Discussions, incl. M Wright & A Peruzzi on Lawvere's paper (contd.)

0:00 You have bus stops, but the problem is, suppose that you have this bus stop here, and here you can, there is a bus that arrives here, this number one, and there's a bus that arrives here, this number two, and there is a place where people stay. But perhaps this is an object that is not stable, and if the variation is subject to, reaches a certain intensity, at a certain point, it's swift. And so people that were here, that would like to take stop at bus 1, will take bus 2. People that take bus 2 will take bus 1. This is the idea under the notion of character. There is an intrinsic instability of objects. Hence, of course, underlying the ultimately topological origin of the notion of object, even in such a general case. Yes, I agree. In other words, there is a very slight anticipation, it seems to me, of these very profound deep metaphysical grapplings with the... With that dimension of generality that even a profound mathematician like Bill, I think, has not yet fully confronted what is at stake, what it is about the people whose dislike for the status they think is implied, as I say, in the bookkeeping axioms, what it is that they have in mind. In other words, he says, as if it closed the debate, how disembodied would be processes without states, but, we've already discussed it, there is, I think, a very slight anticipation of some of your concerns in what Dummett says in his remarks, in his debate with Geach about relative versus absolute identity, about the understanding of the domain quantification.

2:30 Just the difference between the picture of successive articulations of something which is amorphous is to be thought of as having no intrinsic structure, no structure in itself prior to the articulation. In the case of the debate over relative identity, it was simply by the choice of a sort of, gave one criteria of identity, related to debates about the terms of natural language. But it seems to be an idea of much greater depth, and immediately, of course, one sees them. The possible connection with topological semantics for quantifiers, topological semantics for variables in general. As you say, the different metaphysical components in the notion give one the ontology of the three variables. I think they're very, very deep. But I would love to understand exactly, because it seems to cut against what he says here, The category of maps which transform the objects by means of which they are to be distinguished, hence, obviously, the whole question of the existence of inversive matters, the understanding of extensionality in terms of uniform separability conditions, whilst, I mean, that does seem to me to be in tension with what he says about the absolute indispensability as a framework for rational thought of the bookkeeping axioms. And this seems also to me to touch on whether this account...

5:00 Rightly, of the constrained evolutionary complexity of systems and, yes, obviously of actions and systems. In other words, whether cohesive active sets are really to be thought of as starting from sets, are they a generalization of sets? Or are sets a specification of cohesive active sets? Or is this, in fact, a dumb philosopher's question because there is a closed circuit of ideas here? I want to say that the constant sets actually emerge from the cohesive active sets because of the, as a reflection of the structure of the world, the ultimate, it might turn out that we did have, or didn't have a global factors through. The point is that I think that we can continue this discussion by going to the rest of the slides. I'm sorry, I'm sorry. No, no, no. No, I'm, besides, I'm burbling. We're getting nowhere. I'm just burbling. What I, sorry, can I just have a very, very quick look indeed at what he says here? It is interesting, he does relate this Quotients of Decidable Objects paper of Johnston to his own paper in Boulder. I'm, if he gropes to clarify on this paper, I just grope. But it is interesting that, and obviously we're on to very important, so we'll leave that for now, but again I keep coming back to this suggestion that it is by thinking about really the, as one does, very quickly, in other words, the semantics of classical quantification theory forces you to think of the domain as having an absolute identity defined on its object, therefore any equivalence related. ...as naturally of lesser strength, hence the very notion of partitioning and of the partitioning domain of the domain under the absolute identity relation as just being the thing that gives you the structure of the domain as a domain object.

7:30 And this way of thinking of equivalence relations as in the first place equivalence relations in topology and the relationship between the local and the global. I mean, that, it seems to me, goes to the core of the difference in the way that the set theorists, that seems to me to go to the core of what it was about set theory that so naturally gave rise to the metaphysics of extension and what it is, what is it where the naturalistic terms of the arise from some structure in the real world one has in Anyway, I don't know how you'd agree with that. You are pointing to a great problem. Yeah. Well, I'm sorry. Well, at least pointing to problems is the first thing. All the random maps, all the time, that feels strange how this can be fixed. And you have first and least proof that all these kind of random maps are identical. And this is exactly what Bill was saying at the conference.

