Conversations incl M Wright, J Mayberry, A Peruzzi, C McLarty, J Bell

0:00 I can't make any... That's another thing I don't understand, you know, philosophically. Why around 1900 didn't people start screaming about that? Why didn't they worry about junk like Russell's paradox, where the piano curved it? You know, that should have been a really... Yes. Well, that was largely the influence of the people who ultimately gave birth to linguistic philosophy. Yes, because they failed to understand how deeply the roots of language are grounded in levels of topological, geometrical structure in the world. Objective requirements of domains of variation for structures to parametrize them, which might be different from the discrete case, the case where you have what you need to parametrize discrete quantities, and also I think the influence of atomism, that juncture played a part, which was taken over into language, into the kind of platonic atomism of Russell. But I don't think there's a lot of, the actual problem, the actual idea that there would be, it doesn't come with the Greek philosophy, the idea that the world, that the discrete universe, the universe could contain infinitely many objects, doesn't appear to have arisen as an objective. I don't like the term. As a question about the real world, you have to look at it when it actually was taken seriously as a question about the actual world, taken as a physical, taken as a whole, but by Newton's time it definitely had to have been because it's part of his physics. He talks about the idea of the things, you know, infinite space and time with these things uniformly, you know, with actual blobs of matter uniformly distributed. That implies an infinity.

2:30 He writes very nicely. You know, you take an infinitely extended continuum, which he clearly saw. And you have these blobs of matter everywhere. It's an infinity. Of course it leads, delightfully, it leads to an actual contradiction in practice. Albert's paradox. Albert's paradox. But the point is, it doesn't have to. We might have been living in an infinitely bright world. Who knows? But the point is, you can't deduce from that that there's something wrong metaphysically, you see, with the idea of a world with infinitely many actual discrepancies. I mean that the whole notion of an actual, the whole notion that you can actually, you know, that you can actually have an infinite discrete realized collection in some way is simply something that has to be rejected. Or at least it leads to some sort of dialectical contradiction, as it did in Kant's case in the antinomies. And in Hegel's case, well, we know that there were various refinements of this idea. And Hilbert seems to think that the whole notion of a completed infinite, in any sense, is some kind of contradiction. He doesn't work it out, but he says so, it's on the infinite. But then he draws... So he's followed the idea that if it's discreetly true, then he also thinks it's infinitely true. Well, he may have done. I think that's true. He may have done. He went, you know, you're right. He went too far with it. But the point is, the whole idea that you can then say, because we now find the universe, they have finite models of the universe, this confirms in some way his idea that the completed infinite has to be rejected, the completed discrete infinite. I don't believe, I just don't, I just, for me, this world is very likely to have infinitely many objects in it. Oh, that's a good idea. Do you mind? Well, I'll do it. Do you want some help? Bill! I was having a very interesting discussion. I'm sorry I couldn't have heard the first part of it. You were talking about subjective infinities.

5:00 Yes, one of the things I'd love to understand more deeply is your ideas about counting. I was hearing a little bit about this from Colin the other day, but I just am not sure whether he's given the correct exposition. You think that the... Piano axioms actually involve a kind of subjective, I mean, this distortion of the counting process. I wouldn't say distortion so much as simply, when I call it subjective, I mean it's a model of a subjective process. Sure. Which is, yeah, sure. We wouldn't understand anything at all. They've been both in idealization. In other words, Brouwer's interpretation, I think, is correct. I don't see a Brouwer in most things, but he says that we can think the next thought. That's really what the content is. Rather than if there's a next thing. Yes, the notion of structure, the structural progression, or the structural characters of the progression. So therefore to imagine a completed model of that, it could then be an objective model as a subjective model. Yes, is itself to invert the... At least there's something distinct from something objective. Yes, the model has a reflection, a perfect reflection of objective structure. Yes. Yes, that seems to me to be right. In particular, the fact that the subjective is subordinate to the objective, which is a reflection of the fact that thinking is subordinate to being. Reflection within the subjective, or I prefer to say within thinking. It means that one should be quite careful in thinking that things constructed by using countable infinity could be inside the real space. For example, Keanu's space-filling curve is precisely something that's very dubious, which we've been persuaded for a whole century that we must accept as part of space.

