Discussions, incl. J Bell & M Wright / Discussions incl M Wright, J Mayberry & A Peruzzi

0:00 I do want to ask you a question. I would like to know more about the fact that it really is very interesting how it's kind of different from geometry. It makes it space and go over and come over into a decomposable space. And there's this notion. It's philosophically very interesting to have the idea that the continuum had no detachable. This is a very strong version, in some sense, of what Aristotle was saying. I mean, it's not quite the same, but there's clearly some connection. And the complete reverse of what happened with Setter, where it was made... All geometric structures then thought of as being imposed extrinsically on points which were already there, which is the exact reverse of the way that the 17th and 18th century thought about the continuum. And the way you think of the intrinsic geometric structure of objects in a topos, of course, is really the old-fashioned pre-set theoretic heuristic, redistricting us from this more abstract kind. No, there's a lot of work to be done. It's not very commensurate. There's some very good, there's some state philosophical insight there. No, no, there's some... There's some short and forth, technically, but there's not much philosophy there. And, uh, of course he was saying, Gonzalo was saying about these books, you know, ignored by the, by the people of different geometries as they go. On the other hand, it's a difficult book. Yeah, yeah, yeah. I've tried looking at it. It is a difficult book. It's obvious that Reyes at one time was, well, still is, obviously because of the influence of Louvre, but very interested in philosophical underpinnings of this stuff. And that paper that he gave on logic and category theory in the... Now the ILPSM, the big jamboree one, has got some very interesting side remarks, which I'd love to see him unpack a bit more.

2:30 I always like this very geometric way of thinking of the way that, you know, the set of theoretic axioms are motivated. You know, when he talks about the substantial axioms and sets as... I was talking with Bill about how... It's a very, very geometrical and intuitive way of thinking of mappings, and the points functor is... It's tremendously difficult to actually get these models, and yet internally these are completely natural. And it would really seem that in order to use them, you have to perturb, in a way, you know, some classical property. I mean, for example, what you start with is the classical... ...category of manifolds. And you use that to get this new structure. Now, really, it would be much more satisfactory if you didn't have to go through this process. Yes, sure, if you had an autonomous point of departure. But that's a problem to find here. Well, of course, it's what Alberto's trying to do. It's a sort of homotopy-theoretic structure that's forming a kind of pre-category. What maps do compose is actually given you by the additional structure you put on this three-category part in order to get the conditions on domain and codename. It gives you a... I mean, it really does pick up, I think, of all formal notions, ultimately arising from psychology, that bills, figures, are actually topological rules themselves. I think that's an extremely impressive project. It's very well worth a see. It seems the best bet for a naturalistic systemology for that. But obviously it's a huge amount of work that we've done. But it would answer, you know, the question that John Baybrick extracted, which is, well, you know, what is the naive metatheory for all this stuff?

5:00 Should we put another chair here or are you going to move? Oh, that's fine. Thank you very much. Oh, we were going to stop, weren't we, to get some... I started off with this bloody lurgy. I've finished it. I mean, I've got rid of it from here. It's gone right down to the west side. I thought we'd do it from the back. No, no, it's okay. My wife is done. Yeah, but a lot of philosophy and math troubles are out of you. Yeah, but I always hit the logic problem. Doing logic? Quite right. I think I might have a beer. I think I'll have a beer as well. Well, we have two exports. No, I really care about the idea of a proper discussion. Also, historical discussion. I would love to understand more about this way that you handle the natural numbers incidentally, the differential geometry, the thing which Gonzalo was talking about in passing in the discussion. Yeah, that's a part that I'm not terribly familiar with. The, of course, you know, you can get, when you use the model, it just has a natural number after it. Yeah, yeah. Yeah, sure, but not that. It's just a, you know, because you don't appreciate topos. But what I'd like to understand is why Bill regards that as, in some sense, you know, an expression of what he calls subjective idealism. Well, well, it's... And how, you know, the topos without the natural numbers object is, in some sense, the more, you know... In the case that gets nearer to the way one should think about real world variation and how the ideas of counting arise.

