Afternoon talk (contd.)

0:00 In fact, there's Galois, and it seems that there's almost somebody out there who may be surprised to notice that there's a lot of Galois in many classes, and that's because he's from the class of T, and he's insane, and he's a problem of what the book, of what mathematics is, and he's a problem of the outside category, but definitely very different from that. And indeed, there's a sort of map between these two. There's in some sense the whole analysis of it, I suppose, between the two.

17:30 There are a number of different types of physics, and I'm not going to go into too much detail about them, but there are a number of different types of physics, and I'm not going to go into too much detail about them, but there are a number of different types of physics, and I'm not going to go into too much detail about them, but The actual thing, of course, has interactions. You make it hot enough to pretend that it's verified gases and what substances it is,

20:00 but the substances are not the form. Sometimes the alpha is the form. You see all the combinations. Remember that the axiomatic belief in the ABC is definitely not a make-up. This is our faith. It doesn't have another name. And most of them have to be done again and again and again and again and check it again and again.

22:30 Compensation? When you condense, it's not for the dots, it's for the one dot. No, no, no. That's the right answer. That's the right answer. He said, I will be. He said, I will be spent. But not... Yes, that's right. That's true. He's not getting the support of it. The components were interpreted in the more archaic sense. Some characters, no one could have followed up on them. It's a kind of naive course of approximation.

25:00 The idea that, I mean, it's clearly, it seems to be conceptually absolutely fundamental in understanding his whole take on.

35:00 The position of logic within vis-a-vis geometry because the idea is that there is this strong unity between the conditions for the objects in the topos to be decidable, which is the logical aspect of having a complement of the diagonal, separable in the technical sense of the topology, and also this other condition which is to do with, in algebraic geometry, to do with... Splitting of, splitting of extensions of solutions to equations. And they all turn out effectively to be, really to be the same condition. There it is, isn't it? That's the place I'm talking about in there. That's where they get, that's where we, that's where they definitely mean meant to go. Because that's where we were this morning. That's where we went for coffee. And that's where we went, that's where we went yesterday for coffee as well. That's quite definitely. Mitesh just found it when we were walking up here one day. Yeah, that's the place. Yes, it is still open. That's where they are. And... But this whole business, as I say, separate, unified, unramified, decidable objects which all amount to the same thing, are the things which naturally live in the petitopos, and specifically when it's one of these quotient decidable toposes. Right. And then you have this, as yet, insufficiently explored, more general context, where there are non-trivial evidence in the science, which is what? ...connects with these ideas of cohesion and variation. Now obviously they didn't stay here because it's closed, but this is definitely where they came. Oh, no, you know I can see this guy.

37:30 Yeah, unless they're sitting around there. Well, I just followed them in here. We've come here, in fact, for a little sort of sandwich breakfast the last two days. It's quite nice. The people are very friendly, too. This is the old social park. Yeah. Well, I've never seen it before, but, well, I guess in that case they have probably gone down the hill to, in the direction of, of the vaults. Well, I can't, I can't think where else they would have gone. That's definitely where I, where they said they would go. Okay, well, let's go up to the vaults anyway and, yeah, because I'm sure we'll run into them there. It's still quite early, isn't it? Now, what is it about? Yeah, because we're not meeting John until 7. No, he wasn't able to get there this afternoon. He's obviously having big problems with sorting out all this shite with his car, with his brother and with Ben and the power of attorney and the rest of it. I guess we'll go about that way, don't we? Yeah, that must be where they've gone now. Which is the original motivating case of a Petit Topos, which is also quite undecidable, yes, the idea that in that case it's a generalized space in Grothendieck's sense, but of course Grothendieck himself in fact insisted early on on this distinction between, in fact he introduced this distinction between the Gros and the Petit. Because one of the very first things that he, one of the very first toposes that he defined was this thing which he used in the construction of nuclear spaces in functional analysis and that doesn't have the property of being like the topos of shoes on a site. So you've got lots of people who still propagandize through this viewpoint that that is just the definition of a topos in general. The original topos is that Groton, Deacon, Giraud axiomatized was the topos of Schizo on a topological space,

40:00 and then more generally on the notion of Groton-Deacon topology for a site, which was a further generalization of that. To take care of the Zariski topology and other, it was a kind of inverted commas, pointless topologies. But Bill of course insists that the motivation went much further than that, was far more general, and was already even at the beginning in 1960, connected in Grotendieck's mind with this crucial distinction between the gros and the petit, or the general and the particular ones. He just filled in the corner of the picture that deals with logic. With his axiomatization with Tierney, which of course was aimed at capturing the way that logic and set theory would fall into place in this framework. So you obviously knew all that. Well, I mean, I didn't actually know that the Grundig history, but... Yeah, the Grundig history is very important because the whole business about, you know, the sites as carriers of cohomology, which is terribly important in the way he understands the logic as well. One of the things that's really illuminating is, although he doesn't seem to share this, because maybe I'm completely barking up the wrong tree, but it's to understand how you think of choice and extensionality conditions in the setting of a topos, because it immediately connects with this great and deep perspective. That as the site for a, that it should be the site for a non-trivial homology, that in other words, because the case of choice, no, sorry, sorry, sorry, other way around, this is exactly obviously what I meant to say, the case for it to satisfy the axiom of choice is that it should be the site only for the, no, for the trivial homology group, because there are no obstructions to the existence of inverses of maps. And the case where you have no obstructions to inverses of maps is precisely the case where you have choice, because you have stable global section. Whereas typically, the whole of homology theory deals with conditions for there to be obstructions to inverses of maps. And the case where there aren't any is the case where there is no non-trivial homology to speak of.

