Talk (partial) / lunch conversations / talk (partial)

0:00 And there's this sort of failure of exponentiation to the situation that you'd have in the toposystem of synthetic differential geometry, where you've got a further right adjoint to exponentiation which allows these fractional exponents, which allows the kind of prolongation system. It seems to me that there's a common structure there. Uh, interesting, oh, okay, well, yes, well, yes, coming from a complete, you know, tarot these things, I, I, I, you know, I'd like to get a, yes, it's surprising how, you know, one gets sucked into this stuff. It's absolutely fantastic. And all this, uh, obviously John is seeing it very much from a, you know, model theoretic point of view, but I, I, um, yeah, I'd like to ask, uh... I guess it might cost more than that than anybody else who knows about SDG. Hi, I was just going to ask you about this stuff of John's. It occurs to me the fact that you don't get exponentiation in this setting and that you get these systems of different lengths. Is there a possible connection with the... Is there a possible connection with the situation that you have in synthetic differential geometry and, you know, topos and smooth spaces, where you have this further right adjoint to exponentiation that allows for the existence of these factional exponents that Anders Park and Bill have both written about, these prolongation systems that are involved in the way you treat... You treat all the different actual equations in the topos and smooth spaces. There are these fractional exponents that are finished by this further right adjoint. I'm not sure I know what they are. Is it described in the book of John Bell? No, no, no, no, no, no. In papers by Anders Koch and also at least one paper, I think, by Bell, that I must ask him about.

2:30 No, they're called prolongation systems. Oh, you mean exponentiation with some infinitesimal object? That's right, yes, exactly. Because that allows fractional exponents. You know, it's the thing which allows the fractional exponents. But not real fractions. You don't mean one-out. It's just because an infinitesimal object may be a little bit more than just one or something. Well, yes, yes, yes, yes, yes, but it's something, it's that additional, it's the further right agile in terms of exponentiation, which is what captures that, the fact that the, as you say, the right agile. Yes, further right agile exponentiation. Oh, yes, right, yes. And I'm just wondering whether there isn't something similar going on in the case of this, these simply infinite systems of different lengths. It would be nice to see it done in the topos. Yes, that's what I'm saying. Can I ask you again, you know when we were downstairs yesterday talking to Pierre Cartier, or just after we finished, before I asked you about unramified and separable and decidable objects which you explained so wonderfully clearly, we were talking a little bit about, we were talking about C star algebras I remember, in the context of the... The Comte Programme, but don't call me to do geometry, and some of them... I mean, I'm trying to remember now what it was that we were... that you would say... There were the sections of communities and quantum mechanics... Yes, that's right, yes, yes, that's right. ...that maybe the basic category underlying the foundation of quantum mechanics may be like a linear category in the sense that it's a bit more like the category of heterospecies. Yes, that's what we were saying. Remind me what you were saying about that.

5:00 Well, it's because particles... No, no, no. It's all clear what they're saying. Should we think of them as... Well, clearly not. Clearly not. I mean, if anything, they have all... I mean, they support all these interesting symmetry properties, which clearly have group theoretic structure, a great deal of which... But they support these internal symmetries, and of course there was this attempt to unify the internal symmetry groups with the external, with the space-time symmetry groups, but that fails because the cartels prove that you can only ever form the direct product of the... Let's say on R3, it's supposed to function with very complex numbers. For an electron, it could be a function with very thin, but for certain types of particles, it could be a function with very complex numbers. There is another one just upstairs there, I believe, if you don't mind. He won't be there, sir. No, no, he won't. No, well, I was thinking in case you had a desperate need. No, no, no. Sorry. It's very unfigurable. Unfigurable. Now, if you take such a function, there is a probability set of implications here. Which is that if you want to compute the probability...

7:30 If the particle is in a certain region of space, of part 3, then we integrate the square of the function. Assuming that the norm of the function is 1, that is the square of the function, the integral of the square, then the integral over a smaller portion of space is integral to the particle there. Wave function is supposed to represent the state of the particle, and what is curious is now that you, if you take the Fourier transform of this wave function, it's also an L2 function, a square, the Fourier transform is a mapping of a square of x, it's exactly the function that you use in a certain region. Yes, yes, yes, I understand all that. So that's a very strange thing. Very, very strange. I certainly see your point that there's no natural room for sets in quantum mechanics. I mean, in the mathematical structure of the theory, it's not at all... I mean, the basic objects you're dealing with are... The basic objects seem to be linear operators and vectors and things of that sort. There is a way of constructing maybe something like set theories. I mean, it looks a bit strange, but maybe there will be a way to construct set theories from vector spaces and maps. Yes, has anybody attempted to do anything like that?