7:30 That seems to be... Have we? Yeah, yeah, yeah. Unfortunately the map's not in there, it's in the carrier bag. We don't need to get it out here now. Yeah, because we need to have it out. If we get on the road, we'll have forward stop. Well, okay. I'm good. You're going to have to understand it, you're going to have to do it, you're going to have to go and see what happens. Well, that is exactly the position of time you've got. You're going to see something, right? Sure, that's right. But it was regarded as a dead option in the math degree at Oxford at the time. I mean, it was precisely that view. It was very nice, but nobody could, you know, you wouldn't want to make a career in math if you don't study that. That was the view of the time. And then of course it all changed because of the discovery of cosmology and the whole development of quantum and also quantum cosmology. Well, that was considerably late. No, but it did put the point in. I mean, Joe wrote to Mr. Cosmology at the Wittigan going very rapidly. But the point is, in steady state, when Hoyle was in trouble with 3 degrees K background radiation, which made the whole thing not just a kind of purely theological speculation. Hoyle's theory was very pretty. It was much more like theology. Very nice. I have the notes of his lectures. But of course, cosmology has gone that way again now. I know. I know it has. Cosmology. Well, there is some observation... It's more theological than ever. But there is also some observational evidence. Well, of course, you can go to London to see it. And that's what you've got to do. You've got to go to London. Yeah. Mimi was the secretary in London at that time, and she was joining the party. I was thinking this morning, if I closed my eyes and read a thing that you've looked at.

10:00 I see you as a man of fact. Dressed in a leather jacket. I thought that as a man I would be in contact with that one thing and talk to that another. Since then he's finding out the way of... Yes, I think, absolutely. But I don't think it's just a question of his finding out a way now. I think, from what Cohen was saying, that he always did think of Topos as very closely related to what he had been looking for before he came across Gordon G. Topos, which was a nonlinear generalization of the abelian category. Yes, it's extremely interesting, that stuff, because on the face of it, topos in abelian categories seem to be very widely separated as structures, but clearly it's the way that he sees, it's connected with the way that he sees the petty topos as sandwiched within the construction of general topos, or gross topos. I think that's connected with the way that he... I think some of the connections are in his paper in Milan, the Milan paper on metric spaces and generalized logic in closed categories. But of course, Jerry is very interested in this as well, because a projectile is an abelian category. Hang on, is that right? I don't know. Closed category, but the unit and the tense of products is not necessarily the, it's not necessarily, but they act, and the way that Jerry thinks about projectiles is very much in terms of a logic of inter-involvement, the natural progression is from a logic of points, which is the Boolean topos, to a logic of regions, which is the Heiting algebra case, to a kind of logic of directions.

12:30 If you're going north by northwest, you're not going north, but you can't tell I'm going south either, and you're nearer to going north than you are to going east, rather than you are to going west. You can say that you're going north to the extent that you're going north by northwest. I mean, it's a fairly trivial motivating example. But certainly the way that he thinks about the negation in quantum logic in the setting of the projectiles construction is a very geometrical way to do with the exclusion of regions, intersections, the geometrical entities, but the entities actually involve a vectorial aspect, a component of direction. And that seems to connect up very much with the way that Bill thinks about the way that topos are related to the Helium category. As you say, there are ideas that I'd love to find out more about. Colin obviously knows quite a lot more about the history, because he was making exactly this point, wasn't he, in the talk and also in what he was saying in Cleveland. But in fact, Bill... Okay, he didn't hit upon hadrobuses because they came out of Grosvenor's work, but he always saw them as much as fitting into abelian categories, not quite fitting in, but being much more related to abelian categories than most people did. Most people who worked with Tobos thought of them as being very different kind of categories from abelian categories.

15:00 They seem to have lots of roads named after philosophers in this country, don't they? Did you see there was a Bradley Road back in London? No, it was just called the Putnam Road. Probably not the people we're thinking about. No, I suspect not the same people. Yes, but you do think that the categories that were carved out by the way that the expressions of the artificial language worked or wouldn't have worked, were, were, yeah. I think you really did have something, yeah. This whole ontology of concepts and objects is, after all, goes with the workings of the function, you know, such a function-theoretic calculus I bring up. Thought would turn out to be the language of pure thought. I think what's happening is that the logic only comes into play after you've decided. You mean after you've addressed the problem of kinds? Kind is a unit. A unit is one of the things. Yes, well that certainly is a much more... Oh, it's certainly much closer, much more convenient to me than a physics panel then. But then the question arises, you know, where does the completed kind of kinds come from? The completed kind of kinds of abstract objects, which you need to do mathematics. You posit your units of the models you're on.

