Discussions, incl. P Mancosu / discussions, incl. J R Brown on Lawvere on categories of space (& others)

5:00 Did you say there are a couple of them? Please do, please. I wish there were. I start with a reflection on this very issue. Yes, yes. Surprisingly, something like Longo, who actually came out of that very traditional... Well, it's very often the people who did come out of that tradition who reacted most brilliantly against it. Correct. It's the classic psychology of condom. Perhaps I should say the psychosis of convert. Yes, certainly. I could do with... Well, I'm overstating my speech. I haven't had any speech at all. You see, that's why I can do very good at math. Let me say something wrong about that.

7:30 For more information, visit www.fema.edu. Thank you very much. Thank you for watching. Thank you for your attention. In my case, there is usually a short circuit in my brain when I do this topic. Thank you for your attention.

10:00 Which, again, I take from the long-ago ideas of Professor Pritchett and the other co-workers of the U.S. Institute of Technology, for a visual source for all, but even for both. Number seven is somewhere in one of the trinities of mathematics, that an awareness, an inquisitive awareness of a closed dynamic, closed to a complex connected space would have to be a place where one could have the notion of a second, of being a member of a second. There is already a primitive visual component involved in grasping that notion. And I think in the case of number also, there is a suggestion that, at least in introducing the concept of number, after all, a teacher doesn't. In the course of the course of the course of the course of the course of the course of the course This is so he's in connection with the plane that he said that every group of cognition is part of this transverse plane, and so when people got the office, he tried to point about what about us and the grass. There's a concept of number in the trans-categorial frame instead, and I've heard Longo come back with this. With this response, in this case, you have the graph to, for instance, have either an action or an order quotient. So some sort of cycle is always supposed to be done. One has actually to pick stuff up in a certain way that you can't, but it's very difficult to answer the question of how many. Well, we know very little about this stuff anyway. I mean, there are some studies about, you know, numerical knowledge. Pat Marcus works on that area, but we were talking about it yesterday on issues of visualization, for instance, specific to mathematics. There's a lot about visualization and mental imagery.

12:30 The course means that they measure really big, but very little, nothing at all, on the specifics of mathematics. There is, I think, a very good paper by Albert Abruzzi on the connection between schema theory and mathematical epistemology in a recent book, it's in a book on... I think it's actually called, the book is called on the notion of schema. It's about the Sirlian themes in the course of mathematics, but not narrowly Sirlian themes. It's schema, not schemans. No, no, schema, S-C-H-E-M-A, which is a term of after a... From cat. From cat, yeah. Well, no, in this case it's more... Thank you for your attention. It's edited by Albert Hansen, A-M-B-E-R-T-A-Z-Z-I, Lillia, or Lillia Albert Hansen, she's just the editor. The long paper I've got is by a guy called André Ferrucci, who is a friend of mine in Italy. It's actually difficult to compare it for written, and he's done, of course, the mathematics, And so on and so on and so on and so on and so on and so on and so on and so on and so on. Thank you for your attention.

15:00 There are nested terms, and there are nested terms. Adjoint structures, which are kind of pointed out in a nested design. But there is this program which is used by people like me, and I've tried to use it. It's the same theory, it's the same physics, it's the same way. It's not one bracket, it's the same geometry, it's the same thing. It's also the description of the whole. That's why it's a structure problem. Yeah, all mathematics in terms of the rest of the literature is a whole other topic there. Yeah, it's a hugely ambitious, hugely ambitious program, and one which is not, unfortunately, doesn't have much in the way of an accessible literature motivating it, giving it work. Thank you very much for your time. You won a whole fucking place at this day. I still am. Yes, yes. Oh, it's nice to meet one. Marcus Cascade gave an interesting talk the other day about places. In London, which unfortunately I missed, a professor sat me down in Spain on this, talking about a series of seminars at the LSE this term. It was called, How is Mathematics Related to the World? I think that's a wonderful subject, and the only person who's talked about it, by all discussions about it, has actually just been focusing on Hartree-Peele. And, uh, you've got the class, right? No, no, we've got, um, there's so little literature on the slide now. Well, I'm glad to see, uh, some of the... Well, it's organized by Michael Beckman, the professor of quantum physics. Yeah, yeah. It's very interesting. I mean, it's the hottest place that it's ever saw.