7:30 Well, of course, because we've been saddled with the idea that geometric structure is something imposed extrinsically on sets of points. ...rather than arising from the intrinsic structure of Chosin and Perpessor. ...continue to have objectives in the same way that geometric structures, like extending lines, work. We can actually separate these. I mean in the usual set theory these things are quite mixed up. We can start to see them separated from each other in the topos, deal with the issue perhaps through over-determination, an example of how the separation could take place. Well, that's the point. I want to understand more deeply. Try not to be dogmatic about it, you see, in a sense. I think it needs to be seriously investigated and not just... It seems that a hundred years ago, people just somehow were persuaded to ignore this, didn't they? Yes, yes. Maybe a few did. It is, it is, I mean, again, no, this question of separability, this question of the topos as the right setting in their paradigm to investigate the relativization of the separability, the understanding of space as the real variation seems to be very important. One of the things I'm trying and struggling to understand more deeply is in fact how traditions like extensionality expressed in the topos setting, how they already reflect not just the relativization of the notion of element, But also topology has quite strong separability conditions, the way that the axiom of choice is really equivalent to the uniformity of maps.

10:00 So the maps from one space to another have that condition imposed on them. They become maps between spaces which have uniform separability conditions, so that you have the recovery. There is a constancy of variation, but of course one must have deeper variation, which didn't allow the simple... And of course I'm struggling also to understand your... There's a simple example published by Murray, I think it was, in the machines of the Cambridge Philosophical Society a few years ago, Topos, rather easily and constructively. Which doesn't admit any geometric morphism to any Boolean topos at all, let alone to sets. It's very highly non-Grodendieck, but not even in some relative sense could it be Grodendieck, I mean, in the sense that there's no map to a Boolean of any of these characters. Yes, let alone to the one point space. Yes, that's right. Yes, yes, that's extremely, I should like to understand that. Where is this in Proceedings of the Cambridge Philosophical Society? Who is the writer, are they all? Bob Paré. Paré, yes, sorry. I guess I've come across some citations of him in your paper. No, I must read that. I must study that. It's something I've got very, very interested in. And I also want to understand more deeply the remarks in your... No, I thought a little bit about this. Colin and I were talking about this in Cleveland two days ago, before coming up, in the context that I was... Asking him about ways in which one might relax or generalize some of the conditions that give you sets as the decidables of a topology. The thought of as a condition on the intrinsic of the topology. Connecting those with some ideas about relativized notions of individuality of systems in physics. One of the remarks you make in the paper, the San Sebastian paper, the categories of space and quantity paper, I think you've revised that since, because I've seen many of the things that you say there about the topos, the Boolean topos, as it were, the zero level of construction, about the behavior of the kind of triple-actual construction which relates to the...

12:30 You say, towards the end of that, that this is important for understanding philosophical problems of distinguishability and indistinguishability and Cantor's grasp or Cantor's understanding of... Now, this is something I very badly want to understand. Is it possible to give me more motivation? This was part of my Cambridge lecture. Yes, it was, it was indeed, but I still... Your tapes have been combined with... Yes, John was telling me this afternoon. ...by John. By John, well... ...into a slide. Yes, into a slide. However you slice it, it will still be wrong to quote the basic theorem of topology. That article, I mean, have you seen the draft? No, no, I haven't, I haven't. In fact, John said he wasn't going to show it to anybody. He showed it to you and discussed it with you. But I hope before we part, we perhaps have a look at it after you've had a chance to approve it. Talk more about this business about the... In the setting of the understanding of spaces and sets as... The sets of points of space is the limiting case of constancy. That's something I really want to understand. It may be a point that's so fundamental and so elementary that really there's no great mathematical depth.

15:00 Mathematics really starts to flow out when you consider the development of this. The distinguishable case is just to be understood as the case where you do have a mapping to the one. One way of saying that they're definitive is that they have a map of the Earth's base and the Earth's base, so they would have different values and so on and so on and so on. If this space happens to be a discrete space, then you don't have such a thing. So that gives a specific sense to the idea that when the cardinal is realized as the code of speed, that everything is indistinguishable. Right, yes, completely, because everything can trivially go to any other point in a completely chaotic way. Right, that's part of it. What I just said is slightly different, namely you can't map it into a discrete space in such a way as to cleanly separate, so I'm mapping into a discrete space is the idea of cleanly separating, or assigning values to both. Sure, sure, sure, sure. The idea that the truth values are distinguished. And this is also connected with the idea that... Yeah, absolutely, and also connected with the idea of, I think, even the old pre-Socratic worry about the primacy of stuff or things, and the separation, whether you have a clean separation of the notion of the state of the system and the dynamical transformation of the system, or whether one does have a complete separation between the state of the system, the notion of the state of the system, given the dynamics of that system, so that...