10:00 And you have the possibility of, say, of making and do these arrangements around the maps, a kind of, you obtain a wider class of compositional maps, composable maps. Since you do this, come around the maps, you can achieve this after. To this and this G in such a way that that becomes compulsive. This gives you a wider class, but of course one should work in order to find which could hold the right amount for certain purposes, you know, that's one part. Sorry for this long prayer. No, no, no. Don't say sorry, it's the most important. Nothing you say is... Can I just say very swiftly, I don't know. You're right, it is fascinating to me how you and Bill Laubier do think so very much alike, and how you do often anticipate, philosophically, many of his ideas, and how philosophically his position seems to, I think, lack the, obviously, the depth of yours because of the, I don't mean that he doesn't think, but deeply, he obviously thinks more deeply than anybody else we know. But he doesn't have quite the historical reference. He does still have... I think whatever strengths and weaknesses come from his referring philosophical thesis to dialectical framework, but he is astonishing how closely convergent your philosophical understandings are and how much they are what I have been groping to all my life, when all my life has certainly been the last 50 years. Let's try to begin to understand things like these things, obviously without... Anything like your philosophical gifts or your technical equipment, but very, very much the way that our thoughts on all of these things do converge. It would be a marvellous development if you and Lorvier were able to collaborate fully in writing the book, I'm sure, and it would obviously ramify the so many fields.

12:30 I don't know, there was a time about two or three years ago when he was in Cambridge, when we were talking to him, when Bill was participating and he tried, tried to find a position here, a permanent position here, but nothing ever came, coming back. I know that Melonian people were trying to, were sweating to come down to Milan. Since the, how do you say, the place where you put money? Yeah. The universe. The bank? No, the... Oh, your pocket. The pocket, no. The pocket that the Italian government now has. But they always have it. It's the only country here. It's just that they're no longer able to keep the doors open. Well, would you be interested in a sabbatical in a year at Buffalo? And I feel myself. If we expect to have good mathematics, it seems that it is going to be difficult at the moment to sustain the level of discussion and the people, but probably is going to be. We are going to make that up now, this coming few weeks. We are going to go down and spend the time with Lord Eyre after the meeting. We have the invitation to go down and spend three or four days. I would like to have some, not super-sized, not the square. But there are so many things I have to follow here and I would like to prepare very carefully this period of reading. This is where I feel guilty because I'm not able to keep my mouth shut, believe it or not. What? I'm going to keep my mouth strictly shut, except insofar as you can help me formulate perhaps the most three or four questions that would not ramify too deeply, but one certainly is this remarkable relationship between the...

15:00 The understanding of the double nature of the inclusion of the discrete codescreen level. I think this must go to the core of our understanding of the ontological position of cohesive acting, again, in relation to I think that must go, I think, to the core. So I would like to hear and expand on that. I suspect we're obviously pretty well on the lines that you've already found, but I would like to hear specifically on that. And that's probably the most important single thing. There are perhaps two or three other questions that I'm willing to answer. Anyway, I'm sorry, I interrupted you because... There is a lot of specificity in all of these. There is conservation in all of them, and all of them are very, very, very, very, very, very, very, very, very, very, very, very, very, very, very, Thank you for your attention. What do you think is the point that you are making, not connected? Negative, the behavior of the separators of a pair of arrows in A to the 2, vis-a-vis the behavior of the pair of separators of an arrow in the category set.