42:30 And that, it seems to be conceptually, is very illuminating in terms of seeing the way that the logical notions fit into place for him inside this wider geometrical framework. But that's as if we're only getting started because then there's all this business of cohesion and axiomatic cohesion which has yet to be fully worked out, which is what we were talking about today in exploring this notion of the Grotopos. But the way in which he sees this condition for the objects to be... All of this is separable and decidable, and incidentally unramified, although we didn't talk so much about that aspect, but there's a complete unity between those conditions. This is the way that you see the world of sufficiently discrete spaces as falling into place in this wider geometrical picture. In his Eilenberg-Festschrift paper, which is 30 years ago now, 1976, but still a perfectly fascinating paper, about there being, once you've grasped what Grassman's ideas about intensive and extensive quantity are all about, and how there's still an absolutely, for him, an absolutely fundamental opposition which allows us to understand how the whole of mathematics can still be understood as the theory of... ...quantity and space, in, of course, greatly extended, refined, more abstract senses of those notions, but still, you know, they should be seen as the natural descendants of our older notions of quantity and space, and that the view which says that those notions were just completely discarded with the rise of the automatization of analysis and the modern structuralist framework is just dead wrong. They are, in fact, the fundamental notions which we need once we've understood them properly. I mean, this connects with this claim that there are two ways of analyzing the structure of domains of variation. And the more general one is in terms of this notion of quantity, which can be captured in terms of covariant and contrivariant functoriality of the maps into and out of domains, co-domains, in order to be made completely functorial.

45:00 But the correct analysis in general is based on lattice homomorphisms from parts of quantity to parts of the domain over which quantity is varying and the case where you have an analysis of the domain which just consists of Where the quantities just take values of points is the case, what he calls the first but more limited way of analyzing the structure domain of variation, which of course works for traditional logic because domains are supposed to be composed of objects. i.e. the values of variables in the sense of Frege and Quine. Now that I think was an absolutely fascinating idea, and I'd really like to get him to cash it out and talk a little bit more about it. But we actually got much more philosophy done today than we've on any other day, so I'm happy. Right, what are you having? I'll get these, make sure we find those. Well, I'm not sure it doesn't look like they are here yet. Yeah, let's sit and have a drink anyway. If they're not, I suspect they'll find their way here. I don't... I don't think Mitesh is going to be out here. He doesn't like the cold, so I can't see him being out there. No, no, they haven't arrived yet. I'm sure they'll come. I'll have a pint of... um... Oh, actually I'll ring the changes, I think, instead of having the Cornish, I'll have one of the others. What, were you having a brew tribute? Yeah, I was having a tribute, but I've had that each day. It's a nice beer, but I might have something different. I'll have the Young Special, is that okay? Yeah, great, thank you. Thank you very much for your attention and I hope to see you again soon. Again, which connects, of course, with this whole framework in which he claims the notion of quantity can be made both precise and sufficiently general to be together with the theory of domains of variation of quantity, which, of course, is this business of extensive categories and things which are good. Which have the right properties in terms of distribution of product over coproduct to be categories of space, have the right exactness properties that are sufficiently flexible to accommodate the whole of mathematics to be a framework in fact for everything and in connection with that he also he has this saying which he repeated to me several times and indeed which is in print in a couple of his papers that that logic

47:30 What he likes to call narrow sense logic, because he does love this Hegelian terminology, narrow sense logic, i.e., what do you mean, logic, is actually, properly understood, is the theory of the roots of extensive and the supports of intensive quantity and the variation thereof. Now, obviously, an incredible amount. Very deep ideas is crammed into this slogan. When you unpack it, it is very deep because it's all to do with co-variant and contra-variant functional reality of maps and keeping track of the right properties, domains and co-domains. It's sufficiently general for him to see. Logic is just fitting within. But it's also connected with this other extraordinary idea which we were talking about at the dinner the other night, which I'd really like to get him to talk about. He did talk about it a bit yesterday, in fact, which was this talk that Grothendieck gave. In Buffalo, the Mathematics Colloquium in 1973, which is one of these precious, incredibly precious recordings, which I have yet to listen to. No, I've heard some of them, but I haven't heard that one. I haven't heard that one. Well, that's one reason. I don't want to push you too hard. You push things too hard, Bill gets scratchy, and I know not to do that. But you always have to work. Whenever you want to get anything out of Bill, you have to... You know, you have to be crab-like, or dare I say dialectical in your method, but Grendy gave this talk which I had not appreciated in fact until he was telling me about it the other day. He was there. I had thought the first time he described it to me that he was there, and then later I thought, no, no, I got that wrong, he wasn't, because he was in Perugia for most of 1973, he didn't join the Department of Buffalo until four or five years later, and I don't know why, but there was something he'd said which had given me the idea that he had come back to Buffalo to attend one of these talks, but he hadn't been at that one. However, I was wrong, he was at that one, he did hear it all, not only did he hear it all, but he was there at the...

50:00 The famous dinner that they had afterwards in a Chinese restaurant where Grozny spent the whole evening covering the tablecloth of this large table of about 18 place settings completely with these notations, explaining how he saw this relationship and it was in fact very strange because the first time Bill told me this I retained this very strong impression from what he said that the diagram had actually consisted essentially of a... All of these structures were listed clockwise around the edge of the tablecloth and then there were spokes running across and the places where they intersected the center was it was it was the the guiding idea was that The most fundamental notion in terms of which we should understand the relationship between all these theories was the notion of the of the classifying ring of a space which might in the case of the structure which can be expressed in terms of logical notions like a junction and disjunction and relations would be a topos because that's what the sub-object classifier of a topos does but the sub-object classifier of a topos is only one special instance. And so there's more general, there's more universal idea of the classifying ring of a theory. Bill wants to go further. Bill wants to say that in fact they're all the classifying rings of toposys. But Grotendieck specifically didn't make that claim and probably wouldn't have endorsed it. But the relationship between all these different structures, most of course of which are obviously algebraic theories, but you're getting the whole of analysis and geometry as well. All related via their ring classifiers. The ones which have a sub-object classifier are the ones where logic naturally is, but you can say all of that without ever talking about the representation of a theory in terms of what we think of as logic, without having to say anything at all in terms of quantifiers or relations or conjunctions because the whole thing can be said in terms of the limits and co-limits. ...of the diagrams that are related by the appropriate classifying rings.