17:30 You want to talk about the colors. Somebody will tell you the colors aren't really... But all that postulating of units is just that, it's postulation, it's psychology and obviously nothing to do with, it's before, it's what happens before mathematics gets started. It's what has to happen before mathematics can get started. The question goes on to draw the positive view that you're trying to block. It goes on to prove theories. Yes, both Plato and in this very different way, Lord Weir, really do think of them. But you're right, it has been a very stimulating week. I've enjoyed it. Yes, I wasn't actually there when you were arguing about that. No, you were there. Your head was down on the table perhaps. Ah, well, that was it. You were there and bothered about it. Not in mind. Oh, that was the thing you came in the next morning, gave a resume. It was when, well, whilst Alberto was lying from the death's door, and I was, yeah.

20:00 Well, let it pass me down. Hmm. Yeah, yeah, yeah, it's very impressive. But then, yeah. So how did people, so as it were, how did the completeness proof get misused? People have not seen it. It isn't misused, because it swims in the naive set theory. I think it would turn out to be another plug, yet another plug. These definitions are absolute indices. At some point, you've got to stand outside your logical systems. If you're in an axiomatic system, your reasoning is poor. You draw the consequences of axioms. We can do it mechanically, but we're worried about models in the act, isn't it? Right. So, to formalize the name of Frankl, then it's not that simple.

22:30 Why did you name him Frankl? Right, right, yes, I see the point. But where do these models come from? At the same time, you could say, what is the first door? What is the first door and what is the second door like? Oh, I think they're really... Because you just define in a set code case. It depends on the method theory you accept. I mean, I can, my universe of set theory could be such that I can individuate answers only if you take this big cone. It's going down at a certain level. Yeah, but if you only... If you slow down this level only to plus 10, you're okay, because these results are absolute as they are. My point is, if you like, you're meditative if you talk about derivation, but then I dare say you're more built upon slightly. The curious thing from the categorical point of view is that the simple, which is a simple... I think the notion of categorical model, say a topic, well, not say a topic, of categorical model for some language, which is the logic of the language such that the notion of a model seems to be true.

25:00 Well, the answer to this question is that the proper categorical notion is true. And the linguistic answer you get is the entire type theory, not just the logic at one level. When you try to say the twistor of the logic, then the categorical models you get are extremely... Oh yes, we're going to get these, right? Have you got enough funds to run for that? I've got some change. I have. Okay, because I'm almost out of Canadian money. I thought I had some. I'm a generalist. I like sports. I teach at MIT. You do it? Yeah, I do it. I mean, it's a brilliant idea. And they're already talking about speculating that there may be a universal world. But you know, for example, I saw a recent issue of the Gallium Version of the Scientific American, and there was a paper from which I had this information that the etruscans, just through this kind of...

27:30 All these terms have been associated more strictly to a population near the Finnish border, of the Euro-Finnish border, than the other Lenin and Italians. I was astonished by that. Well, how do they know? Maybe because they find these genetic traces in Tuscan. But what is the problem with Etruscan? Why is it so... Well, I thought there were, but I mean, obviously, if there were no written instructions, the problem wouldn't arise. I mean, it would be meaningless to speak of being able to translate Etruscan. Oh, I see. There are no papyri or anything of that kind. But why, if there are only two inscriptions, why would expect it to be, yes, it's fast track, isn't it? Why would expect it to be rather easier? Well, exactly, exactly, or proper names. This is what I find... It's a different script. Yeah. Sure, but I mean, the case with... But we have no idea, even though there was a phonetically stressed language, you know. Well, I thought that it was some kind of Etruscan indulgence in odd cult practices, like burying wives with their husbands or something like that. Something unsaved from what I remember, right? I can't remember exactly what it was. Well, our husbands should have been left in peace and enjoying themselves, you know. Well, I don't know if that was intended, but he sacrificed babies. You know, babies buried under the door, the door jam, the kilts, because that's what we're talking about, isn't it?