17:30 As you pointed out in the case of homo, in the case of a godic theory relating it to the case of a non, of a trivial, which is both locally and globally a space, the action, the monoid or the root, that is acting on the space keeps everything nicely separate because any neighborhood of any point is moved. Okay, and in the more general case. Well, we'll not have that because of the mixing up of stuff and so on, things in a very direct sense of the primacy of, in a sense, the police are perfect, primacy of stuff over things, in other words, the generic, to speak in the loose sense, I don't mean in the technical mathematical sense of generic, but the generic notion of thing or object, which in the classical semantics of modification theory. This gives one the notion that there must be things there to be the values of the variables, thought of as being there and having absolute identity conditions, which is really what gives one the whole building up of structure in cumulative hierarchies, which allows one to see that as the outer limit of all structure, everything must sit inside a structure which is already completely parametrized by discrete things. The generic notion of object is actually seen as the result of this idealization, subjective idealization, or the reflection of that subjective idealization that's already taken place. So we have to have some way of thinking, and in introducing these ideas philosophically to magicians, we have to have some way of thinking of the structural domain in a way that is prior, that lies underneath, that's more fundamental than the Freudian, you know, Zermelo notion of object. Well, in the set theoristic case, the notion of collection and extension has to be determined by its elements, and since the notion of extensionality is absolutely this theory, it's not. So, every notion of variation is a restricted case. It can be modelled within constancy, which is metaphysically not the case. It's really a distortion of the proper relation between being and doing.

20:00 ...object from the deeper understanding of the structure of the domain, the variation, but one sees that it is the product of a subjective idealization or distortion of the way, I'm sorry, I'm not expressing this clearly at all, the fact that the notion of identity on the domain, which gives one this generic notion of object, which is, of course, reflected in the... The Bible is an assumption that there are only two possible truth values. That has to be understood as the limiting case, a deeper way of thinking, of structuring the domain in terms of the tools given us by the theory, and obviously the understanding of the dialectic of local and global equivalence relations in the topology and their behavior in quotient, the behavior in the... When one forms quotients in the appropriate topos, that is the more general case. It's the case where one recovers the classical, extensional notion of point or element from that, seen as involving the imposition of the limiting cases. The case where you have, as you just said, the map. That is simply what gave rise to that notion, I mean that was really what gave rise to that way of thinking of the free and its value. That strongly platenist and idealist way of thinking of the values of a free variable such that there was a notion of object which already fixed this notion of constancy and therefore led to the idea that the domain must be a domain of objects the same or different absolutely, which was a notion within which real variation, real space, this absolute notion, it's interesting to see that as a reflection of precisely that inversion.

22:30 I'll maybe that's yes exactly yes the physical universe the physical universe the natural universe is just this little blob of structure that happens to sit down there at the bottom but then you've got this model the problem yes exactly but the thing which gave rise to that extreme objective form of objective idealism which as you say matter is just a little blob Sitting at the bottom of the Ordinal Code or whatever other way the objective idealist thinks of the structure of his eternal realm. But obviously that has, I mean, the great distortion which crept into mathematics between, as you say, around the uniting countries. It all comes out of that in a very interesting way, and I'm very interested in trying to win a deeper understanding of what happens philosophically in the minds of people like Frege and Russell when they rise to this illusion that there's a fundamental error. There's also an interesting question culturally and politically because it came at a period of reaction against. The achievements of the materialist understanding of the world in the middle part of the... Yes, yes, yes, yes, I mean, no, I'm... No. Well, it is obvious to somebody who thinks, with your amazing insight, that these interconnections, because it's not obvious to a... No, just the term states, right? Oh, no, you see, we're adjusting the chronology. All right. Now, I thought you were saying that you could actually see... Relativity and quantum mechanics. And the so-called Foundations Crisis Paradox of Mathematics, the discovery, the development of set theory, yes, all came together dramatically.

