Discussions, incl. FW Lawvere, G Khatcherian & M Wright (contd.)

0:00 Something very interesting has come, Bill. You know Moishe's letter? Thank you. Thank you very much for the reply. John Mabry's reply is not intuitive. It's level one. So you better read that. Well, before you do, there's a matter of fact. It's one of the things I'm going to get in front of you. If you have one, you can't believe the name. I thought the extensions of John Mayweather and Hobbes were very useful, and I think he's the movie writer in quite a long way. Do you know these two things? Right, they all came in 12. Oh, they all came in 12. I think, don't disrespect the quality of promotion, but not an entirely fair source. Right, this is the first one, and this is the last one, so you should be able to see the last one. This became the galaxy in the 50s. And this is the, this is actually the motivation of the game, of the game, of the game, of the game, of the game, of the game, of the game, of the game, of the game, of the game, of the game, of the game, of the game, of the game, of the game, of the game, of the game, of the game, of the game, of the game, of the game, of the game, of the game, of the game, of the game, of the game, of the game, of the game, of the game, of the game, of the game, of the game, of the game, of the game, of the game, of the game, of the game, of the game, of the game, of the game, of the game, of the game, of the game, of the game, of the game, of the game, of the game, of the game, of the game, of the game, of the game, of the game, of the game, of

15:00 I'm trying just, at the moment I'm just trying to go by four minutes. You can actually, projections, you can actually recuperate and and all, in the case of the author. You can define and in terms of the, a bit more, there are my, ah, there would be interesting. These would be interesting. Let me show you one or two nasty ones, okay. Things don't just come out because we... You can see one of your... I was viewing some extra information that sides up with those... Oh, sure. I'm sorry I inflicted this on you. It was very selfish. I just wanted your reaction to... It's odd that these guys never sent me any of this. I find it bizarre, yeah. I mean, considering that you're the kind of lynchpin of his whole... Exactly. He talks about my... Well, I think that's in the BJPS. That's indeed in that one. he also seems to think he knows a lot about what was going on in your head yes but the idea of uh what was it but this idea never occurred to law here

17:30 that's right yes the idea of designing intuitive schematic methods simply never entered his head But he says you wrote in thinking. His neurons were, you know... Yeah, I think it's strange. Well, now I think you do, yeah. Yeah, well, is he a big guru? No, no, he's not a guru at all. He's a logician at Bristol. He's done... He's really just a straight logician. Well, he's actually in the math department, and he's published quite a lot of stuff in JSL and mainstream logic. If we attend... He's quoting me in a minute and says, if we attend to the literal meanings of the words... Rather than really allowing them to roll sonorously off the tongue, we find that the statement as it stands is simply quite false. This is your words in... You're referring to something. Yes, yes, yeah, yeah, yeah. Yeah, I think it's probably slightly unethical that he didn't send you that after all the time for this. But having said that, I think... Well, why wasn't I invited to this discussion in 1980? I was in... Oh, this is the discussion he had with John Bell at Chelsea, which I missed, unfortunately. Yeah, which I really, which in fact I didn't even know about. I was mad as hell that I missed that. Yeah, I think that's disgraceful that you weren't invited. But anyway, at least I got him to Cambridge. Now let's get back to... But you see how it all comes out of the ontological, these understanding, quick understanding of these ontological categories. Bell, Mayberry, and Scott were the three All-Americans. Yeah, and you. It's very strange. Oh no, this is the, yeah. This was the thing, you know, I was talking about when I was saying that you... Well, this is the non-associativity, right? Yeah, this is the non-associativity. If you take the case where X equals A... We can solve that except that I can't... That's the case X equals A. It is on hold, though. This might be stupid, it's just a... Yeah, at least you react, you sit down and you play around with it. This could be important for you to know how these adjoints compose.

