What are continua? (contd.)

0:00 The one line of poetry which keeps coming through my head, sound, you know, really, really excellent poetry, and here it's being presented, you know, as necessarily attached to this completely wrong-headed view of propriety, syntax, that there is nothing else but syntax, and in which case, what can mathematicians do when they propose new axioms? I'm going to say, they're not even doing mathematics on this. It's absolutely weird to do, and I cannot believe that you've told it. I just think it's a lot of rhetoric about... Michael, I don't know that you know my friend. I don't know. Very nice to meet you. Michael Wright. Very nice to meet you too. We did meet very briefly in Bolzano. Have you been to Bolzano? Bolzano, that's right. Three years ago in Bolzano. It was a very good meeting. I enjoyed your talk very much.

2:30 Thank you for your attention. Thank you very much for your attention. Are you all the speakers expected to lunch together on the web, or are we going to be able to do that? Professor Beattie, sorry, well, I had the chance to ask you. I gather that Bill was giving a talk at your invitation in Milano recently on closed categories, on closed backgrounds. Did you write a report? Did anybody make a record? Yes, of course. I'd be in... I'd be immensely grateful if I sent you my details in English and obviously whatever cost was involved. Would it be possible to make a copy? Well, you're extremely kind because unfortunately I've messed me up. It's just a partial sort of thing. Well, it's still extremely interesting, you know, even if it's only, you know, partial. Well, just what you just recorded is very interesting, indeed, because, you know, the whole program. Okay, well, if I can give you, before we part, if you can...

5:00 Can I contact you in a little while? Are we going to be able to get lunch together, or is it... I have no idea. You're not sort of... If we're going to be able to get lunch at all, I haven't even resolved that. Well, I think... That's true. Well, I think... I'll make a leap of inductive faith on that. We can get lunch. No, what I meant was they're not going to... all the speakers don't have to go off together to lunch. It's not a rule of the meeting. It sometimes is, yeah. They're less formal about these things than they are in France, I think. In France it was very much, you know, the speakers have to go off to the con and... Thank you for your attention. Uh-huh. Yes. I was thinking very much of your view. I always wanted to say to him, oh, but there really is no global separation of formal content, but his whole view is that there is complete global separation of the formal, of the syntactic aspects of Ethereum, as if syntactic categories were things which just existed without. Foundry conditions, or conditions to produce them, something completely in advance of the world, and of course one can see how if you really remain with that view as the absolute priority, you then, quote-a-muir, the semantics is going to come back in somewhere, it's going to take its revenge, it comes back in, of course, in the view of the kind of medieval view of symbolism, that everything is as it were a symbol of, the symbols in the end, since the symbols alone have reality, then they must... The lines of structural intelligibility of the position are leading towards the form of the evil. It can work actually well. It can work in two directions.

7:30 I think the... The world's computer program. That's a wonderful title. When are you giving this score? Oh, you already gave it? The previous art, right. Oh, well, when you do, please send it. I'd love to study it. Maybe I could even, you know... Are you going to do it in English or...? No, in English. Oh, it's in Italian. Well, can you send me the Italian version and I'd like to try translating it? Thank you for rendering it into English. Well, it may not, but even sending... saying the same thing, you know, in a different format might often... No, but seriously, I would be immensely grateful if you would send it. It's a lovely title, too. I still haven't got around to reading the talk you gave in Castiglioncello on the category of lifting as you put it onto College for Syntax. I was hoping to ask, what have you done here? What have you thought of that? Indeed, I was asking about Alberto's talk last year at Castiglioncello on his ideas on the relevance of categorical construction, the notion of the lifting of the symbol and the connection between topology and logic.

10:00 That's very interesting and rather in line, I would think, with your own. Yes. Where does syntax, in fact, come from? I was just thinking in the light of our last speaker, who obviously believes that it precedes everything else, you know, conceptually has no, as it were, exists without any kind of context or boundary conditions of its own. Yes. Exactly. That's my point. That's my point. Exactly. Yes. Where does syntax come from? Does it just exist without boundary conditions or, you know, it's not just existing? Yes, exactly. Well, from what I gathered from reading it, I thought that the paper was at least an attempt to address that question. Where does the syntactic factor come from? Indeed, where are the symbols? No, in that they are symbols. There are also a number of fields of study, such as quantum mechanics, quantum mechanics, quantum mechanics, quantum mechanics, quantum mechanics, quantum mechanics, quantum mechanics, quantum mechanics, quantum mechanics, quantum mechanics, quantum mechanics, quantum mechanics, quantum mechanics, quantum mechanics, quantum mechanics, quantum mechanics, quantum mechanics, quantum mechanics, quantum mechanics, quantum mechanics, quantum mechanics, quantum mechanics, quantum mechanics, quantum mechanics, quantum mechanics, quantum mechanics, quantum mechanics, quantum mechanics, quantum mechanics, quantum mechanics, quantum mechanics, quantum mechanics, quantum mechanics, quantum mechanics, quantum mechanics, quantum mechanics, quantum mechanics, quantum mechanics, quantum mechanics, quantum mechanics, quantum mechanics, quantum mechanics, quantum mechanics, quantum mechanics, quantum mechanics, quantum mechanics, quantum mechanics, quantum mechanics, quantum mechanics, quantum mechanics, quantum mechanics, quantum mechanics, quantum mechanics, quantum mechanics, quantum mechanics, quantum mechanics, quantum mechanics, quantum mechanics, quantum mechanics, quantum mechanics, quantum mechanics, quantum mechanics, quantum mechanics, quantum mechanics, quantum mechanics, quantum mechanics, quantum mechanics, quantum mechanics, quantum mechanics, quantum mechanics, quantum Thank you for your attention.