52:30 So you get the whole of logic for free. And Bill called it Grodendieck's, except he gets pissed off now when I repeat this because he's changed his mind, Grodendieck's programme for bypassing logic. Now he says, I never called him that. Well, actually he did, but I wouldn't dream of upsetting him by arguing. So now I'm always very careful to say Grotendieck's purported or implicit program for classifying logic, or the program which Grotendieck presented. Implicitly, in which some people have seen, as part of his motive, the desire to bypass law. It's a ridiculously long-winded way of saying it. It's the only one which now keeps Bill happy. But the idea is obviously incredible, and only a genius of Grotendieck's level could have had it, that one could in fact see the whole of logic inside geometry in this. In this vast, visionary way, everything could be expressed in terms of limits and co-limits in the appropriate diagrams, in other words, in terms of universal properties, without actually having to use logic itself directly at all. And, this is, of course, what is really interesting, Grothik then proceeded, the rest of the evening, to say what he thought this was telling us about the epistemology of Gauss and his metaphysical status and epistemological sources of logical concepts and, you know, how they actually should be understood as part of the make-up of the world. Now that's the bit that I really would love to understand, but that's the bit Bill is a lot more reluctant to speak about because, apart from anything else, I don't think he does, he doesn't endorse it, and maybe he's, even he, you know, possibly switched off at that point, I don't know, but the tech, but the idea is utterly fascinating, and it obviously connects with this idea that goes back at least to Grassman about how one can still think of mathematics as the science of variable intensive and extensive. And also this other notion of extensive quality, which is what he was talking about this afternoon in connection with this business of the Leibnizian monad, which is illustrated by the graph that just consists of loops. The graph only has points, it's entirely determined by its points, but at the same time the points are not. Because they are precisely the loop spaces, there's obviously something more than just, even though it's disconnected so that each component only has a single point, it's more than just the point, it is the point plus the motion of the point, even if that motion is just simply, as it were, the identity map for the point.

55:00 Supposed to capture the idea of the Leibniz and Monod, how good a model is, I don't know. The only problem is that it turns out it won't do what he wants it to do in terms of finding the representation theory for these topos that contain the germs of motion for second order differential equations. I don't think he's got as much sheer technical power as Olivier, but I think he's got much more of a feel for the conceptual scaffolding, and he raised a number of objections to that, all of which, you know, after working the examples, he worked about eight or nine examples, Bill was forced to accept, we do not yet have a model that does what I want. So we may have to go to the Euler. We may have to take a more connected object than the Leibnizian motor. We may have to take something in the Euler reels, the sort of T to the T, where you've got, where in fact the connected components are more than one point. It is actually two points essentially in the Euler construction. And so in fact it's not an atom in this sense, which just stands for what is absurdly tiny. It's actually going to be something a little bit a little bit more connected than that but still the ideas are fascinating but the ideas as you know coming from this Glassman theory of intensive and extensive quantity and connecting via his own work. This is all terribly broad brush, I know, with this incredible program of Brodendieck's, to which he sees all the working categorical logic, you know, Mackay and Reyes and all this stuff. As having been essentially really rather fumbling and preliminary attempts to make a contribution to this, but so far the program has not even begun to be pushed significantly in the direction of Grotendieck's final vision.

57:30 Yes, it's clearly a very strong orienting principle for him to see how far categorical treatments of logic do conform to this idea that Grotendieck presents. He clearly regards that as his load stock as far as the working modules are concerned and tends to evaluate. ...work in logic by how far he sees it as contributing to that program. So getting clear about the whole conceptual motivation of that program would itself be fascinating. And he loves to talk about this stuff, so I think there's no reason why we couldn't do that before Tuesday. I completely retract everything, every unkind thought I had about Mateusz. He's really done a fantastic job today, and he really wanted to bring... I can sort of see, you know, if you don't come from a logical background, why you would wish to buy that. You know, logicians can sort of say, well, we can buy that. Well, of course, you know, Bill is totally opposed to that point of view, as you know, because he thinks that just completely gets everything the wrong way around. It's not, and the problem is that there's so much of... That's the take that people like John Bell and other people who came into it as... And then he took up topos effectively as a piece of conceptually, you know, new and very challenging, jazzy, new way of thinking about models. That's the viewpoint they tend to have. But for Bill it's much, much more than that. Because I remember when he just, when he came to stay with me in 1989, the first time I met him, after the conference in Cambridge where he gave these lectures,

1:00:00 the thing on discrete and co-discrete space is you've read all that stuff and can't basically study it. He came and stayed with me for a few days in that chamber where I then lived before going up to Bangor. John Bell, who was of course at Cambridge and who was a good friend of John's, that was where I met John for the first time, but John Bell I already knew and had known for many years and he had just published In fact, I think it was that year it had just come out, his book on prophecies and local setbacks. Which he wanted to call Tractatus Logico-Toposophicus, but the OUP wouldn't let him. Shame on Bill. No, I think they had some copyright issues with the Wittgenstein Estate. But anyway, he published that, which obviously you know, I'm sure you must have read. And Bill read that while he was staying with me, and gave me the benefits, which I recorded of course, ...of his take on that. He basically did not like the book, because, you know, there's nothing technically wrong with John's approach, but for him it is far too much the logical, the logician's take on the subject and seeing it as, you know, an extension of quantum theory, by taking his insights about functorial semantics for algebraic theories and just seeing them as a... I mean, this is sort of my point, but... How can huge-pronged geometry and logic can be incorporated into it, and how can huge-pronged logic and geometry can be incorporated into it?

