Morning Discussions, incl. A MacIntyre — agenda / FW Lawvere & A MacIntyre — SUD objects & QD topos (& others) (contd.)

0:00 Well, what is cohomology this long term? It's an easy calculation to get into my face. And that's a good question. I don't know if that's a good question. I don't know if that's a good question. I don't know if that's a good question. I don't know if that's a good question. I don't know if that's a good question. I don't know if that's a good question. I don't know if that's a good question. I don't know if that's a good question. I don't know if that's a good question. In fact, these don't necessarily understand much of the difficulty of quantum physics, and that's something we're playing, and I won't say it has to take care of itself, but, you know, it's something that I feel like I'm going to have to learn all this stuff up from these, so I'd like to be able to fill that in. I'm not confident in it myself, really, but at least I will know pretty accurately what it is that needs to be figured out.

2:30 Okay, let's get the cheese and stuff, and we can try to encourage Bill to come out with... You know, how it was that he made his vision of the logical structures lying within the algebraic geometria, within the algebraic geometry of Christendom.

5:00 And something that's clearly very important there is recognising the geometric meaning of choice and extensionality principles. They connect with principles like relative uniform, separability of spaces, and there's obviously stability of both the sections in the case of choice. That would be, that would be helpful for you to discuss with me. Sheer bloody stamina. I mean, you can see why he was such a great man. Well, he is a great mathematician, but, I mean, you can see he's just absolutely incredible. And he has the, he has the, the enthusiasm and energy of a, of a... Absolutely, yes, he must have been like it. No wonder he was the star, if it hadn't been for all these great people like Ren in the 17th century. Ren and they just had the misfortune to be the contemporaries of Newton. They just had them all. These guys, quite the loathes, but obviously were so enchanted. Or in any other generation they would have been. That's true. It's something like Einstein's contemporaries, you know. Except that with Einstein it was much more a bit of luck written in Einstein, because after all, relatively, it might, Poincaré and Hilbert certainly understood the math far more than Einstein. That made Einstein's achievement only more impressive, I think. Well, yeah, it's not quite as good as Einstein's achievement. No, no, I know. There's a lot of time in public relations, not on his part, but all the time. Well, that's true, and that's a genuine activity we were putting in the campaign. But only had Hilbert got them. Of course, Hilbert was going by a completely different room, I'd like to understand that.

7:30 But had Hilbert published the equations first, and had the eclipse expedition confirmed, then Einstein's name would probably never have come down to the general public. He would still have been a very major figure in physics. It was very good, very good. And the Marxists too, good Marxists. Yes, I know. I like it, I like it, it's great. That was one of the major figures in Einstein's studies. Yes. He came along to give a talk at the Observatory of Einstein's, obviously, and his office, which was a, well actually, he gave a much better talk a few days before the technical conference, still a good talk. He got this false There are four lines between Frank, but just sufficiently well-equipped for him to be, who has got this website, who has published the book, trying to argue that Einstein was a very funny playwright who stole every single one of his ideas. I mean, some of the stuff he comes up with sounds exactly like the kind of stuff that Frank, as an arctic scientist, used to do. But, you know, there are these lines. He was really a bad physicist. Absolutely. I'm not sure if that was his real reputation at all. He stole every decent idea he'd ever published. I mean, it's a bit of traditional stuff. I mean, Stakeholder didn't trounce at all because the guys didn't know. He made this big issue about the, which I have heard in his business, about the updated publications of the general calculations, that I started to see this manuscript of Hilbert be published. But, of course, Stakeholder sort of said that the whole point is that I think that Hilbert has been working by a completely different route. I mean, arguably, the path of mathematics is a route which is subject to the overall development of the world. I mean, Pius, after all, in his book, I mean, it's a really major, you know, in that way, I mean, it's a really major, you know, in that way, I mean, it's a really major, you know, in that way, I mean, it's a really major, you know, in that way, I mean, it's a really major, you know, in that way, I mean, it's a really major, you know, in that way, I mean,