12:30 Just give us a look at the book on Ricketts, there is a certain thing about the pattern of the Ricketts' making, and the rich contrast of that. Oh, the League of Nations, the time of aggression against Abyssinia, yeah. Well, given the fact that Britain is such a great imperialist nation, and Russell answered that he didn't know exactly, but he had the idea that the British government took such a... Yes, they supported the saying of a very strong line. Strong line against the Italian faction because they feared that the population, the local population of warriors could be used by, since they were considered to be very good warriors, could be used by Italians. Which, of course, in fact, they were when they moved around British Somaliland later in the war.

15:00 I don't know. But in fairness, I think Russell was just evading the question, as he often did. Yes, he was evading the question, but he was trying to. Yes, it's a very strange answer. Also, Enrique's question is a strange question, because I cannot believe Enrique was so politically naive as to ask this question seriously. Thank you for your attention.

17:30 What concerns the Cantorian, that aspect of the Cantorian program which involved, as it were, taking back Euclid's common notion five about the whole necessarily being greater than the part, because it does seem to me that internal set theory, in a sense, have a, it's really quite Euclidean, as I think, that you can have this beautiful, poetic language that can climb infinitely far up any one branch. ...without having to quantify over a completed... ...to the diagonal, diagonalization. I didn't hear you. What was your question? Well, I'm just being absolutely off the top of my head, but I wondered whether it had any consequences for our understanding of diagonalization,

20:00 this internalization of the world, as having a kind of internalized notion of an infinitary. So that you don't any longer have a completed infinity, again, it's just like, in a sense, it is a reversion to, in a sense, it does involve a kind of reversion to the Aristotelian view, that there is really a potential for the mass of the universe. Can we convert mathematics consistently? That sounds to me like Brouwer and Witten. And yet, you say that they have been already disposed of, and there are no of them. Who's not concerned with consistency and truth, then? Actually, I think we tend to get Brouwer wrong. I'm not so sure that Brouwer was. As much concerned with the epistemological issues as his kind of crazy subjectivist philosophy would suggest, I think that a lot of his ideas actually came out of his demathematical abilities as a topologist. He thought that he was very dissatisfied with the Cantorian view of continuum and with the nature of continuous functions. I have a suspicion that many of his ideas, particularly in the BAM theorem, came from that rather than from his idealist epistemology, but that's just a kind of speculation. Can I ask one quick question? Are you aware of the book that came out last year by John Mabry from Cambridge University Press, which has got a whole appendix? It's almost one third of the book on your work on your programme for internal set theory. It's a lovely book. It's called The Foundations of Mathematics in the Theory of Sets, but the book, the content of the book rather utilises title. John Mabry, he's a logician and model theorist at the University of Bristol who is also a...

22:30 Anyway, he's a very great admirer and student of your work, and the last part of the book does contain quite a detailed discussion, particularly the consequences of complex arithmetic, doing certain theorems about the complex. Thank you very much for your time. The first two chapters are very expositor and philosophical and very interesting. I don't disagree entirely with his position. I have trouble with that, I really do. Well, his view is very difficult. He writes beautifully, in a very attractive style, but what he's saying is... Yes, he really thinks of modern set theory as the latter-day version of the Greek theory of arrhythmia, of assemblages, of finite pluralities, it's just that Cantor changed the definition of finite, like, as it were. Abandoning Euclid's common axiom part, common notion part rather, and postulating that holes could be in one-to-one correspondence with subparts and that. I mean, it's a bold speculation as to the framework in which you ought to think of set theory. Of course, what this means is that set theory in this sense is already the underlying logic. He's not saying that that is the right way to think of the foundations of mathematics. He's just saying that historically... You know, that is a more illuminating way of thinking about what happens, and particularly what underlay this kind of structuralization of mathematical concepts in the 20th century than the conventional view of thinking of set theory as just one further theory to be formalized in the pre-existing first or higher order logical system.

25:00 I don't, I haven't really had a time, maybe even the ability to understand completely, so I have to try to read those first two chapters, and... You're right, he does write very clearly. Yeah, but what he's saying... Lino Bianco, si. May I pour water for anybody? You like fish cooked in white wine? Sure. Sounds great to me. I don't know what that would be, but I'm sure around here it would be something good. Exactly. I don't think it's obvious. No one can question or argue with having truth as the basic thing. Everything in mathematics has to be true. That's the whole point. And of course, that's controversial. People can disagree with that. You do, I know. Famously. So, you know, he can say that as... What he wants to argue or justify, but he doesn't do that. He just says it as if no one could ever disagree. He states his case provocatively. The business of mathematics is proof, and proof is about establishing truth. And establishing truths indubitably from first principles. I mean, that's his clear view. That's his view, but he thinks it's not just his view, but something that everybody has to accept. Yes, we've got the way to write it. I haven't driven off.