1:02:30 I mean, I love Anders deeply. He's a very fine mathematician. His ideas in SDG have been beautiful and deep, but I mean, nowhere near as deep as Bill's. But he is much clearer in putting them across and putting across the philosophical motivation. And the whole point is that logic for these people, because it is to do with, essentially, it's a very old-fashioned naturalistic view, that... Logic is about the analysis of cognition and therefore about what they call the subjective aspect of it. It is inherently for them to get things the wrong way around to see geometry which for them of course is about the structure of the world in itself. I mean, you know, you say, well, a lot of people, when they hear this, of course, don't appreciate the incredible depth of Bill's ideas of mathematician, just to dismiss him as a philosophical naïve. They think, oh, you know, just taking all this stuff from Lenin, I mean, he could never have read Kant, he could never have read any really serious philosophies, so he's Kant, if he can talk, you know, if he thinks like that. Well, actually, Bill has read Kant, but of course, he does read them with a very... It has to be said, I mean, it's a sort of doctrinaire Marxist viewpoint and, you know, if Lenin had only had the time to really become a great philosopher instead of having to make revolutions, you know, he would have sorted all this out and, you know, the geometry is really to do with the objective structure of the world and logic is the reflection. ...this in human minds, therefore to make logic the primary aspect and to make it the foundation is to risk, at least risk, falling into the trap of either subjective or objective idealism. I mean, that's all to take. I mean, obviously that's super rapid and naive, but that's the orientation. I mean logic has to be understood as a part of the makeup of the objective world as reflected, so therefore it's subordinate to geometry, but of course if you were to say so. You know, the notion of a foundation has to comprise ontological considerations as well as epistemological considerations.

1:05:00 Bill will scrooge and rub his hands in horror because the very mention of the word ontology immediately... But then that's because the word is absolutely freighting for Marxists with anti-dialectic, with a kind of anti-dialectical position that stems from trying to produce a... And finally, the true theory of being the first being will be something static and fixed. There are all these strains of trains which come from... It does become really... I mean, this is the part of Bill's rhetoric where you really have to... I had this problem when I first encountered him, where you really have to form a just appreciation of his really quite incredible depth conceptually as a mathematician. To understand that you just have to bear with the want to a philosopher who hasn't yet apprehended the incredible conceptual and technical depth of the math will be inclined, unless they know a great deal more. ...to dismiss as just all the evidence of complete philosophical naivety. Oh God, here we've got Lenin, kind of re-warmed Lenin 1908. Heraclitus, good. Forelegs, good. Heraclitus, good. Zeno and Parmenides, bad. Maybe not so bad, maybe they were just setting out to provoke, to get the dialectic going. But Plato, definitely, very bad thing, very bad thing indeed. I don't think there's anything naive about it, partly because he's actually engaging in problems that are largely just left behind, as it was sort of too difficult for their research work. You can't publish a paper a year on the nature of reality, well, unless you're Nicholas Rescher or something like that, but we are dealing with... Definitely with a mind of considerably deeper and greater depth. So, I mean, I don't think of it as naïve. No, I agree. Well, I know only too well that it's absolutely not naïve. But it's very easy to get the first impression that it is naive and to think that this is a kind of, you know, Norman Vincent Peale mixed with Lenin in 1908, you know, sort of general theory of reality type, you know, the kind of thing one left behind, well, if one's a...

1:07:30 I'm not saying that I think it's silly to hold it at the top, because I'm on the c***. There are all these problems with being a Kantian, and if you respond to those by taking economics, which I think is always one of the flaws, but you've got to weigh up those flaws, then I don't think that's similar at all, but I just, I can't understand, and I'm trying to get directly really what the problem with logic is, I'm not in business about the objective. Well, maybe we could press him on these precisely, not directly on the philosophical. Not in that form, because you won't get anywhere by pressing that kind of question. You won't be getting anywhere by saying, well, what's wrong with logic? That's not going to work. But by pressing him on this business of intensive and extensive quantity, Glassman and Grosnick's programme, we will arrive crab-like, I think, at a much better understanding of what it is that he thinks is wrong. It's not a question of there being anything intrinsically wrong with logic. But logic as it has been developed as a branch of mathematics, a serious bit of logic, which is essentially the algebraic division of the thesis stemming from Boole and Schroeder through Tarski and Loewenheim and well, actually through Dedekind as well, directly through... through Skolem and Tarski, and algebraic logic, and then I suppose actually Halmos probably deserves to be mentioned too, and into his own work, well, okay, it's a very minor play, but then of course into his own work of post-1963, which he sees as the kind of traditional logic within maths, I mean, what he completely rejects is the idea of a...

1:10:00 The logic of magna that should be put underneath mathematics as providing the kind of foundation of Frege and Russell and Klein. And I think that's right because I think it's quite clear that mathematically that led nowhere. I mean, that's really, in fact, it's a complete, complete dead end as far as... There's a problem with things that have been said over the last couple of weeks, the last week and a half. There's a lot of emphasis on what's good for mathematics, as if mathematics by itself is somehow impregnable and unquestionable. Methodologically, you of course are, but of course, I completely agree, that is the method... The logical Achilles heel of the whole, because... You wouldn't expect the mathematician of Bill's power, or even Grotendieck's, to think any other way, would you? Well, if you become a little bit more reflective... I agree, though, that if you really reflect on what the justification for that methodological take on the status of logical concepts as against other mathematical concepts is, then... Of course, I agree. I mean, I've seen a lot of people in this sort of category of abstract sats business that are phenomenologically more mathematicians than they think like that, in terms of the hierarchy, and yet, when you come to reflect on what you've done, you want to know what you were reasoning about, and to be told that there are these things that are collections of... All property-less, structure-less dots that have no properties except their distinct mass is borderline mystical to people's understanding of what we're doing here, and I just don't believe such things exist. Just on it... Yes, in that form, but then when you... But then, again, when you see that in the categorical setting as actually also carrying with it the idea of the codistry category, where the things are... If I can be as, as it were, maximally cohesive and understand the way that this kind of cylindrical construction of the adjoining functions between the two categories goes, then it seems to me to be much more illuminating.

