Notes & reflecs / pub JM

0:00 I'm not going to tell you on this point, but in general, it's a pure statement of religion. For a convalescent, a convalescent is not allowed to talk about pregnancy and cancer. These guys, for ordinary people, you know, they've never even heard of these guys. No sense to argue with these guys every day. So, properties, you see, are consciously at the domain of the bigger annoyance of formalization. You all know when to stop it. So all properties are consciously at the domain of the bigger annoyance of formalization. That's not the way it works in practice. In practice, first you have to have a kind of structure, and then that can have property. And there are lots of different kinds of structures. So basically a concept is structure plus property, rather than just property. Well, it's not just structure either, as Colin has been arguing lately. At least that's not the way they normally use property. Structure is part of the presentation. The property that you tack on, the limit, is not the concept. No, there is this school of thought at the moment very influential in so-called philosophy of mathematics to try and assimilate the conceptual entirely into the formal.

2:30 To make it, to say that the formal presentation of the theory is half-stored, there's something to be said for being here today, because you aren't confused by the technical developments, you're just, you've got a powerful mind working on the problem, on the important aspects of it all, so it's not so scary. Yes, but there's also something to be said for being aware of the technical developments in all of their interlocking complexities, so that you can judge whether the powerful mind got off on the right foot to begin with, or whether, like Prege, possibly got off completely. It's curious, what sort of odd, too, is the difference between Europe and the United States? Europe is a different place than the United States. It was an intellectual engine, Europe. It was one of the biggest. I reckon. The unification of Italy and Germany both is a fact. Nobody disagrees with the interpretation of Germany, but I think the interpretation of Italy is a bit different. Unless I can't see that, because the greatest intellectual achievements of science in the German-speaking world were really between 1870 and 1930. I'm sure that's true. Okay, there were great achievements before that, but I can't see them. That's true, that's a lot. I agree. Certainly that's true, and I take the point about Gauss and Riemann. Yeah, that's true. 1858 was... But do you think that that was, do you think, yeah, yeah, okay, all right, all right, torpedo running as we said about, torpedo running as we said earlier this evening about Holland and Plato, but do you think that it was something to do with the institutionalization of what happened in the German universities after the 1870s?

5:00 I know, but that's ideal. I know. That's what I'm saying. That was not typical. It was culturally and intellectually used, but not politically. Right. Thank you. Thank you. Thank you. Thank you. Thank you. Thank you. Thank you. Thank you. Thank you. Thank you. Thank you. Thank you. Thank you. Thank you. Thank you. Thank you. Thank you. Thank you. Thank you. Thank you. Thank you. Like Italy in the Renaissance, but of course... If you think that the Risorgimento is a good thing, I'm going to take you down and show you a statue down on the Arno, which is celebrating it. This is the worst piece of kitsch in Florence, I'm telling you. I'm not going to try to infer 19th century Italian statuary or sculpture, or even 19th century Salento. Well, I think you could make a qualified acceptance of the first part of the statement without agreeing with the second, because what you have to take into account is that 19th century bourgeois nationalism of the time that drove the unification of Italy was the only ...focal point that the progressive forces had at that period. And are you really going to say that the Habsburgs, that the Vatican and the Habsburgs would have contributed more to the intellectual and scientific life of Italy if they'd remained in the driving seat than the bourgeoisie? No, the Vatican, they did, they did, the Sorgimenta did kill the secular power of the papacy world. Yes, yes, some did. Do you mind if I have to remember all of these? Not to mention the Bourbons, and I think you have to admit that there was a branch of the Bourbons that controlled the whole of southern Italy, the kingdom of the two cities, for some two or three centuries before that, and arguably the most reactionary and despotic government in Europe at that time. I think there's an uncomfortable connection between the Risorgimento and Mussolini, and that was pretty rapid.

7:30 What's the time frame? Well, you say it's pretty rapid, but the Risorgimento was completed in 1870 and Mussolini came to power in 1922. It was then part of the general pattern of imperialism at that epoch in the mid-1940s. What about the Sacro-Igro-Ismael opinion? The Italian, of course there was a, of course, Italy as a 19th century nation state had imperial ambitions, there was an imperialist project as there was, in Italy more made accentuated because of the overpopulation, the relative economic underdevelopment, the geographical location. I mean, obviously I'm not setting out to defend, you know, the imperialism of Italian bourgeoisie, but I don't think you can draw a straight line from Garibaldi to Mussolini. I think it's, there were many progressive forces inside Italian nationalism as well. There was a radical, there was the radical nationalism of the progressive petty bourgeoisie, as one saw also in France with the radical caste, if them and so. They were compromised by... You certainly can, yes. They had the problem, of course, of finding themselves in what was a largely rural, underdeveloped, and profoundly- Another country with a lot of athletes now, isn't it? Looks like it, yeah. I'm sure they've got somebody lined up to take his place. How are you doing, Daddy? Oh, I think you should do this. Possibly spend tomorrow, a long sleep in tomorrow. And a leisurely day. Not to be bothered by mathematics, at least not by us anyway, that's for sure.

10:00 I think he's coming tomorrow. He's here already, I think. I think he's here today, yeah. I've met him a couple of times, yeah, I met him at the Chris Eisham's Topos Theoretic Museum. I've seen him about in September, he was reading a book about the military. Oh yes, I remember telling you about it actually. I remember you telling me about it. I mean, has he got anything really interesting to say, or, I mean, no, I didn't think, I know you thought, I know you thought it was a mistake to invite him, but there we are, we, we can invite him. Well, he's played, he's played a, certainly does, because he wrote a review of, David Caulfield's book, yeah, I missed meeting Caulfield, yeah, and he and I did, we're at, David Caulfield was going to come here, but he had to cry off the concert, just teaching, just too much heavy teaching. David Caulfield is a philosopher at Cambridge, well he's actually at Oxford at the moment, who did, he's a close friend of Colin McClarty's, he did a, he was a Cambridge mathematician who switched over to the philosophy department, and who... He has published a book called Towards a Philosophy of Real Mathematics.