25:00 All those people. Lord Salisbury? Do you mean Lord Salisbury or do you mean Lord Rowley? Salisbury, the British Prime Minister? I didn't know that he had contributed anything to set theory or to physics, I have to be honest. I know that he was, in his youth, he was a serious Tory philosopher, Tory propagandist, and that he wasn't intellectual, but I have no idea if he did. Me? I knew his nephew Balfour did this, I don't know why this sounds pretty. Oh, Balfour did as well? Certainly, yes. Balfour wrote a large number of books on philosophy. Balfour was in fact one of the founders of a group of idealist philosophers, British and mostly Scottish idealist philosophers. They call themselves The Self, and they met from... Oh, you wrote a book about it? It is Balfour. It's not at all a bad distortion, because he was Salisbury's nephew, and he was very strongly influenced, and his uncle had been an active intellectual propagandist of tourism and of Christian theological ideas much earlier in his youth, but I think by that period, by the 1890s, Salisbury was too busy being prime minister. He became Prime Minister in 1902. He succeeded, sorry, he succeeded. Salisbury had been Prime Minister throughout the 80s and 90s, Balfour succeeded him and was Prime Minister for four years, until his government fell over the debate over imperial protection, making the British Empire into a territory protected for trading purposes. Salisbury is actually... Sorry, this is, this is, this is a silly, uh, I mean, um, Solis Bay is one of the ways that you can easily fault people who like, enjoy answering quizzes, if you ask them, who was the longest serving British Prime Minister to have held office in the 20th century? And they will all say Margaret Thatcher. They haven't listened carefully enough to the question. Not who was the longest serving British Prime Minister of the 20th century, but who was the longest serving British Prime Minister to have held office in the 20th century? The answer is, in fact, Salisbury, because he about served much longer than Fathio, although Fathio resigned only, he only served about 18 months of his term in the century.

27:30 It's just the sort of thing that you, if you have a real know-all and you want to win a bet with them, you might make a small play. It isn't. Actually. Well, they had a great deal of wealth and privilege, and if they had reasonably good abilities, they were able to pursue these different jobs. I think it is impressive just how much... yes, I agree, it is extraordinary, but then look at... but then I think it's generally true. In the 19th century, people do seem to have concentrated their energies and time in a much more effective way. It's equally true of Marx and Engels. It's true of... But it's true of people like Redstone. It's astonishing how much time they found. They certainly contrast pretty impressively with the people who represent the... They're the ruling class today. But then of course the point is that those people are now just frontmen, they're not the people who really take the important decisions. They're just effectively puppets. Thank you very much. As I say, it's very good to have had that clarified. There are lots of other questions I want to ask, but I'm not going to get in the way of all the discussions that I know you want to have with John and... When you go back to Buffalo, are you going back on Monday? We actually have many questions about the study this year. Oh, well, please go ahead. Lots of things to know. As I just demonstrated, I'm somewhat out of it. I mean, it's about nine months since I thought about it. Well, that's all right. Well, anything, anything that you ask me that I know, I'd be happy to answer.

30:00 You know, I do worry about how best to introduce the depth of your insights into the correct connection between understanding categories, such as those of distinguishability and indistinguishability, space and quantity, to philosophers who may be, you know, certainly bright enough to understand them, to benefit from them and to use them as weapons. We've struggled to clarify our understanding of the world, what the world is like, how people whose whole perspective, whose whole framework has been so distorted by what has passed for logic in this century, think of logic as codified by Frederick Russell and the development of that as the ultimate starting point. Investigation in the theory of knowledge or being. How one can really track back the distortions that have set up these, those understandings of the direction of bit of structure that reflect objective idealist or subjective idealist distortions to their origins in the concrete history of… Our subjective instruments, namely mathematics and logic, and to see them from the perspective that you have done so much to create. It seems to me to be such an important task, but it's very difficult to know what other correct pedagogical weapon. Because your papers are very difficult to read for somebody without a good mathematical training to follow. The hope is that these will be slowly... They will be slowly... You sent it to me. You very kindly sent it to me. And I thanked you for it on the phone, and I've been studying it, and I think I've learned a lot from it.

32:30 I mean, obviously some of it is very, even for me it's very simple, but even the simplest things, when the real understanding in depth, even of what it is about maps, which I mean, concrete, finite. To really understand sections and retracts, the level of generality to allow them to fit with this philosophical understanding you just showed now in what you were explaining to me now about distinguishability and indistinguishability and the behavior of maps from space to see this all within that framework does require a great deal of... They're dismantling of faulty training and obviously a great deal of clear exposition. What I thought was marvellous about that book was that you did get so far with the categories of graphic monoids and introducing the notion of gahee's inactive set and with the idea of a very strong... I challenge our ideas of how one really ought to think about Gödel in a first-year course. It was wonderful. I don't know why you had to apologize for not having gone as far as adjoints in the book, because I think it's actually very difficult to introduce adjoints to a first-year audience, because adjoints... I think what MacLean says about adjoints is right, that you have to learn... You have to understand, you have to have understood enough mathematics to be able to see the distinction between generality and the correct generality, the brief generality, before you really understand adjuvants, and that actually involves having made quite a lot of progress in that way. So I don't think... Although a lot of the sort of category theorists who work in computer science don't really care very much about the foundations of our subject, we'll probably say, oh, well, this book can be used, this book can be used, but he doesn't even introduce algorithms. No, no, no, I think it's a lovely book. I enjoyed it beautifully. I gave, I made a, I hope you don't mind, I made a photocopy of it myself and gave it to Jerry Kacherian to study, and he would send his very warm regards to you. Unfortunately he can't be here because he's had to go back to Paris to look after his mother, but, er... Well, we need more expository efforts.

