What is the true relation of logic and set theory? (contd.)

0:00 Why should, for things that are compatible with the type of thing we experience, why should that be taken as some kind of guideline for what we might want to do in mathematics? In mathematics, we know historically that there is a mathematical society in the human community. It seems to me we need not be confined by it. We don't have to deny it in order to also accept perhaps unusual possibilities. Contemplate all kinds of possibilities. Why other things than that? Unicorns. The question was, the question I found was, does this concept of mythology and mathematics make sense? Which I interpret to mean, does there somehow exist a predicate that, through speculation, we could define, which could be applied to the whole realm of objective concepts, which is fantastic, as you can imagine, and more so, which would single out the quote-unquote ones. And I'm saying, in general, no. I don't believe that there is a clear thought way of doing that. On the other hand, within all those, within all those, within that richness of concept, we can see these two aspects. We can see those general theories which are trying to idealize nearly our subjective process in getting to know things on the one hand, and we can also see those theories which are about the construction of the more and more precise images, and we can note that there's a correlation.

2:30 Between things that seem so fantastic but seem unreal, like space film curves and measurable cardinals, and the origin in the idealization of the subject as the object. So there is a kind of logic that's partly determined by that distinction. Things that are mainly subject and mainly object. Logicism. This work of Peter Johnstone, building on Peter Fry, about how all logical distinctions can already be made into what he called QD topos, namely those in which every object is a social and decidable object, precisely excludes this sort of thing. So in other words, logicism does not allow for this. It does not recognize it. It's definitely beyond logic. And can't one argue that isn't there a deep reflection of this in the fact that you can think, as Collins clarified in his work, of sets, constant sets. And this notion of object as it goes with the semantics of classical quantification theory, the thing which has the Platonist ontology built into it, then actually comes out of these assumptions, well in fact can be seen as a reflection of the imposition of this QD. In fact, extensionality isn't extensionality just a reflection of that, in the sense that it does build up this uniform separability condition, and hence imposes properties on the maps.

5:00 When you extract citable or QD or something, double negation, she, there's various things. Or when E is defined over sets, and you've built in deep sense. You have the inverse image functors. They will, of course, put a copy of sets in this dark part. In other words, it will miss the particularity of the grossness of the general or beingness. Two copies that you have both discrete and co-discrete. The discrete and co-discrete sets, they will overlap in finite. Finite will tend to be... But it's the complete absence of instruction to inverses of maps, which goes with that absolutely arbitrary notion of object, that of extensionality, that is seen as introducing this, what you call, ontological metaphysical false generality, and since the title of your talk was The True Relation Between Logic and Set Theory. And it's an absolutely wonderful talk, but I don't know how many logicians or set theorists would actually recognize... You mean the logic for the set theory? Yes, I actually recognize either the logic or the set theory. Oh, yes, the connections, of course, are profound, but I'm just saying they might be rather difficult for some logicians and set theorists to grasp. I just think that's perhaps the point that ought to be brought... They've got 30 years to work on it now. It's high time they started working on it. I couldn't agree more, but... Perhaps that last point is a connection that should be brought out more strongly. I mean about the QD topos. But also the origins of this notion of object in false generality. The idea that the category of objects, the category being the ontological width of which is such that a more restricted notion of physical object or concrete object just sits down, so hence the Platonist picture which I was discussing with Owen last night, that the entire physical universe is a little blob of structure sitting at the bottom tip of the ordinal curve.

7:30 Objects and functions, I'm afraid. In the logicist approach, obviously, you have to bring in the language-based approach, you have to introduce another ontological category as well, but functions in the ZF-based approach as interpreted in a strongly ontological, Platonist way, you just have the one category, but either way, the way the structure and unity is built in in that approach comes out of this way of thinking of extensionality that you have in logic and set theory. Or rather, the way that extensionality is built in in logic and set theory without seeing its roots in the conditions that are expressed in this way, in the toposetting, in their topological aspect, and I suppose you would say as a reflection of objective variation. Would you agree with that, or is it... I think so. I'm sorry. I didn't... I'm sorry it's very badly expressed. We must thank Professor Mojere for his time.

