FW Lawvere

0:00 Now the place, I mean if you want to go there I can always ring up. It's probably about 10 minutes but I have to say it's worth it because I think it's just about the best family. And it's run by this really incredibly nice Egyptian guy, Bibi and his wife, who are really good people. And he's a great cook, a great chef. So we want to go around this way since we can't cut across. It's all closed now, so we'll have to go round the gardens. But it's not that far. Before I forget, another reason why it would be good if I got an early night tonight, I've still got to get down a few words. Don't worry, I'll keep it really short, but I am going to have to speak in French. So, yeah, yeah, that's what they asked me to do. So, I just wanted to check one thing with you. When you worked on the cyclotron, okay, that was, okay, well, right, okay, so that was before you started your studies with Clifford Truesdell. That was when you were still a kid in school, in fact. No, no, I graduated from high school in 55. Right. But I met Truesdell then because he was on a scholarship committee. He decided to give me some money so I could study. And then I met him briefly but extremely importantly. You sort of outlined what I should do as an undergraduate three years, then he went off to Bologna for three years. Ah, okay. Well, I won't pull in all those details. I just wanted to go ahead with this initial, extremely important guidance, but I didn't see him again. For three years, yeah. And, in fact, during the time I was a, you know, we have this thing called major in universities. Yeah, sure, sure, I know. So, I was a physics major. Yeah. He came back just when I decided, well, actually, I should be a mathematician.

2:30 Because you had to get to the bottom of math before you could go back in and sort out the physics. Somehow on the wrong track, you know. Because my initial interest was indeed... You know, electronic engineering, really, so to say. Yeah, I knew that your original interest was in engineering. Then you got into physics, and then when you realized how badly physics needed to be sorted out and cleaned up and was on the wrong track, you realized you needed math to do that. Then I actually formally changed to being a math major, as well as started taking graduate courses. I was still learning, I was taking graduate courses that Fusco was offering. Right, so that was when you started, really, continuing mechanics and continuing physics. Yeah, yeah, that's right. Three years. But your stint on the cyclotron was um was in what we would have yeah possible this is in the that began in the summer between my high school right and undergraduate in other words i make it made some money the following year because my major contribution was already right and then the reason i could get this job was because of my I don't think he had anything to do with this, you see, because I don't think so. You know, I mean, oh shit. What's the matter? I get mad when I think about the disinformation in Wikipedia. Oh, God, yes, Colin was telling me about that, that he's cleaned up some of it. Actually, I looked at your entry about a couple of weeks ago. Well, it's much better than it was. It's still got that damn Bayer's entry in the bibliography. Because somehow Colin has made unity with Bayer's, you see. We went to some meeting in Greece by some Greek billionaire. Oh, yes, I know all about that. I thought you were there too. No, no, no. No, no, no. I was certainly not. Oh, I don't get invited to meetings by Greek billionaires. No, no. No, no. No, no. Barry Mazur? Mazur? Barry Mazur was there. Yeah, he was. But that was really why Colin wanted to go, because he wanted to find out what these guys were. I don't know much about Barry Mazur. I've always admired his mathematics.

5:00 I felt compelled to condemn one of his philosophical writings recently. I don't know whether he's simply misled or whether he's part of a conspiracy. Probably the latter. Well, Colin was certainly very, I mean, I'm sure Colin's motives for wanting to meet him and talk with him were entirely to do with the excellence of his math. I don't think there was anything. Yeah, but the reason that the three of them were writing. Yeah, they were, and also David Caulfield, which makes me suspicious. Well, he, I'm afraid, is completely sucked into this Bayer's thing now. I mean, it's just really sad. All he ever does is write about M-categories as if they were the... The only two philosophers with even the slightest resistance give even the slightest resistance. Yes. Yes, I know. That's why I found it so depressing. But I don't think you have to worry for Colin. I think he's got pretty solid intellectual muscle. He's definitely very solid, but at the same time, I asked him specifically to move that. Well, he may feel that's a kind of ethical thing, that you can't really alter somebody's bibliography without regard... I don't know, I don't know, I could see... Well, quite rightly, he took out the rubbish and put in the correct information, but I don't know. I guess maybe it was because... I think the exciting part is the self-disinformation, because if you go through those websites, you find lies about him. Oh yeah, I know, he's always pernicious stuff. Well, maybe it was because he was going to be sitting by, you know, I don't know, I don't know. It was afterwards, no, it was before. I was, yeah, I guess it was afterwards. Anyway, anyway, why don't you, please, I bring up this whole thing. Sure. Is it the original version? I don't know who wrote it, actually. I think it's possible. I think it got started before Baez joined it. I'm confident enough that there was about 12 stages in the development of that article, even though it's only a little paragraph. All of these people actually writing about me, a living scientist, without even bothering to get in touch with you. It's incredible, absolutely incredible. It's really incredible. I agree, it is absolutely incredible. It doesn't surprise me in the case of Myers at all. But as I say, having now Colin has edited it, it seems, well, you're the one to judge,