17:30 This is not a reflection of the fact that the separator of the pair of arrows and atons is no longer the identity in the straight-ward wave. And he said, what do you mean? What do you mean by a separator? I don't know this term. Well, he didn't say I don't know this term. What do you mean, a separator? I said, a separator is a pair of parallel morphisms. And he said, oh well, this is a term that is used for, he said, ah, you were asking about the generators. I said, well, I'm sorry if I've used the wrong term. Yes. But I was surprised because I otherwise thought it was effectively the same. It's important not to read too much into absolute chance. You know, ways that exchanges go in the course of a short conversation. It's not as if one has to place enormous weight on it. It's not as if this was a considered formulation that he had written in some paper, public paper. Outward space, linear operation, the operator, that's called the generator, the calculator, that is the controller, and the DSP, 0, 0, 1. These have the calculator, the outward spaces. Then you always have, I mean, the world's long pyramids, and things of the era, and who the human beings are, and what they do. This is a property that was shared in the internet, or probably on the internet, since the time of the student of the era, only today, these days, not that, even if it's not the human beings of the era. He is the co-director of Innis Science.

20:00 What is the passion that is the interest in myself in going back to my research on mathematics? I want to talk to you about that separately, tomorrow perhaps. Thank you very much for your attention. In this case, we cannot express the notion of a generator, since in general there isn't. Whereas there can be very easily a quantum motor, still. What happens in the case of sober spaces? Is there simply a co-generator? Shutter space is an abnormally high extension of the 13th century. But is there in their case a generator and a co-generator, or just a co-generator? Well, they have a generator. Yeah, that's what I was going to say. That's what I thought. Well, in fact, yeah, as far as the generator goes. There's two beyond the other. Of course, there's two beyond the other. I think also, actually, that the consideration about server topologies is quite an useful point of entry to the general network of ideas. Now, the point, my point was, it was trying to express this that more of the topological spaces conceived as defined in the secular record, you kind of dualize the situation more around those spaces where you have a culture break.

22:30 And dualizing, taking the social category, you would have a generation, right, since we take the outdoor spaces often, so it becomes too much, but since these are no more. No more continuous punctures, no more continuous punctures. From the point of view of splitting punctures, so there are no punctures. But if you deal with topology, you have the duality between the fact that the proposed category of the powers is the category of frames. Right. And this has all the same properties, even if dualized, of the category of the class. So you have a co-generator, the fact that it is the entire group. So you obtain a generator in the category of the frames. The question is, which are, which is the widest, the most wide category of the topological, pointless topological spaces, which have a co-generator, since in this case we have a generator for the dual category. That's an extremely interesting question, and this also would seem to be very naturally related to our understanding of the range of weakenings. ... of extensionality, which we were talking about a little bit yesterday, in the context of what dictated the stability of the global sections.

25:00 Again, I would love to understand more, again, about how the range of weakenings of the stability of the global sections, in the general context of trying to understand... Both sections and retracts of maps in a more conceptual way, particularly as maps from spaces to spaces with or without the That is to say, the conceptual level at which we approach the notion of mapping, we've been discussing how this would show up in one's understanding of the technical behavior and the fallback of the abstract sections. I have a few ideas in that area, but they're much too diffused to be worth talking about. Did you know that Bill and Stephen Shannon wrote a pedagogical book. I look forward to giving this to you when we get to London. I wanted to make a complete copy of it before I came. It's a big book, it's a duck. There are 400 pages, and it's privately circulated. He sent a copy of it to me last year, well it was about April last year, with a very nice note attached to it, and it's called... The course is called Introducing Category and it is the lecture notes of his first year, his freshman course in Category for the students at Butler. It's actually the freshman course as given to the class which contains Fatima and Danila as students and it's an extraordinary course. Oh, that's right, it's called Conceptual Mathematics, that's right. It's called Conceptual Mathematics, the subtitle is an introduction to conceptual mathematics, but it's a very elementary book at the level of the, well, it's not really elementary at all, it's very elementary at the level of the result that it presents. Extremely elementary at the level of the motivation, but when I say it's elementary, it's only to, well...

