FW Lawvere, M Menni

0:00 Well, it's not very easy to speak, but on the pure quantity level, suppose you have a solution by an approximation method, here's a canonical map from art, so this is canonical, this is the one that's found by approximation, say a human song, so you can make an integral by forming a human song, you can do that with parameters and you can do... So we construct primitives by remap sums, as you know you have to do, and we know, for this particular case, that we can actually lift x on the primitive of x. Does it lift? Do we have a primitive not only as value in the pure quantities, but as value in the smooth? It's similar to... The power series converges, and often you get something like the lower level, just to take the power series, and suggestively, you get a different kind of information. There's the dynamic, and there's the power. We have the code streets. This is the way over to our campus, otherwise this is the site itself. This cartoon is reposed. So 2 to the 2 is an atom even, not merely connected, but projectively connected. So maybe that's a good example of something. Even intuitively, this t-discreteness, when you calculate pi-zero of something.

2:30 What you're doing is saying that two elements that are infinitesimally near are the same, but you don't expect zero and one to be infinitesimally near even following a finite number of steps, right? No, no, sure. So, you know, pi zero in that sense is too strong. Pi zero of r shouldn't be. And it's not in depreciation. This in a way, this is really the Achilles problem, you see, to say that these little steps never achieve the big step, right, and so if your model is too simple and merely combinatorial, that would be the case, that you'll never reach it, and if it's very continuous, then of course you'll reach it, but there's sort of the intermediate step where You have these weak equivalences, or you can think that the ensemble of these finite approximations is at least sort of dense in 0-1. It's the idea of solving differential equations, you see, that differential equations for a path from A to B is just... And so on and so forth. See, somehow in the infinitesimal case, it's not true that the finite repetition of the infinitesimal ever achieves anything non-infinitesimal, and yet, see, the solutions of a differential equation are not really such an iteration, they are just morphisms that satisfy a condition of being compatibility, of being compatible with infinitesimal extension.

5:00 So that the path that goes from zero to one is required to be compatible with the steps. But that isn't the same as saying that it's constructed by the steps. Although again, Peano's existence theorem for differential equations, taking advantage of the set theoretic framework and so on, does construct solution out of little pieces. Right. But that's a certain context where you can do that. And of course, again, it's not really a finite number of pieces, it's passing to the limit, this magical thing known as the limit. Yeah. Yeah. Several proofs about differential equations are based exactly on this. They first discretize and then make a conclusion on the limit. Yeah. Yeah. Yeah. But somehow, I think, the underlying idea that you're trying to develop is amorphisms that are compatible with this, rather than being constructed from it, only in a special case will they be constructed from it. Yeah. Oh, so compatible gives the same result, in a sense. Yeah, by compatible, I mean literally satisfying the differential equation. Ah, yeah. A lot. Yeah. The concept of that is independent of whether it exists or not, or whether you can construct it or not, and the uniqueness. You have a better chance of uniqueness than existence. All right. See, because the uniqueness depends on sort of density of these little steps in something that doesn't just consist of them.

7:30 Yeah. We'll have to have more of this kind of conversation, I don't know. Yeah. Examples. Good, good. Yes, all right. Just think about it. This is Manuela now, we'll see her later on. Oh, she's not, okay, does she know, do you know how to get to the physics department? Oh, you saw it, good, good, good. Okay, that's right, Mott Theatre, yes, so we'll see you there. Just be all sleeping now. Yeah, today, tonight has been, we took every precaution. We touched the air like dynamite. Was she having a bit of problem sleeping? You had a sleepless night, oh dear. I don't know what to say, but perhaps we came back to normality. Well, it's not a disruption for a very little girl. Much better. Yesterday was... How did it work when you went to Paris? Did she adjust rather quickly? Yeah, much better than now. And of course you're quite right, we should be doing more of this, and particularly... That's quite a bit of the conversation that I had with Anders yesterday to say about the initial question, but bringing it directly into connection with this point about the inadequacy of the dedicated determination of the continuum to really effectively track the transition of real motion. Do you want to... I told Richard that we'd meet at the vaults, but that doesn't mean that we have to go in there for lunch if you want to go somewhere else instead. Okay. It's quite, uh... Do you want to go in there again? Okay, fine. The thing is, they'll have the, um, the kitchen on today, so...

10:00 Hello again. We're okay, we're in time. Okay. Well, we're going to have some food. This time, you've got the full menu on, haven't you, so you guys decide what you like. Oh, I'll have the new beef sausage and new potato casserole. Sorry, is that beef sausage and new potato casserole? Yes, that will do me fine, thanks. Yes, I'm going to have a pint of the... I've got into the corner of the tribute and I've suddenly realised I've still only got this. Can you just give me a moment to go to the car because I've only got, I've still only got Euros. I should have done something about it. They claim, so both of the main articles are by Templeton fellows, but one by a Templeton prize winner even, and the other one by a Templeton fellow with his mathematical collaboration. Yeah, Simon Cochin, I never realized. I'm really horrified by that because he did very, very good work in, did he not, in field theory? I mean, in algebraic theory. No, he did one thing with spectra, which is very respectable for a logician. Which is again the same sort of thing. No, I know the so-called Cochin-Specker theorem, but he did a lot of stuff in algebra, in mainstream algebra, in pure math as well. He did one thing, Axe-Cochin result, which is the isomorphism of two different alter powers, so the two different sequences of rings were sort of almost the same. That's what I was thinking of, yes. Almost all values and primes were the same. Although they couldn't isolate which kinds they weren't, but anyway that was considered, it was one of the few results of the intersection between logic and algebra that seems to be non-trivial. Yes, that's what I was thinking of. He did a few things, and also the quotient-spectrum theorem, which is rather trivial.

12:30 I mean, claimed to be. That's the mathematical proof of the ideological content of quantum mechanics, as I know, during and I shall certainly make great play of it, although it's really a pretty trivial result. It's just to do with the non-existence of global sections. It's not like I shall have ever been published in the AMS. You might be next year, who knows? This is a leading journal, supposedly. These are the only two main articles on that particular issue. I was really staggered, especially since I met Cochin in Paris last year, about a year ago, in fact, now, at a conference he gave, it was a philosophy of physics conference, he gave a talk about the Cochin Speakers, it was a purely expository talk, you know, where are we now. I didn't say anything that struck me as being sort of ideologically... I talked to him afterwards mainly about this early work, as you say, the axe coaching. And he did actually express great admiration for your work. I know. I've not met him since 45 years. He said, I would love to meet the little girl again. I have always been an intense admirer of his work. I admired what he did. He was exposed by Spectrum at the 1963 model theory in Berkeley. I think that was... ...the first time to have been left for public. But this thing was always repeated, although mathematically you could say it's a trivial result, but it's always been repeated in the course of theological context. Sure, sure. But then I don't think that, you know, that can be laid entirely to Kirchner's door, that it has been taken on by the... Exactly. I wouldn't have thought so, but now suddenly I see... Yes, I, as I say, I'm very, very shocked to learn that, yes, very, very shocked to learn that, yes, very, very shocked to learn that, yes, very, very shocked to learn that, yes, very, very shocked to learn that, yes, very, very shocked to learn that, yes, very, very shocked to learn that, yes, very, very shocked to learn that, yes, very, very shocked to learn that.

15:00 Well, I have been trying to ring the alarm bell like hell, I promise you. I think I've certainly got it across to Cartier now, to one or two other people. Cartier had his own bad experience. Yeah, he had his own direct experience of them, exactly. Somebody came to Cartier wanting him to participate in some of the propaganda after promoting it as a popularization of mathematics. And then he came to realize that it was a standard, and then he realized this guy was an actor from the Templeton Foundation. I mean, he had an experience of that type, which makes it easier for him to understand. It's really scary. And last year's Ray, who you met, sir, when you came to speak in Paris two years back. The chap who was introduced to you, the chap whose seminar you spoke in, he's the physicist, the cosmologist, he had a much more direct experience than Cartier because in his case they sent somebody when Sarkozy came in and they first announced this plan for... Essentially you're breaking up the CNRS and introducing these research groups and competitive tendering and all the rest of it in order to inject private funding into French scientists. We can no longer afford, and besides anyway we should reject even, apart from the fact that it's too expensive for ideological reasons, this whole traditional state top-down model of no research in France. It's much more known with it being more like the Anglo-Saxons. That's just the rhetoric, you know, Sarkozy's camp. This of course was before capitalism came up in their faces. This was like two, two and a half years ago. And shortly after that, Cartier's lab, oh sorry, I beg your pardon, Lascia Ray's lab, which is called the cosmology and astrophysics lab at Saclay, got a visit from this guy, who didn't say anything at all about the temple, but who was from I forget what. The foundation was the Cover Foundation and said, well, we understand that the research groups here are all within the next year or two going to lose quite a large part of their CNRS funding because there isn't even going to be a CNRS and we don't have to join this research consortium and look for money.