12:30 This is a bit disjointed, it's a bit scrappy, it reads more like a collection of several essays than others, but it is on the whole I think a very good book, and his line is that... Philosophers have spent far too long concentrating on the period in which methodological and foundational issues were very intimately intertwined in 1880-1940, but it's about time they realised that the history of mathematics has gone on since 1931, and particularly that they ought to go out and learn category theory. And particularly that they ought to go out and learn category theory, and he has a chapter in his thesis which was about the role of analogies in the development of mathematics, and about the conceptual aspects of the unification of mathematical theories and mathematical explanation, which is largely about Dyle and takes it up to, just about the column, takes the story up. He stops around 1953, around the period of the vial conjectures. But he's also a great fan of any categories. He has a couple of chapters about computer generated proof. You know, it seems to me a bit scrappy on the side of the pie, but what he says about category theory seems to me to be on the ball. Well, someone wrote to me about Myrtle and the Berenites, who were corresponding in 1863. In fact, right in the month I was writing my thesis, and I completely forgot.

15:00 They were talking about some problems with the plane and the poles. But they, they were, they were quite willing to, the most natural thing would be finite types of all of the classes of physics. So, there was this, I think, I think that, you can correct me if I'm wrong, I think this convention that, you know, I said is something that's a member of something. That's an interesting point. I'll come back to that. This idea that you just sort of throw away the rest, throw away the final types over the classes by sort of cutting it off. It's sort of a cut-off thing. So they were quite, in other words, they clearly viewed that, well, this was a convenient trick at the time. For all the categorists who are afraid of set theory, I'll tell you, I'm going to anticipate a remark that you'll make when we talk, but there are only two first-rate mathematicians that have ever involved in a seriously important discussion of the significance of this. One of them is you, the other is me. Thank you for your attention. You haven't read my paper in the bulletin of all time. The late, late... Oh, yes. Well, I mean, the referees or editors insisted that I explain more fully some remarks, which is the basis of the remarks I made in Cambridge in 89. So there's a couple of pages, two or three pages, about how really significant... But it's interesting that the only two guys that have gone public with this are you and Herb, which is, I think, really interesting.

17:30 Oh, the reason that people wrote to me was because in this correspondence in February of this year, a verbal says, you're just someone who's trying to base mathematics on categories. Well, that was the question, was it me? In fact, Peppermint, in a quick note... It's quite possible that he heard from Scott, that he was in contact with him. Yeah, I thought maybe that probably it was through Scott. He was certainly in contact with Scott at that point in 1963. Peppermint had a whole story from Kafka in a quick note, you see. My paper was published in the Procedures of the International Academy of Sciences. Thank you very much for your time, and I look forward to seeing you again soon. I'm as bad as that. I said to Mike, I said to Mike, we were talking about the Bolzano conference, and I'd located 68. Oh, come on. We're not that bad. We're not that bad. Old age is a terrible age. I wish, I wish, maybe we'll get there. Well, you wish the 90s, well, you wish the 60s would come back. Oh, yeah. But you weren't born. No, but I mean, when you look at the science that was going on, you look at the politics that was going on, you look at the personal stuff. I don't know, somehow I had the feeling I missed the 60s. I mean, I left university. When I left university, when I finished being an undergraduate, We're considered, you know, dangerous, you know, degenerates. Two years later, three years later, you go into the student dorms, the newest Illinois models, and you could get high on the stuff without even, you know, it was just a complete sea change.

20:00 Attitudes towards sex, towards drugs. I think you were on the right side, actually, John. I think you were on the right side, looking back now. I'm a fifties man. The fifties lasted until the... Well, depending where you work. Depending on what level you're talking at. Politically, the 50s lasted until 1989. No, I have a Sudanese boyfriend and I can't believe the comments that people make about him. You've got a what? My boyfriend is Sudanese? The comments of interracial relationships, the comments of interracial relationships. Really? In Buffalo? Yeah, yeah, sure. It is. I'm amazed by that. I am amazed. Maybe they don't care that he's black, that he's born. Do you think this is a post 9-11 thing? They don't even know that he's born necessarily, you know. Do you think this is at all related to 9-11 or do you think it's more... I don't know. I mean, in the 60s and 70s, there was lots of interracial couples and then it just... I watched a really awful movie the other day. I'm surprised about it. I'm sorry. Just out of, you know, just out of inertia, couldn't get up out of the chair. It was called The Pelican Group. Oh, I've seen that, yeah. Okay, so it's Denzel Washington and Julia... Julia Roberts. Okay, now, there was, I mean, Denzel Washington, come on. These are sort of carried throughout the 80s. Black and white. Serious. Yeah. Yeah. Okay. And so the whole thing, of course, you can see there's this incredible sort of intersection of the person between the two and the movement. And there's only a sort of feeling that there's a kind of negative kind of reaction.

22:30 I mean, that was made in the 80s. No, but I think things are worse now than they were 20 years ago. That's why I say I would love to go back there. You're asking about 9-11. Well, there is nothing but 9-11. It's become a kind of version of criticism. Why the bosses of Enron are allowed to get away with billions and billions in scam and corporate scams and, you know. Luckily though, Arnold Schwarzenegger is going to sort that out. Oh, sure, sure, sure. Are you talking about the blackheads? The commerce NIH blackheads? Yeah, you mentioned the blackheads. No, not me, my mentor. Your mentor. We have actually put forth 150 scientists from the National Institutes of Health Related Sciences and are systematically taking away our grants from more and more academic institutions, but they do most things in order to alleviate stuff, sexuality stuff, you know, so... Thank you very much. I just had a grant that was completely nixed, and one of my mentors thinks he's got a good knowledge, you know, and he thinks he's one, nobody knows who he is and he doesn't know a lot about it. Oh, they don't even publish the names! Oh, it's even worse than I thought, it's a blacklist, it's in, it's literally, it's less worse than Hollywood and McCarthy, I think, is what it is. It's what I've been trying with. That's all. ...category theory, of course. It was part of the justification.

25:00 Yes, yes indeed, yes, alpha, alpha. ...philosophical doubt. The Philosophy of Doubt. The Philosophy of Doubt. The second book was How to Reduce Reaction.