17:30 I did enjoy your book, though. Oh, well, thank you. No one agrees with it, but at least a few. Oh, I don't know. I think there's probably quite a few. I have to confess I don't. But then, yeah, you already knew that. It's disrespectful. Thank you for your attention. Yes, no, I remember you very well, this is the first time I've seen you earlier. Welcome to Yale. Thank you. It's an honor to meet you. It's a pleasure. Thank you. Sorry, second time. The first time I didn't know who he was, he made that up in the back, and he made a long list of things. Oh, yes, yes, yes, yes, yes. It was also in London. It was when he was a beginning graduate student in 1975, or 76, or 75, or 75. And it was the, you know the big genocide Olympics? You know, every four years it moves around. Oh, you mean the International Congress? The International Congress of Philosophy and Methodology. Exactly, and it was in London. That year. And, uh, Wieler. The next speaker was Johnny Hewitt, was giving a talk about black holes. And this guy at the back, and Wheeler, he didn't have a call, Wheeler said something about black holes destroy all the properties of all objects. That was it. Speaking of destroying, that's destroying the state. He's arguably, I mean, yeah, conceptually the most important living mathematician. I mean, how about people with... Credentials of the type that you are in full respect, Tim Lambeck, who would say that I am a left-to-left product.

20:00 It's got nothing to do with what happens in the classroom. But I do have a left-to-left perspective. But the bell tells me that I am sort of a left-to-left product. I think that's what he has this program, which stems from these ideas of burdening. Everything I've actually seen is set out in one piece of linear exposition, and it's very, very, very difficult because the mathematical ideas are so deep and involved. There is a great deal about algebraic topology and homology and homology theory, and a great deal of category theory, much of which Cooke has created, is in the survey paper in the Birkenhouser volume, The Shapes of Westia Centra and Atlantic. A huge, great impact is on mathematics. Well, there's a survey paper on topos theory, and about 60,000 scholars came out on it. And that's about the most, I won't say the most accessible, but it's the most complete exposition of how he sees the overall shape of the field. And from what I've heard so far, it actually seems to be being played through line by line by Colin Clark in our Bernhardt group. We had a long session with Florence in the Spanish Open Conference last year. And she's listening to this now. Thank you very much. You can't put these things in the sound right, but essentially, re-description is about the whole of the mathematics based on concepts. The set theory and the whole of the analysis will turn out to fit inside the geometric picture, and the geometric picture itself will turn out to fit together in terms of a radically transforming leap of understanding.

22:30 He spent roughly on the nested levels of a common kind of mark and what he called spigots, which is sandwiched between different levels of hierarchies and is connected with ideas which he mentioned at Galway, and which involves coding levels of what he called cohesion for China, which is a consistent theme of his writing, which I never understood, but China's got a lot more out, but certainly I think I much, I thought I had a much better fix on it. There's a lot going on out there, including the test box, so that the spaces in Europe, where we're in a group of dust points, have a natural meaning. That was the lowest level of instruction. And this connects with ideas about how... Thank you very much for your time, and I hope to see you again soon. Thank you for your attention. The categories of both of these types come together with a third type of category, which is linear, and the extension has various forms of properties, which is called Hamlet space. The three of those together, I'm just trying to reflect the flashes of distant scenery that I got from this one.

25:00 Thank you for your attention. Which I think is followed by the development of some sort of nonlinear set of equations, which led way back home to Harry Malmsteen, which has been used at the very end of his career as a sort of general company. And, of course, that's the past. And also, of course, there would be a past, because it's all about mathematics and motion. Yes, I think that's the... I think that's the... I think that's extremely false. I may have got it all wrong, but this is what I took away. It's a good story. But it would be fascinating. And, of course, even in terms of... There's discussions, those conversations with Jim Brown, J.R. Brown, of the University of Toronto concerning Lorbeer's programme, a recasting, a re-description of structure throughout. All the framework within which Lovier sees his, as it were, re-description of structure throughout much of mathematics as fitting together took place during the coffee break between Paolo Mancuso's paper, Paolo Mancuso's talk, and Marcus Giacquinto's talk on the morning of the 1st of November 2001 at Roskilde University.