17:30 So we'd be very interested in offering money for various projects and research and we're here to help. And you just tell us the kind of thing you're working on. We're only too happy to submit it to our committee, so it's very suspicious in any way, but I went out to lunch with this guy and he was pumped on what he was working on, which was mostly on a kind of little fiber bundle, which was a relativity attempt to make the... Marry up the formulas of relativity and QFT and purely mathematical stuff. But you are a cosmology lab, aren't you? So do you not work on problems in cosmology? Well, actually you do need good mathematical tools and serious cosmological theories. But yes, we also work on... we gave a list of the things we'll... Yes, that's all very interesting. That's the kind of thing that we'd be interested in supporting particularly. Do you have anybody working on the anthropic, the problem of the anthropic principle? Well, why is that? Well, because I consider it's a complete crock, you know, a complete total art of shit, you know. The, what's the, the completely ridiculous, the CRAP, the completely ridiculous anthropic principle. So, oh, I see, well, I'm sorry to hear that because that's really the kind of thing that we'd be very interested in. Providing some research for you. If there are any philosophers who would like to come into a research group and talk about those broader issues, particularly things like, you know, the anthropic principle. Well, if they do, if there are, I don't want them to be part of my research group. So, of course, it didn't go very well. Then later he did a bit of digging and he discovered that this guy is indeed the director of Templeton for France. So it's all, you know, it's an open book if you have the will to read it. It's all done in terribly sort of gentlemanly way and well what kind of things are you working on? Would you have any objection to doing a bit of work on it? Yes.

20:00 How many would respond to the affirmation? Well exactly not many, that's the thing. Well I say well I don't, you could just imagine sort of something like Roger Crowe. I don't actually see it as a very promising idea. I think there's a lot to explore. Of course, you know, let a hundred schools of thought contend. I mean, yes, there are people who want to work on that kind of thing. If you're going to give us money, I mean, who am I to argue? That's how most people would resist. And then they'll go around saying, I don't understand why these people are so obsessive and paranoid about the Templeman Foundation. They don't really dictate any kind of research agenda. They just, you know, it's all quite harmless. That's a level that, I mean, if you look at the list of the recipients of the FQX thing, they actually, by as you know, it's a $30,000 grant to run an in-category cafe, and probably there's about even more than that. Crane has got some now. Crane has got some. Not much, but a little. Crane, Coach, and Hicham, you know, a whole list of names. But anyway, they actually, the little abstracts of the proposal that they made to get this grant. And if you read them, you see precisely already there, they understand that the quality is supposed to be fuzzy and low and bombastic and whatever, and not serious science, so that reaching that standard seems to be your first goal, common goal. Yes, yes. I'm so sorry to interrupt. Can you just get your map out? I mean, we're certainly in the right street, Tyndall's Park Avenue, but I have a horrible feeling...

22:30 It didn't go the other direction as well, did it? Is the physics department not on there? Oh, it's Tyndall Avenue. Shit, we've gone... I'm sorry, we're going down Tyndall's... No, it's okay. We haven't gone the wrong way, but we've got to go... Is this Elmdale? Yes, it is. We've got to go down here and then left. I'm sorry. I was going down Tyndall Park Avenue and it's not. It's Tyndall's. It's Tyndall's. We're going down here. I didn't realize that there's Tyndall's Avenue. I'm sorry. Yeah, the short answer is I have taken you the wrong way, but it's okay, it's retrievable. The back is very near the department. Yes, I know. So in fact, we should have, you were quite right, your instinct to turn there. Yeah, let's go up here and then left. It'll take us back there. Just means we've got to go around the block. I'm sorry, that was my mistake. Well, no, it's a bit more than that because we've already walked much further down the Tyndall's Park Avenue than we should have done. It's okay. We still have plenty of time. I'm sorry, don't let me interrupt your... Discussion.

25:00 No, no, no, no, no. We've got to keep going on now and then take the next left. As I say, I made the boo-boo, I'm afraid. Yes, it is. And the physics department is on the next street along, which is this one along the next left. Extraction, to go back and forth. Extraction, a million groups, yeah. Yeah. Oh, is this strong?

27:30 What was the motivating idea of Aurelius' construction of this business of a ring without a unit? I don't understand that at all. Well, I mean, if you consider the case, if you look at the map decay, retraction onto the constant, which geometrically means a point. We're talking about pointed spaces instead of spaces. I see, I see. But then, look, it's up that way. It's that building you can see from here. Oh, right, so you can see it from here. So that's... Yeah. That's already interesting. They do exist, but they certainly don't characterize them as morphisms. They're two specialties, but in general, it's kind of an independent thing that comes out of it. You don't ask for it, but in a way you can say, well, zero is just a particular constant. What's the whole possible constant?

30:00 Yeah. So, a k-rig... Any k-rig A, by definition, has a map from k. So suppose A is actually a quotient rig of another rig B. You have any surjected map of k-rigs. This is the data. There is the fixed data. There's a standard map from K to quotient. So if you pull that, this will be... This is somehow the data that we need to... That's what I was trying to remember this morning, I couldn't quite say so. The thing that you get is another actual rig with a unit. It has now by construction a map to K, so you can take a push out of that and you recover A. That is to say, in the case of actual rings, you would recover A. For Riggs, you will more often recover A than if you merely looked at ideals in the sense of zero. I have to write this down. You're just looking at the part... Well, shall we do it when we get to the physics department? Is that possible? Because I'm afraid we're probably running a little bit late now. What is the time? It's two. Oh, it's okay. We still have half an hour. Geometrically, quotients describe closed subsets. So the general closed subsets of an algebraic space can be described as the zeros of functions. Instead of just the zero of the place where some functions take on certain strange sorts of values or something, strictly where they are zero is enough to tell you...

32:30 See, so this is entirely in terms of actual spaces in math, or really of actual homomorphisms, but it just brings out the fact that there is a sort of special role for points, and as I say, for Riggs, you can try to, this hasn't been developed, but you can try to develop this idea that for Riggs, this is still a better approximation to the quotient than mere zeros, because you're taking into account all possible constant values. K is included into this quotient thing, so these are the constant values, so you can ask in the other ring, which you're taking the quotient of, which elements under this point take on this value, which ones take on this value, takes on this value, so that whole family, that whole vibration, if you like, of subsets... Where the variables are taking on specified values at this particular point that you're evaluating at, that should come closer to describing the typical closed set. Or you could say these are somehow special closed sets. That's right. Among closed sets, there are some more that are even more closed. Any closed set has a sort of hull of this type. And then pull back and then push out again. The push out of the pullback will be an approximation of what the starter was. And you'll have that effect of being sort of an improved version of it though set. Okay. I'll have to do this on the blackboard. Or I'll force you to write it down. Okay. Maybe. Yeah. It's more of a... It's sort of a smaller reflexive co-equalizer than just taking the congruentialization, you know, taking the kernel of the kernel, which will give you the whole congruentialization.

35:00 Yeah, yeah, this is a narrower thing. Rather the smaller one gives the same result, taking more into account the nature of the category that somehow points you sufficiently. Yes, yes, as you say, it points obviously very... I think that's the physics building ahead of us I think. Let's check we are in Tyndall's Avenue. Sorry to give you the workout like this, mind you after that good lunch, probably a good idea after that lunch. Yeah, I'm pretty sure this is the physics building here. It's ahead. Queens, Wills, one more. Oh, hang on. Wills, one more. Oh, blast, hang on. Sorry, I've made a boob. Look, I'm so sorry. Can you have a look at my map again? I'm sorry about this, but I thought we were in Tyndall Road. Sorry. And we're getting a bit close to... This is bad. The physics department is there. Tyndall Avenue. How the hell have we managed to get turned around. This is Elton Road we've just come to the top of. Oh, it's straight ahead of us. It is, I thought it was, yes. No, it's okay. I was right. It is that building. It's just we have to go. It's okay. It is, it is, it is this building just ahead of us.

37:30 That's all I thought. For me, this is a terrifying step across the road. Well, yes, because they all drive on the wrong side of the road, you know, it creates... They're not sure that I'm on the wrong side because I'm driving faster than... Well, for me, I'm in danger. Well, yes, because Americans are very safe drivers, actually, very safe drivers. I have to say, by the standards of Italians or, dare I say it, I suspect Argentinians, I've never been to Argentina... I suspect, sorry, don't want to stereotype, but I suspect the Brits are not that bad drivers. They might not be the best in the world, but, and they've been driving on the wrong side of it all these centuries. Just because, just because Napoleon never straightened them out, you know. Thank you for watching. There is no need to encourage, to escort students to any class. This is what I need someone to say. If you're a graduate, you have to do this. If you're a graduate, you have to do this. If you're a graduate, you have to do this. Even if SATs are how I make money. Even if SATs are how I make money. Even if SATs are how I make money. Even if SATs are how I make money. Even if SATs are how I make money. Even if SATs are how I make money. Even if SATs are how I make money. Not to mention things like algebraic topology, Dan. So may I ask how long you... I've been in Bristol and you've told us obviously you work on you work on on finitary arithmetic which is I understand an incredibly rich subject I don't know I don't know very much about it I mean my background such as it is is actually more in this stuff in category theory and I'm more a kind of historian of mathematics these days but obviously I know a bit about Paris Harrington and I realize it's an incredibly rich subject.

40:00 What particular kind of problems... You're essentially a number theorist. Would that be right? No, no. I was exaggerating in my question because I wanted the answer to be in a certain... Oh, I see. You were a listener. Yes, I do Ramsey theory and I do prime numbers, but as side dishes to the main subjects, my subjects are unprovability. So my research is exactly the core of my subjects. Oh, that's interesting. So you spend quite a bit of your time talking to logicians? Sorry? You spend a fair bit of your time talking to the logicians? Yeah. Yeah, okay. You must know John, obviously. John Mayberry. Yes, yes. I've hardly met anybody in Bristol. I was teaching the whole year. I was teaching set theory, axiomatic set theory, and now I'm teaching logic, the standard logic course in Bristol. Right, which is what he used to teach until he retired a couple of years ago, of course. It's just that he and I are very old friends, and he and Bill are the main reason I'm here today. Okay, very interesting. And whereabouts in Russia are you from? I take it you're Russian? Yes, I'm from near St. Petersburg. From near St. Petersburg, yes, okay. And whereabouts are you from? I'm English. I was born, I was actually born in England. I lived in England until five years ago. I now live in France. I live in a little town in Brittany called Brugere, not far from Norman. And nearby, if you had a boat, you could sail. Well, it's for various reasons. It's because I'm involved with this project, this big mathematical archive project. It's a long story. I won't bore you with it. I'm British, but have now been living in France for the last five years, but have lots of connections, mainly in Cambridge and also here in Bristol. What kind of history of mathematics do you have? Well, mainly a history of 20th century mathematics, and particularly algebraic geometry, algebraic typology, and of course, category theory. A great deal of which, of course, was created as machinery, as it were, along the way in the course of the development of algebraic geometry, particularly.