20:00 Because either the composition is represented by another element or else it's an interesting new adjoint. I see, I didn't, I didn't, I haven't pursued that to be frank. So if it came of this form, well it's tidy but not all that interesting. Well, but in any case it's another adjoint. So we first project onto B and project onto A. I don't have to sit down and draw a few figures and things like that. The funny things happen once you project. Unless you've got orthogonal projections, strange things, that's not fun to live, okay? Then you can come up with a theorem like this, which again, I am missing maybe one more clue. Where was it? I couldn't even read my clue. This is sad, but in the end, this automodularity comes out in a statement like that, a vector, or any subspace actually, could be split into its... I don't know whether there is any collection with Fontana. I've never looked into Fontana. I haven't. It sounds like a form of something they're studying, while this at least has got a connection with the geometry of it. Okay, I'll leave it within, at least.

22:30 Well, no, I think this is really a very different approach from the quantile approach, isn't it? Yeah, this is much more geometry-based and, I think, much more appealing from what I've seen of it. Not that I've studied the quantile stuff yet, but from what you were telling me about the quantiles. It's true, the quantiles. We don't, we can't throw this one away. But, I mean, you can, maybe you can alter a little bit of this stuff. I tried to relax, but I couldn't. I've spent a few afternoons. Okay, at least it sets my heart at rest because I've spent a few years in this damn thing and I'm not gonna pursue it. Why the hell not, Jerry? Why the hell not? Come on. We're not going to ask you to get on with it. I mean, you people who can do these things, it really makes me... I can't tell you how it upsets me, somebody that would so much like to be able to, when I hear somebody say that. Read something, discuss it, see the various points of view. I can only see my own tunnel window view, and I take a long time to remember. I'd swap places with you any day, Gerry. No, no, you'd swap places with him all night. Well, that's true. I go for one week to teach a package. In computer, so I read a manual. This is gone. I mean, it's blank tab. I mean, you're not that bad. You're a bit better than that. There's not more information. Now, this one is on co-hating stuff, yeah? It's on the fact that hiding and co-hating, then the relationship between them. Yeah, well, obviously he couldn't have called it that in 1887. He had it already. He didn't have hiding. Yeah. Co-hiding had not yet been born. I was going to ask you. So this is a very interesting paper in both respects. He shows clearly what at that time was the procedure, and he has the same computation.

25:00 There's another thing that they attribute to me about if you have both subtraction and implication. Oh, it's in there. Go on. If you have both subtraction and implication, you have... Well, then, you could have examples with both. For example, any linearly ordered set is automatically both, both hiding and colliding. But on the other hand, a linearly ordered set is Boolean only if it has just two elements. But more generally... More generally, if you have both subtraction and negation, so you have two negations, A implies false or true minus A, which is just. So these things are equal if and only if it's Boolean. So there's a nice way to see Boolean algebra as a special case. And of a special case, essentially, of a geometrical structure. Yes, everything defined with adjoins. Everything defined with adjoins. Well, not just everything defined with adjoins, but everything defined in a kind of intuitively very geometrical way. Yes, they have come up as an example in this. I think you've cited it here. No, all I was saying is if you want to look at those categories, you want to look at the metric spaces paper, because we don't know if they're living here. Oh, yeah, that's what I've been doing. That part, I don't know if I can tell you if it's living here. Yeah, yeah, this is tough for beginners. Metric spaces paper is very good, very helpful, very, very helpful. It's very formal. It's very helpful. If you go through the first two series, then you can start to see some examples of that type of mathematics. Well, excuse me, it's all of that, but it's a heavy going... Just start with the examples. You know, you already know vaguely what it should be, so look at the examples, and then you can understand the axioms. Yeah. They have several interesting examples. The one example is this that they attribute to me.