7:30 because after all you are the subject of it, but it certainly seems to me to be scientifically much better and clearer. Also, it's got a link to his other article, I assume it's by him, I can usually tell his style pretty easily now, on Genesis and development of topos theory. Oh, no, no, no, no, no, no, no. This is not by Colin. It's not? Oh, sorry. It is disastrous, terrible, terrible disinformation. Oh, oh. In fact, the main three articles in Wikipedia, adjoint functors, category theory, origin in Genesis, and another one, they're all... Totally rotten and totally through and through and through with disinformation and lies. It's really unbelievable. Oh, well, I must, listen, I want you to explain this, because I completely see what you mean about the adjoint functor article and the general category theory article, but I hadn't, I'm sorry, that's my fault, but I hadn't spotted that with the genesis of topos theory article. Oh, well, you must tell me what's completely... I mean, these things are mixtures, so he might have contributed something at some point, I don't know. But the crucial point, the crucial... Oh, oh, well in that case I will... I certainly won't rely on that in anything I say tomorrow then! I'm waiting... I'm so busy. I'm going to do something. I'm going to ask the categories list people to get in there and correct something. Well, you must tell me what the... Wikipedia is such an incredible possibility, and yet it's very little used in a positive way. I mean, no, it's used... What I'm saying is that the possibility of revising this information... ...is somehow hard to do if people don't... Well, I don't know how to do it myself. Because in my case, if you take the article about myself, I guess it would be against the rules. Yeah, they are, yeah. I don't know, I don't know. No, but you must tell me about what is wrong with the Genesis and Development of Topos theory article, because I must confess I hadn't realized. I read through that and it seemed, well, that shows that I still don't understand it. The history of the matters I should do. Well, I think, I'm sure it starts off by saying toposes are generalised spaces, which is what I will argue tomorrow. Sure, sure. Of course, they are. All right. I'll argue, I'll illustrate. All right. No, but, sorry, I was off and... No, I completely understand, you know, you must put me, absolutely correctly, you must put me right on that. I think it does say toposes are more than just generalised spaces.

10:00 It does say they are that amongst other things. I had naively thought that was... Everything that's happened since Groton League is mere logic. What is in AMS? Ah, that's... Realize that this whole What Is series was started by the person who's not a mathematician at all, one of the editors of AMS. Alan... What's her name? Anyway, there's a recent article, she brags about how she started it, you see. And now again, how they cut down on definitions and blah blah. Oh yes, I know, that stuff is pernicious. What is a sheaf? What is topostheric? No, I know that's garbage, because I've read there, what is a sheaf article, and that's really bad. Yeah, yeah. What is a sheaf? I see him and ask him what the hell he's up to. Sorry, I don't know what he's up to. I mean obviously my chief authority for the history of topos theory is your article in the Birkhauser. You know, the second volume of the Burkhauser history, that's what I've always taken to be the clearest and the deepest account in a relatively short, comprehensive, common theory. Why doesn't the Wikipedia article refer to that, for example? Actually, you're quite right. I had seen that. I wondered why they hadn't given that as the main item in their bibliography. That's true. And it requires two columns to a column, but I mean... Okay, well, I withdraw and apologize for my very erroneous assumption that Colin had had a hand in writing that. He might have gone ahead of hand because it is a mixed thing, right? Yeah. Sorry, the reason I got off on you, Richard, is because of this bit about the sound bite from Truesdell. Le Verre is more of a mathematician than a physicist. That's the sound bite that explains why I went from Bloomington to New York City. So I told Colin to take that out and he did. Good. But you see, the whole point is physics and Treesdale, right? First of all, Truesdell didn't think in soundbites, but even if he did say that, you see, it is completely misleading to anybody who doesn't know what does Truesdell mean by physicists, what does Truesdell mean by mathematicians.

12:30 Yes, exactly. For him, Euler was the greatest mathematician because he was the greatest, what other people would call, physicist. On the other hand, when he used the term physicist, he often was referring to the 20th century cult of sloppy math and no rigorous proofs, basically the kind of thing you were talking about this morning. And of course, that part of the physics to which Jaffe's line applies, plus groundless speculation in the physics, which of course is what I was hearing about this afternoon in this cosmology talk. Anyway, so just to complete it, just to get this straight about the cyclotron, because it's important in my heart anyway. Well, that's why I wanted to mention it. I started with an interest in theoretical, but coming from electronics, and so I was actually somewhat qualified for this job, in spite of having no degree, okay, but it's not a great thing in that sense, because, at least in those days, a lot of people did electronics with soldering irons and stuff. It's very hard to do now. My dad did. It's very hard to do now. Yeah, we discussed this before. Yeah, very much so. Anyway, so that's what... And I didn't become disillusioned with physics, actually, so I started taking physics courses. Uh-huh. I mean, physics has cultural phenomena. Yeah, uh-huh. It had to do with the, uh, I was operating the plexotron. Yes, yeah, I knew you were steering, you were steering the thing. Steering the magnetic field, it was very exciting. And this was where you first picked up on the, uh... You know, the tremendous importance of Maxwell's insight about, like, screwing up and screwing down, the constraints of the, you know, the metaphor that you often use of trying to create a more and more perfect vacuum is a kind of nice metaphor for what's going on. Because it was the chief of this laboratory, he was a marvelous man. The street in Bloomington where the new cyclotron is located is named after him, the Milo v. Sampson Lane or something.

15:00 Got it, okay. We want to come over here, actually, because this is where we... He taught me, in a way, more than just the theory of physics. Oh yeah, that's okay. Yeah? Oh, huh? In fact, under a particular project, I've had a fantastic... Ah, exactly, in the true Maxwell spirit. He pointed out to me, yeah, there's one of them that's obvious. It's a plane doing a sphere of infinite radius. I've never heard of anything like that. It's like this is a practical thing, you see. I love it. Well, you know, concepts in correct generality are tools for the transformation of the material world. It's great. As I say, I'm very much in the Maxwell spirit. You know the wonderful story about Maxwell as a child always asking his father how things worked and, yes, yes, father, but I want to know what is the particular go of it. I've always loved that story. Be careful, we want to... Oh, yes, we're on green. ...discovered in the years of interference with the radio communications, and the bursts were so sharp that they couldn't... Hang on, we're still on the radar. I think we should wait. Direction finding equipment was more primitive in those days. Anyway, they had it, right. So they finally figured out where the source was, and it was the cyclotron. It was the cyclotron, yeah, yeah. I think you told me this story. Actually, we should really wait. Sorry, I'd say we're still on the road. I know these people are walking. 10 megahertz. We should call it 10 megahertz. I don't understand that, because that line is red. People are walking. I'd wait until it changed, to be able to say so. So, you know, at that frequency I had waveguides, you know, like microwaves, but huge, you see, because they were 10 meters, not 10 meters. Okay, I guess maybe we shouldn't walk. Oh, it's okay.

