Discussions, incl. FW Lawvere — various topics

0:00 I'm sorry, I'm afraid I spilt coffee on your working notes.

10:00 That's fine, no problem. They're still quite legible. I was going to ask you another thing, if it's okay. Sure. This is absolutely fascinating. Ten minutes' tutorial from you is worth three years' study at almost anybody else. The business you were talking about at the end, in what sense? Maybe type table and power types. There's obviously an infinite intersection available in calculus with, you know, one more thing, which any methods cannot subject. I hope so I can see, you know, where that comes out of me. Now, what was this business that you were talking about six years ago now in Florence about Grassmans?

12:30 Precisely, you know, put in this setting, seems to avoid this. Grassmann's definition doesn't define it to a challenge, but it defines it objectively as what an arbitrary natural number ought to be if something is reached just by what it does. Right. It's very similar to church knowledge. Oh, I see. It's just the idea of it. Well, I think Grassmann didn't put it this way. Well, giving it a 20th century spin is no problem. The sort of thing he had in mind is that in any category... Any object, any endo-map of that object, you could imagine assigning another endo-map on the same object, a process which is defined for all categories, for all objects, and then an endo-map for those objects. So there's a sort of two-fold naturality that functors from one category to another, maps inside the category that are compatible with the given, and should be compatible with this new assignment. So precisely, functorality... Well, it's kind of a natural order, but you could also say it's functional as a process, so such a process is a natural number. Obviously, 3 or 5 or 37, these are such things because you can iterate and then go back to the entire thing. On the other hand, this is an objective definition, which doesn't say the thing is real necessarily in a particular way.

15:00 Thank you for your time, and I hope to see you again soon. ...Iteration time. Yes, essentially. Capturing the essence of iteration, which is what you need to do, and that's what Dedekind does also, because it's fundamentally illusory, there is no such thing as what the constructivists claim is the only thing that exists, precisely what doesn't exist, there's never going to be any way that you... As is, you know, masterfully, Vatican's very construction, I've always thought. See, really, in a masterful way, and he shows it, but people never take it seriously. Intersecting all the things that contain this one, all the closed things that contain something. It's defining it by squeezing on it. From above, conceptually, from all possible other objects, it's got nothing to do with the building. You just sort of imagine that it does. And the fact that this is a mere imagination is shown, as I said, by the existence of non-standard things. If you change the categories, you can cease to be universal. You get the infinite intersection constructed within any given category, but the categories themselves, the things in the categories may not be the same. So there's no unrestricted notion of common equivalence. The object that is the alleged completion of the subjective process usually would be defined in a given category. The categories can be compared by using elliptic morphisms even logically. I think that constructivism is a grand illusion. Even topos theorists talk about how topos theories, constructives, set theories, etc. Just in order to cash in on the alleged ocularity of this slogan, the sense in which they are using it, constructives, is totally different.

17:30 Even, you see, the confusion was that the whole idea of hiding the algebra itself has nothing to do with mathematics. That was just the first example of something which is, you know, the thought is moving. Then you're going to get hiding algebras instead of Boolean algebras, and that's an example which gave rise to the notion, the notion itself had nothing to do with it. Yeah, yeah, it captures the specific determination of variation. Yeah, and the result of variation is not specific. So there's this whole... No, I completely take your point about... ...improvisitivity and all these slogans and so on. And again, they cash in on the computer or something, well, this is closer to computers than the canonical structure of the universe. Because computers, even if brains can dream about it, computers can achieve it. Thank you for your attention. Well, it goes back to the introduction of the quote-unquote natural numbers. Yes. This is actually a point that John Mabry likes quite nicely in his book. Ah, really? Hmm, hmm. Calling them natural. Yes, yes. But on the contrary, they were pretty much Shindenzen. Then what's the product of the adventure? Yeah, I mean around the turn of the last century there was Borel, for example, a French mathematician. There's a whole circle of some kind of way on the vague side of countable relativity, but somehow you see that the countable is more real than the countable.