27:30 So to a mathematician reading it, skimming through it for the first time, not knowing forbear, it would seem very elementary at the level of motivation, examples, working exercises, and results, and material quality. But such a reading would be very... Because if you actually bother to go back and read in between the lines and in the side remarks, as you might expect, there's quite an astonishing amount of depth. In fact, he, in, I don't know how long, it's about 280 pages long, maybe 300. In all of that, he doesn't even get as far as the academic literature, doesn't even introduce algebra. So he stays at an absolutely introductory level, doesn't even get as far as algebra. That is to say that the main, the line of development of the main idea of what he introduced, in fact, he, by the end of the work, has introduced, he has introduced gross and petty topos, he has introduced much of this material, yes, much of this material on the category of graphs, and on the, no, 17th. All introduced at a level which is actually for the brightest students assuming that the text is a genuine... Transcription of the of their reactions is made accessible even there. One of the things which I really have not understood until I read this book is which and which he would not I think take away from listening to him speaking to a general mathematical audience or to a philosophical audience such as he spoke to at Cambridge. One of the things I really have appreciated is that there is a marvelous feature. Really, very, very, very good teaching. How much of the actual... Exposition, very introductory, very, very introductory exposition in is Lorbeer and how much is Chagnon, I don't know. I expect some of it, quite a lot of it may be Chagnon, but even so it's a fascinating book. I must give it, I'll make a copy when I get to London. The problem is it takes about half a day to make a copy because of the size of the book and to find it.

30:00 I thought he might have sent you a copy. Maybe he thought it was too childish for you, too elementary. It is a very elementary book at the level of the Victoria College, but even so, the cyber-mask, and particularly, as I say, on retracts and sections of maths, are really, if you bother to read, you realise the depth at which he is thinking of these... When I was trying to get him to look at it, and he read it and put it aside and said, well, it's a children's book, why should I be interested in this? And I said, I'm ashamed of you Jerry, you've just not bothered to read the book. Of course you've read all the results of it, but that's not the point. I don't like this mops and matic style. I don't like this mops and matic dialogue style of teaching mathematics. They've got to kill me at a softer pace. I'm sorry, I think it's true, it's not. I think that we, as people here, are very... I get what you do. So do I. Which I really highly appreciate. I would like to see what could be a last lesson for students at Capital Three. I have to confess that, for my experience... Well, you would actually have a very good impression of just what would be the experience of just such students by reading his book. In particular, since two of the people who teach him were his wife and his son, you can imagine that he was going to be very scriptural about how he taught. Thank you for watching. This is a wonderful piece of work. This is a lovely piece of work. Very good indeed.

32:30 Oh, one thing I was wondering. I don't know how much free time you have tomorrow, and obviously I know you're going to be getting ready to go on Sunday. And if Francesca's very kindly said that she'll meet Gary and the Philadelphia one, I really would like to show him, if there's any, from where I can while we're here, there's a bus service on the course, and I just wanted to know. I would like to take him up to Piazzalee and show him that in, you know, um... Well, yes, if possible. Well, no, no, because we're not meeting until 1. Um, what about... Perhaps in the afternoon. Is there a, um, is there a really nice restaurant in Piazzalee? We'll meet you in Madeira tomorrow evening. Is it that I should come to you with my brother for dinner this morning? Was that the plan? I would love to, but I mean, only if you... I think that you can't come. Can I bring Gary in that case? Well, you know, it's not a worry. It'll be his last evening in class. And I'm sure he's a nice boy. He'd like Francesca very much. No problem. No problem. You're sure? OK. And I don't particularly like leaving him in Florence all on his own when he doesn't have any account of it. You didn't pack a car with us in these days. Oh, no, no. Much better. No, no, no. He would have been more stiff, but he has no intellectual interest at all. Well, I would say that. He's, you know, he's, as I say, he works for me at my business, you know, as a driver in the firm. I thought it good that he should see flowers. By the way, how is your mother? My mother? You were speaking to her earlier. Is she in good form?