17:30 Oh my god, oh well that would screw up there, shortwave break, yeah, no question. Thank you for watching. Speakers include mathematics, geometry, algebra, analysis, quantum mechanics, physics, 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, Thank you very much for your time, and I look forward to seeing you again soon. They want to show that we have to bypass. It happened that we were going to do an radio edition, particularly in practice, of that, because the heaters heated the cathode, and the wires were heating the cathode, and they went up along the back, and the heaters were on top. Right, so they're actually just like a broadcasting antenna. Exactly, exactly. Even an approximation of a cracker wavelength.

20:00 Yeah, so they're just like an antenna, just like a broadcasting antenna. Just like an antenna. All right, got it. And so I had this idea, I was learning engineering sometimes. I do so wish you'd known my father, he would have loved so much to hear you talk. There's a huge sheet of Teflon, because it builds a new metal case for the whole thing. So against the sheet, against the metal case, there's a sheet of Teflon, and then instead of wires, slabs of copper, which acted precisely as bypass capacitors, you see, slabs of copper, Teflon, metal walls, wonderful, right? Yes, yes, yeah, fantastic, so it would all work. The copper tubing made into these tub coils, connecting with heat-heating wires, tubing, and of course the copper tubes. This he taught me, I didn't quite realize this, because they would melt instantly with the current if they were just the tubes, so they were stuffed with copper wire so they could carry the current. Otherwise it was my design, these coils and bypasses. I bet, I'm still rightly so. It actually came to my little student's hubble. Because I had my own shortwave receiver, so on the telephone he would tell these collaborators to tune the cyclotron, you see, and I tuned it, you see, it's gone now. The signal is gone. Great. Yeah, well, of course, obviously that would, you know, of course, it would no longer be acting like an antenna. Exactly. It was going where it was supposed to go. Yeah, exactly. Fantastic. That's a wonderful story. My best? 1955, I've got that, okay. In what we today would call your gap year. Gap year, yeah. Well that's what they call it in England these days, but you know. Well, I don't know if the French know that expression. Yeah, the year between high school and college, they call it the gap year in England. I don't know if the French know that expression, but the year between high school and college, they call it the gap year in England. I don't know if the French know that expression, but the year between high school and college, they call it the gap year in England. I don't know if the French know that expression, but the year between high school and college, they call it the gap year in England. I don't know if the French know that expression, but the year between high school and college, they call it the gap year in England. I don't know if the French know that expression, but the year between high school and college, they call it the gap year in England. I don't know if the French know that expression, but the year between high school and college, they call it the gap year in England.

22:30 Oh, and by the way, I borrowed from the physics lab the batteries. I built an electronic device myself, but I needed batteries because I was going to use this in this field to listen to whistlers. I have a paper, my first paper, my first paper was actually written for an English class, you know, you had to write the term paper, yes, for English class. I do know, I did know what whistlers are, I'm racking my brain. I'll tell you, that's all right. Remind me. I'll tell you, okay. I certainly know the expression, I know that bit. That goes in story. Eccles and Storey were English physicists. They were one of the earliest investigators of this thing. So for an English course term paper, I did some experiments. This is the summer of 1956 now. But I borrowed some of it. I built most of it, but I borrowed some of it from the physics lab. It's so fantastic. I love it as well. I understand it's now become something that high school students do routinely. It's a good sort of thing for a project. Well, it happened in the First World War that Germans had long telephone lines. Sometimes there was whistles in there, you see. Strange whistles. And it was figured out later that actually the source of this was very fantastic. It's due to lightning flashes, which are, you know, have all sorts of Fourier components, just crashes, right? But these send out radio signals at audio frequency. Radio signals at audio frequency. Along the magnetic lines of force of the Earth to the opposite hemisphere, like, say, from the U.S. to Argentina or from here to Cape Town or whatever, right? And passing through the atmosphere, where the different frequencies propagate at different speeds. Yeah, which is what Heaviside, of course, worked out, wasn't it? It was going through the Heaviside layer.

25:00 The Heaviside layer was all about that, yeah. And coming back through the Heaviside layer. These crashes have been lengthened out into descending whistles. Higher frequencies come back first. Yes! Well, yes, of course. Now, you can also hear those down there. Where lightning crashes on the opposite end of the earth can be heard here in a radius of maybe 500 kilometers or something around the two points. And you could even, the arithmetic comes into it because let's say it's a lightning crash from here, then the ratio of successive whistles will be even. Whereas if it's a lightning crash from the opposite end, it will be off. And I can actually hear these things, trains of three whistles. It's fantastic. And you see, you hear these things, it's a radio thing, but you don't need radio. You just need an audio amplifier, and you put your antenna straight into the micro jack, you see. And you hear them. And you can hear them. There's only one problem. Since these are in the, you know, 60 hertz range as well, if you're in a city, you're screwed because of power lines and such like that, so that's why I had to go out in the country, southern Indiana, in order to do this, and I needed batteries to operate my, oh, actually the thing, I borrowed this from the chemistry department, I built this in the chemistry department for measuring resistance. The whole thing, right? And then I got batteries from the physics department, and I built it not on my own expense, but because the chemistry department needed an audio amplifier, because I used a special, special audio amplifier for, you know, I mean, it measures your distance by listening to it. Yeah. You know, I, I, I, not only does it ring a bell, but I do seem to hear, I seem to remember hearing about whistles, whistles from my dad. A long, long time ago. He would certainly have known about them because he rejoiced in them. He's a very, very serious student of radio. Hang on. It's still on the red. I'm going to wait for that to change to green. It's all right. We're almost there now. It's just down by the side of the panel.