27:30 How to Reduce Reaction. He understood this very well. He got a lot of information. Yes, he got a lot of... Is it the same Balco? Yes, it was indeed. Yes, yes, same guy. He would have been prime minister. Collins, his work, when I was in a similar conversation with him, can you name a theorem that Descartes proved? Did Descartes change the whole direction of mathematics? Sure, sure. I mean, he did do exactly in fact, yes, and as a matter of fact, you couldn't actually say the same of Grotendieck, but you could say something a little like it, but far, far less than, I mean, given his importance to subject, than he might have done. He gave up on that, not because, as it were, he lost his nerve, but because he thought that he had. ...developed the general machinery, which would make it straightforward for somebody else to do it. Somebody else did it. Well, yes, David did it. But he, I mean, it was really... And, um, okay, he did get the Fields Medal, but, uh, I mean, he was far more, he was certainly far more pleased with having isolated the topos. Fields Medals and Nobel Prizes are not... Yeah, he was far more, um, he thought of the work he'd done in generalizing the notion of variety and producing the, you know, the correct general framework for scheme theory. The development of the concept of topos was much more important than the truth. And I think Bill probably thinks the same way about what he's done. Well, I would be inclined to say, of course, as an outsider who's equally incapable of either, it's more difficult for me to judge, but that certainly is my very strong impression. Well, I'm an outsider in this. I mean, I've never been caught up in the competitive aspect of it. No, no. And it's not that I'm, I'm not as nice a guy as Bill. I'm quite prepared to order from people I think probably. I just never had the opportunity. I have to say, you said this about yourself, I think I see what you're getting at, although I must say, you helped me check the mark a bit, well, I don't think I'm not as nice a guy as you are. Maybe I am, I don't know.

30:00 I think you are pretty nice, but what it's worth, I think you're a pretty nice guy. Let's be real, I think you are. Does that mean you won't have another bloody cigarette? I hope not. Do you want to have other answers? Uh, no. An intellectual... Eddie Roosevelt is a publicity seeker and... Subtitles by the Amara.org community Now, the one specific project that I would like to... I mean, you're talking about, you say you'd like to understand string theory and all that. Well, that's, okay, of course... I mean, there's no one that does understand. I mean, that's... Well, yes, there are. There are probably about 30 or 40, 50 people who understand it. No, no, no, but I mean, if I have one project I want to pursue, it would be understanding what's really going on, this, as it were, subsuming set theory and logic in topology and algebraic topology. But I wouldn't fancy Colin to speak. Colin kind of wants to speak. Colin has understood the history of it in great detail, and Colin's at work in the site of the stages of the project. And Colin probably does have a pretty good qualitative understanding. Colin could teach the subject, and that's always a pretty crucial test of understanding. Though he might not be able to reconstruct the details of all the major truths in his head, but how many mathematicians could do that without him? I think Colin's not going to be able to do that. Colin's more advanced in mathematics over you.

32:30 Of course, of course, I know that. But on the other hand, Colin, you see, is somebody I could really learn from. I think I'm close enough in ability to be able to learn seriously from Colin. Whereas I couldn't really learn directly from Bill. The most I can really hope for from Bill is just... To have made myself useful in a small way, and to get the occasional glimpse of the snow-capped mountain scenery. But from Colin, I think I could learn enough of the subject to satisfy at least an important part of what furnishes this gnawing hunger. Of course, what Colin would tell you to do is start with calculus. It's really hard to learn mathematics when you're at this age. At any age. The reason it's done, it's done as apprenticeship. And that's why mathematical bloodlines are interesting. Mathematicians do talk about it. I'm sure who you teach was. It's absolutely fundamental. What's the guy that built a study with an... Truscott. Truscott. Yeah, now, you know, built kind of hero work. Absolutely. I mean, that's the...these are...this is an indication of how these things work. So you can't...you can't, as it were, you can't do what you want to do unless you're at the feet of somebody who's doing it. And you're not going to be doing that, are you? You're not going to be...52 isn't the problem. Is it? The problem is there's a certain kind of mind that's fast at mathematics and that's not you. That certainly is. I realise that. I'm obviously entirely reconciled to that. But you have a fine mind. It's just not that kind. I know. So it's curious for you. But I'm driven so much in that direction. No one ever, you know, beautiful women lament the fact that they aren't clever, and clever women lament the fact that they aren't beautiful. I mean, you know, nobody, whatever you have is, you know, it's too close. Oh, sure, sure, sure, I know, that's a problem. It's just a normal, just a normal, um, the, the, the, the, the, the, the, the, the, the normal, um...

35:00 There is a component of humans of dissatisfaction. Dr. Johnson said, yes, the reigning error of mankind is not to be content with the goods of life and the terms in which they are given. I understand all of that, but it's easy to say. It's just that I feel glittery. I feel it just as it were. Clicking there beyond my grasp, at the level at which, as you say, something like Colin, I could really learn from. See, Colin's never been a creative mathematician. No, Colin's proved some interesting, relatively minor, conceptually minor... Are you talking about somebody on the level of Bill? No, no, no, of course not, no. As Colin would be the first to acknowledge. I mean, Angus isn't on the level of Bill, but he's very, very good. Very good indeed. No, no, I don't have a desire to be, because that is clearly absolutely beyond my... I mean, by a couple of orders of magnitude, two or three orders of magnitude, whereas colonists have just one order of magnitude. Listen to Angus's conversation. Well, maybe you've never talked to him. I mean, don't do him as a mathematician. I'm just impressed by the immense breadth of his grasp, his detail, of this requirement of mathematics and algebraic analysis. Geometry, you know, I mean he's just, it's partly because he's been focused. To do that, I mean, just think about what's gone into that, you know, years of study with Robinson, teaching at Yale, teaching at Oxford, teaching at Edinburgh, supervising grad students, you know, okay, so he's a guy, he's 10 years older than you, but he's a guy that's, his whole life has been... And it's often... And Angus couldn't do it. He couldn't. Hang on, what did you think it was I said I wanted? You said you were interested in string theory. Oh, no, no, no, no, no, no, I'm sorry, okay, we're at cross purposes. I said, no, the list of things that I would love to be able to master, that's just the level of...