27:30 I'm not so sure about that. I mean, I'm a naturalist about truth, and I'm also not at all sure that we can separate. But I'm much more, much closer to his position about the nature of mathematics than I am to his. What I take to be your view from having read the book is that I have to... I had no problem with his basic thing about how the Greeks had it right. What did the Greeks really say that was right? Can you tell me that? Well, I think what he's saying, and this is one aspect of his business I don't agree with, what he's saying is that the Greek notion of arithmos... This is essentially the, well not essentially, it is in fact, you know, it turns out to be, once you've understood what Cantor really did, you know, from our understanding of sets, it turns out to be exactly the notion that we now have in modern sets. ...of a set as a collection in extension, an absolutely determined plurality just constituted by what its members are. Now this I think is a fundamentally wrong notion. I don't in fact think that... I think that that notion itself actually comes from something deeper and actually already involves topological and geometrical... So, so, a number is really a set. Well, what do you accomplish by that? Why can't you have numbers as sets? Talk about sets as sets and numbers as numbers. If you say a number is really a set, I don't see where it gets you. Why say that it's as pleasant as I remember it is? Well, you can have the kind of construct really that all structures are equal and some are more equal than others. Or you can have the view that you're achieving clarification or explanation by ontological unification. I'm, you know, you're putting me in the devil's advocate position here because I don't share John's views. I'm not telling you to share it. I'm telling you to keep blanking on his views. Well, I suppose he does have a view, but unlike, unlike.

30:00 There is no one today who believes that semantics is just obfuscation and poetic confusion, but in fact you do actually require alchemists. Ontological quantification of the most basic notions, of the notions which actually serve as the ultimate source of ingredients of the definition of all further concepts, and he has, which I think is a wrong view, he has this view of mathematics, it's a fairly familiar view, as essentially a skyscraper in which the concepts which operate at... It was a very firm foundation, it was a very firm foundation. Sure, of course it does, of course it does. It's exactly what it implies, which is why I'm not a foundationist parallel neighbour. I have very strong reservations about that view. I think mathematics is much more, mathematical concepts fit together much more in the fashion of somebody's used time to get to it whilst using the analogy of, you know, that's a revolving space station that gets reconstructed whilst in orbit. And it doesn't, as it were, have to be tethered by a fixed once-and-for-all, you know, given once-for-all and forever, and to the extent that you've got it right now. Yeah, no, no, I mean, I'm inclined to take that view, that systemic easiness is a better model than the skyscraper analogy. And within systemic easiness, you do have a constant reworking, I would say, a very subtle dialectic that seems to operate over and over again between... Very broadly, logical arithmetic concepts and what one, again, very broadly would call geometric concepts, particularly, obviously, in our understanding of the continuum. Well, you know, instead of a science paper, there are many, many other metaphors one can think of. And one of them has to do with the book. All metaphors are obviously systematically misleading in one way or the other, but I certainly don't buy the skyscraper view, which of course is very much John Mason's view, because I don't think that there is just a once and for all, you know, fixed forever, closed.

32:30 The set of ultimate ingredients is a definition of all four of the concepts, and in fact I think that the way he thinks about the sets as determinants given the extensions is of course exactly right, that is precisely the semantics of the standard model of ZF, but that it seems to be replaced on the fact that the standard model of ZF is very good for understanding the way that... There are certainly concepts, notably those of order and cardinality, fit together in mathematics, and there are a lot of different aspects of mathematics, notably arts and space and structures, which would be precisely the defining structures of mathematics. I mean, I think his book is pretty clear and vivid, but I'm not sure. I think they're pretty clear and vivid. It's just that I think they're not very clear. But I thought there was an interesting part of the book that was, in fact, quite an extended discussion. In the last third of the book, all of them have the same idea. Now, unfortunately, unfortunately, his wife would need to be very ill for all that he has, so he's not going to retire from the club. He would very much like to come here for this meeting, but it's too late. He's a very old friend. I don't know if I'm spreading myself hands all day, but unfortunately, you just can't leave a need in the world of physics, isn't it? I don't know if there's a chance we can't get up and down physics, I don't know if it's in a lecture or in a seminar.

35:00 Obviously it happens to a lot of people. It's a very rare problem which is problematic. It's cancer of the liver, but it's a bit of a problem. This is a very difficult subject. Some people die within a matter of months, and other people live for years. But it's a very small example. Anyway, she's happy. She's hanging in there very well. It's the last time I saw her. But obviously, it's difficult, and certainly she's not able to travel. In the course of the course he has to find the analysis course that he teaches, which is the last. He's almost the last mathematician in the department. He's in the philosophy department. No, he's in the math department. So why is he the only one who teaches analysis in the math department? Well, I'd say he does teach the first and second year, and he's not the only one. Well, he's the only one where he teaches it in full regularity, and it tends to be taught, all the other courses tend to be taught by the applied mathematicians, and he has various difficulties with them politically in his department. He has a hell of a fight. I think that's part of it and part of it is just the general problems that they've got at the moment with pressure on dilution or pressure on university students to dilute standards and to drop grades. I mean, you'd have to ask him yourself. I'm not a colleague of his. I only hear the one side.

37:30 I wasn't trying to avoid the question. I can only say that I get a lot of feedback from him. He's a very outgoing, warm, sensually extrovert. It's essentially a very well-adjusted person. He's not somebody who spends a lot of time in whinging or in able-gazing, and the fact that the last two or three years he's just been consistently miserable with life in the department, and I think... It was a very good department, too, at one time. I think it still has got very good people in it, but it had a lot of very bright people, and in fact, Armstrong was going to come as professor a couple of years ago, and then that all fell through. I don't know why. Some idiocy on the part of the administration, the last chancellor. So there was the chancellor, you know, getting the field medalists and a couple of other very bright guys, and since then... My impression from what John has told me is that it's been steadily down the hill. Brian Robinson? Stephen Perner is dead. Well, Stephen Perner is dead now. Stephen Perner died about... Stephen Perner lost the head of the philosophy department many years ago. Stephen Perner died, I think, about two years ago. Thank you for watching. He was a lovely guy. He would have had, I think, a lot of sympathy with your new boy. He was in his 80s. Yeah, he must have been one of his 80s because it must have been 20 years at least now since his dad. And he was a refugee from the Nazis at the beginning of World War II, so he must have been. He was a lovely man. He and his wife, too. I didn't know them intimately, but I knew them through John and through other people I also knew in Bristol, both in the philosophy department and in the math department and in the history department, so I had quite a number of friends. Were you working in Bristol? No, I wasn't. I did my first degree in Cambridge, and then I was in London. But I had an awful lot of friends who did work in Bristol.

