don't think anybody, at least to my knowledge, may well be the toposthos already and have all of this completely figured out. But if they do, I've not come across a place where it is figured out and presented as a unified package. In terms of understanding precisely what the geometrical and topological corollaries of various logical principles are, I think that will be a very interesting exercise for the philosophers. Extensionality. Yeah, yeah. Just sorting out what the topological and geometrical corollaries of these various versions of extensionality are.

2:00:00 For a general philosophical building. I was going to, but he's thinking he's so busy that, you know, talking research topics in topos theory, I think it's quite out of place for me to be pestering about this. No, I think it's quite interesting, because in terms of sort of technical detail, once you've got a specific definition of what's going on, you want to know exactly what follows. Yes, which follows. Yeah, sure, sure, sure, sure. But I just wanted to ask you... Well, I will, but when an opportunity presents itself, because I don't want to cut across when he's discussing with people who are obviously in a much better position to benefit, actually, from discussing with him. I mean one of the things as a philosopher I'm interested in is just as it were way absolutely crudely in terms of I mean you know where logic comes from is it an aspect of the makeup of the world or is it something which is it where it comes from an absolutely transcendental ...level at which, you know, it holds what's, of course, the traditional way that people think of logic holds true for any possible world. What do you mean? What the hell do you mean when you talk about any possible world? Well, that's something, yes. Exactly. I mean, there's various things. You know, there are all sorts of... Like possible worlds. Yeah, which, of course, is one of the sort of things that the logicians, the philosophers, rather, tend to... Here we are. Yeah, no, that's our department. You can go across the lawn. There's a path. Yes. Well, that's the sort of thing which makes me squirm. Well, also the very notion of problem is obviously such a portmanteau word. There you go. That's the kind of remark which does make me squirm, I'm afraid. The bit about reinventing yourself.

2:02:30 Yeah, it's well. Yeah, it's well. We've got the wrong system of keeping score.