10:00 I'll be sure to go and correct you. Oh, is that going to work for you? It probably will. Oh, on the contrary, I'll grab it out of your hand. It's worth 20% more than the dollar. And I'm really too glad to take it from you. Okay. What do you have here? That's very nice of you. Thanks. I'll have a beer. You haven't told me what you want. Ah, I think I... You can see the physical components that actually travel on that curve because it doesn't have time. I agree. I agree. But, but, but, so, so, but, but, but, but, but, but, but, but, but, but, but, but, Nothing's stronger than that? We've got everything down, I think, on tape, but not as poorly organized as my question, I'm afraid. I'm sorry, it was very feebly. Well, so did mine. It wasn't a clear question. I do think there is a very interesting point about it in introducing this term. People steeped in the orthodox way of thinking about how the ontological width of their notion of object came out of this, and that false generality arose, and particularly how crucial upon the understanding of extensionality in the topos.

12:30 I guess it connects with the very simple straight ideas that you were talking about last night, about how this is the correct way of understanding distinguishability and indistinguishability. Cheers. I mean, for instance, the question that you've got asked by the man over there who's talking to Colin is the wife, sorry, the departmental husband of the chairman of the department is the man I'm staying with. I think he's got an extreme ontological place in it, and the question, well, apparently he asked you the question about identity, but then, you know, what happened to your identity condition? You see, it's the way he thinks about identity as a relation which comes out of that, and the answer that he brings in, which was entirely intelligent. His name is, I'm afraid I haven't caught his second name, it's something like Cusack or Crusack. It's Jim, I think it's Krupa. He's actually, he was actually about to publish a book called Pie in the Sky, but except unfortunately, except John Barrow, the English astronomer, populariser of mathematics, populariser of bad philosophical views of mathematics too, and that he co-authored the book on the anthropic cosmological... Thank you for your attention. I see the name very frequently in the context. Is it always the same person? Ian Stewart is an editor of the Mathematical Intelligentsia. He offers a book about mathematics and God.

15:00 There's a man called Paul Davies who is always writing books about God and physics. I've come across the name. Yes, I've come across the name. I think that Colin would probably know the answer. Bill was quite about distinguishability and indistinguishability. The correct way of thinking of the distinguishable object is in terms of matter and discrete, but actually seeing, understanding identity, I mean distinguishability and indistinguishability are usually thought of as equivalence relations less fundamental than identity. The notion of absolute identity on the domain, with the classical semantics of quantification theory, which has absolute identity built into it, and gives you this one, the placeless ontology of objects, kind of platonic atom is metaphysics, which is already there in the origin of the linguistic term of the 20th century analytical philosophy. The classical Egyptian view of identity as a relation should perhaps be correctly understood in the setting as the limiting case, the kind of idealized case of distinguishability and indistinguishability, or distinguishability understood in this way.

17:30 In the course of the course of the course of the course of the course of the course of the course of the course of ...indiscernible relative to the stock of predicates of that language. ...that captures everything relative to the theory expressed in the language from the point of view of this participatability of sentences in the language of theory. You can always make... A normalization in the terms of the model, such that we obtain in the last, with respect to the meta-theory as set theoretically, that you have changed the equivalence relation into an equality. Sure. You can always do this. You can always recur, yes, of course, in that setting, yeah. When you take the interpretations of a given theory into different categories, such as topics, the quotients which give you the equality are very different in respect to their properties in different topics, different from themselves, from the case of sex.

20:00 So, you fix the theory, you vary the models using different quantities of discourse, but this amounts to say also to use different meta-theories, so that your theory could always be embedded in the corresponding English portion of the language. So in this sense, you have a wider viability and a different sense of reassurance. It cannot be said that the notion of equality... You know, we have to pay attention. If we remain anchored to the idea that equality is equality in a classical sense, of course TESS has to speak only about equivalency relations. In this case, the classical notion of length is split into many, and which of those is entitled to be named as a mathematical notion? Yes, perhaps it's better to say in the classical setting we have always The notion that an equivalence relation induces a partitioning domain and the notion of partitioning already presupposes a domain of individual items on which you can have a more or less refined partitioning and obviously the absolute identity relation just gives you the finest partitioning and that's the And that's what is relative. And that's what, as I say, relative is. Relative is what, of course, both you and I want to understand is perhaps the deeper and ultimately topological origins of that equivalence, or way in which equivalence relations mesh.