30:00 That's true, yeah, but... That's what he said. He said, of course, you're right about that. But you're missing the... he told me about his experience, not Walter. He did his best work, he said, when LaMier kept telling him to follow the dialectic. So he says there's really something in it. He also said that MacLean got misled by this stuff and failed to give an accurate proof of what he said there. But I mean, MacLean wasn't misled in that by, I don't know, by Voltaire. Sure. MacLean was proved, yeah. He's a very interesting guy. I'd like to know more about the way that he thinks of what he calls the substantial axioms in the context of topology. He gave some hints in that paper on logical category theory that he published back in the, which was a general sort of survey paper for philosophers. Alberto, you know, I mean, you know the literature, I wouldn't say much. What was Benabou trying to do with his theory of vibration? In other words, that intends to be a more general theory than category theory. It's more of a general notion than topology is. Oh yes, but I've never looked at that stuff and I just wondered if you... You could say that, well, it's a long story.

32:30 Yeah, I'm sure it is. You could take it as a special topic that is a... There is a particular kind of learning purpose obtained by taking topological spaces and for each open space of topological space you assign to it the set of the continuous function all on that open set, right. This actually is not a focus, it's just... A collection of entities which varies continuously on the base. Right. In order to get Sovlas, you shift to the following notion. Take a topological space, x, e, v, and take another topological space, x, and call it this x, s, base. Now take a local homeomorphism from E to X that is a continuous puncture which is an homeomorphic for any given neighbor of any given point. Any point has a neighbor such that it has a homeomorphism from that neighbor to some neighbor within the base. This is a topos. This gives you the basic notion of spatial totals, right. Now, in this case, what you actually have is that you have an aching algebra that corresponded to the openness of the base totals, and over each element of this aching algebra, you have a category, right. Now, this is, if you wish to generalize this, the first trick is to left. This, in 18 algebra, is a category.

35:00 In 18 algebra, left. Left. Yeah, okay, yeah. It's a category. Right. Right. But it's only... Push the on button there, so that... Right. It's only a poor category in the sense that between any two objects, any two objects, there is only one error, the most, corresponding to the inclusion. You enrich the basic category, taking it not only as a thinking article, but as a general category. To reach the object of the basic category, you assign a category, which can have a lot of scratches. And if you assign conditions such that when you give an error in the base category, you can recover in a unique way that error in the upper category in such a way that certain conditions related to pullback and push outs are satisfied. He was trying to use these notions to provide categorical foundations in terms of vibrations, such that the cardinality conditions, since here you can understand, grew very quickly in trinity, trinities of different sizes. Since to any object you assign not only just one small category, but it can be a large category, it can be even a topos or a class of categories, and so he tries to assign to each of these size conditions and structural conditions in categorical terms.

37:30 But I didn't study that stuff. This is simply the basic idea. That is something that I didn't investigate. I don't have, well, we could give a look to the paper and after one hour we have a clear what was going on there. I don't think anyway that these people like Benavoz and others. I'm interested in relating their topics, their points, questions, classical questions concerning the foundations of mathematics related to type theories or set theories around the circle of rank and order, and so on and so forth. Just listening to the speakers is complicated, but that can be a foundation. Foundations were used for telling somebody from scratch what's going on. Yes, I think the sense in which these people, like Renabou, use the term foundation is obviously just in the sense of the right framework for organizing the subject. The right framework for organizing... Actually, the first time I read that paper...

40:00 I learned about, through it, I learned about vibrations. Yeah, which I would like to understand, I really would like to understand. I couldn't say myself as John is pointing now that I could relate what was there in that paper to what I could think is the foundations of mathematics, what has been at least for one century. In a way, they should be at least be so gentle to let me understand the relationship with the one century of discussions about mathematics. They don't. I have reason why I pressed with Gonzalo and Bill on this point about understanding set abstraction as a special instance of discrete vibration. I think that to just separate mathematics from our cognitive life and conceptual life, since long we have done this, then it's difficult to ask for something more than an axiomatization which is sufficient to cover what your needs are. You can't justify it on a more or less strong basis. I'm not sure that that can be... This is my race against the... I'm curious about things like when I was walking in the Kaiz Hall yesterday. At one point in the discussion, you talked about the capital.