37:30 Casual talk. Yes, of course, I'd love to be able to do that. In the sense that we'd all love to know far more than we do know. But the thing which I was saying really drove me with this kind of visceral glory is specifically to understand what's going on with this programme for the subsumption of all this area. ...into this framework which is essentially built up on the basis of topology and the key concepts of algebraic topology as a framework. Because to understand it is to see how it fits into everything you say about the synoptic view of all of mathematics. Well, I think it is. There is implicitly a synoptic view of the whole of mathematics there. That's what I really would love to, that's what I'd love to write. It's not complete. It's certainly incomplete. If only because it deals with the horizontal connections and not with the vertical organization of the subject. He has serious intellectual limitations. But they don't get in his way in doing what he does very well. So... Even Einstein, even Einstein got out of it when he got into his headspace. It's partly a matter of energy, partly a matter of flexibility of mind, partly a matter of, as you get older, you form convictions that are harder to shake off. I mean, I don't think so. It doesn't matter. It really doesn't matter. Because this is not going to stop him from doing what he does superbly well. But that's what you want. You don't complain to the guy that's a good high jumper and can't master the fold. So, I completely understand what you're saying. I think I may not have got across what you... You'd like to read these papers and understand. I really don't have the ambition, because it would be completely crazy, I would be in need of a straitjacket of being a research mathematician. I know perfectly well that's probably one of my powers, but just to be able to read and understand...

40:00 It's not. Mathematics is a subject where that's the only thing you can do, in a curious sort of way. It's not... Yes, I suppose that's true. I think how wonderful it would be to run the 100 in 99.9 seconds and so on. But you can't really understand, you can't watch and fully understand an application without being able to, yes. So, in some sense, that's not... I don't know how to explain it. I take your point about it being an irrational impulse, an irrational hunger. There's nothing irrational about that. It's forming a reasonable assessment of what you can know and what you can't. I mean, I see things... I mean, I've been doing a lot of work on stuff that I don't really know, out of necessity. Yeah, I mean, you're saying that you were having to learn an awful lot about probability theory in connection with this stuff. I didn't learn a lot about it. I'm just hanging on the fringes of that. But I think, in some ways, that's kind of an advantage. Dick and I were talking about it. What you need is some sort of brutal, sometimes what you need is brutal synthesis just, you know, not to be distracted by what's going on in the profession. You just, certainly you want to know, just find them out. But I did read an awful lot of groups here and I found it difficult. I found it really difficult. And, you know, I get to the point where I say, well, now, if I just spend six weeks on this, I might really master it. But I didn't have six weeks to spend on that. So I was picking my way through, you know, I was trying to cherry-pick stuff I needed. And then as I cherry-picked, I realized I had to back it up. So I had this sort of feeling of what I needed backwards.

42:30 But that's the way one learns any subject. Except, I guess, at the very beginning when you're... Yeah, I guess that's true. Especially if somebody's already a research mathematician. The point is, the point is with mathematics, you can just not understand anything. And then all of a sudden it all, I mean that's not going to happen to you as a historian, you're going to know, you're going to have the date of the Battle of Waterloo in your head, even if you, you know, so you already get a picture, even, you know, and this is why mathematicians tend to, I mean I just had a long argument with Tom about Darwin. I mean, he thinks, you know, he doesn't think Darwin's accomplishments was all that much, but of course he has really studied the character. The fact is that Darwin... I can understand why a lot of mathematicians think like that, because they... Darwin did things I couldn't, I couldn't do. I remember we're having an argument this conversation, right? I completely... I'm sorry, you should have brought some more. No. We're okay. I'm just nervous that you didn't even move. I want to thank Barbara for this morning. What happened when I was reading this stuff on Groove, I had to go back to a very elementary text, which was a textbook on group theory, but it was centered around these space groups. That was going to be to pay off this stuff, so I went to page 99 and started reading on space groups. So I went back and read the first sections, worked the exercises, and bang, the penny dropped. The problem is to understand, in this particular case, what I was stuck on was the difference between inner and outer morphs and why the inner ones are important. ...and why they're so different. There's subtleties in the stuff that you just, you have to bang your head up against them before you realize, you know, it's like this is the point, you know, and Dickens saw there was a hole in this argument, but I couldn't understand his objections and he couldn't, and he admitted he wasn't really, it was a kind of fear.

45:00 So I went back and actually was this all in connection with your attempts to kind of get the Euclidean metric out of this? Well, no. I have to say with a few hints from me about how I thought it ought to be. And Dickens was his second year course in probability theory at Oxford. He did the calculations and he hit on this idea. I'm sorry, I didn't have the technical capacity to do it. Yes, this is the idea which I think is in Finkel's work, which I still haven't been sent to as I kept promising. Well, okay, but this stuff about the roots, I had the idea to go at the minimal parts graphs via their automorphism, That's the key to doing it. And I laid out the axioms for the graphs themselves. If you get the picture, space is made out of minimal parts, like the television picture is made out of pictures. Given that, because it's a street, the central point is the nearest neighbor relations between these minimal parts. So I had the idea. I was going to impose very strong symmetry conditions to make it tractable. The only thing I could think of was counting. I couldn't even get the dimension. I couldn't even figure out how to define the dimension because I couldn't get the counting. I think it could be done. A probabilist could probably do it. So I built in all these symmetries to make the thing tractable. No, so I had the idea that...