22:30 But this is to bring back logic to, as you put it, you know, this is to take back that... The division of logic, cognition, and physical world that one has in analytical philosophy, which I see all of Bill's ideas as a fundamental contribution to, but I just wish I could express things a little bit more sortly. Sorry. Yeah, sorry, yeah, go on. I was thinking, you know, I was still thinking about what is... Yes, there's so much that I shouldn't have put up with you when you're thinking about such a deep subject. In each presentation, the notion of logic enters the moment of the subcategory of objects, of the economy, of the agronomy, where there is the notion of classical science. It comes from the situations in which you are fully attributable to just mankind. You unify these two situations when you have just discrete entities. The purpose of discrete entities coincides with any function to two. The same thing as the object you could obtain through mapping from the symbol.

25:00 This is the situation. What I understand best here is, from the point of view of logic, I would just like to be there to hear his answer. But how, I mean, how profound a shift of this involved and how difficult for somebody trained in mid-century logic and analytic philosophy to appreciate the depth at which the subject is reorganized. As you said in the paper, it's not just a question of redistribution of weight between... I find it's a little like a group of Aristotelian physicists and Ptolemaic astronomers suddenly finding themselves transported and having to sit and listen.

27:30 To Lagrange's lecture on celestial reality. It's about that, I would say, it's about that. It's not Aristotelian, it's not the brightest minds of the 15th century having to listen to the minds of the early 20th century. But it's about, I think, their having to listen to the minds of the 18th century. Of course it's a huge problem to solve. All right, OK, I've been sitting for long enough. You sit down. How are you feeling, by the way? Lousy. Dreadful. But I think that's the effect of the antipartic. Antipartics do make you very, you know, it's very, you know, sleepy, I feel, flowy, woozy, the word, I don't know what it's in Italian, but woozy. Like in, you are in a, almost in a dream. Like a waking dream. I think it's unlikely. I'm sure it's a bacteria. I'm sure it's just... it's just, it's just, it's just, it's just, it's just, it's just, it's just, it's just, it's just, it's just, Let me get you one. Have another one of those.

30:00 Yes, but, of course, the classical logisticians, the people who believe that mathematics is about tracing Pre-existing liaisons in truth, but the deduction just is they're tracing pre-existing liaisons in truth, that there are only, we'll do two cases, we'll say well this is just an algebra, this is just an algebraic formalism, it cannot be, until you show us how it can form the basis for the semantics of a language, until you can actually show us how it can form the basis for semantics for a theory of meaning. You know, we don't recognize this as a semantics, and any logic must be based on a semantics, and I don't agree with that position, but I'm sure that is what people in analytical philosophy, people like Dummett, would say if confronted with this stuff. It would obviously be fascinating to reflect how on earth people like Dummett would react confronted with Mill, but I think the trouble is it would be just like Harris, it would be just like... An Aristotelian scholar of the 14th century being confronted with, being confronted with, I can't think, being confronted with Lagrange or something like that. It's not impossible. I tried, for example, to let Fatman understand what it was. The answer was just indifference. Sorry, say that again. I want to listen carefully. The answer was just indifference. What, the answer on the part of? Of Putnam. Of Putnam?

32:30 Well, I'm sorry to say, I think that's a disgrace on Putnam's part. I think Putnam has become lazy and flabby and too easily, you know, he writes his articles in the New York Review of Books and he has his established reputation, and he's too old to learn fundamentally new ideas. It's like asking a very bright 16-year-old autonomic astronomer who knows all about calculating, I'm sorry I've forgotten the vocabulary, what was the... Ecliptics, it's all about calculating ecliptics and to try and understand early, you know, trying to understand a book that depends on calculus. Now, if he was very clever and hadn't lost his, you know, his ball by the 1860s, and if you gave him Newton's Principia, it might be all right, because after all, the proofs of Newton's Principia are in Euclidean geometry. You know, he wouldn't recognize his subject in an 18th century treatise on mathematics, lately in the 18th century. So I suppose, but that is one of the things, when you have a, I don't know how fundamental it is, I suppose it is very fundamental, when you have a fundamental discontinuity, discontinuity which has also returned in many ways within thought about foundations of mathematics, deep return. You will lose, you're bound to lose, a very large part of your audience. Just as Cantwell found he had lost a large part of his audience. It's really small, but still remarkable. I found it very small, but I mean, if you had, if Bill had given that talk in London, England, he would not have had, well, had he remained entirely at the level of, you know,