42:30 Obviously, particularly the parts of the Groton-Dickery and these school-created, though, not only those, but, yes, I actually have a very good friend in, well, he lives in London now, but he's not a mathematician, he's from St. Petersburg, but, as I say, he's not a mathematician, he's a, well, he was until just recently a lawyer, I'm afraid he's, like a lot of people, he's lost his job in the pleasant, you know. Rapid implosion of the world financial system. I've heard the name, certainly, yes, I've heard the name. He's in... Oh, I do know, I know another Russian mathematician. But he's not a... he works mainly on dynamical systems. He's at Imperial College, Ari Laptev. He's quite feminine, he's just been appointed head of the... He's actually the president of the European Mathematical Society for the next two years. That's probably why you saw the name. But he works on very, I mean, about as far away from the finite sets. There are a number of theories you can get in. No, no, no. Dynamical systems are very close. I suppose that's the truth. Since Kirstenberg's theorem, you know, dynamical systems used to prove... What's all this mathematics? I don't know anything about this at all, I'm afraid. Dynamical systems and even ergodic theory is really very much part of the subject nowadays. Really? I have heard there was a connection with ergodic theory. You didn't know that dynamical systems were now part of... Actually, I went to a seminar which I have to admit I didn't understand. Hard to understand a word of about arithmetic geometry, Arkelov varieties, which they also use dynamical systems theory in the context of, I think, actually trying to understand distributions of primes, things like that. So there is a connection, I imagine, with that. Okay, well that's okay. So I know, yes, I went to a seminar about six months ago in Paris about that. I'm afraid. You know, I maybe got an occasional little bit of a glimpse of the landscape because I have no background in that area at all, but it certainly sounded very exciting and interesting stuff. Arithmetic geometry, I think, was the title of the... It's an interesting idea how you use higher principles and follow the same natural numbers, and they might actually possess some logical terms, which is why some of the points are quite interesting.

45:00 Is it that you're using those methods, like ergodic theory or... Does it mean you're using stronger methods, or is it just a conservative encoding of what you could do by elementary methods anyway? I have no idea, of course. I'm not confident to answer the question, but it's clearly a very interesting question. The question you put to Bill, or at least a question which is perhaps at least related to the question you put to Bill, ...to these problems about prime distributions, for instance. I understand there is a quite striking connection with, I mean, this arithmetic geometry. They use a lot of the sheet theoretic machinery to try and study, you know, properties of prime distributions via the localization of... Localization in order to get stalks of sheaves, in particular constructions in sheep categories and pre-sheep categories, they bring quite a lot of that machinery which originally came out of algebraic, was originally devised in the context of algebraic geometry, is now directly relevant to these. ...studies of these vibrations of spaces to understanding properties of prime distributions and just kind of localize the primes to get the stalks of your sheep. That's the idea. I certainly am not the person to explain it to you, but I conjecture it would provide at least part of the answer that... You know, if Lovie had had more time, he might have provided in developing in more kind of mathematical detail the way in which ideas and machinery from category theory and topos theory would be directly relevant to what you're doing here. Yes, you never get exactly what you expect from Bill. You always get something a little bit... The only thing I know about Douglas is that I read Goldblatt's book when I was young. I know some people criticise it because it was so simple and it didn't go into a group of three sheep's sheeps or anything. No, and it gives people a very, very distorted idea of the history of the subject because they think it came out of logic, which is, of course, absolutely not the case. Well, Goldblatt, he is a monologician. He wrote his take on the subject. I'm not criticising, well, I suppose I am criticising, but I don't criticise him.

47:30 I'm doing what came naturally to him. It is unfortunate, exactly, the most accessible, but it is a little unfortunate that it has become the standard. There are others. Well, I'm very impressed. It's not that bad a book. It gives one a good flavor for the subject. And it gets across the fact that this does provide a very... Well, I'd say a natural framework in which to show how geometric, algebraic and logical aspects of the structure are naturally kind of connected and shed light on each other, but that's not to say much except that's a pretty hand-waving remark, but still it gives a good flame of why the subject's interesting and arguably, you know, very important, because Bill claims it's much more important than that, but it really provides a kind of... A pass-key to the whole of mathematics, which will be very interesting. Once the ideas crystallize and form some kind of formal axiomatized theory, then we'll be able to convey it with other theories by consistent systems. Well, it has been an axiomatized theory since the 1970s. I mean, since the 1960s. The original Groton-Digereaux axioms for topos were... Actually, presented in 1963. But then, of course, topos were thought of really as just good sites for homology and cohomology theories. Although it's interesting, Grote and Dieck, Grote and Dieck being such an incredible universal genius, did actually see, even then, that they would turn out to be very important for, across a much broader range of mathematics than just algebraic geometry, than just, you know, just as... Machinery for those good sites for homology and cohomology. He thought even then that there were other kinds of prophecies other than the so-called localic quotient-decidable prophecies, which would turn out to be very important for a deeper understanding of the notion of space in general, particularly understanding of the kinds of spaces with which functional analysis deals, mannock spaces, protoid spaces.

50:00 His own development of nuclear spaces and more traditional differential geometry and he thought that there were, in fact this whole business of cohesion which Bill Banks on and on about, he really got from Grozny. He's got involved with his own coinage, but the idea, the guiding ideas, are very Grotendiecki. They're very, very much taken from Grotendiecki's take on the subject, there's no... I mean, he's always quite open about that, he's said that to me many times, but the... I mean, it turned out the original Grotendieck-Giraud construction, on the face of it, didn't seem to have any obvious connection with logical set theory, it was purely, as I say, these were sites, these essentially were, you know, pre-sheep categories which, because they carried the right Grotendieck topologies, were very good sites for doing, you know, for developing cohomology theories, but... It turned out that then Bill came along, this was the thing for which he really became famous, in 1969-1997, he'd already done this work on the categories, and in his thesis he had done this work on... Functorial semantics for algebraic theories showed, for instance, that, you know, Kahn extensions that quantifiers, universal existential quantifiers, were just right and left adjoints to substitution. You know, it was the most algebraic view of logic that anybody had really produced since, well, I suppose one could go back to Boole and Schroeder, but obviously logic had developed an enormously long way in the 20th century. He really took the whole subject and, you know... Put it back inside algebra, I think you like, with a great deal of... Geometry is well attached to the algebra as well. I mean that was what was so dramatic about his approach. But he also, a few years later, about three or four years later, went back and visited the, revisited the toposaxons and he saw, which Grothendieck said afterwards, was the greatest missed opportunity of his mathematical life. I mean the only thing he should, he was really kicked himself afterwards that he hadn't seen it, that there was this natural, the so-called sub-object class. ... inside a topos, when the axioms were simply slightly simplified and rewritten. Well, both generalized and simplified, and there's a version that he and Tierney produced in 1970,

52:30 which allowed one to get the whole of logic. It's perfectly natural, because it's essentially truth-value objects, it's just interpreted subject, classifying the two in the case where it's two-pointed. It's just the truth value in the theory, so from this one can encode anything which, as it were, deals with relations and properties and other logical notions very naturally in the toposetting and bring the geometrical tools which, quotently, get already developed in the context of, you know, theories of sheep cohomology to bear also on the logic. Hence showing all sorts of interesting ways in which one might Modify, generalize, or indeed you think of the logical concepts as subsumed in this more geometrical setting. Yes, okay, I'm sorry. I don't pretend to be able to teach you anything about the subject. This you already knew at 17. This you already knew at 17, so I'm sorry. Actually at that time I was fascinated by the idea to build mathematics without points. Which of course was also the Grote and Dieckian problem. Yes, of course. Hi, Dan, very, very nice to see you. I look forward to seeing, I'm going to be in Cambridge for the thing with Peter Johnson and Martin this weekend. Are you going to be there? No, no. What a pity. And I'm going to miss you in Paris because I'm still going to be over in England when you have the meeting unfortunately. Because I'm supposed to go up to Scotland to record for our archive, to record the Atiyah Fest. You know they're having this big meeting for Michael Atiyah's 80th birthday in Edinburgh at the end of the, well I think the 20th, the 20th to the 23rd. I've got to stay around for that, and then I'll be going back to France straight afterwards. But I've gathered from something that I overheard that you're going to be in Paris again. I'm going to Paris in May. There's a meeting on... Fantastic. Yes, with Steve Audi, yes? Because he's also speaking at a little meeting that André has... I think it's a meeting. Well, it should be, because André there has arranged a little mini... A little sort of one-day symposium with Steve Audi and Colin McLarty in Paris on the 13th of May and Colin, yes, and Steve is speaking at this thing for Fair Martin Lurft's retirement in Sweden and told me he was going to be in Paris afterwards so I assume he's going to be...