40:00 And there was a very, back in the 70s, the late 70s, there was a very good group down there, very interested in the philosophy of math, in foundations, and particularly in calculus. And John was absolutely opposed to the idea of a foundation, having any claim to be a foundation, and as it were, dare I say, deconstructed that claim at great length, and I disagree with everything he says, but at least I get a very good feedback, I'm forced to think very clearly, by, you know, to address his arguments, yes. That's exactly what I'm saying. He's the group in the audience. But sadly, a lot of his colleagues are like Brian Rothman and like... Was Brian Rothman there? Yes, he was, yes. I thought you were going to say that. Ah, well, yeah, you're right, he actually did well. But I was thinking, because when I used to go there, he used to come to his meetings. He's a very good skipper. He once stayed in my apartment. It's a 10-minute skipper job. I put him up in my apartment in Albuquerque. He passed through. This is very congenial. Did Memphis, then he was in Louisiana, and now he's in Ohio? Is that right? Really? I don't know. I don't know. John would know. Yeah, right. Well, I gather his wife had a tenured-tech job at Memphis State University, and he was there. But then I heard from him at different addresses. Baton Rouge, Louisiana, and his latest book, which... ...which I haven't heard of, which I haven't seen. I didn't even know this would come out. Well, this is more recent than the one he wrote, told of him very much. Yeah, it's about Stafford University. It's not an ass of philosophy, it's some other series of humanities published in Stafford. Well, he has now, as I say, drifted over to being a...

42:30 I think his first article on that semiotica was really interesting. I haven't read it. I've read his book, the one about zero and the one about the second one. The second one is good, it's stimulating. I can tell you what the semiotica article said. It's strange to me, it took me a long time to really see that it was okay, so the first day was really too weird, because I had no preparation ready, but it described three different aspects, or three different sonai, or whatever, of the mathematician. I don't know how many terminologies there are, but what I would call the abstract definition of mathematics is we. And this is an abstract person who walks around in a new world. But they don't normally say we, they just say let, as in, you know, let there be a complete ordered field, you know what I'm saying. And then there is an automaton to actually, to imagine, does calculations. Proust theorem, yes. And then there's also the real person who signs the paper, and that's Witten. Number two is the purely syntactic engine, the one that Ed was asking us to believe in this morning. That was really interesting, that description of the three different guys for... And the third guy is the one who's wallowing in all of this poetic obfuscation. He's the one who really is alive. Yeah, sure. He's the one who actually has the ideas. ...who actually has, you know, underlying reasons for proposing new axioms that might just...

45:00 I think it was really interesting. I think... And I must read it. It sounds like an entertaining book. It was in semi-aragum. Uh-huh. Which I never see. I have to show you. I would be very interested in reading that. If I give you my address, could you send me an offering of a photocopier? I'd be very interested. I know how to get hold of my mouse. I can do that. That's what I can't do. I have the same problem with your files as I have with mine. Mine are in a pretty good state, a pretty chaotic state, can you believe it? But I could go to the library. Well, I don't want to put you into that kind of bother. No, okay. It's semi-optical. If you regret that, I'll take care of it for you. My problem is not being in any kind of institution. I'm not in academic life anymore. I haven't been for 20 years. I was in the cruise. I'm looking at Tom Lear and some of the others, sliding down the rails of great device, and have I run a small travel business, that's what I live in, and I must try. I certainly enjoyed the, I certainly enjoyed both your contributions and those of quite a few other people, Vaughan Pratt and Bob Tragesse in particular. Do you know Bob Trapezoid? You're sending me an offensive spot. Oh, sorry about that. I never had an offensive spot. But, my good friend, John Tyler Rutte, was also a good friend of Trapezoid. Oh, I see. I had heard of Trapezoid and read about it in books. I've never met him personally. Well, let me tell you the story. Thank you. We're looking for a job, and he applied to where I worked, and he was an inspector, and he was interviewed there, and I didn't meet him, but I know that he was in the math department there, and he would ask the math department of the fourth floor of the anatomy school, the philosophy of the physics school, and he had agents, he told me there are agents in the math department. I was actually attracted to that.

47:30 Thank you for watching. They didn't really care, but that's not what you're going to do. They hired somebody else, who was in the grand block of the book, who was a different guy, who would give them more of a personal image. And the weird thing was that the chairman, Russell Blitman is his name, ...struck him as a real philosopher because he could see he was a deep thinker. Well, I think they didn't hire him. Well, that's probably the real thing. I hate to say it, but it doesn't come as a surprise. That's just the beginning of the story. Okay. So, he's on this long list, which I am on. And although he had earlier written to me that he liked my book, that it made him think, he was wondering if he was going to attack me, call me a terrorist, actually. Actually, a terrorist. I'm sorry, that is so far off the law. I'm a terrorist. He wasn't just using a very extreme metaphor. He said I was the guru of the dumb-downers, the people who are dumb-downing mathematics. I do recall the exchange, I'm afraid. Yes, I do recall the exchange. But once it came from Robert Augustine. I'm sorry. That was the only thing you said in your name. I was really offended. Well, I can imagine one would be offended by that kind of name-calling. I mean, name-calling is not as much name-calling in many ways. Well, I'm very interested to have that information. I have never met him personally. I've read a couple of his books, and I liked the line that he had, particularly on the significance of category theory, and potentially, I mean, amongst its other claims to, highly dependent claims to intellectual interest, you know, both from the point of view.