27:30 Well, Lagrange is buried. And a few other good guys. Carnot. Carnot, yeah. Yeah, yeah, that's right. Yes, that's right. I only knew about two of them. I'd forgotten about the... Well, this one. Well, maybe I've made it too long, but you see the context of it. No, no, no, this is absolutely fascinating. I mean, you know, this is great. I mean, I can't use this, obviously, in your introduction, because otherwise I'd end up like a loony. It would be infinitely more interesting. Well, it might be, but it would still be unfair to you to take up that much time. I want to keep it short, but I just wanted to make sure I had the chronology right, and that was what you did in, what we call, Jogapia, exactly. 1955, that's when I can even mention the year. Because I know that you're going to say something about the, you know, Maxwell's insight about, as you say, the screwing up and the screwing down, and I just want to mention that when you cut your teeth on the cyclotron all those years ago, picking up from that experience and from the engineering interest, ideas which were going to be fruitful, very fruitful later. No, but I just, I'll keep it very short. I'll run it past you tonight, and then I've got to turn it into French, of course. I think since it is, after all, the seminar at the École Normale, the audience will be... Although, of course, they'll be... Oh, they'll all follow. They'll all understand pure English. I mean, they'll speak perfectly good English. But just as a matter of courtesy, I think I ought to introduce you in French. Or maybe I should introduce you for two minutes in French and then two minutes in English. Oh, well that's a wonderful, that is wonderful. I'm so glad you told me that story because you had told me about the problem with the police shortwave radios and how the thing was acting like a huge radio antenna, but I hadn't heard about the whistlers before. That was the other year. Yeah. Practical jobs with chemistry and physics. Fascinating. That's right, yeah. Bypass capacity. Yeah. Of course the cyclotron...

30:00 So I've had a workshop with two very skilled workers who don't fulfill anything that Lionel D. Sampson has to do. I was really very impressed. I had already seen the last plot that happened before, the last shooting that happened. So I had a meeting with Lionel D. Sampson. Oh wow, their eyes must have glazed over when they came back. Well, he sounds like a marvellous man who has had quite a lot of work for his indirectly. Milo Sampson. I don't remember that name actually. I doubt he'll be able to publish it. Let's go down here. Wonderful. And how did she learn him? Oh, wow. But she recognized him through the script, right? That's wonderful. And one of the things that it says in... Ah yes, yes, this is where I remember reading it I guess. This is the one about foundations and in connection with pedagogy, foundations of pedagogy being also foundations for the exact title of the article, but it's one that makes the point about the absolutely crucial, that one should, that one of the most fruitful ways of approaching foundations is through pedagogical issues and understanding how these processes are understood.

32:30 Well, the point you were making this morning is about the market, of course. I'm sorry, there isn't much to talk about. No, no. They wanted me to explain the convenience of my project. Yes, I remember that. Yes, in the sense the cancel meant it. Yeah, yeah. Well, I felt I had to explain it a little bit. Yeah, and this is where, of course, you use the analogy from the back of my... The analogy from when you build it up. Yeah, yeah. You must build something new. This is our way. Okay. Good evening, how are you? I'm good, thank you. This is my friend, Bill. Do you have a pen? Yes, yes, yes. I reserved the table for you. Okay, that's good for me. Okay, I hope you like this. Several determinations of the whole point about having many different levels of different levels of different levels of cohesion or in order to get relative constancy It can certainly be compared with the perfect vacuum. Yes, the absolute case of no coheat or whatever.

35:00 Approaching that, you needed higher technology. I think it's an absolutely marvelous analogy. I don't know if anybody understood it besides you. Well, I did understand it. I will say that. I think I can claim that. I think one or two other people have understood it. I think I managed to make John Stachel understand it. I explained it to him and he seemed to get it. I will tell you what is really good. They have a set menu at only 17, which is the thing which I would recommend is the menu at 24, which goes out just a little bit, which is on page 2. And what they have, which is absolutely, it's the confit de canard, the duck, with honey and almonds, which is just sensational. The great thing he has, which really kind of freshens your something like that, which is pretty rich, is the soup, which, I mean, you don't normally think of having soup as a dessert, but he has this fresh strawberry soup, which is kind of chilled, or the melon soup, and those are kind of sensational. Do you want a pair of teeth to start? Here or something like that? No, I don't want to. No? Okay. No, that'll be neither. I'm just going to stick to one glass of wine between the two of us. Just the basic house wine is fine by me. No, I... Yeah, well, you do have to work tomorrow. On the other hand, well, yes, that's my problem, too. Yeah, exactly, I know. Well, no, that's because we've got to get you on the line. But I would think a half. I'm not a wine snob, I just drink tea. And Bebe's perfectly reliable, I can just have her. And if I was starting off, well, the really sensational thing is the verluce foie gras. But if you're having the confit de canard, that might just be a little too rich as well, I don't know.

37:30 Uh, yeah, I was going to say that might be just a little bit too much, so, um, any of those starters, if you have any lunch, I may be able to have them cooked, or the onion soup, if you stand by, the salad, if you want, something to help, yeah, the gourmet salad is great, I have that, uh, yeah, yeah, that, that will be good, okay, that's fine, good, yeah. Uh, it's absolutely fascinating about the cyclotron, and, um, yes, I have understood the point about the lascum and the analogy of the girdles, you know, the model of the people, well, I've got the six of them, so, you know, that's fine. If you sort of come from a set theoretical background, you'd think it's pretty much the same. Yes, yes, you'd think that everything is in terms of that silo. So the fact that a category is such that an example of such a category can pop out of German construction, but it doesn't retain all of the... it's like a cumulative coordinate system, an arbitrary presentation. Yeah, that's right, that's right, yes, yes, well done. I learned about that from Mr. Merriman. The actual thing I was working on, other than steering the magnetic field, was the nuclear magnetic resonance. Yes, of course they use something now, because they use it to image. And body it. Building down their business.