55:00 In May that I'm going to, the main event is organized by Pierre Wagner. Yes, I know him by name, but I don't really know. Where is that? As we say, there are so many places in Paris. I think it's the... what's that thing on the end of the board? Oh, that's IHPSD. Yes, IHPSD. Yeah, yeah, yeah. Yes, okay, yes. Oh, well, I'll certainly try and get along with that. Do you happen to remember the date or dates? Yes, yes, it's the 14th, 15th and 16th. Oh, no, in that case, it must be the thing Steve, I must be talking about, because he's speaking at our meeting on the 13th. So he must be speaking there as well. Well, if there's a thing about Carnap, it would be very surprising if he wasn't talking, given that he's just edited his presentation. Actually, he gave a very nice talk about Carnap and Maclean. We had a workshop at Oberwolfach in February on the subject of, essentially, history of category theory and related fields since 1945. And Pierre Cartier, Colin, Steve... Jean-Pierre and Marcus and various other people were all involved. But Steve gave a very, very nice talk. Actually, it was really a kind of pretext for giving a talk on the Tarski problem of characterizing, providing complete characterization of logicality, the logical constant. Which he claims his research group at Carnegie Mellon, you know, his little group of topos theorists has now cracked it, you know, we now know exactly how it should be done, you know, what was missing from Tarski's definition. I'm not convinced by that, the sales pitch, but on the other hand, they certainly have made some very remarkable strides both conceptually and technically, I think, on the problem. I think you'd be very interested in, well, you're going to be talking to him about it in Paris, obviously, in May, but along the way, he did, of course, the first half of the talk was a historical talk on what Carnap's relations with Maclean had been. They were much closer and more frequent and intimate than I had ever realized. Oh, yes, yes, they saw a lot of each other. No, but even earlier, because Maclean was in Göttingen. Oh, right. Maclean was in Göttingen when Carnap was still in Vienna. And he went, and in fact, Maclean went down to Vienna to visit Carnap, shortly, I think in 1932 or something like that, when he was in Göttingen, and then of course they took up again when they were both in Chicago, but also Maclean was a very close friend of Quine's, and they spent a lot of time talking, and this is the most fascinating thing, Carnap,

57:30 Quine, McLean, Tarski, and one other guy whom I, oh and Russell, were all members of this seminar in 1940 in Colombia. I think it must have been, was it in Colombia? Or was it in France? I mean there's a it's a very well I know I could I could be wrong but there was there was a conjunction and and the point is it was it didn't last very long but somebody one of there was a Polish research student there who kept very very good notes I mean extremely detailed notes and it was there actually At that point, Tarski was very dismissive of the whole program to try and characterize logicality. He came around to it, later became a convert to it, but the story of that seminar, which Steve told in the course of his talk, is exceptionally interesting. But the second half of the talk was more on the... This is the progress which, using topos-theoretic methods, they've made on the characterisation of logicality, which is a very interesting issue, via the notion of automorphisms of the language. But it turns out that you need something rather stronger than just automorphisms. You need automorphisms in a particular type of continuous transformation to really do the job. Which the model theory, the traditional kind of task of trade model theories would have been very difficult for them to see because you can express it rather much more naturally in terms of maps than you can in terms of models. Well, Tosca's a very, very smart guy, but, you know, everybody has some limitations. We all know Aristotle and Euler were incredibly smart guys, but, you know, somebody's got to develop the concepts however they are. I mean, usually it's those guys who do it, but we can't expect, I mean, even a genius like Grotendieck was only able to see maybe 300 years ahead. I think Tasker did pretty well. No, I'm not doing anything at all to minimize Tasker's achievement.

1:00:00 But anyway, the claim is that they have made some really interesting progress on this. So I hope you get a chance to talk to them about that in Paris. I'll be there for that meeting, certainly, the one in May. Yeah, great. Love to everybody in Oxford and take care. Oh, please give my very best wishes to Simon and Harvey if you speak to them. I'm supposed to go and see Simon on the 23rd of April. And I thought I might not be able to because of having to go to record the Atiyah thing, but in fact they're covering the recording of it in the last day, so I will be able to get to Oxford for that, so that will be a bit of luck. Have a safe journey back anyway. Take care. Cheers. If you add up the ability to apply them, talk about the difference between the equalizer of zero and Z and the equalizer of zero and Z. I don't know. But if we're not there yet. Right, exactly. Okay, so then you have three of your products to equalize. Then you say, well, gee, what if I have an exponent? But you're coming out on the 10th of March. Yeah, we'll be there. Okay. Mattias, I'm going to check out tonight about the, um, you're not coming to dinner tonight? No, I don't even know that. I think Bill wants to go out to an Indian. I don't think we've actually got anything sorted. I'm going to check out tonight about the Granhar, but from what I've seen so far it doesn't look as if it would. I think it may be cheaper to take the bus after all, but I'll check it tonight. I certainly should know by tomorrow morning. Okay, if tomorrow turns out it's not good enough then I'll just... We'll see. If it really is worth it financially, if it means that we all save a bit of money, then it's worth doing. And plus it's obviously a bit more convenient to be able to drive directly to Cambridge. The worry that Bill had about going through London during G20, Peter Johnston says, is really quite irrelevant because it's going to be finished tomorrow afternoon anyway. So if we go on Friday, the big shots will have been gone, the demonstrators will have dispersed, so it shouldn't affect us at all if we do go on the bus.

1:02:30 You know, it would be nice to go cross-country on our own, under our own steam, but I'll check it out tonight, anyway, you know. Okay. I'll see you tomorrow. Did you do reminiscences, like, in philosophy or mathematics that she's doing, or...? She was talking about... I heard the words Caley Graff come up. It's kind of interesting that he proves using diagrams. Oh, okay, that's a topic of interest. Oh, you're talking about Arisha? Yes, yes. Yes, she studies particularly, you know, the role of the diagram in Euclid and in Greek mathematics in general, about which there's quite a lot of his... There's quite a lot of controversy amongst the historians of mathematics. Oh, is that diagrammatic? That's a rather new diagram. Oh, sorry. No, no, no. Well, you know, it's all... As a topic in the history of mathematics, you know, the role of diagrammatic reasoning is quite a... It obviously has a long history. I mean, there are people who try to connect the role of diagrams in category theory and in... ...scheme theory and sequences with what people did in diagrams in a previous epoch, but she works on the classical stuff, she works on the 17th century and the Greeks, you know. That's her stuff. Well, she's having a, dare I say it, Bill has always been... I do adore Bill. He's always had, he's always been one, you know, eye for pretty, you know, for a pretty girl. Is that right? Oh yeah, he's been closely with her the whole, you know, not that he's, I'm sure, not that he's ever for a second been unfaithful to Batman, but he does. He does like to theory. He has an eye for propriety. He has an eye for... He has an appreciation for... Good appreciation for... I've shot him, let's say. It's a good appreciation for him. I've seen him flirt outrageously in Italy with, you know, I've been down with him with, you know... Subtitles by the Amara.org community

1:05:00 Well, it's just, I mean, nothing remotely sleazy or anything, it's just very, very charming and very French. Oh, in the middle, in the middle. Wait, wait, wait. I teased, I put his leg about it afterwards and he quoted to me the words of a... A French general, the reason was because the guy was from Alsace, from Strasbourg, which of course is only just up the road from where we were in Oberwolfach. Who the hell was it? Anyone wanted Napoleon as Marshal. And he said, you know, whoever in a room stays more than... More than six inches away, it's more than 50 centimeters away from a pretty woman is either a sodomite or an imbecile. Can't equate this with great approval. I couldn't help noticing this, I actually got... Roger Scruton was my supervisor for Kant as an undergraduate. He supervised me. Yes, I cut my teeth on Kant with Roger Scruton when I was 19, 1970, 71 actually. He's an extraordinarily productive and versatile Roger. The amount of stuff he's churned out in his career. Whilst also finding time to be, you know, to have actually qualified for the bar and be a practicing lawyer is quite amazingly energetic and I think that's not a bad book at all. I think it's remarkably insightful. I mean, to find something original... To find something genuinely striking, to find something original to say about Kant after all this time and in a short book is remarkable. I agree, I think he does manage to say something.

1:07:30 No, no, no, no, no. Well, given the brief that you have, I think it's some rock. I think all the things in that series that I've read, it's one of the very best. And, um... What are you reading with it? Of course, on Kant's first critique in the third year, it's next year. I'm trying to read everything there is about... out there, of course. Well, you'd be dead before you did, of course. I mean, the secondary literature on Kant must relate to the many thousands. Oh, yes. Yeah, but a lot of it's not very good. Of course. Sure, sure, you have to be... That's the whole problem of the noise. Well, I think... Really good, really good. You're quite right, the noise-signal ratio is pretty... I learned something absolutely... Yeah, that's a disadvantage, obviously. But at the same time, if you're going to... At least it simplifies your problem. I'm not sure... I'm not to the conclusion it's not a disadvantage. I mean, advantage, because most of the... I think that's right. Which of course the most wonderful example is, you probably know, the very first translation of the critique into English by that weird, weird Scotsman. 1797 or when it was. Certainly when Kant was still alive. I can't think of his name now. I mean it's never been reproduced. It was historically the most comically awful translation of Kant ever. And his translation of the passage on Inconger and Counterparts, you know the business of that, in the Prolegomena, the passage on Inconger and Counterparts, he actually describes this as snails curled up contrary to all sets. That is sublime, that's so good. Yes, much beloved of Simon Saunders, really. Snails curled up contrary to all sense, so whenever I see him, I speak to Simon Saunders, I end up with a hand like this, going, well, but, but, Melvin Bragg's program, with Simon and half of them, and Basil Hiley, well, since all three of them are trusted.