10:00 I wouldn't get much more credit for Gino than he gets, which is a fair amount of his own credit. Well, that may be true. But he wasn't such a cheater. No, I'm not. Okay, how is it? No, I'm all the same. Well, it won't go up there. Let's put some... How do you recall that when your legs go numb? There's a name for it. It's been publicized recently that people often get it when they go to a Atlantic flight. Oh, yeah. I have to have eyes to see that. Tenurism, maybe? Tenurism, right. I'm sorry, because there's some patient earlier than that. The aneurysm is, of course, right. When you actually have a blood vessel, which plays up after I sit down. When it actually births. When it actually births. That's in fact what Paul Michael Redhead had. And then he killed him. He killed him? No, he nearly killed him. Redhead survived. And he's much better now. But he's still, yes, he is still British. There we go. Some more chairs out in a moment. Oh, I've got them.

12:30 Do you want to get a couple? Um, yeah, actually, we could just take these out, but I can even use that little one. I could have dragged them down from upstairs into the paint. So you get this surjective geomorphism from the original one to this new one, so he has a paper some time ago where he exposes this idea, and these toposes in many ways play the role of generalized spaces. Because he characterizes the torsion sites, you see that if you take the appreciates on the category C, and then in any case, they call on you to change the point, this will be such a key element, in other words, all of the solutions are decidable, if and only if, in the category C, it matters enough you want it. It's a cancellation of that. Now that means, in the scene, they usually won't be able to work as effectively as they usually do, relative to other objects in the scene itself, if they really have the same relation to one another.

15:00 But that's a problem, if you just decide to have this problem. So clearly... Clearly, the classical spaces are where the ones are actually going to set, but trivially, if you have only one map, I'd have cancellations in between two of them. On the other hand, a group, every map is invertible, so in particular epimorphic. And that's sort of the other basic example of a generalized space, the G-sets, because... So, and then some of the general covering theorems and so forth that were true for the so-called locale or preset case were better formed, so whatever it was. Anyway, so from those two examples, namely post-session groups, you can see that it's a reasonable idea of a generalized space. Different from the other one, Etan Du, which actually includes both cases, though, is to ask for a little category of all-national monomorphisms. But that turns out to be inadequate in many ways. Anyway, so anyway, so now the point is that the key etalto pose to the scheme is a human of a mathematical scheme. You can take as a site all the connected etal maps to the base, and the fact that you take them connected, they do have this property, that you have two maps over the base, but when you agree on a point, they equal each other, and if you agree on a point, it makes them equal. Thank you for watching.

17:30 I don't know. I don't remember. Well, last night, I did put it off on points, apparently. Well, I don't know. So, how do you... Such as B-slash-S. Mm-hmm. Skip Tuesday and generalize. And then Thursday we go out to Miller? Mm-hmm. This is the property of the... I see that. I see that. And so the QDs, unfortunately, don't have that stability, but they could easily extend to something which wouldn't be a big disadvantage because half these actions are focused off spaces, what we call growth, and so one wants to generalize the space, particular spaces, enough that they don't intersect that. It's a kind of natural... There is now at least one rational definition of generalized space, and it happens to be decided in that sense. All of the above can be found on our website at www.magnus.com.

20:00 No, no, this is one of the slightly other topics I thought we would come up with, but very much so, it's imprecisely the topic we're going to be talking about. Well, you're doing a bilateral, you're doing a bilateral, but I knew I couldn't stop you from doing that. I'm not going to be taken out and shot. Bilaterals are discouraged, but not, you know, they're not a thing. As long as the really interesting stuff gets, you know, served up for the others, you know, while the recorders are on. From the very beginning of the topic that I've been studying here, I've been trying to establish a certain way of how the geometrical concept thereof seems to work. It's actually Alberta that we're in now, which is one of the ways that I'm deciding how we do it. Good, good, good. I know, I know. You haven't cut yourself yet. No, I've cut myself. It wasn't a valuable family heirloom, I think, but... Give it up, but it wasn't valuable. Yes, yes, yes. What? We saw hardly a ribbon. We might have had to clap. No, no, no, exactly. No, no. No, no, I apologize for that. But now you've got the corkscrew. No, it's all right. I'm not worried. I really only like Italian wagers, corkscrews, and their lines. No, I like the other kind. Just they're easier for me. Well, maybe I... For me, those are, you know, those are canonical. But you have to have a sort of, you know, a mighty arm to use them sometimes. It's the wrist. It's the wrist action. It's all right. Whatever we do, I promise, we're not going to...