20:00 Because that way you'd be able to get the structure of all the large things inside the small. Yes, and so you're thinking of imposing structure, as you say, on the continuum by giving this condition that you've got in the intersection layer, which you obviously have now, as we've passed that, but you think of that as coming from the imposition of a particular structure on the, or emphasising a particular structure for the geometric... Subjects, things which in fact have an intrinsic value. Well, indeed, if I may be permitted to ask, the philosophical terminology, which means that the bad infinity is an instance of the objectification of the subjective. You're making an ideal model of the subjective process. This isn't a bad idea, necessarily. No, no. You need objective models. But pretend that that was the objective. Yes, then make that into something. What I might say is the objective in itself. Making objective models of subjective processes is a natural and in fact an absolutely unavoidable aspect of the idealizations we have to make to deal with. And you've made yourself fun of recursive. Functions of, for example, objectification, but the point is, is the objectification of the subject different? But you see, there's an even more basic subject of truth. So if there's a truth value object, at least, you know, with the natural properties that that should have, if it really existed, well, then you can construct, assuming this geometrical fact, an endo-map that's going to be on it. And you can construct like, see the infinite intersections are inherent in the universal property of the truth value. So, so, so, so, so idealizing to make an objective model merely of the idea of truth sometimes implies... Subtitles by the Amara.org community So we're always making idealized models of objectivity. Objective acting on itself. So my basic view is not anything about ontology in a way, but just that the idealizing objective on the subject is.

22:30 Certainly further out than objectifying the agenda. So one shouldn't reverse the order or think that one is as good as the other. In other words, the results of pain about recursive functions, for example, are likely to be rather far from anything useful. Yes. About the actual subjective process of computing. Yes, yes, yes. Then, some objective model, say, of motion or something like this, might be in terms of, in terms of... Yes. No, I see that point very strongly. I have to say that's the one point where I think that people who are... Pragmatists, or methodological pragmatists in their attitudes towards mathematical structures. I have to say I have Colin in mind, he's a great guy, but I think sometimes when I discuss this with him I do have the impression that he has a sort of opportunistic attitude to how structural mathematics works, it's all a matter of virtue. Those were no points of view from which you can learn as it were. The lesson of category theory seems to be that there is of course a reflection of things, both of objects and subjective aspects, but there's no There's no preferred starting point in deciding which of the various structures, whether one's categories of space or sets of entity one starts from. And it will just be determined from experts when we look back from the far future to see how the child got through, what led him to a deeper unification. There is, as it were, no prior way in which we can orient ourselves with respect to the key strong candidates for being the objective or subjective concept, and this, of course, is our proper concept.

25:00 Although in almost every other respect it seems to me that his point of view is very, very sound, certainly far more sound than those of the people who want to build mathematics on some forever fixed foundation, particularly obviously a fairly fixed foundation of global and internal. I think that there is a slight kind of pragmatist error in this overall feeling for the way in which mathematics fits together as a systemic and a cohesive whole. Very shrewd. Not to say that he isn't a very good math. No, no, no, but you see it's a sort of thing that we receive. But for instance, whenever I get him to try and argue, to discuss whether we should take a geometry or... ...as reflecting... ...you know, he refuses to, he refuses to apply inference to that. He doesn't see the distinction. Well, no, there are a lot of smart guys who do this, and it produces very, very interesting structures, which then turn out to connect up in all sorts of different ways, and we'll just see how it goes. And that's fine. I mean, foundations do grow, as it were, from within real mathematics, and grow from the inside out. And what may be candidates for, at least for the time being, ultimate, but not in any real sense ultimate. Ingredients of definition of concepts only come into view as it will come into focus, you know, working from the inside out, but I think that's not to say that there's no point of contact between this and an overall programme for understanding the world. Otherwise, the danger is that it just collapses into some form of pragmatism, mish-mash, and some kind of structuralist philosophy of the kind that these sort of ontic structural realists are now propounding, which really is just another variant of Platonism or Purgative Idealism, with a different name. There's a rather more sophisticated methodological package attached to it, but ultimately still just as detached from the real world.