35:00 Yeah, well, that's something I really want to try and give my time and energies to in the next year or two. Well, I hope so. I hope so. I don't see them coming from the ranks of philosophy. It needs study of history to a degree which is more serious or more directed, let's say, than usual. The way you study history is... The idea of most people's idea of how you study history is just to internalize a lot of unrelated facts. Or else just to pick a fashionable thesis or position to attack or defend in order to make a reputation, it's really, yes. History is in a very depressing state at the moment. And the standards of, as you say, the standards of what passes the standard of truth in history and journalism, except it's got even worse since you wrote that. Which I wouldn't have believed possible. Oh, yes. Well, exactly. Even when they speak, even when they look. Well, in things that have been left, things that could possibly happen to, you know, because they happened long after Stalin had died. I think actually, you know, Stalin has now ceased to be a very effective excuse for the, you know, the journalists. You know, the pseudo-historians, in the sense of the... I think it has... I think it now has to escalate. Well, it has escalated already in the sense that obviously Lenin has now been... But that was a very obvious progression. It will be more difficult if things continue as they have been in the former Soviet Union for another year or so, and if they go completely flat, then will they blame that?

37:30 What I find particularly extraordinary is the way that the, I don't know if it's so in the United States, but certainly in Britain, Insofar as there is a solid remnant of the CBSU, it's still a sort of regroup for themselves, these people are always described as the right-wing, the extreme right-wing, and also, it is always said without any... that they are in league with the extreme anti-semites and nationalists. Whatever they call them, and Shavarevich, because they're obviously all there. Because if DuPont Chemicals say they are, they must be. Do you think DuPont Chemicals are the one to apply this? Well, I think, no, I just say specifically DuPont Chemicals. And what I would really like to understand is what is happening in Central Asia at the moment, in places like Kazakhstan. I suspect some very, very terrible things indeed are happening at the moment. The whole community is liquidated for the benefit of the exploiters. But I think taking a global view... The perspective for socialism is probably no darker now than it was in 1900, just before the 1905 revolution. The process of regrouping will be... It's quite dramatic. One can't see how effective the regrouping will be or how soon it will happen. But I think one of the things which will enormously contribute to the regrouping of historical forces is precisely...

40:00 The stream of poverty, the shameless absence of any theory or understanding or department at this point. At the moment, actually. They have almost made an ideological fetish of glorifying the fact that they have no understanding and that there is no understanding they have. It really completes nihilism in the theory of knowledge and, well, I see it also in philosophy, nihilism in the theory of knowledge, nihilism in the theory of being. The idea that all meaning is a play meaning and spun from where it goes. Anyway, there's much work for us to do. And I do hope to learn a lot from you in these few days. When you go back to Buffalo with Fatima and Danilo, is it possible for Alberto and I to come and... No, I don't mean to impose on you, but just to stay nearby for a day or two on our way back to New York? I would think so. That would be very good. We'd hoped we could. When would that be? Well, it would be, if possible, because Alberto has to go back to France on the 24th, and John has to go back to Bristol, so I got their return tickets from New York on the 24th, which is the Wednesday evening, but it's difficult. If not, I could perhaps make it a little bit late, because I'm going to be staying in America until the 9th of March. I have to go and try and make money by going down and concluding various business contracts. Anyway, I thought you were all very much interested in this. Yeah, because you know that I will use it for good, sir, I hope. It's all going to be spent on supporting category theory, I hope.

42:30 Well, I'll come tomorrow morning. You know that I have some... Did you win your firm back? Yes, I did, I did. Yes, unfortunately... Maybe you taught these people then. Yes, I did, but I have to go down to Oklahoma, in fact, next. I did finally manage to get most of what I was owed, not all of course, but unfortunately the man who had been my business partner, who defrauded me, went to prison for three years. Insofar as there was any satisfaction in that, I mean, the money was never recovered. But he went to prison for three years. Not just for what he did for me, because he had actually also done it for about four or five other people who had merged at the time. As usual, a great deal emerged that I had never known. Look, I think I'd better go, because I know that John has arranged me to stay. Have a good stay here. See you tomorrow, it's good. Good to see you. See you soon. Okay Alberto, see you tomorrow morning. Bye. Cheers.