1:10:00 Well, it was a bit unfortunate. He didn't really have a... Well, I think you'd be surprised just how... He may not have come across all that well on that, but then he was kind of stopped dead from... Look, he was asked... He was asked to do... He was asked... He was asked to do some extremely basic scene setting about the two-slit experiment for a general audience, and he probably did sort of take a bit too long overdoing it, but all the same, he was kind of stopped halfway through. Hazel Kiley, believe me, is a very deep thinker. I think he's a remarkably... I mean, he doesn't come across that way to a lot of people, but I've known him now for... actually, I've known him now for even longer than I've known Bill. I think he's one of the very deepest thinkers in physics. He's really thought about the issues in... Well, it was a wasted opportunity, but that was partly because Melvin hadn't read his cue cards and hadn't done his homework, and, well, he does sometimes read his cue cards, sometimes very good, but on that occasion he hadn't read his cue cards, hadn't done his homework, so make them spend 20 minutes doing the absolutely basic setting of the stage machinery, which left them with only 20 minutes to actually discuss the issues, which left them with just enough time to set out. ...to kind of triangulate dynamical collapse versus Everett versus Bohm. And then, you know, the curtain fell. That was it. They went off to a little cafe across the road from Broadcasting House afterwards. Simon told me, Basil also, and spent three hours continuing the discussion, which would have been fascinating to listen to. But isn't it typical of the BBC that they make I mean arguably our greatest living scientist and a member of the order of merit to piss off across the road to a sleazy little cat for a cup of coffee. They don't even offer him a drink in the green room whilst they pay buddy Jonathan Ross you know whatever it is you know six million quid a year Penrose! Did you come across, did you really hear this in our time, the Melvin Bragg chat program? Well, purportedly on the measurement problem. They didn't really talk about the measurement problem. Well, they did start talking about the measurement problem. And then they just went on to a kind of super fast survey of basic alternative positions on the interpretation of quantum theory.

1:12:30 It was actually on the 5th of March. It was the 5th of March. No, it's my birthday, so I know that's why I remember it. It was on the 5th of March. I know since all three of them are trustees of the archive, I naturally was rather keen to, you know. Well, I thought it was yet another... I rely on my mother to tell me that easily enough. Well, it was a lost opportunity. Well, I think that on the whole the standard is very high. Of course, it's very difficult when they tackle something in mathematics or fundamental physics where, you know, the conceptual... Thank you very much for your time, and I look forward to seeing you again soon. Screaming at the, you know, the radio set and wanting to go and chuck it across the room was the one on the girdle, which I thought was an utter, oh, so awful, but the one on girdle, which was just so awful, it should never have been broadcast. It was so bad. It was terrible, but the worst they have ever done. I mean, they just didn't begin. Well, Philip was on it. He said very little. Well, I'm relative. Well, relative it is. He was just shot down by Melvin Bragg's utter stupidity. Well, utter ignorance, let's be fair. It's not a question of stupidity. I don't think Melvin Bragg's stupid. He's not stupid at all. You can't expect people... Well, by the standards of somebody who's... He's asking the questions any intelligent layman would ask. He hasn't, he's not a mathematician, he's never studied Gödel, he's relying entirely on his cue cards. Then why does he do this job? Well, I think, because somebody, thank God, at least somebody is doing it. There's got to be someone out there, there's got to be someone. The more you know about the subject, the more critical you are of any kind of coverage of it. But they won't get the audience. Total embarrassment, yeah. But then how many logicians would be able to get up and give a talk about twistor theory or complex analysis or Riemannian geometry at the level that Penrose can talk about it. I agree. Then become amateurs again. No, I agree. That's absolutely true. The thing about that meeting, and I didn't attend it myself, but I heard all about it. You were there, weren't you?

1:15:00 Yes, yes, yes. What did you think of Angus MacIntyre's talk? I always laughed. But he quoted me very nicely. Oh! He actually quoted you. I'm delighted. Yes, he said if a mathematician formulates an improbable statement without sign, and that's actually a paper which he was reading last night, my paper. I read the talk, I read the transcript of the talk, and it was extremely interesting, I thought. I mean, it was by far and away the most interesting talk of the non-technical, relatively non-technical talks of that meeting that I read. And the thing which impressed me was this kind of general... Take on the position of Gödel vis-a-vis 20th century mathematics. He caused apparently quite a lot of ill-feeling on the part of the Gödel Society who organised it because he essentially said, without launching a kind of all-out attack on Gödel's reputation, he did say more or less in these terms, well, actually Gödel's importance in the history of mathematics in the 20th century has been rather inflated, i.e. very much exaggerated. You know, the broad architecture of mathematics has not evolved in the way that Gödel anticipated. The problems that he worked on are... Pretty detached from the rest of mathematics. Incompleteness was not very conceptual. There's a few papers which says it as well. Yeah. It's not the first time. No, no, sure, sure, sure. And I agree with him broadly. You agree? Yes, yeah, broadly, I do. Because I disagree. Ah, okay, that's fine. We've got a good argument then. One of the reasons I disagree is, of course, the stuff we've just been listening to this afternoon. But, no, I think that, well... I share Bill's view about the natural numbers as being, let's say, the root of all evil, but I don't see Gödel's incompleteness theorem as being anything more than an interesting fixed point theorem. No, no, I don't mean the proof. Of course, the proof is of no interest. I mean the fact. I mean there are unprovable statements on whichever branch you go. It will split. For example, once, whenever you agree on whatever you want as your intuitions formalize, then you finish your formalization and there you are, there's your theory, and then it splits.

1:17:30 And it's almost automatic now, especially after the work of Friedman, that we can produce those splits nowadays. So it's this fact that matters, it's not diagonalization or whatever. No, okay. And of course what is more important is that... I agree. What's most important is formulas with many quantifiers, not with one quantifier. Because when they are with one quantifier, then you can have a preferred choice which one is more true than the other one. But it's the actual split that matters. With the many quantifier formulas that indeed, once you formalize all your intuition as your favorite theory, NF or ZF or some category theory foundation, the split happens after it. And they will indeed be neither true nor false. Different branches are absolutely genuine alternatives. But my understanding of what I was trying to say yesterday is actually that in category theory there are other notion of axiomatic methods, as I understand it, that all this, you know, there's a type of theory there and the same point of view. You may still use those arguments. No, I mean, you can just rethink what is axiomatic method and use something else, and in that sense... It's something else which is not available in the axioms. This position existed in the time of Brouwer. Brouwer was saying that you don't have to stick to axiomatization. And even the extremist Platonists are claiming that there will be new axioms. Oh, yes. Well, precisely. That's the only position available to them. If they really are whole-hogging Platonists and ontological Platonists, then they really do believe in the... You know, there is a unique universe of sets that, of course, they must believe that it's at least possible, in principle, that one would find new axioms which would decide the continuum hypothesis. There will be a philosophical argument which will refute real mathematicians. Real mathematicians come up with genuine alternative theories and they believe that a philosophical argument will refute all of them. Well, no, I think that's been, I mean, no, I disagree. I don't think, I think, I think there are plenty of working mathematicians who are, I mean, I don't know what Woodin's position is on, Brayton is, but, I mean, clearly this is the kind of program that he pursues, that, you know, to find stronger and stronger axioms of, coming, obviously coming out of this reflection principle of large particles. Yeah, yeah, and he believes that this will...

1:20:00 As I understand, he actually claims already approved restricted version of the Continuum Hypothesis, but because I think that Bill's right, that this is completely the wrong way to think about the Continuum Hypothesis. The Continuum Hypothesis is, I think Bill's right, that it is trivially, well not trivially, I wouldn't say it goes as far as that, but that it is true if you, as he puts it, enforce the constancy and if you make things sufficiently point-like in the universe, then yes, sure, it will turn out that it holds. Of course, that's only a fragment of mathematics. You know, interesting structures in algebraic… Geometry and topology will refute it straight away. I mean, just in the way that the axiom of choice fails in any way you have a non-trivial cohomology, because it's all about the existence of obstructions to inverses of maps, and the axiom of choice tells you that there are no obstructions to inverses of maps, whereas the whole of homology is just really essentially about the study of the conditions of existence of obstructions. Yes, and actually this is the thing I like about category theory, that indeed the models are, you take this category, you take that category. There's a variety in front of you. It's like those people, I recently learned they're called structuralists, who believe that... I'm not saying that is the same thing, but philosophically, for me, they're kind of... Kind of the same kind of the same thing smorgasbord mathematics what smorgasbord mathematics no i'm teasing the smorgasbord view of mathematics smorgasbord is the um scandinavian term for when you have lots of different dishes on the table yes you you have many many many different different dishes that you can tap us perhaps i said the tapas bar the tapas bar of yes because i was recently studying this theory by martin that It's absolutely amazing and unbelievable, because choice really refutes it strongly, choice allows only finite dimensions.