22:30 Give us a private thing. No, no, no. It's obsessed. It's obsessed. Those are fighting words, Bill. You're a man who goes on about artifacts and, you know, the higher forms of art. I know, no violin, no violin in the world. I wasn't quite so sure, but it's a bit like watching a very nice painting drop. I mean, you just think, you know, it's still a painting drop. So that's what I'll say. Thank you for watching. Well, in this case, just pretend to master it. Yes, okay. Let's hold the empire together. We might not have lost Cuba if we had a similar game. Did you actually play? Yes, I did play cricket. I'm not going to absolutely denounce it. I thought I was no good at it at all. But I did have a certain appreciation. I was only good at things stamina-wise, I think, and being the anchorman of the tug-of-war. Despite my size, I actually could manage cross-country running, too, although I was always used to coming to school. If you were stuck at some point at some mathematical point, then you would call myself or the people and say, can you help me? Can you help me? Of course, of course. Of course, I can answer. Oh, here's a difficult equation, do you know? I know the way to approach the theory.

25:00 And what do you do? Tell me, tell me, tell me some good reasons to do that calculation. Of course, I can answer. And after that, you would say, mathematical, science, blah, blah, blah. And I remember I gave a talk about a game, a game. Magnitude and the resolution of the orbital of the sea. And I can understand how the connection with the electron and the electric field. Yes, of course, yes, of course, but I know you have been working on that system to transform a very tiny electric current into a picture to understand what is a form of material, what is a form of material, it's clear that it's a very big issue, but...

27:30 At the time of Galileo, we had decided to invest while there is a form of beauty of why, etc. Well, that was precisely the revolution that Galileo knew. No, no, we were still at the same point. Now, it's time to go back. Make the simplifying assumptions that have to be made in order that we can approach these problems with excitement. It's good that you can go back. It's very unfortunate to see the difference in temperament. At the same occasion, O. Cohn gave a lecture. Well, that was at the end of Mitterrand's term and, okay, for some rather formal. He asked a number of scientists to give a lecture. ...was fully understood even by people who don't know what's... Then Cohn came and said, oh, you see, you have 20 minutes here. And then, when he was older, when he went back and sat next to me and was begging a compliment, and I said, well, good introduction to your lectures, good introduction. So, pure scientist works. Pure scientists work in non-communicative geometry. Repeat that one for the record. I love that. Excellent stuff.

30:00 I don't understand non-community analysis, and I understand that modules over a C-star algebra are something that's conned about three times. What is a map between spaces? And he gave me three completely different answers. First he said it's a ring homomorphism in the wrong direction, right? And then I said, well, that's really too special, because look at these again. So the answer was, oh, it's a bimodule, mini bimodule. Then I thought about, oh no, that's actually too general. So then he said it's a bimodule that's something like a homomorphism. I concluded from this, maybe incorrectly, that he doesn't actually have a definition of homomorphism. No. He doesn't have a definition of space. Well, even if we accept these star algebras as objects, in particular, because we were talking about diagonal math, There's no way you can get a diagonal map on the C star algebra except with additional structure, like if it's a group algebra, for example. No, but I think it's a problem. When I discovered it, Connie would still explore space. Fine, but does he have an exploring definition of math between spaces? I guess except the competitive nature of it, but that's no substitute for the recognition that in every form of geometry so far known, Whatever precise definitions have played an essential role. It doesn't have anything to do with it. I do. It's seemingly the one you mentioned, but it doesn't have a general definition.