35:00 Well, I think the people in the multiple theory of algebraic geometry would have followed perhaps much of what he said, but they just wouldn't have seen what the payoff for foundations was, and they would just have said, well, this is just a very batty heuristic for thinking about particular constructions, and the interesting ones are things like this, you know, this result in algebraic real fields. And, you know, you don't have to do... Okay, so perhaps we can do without measurable cardinal axioms, but they would express that in terms of the framework of ontological economy, parsimony, it would all be, you know, it would assume the basic framework of mid-century logical perception and analytical philosophy based upon. I mean, I suppose the most acute way one can pose... This is to ask, what would Quine have made of Bill? I mean, I don't mean Quine now, but Quine in 1960. I suppose Bill would say, Quine in 1960, or a very good analytic philosopher in 1960, so okay, Quine didn't help with the range work in the last, he wasn't tasking, he wasn't durable, but he did do significant work. You know, Quine ought to have... Quine ought to have learned about Isbell's results and have thought about that in terms of his framework of thought, but you know you can't expect subjects to communicate the significance of discoveries to be communicated as rapidly as that from one area to another, but more and more I think one does see, and this is the thing I do want to Hold on to the very point I really didn't mean to make. How bad for philosophy, already by, certainly by the 1960s, was the legacy of the linguistic term. If it hadn't been for the legacy of the linguistic term, if Quine had been trying to be a philosopher in the same way as Aristotle, it wouldn't have been possible.

37:30 If Einstein was trying to be a philosopher, then his ignorance of the origins of thinking about mapping spaces and how those impact on a category of objects and on the whole framework of thought about size considerations is set there in a book. A book like The Roots of Reference. If there was that kind of community of concern amongst thinkers, now there's a sociological explanation of why that couldn't happen, that the branches of knowledge are so diverse it's impossible for one person in a life to try to keep up at the frontier unless they're a universal genius. But there's also, as it were, an organic, a structural explanation within philosophy, which is And your paper on the linguistic term is the only place, well, there is now Hilton's book on theology and philosophy, which I must send you, which I want you actually to take with you when you left my house, which is a good book, but, you know, more and more one seems to tell how distorting and arresting for philosophy by, certainly after the mid-century, by 1960s, was the influence of the linguistic term. And this history needs to be just as much as the history of thought about, the history of the relations between geometry and analysis and geometry and logic needs to be written.

40:00 If you have a framework within which you are going to do your subject for the rest of your life. The basic set of materials you need, in that case, it would require, let's say, one year, since you have a literature, but geography, mathematics... In that case, yes, no problem. No, you're right. For any normal scholar, it would take five years. Yeah, if you wish to be, I mean, exhaustive, even if you're not at the most detailed level, because of the various components of opinion to be explained in their mutual relation, if you wish just to... Make the science directly important could be, for example, I think, a very well-chosen option by you, unless you could do this. I need to have a very well-chosen option. It would have to be something. I think I've left it too late. Maybe. I would like to, very much indeed. If I can get myself in a position where I'm not going to be threatened with the gutter, and if I can just get myself in a financially secure position for the next two years, then it's certainly what I want to do with the rest of my life. You know, I don't just don't look ahead that far, I don't.

42:30 It's what I want to do, burningly, as you must be aware, but... Yes, well... Every aspect of the things you like is a challenge. Of course, you have to select the things that are in the right balance between the answer and what you could do in a relatively accessible field of thought. When you look at physics as well, I think that the options are diminishing in number. Yes, they do. And you can reflect on them and be the first one to start with. Well, you have certainly chosen the right project to work on, I think. No, no, I do. I really do, because of, you know, the application, the really deep philosophical application. Category theory and topos theory are, it seems to me that among the deepest applications of those are precisely in the area where you have chosen to make your mark, which is in the borderline between the theory of meaning and logic, the reconvergence of logic and psychology or logic and cognitive science, the theory of knowledge, which appear to be precluded. By the anti-psychologists of the early, of logicism, by Russell and by the linguistics. This is a tremendous turning point in human thought. There is, of course, the more general question, but it's been the question for the sciences really.