50:00 The methodological dimension of foundations or organisations. He had some quite sensible things to say about its potential as the framework for a dynamic logic of whole-part relations. Meriology comes following, which he saw as, as it were, as natural, as providing a framework in which... ...and actually he gets it right because he and his wife... Really? That's extraordinary because Colin McLarty is also a friend of mine at Case Western Reserve. I don't think he has, apart from an interest in category theory, although in college case I have, it's far, far... But in Connes' case, his knowledge is much more technical and informed. There is far far deeper, there is far more rigorous and technical, I don't know if you guys have some good ideas, but I haven't seen what he has developed them in, in detail, and I haven't seen what he's done, but the actual ideas he turns out, I think are frequently very striking, whereas Colin is a much... I'm going to be careful with the detail of the person I'm going to mention. My cousin has actually published a, you know, an excellent book on algebra and physics and so on. Well, it's not actually an introduction. It's called Elementary Mathematics and Elementic Proposals. It's not, alright. He's using an elementary level, you know, technical, internal, set.

52:30 I haven't read it yet. It's a very good book. So when I do consulting for art groups, they pay me with three books. Well, it is an excellent book. I can recommend it. It's a very good introduction. What you two were discussing earlier, a romantic man. The question of the difference between an English gentleman and an American gentleman, the answer is that the English gentleman is never unintentionally rude. Yes, I hope you don't feel I've been intentionally rude. But I know, I realize what you're talking about. You were talking about regresso. I think that's probably right, isn't it? Unintentionally rude. But I guess that would apply both in England and America. It sounds like Wilde. It's a good one. It's certainly an excellent definition. It certainly would have seemed to apply to the gentleman you were talking about. I haven't had the honour of his acquaintance. I just simply like what I read of his postings on the phone. But I'm sorry to learn that he calls it that. I think he may just be, what was Glaston's expression about, a rhetorician intoxicated with exuberance of his own verbosity and the intent of... The reference to intellectual terrorism may have started as a kind of tongue-in-cheek metaphor that ran away with itself without it realizing that it genuinely, obviously, was going to be very offensive to use. Was there a context where many other people were attacking you? A guy named Rhys said I was a clown. Well, there was an awful lot of very pointless and... Unpresent name calling, especially in that period on the FLM list, which I think caused a lot of people who would. It would have been very interesting to have read their contributions on technical issues to just drop out of it and refute, well, notably, Solve Betterment dropped out of it and wouldn't contribute, and I think Vaughan Pratt dropped out and wouldn't contribute, and quite a number of other people, I think, yes, and you had to be at least as thick-skinned as Colin McClarty, and Colin is very thick-skinned, to continue the debate about category theory with Friedman.

55:00 Thank you very much for your time. And when he had actually weighed in on their side, the Freedmen didn't seem to have the patience even to notice. In fact, John came back with an excellent and very, I don't know, very apposite retort after Freedmen had sort of trashed the plot and said, um, you know, in the old black and white Western movies, you know, when John Wayne goes to the swimming bar and gets into the, you know, the real... It is actually conventional to try and figure out which side the guy is on before you smash the chair over his head. You smashed the chair out of my head without even noticing that I'd actually weighed in with Spider going to the sun, which was completely true. He weighed in with a very weak reasoned and terribly worded argument as to why category theory could not be given his definition of what consciousness is. What comprises a foundation of mathematics? Or what a foundation of mathematics must necessarily comprise? Why, in a category theory, could not be foundational in that sense? It just seems extraordinary to me. It's about second-order logic. But he didn't even bother trying to understand the statement. It is quite clear that what he means by that is that he thinks that, given his view of the nature of set theory, it is itself already the source of the absolutely primitive ingredients which we need.

57:30 It comes to work in the 20th century form. It didn't, of course, in its original opinion, than it really was about ultimate principles. That was a lie. That was a lie. Yes, and John would respect his achievement in the axiomatic theory, which is enormous and which is not in dispute. Well, he disrespected it now. Well, he disrespected it enough to say that it was the one true foundation of mathematics. Thank you for watching. There's a rather high emotional immaturity quotient, especially if he has his little ass licker, you know, to deuce my language, coming around. He has a, you know, he's obviously a colossal in my eye, I mean, that's not in the studio. And he has a, to my way of thinking, a highly idiosyncratic view of the world, you know, of foundations. And he has a program which is independently of great interest apart from his work he does about mathematics. Well, he's probably just being a little bit cynical with his own expense. He does have a bit of a subtle opposition, it's just that I'm not clear with him. ...that everything in mathematics should be rearranged. The way, the way prizes are given should be done by ourselves. The administration of the subject should be rearranged, and that's it, yes. Of course, he also, by implication, believes that the actual architecture of the subject should be rearranged, too. There should be a different order of comp-