40:00 The phenomenon itself was new then. I'm sorry, the magnetic resonance is very non-cure. Is it good? Do you have any more? Yes, yes, I think we'll take... Would you take one of the salads, please? Okay. For the French salad, for my friend, and so two confits of canards, and for me... I'm going to order the first one. And for me, I would like to do the dessert, I would like to do the dessert before, the two soups of ice cream, and for the drink, um, okay, and a half of, uh, uh, uh, uh, uh, uh, uh, uh, uh, uh, uh, uh, uh, uh, uh, uh, uh, uh, uh, uh, uh, uh, uh, uh, I think you probably want the vegetables, I think you probably want the vegetables, I think you probably want the vegetables rather than the fruits, yeah, yeah, yeah, yeah, yeah, yeah, yeah, yeah, yeah, With the gas or no? No, with the gas. With the gas or no? With the gas or no? With the gas or no? With the gas or no? With the gas or no? With the gas or no? With the gas or no? With the gas or no? With the gas or no? With the gas or no? With the gas or no?

42:30 With the gas or no? The chef, he'd be the guy, this is the Egyptian guy, he'll probably come out afterwards and join us. He'll insist on giving you a, you know, he'll just eat the nuggets for free and he'll come out and say hello. He likes to come out and say hello to people. Normally I like him after dinner. No, no, no, he's the same. He decides he wants to get some sleep tonight. But no, that analogy was very helpful to me when I was understanding what was going on in this, you know, this campus. So I actually built, using Army's real class computers, which I had a lot of, and which I was already familiar with myself at home, I actually built the magnetic graviton, oh yes, it had, there was a radio signal involved, 30 megahertz, and we were trying to measure magnetic field at 7,000, the precise correspondence, you know, I had to turn, so it was the new Army transcript of the Second World War. They rewired it and reproduced this, because it reproduces the signal, but then there was a special little vial of heated water that would resonate with it. So we put this into the magnet and then there would be a little blip on it, so much of the energy would have gone into the procession. Which is, I guess, more or less how nuclear magnetic resonance devices work today, isn't it? I'm not sure they've kind of fine-tuned the technology, but... Plus, the radon transport. Yeah. No, I knew that radon liquidite was very important. I need to have that. What? Stands. Ah. We got onto that because we were talking about...

45:00 The particular vacuum. Yes. Which, as I say, is a perfect analogy for... Yes, mechanicals. I'm not so sure it wasn't a lot closer to what was in Cantor's mind, too, that he was using the discrete and co-discrete spaces of the tool. Well, the thing is, I think Cantor came upon this recursive idea. I mean, he had it also, if the man wasn't known for that. I'm actually on the subject of the law. You can write down the binomial theorem for the methods too, but it's a subjective process. Well, there it says a little bit like the natural numbers. You can't do that. It can't be completed. I don't do that. I think it's an idealization. It may be useful, but it's further beyond the natural numbers. The natural numbers themselves are dubious categories. The natural number object nature. Even this has to be checked. You can find that universal confidence in a given category. It's there. It's there. This is completely unrelated to mathematics.

47:30 My proposal, I haven't published it yet, but my proposal has those inducted in itself. The category, the basic axiom that tells me this is that there are just categories that represent up to and from concepts like finite and measured. The linear of identity must be recurred in the column. So how? A set in general, that should mean that it's finite. Now this is a very, very categorical area. It means that hom sub 3 is 3 to the x comma 3 equals x. Right, because given a set x, consider any process with, through every partition of the set into three parts, value which one element is supposed to be in it. It does this in a consistent manner, namely any of the 27 permutations of the three elements set. It's compatible with that. If you commute these indices, then the ghostly choice is similarly commuted. Of course, if you had an actual element, this would automatically be true. So the question is, is x such that the converse is true? You can only measure an actual element. That's the definition of the finite set. Nothing to do with building it up. Yes, that's correct, and not a stop from the membership at all. No, no building, no. The axiom, instead of the axiom the natural numbers exist, axiom, there exists a category in an object, in a metaphor. The Compton Equivalence classifies just these sets. Among all the discrete sets, those set of acts satisfy that.

50:00 Consider instead of three, the natural numbers. Or, better, the following connexation. A definite geomathematical object, a fixed one, rather than the endomath in the kernel of the piano, of course it is. It has all the operations and conic sections. See, ring theory is really just conic sections. So you could describe this thing in various ways, but it's basic high school mathematics, considered a geomath object, and it has various endomaths. Instead of Ham sub 3, call it the Ham sub r of r to the x term, and you have some stuff. Is it very nice? Yes, it is very nice. The only function from X to R, just projecting it, think of it as a complex plane. Any map of X in the complex, then this oracle will tell you which real number is the value of the function at this hypothetical point. And again, this should be invariant with respect to all... Continuous, smooth, algebraic. Algebraic is good. Things that preserve a kind of structure. So if you follow your function by any such thing, then specified state points should be similarly moved. Now suppose x is such that there is no such thing except an actual point in x. Then one is tempted to call x measurable because you've just seen how to measure this point. But, in fact, it's the opposite. It's the first, you know, things last in the first motion. Yeah, yeah, yeah, yeah. It's an unfortunate problem. It's a very unfortunate problem. I mean, Bonac had a great idea to investigate whether all subsets could be measured.

52:30 And it turned out to involve not just real numbers, but it was sufficient for it. But immeasurable came to mean that there exists such a ghost point that it's not a point. So immeasurable is like a process of... So again, in axiom there should exist a category, it's going to be a topos, a category in the universe, an actual object, which represents exactly those things, those disputes. Now there'll be others. The universe is Cartesian closed, so you can take the category of small threats to its own power and have the ordinary type of randomness. This is the outright definition of small sets and the assertion that there's no, it's unique. The total of the finite sets and the total of the small sets are both uniquely defined by this algebra, geometry, duality property. Which is absolutely fascinating. So you don't have to worry about all this, this kind of mystical class set that's been postulated by Professor Peirce or Grosvenor University. You're still talking about classes, but there's no iteration. Yeah, exactly. There's not the mysterious. The difference between classes and sets, exactly, but somehow these are... No, no, the thing is, with classes, you can have subclasses of the universe. In fact, we just use it, and some of our classes of x's, and we ask if they can be represented, but we just demand a certain one. No, no, there's always that distinction between elements of the universe and subclasses of the universe. But when I say universe, I don't mean something like D. See, in general, it's very, it's very confusing. In general, then, I accept it. It's not the actual universe of the model, it's an element of it, just an element of it. I mean, the universe of the model of that actual theory, without hierarchical iterations, is completely idealistic.