47:30 I've built in these symmetries to simplify the problem and then the idea came up with all this symmetry I ought to look at the automorphism group and then so I looked at a couple of uh I looked at a couple of textbooks and I ran across this theorem of bieberbach which says that any uh it says that a group A group with a maximal abelian subgroup, a finite index, is isomorphic to a space group. I wouldn't even know. Actually, I don't even know what a space group is. Crystallography. Oh, that's right. Okay, right. I was going to say. So it's in n dimensions, maybe. I thought I proved that these things, that these automorphisms, there was an obvious sort of maximal being in sub-deformations. But Dickin noticed this really peculiar problem about, he called it quasi-spanning. It stemmed from the fact that you were really looking at a module over the integers. And modules, they behave like vector spaces, a module has... You could talk about a group can be an A module, A is a ring, and the group has an action ring. So in vector spaces, the group is an adequate group of vectors, and it acts on the module that acts on the field of scalars. Okay, so I was using the analogy of that, but I was thinking, and I was taking the group... Acting on the field, not the field, but the integral domain of integers. There aren't any inverses. So I was just going on the analogy there. The vector space analogy, and this is very tempting because it allows you to draw pictures and so on, but there's something deeply wrong with it. So Dickens came up with this idea of what he called quasi-specting and proved the result I thought I'd proved correctly, proved the two-dimensional case.

50:00 A colleague, and I asked him to read the stuff because I said the two of us are flying, you know, or whistling in the dark. So he, he, he read it. And so yes, this is a familiar difficulty and it has to do with semi-direct products and that's what you want to look at. So we did all this stuff and he's now got the class. The point is, and does that give you what you're looking for, the abelian subgroup? Well, these things do turn out to be space groups. Right, which obviously is what you were looking for. So I'll get... What do you mean? Why didn't you put in the other? The other was just another five. You have two five? No, I might have got something. Wait a minute, I've probably got some... Oh, I think I can manage it, it's okay, I can do it. It's alright, you just get the next slot, I'll be fine. Oh, it's okay. Well, yeah, and then take the change when he brings it back. No, I... We got this here. It was classified as a three-dimensional group, the three-dimensional in the middle of the parts of geometry, so it probably has got a start on the four-dimensional one, which is going to be the interesting one, because four-dimensional is a novelist in all the eyes of science. But what I'm saying is...

52:30 There's no such thing as understanding mathematics. Seriously. Because there's always something that you're missing. You know, the things about calculus. It's not like, the analogy is not like a guy that eats a meal and now it's inside him. It's not like a guy that climbs a rock face. The first time he does it with great difficulty, the second time it's easier. Once you, if the rock face happens to be elementary, you've climbed it enough. You just know where to reach. But you don't remember everything. No, I'm with somebody. Now I can enjoy music. You can remember. You can remember things, but you can't remember, say, novels. Yes, because it can, you've got to sort of. Well, obviously, the framework is provided for you, possibly, because it's a matter of, as it were, sort of static things. The nature of the subject matter is self-evident at one point. But in mathematics, so much of it is. If you're pursuing a path in the same way you've done it a lot of times, you really do see it. Something you learn that way. If you ask me to prove this... If you ask me to prove the... The formula for the derivative quotient. First of all, I'd have to sit and think for a minute about exactly what it is. I've got a kind of vague idea. And then I know that there's some really messy, as I've done it, there's some really messy inequalities you have to prove.

55:00 So what would I have as a result of having taught the book? What would I have over you? In understanding, well, I've got a great deal. Give me an afternoon, I'd reconstruct it. Yes, exactly, but that's probably what I'm coming to. Probably take me an afternoon, and it's a trick. Most mathematicians say, oh that's a complete triviality, and some guys maybe would remember it. I don't know, I'm not like that. I don't kind of remember. Yes, I mean, even just understanding that much about it. I've taught this stuff on Gretel's L. And I could give you a sketch argument for it, but the details would take me a day, at least, or a hard word. And even that would be kind of a sketch. But I'd have a kind of image of this sheet moving out of the... Yeah, I mean, the curious thing is that I think I could just about... Okay, so that's a really... About the Gödel-Ell construction, the constructible hierarchy, I mean, that seems to me a lot clearer than that, with complete delusion on my part from all that I've read and studied. I think I could understand that a lot more easily than the stuff that you're talking about in functional analysis and calculus. We just lack of familiarity with the material. Well, it looked very nice. You didn't understand it when you first saw it. Okay, smart. And part of it is confidence that it's true. I wouldn't have wanted to be the guy who'd read the paper. I'd have to have done it with sciences. If you look back from the perspective of 40 years or 60 years or whatever it is, it's really about the essence of what's going on, but it's said in a sort of odd way. It was the first time anybody ever used Lohenheim's code for anything that was

57:30 A really non-trivial use of it. It was a bloody subtle thing. I probably understand that. I don't understand it as well as Philip Welch, because he works with that stuff all the time. I understand it better than most mathematicians probably. But I don't completely understand it. And what I understand is something, is like a promissory note. It's not, oh this is like that, and so on. It would be more like... Give me a day or two and I'll produce a lecture on this. Well, that's a hell of a test of anybody's understanding. Yeah, but it's not the same as, it's not the same as, say, I don't know. The feeling, the feeling is out. That aspect of the... It is subtle, yeah, but of course it also has this, I mean, very direct reflection into topos theory. You've got particular restrictive theory, you lose the detail of it. Well... Yes, because I guess, you know, Bill and Co. would say, but you also, of course, pick up on the essentials. You get the essence and you see what's really going on. But your point of view is that it doesn't matter what's consistent with the axioms and set theory. Well, no, that's a bit of... No, on the contrary, it does, because he wants to understand the concept of the axioms and set theory precisely from this topos-theoretic viewpoint. I mean, one of the things which comes out of, as I understand it, from the topos-theoretic viewpoint on B-Casel is that... 13.50 Um, I'm going to get another line. Okay.