27:30 It is many times better than most philosophies. Oh, yes. Very long time. I didn't intend it as a harsh criticism. But in the philosophical world, you see, there is this sort of genesis. John Corcoran explained it all. Better philosophers are billionaires. You can take any two things and you can make a paper in philosophy on it. Thinking and biology. Take any two things, you see. And so that's kind of what he does sometimes. He says, well, let's take the idea that maybe points are points of spaces, so we can eradicate some of that. Which is, of course, a very important insight, because it was very important to see points as points of spaces. There is a tendency, some kind of tendency, to choose those pairs which have a real relevance as part of problems. But at the same time, this tendency... ...is outside of his basic guidebook, which would, in principle, permit any two things. Yes, any two things, that's the problem. Better thinking in biology than thinking in theology, by the way, but even so... Even theology needs to be explained some more. How the hell is a thing like that possible? Well, objectively, I'd say because of the tendency to seek for generality. I mean in the metaphysical sense of the word. I'm probably coming out of this Aristotelian prediction that there has to be such a thing as first being and then the backless therefore being in some sense self-sufficient and attain the real explanation for its own existence in that sort of way that the other categories in terms of which one thinks of the world as formative and metaphysical unity have to have this kind of relation of both of them of being grounded in the further being of something else. And that's the objective source of the Theological Spectrum. And of course, the most important aspect of all is the historical and sociological explanation, which is that it provided for...

30:00 A block on the development of science, of real scientific exploration of the world, and hence a very powerful tool for the pharaohs and the Caesars and the popes and all the other people at the top of the power structure. A combination of those three things, I guess, if I was really asked to say where the hell did something like theology come from. Oh, and also, possibly, to some extent, the fact that, um, no, a fourth, a fourth, um... ...a source which we don't yet understand, which is the subjective experience of so-called mystic experience, which obviously will turn out to be explicable at the level of detailed brain chemistry and things that people have undergone, which have led them to postulate this woozy kind of all-is-one-ism. Which has its intellectual expression in the thoughts of people like Whitehead. I understand there's a big movement in theology now of the so-called process thing. God is not a being, he is a kind of all-encompassing process. He, she, it, whatever. Intelli-key is all sort of built into this kind of way of rehabilitating, but most of it is just straight, very straightforward self-serving fantasy, self-serving from the point of view of people manipulating the credulity and superstition. They're victims in order to cement their own power and all subjected in the sense of just providing a comfort blanket, I think.

32:30 There seems to be a tendency toward pragmatism deeply embedded in the American culture. That's a claim that's often been made and I've never been sure how far it is. It's just a kind of rather lazy sociological... People like Russell were very fond of making it, and I think it probably needs to be examined very carefully. It's a very crude first approximation. There may of course be a specific concrete historical explanation for this, which in fact you yourself have talked about, in that there were people in the metaphysical part of Harvard in the 70s and 71, who had a very specific program that they wanted to launch. So it wasn't... No, I haven't, no. The reason for promoting this, you see, how wonderful it was, it's just unique American thought that you actually know with people like you. I have the book. I read it after my son borrowed it. It has an elo. Yeah, it's an elo. I'm good, I'm sure. Good, good, I'm sure. I'm like, well, I'll get a good... well, you just ask an elo for a good account of it. It'll probably give you an excellent synopsis. I think the bad thing about... Yes, I mean, I would be interested in reading that. I mean, clearly there was... I mean, it is clear that Marxism was very large in the American philosophical movement. But what I don't like and what I think has to be... Alexander Bain, in a way, brought the idea to Britain. Yes, I just don't... Well, all I can say in Anglo, I'm acting. Yeah. But the... Well, I suspect that there was probably also some influence from... But what I think one has to be very careful of, to guard against, is the very lazy-minded, cheap explanation of people like Russell. Which is a quite typical aristocratic British written explanation, which of course is exactly the sort of thing we'd expect Americans to come up with, which is really just literally kind of garbage to Russell Witten. And of course they didn't believe in objective truth because they were all so busy.

35:00 And that's all they are. Therefore, of course, they produce this philosophy. Well, that, of course, is really the most incredibly arrogant and superficial silly explanation. And so that is the sort of thing I answered. But it does have, of course, a connection. But like many... Our early prophets are what we say they are. Oh, yes, indeed. It's true, yes, it's true. It works for you. If it works for Arthur Anderson, it should work for you. And this, of course, Russell is obviously on something there. The nature of objective truth, which is not at all to say that truth in that sense is not ultimately subjective truth, but the idea that what really exists in some sense to do with ontology, systematology, or everythingology, is just The structure is a truth-bound object. That's the objective notion, and matter in motion is a subjective reflection of that, and that of course is the kind of movement that Frederick Russell was going with, which is not to say that Russell was not onto the sense of his work, but in what he was saying about pragmatism and its appeal for There are certain philosophers who were probably more likely to be found in the United States at that particular juncture than in European societies. But of course he dressed it up in this absolutely silly and arrogant, aristocratic disdain for America. Of course, the infamous dismissal of Wiener. You know, when Wiener came to England as a student, as a young prodigy, and his father was seeking permission to go into Cambridge, so it was probably a bit of a mistake on the part of Wiener's father, but he did do it. Anyway, Wiener didn't live to regret it, but I think Wiener wrote a very interesting impression of Russell, not at all flattering.