55:00 Only gods could do all of that, except theorists used to talk to each other about it, that this demon is more powerful than that, so it's true. Well, you were saying that even Grotendieck had fallen into this trap of basing the... There's the universe, the ocean, the universe, the universe, the universe, the universe, the universe, the universe, the universe, the universe, the universe, the universe, the universe, the universe, the universe, the universe, the universe, I really don't like this whole Audi trend, small, in other words, different way. I think it's very nice, but I was really unhappy with the way Alan Murdoch devised what they call algebraic pedigree, a way of talking about pseudo-mur. The quirk does not contain all small elements. The actual set theory is what I'm going to depict. That's no more interesting. If there exists a model of general granite, then there exists a model with the usual full land-decorated properties. This would be a foundation very much in line with, in the spirit of your remark, I remember you making in France.

57:30 ...meeting in, um, I forgot the name of that little place in the Arbenn where... Ah, the Place Beaudouin. Yeah, Beaudouin, that's right. But really, one should think of it as essentially old radiometry. Oh yeah, yeah, yeah. Oh yes, yes. And, uh... So there's this world, so I had dreams, and I had some objects that were called conic sections, but you could take any map, you could have a similar idea, you could take any object like that, and redouble globalization into it, which is... Respect to all its endometrics. I mean, you see, when you talk about ultrasonography, it's the same sort of thing except for these binary operations. It's a general phenomenon, you see, if you talk about a-ary operations for these purposes. You might equally want to take the set B to the power a, and only endomorphism is that. For the monoid of endomorphism, you can always think in terms of endomorphism, you just take the dualizing objects. That's why I took three instead of two, you see. You can use two, but then you have to talk about binary observations. Yes, of course. You have to strain out the real jokes. Yes, so it makes perfect sense to choose something that's not quite as big. And Isabel, I got the whole idea from Isabel. His idea was that you could use countable natural numbers as a test. But again, if you use two of the natural numbers... You see, it's actually a conglomerate of rather adult-stomach observations. You don't need to use all of them. Just addition and multiplication. That's why ring theory became important, why it became important for analysis. Although, in some sense, the natural algebraic theory could use all continuous matters, all continuous canary algorithms, or all algebraic ones, or whatever your category is. In fact, the algebraic ones suffice, even in the Sweeney and Liff-Nulner's theorem.

1:00:00 Liff-Nulner showed the coordinates of the ring and C-infinities on the sense of ring theoretic points. Are the same as the C-infinity theoretic points, but seemingly are much more restrictive than that. Did I say that too fast? Setting up this visualization, the first thing you think of, as an object, is to use all endo-maps of it, or maybe all binary operations of it, in order to test the map connects into it. It may be a non-full, but wonderful thing that Venn Theories are not even your Ambien Theories are actually much richer. For this purpose, so the definition of homomorphism, of course the definition of the ring, the definition of the C-infinity algorithm is much deeper than the definition of the ring, but the definition of homomorphism... Yeah, it's just dependent on the... Yeah, it turns out to be... Just dependent on the ring. Yeah, which is basically the calculation... Yeah, so the algebra, addition, multiplication, and so on, as I said, I think the most shocking way to... Yeah, that's really... And do get in the plane. Rather than thinking of the line. You can think of the line, fine, but then you have to bring in binary terms and operations. Quaternions would be even better. Quaternions are great. Actually, it's a very interesting idea, using quaternions. If I can just track back about, to ask you something about Isbell. One thing I really would like to... Do you want to just have a... This connects with my earlier question about your remark about set theory being, or reviewing set theory on geometry. I'd really like to understand more clearly this issue about adequate and co-adequate subcategories.

1:02:30 And how the category of sets is picked out by the condition of the points, figures, which are the adequate and the co-adequate. ...generating figures at that point, yeah. Can you run that part for me and explain it? ...because the adequacy only makes sense with respect to a background. Yes, yes. That's why I was saying subcategories over there. So it's hard to do natural backgrounds here. We want to talk about the adequacy of a subcategory. Typically it's a small category A mapped into a big category X. Or, this has to do with their being based on some background, both of them, depending on the nature of this x-axis. The really neat thing is you can find s inside f, or even some special sets. There will actually be some very small a, one or two objects, our favorite figure types. Figuring in some x as a map. I'd just say x is Cartesian. If we were to look at those objects s, for which s to a equals s, All figures are constant. So those are the discrete. And that's going to form a background between the A's and the S's. Based on the Galois equation, there's a natural map for the S's and the A's.

1:05:00 The crucial connection is that's an isomorphism. The S thinks that A is connected. So A thinks that S is discrete. That's a good way of explaining it. You can look at it both ways. Well, I don't like... Perseus is better than Trump. No, because, you see, thinking, you might cogitate and figure out something. Witnesses, perhaps, even. It's more direct. Just the direct. Anyway, so, you start with a few, very few A's in that way. This is in that thing you... Right, which, I'm sorry, which I haven't had a chance to look at yet. It's outlined, but it's in another paper that's published. So we don't need any external sections. Provided the extra sort of thing is this amount, we find this discrete part. And the marvelous thing is in algebraic geometry, you don't get sets. You get the Galvatron. She's on the field alone. Over this whole craft about the underlying set of the scheme. There's not an abstract set in the sense of Cantor, not quite Cantor, precisely in the sense of Galois, it's a function on a field. It's a function on a field. There are points on this field and also points on another extension field, and these all fit together punctually. Jurassic sheath conditions enable that any extension of field is a covering. Right, and the case of where it's a set, where you've got sheaths on the one point space, because this is a... Well, of course, in a way, you can correct that mistake by saying, of course, points are never merely points.