45:00 It's always been the question for the sciences, is it really any more the case now than it was in the time of... The Hellenistic civilization or of the 17th century, which is of trying to make our broader audience aware of the significance of these results for the way that they think, you know, what they think the world is like, but this is so difficult because there, I mean, almost every barrier stands in the way. And the most what I've had to do is to present them with things which they may just find striking at the levels of puzzles and curiosity, but that is such a huge project, what will make the cultural politics of our civilization do concern me deeply. I do see now for the first time in my lifetime, perhaps the first time since the... I think that there may be a genuine reason to worry about, well, certainly a very long pause, a genuine sort of a rest in the progress of the deepening and inspiration of human knowledge because of, first of all, the extreme fragmentation. But also the cultivation of that fragmentation by people whose attitude towards knowledge is entirely instrumental in pragmatics, and also the forces in our culture that are hostile to the intellect in either for deep pseudo-spiritual reasons, but in the present context more just for reasons of... A form of commercialized, in a commercial decadence form, the sort of thing that, I mean, it is depressing to me that here in Canada, as much as in the United States and in England and in Italy, everywhere, people in the universities, in the sciences, people that are pursuing these concerns.

47:30 It must be even worse for people who pursue a pure scholarship of a more traditional kind of humanities. It's clearly the case even for people in the sciences, except for those sciences which have strategic and commercial applications. They do seem to be more and more isolated. I mean, less and less cultural esteem to the, put it crudely this way, fewer and fewer people in, far fewer I think than in the 19th century, than in the general educated public of the late 19th century would understand Farley's remark, that Colin quoted over there, when he said, to the man who asked him what news was. The assumption that what you know is just for the purpose of making profit seems now to have become so deep that the only alternative view of the world that has real deep historical roots and potentially It's actually quite overwhelming, power and force, and that I am horribly afraid will win, for the time being, perhaps for half a century or a century or so, is, I don't need to say, is religious. I really think, I do really fear that within our lifetime there will be a huge religious revival.

50:00 A huge popular religious surge. Because, I'm surprised in some ways, it hasn't already happened. It will grow. In some ways, I have a very big attitude to it. I am not, I, because the motives. I understand, for people who have not been given the framework to think about, it is a better alternative, I think, as far as their own individual lives are concerned, as far as their dignity as human beings is concerned, than the alternative of just a society in which the only bond between is the cash. I think if that is, if those are the two alternatives, I'd rather see a revival of religion than, but I, they cannot be the alternatives they look to, you know, but there is such a thirst for popular... You know, books by people like Barrow and Tipper. But it's not only terrible. A book like Hawking is very dreadful. The best-selling non-fiction book in the world, but it's not read by one individual. A hundred of the people who buy it, perhaps not even by one in two hundred, and of those who would read it, I'm glad because they would have a terribly distorted and fitty and silly picture of, well not silly, but nothing would fit together. They would have no hold on what the real issue is.

52:30 Thank you for watching. Absolutely, but we've always had a challenge with relationship and the present typical of the understanding nature are pronounced. It's just ridiculous. It's a completely false view of the way mathematics actually works. I mean of what it's for. And I'm afraid that a lot of the physicists, not Penrose, you see, I mean not someone like, not a great, not someone like Penrose. He has a very demented view. He really does. He means philosophically a bit naive. He really does think of mathematics as the correct actual description, looking for the underlying... Yes, but his philosophy is so naive that the way he thinks of that correct description does lead him into extreme form of objective idealism. Nonetheless, compared with Wigner, he's the same. But Wigner said it was a gift from God that we either deserved or understood. He didn't actually say a gift from God, but he meant, you know, he might have, you know. I think we'd better get it. I'd rather get back and help me than you. Of course. The girl needs to rest. So should we all come back with you? Yeah, whoever wants to come now. Yeah, that's all right. Okay. But I'm sure there's probably some lists. What are the plans as regards the... This evening, we'll be eating. Well, at least we'll start to eat in about half an hour. I'll just start.

55:00 Oh, my place! Oh, yeah, there's an enormous meal there. Yeah, yeah, no, I agree. There's a vast Indian meal all done. I prepared two curries last week. Very good for you. Thank you for your attention. Well, you better take Danilo and Fatima and Bill. Do you have a car? We have our own car. Oh, you've got your own car? Okay, well, if you take me and Colin and John? Yeah, why don't we go? We're going back to John's now. I've got to go back now. It would be better if we didn't come. Yeah, come at 6.30 or something. Okay, so we won't come now. Come in, come in. Oh, tell us if you're coming. You're going to get some rest, aren't you? Yeah, we can't sleep. Good. Well, you just... In Western Napoleon, do you ever get more than a two-hour sleep a night? Oh, God, yes. Well, you make up for it. You make up for it with what you do when you are awake, I don't know that for sure. No, it's okay. I'll go back with... Actually, I think it's probably a better idea if I stay here with John and Carly, because we'll only be in the way if we get... I was just thinking I could help John prepare some of this. Are you going back now with John? Okay, I'll go with John. This way. See you. It's just a way of speaking. It's part of the real world. And then there'll be attempts to sort of reduce all of the mathematics. You know, ordinary things about numbers or stats or functions or whatever. And it's really just talk about possibilities.