37:30 Oh, you knew about it? Well, I only learned about it recently. But there was a conversation between Russell and Wittgenstein shortly after Russell had interviewed Wiener for admission. And the conversation was not, of course, about the actual mathematical context, but the fact that a boy at that age could already be doing. There are such problems as well as obviously mathematically. So it was obviously mathematically far more gifted than Russell, which is, of course, just what Russell's real sticking point was, you know, the kind of intellectual jealousy of the teacher for the more gifted kids, which is always a very sad phenomenon, you know. But it was all to do with this conversation which Russell had with Wittgenstein about the absolute sort of sheer ghastly vulgarity of Americans and American Jews in particular. And if you tell me that Wittgenstein said to Russell, But if you tell me that Wiener is a good mathematician, then I tell you mathematics is no good. And Russell said, well, you still have to understand the problems of mathematics, they just have no standards, they have no standards. Incredible, incredible sort of arrogance. You can really understand, I mean, you can always understand where somebody like George Bush comes from, his hostility towards this kind of dossiating, condescending, aristocratic arrogance. Thank you for your attention. He's from Wien, isn't he? Oh, Wien, I guess so, yes, I guess originally the family probably were, yes, silly, of course I wasn't thinking, yes, of course, you know, must have had some connection to Wien, of course, sorry, I wasn't thinking. No, I'm sure that was the last thing that Wittgenstein was likely to do to his fellow Austrians, although it might have been to stop him rushing to join up in 1914 in the Imperial Army.

40:00 We also, we now know, some people are trying to make a big thing of this, and the only thing that is curious is that we did He did spend a year in the same class as Hitler. They were exactly ten years. They were actually born in the same month. Curiously, Wittgenstein's father was one of the wealthiest magnates in Vienna, one of the three or four wealthiest men in the whole Habsburg empire. Yes, that's right. Anyway, he decided that it would be good character training to send him for a year to a really very obscure provincial gymnasium rather than to the very echt lycée in Vienna, which is where the students, the children of the privileged elite were usually sent. ...because he thought it would be good to go slumming for a little bit with the Peons. Partly also they've been because of the incidence of anti-semitism in the academy. They were quite keen, as I understand, to kind of disinvent themselves, to distance themselves from their Jewish identity. And there's a photograph of him, he was 11 I think, and he spent the Which would have been in eighteen... eighteen... in nineteen hundred. And so he spent the year in the Realtor Guild at Linz, nineteen hundred to nineteen hundred. In the same class as Young Hitler. And there's a photograph of them, it's a small class, about twelve or thirteen boys. And afterwards, this occasionally... It became more and more well-known after 1933.

42:30 He seems to have had an extraordinarily naive, even by the standards of very isolated intellectuals, very naive estimate of Hitler's intentions. He's on record several times in conversations in the 1930s at the time of the Angelus, at the time of the Angelus, at the time of the Angelus, at the time of the Angelus, at the time of the Angelus, at the time of the Angelus, at the time of the Angelus, at the time of the Angelus, at the time of the Angelus, at the time of the Angelus, at the time of the Angelus, at the time of the Angelus, at the time of the Angelus, at the time of the Angelus, I'm told, and I'm glad to say I've never actually read Garslitt's book, which is apparently almost completely unreadable anyway, because it's so, I believe, a bit turgid, so I'm going to sit down and read it, quite apart from the content being so naughty, I think. But he has this remark about, you know, the one good Jew I had, when I was a boy at school in England, I was befriended by this student who was the one... For a long time, until I came to Vienna much later, because of this, I had not understood what the origins are. Now, of course, this is probably a huge amount of speculation. There's absolutely no evidence directly to suggest that this was Wittgenstein. But, of course, since it became known that he definitely did spend this year in the same class as Hitler, Journalistic speculators have published books just asserting that Wittgenstein must have been Hitler's book and therefore his point with Frenkel, but there's no evidence at all that Wittgenstein was the same guy together that they ever were. It was friendly. Indeed, I would have thought it's probably quite unlikely that they were both, I would think, very standoffish and awkward and unpleasant little 11-year-olds, even by the standards of, you know, they were not the two. It's just possible that they were the two outcasts and that they did.