1:12:30 But, of course, he sees it, I think in fact his take, you know, to a first approximation, only a first approximation because there's all this kind of Hegelian background as well. But to a first approximation, I think his view about mathematical concepts, their status, how far... It is perfectly in line with MacLean. It's a kind of network theory of meaning for mathematical concepts, plus, well, you know, people in the opposed tradition would say a naive, I think, not at all naive, I think in fact incredibly deep once you look at them, abstractionist epistemology. He does, yes, exactly. It's essentially going back to Newton and Euler and these people in the 19th century to say that... Yeah well when we do actually understand the notion of abstraction properly we will see that these things are abstracted from the real world but of course abstraction is involves this incredibly subtle dialectic and we are and methodologically I think the answer to the objection would be but of course we have to have a theory of abstraction because we are in medias res. I mean we are if you're a thoroughgoing naturalist which a materialist is then we are the product of evolutionary... Biologists are the product of all sorts of boundary conditions without which there would not be mirrors to reflect on the world, and the world must have sufficient structure in order to be able to account for the Formation of cognitive systems of various levels of complexity and obviously cognitive capacity but the loop closes and there is one remains as it were firmly within this naturalistic framework of course except that their explanations give out because I think one of the deepest of all problems which Peruzzi is I think very very good on is that... It's the great objection to functionalism in the... The philosophy of mind and to structuralism in mathematics of the kind that Bill is opposed to is that it assumes that there can be a global separation of form and content. When we really understand the notions of form and content as not as metaphors but as sufficient depth, then it becomes more and more problematic that there could be a global separation of that kind.

1:15:00 I know against a global, a globally fixed separation, against a global, that was the word I wanted to stress, against a global separation, against a global separation of form and quantity whereby one can say that no one could actually take form in itself, i.e. some notion of pure structure, the kind of notion of structure that James needs to get ontic structuralism off the ground and say that's... The only account of first being, well, in fact, this is what seems to be methodologically significant, and probably Bill has never reflected on this, it's a pretty big assumption to make, but I suspect he has. For a structuralist a la... James. Of course, the whole nature of ontological inquiry has been completely displaced. The very aim of ontological inquiry has been kind of, because we're completely displaced, and has been, you know, has been re- So reconstituted in a completely methodological form because all we have to go on is structure. Let us assume for a moment that there is some. ...notion of purely structural characterization of a theory available, which of course I don't believe there is, because I think that's where the sophistication of Bill's mathematical insights makes me realize that what structurers talk about is that there is not some absolutely stable pan-mathematical notion of mathematical structure to begin with. But if there were, then the structurer's solution to the ontological question... ...would have taken the form of a complete rejection of the terms in which the question was posed. We will never say what there is. All we will have is an account of how a methodologically justified and, as it were, internalist account of how... Successive theories keep track of their successive structures and reconcile them, making them consistent, and some general methodological argument for the justification of a structuralized, inverted commas, ontology for a theory as pro tem all we have to go on.

1:17:30 But this, of course, is to say that... The nature of an answer to the question what is there in the form that Plato and Quine in their very different ways both believe that it's the object of philosophy to provide is just inherently unavailable. It's just essentially to accept the Pyrrhonist. There is absolutely no position that there is, well, except to Hume's claim, there is absolutely no, well, if you accept the pessimistic meta-induction, then, you know, asking what there is is just a lousy question. All that one has to keep track of in any way that gives us, in fact, if the structure is right, if you don't, If you don't accept that you can only track ontological omitments in a purely structural version, then if you at the same time accept the evidence for the pessimistic meta-induction, then you're going to end, it seems to me, in an even worse case, which is a radical... The other wing of, I mean, the structuralist, ontic structuralism is what I would call the methodologically sort of decent and for me acceptable in their terms response to the pessimistic meta-induction. The other way to go, of course, is much worse. The other way to go is... Postmodern is the deconstruction. There are only tropes and bodies of discourse and signifiers and all that shite. That's to play the Lacanian. That's to play the post-structuralist game and just to become completely flippant and, I think, no longer intellectually serious. No, I don't think it is. I think. No, no, no. I said if you accept the pessimistic meta-reduction, if you accept that all we have to go on is this.

1:20:00 Is the, in order to track ontological, that one has to reconstitute the very notion of ontological commitment, and with it of course the notion of being, in structural terms, then that seems to me the only, resting on the methodological justification, that this is the only way that we shall have any account of. No. Of stability and retention and accumulation of structure across theories. So the only way that we'll be able to characterize reality is in purely structural terms. No, I'm saying if I accepted their premise, if I accepted the pessimistic net reduction, then I would say that that was all we had to go on. The only other alternative is much worse, which is just kind of postmodernist abandonment of any notion of reality, but I don't accept those, and I think that actually these really quite naive arguments, but first of all because we're just at such an early stage, you know, the, well, particularly early stage, we have, I mean, how absurd would it be to say that Science has gone on, we've only had science in the modern sense that we recognise as kind of post-Galileo since the 17th century and this isn't even an eye-blink of evolutionary or geological time, it isn't really much more than an eye-blink of human time. To say that we can already tell, only three centuries in, that there's never going to There will be, you know, some final convergence. And that, you know, successive transformations in our notions are fundamental, that we will never be able to have a grasp of the categorical structure of reality, that there will never be, as it were, an account of the world as a, I would be quite happy to say, as a metaphysical unity, as something where, you know, the interrelationship of... Categories as exhaustive and non-redundant in their interrelationships fits together in a way that really does. ...turn out to be stable over the long run. So, given that we're trapped in the world, it seems to become, over the very long run, I guess of course here one's skirting the Persean notion of truth as agreement over the long run, to say that we're never even going to have that.