1:07:30 This was part of what he was getting at. No, no, no, see there he actually meant these two fields. Oh, I see. Well, I, okay. No, he meant actually because precisely the field extensions have automorphisms. This puncture is not something trivial. For every finite field extension it's set. But for every morphism of fields, it needs math. And for the particulars, the automorphisms of all the fields. See, this doesn't have to do yet with... In one sense, there's a lot of the unnatural things about how the grades reach down. The unnatural stuff that we hear the most people talk about how to reach down, it's like not that puncture on fields, but it's direct limits. Oh, that's a point. You mash these together, but since that category field is not directed, and the categorical sense is directed, it is basically the sense that two things can be... I haven't understood it before and I'm just beginning to try to get my head around it now.

1:10:00 Very interesting stuff. I do see the point about the Galois structure inside it being, in general, richer than the case of Stuxnet and that being, you know... If you have the line 10, for example, that says it's pi-zero, preserve products. This should go down to that level, but not if you go further by this direction. Get the good into its property. You need that category of Galois. The case of the pure abstract set, so when you talk about the cases being picked out as the case where you have the adequacy in the cancel set, so the adequate, so that's good, because that's... It's a digression. I'm sorry. Okay. You pick any reasonable category. Cartesian is always casual. And, so in algebraic mathematics, you can take the spectrum with you only, or you can take the lines, which is really interesting, and there's a very few... By the way, please, please, will you have at least one last of that? I can't let you know without not taking that. No, could I please? Oh, I will ask her for a spare spoon. Okay. Because I really do want you to take that. But, sorry, but please go on then. Thank you very much. This is, you know, this is synthetic. Synthetic means that you put yourself in the world that you imagine and try to write down axioms and describe what you imagine, and then you might find that you have to change it, you see, but basically it's starting from an objective vision rather than some building up process. You can later figure it out.

1:12:30 It's like any notion of algebra comes first. I think even the notion of algebra comes first. The abstractality is that we're implicit or semi-explicit in Boole, or, you know, you have this experience leads to this vision, and it's nothing mystical about this, it's a way of thinking where the experience leads to this vision, and then you try to make it explicit, and you thought, oh, it's an algebraic structure, so I can even present it. I thought it was dependent on Boole and algebra, the abstraction of this oppositional concept. Yes, yes, no, just the opposite way around. And that's why there are many different propositions, what is one of these levels, meaning all the objects that we have, according to Bernoulli, Perewitz, Voltaire, it has to be Cartesian code, so-called. Okay, so you're in this setting. Now, moreover, and this is another feature about such a vision, Cantor had mentioned this set, Dedican's set, according to Emmy Nurture, who actually wrote down, did I tell you that?

1:15:00 She wrote down, she was actually obviously agitated by this question, which is after Phenomenon. And so she talked to Bernstein, somebody who actually had known Cantor and Dedekind, what did they think a set was. Fantastic. It's so fantastic. I didn't know this at all. This is very, very interesting. It's been available there for years. This is incredible. But then, adding to Cantor and Dedekind, as cited by Myrtle, there's a postcard that Max Noren sent me, a very treasured, he was obviously sympathetic. To my progress, I just came across this quotation from Hilbert, you see, the point of Zorn's former statement is, to define a set, you do not have to have all the elements first, this is what they've done. You have the real numbers, the conceptions, or any power set, it doesn't necessarily, you have the idea of a power set, and you have some examples of one, you have both. We know a few examples of real numbers. We also know, you know, that it's a two-word field. These are the two things we have. And then, okay, then we can get into all kinds of things. Oh my God, that is kind of whatever, but we have these two levels of objective vision and a few key examples. Yeah. We know a lot of more examples. Even a very few of those are key. 0, 1, minus 1, 2, 3. Anyway, so the same idea, which is such a category, division is a huge category, Cartesian flows, and there are a few examples. Now this might be the objective idea of motion, might be just that. Or you might think it's an interval, or an arrow, an abstract arrow, which is a category. A category is a category, and what's going on with it? Just an arrow. So there's a few, let's say three or four, one or two, whatever. Some things are clearly examples. So you let these be A's. Now, consider all the S's. So which S to the A equals S? It just says A.

1:17:30 That's a subcategory of exclusive spaces. If there's a street line, we're going to use that as a background. Use that as a background. Now, from this definition alone, it has all sorts of unknowns. It automatically sums products and exponentiation, just from the formula. It also preserves any projected limits that might happen to exist, so modulo the adjoint functions there, and it's a reflective subcategory as well. So there's a reflection, it's a pi-zero concept, which automatically preserves products because of the way that they need to function. So there's a reflection that preserves products. Now, Cantor's bold idea was, it's also co-reflective. Every space has its subspace of pure points, you know, it has its pure points. The part of the space without any cohesion, just think of it as another object, it has to be a discrete object, it has to be a given object of a universal purpose. So that's it, that's the other, now this, so now we have a unity of opposites. Yeah, yeah. Yeah, all right, all right. But again, thinking very much in terms of this as a category of space. A category of space. Yeah, so now we now have the set up, considered a problem of attitude. Consider those days that you started with, or any subcategories, or even any functor for a small category. Sorry, to get any more correct, we should take an internal category and a discrete thing to customize an internal functor. Never mind, let's take two or three out, whatever. Two objects. Are they adequate for the whole category? What does it mean they're adequate? Well, given an object X, we can define a tree sheet with value of nothing. These discrete objects on the category of these chosen A's, namely to any A, to every object X we want to associate to a general object, forward space, dimensional sphere, any object, perhaps far from those that we've thought of so far, we define a functor, we take X to the power A, and I'll take the...