1:22:30 And with AD, we are now building a model of ZF plus AD with a student of mine, where Aleph-1 will be, well, we're learning it, we're not doing a new one. Aleph-1, erroring Aleph-1 with Aleph-1, and for me as a Ramsey theorist. Well, possibly an amateur ramsay theorist, but an ramsay theorist. This is absolutely amazing, and this is Tapa's board, and this is a model of Zeteksi, which does it. Yeah, very interesting. I'm fascinated by the axiom of determinacy. Is there any mileage in that, I mean, sorry, because I don't really know the first thing about set theory and that, you know, real set theory, the stuff that people, real set theory, you know, You know, stuff that you're doing, or stuff that's relevant to your work, but is there any mileage at all in that, you know, metaphor of, who the hell is it, who the hell is it who likes to illustrate the acts of indeterminacy by that business about chucking darts into the dartboard? Kai Hauser? It could be, but I actually heard it from somebody else in, I think it might have been, I think it might have been Ed Nelson, but of course it's not his idea, I think he was just expounding. How is it? I don't know. No, no, no. Well, it's the game-theoretic version of determinacy. He likes to illustrate it by this story's metaphor about chucking... Okay, he likes to illustrate it by this metaphor about... People chucking darts into a dart board, but maybe I have, okay. If you don't know the metaphor, there's no point in dwelling on it. I just wanted to ask what you, you know, you thought of it. Yes, one thing from three minutes ago. I, of course, don't mean splitting at the time of, at the level of C-H. C-H, some people think they understand what it means. I don't really understand what it means. The ideal objects which are introduced by those stronger theories, like an F, Z, F, A, D, or the first interesting billion of alternative theories, they may be talking about their ideal objects, but we don't necessarily. I don't say they are meaningless. For me, for some people, they might be. Each of them has the first-order arithmetical fragment. Can you give me an example of what you mean by an ideal object? Because obviously you're using the term in a rather technical sense here, not in the sense of ideal elements.

1:25:00 By forcing you mean it's an ideal object. For example, ZFC is talking about its own objects, ZF sets, ZFC sets. They're not sets, and ZFC of course is not a set of things. It depends where you interpret the theory. No, no, no, no, no. It's a theory. The intended manner of... What does it mean, intended? That's the kind of thing I never... No, no, no, I'm not going to philosophical questions. No, no, no, I agree. I think I understand what you mean by the intended model of ZFC. No, not the intended model. Of course it doesn't have the intended model. It has lots of different... It did at the time that Zemele and Frankl wrote it down. It's not that simple. I think we know nowadays... We obviously know that there are lots of non-standard models. So NF has its own models, lots of different ones. Sorry, did you say NF? Yes, yes. Oh, NF, I don't know anything about it. The F was determinacy, and you can see the geometry of those models. It's the way the notion of determinacy behaves relative to the foundational system. Models of ZFC are interesting models. ZFC plus large cardinals, ZFC plus V equals L. These are all our theories and we have a spectrum of models of each of them. Of course, neither of them, like in the language of arithmetic, there is no intended model. Do you have a standard model for the language of arithmetic? So the same way in ZFC? No, no. So in arithmetic we are talking about the more or less real. I have meaning, I can assign a kind of meaning to arithmetical statements, but then we come to ideal objects, and let's have a theory of Wombles who hug each other, and there's a relation of Wombles hugging each other, we can throw some axioms about Wombles hugging each other, these are ideal objects, Wombles, and this theory will prove extra arithmetical stuff about natural numbers. The embedding of the language of arithmetic, how it's interpreted in the theory of Wombles. But do these Wombles, as it were, have to... What properties do these Wombles have to have to be ideal objects? I mean, you characterize them purely structurally, presumably, in the way that you do points at infinity in projective geometry.

1:27:30 Your axiom is about... No, no, no, this is a... No, no, no. Andrei, Andrei, this is an absolutely relevant question. I am not even going to ask. No, but we use this... But you are using terms which you are obligated to define. Anything which is not in the language of arithmetic is an ideal object. Oh, I see, OK. Now I've understood the criterion. OK, that's fair enough. There are different versions. For example, you may allow the second order language of arithmetic. How restrictive is the language of arithmetic? Yes, yes, that's what I'm saying. There are different versions. You may go into first order language of arithmetic, you may allow second order language of arithmetic, you may even allow a little bit more, it doesn't matter. We may fix it once in a while if we want. I'm not dogmatic about that. Well, if it's first order, you won't be able to fix it because of the, you know, failure of categoricity, and in the second order, you're already... If it's a first order language, then you're not going to be able to fix the intended model because... There's no intended model of any of those theories. But you've just said that these are the real objects and this is what you really talk about, everything else is... So what is the distinction? What is the distinction real-ideal? A certain camp in philosophy of mathematics likes to ridicule the pluralist position by accusing them of being formalists. So let's be those stereotypical formalists. And let's say we're just talking about theories. There's this is no no he says he's just talking about theories just talking about theories just purely syntax yes oh so you're staying at the level of syntax okay yes yes just syntactically okay but what matters sorry so sorry is the Hilbert-Bernice theorem yes okay sorry Richard by the way can I get you a drink No, I'm OK. We're going to talk to the restaurant in about 20 minutes. OK. Which restaurant is it? Indian Run. I'd love to come. Is that OK? Yeah, of course. Have you been back to the flat already? What? Have you been back to the flat and used the computer? No, no, I just called. No? Well, aren't you coming to have dinner? Because we're going in 15 minutes. Which you just said. Well, why don't you look... Well, why don't we go back? I mean, you know, I... Why don't you do it this evening, afterwards? May as well. Is it that urgent? It's about this meeting in 15-16, just to make announcements and stuff, because more and more people are worried about what's going on.

1:30:00 Okay, okay. Well, they're not going to, I hate to say it, but what's the time now? It must be already about 7.30. If they haven't looked at their email today, they're no more likely to look at it. Yeah, but at least you gave me the tutina and the drink. Okay, okay, okay, okay. Well, I gave you the key, didn't I? Yeah. Well, then you've got it. You too. Okay. Yeah. Yeah. All right. Well, piss off then. All right. Sorry. but just but just do close the door carefully afterwards and you know sure just don't disturb any of my papers okay so i'll see you later i'll see you back though okay sorry about that okay i know how to twist this conversation in the right direction no it's very good i know i know how to twist it correctly good go ahead yes yes about hilda let me finish go ahead yes Passing from syntax to models, you don't need any set theory because of Hilbert be nice theory. That's the claim, yes. Yeah, I just don't, well, there's things I want to say about the notion of the priority of syntax when you look at things functorially in terms of functorial semantics, but let's not go there. I'm an arithmetician in many different senses. In St. Petersburg, I became a constructivist. In Birmingham, I learned to build models of arithmetic. Now I'm building models of the taxi. There are many things that I am. So, a Hilbert-Bernice theorem which gives you one formula, given the code of the theory, builds a model of it, and this model building finishes if the theory was consistent in the first place. So this is one automatic process. And you can do this during the building of this model, which is one formula, it's not a set, which is one formula, you can insert anything you want, model theory, recursively saturated, minimal, whatever, and by the way you're building completion of the theory in question, so that's the theory I'm referring to. So it was an answer to this philosophical question, what is the intended interpretation? There is none apart from this one. Okay, so it's purely, okay, I understand that. So it is, in fact there's a technical term for that, isn't there, for the purely syntactic presentation of a theory. It's not, but anyway, it's something distinct. It's not a model in the sense that any of these issues about non-standard models touch.

1:32:30 Okay, so it's a purely syntactic presentation of the theory. First we have a theory, but then we build any models we want. And they're in front of us because they're described by a formula, one formula. All models are in front of us. That is the first place. Oh, I see. I had something very different in mind. You don't need even to bring in any set theory. For example, a simple theorem of Peano arithmetic, even of primitive recursive arithmetic, says that Con implies Con equals L. It's a theorem of primitive recursive arithmetic. You start with the ground model, you build the new model with its L inside, so it's all arithmetized. So for those philosophers who may say, what is the subject matter you're talking about, I will say, I'm an mathematician, I don't care, so since everything is arithmetized, it's not... Well, yes, provided you can do everything in this. A very schematic, syntactic presentation of your theory, then yes, everything is arithmetized. Yes, but also model theory. So the usual accusation, these are formulas, they believe it's only strings of symbols and nothing else. It's irrelevant. This accusation really passes by without hitting me. I don't see why, but maybe I'm just stupid. I probably am. Why does that accusation not touch what you're saying? I'm probably just being obtuse. I'm sorry. Firstly, it doesn't matter, because it doesn't touch on what I wanted to talk about. But why does it miss the point? Yes, that's my question. Why does that miss the point? Why is the accusation that this is just a completely formless view of arithmetic? It's an imposed question. I understand that in Catholic countries and in the Catholic set of mind, the nature of angels, the nature of God, the nature of saints is different from the nature of sin. These were questions which were developed for hundreds of years and they're somehow affecting the people asking us these questions. And what is the nature of mathematical objects and... Yeah, that's not a question that interests me in that form at all. But this is how I see it. These people are asking me the question which is irrelevant. So maybe what matters is the picture I will draw now.