45:00 Who knows, but that's just possible, but it's utter speculation, utter sheer response to speculation. But all the same, intriguing. So anyway, thanks for... I thought McLean has a student, a lady called Geraldine McGrady. You mentioned her in your list of possible invitees to the meeting, yes. I've got her email address all set up ready to send the invitations out. She works with England. She's the main administrative assistant. That's right, you mentioned that as well. Yes, he did. I think it's his very last PhD study that you mentioned. I'm not sure which way we're at, but one of them is there. Steve Hardy, do you know him? Yes, I was talking to him yesterday. He might also be... No, he's also on the list. I've already spoken to him about it. That's what I was doing yesterday afternoon. Sold out. Done and dusted. So get back to Fermi. So she published a book about the history, let's see... There's picked four pictures in the front. That's Wiener and Schroeder, four logicians. So it's at some point about the history of logic that actually... I'm glad Schroeder's on the front. Oh, yes. Good, good. Well, she was a MacLean student, so we'd expect her to have a good view of the... She was particularly promoting Schroeder's contribution in Spain. I mentioned you. Oh, you did indeed. That's one of the things I was going to ask you about. This conspiracy between Frege, Husserl, and Piano to dismiss Schroeder as a mere misguided algebraist who had got inclusion and membership all mixed up and therefore couldn't possibly... Yes, yes, yes. And that insistence on membership as the absolutely basic notion of completion of something so completely confused the whole project. Almost eccentric. Well, these guys worked hard to... But anyway, she has an appendix where she puts the description of Russell, which I'd never seen before. I don't remember the details, but I just saw the book in passing. It was quite interesting.

47:30 Oh, I must look that up. Well, you apparently already knew. Well, no, I knew. I'm sorry, but I knew that I had come across. A reference, I've come across a copy of what it was that was being said about Russell and I can't recall now exactly, but it wasn't very complimentary. No, none at all. And Wiener was, by all accounts, an extremely generous man. He wasn't one of these people like Howard or Warren. I'm just waiting to hear from Colin as to whether he can get away in November. I'm hoping he can. I know he's got all these new responsibilities. The head of the department, it does give him almost 18 months to arrange some things, and he wouldn't even have, you know, it would be a great benefit for him to miss the three days of your talks, which is what we obviously wanted to start off with, the discussion, the following minutes. But the actual invited talks will be coming down in the next four days, since it doesn't look as if we can really hope for more than a total of about eight days for the whole meeting. The responses I got from people who contacted me in 10 days was just going to be too long, so we then had to cut it down to around a week. And I'd like to stretch it to 8 days so that we could just have one-on-one relaxation, private conversations, just people to go off and do visits. Yeah, just to have a day of relaxation. To finally see the academy. You never got to see the academy? What in Christ do you mean? We went to... Oh, no, no, no! We were outside. Yes, that's right. But didn't you... I had gone earlier. Yeah, but you said you had got in, hadn't you? Yeah, briefly. You had got in, okay. Now we went and we tried to write and I was... Typically English. I was saying, but you of course typically were forthright and direct and bold and you just got hold of the guy and in fact had a... The gardener wanted to let us in. Yes, he wanted to let us in. He was really helpful. We went around the outside. We saw the loggia, didn't we? The one which has actually got the paintings, the press case inside.