1:20:00 So now this is a functor from X, the category of all the, our original, our embryonic category, into the category of S-value species under H, and then this F is equal to. Is that a full functor? If so, it's, in other words, can we detect, well you could ask for it through the state, if we have a map from X to Y. I think this is a natural transformation. These two functions, these values are S's and matter with respect just to math between the A's, just to the A's, so this might be faithful, in other words, can we tell if two maps between some interesting spaces are equal just by looking at whether they become, they look, the discrete version looks equal after you test it with some A-shaped figures. Yes, yes. You see? Isn't it clear? I mean, it's so clear. You know, now I've really, I have actually grasped it now. It says by two or three steps, but it's very clear. And I see exactly how the case of, you know, the adequacy coding sort of points, and so it falls into... So then, but then the really, I mean, the faithfulness is... Fall in place. The faithfulness is the accountable thing. This is the... What's stronger is the adequacy condition. Namely, suppose you've got two phases. If you look at the speedized version of the A-shaped figures, I mean, there'll be an induced... If you have a natural transformation, then your naturality is quite truthful. It's natural with respect to the maps between the A's. The maps between the A's are called adequate if, given such a natural transformation, there is in fact an actual map from x to the y to the z, so that we can now go turn around and present the whole thing by defining our category to consist of certain creatures on A with values

1:22:30 Now, co-addictively, you consider figures and incidents related. Consider the geometry of incidents as figures and incidents related. But, by the way, I started actually thinking of it more in terms of vibration. It's the same idea. Geometry of an object X, any map in the category is continuous, quote-unquote, meaning that it induces a map from figures to figures that does not tear the incidents. So the dual situation is, instead of figures and incidents related, You consider function and operation, algebraic operations. It just means for any x that could match x into a, to a and a prime, you may have a math theta, call it an algebraic operation. It will operate on any math from x to a and give a math from x to a. A might just be the cube of a prime or something, so ordinary algebraic operations are included in the commutative triangles and the other sort. So that's the algebraic operations. A math from X to Y induces a homomorphism in the opposite direction, obviously, that's just, you know, I said particularly, on the homomorphism on the function of it. And so co-adequacy would mean that there's A's are not algebraic terms. The algebra and the geometry are both parts of the math of geometry. The co-adequacy aspect is really taking care of the algebra and the homomorphism. So the typical algebraic geometry typically is algebra not actually co-adequate.

1:25:00 These schemes are only locally active. It's in the so-called affine state. The algebra of A will be co-adequate among the, via the spectrum. It will be co-adequate among, not among all schemes. So that's sort of the typical thing that, at least the original notion of figure that you thought of describes it. The original adequate notion will not be co-adequate, although you could always, you know, start climbing. But you'd need to perform. But coming back to this measurability, the definition of finite and small, in terms of free and co-adequate, this is precisely co-adequate. In the category of finite sets, free is co-adequate. In the category of small sets, the natural number is co-adequate. If it's small, it's still big. High school algebra. This is more like Coates. It would be within math, don't you think? That's right. So to answer your question, in some sense... To talk about adequacy within the category set is vacuum, because that's already in the background, but co-adequacy, by contrast, is not at all vacuum, because you can find an algebraic, you know, a system of algebras, of faces that serve as algebras, a substance type, within the category set, but that's now getting into the business of measuring, nothing mysterious, but the idea that you could measure. The word measurable applies to the objects in the category, but the category itself is totally different things. Typically, it's the object itself. Whereas it should be whether the objects themselves are doing harm. The object itself is doing harm. That's the point about co-advocacy.

1:27:30 This is absolutely fascinating. Thanks, Keith. Well, I've certainly got, you know, a pretty good glimpse of the scenery. I probably need to go through with you again, but that's absolutely fascinating. I've certainly understood much better what is going on in Hearsville. I was very much helped, actually, by that very touching, but also very, you know, instructive notice that you wrote after me. You've prompted me to ask the question because I'm not sure I can answer it. And, yeah, I really, I do see the point about the Galois distinction between the case of the, where you're using the Galois constructors as a category and the case where you're just using the abstract sets and the way that it connects with the very depreciate from the... In other words, it's very much in the spirit of the world's traditional ideas, the field of things. Of course, it's a non-physical thing, but to the extent of it, it's all human properties that are buried in the earth. And that's the T. It's not far from there. It's a bar atomic. It's a bar atomic. It's a bar atomic. It's a bar atomic. It's a bar atomic. It's a bar atomic. It's a bar atomic. It's a bar atomic. It's a bar atomic. It's a bar atomic. It's a bar atomic. It's a bar atomic. It's a bar atomic. It's a bar atomic. It's a bar atomic. It's a bar atomic. It's a bar atomic. It's a bar atomic. Speakers include mathematics, geometry, algebra, analysis, quantum mechanics, algebra, algebra, algebra, algebra, algebra, algebra, algebra, algebra, algebra, algebra, algebra, algebra, algebra, algebra.

1:30:00 I'm sorry, I'm sorry I never got to taste the pro-al, but I, I am, yes, I think I'm going to. No, no, because that would be, you'd have to have, all the other things would have to be projected, wouldn't they? In fact, doesn't the case where it does have to find, try to pick out the abstract things? Yeah, yeah, that's good. I mean, the algebraic and cohomology case. Yes, exactly, which again makes your point about fit theory being a kind of branch of algebraic geometry. Yeah, yeah, yeah, that's right. Yeah, I think this is extremely deep in the way, well, it's the way one should think of sex. Oh dear, if we could only get the message across. That was awfully fascinating, Bill. Yeah, so, well, it's two double negations. First of all, it's Cantor's negation, the great cohesion. Yeah. Well, Scepteris had ignored it, and he retreated it. We could have analyzed it further on some more objective basis than we have here. Yeah, merci. Oh, that's hot, by the way. Yeah, that's fine. Merci, ma'am. Yeah, it is.

1:32:30 OK, the vegetables are going to come soon. Oh, so it's...