27:30 That is, with both the vocation and subtraction, you have only a two negation degree. So I was very proud about this, you see, for many years, after all, I find some discovery in my book. Then last year, when studying Grassmann, I looked through Mathematica in Ireland, and just by chance, I found the paper of Schroeder, who refers also to Grassmann, so that was interesting for our research in that part. He's got exactly the same calculation that I did in 1963. But all the same, history of logic footnote, how revealing that this... The work of Schroeder is so neglected because, you know, Frege stands into the historians of logic as this great figure blocking, completely distorting their view of the history of logic. Completely destroying and distorting the history of logic. Overshadowing Schroeder. Overshadowing Schroeder regarding, you know, Schroeder as just a kind of Pali-German imitation of Bohler, people working on completely the wrong, because they were trying to get logic from mathematics, and that wasn't the way to go. It's Frege who sets the problematics, defining everything in terms of... No, and this is the attitude. I mean, the fact that people like, people, the fact that people like Crispin Wright still hung up on this Neo-Fregean program, the people who think they're doing philosophy with mathematics today, it's still hung up on this perspective. I mean, really, this is why I got you to Cambridge, Bill. I mean, to get out a book that would destroy this rotten... There's no other way of looking at the history of philosophy and mathematics apart from anything else. Precisely in the Bismarck period, right? Yeah, yeah, yes. Anti-socialism. Well, Frege was a great, I guess you probably know, I mean, Frege was an extreme rabid anti-semite, francophobe, and a great supporter of petty-minded Prussian conservatives. Yes, no, I know, I know, but I'm just saying it was really interesting. So that's a good law, man. The hating algebra is the Berlin algebra. It's the only way is also the co-hating algebra. The notions of negation among mittels, which is the subject, they may coincide. Great, okay, well, Schroeder-Lorvier-Kachurian, let it be known as the Schroeder-Lorvier-Kachurian, isn't it?

30:00 You mentioned the logical operating algorithm. Yeah, this is the thing about the logical subtraction operator of the joint two implications. You must have known that. Well, it's the essence of the boundary operator. Boundary operator is a pure logical operator. It's already there. But there's one remark you make in introducing that. You say that this comes out of thinking of A more general kind of generalization of domains of variation than that are not determined by a single topos of variable quantities. Now, what was behind that? And you actually make the comparison. We say like rather like like Grassmannian manifolds that are not determined by a single affine line. Had you already been looking at the outstandings layer at that time? No, no. You see, I mean, the status of Grassmann is that geometers use thousands of times a day the Grassmann algebras, the Grassmann manifold, without ever knowing where it came from. So it's just a common... it's the basic example of a manifold which is not affine, you see. Yeah, yeah. There are many such examples, but it's the most basic. Projective space is a special case of it. It's simply this. You take a vector space and then you consider all subspaces of dimension k for a fixed k. So the set of all subspaces is itself a space because you have a notion of smoothly varying. One k-dimensional space can be smoothly varied into another. So if you have a notion of smooth variation, you have a space. It isn't just an abstract set. It's another example of the fact that if you give something a so-called set theoretic definition, those things that satisfy a certain property, such as k-dimensional linear subspace of the given linear space, it's automatically not an abstract set, but again a space. There's a notion of tangent vector, there's a notion of motion, and so on. So that's the Grossmannian metaphor. It is just the...

32:30 Yeah, but it was the fact that that prompted the analogy with... So for K equals one, you get projective space. The space of all lines in a given individual space is an n minus one dimensional projective space. If you want an example of a non-affine space, then simplest one is projective space, but Grasman is slightly more general because the kind of subspaces that you take could be K-dimensional instead of one-dimensional, So in algebraic geometry, this is the first example, you know, of a non-math plan. But the point is, the basic point is that the topos are found somewhere lately. Well, if you need to... In my notes about logic, no, sheaves large and small. Did you have that? No, no. Oh, God, I want to get held back. The lectures that I gave for topology... No, no, no, you never... I never heard about that. ...in April. So please send me, but go on. Well, go on with what you're going to say. The point is that topology, education, topology is now so fragmented, you see, they're brought up to work only on hard problems and become great, you see, but no general theory at all. So we have very good topologists, young ones at Buffalo, who work on three-dimensional topology, bunker A conjecture, a very difficult thing, but they don't even know what a sheaf is. The basic tool of algebraic topology, well, they heard about it, but they never learned it systematically or so. So they were having a seminar in which they found they needed this, so they invited me to give an explanation of what they achieved. So that was the origin of it. I wrote it up then, and so it's a set of... I gave a copy to Tierney and he didn't give it back, it was supposed to come. Well, but the point is it starts out by saying that a topos, what is a topos? A Rotundi topos. It's an algebra of continuous functions, continuous set-valued functions, on a kind of situation that's called a topos.