27:30 I hope, outlining, Laugier's ideas are recoverable. It's now the morning of the 2nd of November 2001, the hotel in Copenhagen to catch the train to Roskilde for the second day of the meeting. I'm just so sorry I can't get to those. You won't be able to get to many of them. No, I don't think so. I think the very last one. I think the very last one I might be able to get to. Otherwise I'm going to be away unfortunately compared to almost the end of November. But I'd love to. It's a bit unfair to ask you to give a seminar a second time to one person, but I really would be interested in hearing your views. I'm sure you've had lots of new and interesting ideas about it. Well, actually, I've been thinking about it for a very short time, because Michael Redhead phoned me up in September to say, would you contribute to this lecture series, although I hadn't thought about applications. There are a number of different types of mathematics, and I think it's a very interesting topic, and I think it's a very interesting topic, and I think it's a very interesting topic, and I think How philosophy of mathematics takes into account that. It takes into account the very fact of applicability. Yeah, that's right. So, yeah, it is applicability. And what I was trying to do, I was asking myself the question... Is there a problem, an insuperable problem for Platonism?

30:00 In the applicability? Yeah. Well, I'd really like to talk to you about it, or I'd rather like to listen to what you think about that. Is there a chance we could meet and chat about that? Yeah. Okay. Maybe over lunch? I don't know what your commitments are. The only thing is I don't want to miss out on the discussion. No, no, no, no, exactly. Well, maybe a little bit later. Yeah. Well, tomorrow afternoon. Yeah, but let's stick together. Yeah, I will. Well, now I'm feeling a little bit gruesome. I certainly, you know, think I'm wrongly acting. Okay, well, good to see you again. Thank you. Cheers. Yeah, no, no, that's what I mean. Sorry, forgive me, I'm sorry.

32:30 There's a great deal of, which I would love to understand, the continuum. To understand what he meant by a general of relation, which, well, somebody says that the continuum is a general of relation. This is the reason his rejection of Cantorian, his conviction that the Cantorian approach would never resolve the continuum problem. The continuum problem is not even well posed because... Well, of course, his metaphysics, you know, the continuous ontological procedure.

35:00 But I would love to know more about the, you know, the background of the various... Yeah, I think, basically, I think this concept of continuum is at the teeming, and I'm... Well, it's certainly very influenced by Aristotle, but one has the impression that... He doesn't think that the continuum exists at three given points. Well, he certainly doesn't think that, does he? He thinks it should be taken as a primitive norm. Of course, you'll all be able to indicate actual points on it, but that doesn't bring to see the fact that it's consensual. So, of course, that leads him to pass the continuum as sort of inexhaustible. Yes, and within that, I mean, one has at least no local failure of well-ordering. Yeah, yeah, of course. And presumably all sorts of interesting conceptual possibilities involving, coming from that, from local failure, well-ordering, involving entanglement, and connections between the local and global structure of the continuum, which would connect with his insights into topology, and particularly... But as I say, is there any evidence that Brouwer read or was in at all influenced by Peirce? I don't think so. No, just simply a... I don't mean to. I never heard of him. No, it doesn't seem likely given what one knows of Brouwer's very... These are all idiosyncratic and strikingly isolated in intellectual formation, but on the other hand, there's a very clear convergence in some of their ideas, it seems to me, about their reasons for the rejection of Kantorian approaches to the continuum. I've been very much interested in the possible relation between Kurs and Husserl. Really? No, no, no. In fact, the other, but I don't think he read the other very much. I mean, you know, there are certain references in first to, but what he says about it proves that he didn't read it at all, because he played work of typical German psychology and logic and things like that, you know, that's a complete representation of the logic that he enjoyed, but I think personally...