1:22:30 On the basis of the pessimistic better induction as it run run backwards over science since the century sounds always always struck me as a little bit like reverse weather forecasting you know we know that our models of our models of we know that climatology is an absolutely exactly completely stable science we know how secure it is because we can run the computer models of climate backwards over and some of them with a bit of curve. With a hell of a lot of curve fitting, actually, with the most monstrously outrageous curve fitting, actually, you know, tell us the story. Well, I think that the claim that we're absolutely stuck with a pessimistic meta-induction is a bit like that. You see the analogy. And I'm just, I can't convince. I'm not, I'm being a bit of a coward because I'm not convinced of the opposite claim as well. Which is, I'm not convinced of the opposite claim as well. Which is essentially, if you like to be flippant about it, Bill's claim that Comrade Lenin was right and that of course we can speak of the reflection of reality in concepts in our minds and say that in some sense these do attain, come closer and closer to representing the world. Um, I just think it's far too early to tell. And, uh, but I... My problem with James' theory is just that I just don't understand what some kind of structure is going to be. Nor do I, and I think it's more and more apparent when you look at what real mathematicians, especially mathematicians, obviously think of the depth of buildings. Um, but you don't even have to go to... A mathematician who builds conceptual and technical depth, really, you know, you can do the job quite cleanly a long way before you get to that level. This notion, the general notion of mathematical structure is just a completely vapid notion. There is no such thing. There is a fantastic variety of structures adapted to different purposes and arising from different sources.

1:25:00 The incredible depth at which... Using the term in a restricted sense, in which structural analogies between different portions of the structure do allow the whole thing to hang together in a remarkably stable way, although not in the absolutely stable way that Platonists would say that it inherently has to because it is just a body of truths about ontologically determinate pre-existent abstracts, because there are clearly bits of math that do get... One of the things which of course is particularly fascinating about Bill's programme is this business about the natural numbers, that he doesn't really believe in the natural numbers. He actually defines the natural numbers in a topos, in a completely structural way, in the natural numbers object, which is almost indispensable to getting analysis, arithmetic let alone analysis, going in the time of setting. They've been completed to what he calls completed discrete infinity. That's for him obviously an idealistic notion. So we need something else. So, in fact, this is why I think he's really interested and motivated, he's motivated his interest in, in, um... But then I'm not, I'm not surprised that it has really taken... I mean, I have been trying to sort of address some of these questions about the natural number object. I was talking to someone about this in the office today and... Yeah, yeah, I was listening. The natural number object is immediately and uniquely... Well, the point is that adjoints have to be unique up to isomorphism. Well, take it just as an original definition of it. Oh, certainly its original definition is unique up to isomorphism. Yeah, of course, but then so are most models of the natural numbers in all. But then, why define it in such a way that you guarantee you need that straight away? Well, the lazy answer is because that's how most of the time, for most of mathematical history, it has been defined. But he doesn't believe in these completed abilities, does he? I mean, I think he's interested in these sort of finite numbers. Yes, because all of this is provisional. I mean, just having the natural numbers object in the topos in the first place. When you're sort of pulling back and looking at the big picture of the Zussmanian math, it gives lots of things which he absolutely doesn't want to be there, notably all the pathological functions of the 19th century, the Sierpinski snowflake, the space-filling curve, the everywhere.

1:27:30 Continuous, nowhere, differentiable, all of the pathologies of 19th century analysis come out of the assumption that the endomaps in the piano category are... The correct end, then, that's on which we give you this, well, this kind of monoid, which is equivalent to the existence of which is equivalent to the piano axioms, and therefore give you the natural numbers of it. He doesn't want any of that. He absolutely rejects that stuff because he thinks it's totally ungeometrical and is a reflection of this privileging of the, as he would say, of the subjective, of the arithmetic notions over the axioms, over the geometrical. Of course it's indispensable to huge bits of very well-developed mathematical theories, of which he was, as a mathematician, obviously wants to be able to give an account. But in terms of what he thinks should be the lasting components of the big picture, he doesn't want it there, so he's got this strategy, which I don't understand because it rests on some very technical topos theory, which I can't pretend to follow, which is connected with this program for tamed topology, the Grotenbeek program, which is now pursued by these people who do finite sub-analytic sets, and the whole tamed topology program. Which is Angus MacIntyre and Mackie and all these other guys in in Edinburgh, actually in Malin, but in other places. There is this way of getting, actually I was talking about this today as well, but in a different context, but unfortunately that was when you weren't there. Why he dislikes non-commentative geometry. Steve Schaniel, who is actually brilliant in art, I think almost fills in for him in terms of conceptual depth, has got this construction within a topos where they do the spectrum of a ring in a way which allows this construction to be kind of non-commentative in the topos setting. It gives you this construction which involves triangular matrices whereby the only thing which comes out are analytic functions and you don't have any of the, you know, it really demarcates the geometry from the algebra in a really clean way.