35:00 In other words, the situation is the topos, but for some, by some confusion, we call this algebra of functions also topos, which is really just B. The only thing is we don't know what is a topos, so we work only with the algebra. But this is exactly the situation in algebraic geometry. You have the ring of complex value functions, and that is the way that you get at the space. So more general spaces are gotten by taking the dual of the category of rings and looking at sheaves on that, and largely do the same with toposes. Take the category of all toposes. Sheaves on that. So you get some generalized topology which will, for example, represent or classify more general kinds of theories than just geometry. But the ordinary topology... So that's what I was... I'd very much like to see that. So there's this general speculation, but then there's also the idea that there may be some examples of theories which are not geometrical theories that could be represented that way. So the idea would be that you have some kind of, first of all the topos is some kind of concept of models or something, a situation, but now you could have two of those glued together to form a more general space, which was not itself determined by a single global function. Topos functions on this part, that is central to functions. Topos have set value functions on this part, and the knowledge of how to glue these together, getting a third smaller, a fourth smaller topos, but there's no single big topos which is containing all that information. It's certainly... Just as, just as, just as projective space... Yes, yes. Or more generally, the Ashman manifold is a, is a union of two affine spaces. But itself only has constant global functions, so the ring of global functions is no good to determine that space. But, you know, how do they do that? Say the projective lines, it's described by AX plus BY, where A or B is not zero.

37:30 So there are two parts, where A is not zero. So each of those is an ordinary half line. But then they overlap, and both are not zero, and so that gives a point of infinity. So you have these two simple things, which are determined by their ring of global functions, overlapping, give it a new space, which is a real space, but it's not determined by its global ring of global functions, because in fact they're all constant even. Because in algebraic geometry the functions are sort of polynomial functions, and there's no polynomial function that is global, except constant, even analytic function, there's no, there's no, this is speculation, but maybe there's some example which would fit this, something like... Yeah, with topos. Yeah, yeah, if there is, then it really finally lays to rest. Good question. I mean, this is a very minor corollary, but I think it really does give exactly the construction that these people like Griffin have been trying to find a semantics-relative identity. I think so. It finally lays to rest the idea that you have to have a domain of variables the same or different absolutely in order to have a decent semantics for the quantifiers. Well, in a sense, what would quantifiers, which are after all just left and right can extensions, be over a space like that? How would you define it? I mean, I'm trying to think of it as a kind of logic. The example I had in mind was Peter Johnstone's idea that he didn't really know the synonyms of these things. Yeah, this is a very interesting speculation. You can say that again. I mean, I can say that again myself.