57:30 But in Putnam that was explained strangely enough in mathematics. In Putnam that was expected to have some leverage. You were supposed to be able to sort of substantially weaken your beliefs about what there is. Well, that was because he had a concrete geometrical model. But there are worlds with any given finite number of them, and so there are possibly any finite number to the world. And that's the stuff that's ruled out by Hellman's result about concrete cartesian product spaces. You just can't get enough. But if not, isn't it? Well, no, because he thinks you can't be that way. Because he actually shows that you can't be that way, because you can't get enough math out to do anything. Yeah, well, you could even have... Well, you could have space time just full of points, so you've got, you know, the continuum right there, so you've got space time full of things, you've got the continuum, but that's very much, yeah, and besides, you know, there's just the, yeah, but Hillman points out that you can't actually get enough math to do Hilbert space, you've got to get enough math to do the, You can't get enough math to get Gleason to say the math of that, so you can't even get enough math for what appears to be quite an indispensable result for the needed quantum theory in this book. No, I think that's entirely the wrong way of thinking. I think, yeah. Oh, well, it's got more phases than Stonehenge or Bertrand Russell, doesn't it?

1:00:00 I think at the moment we're on 7a-3-sub-6-d, aren't we? It's a little bit like the line in George Smiley, in John le Carré, Who is speaking here, Henry? Or Hilary, rather. Who is speaking here, Hilary? So we're talking about how many Putnams are there, you know, who is speaking here. Is it Putnam 3, Putnam 3C, Putnam 7? Explicitly. That's crazy. As you say, that's just like you go through every math book and put a little diamond in front of all the stones. It's crazy. It's crazy. Well, that's what a lot of people said. I know the difference between there being a box in the front of my car and there maybe being a box, but what's the difference between there being models in the off-air parking lot? It's completely crazy. And you know, he just gets away with a primitive operator of mathematics, a primitive modal operator of mathematical possibility. And he actually more or less gives the game away when he says at the big, just at the point where he says in the book that, okay, what he calls concrete Cartesian product spaces, which is an attempt to embed everything in a geometrical structure, which in some sense rests on a nominalist oncology, like Putnam did in the original 1967 paper. He's talking about dots and it's between dots. He does say... Well, when I make, you know, having made this concession, you may well think, you know, I have straightened out and swallowed a camel, that my ontological commitments are in certainly no less than those of somebody who's a full bloody ontological platonist about to say something. But, but, I still retain the purely logical, the purely logical residuum in the notion of object, which is the fact of something which is distinguishable from something else. I mean, I just put a little diamond in front of that, too, and that makes me feel much more secure.

1:02:30 So, how many people have discovered an end running out of theory? How many people have refused to do what Quine suggested? And of course Quine refused to do what he suggested. Yeah, but the problem with that is, I agree that's what Quine said he was doing, but in fact Quine did saddle himself with a very full-fledged social platonist ontology. No, it's not a theory, it's a proposal for how to sustain it. There's people marching straight line, and they propose a lot of things straight line. There isn't, yes, but in Quine there's less of a difference than... Yes, logic is a grand logic, what, what, what... No, no, no, no, no... Yeah, cropping punters... You could eliminate the quantifiers by using abstract logic. But Klein doesn't say, ah, you've found a way to avoid ontology. He said, you've found a way to not do ontology the way I suggest. But if you would do it the way I suggested, you would find that these theories have the same... But the point is that in order to do ontology, you have to have, you have to equip yourself, you know, you have to assume this notion of object, which is derived from the Fregean notion. Isn't that the causal for how to understand ontology, understanding in terms of these objects, how you can interpret them in a way that you see the problem? Yeah, and that already, as far as I understand, builds in a very strong ontological position, whether or not the bill or the bill is attacking itself, it is. Well, no, because then you could proceed to say that, you know, that theory of the world, that the law and logic are the same thing.