1:30:00 Which he thinks that the Conn program and the other versions of so-called non-conometric geometry don't, and what they're doing is just algebra. It's not geometry, quite perfectly. Well, it's okay as algebra except, of course, that it rests on various constructions and rings which haven't taken account of the way that classifying rings really ought to interrelate. But, you know, he would accept that Conn is a smart guy. It's not going to provide, you know, it's not going to give a non-cognitive, he thinks geometry is inherently cognitive, just has to be, given what he thinks about, you know, the requirements on categories of space and categories of quantity, what they have to be. And that's kind of non-negotiable for him, and... But as far as the, that's a separate issue, but as far as the business about the natural number object is concerned, he would like to have something much weaker than the natural number object, much weaker than the category of the piano endo maps, in order to do... Well, in order to be able to do as much analysis as the geometry really allows to be done in terms of the notion. Where the map spaces in the categories are what he calls geometrically reasonable maps. So he really rejects the whole term that mathematics took in the 19th century. This is something which methodologically I think is quite important in getting a fix on his whole orientation, the whole way he thinks. He has often said... To me, the foundations crisis, the true foundations crisis, had nothing to do with Russell's paradox of Raleigh 40, which are really, from him, almost completely trivial results of fixed point theories, which are just not interesting at all. The real foundations crisis came 30, 40 years earlier. With the pathological functions and the reaction of Verstrasse and Kronecker and all the other people who were pursuing the programme and to expand the notion of function in a way which would admit all these pathologies.

1:32:30 That was the true foundations crisis, that was the fundamental foundations crisis and that's the point at which... From his point of view, you know, the wrong turning was taken and the so-called foundations crisis and the paradox of set theory was just non-issue for him. And I think that's very interesting. I mean, you may disagree with it, but it's important, as it were, as a fix on his... Conceptual orientation. And he wants to go back and revisit those issues precisely to sort out what he thinks should be the true relationship between the geometric and the algebraic and the analytic and arithmetic concepts, and to rethink the analysis and even the arithmetic from the point of view of what he thinks is really the correct geometry, which centers on this same topology notion, which incidentally... All of this was christened by him but thought through by Grotendieck. This is one thing he took directly from talking at considerable length to Grotendieck. And Grotendieck himself, and I don't think Bill would say this unless he was quite clear, Grotendieck himself, he claims, is exactly how Grotendieck thought. Grotendieck really did regard... The geometry is fundamental and the algebra is flowing out of the geometry and that therefore the taint apology program for him was a tremendously important conception in getting, basically reintegrating the whole of modern mathematics on a basis which essentially went... Backtracked to the 1860s, 70s to the point of which the pathological functions and reintegrating the whole understanding of the relationships between geometric and arithmetic concepts and the concepts of analysis in a way which... Yeah, gets rid of them. I mean, he'd be sick if I was to even mention Lakotosh's name in the same sentence as his, but yeah, kind of, you know, classic bit of Lakotoshian monster barring. I'm not saying he's right or wrong, but I am clear that that's the source in terms of conceptual orientation for the programme.

1:35:00 The 1860s screwed up those intuitions, got them kind of wrong, and made John McShane a little bit taken away, thereby robbing it of its intuitive content. No, it's not just a question of robbing it of its intuitive content, although that is an aspect of it. The geometry could never again be a foundational part of mathematics, geometry could never again be seen to provide the foundation of mathematics in the way that it could for the 17th and the 18th century for the Greeks, because if you accept the results of the term that's represented by the arithmetization of analysis, far more really by the arithmetization of analysis than by... ... and epsilon and delta methods. I mean, set theory, as it were, is almost a kind of footnote to that in terms of... Yeah, exactly. After all, think where Cantor originally was coming from when he wrote his first papers on set theory. It was all to do with trigonometric series. But given the turning that was taken by analysis, then... You are saddled with the view that of course what John calls set theory in the form of the modern version of arithmetica universalis has to be foundational because that is the only source of really rigorous and mathematically tractable and properly exactly defined in terms of ultimate ingredients of definition structural notions and therefore the only thing which provides a proper rigorous And of course, as a result of this structuralization, a vastly deeper and more diverse and richer geometry, but a body of theories and concepts which, in terms of its subject matter, could not any longer be taken to provide in any sense a foundation.

1:37:30 And that's, I understand that part of John's, and that makes perfect sense. It absolutely makes complete sense if you accept John's premises. Bill clearly rejects those premises because he thinks that when you revisit what happened in the 1860s and 70s, this was all based on the rejection, well, he thinks there is a notion of... Geometrically reasonable mapping, which, if you think about it, means that you must think that there is an inherent notion of the subject matter of geometry in itself, which allows one to make sense of the claim that geometry, in this extended sense, is in fact the true foundation for mathematics. Although the notion of foundation itself, of course, has now changed a great deal from John's version. I tend to agree with, this is not my coinage, I mean it's something that Jean-Pierre Marquis said to me a long time ago, but I think it's pretty correct, that it pretty well sounds, the notion of foundation is obviously a, in itself is an extremely complex notion, and it's not, the sense in which one... Any ingredient of definition of a concept or one concept in this form can, and not just burbling, I am actually trying to indicate an important distinction by that form of words, can be a foundation for another, comprises at least Three and possibly four immediately obviously distinct senses, probably many more. I mean, the first one with which mathematics doesn't normally do is the automatical sense. One is the ontological sense. One concept is simply prior in the order of being. It gets at a more fundamental aspect of the structure of what there is than another. In terms of the total account we have of the world as a metaphysical unity, it is something which is more fundamental in understanding the way that the other concepts within the conceptual world of metaphysical unity fit together. That sounds wildly fuzzy but again I think that can be made precise. So the notion of something being ontologically more basic than something else is something with which philosophers should not.