40:00 But that, you see, that is a very natural setting for thinking about the relative alignment of it. You remember Applegate Piano years ago had a paper on modern... Simplition, topology, the model objects and common myths. In the category of... I have to read this because I can't remember. If it's the K-model and K-linien maps, the model objects are you grasping the theory of. The model objects are... that was the point I wanted to make. Say that again, Jack. Well, how long is that? Is it a long passage? I don't know. I had the brain... ...applicate an attorney to find a gross implicit role. Was the same thing just with the exterior algebra of differential forms or not? You're not in simplicity of topology. No, I don't. I don't know any simplicity of topology, I'm afraid. The triangulation of the spaces that are all going together. Well, yeah, I mean, very qualitatively, only very roughly I know the idea. It's a similar procedure, where you start off, you sort of model objects and then you do them. The monomodal objects are also... So, modal objects, yeah. Well, what shall I call them? You probably... Generic figures. Uh-huh, uh-huh. Yeah, okay. Yeah. Well, I propose that the last one, exterior algebra, is all the underlying modules. Yeah. The category of K-mobiles. The underlying modules of the algebra. Actually, there was a question I'd like to ask you. You made a commitment that algebraic theories and all these things come back to hunting. Yeah. Well... You made a commitment that click with algebras... ...obtainable by joints is only to generalize the idea of the inner product into...

42:30 Yeah, yeah. ...the ring of an algebra, not necessarily the spirit, yeah. I spent a few evenings and I couldn't work out... I'm not saying that... What I was saying, do you have any details, Dr. Clarkson, of the way in which... I don't know this paper. It was a... I mean, you see, this paper is mind-filled with comments. Yeah, I know. This man throws out so many ideas. Nothing left unsaid. By this time he's moved on to... So I want to draw one, just to get some satisfaction. I just want to not work it out. Is this complete? It's a stupid remark, but I'm talking to the... Oh, yes, this is one thing we must talk about while we're with Gerry tonight. Look, sorry, I have to say this, because while you're here, you can tell us much more about this. While we're having dinner, and I don't know about you, but I would quite like to go through the questions. Yeah, are we going out to eat? Yeah, right, okay. Well, when we do, we go out to a good place in Hatch End. Well, you can come back with a year afterwards. But what I was going to say, there is a... The derivation of the CCRs, this whole intensive-extensive quantity, is pulling forward and turns onto an extensive quantity, and the various instances of that constructions, canonical commutation relations, just like the Heisenberg relations in quantum mechanics, this is the standard expression for this. Okay, right, well CCRs. Well anyway, this is just, Bill is saying in his Cambridge lectures, And again, he explained it to me very carefully the other night in Bangor, but I was too pissed to take in. I was falling asleep, I'm afraid. But anyway, and so he very kindly promised to go over it again with me, and I've got the notes he made then, so we'll go over them. But it occurred to me, and this may be, it's probably, this is just... An idea that's based on the insufficient understanding of mathematics, but that what Basil Hiley was trying to do a few years ago with Frescura, this taking co-gredient and contra-gredient algebras of what?

45:00 He proved the theorem for all n, while it was proved for all n by someone in 1955. But you sidetracked. Okay, so the guy's an unscrupulous fascist. He's not only a fascist, but an unscrupulous fascist. Okay, but I wish I hadn't even mentioned his name, but he just happens to appear as the co-author with Basil, so, you know, so I give credit. To get to the point, was he? Okay, I believe you. They may still be on to something. But it seems to me that what they're on to is exactly an instance of this construction that Bill is talking about, the underlying idea of pulling forward intensively onto extensive quantities, and the Burnside rig is an example. I want to have a look at that, and I want to try to understand it, but they did it in terms of algebra of differential forms, They did it also in terms of left and right-sided ideals, and they suggested that it was a model for, well, actually, I think the whole point was that they thought they could get CCRs out of it, because the creation and annihilation operators actually come out, you know, pop out of this as an anti-communication. Yeah, exactly. Well, supersymmetry comes out of it as well. I mean, it is itself another version of the supersymmetry algebra, but the thing is, it's done in terms of differential forms rather than in group theoretic terms. And they related, I say they, I mean, Basil Talley, I think, who was very interested in the connection with Grassman, because he published that historical paper about the spinner and algebras. Well, it was interesting because he was making the point that everybody thinks, because of the late 19th century, because of Klein and Gibbs, everybody thinks of Grasmann algebras in terms of vector.