50:00 I've seen photographs of it from which Cosimo used to enjoy addressing the peons. ...together with all his little prosonic acolytes, to buy intellectual credibility for... Yes, next time we can... Well, they have some interesting paintings inside. Yes, I've seen reproductions, I've seen photographs of the paintings, that's what got me so interested. Well, if Alberto is able to get our Scalilea is Vila, for at least for the first... Ah, that's on the other side of the... That's on the other side of the... Oh, no, that's not the... Well, that would be where we would have at least the first, maybe... I had the impression when he took me there that Dylan was rather in disrepair. Is that a wrong impression? Have you been there? Yes. It's not the problem. Maybe I'm just... Well, if it's the same place that I'm thinking of, the observatory of the astrophysics department, yes, that's right, yes, yes, yes, beautiful city, oh, absolutely beautiful, well, if for any reason we can't get that, then the alternative is what they call the Galileo Tribune, which is a particular hall in the university, which also has associations with Galileo, I think that should have been the place where he was. Interrogated by the Inquisition. I'll have to find out more about that. So this will be very appropriate for you to give your talks there and then have... It does move! Well, very appropriate, especially when we're coming up with something of atoms. But the points do move. Placement and displacement. It's not all fixed and eternal. Anyway, Alberto has promised that he can for at least the first day, hopefully the first two or three days, which would be the part of the meeting with your thoughts, and then hopefully a day or two of questions and a month or so of free time. There, and then the invited thoughts we'd probably have somewhere else for five or three or four days. He doesn't think he's going to be able to get it for the whole week, but he should certainly get it for two or three days. More to the point, he's actually given a commitment to support the meeting, which means that with the private funding that we already have, that we should have enough to bring at least 10 or 12 people. It's beginning to shake up, and I'll be working on it for the rest of this summer, so I'll report back to you in, I'll see you in October, which I think we've, hopefully we'll have got things really nailed down, invitations, and we've even got a few, yeah, well we already have titles from John and from Alberto, from Colin I should say, and Alberto and a couple of other people. And I was very, very pleased to learn Scott gave a very positive reaction.

52:30 I think he would enjoy the opportunity for spending two or three days talking with you. I think so. Since 40 years. Well, since Stolzky. Well, you've done an awful lot, both of you have done an awful lot since then. Actually, come to think of it, I'm trying to think, there's not much between the two of you that has been done since then, well, slight exaggeration, but not much, that hasn't been done by you. Yes, it will be very interesting. I was glad he was so positive because he has the reputation in some quarters of being a rather difficult person. I've never understood why. I mean, I've never found him so. He didn't give any evidence. No, no, not at all. And anyway, he's very enthusiastic about that here and has said definitely yes once we've let him know. That's very good. Yeah, yeah. And Steve Adams also said he liked chemistry. I think he showed you a lot of nice things. You haven't written to Geraldine Brady yet. I haven't written to anybody. I've only spoken to people informally because I hadn't yet got the dates right by the middle of the hour. Until we had those, it seemed pointless to send out emails. We were only going to have to send out more when we had them. But now that we have the... I'm just waiting to hear from Colin. And once he tells me that, yes, he can get away in November... Then I'm going to ask Albert to make the arrangements with the University of Florence for the, probably looking at the third week in November. It needs to be far enough in before Christmas. There are a lot of problems with the airfares if you get into December, because as you get closer to Christmas, the airfares start shooting up, because that's when the airlines make all their profits, and people have to fly home to speak with them. So we want to try and keep it in November, and that's usually a very good bag, and also, of course, for accommodation.

55:00 I know you enjoy warm weather and you like hot weather, but I think even you would have probably not enjoyed Florence in June when I was hanging out with John Bell. The temperature was in the 40s, 43s one day. And of course, being in that valley of 40s, incredibly humid with it, it was very unpleasant. Unless we had to spend the whole meeting in air-conditioned rooms, it would have been very, very wearisome. In fact, right now, Lawrence is pretty mild. It's just an exceptionally hot spell, but I think having him in November does give us that more time, that much more time to plan. So John Bell, he's interested in too. Yes, I've invited John informally to give a talk on, just on distant views of the continuum, from Brouwer versus Cantor to the present day. And so of course he'll want to speak mainly about intuitionistic analysis, I guess. But he might want to speak a little bit about synthetic differential geometry as well. But obviously we're going to have Anders and himself there. But John is always... It's worth having on board for general philosophical discussions. It's so interesting actually, you know, he was saying earlier on this morning about stern duality for Boolean algebra. It's just being, as it were, the logician's trivial example of stern duality. He's not the one who's doing the really serious mathematical work. This one has the examples of functional analysis. John gave me a very, very excellent tutorial on certain duality while we were walking around Florence last night, and I felt I had a sort of great deal more of listening to him than I had before, because a lot of his early papers were on. That standardality is obviously in the mathematical context, or even though I know it's a much less rich mathematical example than some, there's still quite a lot of subtle things going on that connect with the independence proofs.