10:00 Very nice. Was it intelligible? Of course. Yes, for everybody. But I would like to read a paper if you have. I've got a book out on it. Oh, I see. Which I doubt that you can afford, Audrey. Well healed though you may be. It's published by Cambridge University. But you can order it. Yes, yes, yes, sure. No problem. But sorry, do go on with what you were... It must be that you failed. Okay. It's hard to get through. Well, that's interesting. Very interesting kind of procedure. What I'm aiming at is a map. Yes, it's just down to the corridor to the left. All right. That's basically what I'm aiming at. Yeah, yeah, yeah. I'm still intrigued by, as I was saying to Andre at this point, that Bill raised in Bristol, about whether there might not be a connection with the... I'm still intrigued by, as I was saying to Andre at this point, that Bill raised in Bristol, about whether there might not be a connection with the... I'm still intrigued by, as I was saying to Andre at this point, that Bill raised in Bristol, about whether there might not be a connection with the... I'm still intrigued by, as I was saying to Andre at this point, that Bill raised in The additional right adjoint to exponentiation that you have in the topos of smooth spaces that allows the very infinitesimal... Yes, yes, yes. I have to think about this because... ...prolongation systems... Yeah, I forgot a little bit what these right adjoints were. I remember... It seems a very small time ago I knew what they were, but I forgot. It just seems that you... I have to rethink about it. Yeah, I can't think of any names that you may have described. There's just two things very quickly. First of all, okay, what you're going to say, has anybody attempted to do any kind of a set theory on, you know, using that idea? I know there's something that... A set theory using the quantum... Yes, using this quantile. Not necessarily quantile, but the... Yes, quantum... Well, using those kind of concepts that you were talking in this category, category of values. Takeuchi did something like this, and there was some connection with the Provence-Scott work on Boolean value models, which I'd like to know more about. But he essentially took operator value terms. Yes, Boolean values are kind of familiarities. Yeah, yeah, it's obvious. Yes, sure. And the other thing I want to ask you is the thing about Feynman diagrams. You know what you were saying to me yesterday? Sorry? The thing you were saying about Feynman diagrams. The Feynman had, you know, the generalization of the notion. Okay, well, not now, but...

12:30 Okay. Okay, well, I want to run that over fast a bit again. Hi. Do you have an envelope? An envelope? Do you have an envelope? Yeah. Great. I just need to give Martha Dunga her money. I wanted just to give her some in a clean envelope. Thanks ever so much. That's all I need. Thanks very much indeed. Yeah, I gather that the office is... I have done a registration fee in two weeks. Great. We'll go over that in a minute. Thanks. Sometimes when there is a present at a convention, it is rather foolishly preserved in a formless sense. When there is conventions referred to us as living beings, psychological entities, as the theory says, are our faces. Action is based in our faces, and my aim is to work at conceptual frames which fit, or perhaps even...

37:30 And of course, that's the definition of a morphism of architecture. It's basically what it is. I started with no idea of what it should be and tried to figure it out. The idea was let me do a simple ideal embedding, base E to the segment below E. I'm just looking at how it backs on arrows in the topos E. The segment below E, that's just everything down here, inclusions. And when I do it, I hit the top, and I'm trying to find out, I'm trying to define... Morphism of three principal ideals, where I take E, but let me read this way. For every sub-object here, A, I compose with F, take its image, that's of A, that's a sub-object of F, so it's in the ideal below F, and this, then, is the component.

40:00 And now, basically, what I have to do is I have to go through and verify the conditions C, S, P, and U on this category. So, I'm going to try and do all of that, but I'd like to show you just a couple of the cases to give you kind of a feeling for the categories I use as kinds of natural appealing instructions from the groups the first time I've tried to do them, because they make a lot of sense. I'll tell you what the product is going to give. What I'll do is I'll take the downward closure pairs of things, and that's certainly going to give. Let's just see how the projection works. For example, the projection down to A works like this. If I take some C over here, how do I get it down to something like this? Well, what I can do is, I'll put its limits in some A cross B. I take the projection down to A. I factor out its image. That's, so here I'm trying to find pi 1 on ideals. So I get here pi 1. This is pi 1 on ideals. The definition of C is either unique epi, inclusion, vectorization, and this is probably one of them, and now, okay, this doesn't depend on the complexity of the object, it depends on speed and everything works out right, but essentially this is the kind of construction that comes up again and again in determining the structure on the category of ideals. But epis, for example, it's good to know, you may need to know what my different problems are.

42:30 In the category of ideals is the right epi, if and only if the mapping part is an offset of C. The small maps on C being, well, first I'll do the definition. The mapping part F has a right edge joint. After all, the mapping part F, this is F support, it should have a right edge joint. That's equivalent to requiring, whenever I take a pullback here of a small sub-object, You know what the small sub-objects are supposed to be, so let me say, whenever I take a pullback of a principle, I have a principle idea. Let's see, what is the small, to the terminal line here, well, I have to have a right edge.