32:30 Well, actually, this was precisely one of the topics I was hoping we were going to come on tomorrow, just before you leave. Expose on proton decon points and the program that has led to this. When the recording is running, we could discuss it again tomorrow, since it's a very important piece. Yes, right. I mean, so this is one of my reservations about your article. This is definitely one to be kept for the open sessions, I think, well, it's not a bilateral because we're all here, but I agree this is a very important topic systematically. Shall we just take five minutes and then we'll reassemble? Upstairs? All right. You're not clear at the table while you present. It's about 25 past now, so... And that's why you came. And it has made a difference in my life. But it has not made me. My second cousin, twice removed, was the wealthiest woman in the world at one point. Wow! You can inherit without inheriting a very great deal. Sorry, I've been incredibly obtuse. I was wondering, why is Colin mentioning this interesting piece of family history? Because, of course, the reverence that Colin's got. Yeah, of course, of course. I'm sorry, I'm very sorry. I really am so, so, so sorry. You get a huge number of emails coming in all the time, I mean, the problem is, in the course of my message, I usually am interrupted about three times by, you have three new messages coming in, oh, it repeats, no, no, no, no, no, it repeats, the three new messages on the internet server, which I only get about,

35:00 Well, they just choose to repeat that until you have actually accessed it and read them, and then they tell you what it is. Is that one of those points where you don't try and look now before we start? What I do get, however, is every few minutes I get one of these bloody, from my old server, groups that they will not take on board the fact that I'm no longer on that server. So I haven't disconnected yet, but I think I'll just go back and pull it and do it, because I just get... This is why I now use the U2, because that's got much better anti-spam than anything. The answer to the question is that the answer to the question is that the answer to the question is that the answer to the question I explained, I gave an introduction to imperialism, a historical and a logical introduction to imperialism, and at some point I referred to the book of Lazare Carnot, and I said, well, look at my copy, look, it was printed in 1907, no, I mean 1797, and remember, it was a time Lazare was a member of the French Junta. And then of course, what I expect to do, one student should have said, don't rehash Pinochet to write a book about mathematics.

37:30 But just to show that people felt free to speak. In the last years of time, well, in the beginning everything was the same, but in the last year, and I was at this visit, another visit, I mean, just about a quarter, I mean, low. I mean, maybe it's like, but it was sort of forced on me. I mean, I didn't know that. I was there, but there was a different measure. Yes, well, I'm sorry, mathematics, just mathematics, just quantum mechanics, because I didn't know it.

40:00 In fact, it's got lots of them, as long as there's no constraint in it. I mean, yeah, once we've said, are there any constraints on the values for y and y squared? Once you get all those values, then it's all being determined. But it is still, it's just free on y and y squared, right? Free on this. On y and y squared, for sure. Okay, let's, well, suddenly, let's, you know, y equals 3, this is, you know, y equals y. Yeah, there's no hidden possibility of that. No, no. Yeah, yeah. But yeah, if it does drop out, then you don't have these semantics. That's right. This is spelled out again in your paper, isn't it? You say something in this paper. I have a feeling that this is an interesting lecture. I thought I had seen it in here. Set the lighting for me. No, I guess you spoke to me about this thing, and then I read this thing. No, I think it's not. The tricky piece is to me, and the first thing is the circumstances then. Yeah, and then, I mean, if you were leading here with a leading provision, which was a polynomial with x,

42:30 oh, then you wouldn't know what yq is. That's right. That's right. The question is, would it be possible to take that out? If this were an integral, you'd have to assign values to all the powers of y. And then there would be space. That's right. So it's not exactly a relationship. And it could exist, of course. But that would be built back into the particular relations. So in this particular case, I'm talking about the x. I mean, x is transcendent. The x is the book. It's transcendent. It's not this game we're sending there. Thank you for your time, and I look forward to hearing from you in the future. But something that when you, okay, it has to go to something, this has to go to something that when you multiply it by x and add and subtract these, you get zero. The problem is, that tells you what the something has to be in terms of these, and it may not be divisible by x. That's the point. It may or may not be divisible by x. And in the simple case of just one over x, we got it to be zero, which we got to be, but in fact it had to be zero. We're talking about solutions and algebraic equations. Is that consistent with the usual project that you're talking about?