35:00 My mother is in a little island. I thought you said Hintico. Sorry, I can't hear. We'll leave it. Leave it till later. Oh, yes, I know this. There's a sort of games-theoretic semantics for it. It emits a games-theoretic semantics, yes, I've seen it. Perhaps some ideas of information will vary also. Which is a contraposition that is much eased, I think, in the last many years. It would not be out of place in most, I don't say most American universities, but certainly it would not be out of place, I think, now in Harvard, perhaps still in Cambridge for that matter, but in the ordinary philosophy department, not in the philosophy department, perhaps in Harvard. I think we can go on. Sure. Because our... Yeah. What's Partnum's paper? Partnum's paper is on quantum mechanics. Nice paper. Thank you. Can I pay you for this on visa? Or can I pay for this in visa and you settle up? Because I am rather short of cash.

37:30 His ideas have changed apparently very much about quantum mechanics. In the last year or two I was hearing a lot about, you know, he came to be at the complex in LSE. I didn't, unfortunately, hear it. But I understand the mental complex and he now has a completely different position from what he held at the time he wrote the, uh, but not just from the old days of quantum, uh, his, uh... There's extreme realism, but also the more recent papers, which are really from the internal realist position, which provide this position a great deal since 1932. Is the video complete? Yes, it's complete. Pleasure. Well, you're putting me off in this department, aren't you? Well, perhaps we can have a look at it tomorrow. Now, I must go and make copies of those papers for a McLean tomorrow. So, how many copies will you produce for McLean tomorrow? Will it be open on Saturday? Half-past nine in Europe itself. Fine. I'll go and do that tomorrow then. So, what are our plans for tomorrow? I'm meeting Francesca at... About 1 o'clock at the Uffizi. And then what time do you want to meet in the evening? I think that I will be at the flat where you are. Sure, right. So just to arrange it with...

40:00 I will be at the 25th. Sure, okay. Probably about six or so. That wouldn't be too early. At that time, Umberto. Oh right, okay. Perhaps I could spend a little time with Umberto. Yes, I could see that already from our talk the other day. He's not able to express his ideas in English. No, and unfortunately I'm not able to express mine in Italian, which is a fault on my side and I must do something about it. So, such a satisfactory scientific language of today as Latin was, is it really? Yes, of course. Yes, of course it has, but I... It's bad John, for example, and before that John, the combinatorial magician, Hindley. Hindley? Who wrote that book, with other two logicians, the famous book on combinatorial logic? Roger Tindley. Curious. I'm afraid I don't know the name at all. I thought he was a... I mean, the only combinatorial logicians I can think of are people like Fays and... Yes, he was a... You wrote it with Fays and... I think, yes. What? Fitch, Fays and Fitch? Fitch, yes. Parry, of course. Well, the earlier generation. I'm afraid I don't know the name, that's surprising. I think he was in very good relation with Umberto. And then John too, since they stayed here some more days than you came. Yes, sure. Umberto is basically a logician by training, but he's also very interested obviously in foundations of geometry and mathematics in general. Mainly the part of mathematics that he was interested in was linear algebra.

42:30 That's interesting to have got into the philosophy of geometry from linear algebra, not a natural line of approach. Oh, I don't know perhaps so much, though, because... It's often occurred to me that thought about the foundations of geometry is perhaps not entirely unconnected with some topics in combinatory logic to do with... No, the properties, the topological properties of spaces of functions. Of course, the simplicial topos will manifest this. Exactly. Oh yes, now, now that we have topos theory, of course, the connection is much deeper. But I was thinking that even before topos theory, even in the... I was thinking long before topos theory that ideas, that fundamental metaphysical thought about relations, such as the problems which posed themselves for Russell at the turn of the century, were already pointing in some of them towards concerns that gave rise to... ...to combinatoric logic, particularly, obviously, the analysis of substitution. Just not taking substitution as an unanalysed operation. So look, Mike, the options for quantum problems... Yes, I remember it from when I was here before. I remember it. I remember it. Okay, Alberto, thank you very much indeed for a very nice evening. Thank you for this evening. Oh, thank you. I only hope I haven't strained your voice too much. Will you go back and get some rest? I'll see you tomorrow evening. Tomorrow evening. Okay, thanks. I'm just going to go for a stroll before going back to the flat. Okay, see you. Give my love to Manuela and Francesco and Gregorio.