1:00:00 The book was published by his fans, of course, because they wanted mathematicians to be aware and know the physics. That was extraordinary. Yes, well, that's an extraordinary claim. I hate to say it, but I would think that the field's... The Nobel Prize winner will be nominated for a Nobel Prize in Mathematics, Physics, and Geometry, and will be nominated for a Nobel Prize in Mathematics, Physics, and Geometry, and will be nominated for a Nobel Prize in Mathematics, Physics, and Geometry, and will be nominated for a Nobel Prize in Mathematics, Physics, and Geometry, As yet, at least, well, Ed will tell me different possibly, but I don't think that, we don't think that they award the Fields Medal or even the Nobel Prize as yet on quite the same basis that they award the Oscars, i.e. Lobbying, intensive lobbying by a little clique doesn't necessarily or usually will not do you any favours, it's not just the opposite. Rubin was saying... What Rubin was saying, this is exactly what Harvey Friedman's friends did. This was censor circular to everybody, kind of listing all the reasons why he ought to have been shortlisted. I'm surprised why he didn't give me that. I have a question for Mr. Bartle. In the introduction, you were saying that it's unfortunate that this guy's wonderful work has not been done as a student, but the mathematics community is not sufficiently aware of what Harvey has done, and so we're actually publishing a book and giving it to you free, and you came in the mail with that. I hate to say it, but that's not good at talking to people. Publishing is not an advertisement. In fact, it's very counterproductive because there's no question that his work is of a great distinction and he is a very brilliant guy, even though I don't happen to think that it's of enormous conceptual significance for the foundations or organization of the mathematics that he claims for it. But it's certainly deep and very careful. There's all sorts of ramifications come out of it, including a solution of n equals nt and other deep problems.

1:02:30 He's a very smart guy. But I don't think he's very ill-served by his friends or by the little clique of slavish admirers that I think has kind of gathered around him. You were very insecure at first. Well, that's my impression. I never met him. It's extraordinary, I mean the guy's clearly brilliant, everybody knows he's brilliant, he's acknowledged by everybody, and he was given a professorship. Well, you still have to give the Guinness Book of Records, you know, for the age of about 17. I mean, why, how the hell the guy can be that insecure? Well, I think it has a lot to do, frankly, with emotional immaturity. I mean, I'm sure the guy just never basically... Maybe it's because he's short. Has he? I've never actually seen him in the flesh. You know, in this controversy with Bob Short, he's short, I mean, you can't be... ...needling him to be short. Oh. Well, that was... But they say short people tend to be edgy about being called short. Well, they tend, yes, because famously they tend to be driven. I never knew Alexander the Great was short. But I didn't know. I mean, we'll be right. Well, go out and conquer the world, in the case of those two gentlemen, but don't fight for displacement activity, will you? Yeah, absolutely. That may be part of what's happening there. I mean, there is obviously a massive amount of emotional insecurity there. Which unfortunately is backed by people who make themselves into little acolytes. That's always a very bad thing. You know, I did quit finally, but it was my second time. The first time I tried to quit, I got a message from Simpson, oh no, you can't quit so-and-so or talking about your work on the list. So I relented.

1:05:00 You did find they quit, didn't you? I know a lot of people did. I think you have talked about that. That is considered that kind of mathematics, isn't it? Yeah. Well, it depended on what they were talking about. Some of the debates, I thought, as I said, particularly the postings that they had on Continua von Pratt's postings about Continua and about, for instance, in the context of category theory, what concepts should be regarded as... I never understood the point of that. I really didn't. Free but explicitly acknowledged category theory is useful. Why don't you go read some? You can do that some for yourself. Thank you for watching this video. Very tolerant and pluralist view of foundations. He thinks there has to be one true all-encompassing foundation. What do I call the athenaeism? You can't prove that! What about there? For a while I had a point plan file for people using the main language to get the message. My plan is to work like a pyramide on the foundations of mathematics. That's an excellent, that was your, that's splendid, that's splendid. You must copyright that, because I shall certainly use it quite shamelessly, just like Oscar Wilde. You know, it only occurred to me this morning, in my thoughts, that I bring up the question, why, what do you mean by the word foundation? And the fact that this is in the school of architecture makes it so relevant. Well, we all do speakers. That's right, that's right. But I had been to that, so I heard of it too. Well, and of course, the very metaphor of a foundation does... If one takes it too literally, it does tend to lock one into the John Mayer appeal, as I said earlier.

1:07:30 Mathematics is that everything must be built up in an orderly and essentially lineal fashion. Except common phrases without one single word. In fact, it's my impression that too many people, in the history of mathematics, they automatically... I think that attitude is changing. I think as a result of people like Colin McClarty and other people, there is much more of a focus now, for instance, on mechanics, than when people say yes. Well, a lot of people, I think, especially math departments, do, I think, still make that assumption. It is shifting quite rapidly. I think a lot of people now are aware that there are a few other dimensions of philosophical interest in mathematics, including, for instance, the nature of explanation in mathematics, the various models of explanation that have been proposed in the sciences, physical sciences, broadly in the sciences. Are the various components of those models, such as unification, reduction in the number of basic concepts, can they be taken over, more or less, in fact, into mathematical field? To what extent do they have to be modified or changed when we look at patterns of subsumption of our mathematical corpus? The theory of concepts, I say more than just one specific theory, one or another. And this obviously connects with what we were talking about earlier, is with the metaphor of systemic cohesiveness more broadly than that of the mathematics of the skyscraper. You might be interested in these little blisters that I do really think that people can enjoy. The story of mathematics. Do you want that? Well, unfortunately, I haven't had a chance in the last year to really look at it very often, but I do. I look at it regularly, and before coming here, I put a question on it. What, why, where did this medical come from? Foundations of what medicine?