1:00:00 Sorry. I've got to indulge my voice. Excuse me. What I'm getting at is, I'm not sure what you haven't got is a potential mathematician's idea of what understanding something is. No, that's probably right. Maybe your vision of what understanding something is is better. That's why I was telling you this stuff about knowledge. Yes, it could be. I mean, I do think that in the case of V equals L, I mean, one of the things that the topos-theoretic kind of... The final version of the argument brings into focus is just how foreign the axiom of replacement is to the topos-theoretic way of thinking. These are kinds of trying to get everything that you get via the axiom of replacement in the standard set-theoretic way of thinking, which sticks out of what essentially are these constructions on the function spaces in the algebra. Defined via the appropriate limits and co-limits in the topos and these conditions have a value. I would like to understand, I mean, hey, you can see how far I am from the mathematician's sense of understanding with these kinds of hand-waving marks. This is why Bill Hangs put so much importance to the vehicle's old construction, because he's got... No, no, no, I was just talking about two different things. There's the Tierney proof of the Fortham construction of the independence of... The topos, which is a sheath theoretic argument, which hinges on this way of looking at quantifiers as adjoins, which is very subtle and looks too Mexico, as I say, once you've understood a fair bit of topos theory, relatively straightforward. And then there's the V equals L stuff, which is quite a separate issue. Well, Bill did it, didn't he, correctly, in his...

1:02:30 ...beard him in his den and ask him to explain... Not tonight, no, but maybe in some time over the next week, but he's... I mean, this is one of the things he repeatedly banged on about, but the Wiesel argument is... he sees Wiesel as the condition that enforces constancy on the... Ah, so he's doing something funny with the logic. Yeah, he enforces constancy to the maximum possible extent that it gives you... The essence of the argument is to prove an axiom of reducibility. Okay, so then you bugger about with the logic, and the axiom becomes a different thing. Yeah. But the actual argument Gödel used, which actually established a rather important fact, which was that there was nobody going to prove the axiom of choice from, or the continuum hypothesis, from the Zermelo-Krempel axioms, and that means, practically speaking, There's a strong conviction, it's not true by definition by the way, but there's a strong conviction that there's not going to be a proof accepted that isn't formal, that sort of thing, that isn't formalizable in ZX. That's the source of the prestige of the result. And Gorotovsky and Hilbert and all these guys had a go at it. All had a go at it and all thought it through. And we now know that there's nothing that they couldn't have done that they were trying to do as a result of Gerard's work. So the action-reducibility aspect of it is critical. So from that point of view, it's no good to say, well, if you take the essence of the argument, you can really just bugger the logic about it, and some of these constructions continue to work, and they're much clearer that way. That's not what the result is. The result is that you can't, that these, I mean, the thing that's surprising about it is, here's a problem all these really clever guys worked on, possibly thought there was an answer to, and there isn't. In the terms they or we would accept. Now that's a deep result and you won't get it out of Thomas there.

1:05:00 Well I don't agree. I think that's precisely what you do get out of the way of looking at it because it brings out the condition that you're imposing on the... ...on the objects in this, from the point of view of the kind of geometric structure of the objects in the topos, in order to have the, well, in order for the acts of a choice to help, for instance, is a very restrictive one in geometrical terms, because all the objects are projected, subjective and injective mass, composed in this particular way, which has this meaning in terms of, you know, coverings, for instance. I'm not denigrating the results, but what I'm saying is they aren't what Gödel proved. What Gödel proved was that these people couldn't have done what they wanted to do. Because what you're overlooking is the fact that ZF formalizes the actual methods mathematicians use. It's useful to look at it as a structure and look at these analogous structures and so on. That's a perfectly valid way of looking at it and fruitful of all kinds of interesting consequences, but it isn't what Goebbels proved, and it's not why it's famous. So what he proved was that Hilbert and all these guys were barking up the wrong tree. There was no way they were going to go. Now that's amazing. And that's why people were amazed by it. It's just, I mean it's the first time anybody, I guess it's the only time. The Gödel and Cohen results combine us with the only case in which people showed that there were classic problems that first-rate mathematicians actually tried to solve, that they couldn't solve and that we can't solve. So where do you go from there? Maybe there's something wrong with the whole picture. Well, that of course is what I guess the topostheoretic take on these things is suggesting, that there is something wrong with the picture. Which is that the difference is that you're building in an awful lot of geometric assumptions on the structure. Of course, I'm arguing against what I actually think. But let me put the devil's case.

1:07:30 The difference is that sentence, although it can be looked at, it can be looked at structurally. That's not what it essentially is. I understand your position on that. It's not what it essentially is because it is the portion of mathematical thought which makes available a rigorous notion of structure in the first place. So one's simply confusing levels of discourse in trying to produce a completely structuralized account of this and then just show how it's related to other portions of mathematics. A network of ideas or, loosely speaking, of mathematical forms, none of which is ultimately... ...in any final sense prior to any other, but all of which are simply interlocked in this freestanding way, and it's just the conceptual organisation of this evolving network that's dealing with it, but that's the only notion of foundations in mathematics we have, and what he's lost, I understand what you're going to say, I understand precisely your objections is... There is at some very deep level this deep insight, which does seem to me is correct, about the nature of the conceptual organisation I develop in mathematics, and what I call the kind of horizontal organisation of the subject, has got to be brought back into contact with vertical organisation. But the vertical adaptation isn't a matter of slotting something in underneath an already going enterprise. The enterprise itself is built up this way. Or maybe it shouldn't be. I'm convinced that it shouldn't be, actually. But that's what the enterprise is. That's where it came from. That's how it's taught. That's where it's at. Looking at it in this curious way is fruitful in a lot of ways, but it can't be philosophically sound, because it just simply ignores what's really... what's just basic, you know, you've got to prove things.

1:10:00 I mean, we make our students prove piddly little things, so they grasp these concepts. If you can't grasp what a structure is, that's very difficult. Yes, I agree. People, clever 18-year-olds, sit there with their mouths open and don't understand what you're saying. And don't get it for a long time. They never do. It's not made easier by the fact that the primal physicians get to them and tell them that they don't need to bother with all that stuff. It's just a cooking book of recipes. There's a lot of tricks involved. For some purposes, that's what it is. Because it's one level of organization of the applications and stuff. It's an enormous subject. And you can get by in large parts of it just by the recipe. We get by in all of it. Because even the guys, the young bloods that Colin was talking to at Harvard... I didn't understand the fundamentals of what they were doing in the way Collins thought would be necessary to really... he came away quite disillusioned with the whole business. It's curious because in a way I would have thought that that recognition would have been Krista Collins-Mill which is... Given that he is very much on the kind of McLean side of things, he does think that there's simply the conceptual organization of the revolving network, rather than, he thinks that any worthwhile talk of, what I said, using the short-hand vertical organization comes out of, so it grows out of, you know, horizontal organization. So I would have thought he would have been quite happy with that position. Because they hadn't even carried out the conceptual analysis, they weren't even aware of it. They were sitting on top of the shoulders of giants knitting together things that they saw people around them doing and making massive patterns and that those guys and they had no conception or no interest because the payoff is in is in what you weave at the end what is it the cutting edge is it just peeling back the another layer so that's why probably people under April I'm not sure about his theories. Because he doesn't do that sort of thing. No, no.