1:05:00 Yeah, but you wouldn't do that if you know it. That wasn't that crazy guy who wrote about symmetry groups and tried to do symmetry groups in category theory and published a weird paper in the Yorga memorial volume. I can't think of his name. Well, you say that, but then look at a paper like With a Physical Object. Are you saying to me, look at that? But don't dare think about the classical semantics and quantification theories having an ontological position already built into it, because when Quine says in response to Giege, oh, the notion of relative identity is antithetical to the very notion of quantification. These are all presuppositions of the name of objects of the same or different absolutely to be the values of the variable. I mean, that is, in my mind, to build in precisely that width of the ontological category of objects, which comes out of this way of thinking about the free variable, which is, I think, what Bill was, one part of what Bill was presenting the stuff with. It's an attack on that. When he makes this big plug for the significance of Johnson's results on QD top-office, that's a big... and relates it to all the stuff about quantity and space and domain of variation. That's his plug for those that...

1:07:30 But the role of Feynman has a decidable identity role. Why should it? Yeah. Yeah. And you don't. But Feynman does. Well, it's a big ontological payoff for the difference between the oppositions, isn't that? Yeah. So how can you say Feynman doesn't have an ontological position? But that's not on the level that Feynman's talking about when he says the way we should understand ontology. When he says to be is to be the value of the round variable. This isn't meant to be a discussion. This is a proposal for how to regiment the discussion. Fair enough. In connection, exactly. I think at the end of the day, Quine's got to have a category of... And I think Quine's position is not very different from that of John a few years ago, when he published a paper, The New Progressive. I think Quine's position would be very close to his, your ideas about the category of objects, as an ontological category. I know you've moved on since then. Oh, sure. I know you don't think that set theory is a theory of formalized and first-order logging in the way that Klein does, but... Well, I mean, Klein just thinks of everything. Yeah. Formalizing is one big thing. Yeah. And then you squabble about which... Yeah. But whatever you put in, you automatically live that sort of thing. You see, and I do believe that these ideas are not there, but if you tell me to come along and say, ah, I'm going to say there are objects that exist in the purpose, but I only require that every four objects possibly have a product, and I'm going to talk about that product all the time, but I'm not going to, first and foremost, I'm only going to say that they may exist, and I'm only going to say that they may be exponential. So... Well, really, you have to say that.

1:10:00 No, I do. I pay you two opportunities at most, I would call them. What I'm proposing is a purely verbal weakening of it. That's no ice at all. We just don't say the products exist, we just don't say they might exist, they possibly exist, which is better than might exist, because they really do possibly exist. It's not that we don't know whether they exist or not, we really know they possibly exist. And so then I can say, see, I've weakened the ontological functions of calculus theory. How specific are the old science and chemistry objects in this scenario? I'm sure it is a lot different. Well, the difference is that if an organism suffers its defective, it seems to know that the condition is false. So it may be true that if a function is continuous, then its function is positive. That's not a problem. You already knew that. Well, but we don't believe that all kinds of functions have value, all continuous functions have value, but it's possible to argue that there is, and we should. What we do is, we interpret, I'm not sure if I'm telling you, we interpret all categories, I mean, like 50 of those made up of all categories, the categorical ones, like there is a null set, that's the one that you can say, if this is true, then, if the action is complete.

1:12:30 So, how is the way you describe Hellman, how is it different from that view? Well, because Hellman is convinced that it's simply not the case that if there's a natural number object, then all numbers are the same. But Russell thought that could be true, and would be true, but Hellman believes we know it's not true. But we can only know that's not true if we know there could be a set of problems. The discussion of physics in the book is worth reading. Oh yeah, I agree. But I think it's safe to condemn it even now. Screw it. But he does, you know, he does tighten up the indispensability arguments, I think, very nicely. And he makes a nonsense of the, I mean, his results on Cleason's theorem, I think, do make a nonsense of the project that people like David Lewis, a little while ago, well, the so-called nominalistic set theory, which they went kind of using meriological surrogates. Oh, the parts of sets. Yeah, the parts of sets, the parts of classes. But the idea that membership was really, could be interpreted as pseudo-membership. ...in terms of the lowest part of relations. And you could do codings, but you could do a completely geometrical theory of codings. I wish, I wish, because I wish that... Oh, I've read it. I wish it could have been made to work. Because I think it would have given me the same metaphysical payoff that I think Hill is striving for at an incomparably deeper level. But this is where Aristotle says that nothing can suffice as geometry more than the diagonal work of a metric. Sure, it would be great if all numbers would work by one point of view. Yeah, yeah, that's a little unfair to Lewis. It's better than that. I'd like to understand the... I mean, Bill does make quite a number of remarks about numerology in some of his papers. I'd like to understand that more. Obviously, all this business about components and...