47:30 But in fact that's Grassman on a given underlying metric space, but Grassman had this much deeper, well he says in the paper, a much deeper way of thinking of them, which of course he thinks ties up with Boehm's ideas about implicated order, but I don't think you necessarily have to buy that software package to be interested in this construction, at least I hope to God if you do then I'm backing off. I'd like to watch John Doerling's terminology that I thought of another one. Computer virus. Now say to him, thot, which is the first. Promptly hate gravity everywhere it's short-circuited. I think Implicit Order is definitely a computer virus. It appears that it starts reproducing itself in short circuits. Somebody said the difference as between the two MacLeans, Saunders and Shirley. This is definitely more likely to appeal to Shirley than to Saunders. Well, you know that Shirley MacLean actually turned up at that department. Apparently, was it you who told me that? Apparently, Shirley MacLaine actually turned up at Birkbeck to talk to Bern, you know, because she had read about this wonderful thing, the Implicit Order, and Buddhism, and Holism, and she said, well, then here is this world. She'd been told by Newsweek that he was a world-famous physicist. Thank you for watching. Took her out to dinner, apparently. He's a lovely man, actually. Well, that in a sense, he'd make her write out a cheque for half a million dollars for the physics department at Birkbeck while he was about it. All I heard after I left Birkbeck, there was 300 pounds every year allotted to every student who traveled, and I never got a penny.

50:00 I was never told that I needed money. Yeah, I know, I've never met Francis Prescott in my life, I know nothing about him, except what Kylie's told me, which is all bad, that he now worked for the South African government. Yeah, this guy, yeah. He was a Christian Scientologist also. Christian Scientology, I wouldn't believe anything. Well, I never knew him. I don't know how much he did know. Do you have all this? Do you know all this? Yeah, I said to Fatima what he actually needs. What he needs is a soundtrack consisting of a couple of guys with kind of sound mics who pull him around every minute of the day. Well, for that paper, come to Italy, we come and support you. For that paper, it was just a passing example. Yes, I know. I should have, because there are several things. There are a lot of people who write, who have one idea and write eight papers. He has eight ideas. There are people who have one-eighth of an idea and publish eighty papers on it. And there are people who have eighty ideas and he publishes about one-eighth of a paper. Well, okay, leave this stuff. I would like to talk to you a bit about this tomorrow, though, if you've got any patience. It helps me clarify my ideas by arguing against Mabry. I just happened to find that weakness of me a good way of clarifying my own ideas, to argue against him.

52:30 Because I think he is a clear writer. That you can't take away from him. He's a clear writer. He pins down exactly where the, you know, the commitments are, the metaphysical commitments are, particularly what he says about the popularity of objects and how he thinks about them. Let's go and then we can come back. Yeah. Yeah, sure. Come on. Yeah, actually the conclusions seem to be pretty well, he's won as, well there's a big concession at the beginning, I didn't, obviously didn't talk about that. It's, I've reached my provisional conclusions, now where is it? This is the logical point. Yes, Lorvier is not a structuralist in your sense. He has given us a mathematically precise alternative to the conventional Bulbachian structuralist technique of handling abstractions. I do not think it is so much a matter of abandoning Bulbachian structuralism as of augmenting it. But that is to speak of the level of logic and foundations. As a mathematical approach to the problem of dealing with abstractions, it has, as I am now beginning to see, the most revolutionary implications for mathematical practice. Who is this? You. This is Babe Ruth. Oh, he wrote it recently? Yeah, he wrote it yesterday. It's a letter. Well, at least the man is not blind. No, no. Not at all. Ah, Stalin, Joseph. J. Vissarionovitch, J. V. Koba, you don't take on and defeat Nazism and Fascism in four years without the...