57:30 See, I mean, of course you have a choice of dualizing objects, so if you take the Cartesian square of the object you thought you liked, then the unary operations on that contain and reflect the binary operations on the original, so they're matching, say, with complex numbers. Thank you for your attention. Stone duality for continuous real-valued functions, but also so-called Gelfand duality for continuous complex-valued functions, which are, quote, C-star algebra. Yes, yes, the Gelfand duality, of course, and for C-star algebra being precisely what these people doing, the non-community, geometry, want to make such an industry. But you see, again, to call it Gelfand is kind of... See, the commutative case is kind of a trivial issue. I mean, not trivial, but I mean... There's no reason to make it complex, you might as well make it real, you see, so that's another strange thing about quantum mechanics is that the I is everywhere, it's a sort of tradition, everything is imaginable, anyway, so yeah, so basically, but what I, I consider this. If you have an object in a topological category, and then you construct this v, what does this mean? This is the sub-object of v to the power of e to the x. Consisting of those maps, those... Functionals, which commute with every endomorphism in V.

1:00:00 Simply, if lambda is in V to the V, then you're looking for phi of lambda X, or lambda F, which is the lambda of phi of F, or all F in V to the X. And that's the kind of C that one considers. The idea is that if you choose V properly, then the fundamental prejudice of some dwellers and all that is that this ought to be an isomorphism because you put on enough conditions by using all endomaps of this sufficiently big object to strain out the actual maps from the... There are many, many, many things that are instances of this. Math is a particularly complex model in the continuous world, or in the sig-infinity world, according to John Milner's theorem. You get the same result, the essence being the least form. Using small spaces, this will be an estimator to cover the geometry of some kind of algebra, revalued algebra, but it can be just a unit. If you take the ordinary stone draw for logic, if you just take a finite discrete set, and V is a three element set, This is also true. That's the idea. Because you don't really need and. If you've got three values, then merely the endo... The operations on three values, which are unary operations, are enough to strain out every factor in the notion of ultrafilter, an ultrafilter in the case. This is the definition of ultrafilter, too. As I said, in the discrete world, if we use three elements... You might as well consider this to be the definition of a multiple, you know, not some subset, some various closure problems.

1:02:30 It's a map that tries to act like a point, and insofar as all these random maps can tell, it actually acts like a point. So there's a strong tendency for it to be a point, and it's basically not to be a base. But see, even the notion of measurable cardinals, you take v to be an infinite set. Any infinite set. Then, Isbell's theorem in 1960 was that if an isomorphic abstract set, if and only if x is smaller than the first measurable cardinal, in other words, the measurable, the so-called measurable, which in this case is really non-measurable, this is kind of measuring x by means of functions, you see, and so the so-called measurable cardinals are what you can't measure. So in other words that's all these notions which presented in disparate ways are really special cases of this simple idea. And about the ultrafilter obviously in the case of the logic. Yeah, yeah, it's just the finiterings. If he is finite then... You have a question? Finitary. Pseudopoints. I want to do a finitary approach, like points. That's what I'll do. That's exactly stated. If it was a finitary point of view, you'd want to use finite sets. I've never understood the origin of the terminology measure, because it seems to be almost exactly the reverse thing that can't be measured by... Yeah, yeah, no, no, it's because, no, I mean, it has a reason. You see, these functionals are sort of like measures. They are integrating b-valued functions to get b-valued results. Oh, I see. So, x is measurable in the sense of umam, which means that this is not subjective. So there exists a phi, which is a pseudo-point in a strong sense, but isn't really a point. But that's a measure. That's the existence of such a hintily additive measure. A measurable cardinal is one for which there exists such a bad measure, quote-unquote.

1:05:00 Yeah, yeah, yeah, I see that. But, see, the word measure, well, it's just a historical accident, because for the two valued things, to be a measure or to be a homomorphism is more or less the same thing. You can translate one into the other. Because they're not general measures at all. No, no. They are required to preserve multiplication, too. Yes, yes. I see, so they're not. There are measures of variance zero, but not zero themselves. There are probability measures of variance zero, and I'd highly add it, well, no, countably add it to this. If g is a countable set, then somehow this preservation of all these permutations and the values is equivalent to countable additivity. Well, that's absolutely beautiful. Thanks for explaining that again, because I mean, I had, I was aware of the Isbelli's algorithm, of how important it was, and I was wondering at this point about the first measurable part of being the, as well, the first thing that violates this. But it fits precisely, you see, into stone duality and Milner duality for the C-infinity case, but stone duality for both the two-valued case and the real-valued case, and Gauffin, all these are really connected with the same. Same circle of ideas, just a slight vary. Which topos are you then and which object B do you choose? In the case of the Gil-Fan duality, which is, what is the space you're on? These are, these are continuous again, but complex, you see. Ah, yes, the, the, the, of course, the C and the D. Continuous complex, one, yeah. And in all those cases, one, at least one of the finite cases, an infinite case. There's this further fact that only a few of these lambdas suffice. There's a huge number of continuous maps from the banks of the plain, let's say. But it turns out that in addition to multiplication, there are enough to reject any fields that aren't good. I don't know, why did I get off on this? Oh, because I started asking you about stone duality. Stone duality. That's right. Is there any chance I can keep that?