1:15:00 Like what? I mean, you know, Tom hideously gnaws at me these formal theories. I've now got, you know, this is what you guys ought to be doing. You know, who's going to buy that? Well, that was the discussion with Tom, and people kept asking this after he'd written his papers before the book. Tom would get up and say, you know, we've had, we've seen these modal theories, and then we've seen these diptenic theories, then we've seen these structural theories, then we've seen these large fragment theories, but what I'm going to do is take them all at once. And he would say, but, Jeff, these were meant to be devices for avoiding the other ones. He didn't bother. He just thought it was worth doing them all over again. Well, he's not a reformist, though, is he, in the sense that he can do math differently? No, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no. Well, it seems pretty implicit in his research, except he's got an incomparably deeper understanding of geometry than at least this is ever going to happen. But he does actually produce, you know, very profound theorems in topology and quantum theory. But when he says that sets are just the special cases of maths... From one space to another with particular kind of essentially homotopic properties, which is connected with uniform separability, and I find that very... Of course, somebody believes in equivalence as a sort of epistemology of mathematical concepts. It's very attractive. Well, he shows that you cannot get enough math for Gleason theory. Thank you for watching.

1:17:30 I don't know what... I don't... This is not my area, so I'm dumb on... I don't know what happened with what they are using now as problems, but anyway. Well, and that is actually what's... Yeah. Now, well, they're independent reasons to believe so. They're very unreliable. We know that about the community. Well, I think this is a substantial judgment. You're not going to get away with it. Yeah, but are those proofs not mainly, many of them are, I mean, not many of them corollary to the police system? No, the Bell results are fairly independent. Actually, what Bell does is show that there can be local, but there can be non-local. What Gleason shows is that there can't be any variables at all except for the so-called contextual thought. So the property of the electron is actually kind of related to the actual orientation of the measuring instrument. Well, gluten doesn't say anything about electrons or measuring instruments, does it? It's a very, very pure... Sometimes it's called environmental law. Yes, E-lock, what Michael Redhead calls E-lock, environmental accounting. As opposed to all the logical accounts, E-lock and O-lock. Oh, it says that in any public space of dimension 3 or greater, there are no... The connection between that and hidden variables is many steps. Oh, yeah. It's a very tough theory. I don't think I can even take a theory on this stuff, but I know it requires more than three dimensions. There are three or more. And then it says that there are no closed subspaces of a measurable Hilbert space of dimension or something like that.

1:20:00 And then through a long chain of inferences, it means that the damn little electron can't have spin components in all three directions simultaneously. But you can't rule out the possibility that it actually does have a spin number in the dimension that your equipment is just about to measure. Oh, sure, but that's for textual... But you're not talking about... you're talking about this new, more complicated object, electron plus filter. Yeah. And he doesn't rule that out at all. That has a non-local feature. Well, Gleason did completely independent locality. Yeah. So he just says no simple hidden variable, but you can still have the contextual kind, and what Bell shows, Bell says you can't have the local kind, simple or contextual, so that leaves the only logical possibility of non-local variables, and that's in fact David Bohm's program, is a non-local contextual, because the quantum potential is always... ...altered by the measuring apparatus, which is the value of the potential. I think we'd better get going, or they're going to eat all the supper. They're not. They're not even going to start... No, no, they don't even want us to go until at least 6.30. They're not even going to start preparing the supper until after 6.30. I think we'd actually be, you know, doing the right thing by then, by not turning up. By not blundering in. Yeah, by not blundering in, i.e., why don't we all have another drink? Oh, I don't know. Bill, I think you'll be drinking there. Okay, good. Good for you. What do you have? Is that yours? What do you have? Yeah, we've got half each. Okay, let me get these in. Where are these? Okay, fine. Can we have two more halves? Whatever you recommend. Can you take it out of there?