1:35:00 Okay. So I want to shift this conversation to a... I grant your syntactic presentation of a theory as something which is kind of completely self-contained way of getting started. There's no problem with that. Okay. I think there is but let's say okay let's grant this okay so you've got this completely you've got this machine you've got this machine you've got this completely syntactic presentation of the theory where there isn't any there aren't any issues about models or anything and now what yeah there isn't any issues because all models are in front of us Anyway, which is a metaphor which I don't quite understand how you're playing with, but I grant you that. Now I want to understand what the claim is about arithmetic. Okay, sorry, hang on a second, I can give you a piece of paper to do that on. We should probably, I just think you can get your train back inside. Oh, you're going to head off? I think we're probably about to head off. I've been pulled out of the way. Oh, so. I think we're going to go to the restaurant and buy two minutes or so, so yeah. Okay, yes, we're just coming, Anders. So where I am? Yeah, sure. Whereabouts? Well, Indian restaurant I'm enjoying. I like Indian. Whereabouts? I haven't caught such hair like that. Oh, then I will have to follow you then. Okay, well let's go. Well, why don't you draw it for me when we get to the restaurant? Yes, let's do it, let's do it. Okay. I'm not drawing the horizon. You know the picture, the horizon. Okay. I have this picture with his signature. Wow, okay. I look forward to it. It's terrific. Even better. Well, that's my signature. Yes, I'm not sure how Cyrillic horizons differ from anybody else's horizons. Unless it's something to do with all that vast empty steppe in Russia. I think there was an argument that it's insulated. Why Russians are constructivists, why they don't buy the Catholic argument. But I've seen the same argument played the other way. In fact, the reason that Russian set theorists, the Russian set theorists, no, I'm talking about the Russian set theorists back in the 20s and 30s, the Russian school, yeah, no, I'm talking about Lucian, I'm talking about people like Lucian and Alexandrov and Susslin, I'm talking, there is a claim that the reason that those people were so much more prodigal in their

1:37:30 Well, implicitly, you know, in the ontology that they were prepared to, the moves, that they were so radically non-constructed by comparison with the French of the same period, but the Borel and those people. There's a historian of mathematics in Paris, who's actually a rather interesting guy. He's actually a very strange guy, but his name is Jean-Michel Cantor, and he knows a great deal of set theory, actually. Well, I'm told by good set theorists. And he claims that the reason that the Russian school, as they say, Sussling, those guys, had a completely different approach from the French, who were contemporaries, was because they were deeply influenced by a particular heresy, which was very common in the Orthodox Church at that time, called the Name Heresy, or Name Worshippers. I don't know if you know anything about this. Well, apparently it was theologically very influential, to the point that the Holy Synod... In, I think, 1911 or 1912, actually sent an expedition, or they persuaded the government, to send two naval ships, cruisers, full of marines to Mount Athos in Greece, where there were all these Orthodox monks, of course, very typical of Russia, yes, and they sent ashore, you know, about 300 marines, and they arrested About half of the monks on Mount Athos and brought them back to Odessa to be tried by the holy synod for this heresy. And the heresy, I don't of course pretend to understand the details of theology, but very crudely speaking, the heresy was that God ...in himself is so inaccessible, is so beyond the possibility of human conception, that one cannot even adore or worship God directly. One can only worship the name of God. And this was held to be highly heretical. I hate to say, but how Lennon and the Bolsheviks must have laughed at the thought that the Akrana and all these people were spending their time and energies in hunting down these crazed heretics. Anyway, that's the claim. But it is the claim, the further claim, which Cantwell makes.

1:40:00 The other thing is that this position was extremely influential amongst many sections of the Russian intelligentsia outside the Orthodox Church, and that it particularly influenced a number of mathematicians, of whom Suslin and Lushin were two. And that it influenced the way in which they did set theory to a significant degree. That is one of the reasons that they were not bothered in the slightest by problems about constructivity. Oh yeah, it's a very interesting theory and there's nothing I can say about it. It could be true, yes. I didn't mean, by Russian, I didn't mean Russian Empire, I meant Soviet scientists of the 1940s, 50s, 60s, this is a completely different culture. Of course, yes. Although curiously, Alexandrov did carry over from the one to the other, one or two other people as well. Yes, you see, the last people of the Russian Empire tradition. Alexandrov, Kolmogorov, Uryson. They are maybe half way, and already you can see that Markov, who was educated after his father's death. In St. Petersburg University, although he might have belonged to that generation, he is already a Soviet scientist with a scientific kind of mathematical philosophy. And of course constructivists are claiming always that they are the scientists as opposed to the... Reactionaries. The Platonists. Yes, yes, yes. Interesting, very interesting. And it's part of the Soviet upbringing that you are not a reactionary. You seek the truth, but not fantasies of reactionaries. And there's a lot of this truth-seeking in Russian and Soviet constructivism. And of course, formally speaking, they could be easily accused of being Platonists because they believe in the truth. I am holding a public debate in Bristol on the 14th of May. I should love to come. Unfortunately, I'm going to be back in Paris on the 14th of May, because I live in France, so I have to record a conference in Paris on the 13th.

1:42:30 Unless they refuse to be filmed, it will be filmed on the web page of my logic book. Oh, excellent. Might I ask your permission to put it perhaps on the site also of our archive? It all depends whether they will agree to be filmed in the first place, or if they disagree, because one of them already said he might disagree, I want the permission to film but not to put online, so there will be a film which can be watched but not accessible online. Well, I might just ask for a copy of the film. Anyway, it sounds very interesting, very interesting indeed, utterly fascinating. What are they afraid of about being put online? Do they think they're going to... No, no, Sazonov says his English is too bad, he just doesn't... Oh, I see, it's just a presentational thing. It's not that they're... I mean, after all, if they're prepared to stand up and defend their positions in a public debate, one would think that they were not afraid of having people actually hear what they had to say. Well, people have different speeds, and perhaps some people wouldn't want... Some people of contemplative kind wouldn't want to see their failure in the public debate exposed so much. Well, I would have thought in that case they would have been reluctant to debate publicly in the first place. But I take your point. Obviously people are different and everybody has their own take on things. But the point is they're not... They're not frightened of the ideas that they want to advance being more widely discussed, obviously. No, no, I'm trying to find people who are really devoted to these ideas, who would be not pretending but defending those ideas. Yes. Not a debate, but a proper disputatio in the medieval tradition, a real proper disputation of populism. People trained to be politicians or lawyers, people trained to be effective public speakers who are just defending whatever position they get to defend and then will just show how well they can. They can perform just as public debaters, you know, debating society, which is not people who really believe in. Because people have subtleties in their own positions, and I'm asking for something stereotypical in real time,

1:45:00 so they might slip into stereotypical arguments, which they will defend not full-heartedly. This may happen as well. Well, it's also, of course, a difficulty to get. You do have to use some form of abbreviation and shorthand in any... ...debate of that kind, but on such complicated intellectual topics, obviously. Unless you're going to allow things to go on all night. I mean, people presumably have a limited given time to debate in. I will be at Potsdam. I will be with Jerry Springer for a Swiss show. Well, I hope you're a bit better than Jerry Springer. There used to be, I mean, many, many years ago now, on British television, an extraordinarily good programme called After Dark, which was a little bit like this. Melvin Bragg in our time program, which we were talking about earlier with Richard, but much better in that it went on. Instead of being limited to just 42 minutes like the Melvin Bragg thing, they had usually three, occasionally a larger number, usually three, at most four. Very often leading scientists or they might be commentators but they were chosen for their expertise on a particular topic and they would just discuss this topic and the thing ran on and it sometimes ran all night it started at about late in the evening at about 11 30 and it would just go if they were still talking at six or seven in the morning then We had a similar one in Russia until recently. It's recently closed. I don't know the reason, but absolutely fascinating. One of the very last ones they did was actually on the foundations of mathematics, John Bell, in fact. Did you know about that, Anders? No, I don't know John Bell. Of course you know John Bell. Sadly, we hoped you'd be able to come to this workshop, but Mimi is very ill, unfortunately. But no, it's just saying that there used to be this program on British television many, many years ago now, 20 years ago now, and well, it was when John was still at the LSE, it was just before he moved to Canada with Mimi, so that must have been the 1980s. But one of the last ones they did before they, it was, it came to an end. In fact, the circumstances in which the program ended were quite intriguing. But they had these discussions and they were often extremely good discussions. ...which began late at night, about 11.30, quarter to midnight, and which ran on often right through the night, sometimes till 6 or 7 in the morning, usually not, usually they ended after about 2 or 3 hours, but it was completely open-ended, there was no limit on the discussion.

1:47:30 It was Channel 4, it was the independent Channel 4 in the days when it was run by Jeremy Isaacs. It was TV. And they were sometimes extremely good, sometimes they were... It's kind of quite comical. Sometimes people got very drunk because they were fed fairly liberally. They were well lubricated. There was one that I remember which was absolutely astonishingly interesting, which was a discussion about the Mafia and about organised crime. An American expert who was the kind of top, he was retired, but he was, he had been with the FBI for many years, and then he was the top expert in investigating insurance fraud, systematic insurance fraud, and then there was also this guy who was a financial journalist and who knew a great deal about financial, and it was utterly fascinating, I mean, the amount of stuff that came out. Quite quite amazing and just in terms of factual you know things that one learned listening to them it was that was but they also had topics and one of the very last ones they did was a discussion on the foundations of mathematics which John Bell participated in. I see. John Bell and I think Moshe Machova was on it as well. Who? Moshe Machova? Ah yeah. Bella Machova the model V. Model theorist at LSE. And a couple of other people. I'm trying to think who they were. Oh, I think Fred Rowbottom might have been one of them. There was certainly somebody from Bristol. But it was a most remarkable series. But unfortunately it was killed. The reason it was killed, it was actually during the Thatcher time. And it's curious, isn't it, the way that the British use that locution about Thatcher in a way that Russians talk about the Stalin time. We don't talk about any other... We do talk about the prime minister in recent history in that way. Nobody ever refers to the Macmillan time or to the Blair time or to the Churchill time, but we do talk about the Thatcher time as if it was just in the way that people talk about the Stalin time in Russia. It's quite incredible.