55:00 You think it's just a Hanna-Balloon? It's a way to, it's a road toward attacking. You're aware that Lenin had a bit of friction with the man? Yeah, yeah. Is this again blown up? No, very much blown up, yes. You see, this is interesting. He's trying to understand... Ah, but I don't think he has understood what you have in mind by now. What is that idea? Yeah, hang on. The distinction you make between formalized set theory and ordinary mathematics as expanded by Bourbaki is absolutely crucial. At this fundamental level, the distinctions between set, collection, domain, etc. are, as you point out, superfluous. There is one intuitive idea and one intuitive idea alone here, whatever you call it. What is that idea? It is, I am convinced, the ancient notion of arithmos, as analyzed by Aristotle and employed by Euclid, and arithmos is a plurality of determinants size composed of units, monads, and a unit is, according to Euclid, just that in comparison with which each of the entities of the kind under consideration is called one. In my example, a mena The set, in Lorvier's sense, may contain or give rise to Arithmoi sets in several ways, depending on how the unit is chosen. Thus, a heap of shoes contains both the set of six pairs and the set of twelve shoes. In the former case, the unit is a pair of shoes, in the latter a shoe. Well, I think he's, you know, he has understood an important part of Menger, but he hasn't, I think, understood the point about motion, has he, about cohesion, or domain of, I mean, about space. Well, that's because if you take the actual pairs of shoes, you see, that means that they cohere, whereas something apart from two disparate pairs doesn't cohere. Ah, right, and that's, of course, the point about the failure of maximum choice. Well, no, I mean, the counter, you see, you have this... This heap of, was it six shoes, right? Twelve shoes. Twelve, all right, twelve.

57:30 So the maps from the generic pair of shoes must preserve leftness and rightness, you see. You don't get twelve squared maps, you get fewer. Yeah, yeah, yes, I see, yes. Yes, actually, I was, I'm sorry, I was thinking of something completely different. It's a choice of unit, don't you think? Yes, yes. A generic figure is a choice of unit, so it gives rise to, for every object, an arithmos of one case of shoes and the other case of pairs of shoes, which are things together. Now actually this is a good lesson. There is a rival to this arithmetical notion of set as a plurality of determinate size, namely the logical notion of set as the extension of a concept, but as we know Only too well, that notion is fraught with difficulty. It is the notion employed by Dedekind in Verzindeln, Verzollendisalen, and by Frege in the Grundgesetz. And it was in his review of Frege's Grundlagen that Cantor put his finger on the difficulty. Only when the objects falling under a concept have a determinate power, cardinality, finite or transminite, can the concept even be said to have an extension. I'm not sure that's right because I think there are ways of thinking about extension. I mean, certainly when you look at the, I'm thinking of the work, John's work on orthomathesis, it does that. You don't have to have, I'm not sure that's right, you don't have to have an extension in that sense of a concept, or you can recover set intersection, that's the special case, where you have that kind of compatibility machine. There should be an extensive quantity, which... Yeah, this is it. That's what he's saying. Yeah, he's saying there should be, yes. Extensive quantity, which is really... If the extension can't be quantified at all, then there's some doubt if it is an extension. That quantifying would be an extension point. Well, you've got him with pretty good billing here.

1:00:00 Well, you know, he's learnt a lot from that week. Well, he's decided there's more to it than just sonorous words now. Well, a lot more. He obviously seems to mean it. You see, here am I a poor, dumb, woody philosopher. I can only think about problems at the level that somebody like Mabry thinks about them. I can't follow you sawing eagles into their career. What's that? It's very important. Is it? Oh, yeah. Well, we'll come back after... Ross Street is using this for systematic... Does he know that? ...dimensional categories. I think... I'm guessing what one it is, but what is it? It's, it's, er, but it's tea. When we come back it's gonna be tea. I'll be fine. Um, do you want to look down to the station, then? Yeah, sure. Er, well, hang on, wait a minute, wait a minute. Oh, God, I won't have clever, how much are they? What, for cigarettes? It's only 7p. Eight. Right, that's it. Yeah, right. Oh, what I have got is a, is a, is a... Capital card? No, I don't have one. Oh, you have your own?