37:30 I first had a copy of that book, but I don't think he ever read it. Of course, it would have been, yes, and I'm presuming if he had, it would have been very late in life. Yeah, yeah, sure. Very late in life, yeah. But maybe, I think he stole the notes, you know. For example, for Svartan T. Peirce, Ohlin begins to use that word, but after he got the copy, you know, the mature versions of his thought were like phenomenology or phenoroscopy, you know, these other... Yes, he used this other term, didn't he? Yes, but it plays a huge... But to return to his ideas about the continuum, and particularly, obviously, there is a Stelian background, the conviction that they ...whether the continuous precedes the discontinuous, and that the one has, as it were, kind of inexhaustible in the continuum, that it's, as it were, the form of continuity precedes any termination of it by logical arithmetic notions. I mean, what's this kind of... because unfortunately I know very little about works... Can one trace a direct influence from this? On Peirce's ideas in topology, his choice of primitive concepts or axiomatics. The idea of figures as being more fundamental, as Polunov points, as it were being the limiting case of figures as being as it were the defining The structure for a notion of space in general is one which has been revived, I mean, very strikingly, in modern, no category theoretically, based approaches to topology.

40:00 And particularly in geometric Galois theory, there's a whole literature now in which you have this so-called adequacy, co-adequacy of points condition, is the thing which just picks out the sort of zeroth level structure in the Galois hierarchy. Parallel with Percy's conviction that the points, as it were, are the set theoretic dust with which you're left after you've taken out all the confusion in the structure, which is what is, as it were, the ontologically fundamental term, and not at all the generic elements from which one starts in building up any possible definition of structural characteristics in the first place. No, it's just interesting parallels, it seems to me. And it's a striking instance where conceptual argumentation of a very strongly metaphysical kind does connect with the strategy for recasting the structure in a whole area of mathematics in a way which is the connection seems more direct in the case of this particular work than it does in most. Any symbolic presentation will always require continuous intuition, even the most formalist. Yes, there's an original revival of those ideas also in the kind of recent work. I'm very attracted to this claim that every ingredient of cognition is the transpose of spatial structuration at some level, that it has to have some sort of primitive founding.

42:30 Intuition, which has to be in place before you can have the lifted content of a symbol in which, you know, and when we take that as the ontological fundamental, we then get trapped into some platonic atomist metaphysics, which Russell and others, you know, bequeath to us, which goes outside the geometrical framework and regards that as something which has to be reconstructed from below by this... It's clear that this works, and of course there are all sorts of unresolved problems with set theory, and not just at the technical level either, but also even at the level of just the definition of the finite-infinite distinction, and of course the problem of the continuum hypothesis, which Jim was talking about. I want to have a go at Jim about one thing, except he's very busy on his work. Yeah, let me do my email. I'm going to get booted out of here in 30 seconds. Boot it up and then boot it out. Okay, well let's go and get some coffee while there's still time. I think there is. Thanks very much for the paper, it was very interesting. Hello, I feel for you. Yeah, I only hope it wasn't me that gave it to you. No, presumably you were already starving yesterday. Thanks, cheers. See all the structures fitting together in virtue of the way that... The more general structure, more general and ontologically fundamental and holding the key to the understanding of the way that the structure that is possible can be conceived of as candidates for structures and systems in the world fits together in itself as it were already in a way which is ontologically prior and autonomous, fits together in an ontologically prior and autonomous platonic heaven, to use that metaphor.

45:00 See, rather, the way in which the structure, which is a candidate for structural systems in the world, fits together in general as holding the key to the understanding of the manner in which the structures. Thank you for watching. Thank you for your attention.

47:30 I'll be really angry. It's already one year. Oh, well, it's not to me. They present themselves. You know, it's a family. That's a good one. Yeah, that's a good one.

50:00 I hope you're soon feeling better Paolo. I hate to say it, but it sounds to me as if it's not going to go away just because of eating tangerines. I think you probably need to take some serious medication for that, it sounds to me.

52:30 Thank you very much for your attention and I hope to see you again soon. Thank you for your attention. If you're alert to what our students are going to say, this is a guy who has no time for visualizing. It seems like he doesn't have time for visualization. He could have drawn that out. But the presentation is . Yeah. Yeah. Yeah. Yeah. I talked about it. So, um... I'm delighted to introduce our second speaker of the day, Rémi Alnette from Stanford University. He's going to talk about two words and aesthetics of mathematics.