1:50:00 Was the discussion good? The last one was very good indeed, yes. It was remarkably good. It was astonishing. I would have liked a slightly broader spectrum of points of view, but it was still very good. But the reason it was closed down was because they were going to have a discussion with the Adams, who was then the spokesman for the civilian wing of Sinn Fein, the IRA. And they had just had, and they had, they were going to have a discussion about Northern Ireland with several people, and Thatcher had just passed, well, the Parliament had just passed a legislation, because there was a large Conservative majority, which prohibited any... The words of any Sinn Fein spokesman, anybody who was kind of designated by the government as a terrorist, from being broadcast, and rather than submit to this, they just closed the program down, they just said in that case, we will not broadcast this program, we won't broadcast any more either, we just closed it down. Curiously, they realized an obvious way of getting around it a few weeks later, which was that whenever Adams or any of the other Sinn Fein people came on television, they... They simply broadcast them, but their words were lip-synched by an actor. In other words, they gave their speech and somebody else spoke the words, they just literally spoke, somebody who was listening on a headphone just lip-synched what they were saying. But it was an actor, so they got around it on the grounds that technically it was not their words that were being broadcast. It was their words, but somebody else was speaking them, even though they were speaking them virtually in real time. They just had a delay, a relay of four or five seconds, and then they would have an actor lip-syncing the words to the person speaking them. The picture was of the real person speaking, and it was exactly as if they were speaking. Except that sometimes their voice was slightly out of sync with their lips, which you sometimes experience that anyway with. And, you know, you weren't actually quite seeing them in real time, it was three or four seconds behind her. But this became, you know, the whole thing became so ludicrous, because they could get around the legislation so easily that way. And Thatcher was just made to look a complete laughing stock. Because short of arresting all of the directors and the producers of the television company...

1:52:30 Which is not the sort of thing that can happen in a Western European country at this stage in history, or that stage. There was nothing they could do, so the thing just became a complete mockery, and in the end they had to abandon it and retract the legislation. So I'll try to make one of those. Yes. Well, it sounds so exciting. I'd love to come. I really would. I must ask John or Richard to go along. But you say you're recording it anyway. I'm still thinking of a format, how it will be. Will they first give their speech? Then they will attack each other, explaining why, or debate in pairs, and then questions from me, the host, the audience, and the final speech. Do you actually have the people lined up who are going to speak so far? I have candidates. Although Britain is one of the countries which produced some original constructivisms, like Turing or Goodstein. I can't find a British constructivist in Britain. I wonder who we would think of. So, at the moment, I... Have you spoken to John? He must know a few people who are convinced constructivists. Well, curiously, I mean, Moshe's a bit long in the tooth now for that sort of thing, though, isn't he? He's retired now. I mean, he was quite a strong constructivist. He was both an intuitionist and a constructivist. This guy, he was at the LSE, the model therapist who worked with John Bell. Then a Platonist will be somebody from Bristol philosophy department. Is a pluralist just a polite way of saying somebody who doesn't know what they believe, or is it somebody who believes that there are multiple concepts? There are different versions, but there are more or less rigid concepts of where you can end up. One of the versions I will draw to you in a moment. I'm looking forward to this right now. But there are other versions as well. I think we probably want to make a move to the restaurant fairly soon. I think we probably should, yes. Are we going to the station? Okay, I should go with them. Anyhow, look, I may...

1:55:00 I may send you an email at some point. As I said, I'm not a categorist, I'm a working mathematician, but I'm intrigued by all of this and I have a real interest in foundations. Also, Michael McQuillan, you may know him. He mentioned he visited Pasadena Buffalo a few months ago. And he mentioned that he told me that I should... Anyhow, he's a number theorist, geologist, who's a student of Barry Mazur's, but he's highly interesting. I don't know, I don't think I have a category. Anyhow, but he told me, he told me I should not miss the opportunity. Where do you live? Which way to the street is it? Down there. Where? Straight up there. Okay. We went to back in the city of Cambridgeshire. I should go down there. You're in the city centre. It's pretty much that direction precisely. Where have you guys got to get back to? We're going back to Coventry. To Coventry? Well, John is going back to Oxford. Adam and I will be going back to Coventry. All right. Thank you. Thank you. Thank you. Thank you. Thank you. Thank you. Thank you. Thank you. Thank you. Thank you. Thank you. Thank you. Thank you. That's great. And then a short bus ride back to Cambridge. Good. So long. It was good to see you again. It was good to see you again. Yeah, yeah, yeah. Thank you. Sorry, I never worked with you. I'm Daniel. I'm Mike Wright. Very nice to meet you. Nice to meet you. Sorry I didn't get a chance to talk more. Nice to meet you, too. All right, cheers. He's still walking his way to sleep. We learned that from that hour. Though I say it myself, I think I did bloody well to get him there at 2 o'clock. Oh really, was that actually? Yeah, because Matthias just kept about 45 minutes every 10 hours in the street. He'd think of something that Matthias had said to him and that he wanted to... So we're having a discussion about partial, you know, basically the stuff about partial differential equations, how one ought to define manifolds really in the right category for defining manifolds in the way that would really allow you to do all the things with synthetic differential geometry you like to do in, not just in mechanics, but dynamics as well. No, it's just that, you know, Bill was having new ideas every 10 yards or so, but of course he does tend to stop and then of course there's 10, 15 minute conversation. I was just cutting in easy, just steady stages down towards the corner of Tyndall Road, Tyndall Avenue rather, and just hoping that to get him around the corner into the Wills building.

1:57:30 By 2.30, but I think as it was, we did quite well. It's okay, do you think the numbers work? Oh yes, I think it was fine, it was fine, it was fine. No, not at all, not at all, no, no, it was perfectly respectable. I mean, obviously it wasn't a sellout, but it was perfectly respectable. No, it was a perfectly respectable audience. No, no, no, seriously, I'm not, I'm not, no, no, no, you really should not have any worries or qualms at all on that score. In 1998, after we had the big conference in Bolzano with Bill and Colin and various other people, Toro Alberto Peruzzi organized a meeting in Florence, which he had advertised for about six weeks beforehand. There are precisely three of us in the audience when he's got to speak, Alberto, John Bell, myself, and John, four people, I'm not counting myself in it, I'm using myself as the fix point because I don't even count as a mathematician, I mean that was pretty shabby, I think Florence should really be ashamed of itself. Really a shame to itself. And it was actually one of the most interesting talks I've ever heard in Gears. It was all about this business about lexist, lextensive, they're left exact in extensive categories. And again, this whole business about intensive and extensive quantity. And it wasn't, but the trouble is that Italian philosophers, except for Peruzzi, which is what makes him so exceptional, I really do tend to be saddened by their own art. They have these people in their department at Florence who do this incredibly baroque cryptic semantics for quantum logics and modal logics, but they just have no glimmerings of what category theories amount, and they obviously just kind of looked and said it doesn't interest us. The second talk he gave, because it had dialectics in the title, that got a much bigger audience. That got about, you know, 50 or 30 or 40 people. Well, I think it was a great talk and a great discussion.

2:00:00 Yes, it's all captured, all immortalized, all in the great archive, which one day, I hope to God, will be actually digitized and put online. Well, most of it, a lot of it. Well, this is, but the one in 1998 was not actually. I was still using audio cassettes at that time. Would this one be able to go up recently? This one should go up. If you have a site you want to put it on, then there's no reason it shouldn't be done. Oh, it can be put directly on the site? In principle, yes. I can then see why not. Oh, right. I thought you would want me to... Well, I can just turn it into MP3 files. Oh, no. I didn't think technically it would be a way. I wondered whether you would want to... Well, I quite... Well, it's for Bill to decide. It's his talk. If he feels happy with it, and you feel happy with it, of course I'd have no problem with it at all. I mean, if you could just put a little sort of note about the existence of the archive on the site, that would be more than welcome. No, no problem at all. That's the whole reason I do these things. Get them out into the... No, no, of course. No, no. In fact, the sooner the better as far as I'm concerned. I'd be all for that. And I can also give you the notes of the... Because something like that, you really do need the notes as well. I've got an extremely good set of notes, so I'm still... Well, actually, once you listen to the talk, I'm sure I can put them in order. That's the one problem. Bill tends to cover the board in slightly higgledy-piggledy fashion. Hang on, are we going the right way? We are going the right way, aren't we? What? We're going the right way. We're going the way I thought we should. We could go anywhere we want. Well, if we're going to this Indian restaurant, then, presumably... Why is this... I don't think it's a very good plan. It's a lovely evening, isn't it? It is. Absolutely gorgeous. Hi comrade, you okay? Irina has left, Irina left. Irina's not coming to dinner with us? I thought she might come. Apparently not, she went off with Andre. Oh, oh. Oh no, but Andre's coming back later. He's going to the place where I'm staying because he's desperate to use an internet connection. So I gave him the key to...

2:02:30 Richard's flat, which is where I'm staying, because there's a computer there. For some reason he had to send some, he had to do some email rather urgently to do something that he's organising in Paris and various people he has to contact about it. I think he's coming on later, assuming he knows the restaurant where we are. That was great. Was it all right? Absolutely, absolutely, yes, it was a brilliant talk. So it was disorganized toward the end? Well, no, I thought it was actually the beginning that I wasn't terribly sure where it was going to go, but it was after about ten minutes I thought you were absolutely firmly in it. Thank you for watching. ...like a rocket. No, no, no, it was absolutely terrific and on the contrary, I thought they, you know, got more and more superbly focused and it's all going to fall into place again and see why you don't need growth in big universes. You can do everything inside, so you see once you've understood the whole issue about large and small. No, it's terrific. It's a bit disappointing discussion. I'd hope that there might be a more broad-ranging discussion on precisely those issues, on the, rather than, I mean, this young Russian guy is obviously very bright, but he's, you know, he's in the Friedman camp, he obviously does this reverse mathematics, and the young Russian guy that's, that asked all the questions. Yeah, yeah, well. Yeah, yeah, I know, extreme fanatism, yes, that's what I gathered. He's been telling me, he's been offering to draw me pictures all evening about the nature of the horizon.