42:30 Well, uh... You know, if you take the notion of category, then you explain, you introduce the notion of factor between two categories. Well, you can consider each category as an object of a category. Well, I get that. I mean, what was he talking about? But I mean, that's... The category of categories has to contain itself as an element and all that kind of is very bizarre about that. So what does he in fact mean by it? I think that the colleague approached that problem in his paper on the category of categories. But he presented a joke on that before. I was not, I simply wasn't sure that it could grant the consistency of the entire construction. I seem to answer that there is no contradiction unless you interpret the notion of category remaining attached to a notion of set. So if you start simply with the notion of category. When you define a set as a particular kind of object, you call it, you call it a set, characterized by a set of categories, you are, well, it sounds so, it sounds pretty dense. What was Mackay at that time?

45:00 Since now, these people I think that works in a meta theory which is a little extending and compressing. Depending on the practical needs, simply, if you ask them what is the category of gravity, they answer what I was starting to answer you with. But given that they have a very permeable view of the distinction between syntax and semantics, it's very permeable, given the way they think about structure, it's not so surprising. I know you can say that they're just being optimistic. It's a bit like an 18th century calculus. It works. So, go on. Don't bother about what I'm going to ask what you're told to do. Of course, calculus works to produce what's your standard and the necessary proof. Depending on what you want to prove, I mean, what's proof of a genuine hypothesis you might produce. In the case of calculus, you see, you had a... You had the feeling you were rooted in concrete reality somehow. You were actually talking about facts about change and so on in the world. So even if something went wrong somewhere, I mean, the whole thing could be just... But from the point of your feeling, it is just like that. Categories are a way of describing facts about the facts of both combinatorial and topological.

47:30 You just try to understand better and better which are these not logical aspects related to this, that, etc. problems. And then, of course, there is the standard of what works related to the practice of algebra and so on. Well, in the case of most algebras, we're interested in no attitude to foundations at all, but just... Well, I think if you push the line down to the center, that's more conventional. Yeah, I'm sure that's right. It's supposed to be a logical observation, isn't it? I mean, the point is they usually just work in particular structures. This is a point that I would like to understand.

55:00 So I suspect about QDX require another sort of sign, possibly. I haven't quite understood that. I know, but what is the operator's T? Not even after, maybe not this object, but some covering of it. So it's a little like, yes. That's why I call it H, you see. I see, that's a... You have one space of states, another kind of...

57:30 ...no longer, in that case, you no longer have a kind of localization of power. The Pipke models, for example, are terribly important. This also connects, I think, with your side remarks about, what's a Pipke model? You see, you have dollars, slash, subtracted certain elements, you see, that's more than you can go on. So if you mod out by that, it is where did this imaginary symbolic, you know, symbolic representing activities, where the symbols made identifying reference to objects.

1:07:30 But were there in advance to be, whether that idealization of the process of what you made last night about the idealization involved in the natural number, there must already be a reality there.

1:10:00 And then philosophically that is distorted by assuming that the equivalence relation which collapses it back down is in fact the identity of a real object which was there, which has had to be there in advance to give you the structure that is in there. Information retrieved from the equivalence relation that it becomes idealized in the platonic form. I guess this is part of what is meant by the de-application of points. The idea that everything which was given in advance from the world, we better know. He's the guy who supplied us with metaphysics and ontology. That's good, but all these ideas...

1:12:30 No, it's just, it is that, but it's all right. He's, you know... He's all this heavy mob anyway. We could both claim to be previous fans of yours. Yeah, I think that's right. Yeah, but I am a fan. Okay, so I think I'll go... Go with Alberto and... Okay. Well, maybe he doesn't agree, but I know I have. I would like to see them. I could say that. On the other hand, I think I'll probably get more from that. I wish I could help in the class. Well, it's wonderful. What would you like?