1:10:00 And I got some interesting answers. In particular, a set list was very impressive. It was a very careful and clear explanation of Frege versus Rosen. Quite ahead, it's really interesting, Frege's interests were not establishing certainty, as Russell's was, but it was dealing with the discrepancy, the epistemological discrepancy that he saw in the way that Russell did get the answer to. Frege was far more in contact with really deep-living mathematics of his time than Russell. I think that there's a new, I wouldn't say consensus, but certainly a quite new line of thought about Fragot that's emerged recently amongst philosophers and mathematics that has taken that on board, that has recognized that he wasn't just... The protologicist, and it wasn't, the logicist program underwent a lot of shifts and changes from the time that he left his hand, and it wasn't in a book I'd never looked at, although it was in my library, by a guy named Stuart Shecker. It's called Philosophy, Science, Mathematics, and Logic in the 20th Century. No, I haven't got that book, no. That's what this article is about, and it's the only thing that I really like, but I think it's very informative. Well, he's a very bright guy, Michael. And there's also a guy called James Tappenden that he's written quite a little bit recently about Frege. Again, he's made this point that Frege, the focus in the philosopher's understanding of Frege, which is that Frege's primary concern was I think that is right. I think that is certainly part of what happened.

1:12:30 But to see Plato's concerns, especially his concerns with the, trying to establish that the truths of arithmetic were analytic, in the context of the debates about the relationship between geometric and arithmetic concepts, in particular the crisis of intuition. This kind of outputization and analysis is actually more... But isn't it remarkable that people, at least philosophers, even someone as anti-Reagan, writes that before Reagan there was no philosophy of mathematics in the prehistory? And yet, Reagan himself repudiated all this. He said it was all wrong. Well, there was a great deal of philosophy and there was a great deal of philosophically thought on the part of mathematicians, which has just not been recovered by... I mean, not just the obvious people like Heidnitz, but... Why is it that if the guy himself says, no, it was wrong, everyone else... I have a friend in Cambridge who is just supervising a Ph.D. You probably know David Caulfield. I haven't met him, but I've heard quite a bit about him. Well, he has a student at the moment doing a Ph.D. on Clifford's work. The relationship between Clifford's work, both as an applied mathematician and as a pure algebraic, and Clifford's philosophical ideas. But his naturalism is absolutely fascinating. He gave a talk... Well, he was an alphabist. He was incredibly... That's my impression, too. But he did in fact have a really quite worked out what one might call a dialectical naturalist philosophy of mathematics. In fact, a general account, a broad brush account of epistemology of mathematical concepts, the relationship between geometric and arithmetic concepts, which he never published because of course he died. This is something which is very peculiar to Cambridge and very bad. This is based on his unpublished papers. No, no, there's a whole archive, there's a mass of unpublished papers. Of course he was only about...

1:15:00 How old was Clifford when he died? Bill, do you happen to know how old Clifford was when he died? He was very young, he was a good kid. He was certainly 20, I think. I think he might have been as young as 36, but he certainly wasn't older than 30. He published very little in his lifetime, and there is a massive archive of papers which was left to Trinity, and to the disgrace of Trinity, which is shared, I'm afraid, because it took them 300 years to get around to going through a museum, they had some stuff to publish back, so that's how very bad they are. Who's the idiot that's appointing a scholar, just an industrious... ...a scholar, a bright spark, to just edit the unpublished papers. Yes, yes, he does. He's at Leeds now. He's at the University of Leeds. But he hasn't. There's a seminar in Cambridge. ...which he heads up as well, so he commutes to Cambridge to do a small philosophy. Yeah, he was in a real... Yeah, he was, but he's very lucky. He's landed on his feet. He's now got a position in Leeds in the philosophy department there. He and Blimey had a seminar on this stuff. They did indeed. Which I would have missed going, but it didn't work out. I would have loved to have gone for it too, but it was a couple of things as well that actually happened on the plane. But unfortunately, it was my invitation. You had to be a US citizen, which I think really sucks, I have to say. I think they're outrageous. You know, you have to be a US citizen. What was the National Endowment for the Humanities? I'm sorry, I still think it sucks. I would have been quite prepared to have paid whatever the admission fee was. I'd be quite happy to pay Uncle Sandline. It was mainly for high school. See, I wanted to go. I was teasing. I mean, the National Endowment for Humanities has to have rules. They're going to have those summer schools. They're going to have them.

1:17:30 I was a little bit taken aback by that rule, but you do have some controversial people like that who have stupid views. I can't quite sure what this is. Well, I don't think there's any security implications in people arguing about the foundations of mathematics there. The foundations collapsed, don't you think? Yes. It's a national point. Sir, we have a massive intelligence failure. We have that. It's all coming unraveled. Massive unraveled. Yeah, well, they're both personal friends of mine, so I'm quite lucky to know them. Collins is a really neat guy. Very laid-back, very kind of wisecracking, witty, smart guy. Thanks a lot for listening. He's, oh you're talking about David now, you're talking about Dave Caulfield, he's the one who works quite on a lack of knowledge, yeah, yeah, yes, he does work on a lack of knowledge, um, well no, no, it's a PhD, he's got a student, he's got a PhD student who's working on it, it's his student who's working on it, um, but, um, but, uh, well Colling works on all sorts of things, but mainly on, well for instance on the, on the continuum, but mainly on... And he's, of course, very, very, I think, attuned to your own pluralist view of foundations, he thinks, very much. Foundations is just a matter of cross-relative consistency of truths between different parts of mathematics and the whole thing is, as it were, foundations grows from the inside out, you know, it's like... Do you know a guy named Richard Granby? Uh, yes, I get, but not to speak to. Unlike Colin and David, I only know him by name. Well, I don't know him either, but he was one of the people who correctly responded to my request for information. Why, for example, is what Harvey Friedman does a foundation?