45:00 I couldn't tell you how good it makes me feel as somebody that laughs at the original math after four days. You know, just a month ago I learned how to solve cubics for about the third time and I think it's going to... I would say, I mean, fifth degree equations, you can do those too. Fifth degree equations, I mean, those are odd degree equations, so they'll have, even over the reals. I mean, we'll be able to solve on the set and then in the fine lines, yeah, we can. Even if we don't know what the solutions are, we can evaluate functions at them, at least in an average sort of way, by taking the average value of a function over it.

47:30 Yeah, this business about... It gives you so much structure with the metric. You mean about the statistical decision problem? There's a distance between maps. You can ask whether one dotted arrow is more nearly the equation of the square. If you take the parameters, usually called theta, this is one of those fantastic maps. The size of each parameter value is probably the distribution over the parameter space. If the parameter values were so-and-so, then you would know what the... On the other hand, there's a decision space, and this is also given, namely, the delta, because if you really knew what the parameter was, the idea is then you know what to do, and so the question of, sorry, this is not the, you're looking for deltas, you're looking for deltas, what is this called, so the point is, when you compare delta P with D, D was a bad name, too. This is a good decision. So the thing is that there is this intrinsic distance on the space of maps here. This is an example of the G interval. Whatever delta you choose is in the same universe as G. This is a convex set because these are maps with values which are probability measures. A mixture of probability measures is still a probability measure, so this whole thing is really a convex set. In terms of how you could represent B as a mixture of A and something, you see, if you knew you were on the boundary, this would be, you would have genuinely less information, would be given by an experiment, that's the way they usually put it.

50:00 You know you're on the boundary, then you know that, you know, the system depends on fewer variables than what you thought or, you know, you have some very definite things, so. If you can express B as a mixture of one of those things with less information than A that you want to mix it and express it as a mixture in the best way possible and that's that's the distance you made to B and so again a very non-symmetric distance right so that's the one that we use where this is this is the maps in this category. And so the idea is to choose G so that you're as small as possible. And then you could also make a distance from G to this. And then this again is underlined with the kind of the . And here because we're in the, presumably the trajectories that are being talked about in physics would be, would be solutions of some classical equation, differential equation in that case. So it's not really all curves, it's all sort of possible curves and some, some equational constraints. So here we're talking algebraically the various solutions. We can transform any function on the whole plane into, sorry, on the curve, on the curve, any function on the curve, well, hence in particular, any function that's defined on the curve can be turned into a function of this variable only just by taking the average of its values of those three points and calling that the conditional expectation.

52:30 For as long as people feel comfortable not having to talk about another area, just to wrap up some of the main issues, and then I thought perhaps we might even ask Pierre what's the start on his next discussion. We had sort of decided roughly that the last half day he was going to be with us, which is maybe tomorrow morning, was going to be devoted to a discussion of Grendik's points. It has come up quite a bit once over lunch and again briefly now, and it's found on the peculiarity of this. Yeah, well, okay, he admitted as much that it's an advertising slogan. Yes, I'd love to try and get that to work with everything. It's not a subject, that's what I should say. Yes, that's the point. Yes, I think it would be worth spending maybe just half a... Something very self-confirming now. Either now or tomorrow morning. Well, either now, I'm getting it out of the way, or even until tomorrow. Well, there were still at least a couple of questions I wanted to ask you. Again, just going back to the broad issue of the way that one sees the unity between the geometric algebraic and mathematical aspects of math.

55:00 And finally, the unity that you stress between the separable, un-ramified, and decidable aspects of the object. Well, yeah, one of the things you were talking about, again, you were talking about Branger's at lunchtime, and it obviously has some very deep ramifications for the distinction between the growth of the petitoluses and the general nature of categories of space. That's why I say the general nature of categories of space and the nature of categories of generalized space distinguish those aspects and I think it's something which, you know, we should explore a bit further and it's relatively self-contained and probably would make a good topic for us to wrap up with this evening, but it's the meeting anyway. Let's leave it for five minutes or so.