45:00 What is the terminal line here? Well, it's not very hard to see, but that is the down segment of the object in the total space. If it has the right edge joint, well, then let's take one and we'll send it back here. So we take one, send it back by the right edge joint of the star. This one is principled, so everything is below one over here, and so, in particular, everything in this book, in A here, is below this, because its image over here is always below one, and so this one here. So we found that the small objects are exactly objects, which I was writing this way, and that's what the principled idea is, that is the image of E, if you'll allow me to write it that way, principled in A. So that might be a good time to take questions. Let me just wrap it up by saying, now that we've defined these, our prairie topos, oh by the way, it's the power objects. I've done C, I've done S. The power objects are what we think they are. But thinking about its view, of course, finally is simply the whole. It's the maximal idea. Every idea is obviously the maximum. So, here's an example of E, I showed you how to build, and now I will, this third theorem, although I can't say much about it, at least makes sense, saying that if I have any category of classes, I have to find all the ingredients in these cases here and now, for any category of classes C, there's C, E, that is, logical,

47:30 And now biologically, I mean, preserve all of the structure mentioned in the CSTU, conservative, ISOs, so that we really have reduced, as we're at the theory of the class categories here, to that of idealism. Yes, but I think the model for the normal mathematical set theory is in today, and it's a topological inclusion because, of course, one set is included and another set is not included.

50:00 I could have a lot more extra stuff there, but then I could just focus on the third class of the year, so that would be a bunch of examples of... How important is it that the categories of classes are just regular as opposed to exact? We can see in Spindler, in the analysis Spindler predicts, he emphasizes regularity in the numbers. The probability of exactness is all multiplied. Is this still the case? Yeah, there are some benefits to it, as opposed to exactness, but I'm being puzzled by the fact that the categories are really general and exact. They are general and exact. So in that sense, there is a certain kind of experience that I have in my life. Unless I'm mistaken, but I'm glad to show you the articles. Yeah, I apologize, but these are maybe my articles. That's in your sense that the view is fixed for you. You think the idea is released for a lot? Yeah, that gives me things like combinations. So do you agree with that? Yeah. But if you want to model stronger or other traditional properties like foundation, active foundation, you put additional traditional properties in there. Is that correct? You know, in the traditional practice, it's really very nice to take quite a few systems and see what can be derived from this. And that seems like a pretty good practice. I'm adding category of ideals here into the sheet that was over here. And we'll be there for a while to get close to that. I was fascinated recently when I heard of Berlin and Bernice in 1963. I think we just take finite types, classes of classes of classes, and we're not too far down.

52:30 Which is amazing. I think that this principle of anointment, of something small that can move in such a memory, is so unbeatable today. The reason that Jericho used classes was he didn't want to use mathematics, and so he divided the classes with the types of formulas and accepted them, but he was just kind of like Jericho or Franklin, because in the separation axiom you don't qualify all the classes when you're separated, so he sets them in the monograph there. Because of the sort of objectified syntax, by replacing syntax by a large number of operations by classes. In particular, if you're going to find an action factor. The point here is, so why not make, when you say formulas, why not form them as a finite type of a class theory, not just an element of a class theory. And it doesn't seem to really bother the proof theoretic strength, for example. Then, if you did that, then your various sort of pseudo-power sets couldn't be viewed as sub-powers, and as such, they would be examples of extensive quantum powers, truth power, and all kinds of other things. So, it has to come up with your finite-type separation analysis, and that separation then goes through a higher level of logic over the data. And so that is getting stronger. Part of the part of the classes and a lot of separations and events is stronger than the first part of the day. Depends on what you mean by hybrid and planned. You see the parallelism of a version of that. Namely, topos theory, but the elementary topos can be a certain secured object P. Just in the category of classes it happens to us. You know what I mean, topos theory does involve the power set and the power set involved. So, I mean, I just don't agree. It sounds like you're doing higher order logic over universal sets.

55:00 And so that is wrong. I mean, Kelly-Morris is second order logic over sets. And you can go, I mean, a total series higher order intuition of logic. So if you think of the universal sets there and you're doing higher order logic, you're doing higher order logic. And so that gets stronger than that. I like the ambiguity between these and these. Which, for Italian years, is almost identical. It basically is connected with basements, so it gives the meaning of ground floor. So it's this far-sighted attitude. Even more now that we are beginning to understand that taking a great power set action is... Right. So, in many words, with these two people you'll be within between this and this. Like beautiful or... I guess what I would say to your body is, we have to take away the power syntax here, then I'll just, I'll do this and then I'll change it. Predicative. Are there any small objects that are non-subjective?

57:30 No, that's our definition. We give this context to the category of objects. If I've said it, we just mean small objects. Yeah, we should thank the speaker again and... I want a copy of the picture. Tell me, because I have to know who wants it. At what time at the restaurant? What time? I'll be at the hotel at 7.20. And we're going to be there at 7.30. There's a link on my website. Can I ask you a question? Thank you for watching.