1:40:00 It may require a great deal of detailed debate and unpacking but the general notion of something being ontologically more basic than something else is not inherently kind of crazy. Then there's a separate epistemological sense. One concept is prior in the order acquisition of concepts. You could not have this concept unless you had already acquired the other. Then there's the logico-semantic sense. Something is just required in terms of a... In terms of a language, to be a defined notion before another notion can be defined, and the logic of semantic dimension is distinct in significant ways from the epistemological. Then there's a much broader and slightly fuzzier sense in which one can say that one concept is methodologically more fundamental, except now we're speaking about... Fundamental rather than foundational. So I'm not going to bracket the fourth notion, but that in terms of the, because that's much more, that's much more dependent on, as it were, where you enter the circle of definition of concept. But let's just stick with the original three. We can distinguish an ontological and epistemological and a logical semantic sense of... Is foundational with respect to another concept, in characterising any concept. We should in principle be able to do that. Those three ingredients all seem to me to be involved in play in any account of the foundations of mathematics. And John's account regards the ontological component of what it is to be a foundation for insofar as he thinks it can be addressed at all. He thinks that, for methodological reasons, it should not be addressed by mathematicians in the first place. Well, the Greek case of arithmetic is just, you know, the arithmetician, queer arithmetician, posits the arithmetic. There is no metaphysical issue here. You just have to accept the notion of being a unit in arithmetic to get started. And more generally, the logic of semantic.

1:42:30 The logic-semantic dimension of what it is to be a foundation, for one concept to be a foundation for another, seems to be for John to be very strongly privileged. We have to have ultimate ingredients of definition of concepts. I think he would go so far as to say all concepts. In view and clearly in focus in a way which does more or less conform to the linear order of dependence. So in other words, to the metaphor of mathematics as a tower. And we have to have that. There's ultimate degrees of definition clearly in focus in order to say that we have a foundation of mathematics. Once we have that, the epistemology is pretty well irrelevant anyway, and the ontology just if you're going to. Talk about it all, just fall into place around that. Well it should be done prior to mathematics. That's more or less the John's point of view. The mathematicians just don't have the tools to do it. And it's not their job, it has to be done prior to mathematics, because you're asking what the ontology of mathematics is and you can't do that within mathematics, because within mathematics you've got to be referring to the ontology. So that's almost by definition. But my point is this, that it's only by absolutely privileging the logicosemantic dimension of is a foundation for, with respect to concepts, that you land yourself with what appear to be the inescapable lines of force of John's solution, the promise. And when you backtrack from this, and when you take, of course, the much more radically naturalistic view on the epistemology... I'm not even saying one has to go as far as the engagement with the kind of logical issues, then the whole priority of the logical semantic dimension is called in question, because one is thinking, well first of all we focus only on the final sedimented result of semiogenesis, i.e. This is one of the things Peruzzi, of course, we have these fascinating discussions whenever John argues with Alberto, you simply have to come down to Florence and spend some time with Alberto, he's just such a brilliant guy, and such a nice guy too, I'm sure John has told you how dearly he treasures his friendship, and you would too, he's just a great guy.

1:45:00 There is the whole issue of semiogenesis. If one believes that semantic content is, this is the point again about Bill's take on the status of logic relative to geometry. Given this, the writers would come out and say that it's a materialist conception of the world, that semantic content is itself an evolutionary product, more than that, I mean it's both individually and collectively the product of semiogenesis, we do not start, we do not, language is not given to us from the transcendental realm. On what aspects of cognitive capacity in pre-human animals, in other cognitive systems, is the ability to use language too? You know, to use strings of symbols in order to carry the base. I mean, I think that a position like Bill's ultimately, I think, must, and of course this connects with his belief that geometry is somehow ontologically more basic, I think must rest on the view that every ingredient of cognition is ultimately the transpose of spatial structuration in some way. I think that's what I conjecture. He'd probably never come out and say anything like that because he doesn't think like a philosopher. Yeah. Well, one could, of course, one could have both a Kantian and a... Well, actually, in some ways, it's a rather sort of 19th century naturalized variant of Kantianism that I guess I have in mind there. Right. So, but... But they're not too far from Kantianism. No, no, I agree. Absolutely, I agree they're not. But... But then, of course, you see... John would say, but that's just absolutely unavailable, because then you've got to think of the subject matter of geometry as some kind of Urstock, some kind of pre-Socratic Urstock, like... You know, the wheeler of the geometry dynamics people, you know, it's got to be some kind of first being of primary matter, and that's a non-starter given the kind of structuralised geometry which we now have. So the circle of dialectical tensions is...

1:47:30 More to get I'm not so convinced that it is completely unavailable. Oh gosh, we better keep an eye on the time. We're supposed to meet them, I haven't got a watch, I'm supposed to meet them at seven, it's probably already after, isn't it? I'm so sorry, I'm burbling away. Oh, okay, let's go down there. Anchors. We have a continuous space of knowledge and analysis moving on and on and on and on and on and on and on and on... There are a lot of them. There are a lot of them. There are a lot of them. There are a lot of them. There are a lot of them. There are a lot of them. There are a lot of them. There are a lot of them. There are a lot of them. There are a lot of them. There are a lot of them. So that's why, that's one of the main reasons why I'm interested in finding out the concept for a man who can do it.

1:50:00 Because the fundamental lemma of the omnidimensional theory is that we have a multidimensional theory, which is supersimple in dimension one, and that implies that it's not too bad to do the entirety of it, even though it doesn't fit into the common theory. For example, no infinite discrete, that's what it's described by. So that's the precise lemma for that, which is due to the law, the rest in turn of the theorem. Which Ramsey theorem? I've never looked at Gramsci's original paper. He was trying to do a special case of the decision problem for first-order logic. I've never, um, I've never, I've never checked to see what I'm talking about. I have this theory that's right at the end, but I find it very easy. Because, um, the number of iterations of power can be unanswered if they were to prove it. The number of accentuations goes up with the dimension of the color. So, now we have a number of colors. It just goes up as the size of the color even gets bigger.

1:52:30 It is more and more iterations. It's a few more than ever. Theater is fine. Oh, it's fine. I'll be home. Right on the edge of... I find that we're often depending on the problem. So there's no... Yeah. Yeah. Oh, I see. Oh, I see. Oh, I see. Oh, I see. Oh, I see. Oh, I see. Oh, I see. Oh, I see. Oh, I see. Oh, I see. Oh, I see. Oh, I see. Oh, I see. Oh, I see. Oh, I see.