1:12:30 But it's all in sense. Do you remember it? Do you know any theorems of it? Well, the answer is yes, Bob. But the point is that, yeah. Well, I mean, I'd be hard pressed to think. I don't really know what he did. I don't know what, I know that he was worried about equations of the fourth and fifth degrees. I don't know what, I know that he was worried about equations of the fourth and fifth degrees. I don't know what, I know that he was worried about equations of the fourth and fifth degrees. I don't know what, I know that he was worried about equations of the fourth and fifth degrees. I don't know what, I know that he was worried about equations of the fourth and fifth degrees. And getting planar solutions to them. It's all just sort of a jumble in my head. I couldn't say exactly. If I tried to read the geometry, I'd find it really difficult. Well, that would largely be because of, you know, the foreign, because of the language, you know, the concepts, you know, the organizational subject is so foreign. Yeah, because I'd be trying to do it before people had worked out routine ways of thinking about it. So the fact that people rely on these routines without thinking about them is not necessarily... Not the contrary, the point you've made over and over and over again, which is that when concepts become sufficiently fundamental, they're not, sorry, foundational, because you draw back into the distinction between fundamental and foundational. But when they become sufficiently indispensable to getting the subject off the ground, they're not actually, for most of the time, expressed in formal definitions at all, they're simply expressed in the notation. The idea is to make things invisible. Yes, and they are made invisible. The ultimate ingredients of definition concepts are largely... You don't want to... The last part of what I'm talking about is actually embedded in notation. You don't want to see that. And the best illustration of this is general topology. General topology is about getting the notion of nearness. You could tell a big story about this. But in fact, it's the first thoroughly developed and universally used theory of this peculiar sort, modern theory of this peculiar sort, where by organizing the thing correctly...

1:15:00 The concepts you started out with just disappear. And you work, for example, in the case of topology, you're interested in questions about continuity. So there's a notoriously difficult definition of continuity. But when you do general topology, it's usually taught, it can be taught sort of axiomatically. So that you just give the accents for a topological space. They're very simple. And then you start developing things. You assign exercises to show people that certain... And so on. And how you define the open sets. But you don't pursue the provenance of these ideas because that comes, you have to actually master the stuff before you see, and I think probably a lot of people that are perfectly competent at it haven't even thought about these questions and don't understand what's going on. This is an extremely interesting example because there's recently been this debate on the phone about the historical relationship between the arithmetization of analysis and the development of set theory and the development of general topology. There's this big spat between sets. There's a couple of other guys. Actually, it was in the context of the debate about Caulfield's book. I don't know if you picked up on it. Well, it was interesting, and Simpson, of course, wanted to say that without the foundational interests of the creators of Probe and Set-Theoretic, there would never have been general anthropology. He's right. Yeah, I'm prepared to say that he is. And other people were saying, other people were writing to say, no, general topology was in the air in the mid-19th century, there were all of these ideas about continuity and convergence, which would have, which would have happened in the 17th century. It's just that they weren't properly formed yet.

1:17:30 In around the 19th century, some of which of course went on to be developed into other branches of topology than general, point-set topology, like simplicial complexes and so on. Which might have kicked in, which might have provided an alternative had it not been that set theory, the set theoretic foundation came along and that provided the basis obviously for the general concept of topology. Well as a historical point I think Simpson's right I don't think. But of course one of the persistent themes that Bill Hammond's all over about and which I think is really interesting is that looking, and this is one aspect of what I would so desperately like to understand. I don't understand what's going on in this program for the assumption of set theory into the theory of mappings between spaces as conceived in the setting of categorical homological algebra. I mean, Bill's got this point that Steenrod and Fox and all these guys who were the big, big named apologists in the 1940s all stumbled in... In various ways, you know, from slightly different points of view, upon this fact, there was something fundamentally wrong about the way that the subject was organized and that the thing which was lacking were these nice... Map space properties, which one would rather have than to hold in general between any spaces, well, objects in the tattagic topological spaces. And it was precisely the fact that taking the open sets as the defining notion, the fundamental defining notion for a conceptual space in general topology.

1:20:00 The thing that was missing was precisely the kind of, the additional stuff that ...which one doesn't get in a natural way, allegedly, by simply imposing no further structure on open sets, but one has in the case of the path-connected spaces, and that one should actually see the spaces where you could do everything simply in terms of open sets and coverings of open sets as a special and restricted version of the structure that you get from starting with... These are half-connected spaces, and seeing those as it were as just one stage in building up to these more complicated constructions, well, from one point of view, more complicated constructions that start out with figures with higher levels of connectivity. If you don't understand what that means, you wouldn't know, would you? Well, I wish I... this is precisely what I would like to understand, but the fundamental object in... one of the most fundamental constructions in algebraic topology, in algebraic topology, is the components function, this is pio, this thing, this, this, this, the components How the space is connected, how, I mean, say, take the example of a space that just consists of a path and a point into the disk. I just don't understand it. Okay, I'm floundering because I'm trying to recall stuff that I've read and not quite understood. You understand now why I'm so frustrated. There's, um, I mean, you know about Hurowatch and this, the stuff in 1940s algebraic topology. Well, it does seem to me to be, well, that's precisely what I want to understand because it's absolutely clear from... He regards this stuff as absolutely key to understanding what topology is really about and he sees the open set, the whole of general topology, as falling out of this in the case where you've got what he calls adequacy and co-adequacy conditions for spaces which are...