1:07:30 Of course. All right, great. Brilliant. I shall study that with great interest. I think that they're probably hinting that they'd like us to go. Oh, okay. What do you want to do? Because I don't want to take up... I know you've got an early night. You'll lead me to a bookstore. Oh, yes. Well, let me do that. And let me show you where the Worker's Museum is. And the toy store, too. Okay, well, while you're in the bookstore, I'll have a look at the toy store. I didn't look for a toy store, but I'm sure there are plenty. There seem to be quite a few shops like that in the area around where I will stay the night before. And that's also with the workers' museum, which I thought you might like to see. But the only problem is somebody told me at the hotel that the workers' museum is closed on Mondays, but I wasn't able to check it out because I'm not sure about that. I don't, but then I'm an Englishman. You don't expect me to. I just get wet. I think it's actually stopped raining. You can have my umbrella. No. Well, it is literally a spare one. No, I prefer this coat. Actually, this coat keeps me more dry. Yeah, I would have bought my coat, but what I did, I stashed it. All my luggage at the back of the hotel rather than having to cart it, because I knew we'd be walking, so I didn't want to have to cart it. Oh yeah, those are some of the other things I wanted to ask you about, but that'll keep you. Before I do that, can you tell me the dates that you're going to be in Marseille? You said it's October, but do you know the... If you don't, just send me an email. Must be the first days of... I know that I'm going to a meeting in Toronto at the end of September, and I don't have time to go home. I have to go skate in Toronto. So it must be a very good beginning of that. Okay, well, when you know exactly what it is, it's just I'd like to pencil it in. And it may well be the first one further, but let me know. Well, it's probably posted on the Conqueror Archive. Is that what it is? The meeting is actually promoted by the Conqueror Archive? Oh, yeah. We're going to remove that archive as well. Okay. Now, you said, and what meeting about is it on? Is it specifically on Conqueror's? No, it's not. Just in terms of looking it up. No, it's not specifically about one billion dollars.

1:10:00 It's something about a lot of money. Well, anyway, the director was overjoyed at my title. Which was? Was sind und was so und so. Ah, yes, but this is the same title of the talk you gave in Italy last year. Yes, I'm sorry. I'm so sorry I've forgotten his name. The young student. Oh, yeah, Lasteria. Lasteria. Lasteria, in fact, recorded. Oh, he did? Yes, and indeed, which he sent to me, and which I am in the course of transcribing. Bravo. But since my Italian, unfortunately, is not so great. You spoke in Italian? Yes, it's okay, but I can understand, but it's understandable in Italian. Yes, well, that's right. So I'm going to transcribe and translate it. But I'm actually in the course of that project. That's one of the things. Hopefully I will have done that by October. I haven't done it. This is consciously a warm-up. Okay, well, I assume that would have been one of the things you were going to talk about. I should be absolutely fascinated. But this apparently fits in very well with the theme of the meeting, whatever it is. Well, and of course the general theme that you were touching on earlier, of the discrete emerging from the continuous. One of the things I wanted to ask you about was SUD objects. For a second, let me take this off. I'm just wondering if there's any way they'd let me stash this behind the desk. Probably not. I'm not staying here. That's all right, it's not that heavy actually. It's okay, because I'm temporarily going to have to dash straight off. Well, certainly not long after that. You said you wanted to be off. There's a train at 3.47 and it's going to be blown tonight. From where I can get the... I prefer to count by a train when it's getting sharp, so I'm not going to do it, I guess. I see. So you'll be going to... I'll be going this afternoon. But I can get a later train. I can actually get trains later this Friday. So if I do, then I'm going to miss a night's sleep. Whereas if I get the 3.7, I get back to Cologne at midnight. And there's a little cheap hotel right on the station at Cologne, which I use by the light here.

1:12:30 May I just want to use the loo? Sure, sure. Let's go to Sarah.