1:20:00 I mean, it's fine, but why is it a foundation? And Grady's answer was rather brutal. He said it's only because it's a continuation of something that was a foundation, it was not. Damn! I think that's right. So in itself, rarely being called powers, because we have one power structure. I think that's a good answer. I think it is a good answer. I mean, all these people who believe that, you know, ZF is the final framework, and you're really trapped with that, but they're essentially taking something which appeared at a certain historical conjuncture to provide post the arithmetization of analysis, post the... The crisis of intuition and the discovery of the pathological bubble, the definition of the presentation of the space-willing curve and the little air differential continues all the other pathological bubbles, to be a foundation around the early 20th century. And it's on top of it, for me, that it's a foundation. In fact, the way that people like to think of it, and it's broken deep, clearly. They're going to be able to think that they are going to be able to recast, recast the description of structures around them, really the whole of mathematics, in using underlying concepts coming from ultimate topology, not from humanology theory, which is not a foundation at all, because in their view, set theory and logic just fall into place as one tiny fragment of this bigger geometrical picture. The only problem is that Billy is so cabalistic. Oh, you just mean in terms of career paths, wasn't it? Oh, yeah, but just because. But then there's so many things coming about that we want to question ZF as an ultimate foundation. One, there's all of this work in topological field theory, in physics, which is using eight-dimensional categories, which makes it appear that... I mean, may not turn out to be relevant to physics, but it just makes it appear that ZF is just... The tip of the iceberg, from the point of view of this way of thinking, is disgusting, and not on its way up, but I think it may just be revealed as part of the much broader geometrical picture.

1:22:30 Yeah, of course. I guess we'd better... What's that supposed to be about? Oh, that's okay. Well, it's already about 2.24 now. Yeah. Now, do we... Was this provided? Should we not at least leave this or something for the people? Well, I'm going to leave something for Jessica. Well, it's nice talking to you, but just remind me, where the hell do you land on Earth? I work in the Mediterranean. Right, but you actually teach in... No, I retired. Ah, you're one of the... I was only seeing you at high school. Oh, that's a very honorable thing to be asked. That's what I hear about it. That's great. Well paid. It's amazing how everybody starts getting their best ideas after they retire. Um... Yeah. Um... And it sounded perfect. At least you get nice warm winters down there. It's not bad. If the tour is set, it's a place where rich people go to play and have fun. But it's physically not a bad place, if you don't mind all the rich people. All the boutiques and galleries and all that crap. It must be a bit like the Boca Raton or somewhere like that, I guess. Well, I have to make my living by taking these rich people around on their vacations, because I'm in the travel business. I just finished taking around a lot of extremely... Well, get all your money out of them. I do. I do. I try.

1:25:00 Overcharge everyone. Well, within... Within certain limits, that's more or less what we do. So you're a tour organiser? Yes, that's right, yeah. That's where I make my living at. So do you ever take people to Venezuela? I'm afraid not. I don't take people to the US at all. I just take Americans around Europe. But I did... So it was a place that you thought was the most desired place to go. Really? Well, even more than places like Malibu Beach? Well, I don't believe it, but that's a big place. There's no one there. It would be crazy to go there. Well, except lots of rich people and the Jews. Do you know the way? I don't... Yes, but Alberto certainly does. I'm just following him. I was going to say, the Santa Fe Trail is the only context in which I can think of Santa Fe. Well, that's a good bar. But there is a hotel there called the end of the film. Is it near there that they have the Amerindian cities actually built into the cliff, those great... Yeah, there's a cliff going nearby and also going to move away. There's a place called Bangalir National Park. Where are you in relation to Albuquerque, you're sort of above, oh okay, so pretty close, in American terms, absolutely next door, yeah, right here, yeah. I was doing that actually. But when did you actually retire? About five years ago. Oh, as long ago as that, wow, so you must have taken very early retirement. I continued, I continued, I must have what? You must have taken very early retirement. What? I'm 33 years old. Jesus. Jesus age fucking Christ. No, I'm sorry, I'm not saying this to be polite. I would have put you about 10 years younger than that. I would have put you about 63.

1:27:30 You thought you were probably 65 or something? I thought you were probably about that, yeah. I thought you were about to be early, mid-60s, yeah. Wow, extremely good shape. Well, yes, I know myself any too well, but you too, extremely, certainly. Well, Bill just retired last year, too. He's turned 63, I guess he's 24 now. And as they say, he's never been so happy. Bill will be talking to you last night. Many people love being retired. I happen to like teaching, so I think that I'm teaching a hard time. Well, he loves teaching too, just that I think that he doesn't enjoy it as much. Well, you don't have to do it, do you? No, I've no idea what it's like in Buffalo, but he seems to be very well, very happy. Just having the freedom to... Buffalo is the greatest place to live, I think. No, no, I've always thought that, especially out in the winter. Well, I'm glad you think it's hideous. I was a bit worried I might be expressing a very philistine opinion if I say it's a bit hideous. Yes, I agree. I think it's got respect. And also, I don't want to sound like an old fart, but it's in an extremely inappropriate setting. I mean, you've got that, you know, very elegant ionic portico and... I mean, it would look okay on its own in a sculpture park, I don't know. Didn't pass through the Zurich station. Maybe that's the secret of Santa Fe. Can I just come and savor your pipe tobacco?

1:30:00 I did very, yes, I did smoke just for a little bit for about five years, but never very seriously. My father was a very serious pipe smoker and quite a few of my colleagues. I don't think you have anything to worry about with the pipe. I really don't think it's a guy. Well, I'm sure it does more than higher. Yeah, but just think what it's doing for your blood pressure in general and sense of well-being. Well, I guess we had better going. Yes. Well, I just, well, I was trying to find out. I can only assume the organizers, but did we not, well whoever it was, they paid for a very good lunch. I'm sure we'll find out.