1:22:30 ...are just defined in terms of the structure on their points, but this is a very special case in this more general setting, which is to do with the way that you restrict the components... You have a strange idea about spaces not having points. That's right. It's strange from our point of view, but it's not so much that they don't have points, but the case of the spaces which can... the structure which can be defined simply in terms of structure on their points are just a very special and restrictive case of spaces in general, which are really defined in terms of figures. The way that Euclid's actually thought of it is this notion, this kind of generative notion of it. Generating figure is the fundamental concept. It's trying to take that sort of very ancient intuition and make it, make something that really works. This is rigorous of this geometry and algebra because it looked at analytically, trying to say, are not, are no big deal. But that just reflects the fact that analysis doesn't connect with certain aspects of the way we think about things. I'm expressing this very badly, but there is this key point about, okay, if you've got a more general notion than topological space, which topological spaces are special? But the point I would want to make is that this is not some idiosyncratic notion of Bill's, I mean Bill insists that this is the way that all of the great figures in algebraic topology itself since the 40s have increasingly gone, and that the whole development of the subject since, you know, sort of Steyroth, Bott... And that's what was really going on in Eilenberg-McLean. That's where category theory came from. It came out of this recognition that topology had essentially taken a wrong turning in basing the subject on, as in general concept topology, on the notion of open space as the fundamental structure from which you started in defining the way that space... Yes, because algebraic topology is interested in a different class of problems. It's the thing I know least about in mathematics. I mean, to me...

1:25:00 General topology is a tool. It's inconceivable how you would think efficiently about problems in functional analysis without using it. And it's taught in the second year, and students are expected just to master it. The notion of compactness, the notion of all these things. And also, he wants to say very strongly that that definition of space, that one has in general, that when one looks at what he sees as the absolutely central structure in understanding mappings between any two spaces in general, Such as obviously have to be defined to have a good understanding of the category of spaces in algebraic biology. It's precisely the fact that they don't have, that when you base your definition on the point set... General topology. You have failure of Cartesian clauses. There are all of these nice properties that the mapping spaces should have that you don't get in this picture. And he certainly said this to me quite specifically, and I've seen him commit himself to it in print in many places. But this is precisely the great insight which... These guys in the 30s and 40s had, that led to the work on group cohomology and the development of the theory of mathematics. And so on and so forth.

1:27:30 The notion of topological space, that one has the kind of deepest payoff, or at least the early… Ordinary working analysts were skeptical about general topology, and people working on differential equations were skeptical about it until they started introducing solutions to problems they were actually working on. So it was dismissed as abstract nonsense. Which is ironic, of course, because it's just what happened in the 30s. It was dismissed as plagiarism. Sometimes things remain looking like abstract constants for a long time before somebody sees how they are. But certainly the way he sees general topology as fitting, he sees general topology as an interloper. Within the way he sees that, you know, the structures of spaces and sets of discrete and co-discrete spaces are fitting in, he obviously sees the way that general topologists think as... Well, somehow having taken a wrong turning, I'm absolutely fascinated. That's what I would really like to understand. That's the kind of gnawing hunger I was working on. And I think I'm just probably capable if I spent... I've spent two or three years studying all the right stuff. That's why I'm so frustrated, because I'm not so naive, I'm not so childish as to imagine that I could ever be a researcher. But I think I could probably understand what's going on here. I certainly think I see enough of the overall, the salmon hang of this, to see why it's so important, because it does appear to represent the decisive twist in the dialectic that's been going on since the very beginning between geometry and mathematics.

1:30:00 It's driven the development of mathematics. Just have a good... Qualitative understanding that would allow anyone to make at least semi-intelligent commentary on these things, but even that would give me an enormous amount of solace and allow me to achieve something very well worth achieving. Pushing your mind as far as you think it could reach, knowing that there's an awful lot that it never ever could reach or will reach or have the time to reach, given the finitude of our lifespan, is one thing. The source of my frustration is more specific than that. I sort of see what you're getting at, maybe I'm... It's also a metaphysician's worry, because one wants to understand within the categorical structure of our understanding of the world, which concepts are prior, what is prior for us and what is prior simplicity, what is primary simplicity. Your conviction that what is primary simplicity is... Ultimately, arithmetica universalis is a very natural one, but I'm not certain that one might have an understanding of the... And this would ramify in the most profound ways, it would ramify out to physics because of course if this conviction that there is some unifying structure to the concept of space in general is right, then it would seem that an awful lot of the way that physics has gone in the last 70 or 80 years could also be going up a blind alley, which of course is another persistent theme of all this. You know, the unrelenting opponent of non-commutative geometry and why he thinks that non-commutativity is actually coming from somewhere quite different. I'm sorry, but it's not to do with the revision of our concept of space. I'm floundering. That could be completely wrong. I'm floundering because I don't know the basics. I mean, I know something if not on the top of it. I don't really understand the molecules. Well, that's what I would really like to understand. But of course, in my case, it's running before you can walk.

1:32:30 At least in your case, you know what you have to understand before tackling it. Thank you for watching. No, I'm not saying, the project itself is exotic. I mean, the danger in these things is that, the danger is that if you, it's like, think of the mathematics community, mathematics, the profession is kind of like a sort of monstrous individual that's trying to grow. I see Boris Karloff in Mental Gaze, obviously. So, I think the analogy is with Antaeus. It loses its strength when it's not firmly grounded on the earth. So this is why even people who deal in abstract constants deplore abstract constants. The skeptical attitude of analysts and solvers of differential equations and applied mathematicians and so on about this stuff is healthy, but it's also, I mean, it's like the conservative, it's like the conservative part of a, of a, of a, of a, of a, of a, of a, of a, of a, of a, of a, of a, of a, of a, of a, of a, ...and so on. On the other hand, if we try to do what Stalin tried to do... We end up in a mess, too, so... I think that's being extremely kind to start with. Well... Credits him with good intentions and just editing. Well, I mean... But I understand what you mean. Okay, so... So, comprehensive top-down social re-engineering. So the reason... So that's maybe the quixotic element in it. Is it... You're looking at it... I mean... Or the 19th, and all of this is getting hard.