Talk

0:00 Subtitles by the Amara.org community You were actually dreaming about its morphisms, or you were just dreaming about when you remembered when you first thought through the axioms for it? I did a clever construction of it. I did fall asleep during the passage, but it was morning. You must tell me about this. This could be one of... What Vladimir calls dozing, something she's been practicing lately because she had trouble sleeping, so she learned to doze instead, which means you sort of sleep, but you can also think. Well, it sounds as if it might have been one of the more interesting, you know, intellectually productive dreams since Kekula's famous dream about the benzene molecule. Calculus can be a foundation for the category of categories. Okay, well I'd be very interested to hear more about that. What I really wanted to ask you about tonight, actually, if you feel inclined to tell me more about it, is this extraordinary program of Grothendieck's for, as you put it, bypassing logic that was implicit in the talk that he gave in Buffalo, and of which, as you were saying, the talk that this Japanese guy gave on the Grothendieck-Tiefmüller. The group this afternoon is essentially a kind of fragment or offshoot of this. This is something I really would like to understand a little bit better. You're confusing three different stories. Okay, I'm confusing. That's not new for me. Which of course are related. What did I get? Be careful. Well, let's find ourselves somewhere sweet, and then you can clarify my confusion, which will not, as I say, you've done so many times before.

2:30 No, no, no, I understood that. I understood that it wasn't something he claimed or it's something which you see as being implicit in, you know, the program that he delivered, that he outlined. I understand it's not something he said or possibly even thought. Oh, I don't really want to go into McDonald's, no, on all sorts of grounds. There were a couple, I think, up this way that looked okay when I was here. All right. No, it was rather that, as it were, he had no need directly of logic in this. Well, yes. Narrow sense logic. I'm doing an exam here. Yeah, okay, okay. Well, obviously, objectively, logic is what you can encode in the structure of the sub-object classifying a topos. It's notions like relations, quantifiers, implication. Yeah, relations, that's the point. Yeah, relations. You see, there's actually more. Yeah, these do look a little bit basic, don't they? Good-bye, pizzeria. Good-bye. By passing logic, what do we mean by logic? We still need to make the choice, though, because there's a choice from each course.

5:00 How do you say? How do you say? You're not sure? Okay, good for you. Good honest answer. Well, I thought it's a Thai word, so maybe you would be able to help me pronounce it. Okay, I'm going to go for the Caprice du Roi as well, but I'll have the canard crostique still on after, okay? With fried rice? Yes, please. Would you like it? Would you like to drink something? Would you like to drink something? Yes, why not? I think... Do you have... Yes, do you have the Vin en Bichet? Do you have the Vin en Bichet? I don't actually see... Oh, it's there. Sorry, I wasn't looking at the wine list. Hang on. Yeah, sure. Shall we just get the carafe wine because there's not going to be others. Good, good sort of peasant wine. um yes let's do that oh that's very reasonable it's only six euros yes we'll have what would you prefer red or white we could actually have a quarter of each no no let's take a giraffe okay let's hope carafe i'm easy i really don't mind at all do you want to split the difference and have a rosé no no you don't like rosé okay okay let's have a red okay a carafe of red good idea okay that's what i would have gone for too He's certainly long on uplift. He has been elected President of the United States, hasn't he? Fifty? Or fifty? Yes, fifty, but for the two of us, yes. I mean the fifty centilitres graph, yes. Yes, the red, please. Thank you, yes.

7:30 Well, the full text is on the inside if you want to read it now, but... I suppose that would detract from the dignity of the occasion. Yes, he had said that. That was one very positive thing he said. And he also said at one point in the speech that science should be placed back at the centre of the educational agenda. Science should be restored to its proper place in education. No more trimming or… Playing around with, no, intelligence design, intelligent design. In other words, commitment to pragmatism. Yes, that seems slightly to conflict with the commitment to science, of course, but I didn't say that. between science and anti-science yes yes well at least at least he's probably less wired to a kind of templeton agenda than than his predecessors i have to say there's also one thing about that photograph which is quite hilarious which is the expression on clinton's face it looks good kill i think glaring daggers I'm sure he was trying to conceal his feelings very hard. He was looking at something else. He usually is. Burnishing it. He's out of camera shot. I think they cropped him up. He's in an undisclosed location. Of course, I completely forgot that. Yes, he's in an undisclosed location, safely out of camera shot. He's probably still hiding in his bunker.

10:00 Anyway, to get back to the serious matter of logic, Yes, where everything, as you say, can be dealt with. And this, of course, is a fragment of the whole theory of, let's say, of pre-topos or, you know, categories of limits, finite limits, finite pro-limits, and distributivities and so on. But the idea, what I'm saying by logic in the narrow sense is to reduce all that to relational systems, right? Well, he didn't need that. Because they use directly the limits and co-limits without their universal properties and just being able to look at a definition and see if it involves limits and co-limits. Without having to go through the whole machinery of reducing things to relations of order, that's right, yeah, categories of ordered pairs, this was the, so therefore you could even do something which is wholly related to logic in the broader sense of the study of structures, you could construct the classified topos for structures which are definable in the positive logic anyway, or anyway the distinctly positive logic and other logic. The topology is again there in the fact that... The usual first-order logic is too special in one way because it talks only about sub-objects, but too special in another way that it involves universal chronifiers and implications on the right-hand side of entailments. I mean, to me, it reveals once again, though in a very strong way, that the really basic logic, even from the point of view of logician, should be positive logic and not full first-order logic. In practice, they use it. In practice, very often, they're using it.

12:30 They gave it some strange name like generalized atomic formula, or they have very strange, again subjective, i.e. syntactical names, but covering up the fact that they actually use things that models that are generated by positive logic. So, cheers to your very good health, John. Thank you. Well, I don't think I even sang it at this point, but I certainly had a fantastic preparation for it. In the setting of the, in which Grotendieck is directly encoding everything in terms of limits and co-limits and things like that, Excuse me, I'm sorry. Yeah, I see obviously how the, what can be encoded in terms of sub-objects naturally falls out of that as a kind of, as a fragment, but what's the particular role of, you mentioned the role of the classifying topos, but as I understand there was a more general framework for classifying rings. No, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no, no. There is declassifying topos for community of rings, one example. All of the subtoposes are defined by a couple of toposes again, so they classify special classes of rings, of course, under the duality here, I think, and what I was saying was Thank you for watching. I think he was taking various complicated classes of rings. Well, everyone knows there are integral domains, but then there are Japanese rings, and then there are Cohen-Macaulay rings, and he looked in a book on community ring theory, there are all sorts of classes of rings that are studied with good reason, right? Okay. So he was able to, in one, I think it was just in one day of sort of free association based on...

15:00 He ran through all the classes of community of rings he'd ever heard of and noticed which ones were classifiable by chub-toe coaches on the ring classifier, and this without ever using . . . I mean, you know, a logician who knew about the definitions of Japanese rank or whatever could write out a good thinking a while and write out a definition in terms of traditional logic of sub-objects, right? But he didn't do that. He just wrote down classifying topology right away. Okay, I think I understood much better now. He recognized, because he could recognize that the definition of U, whatever it was, could be expressed in terms of finite elements. Right, right, yes. Without passing through the, as you say, without passing through the normal logic, yeah. To have that kind of geometric intuition is just incredible. At one point you described in one of your writings, I think it, I'm trying to remember where, you described that the... The whole definition, or first characterization of logic, as the study of the roots and supports of intensive quantity. That was, of course, truth values, I guess, being a kind of intensive quantity.

17:30 Yeah, right. I mean, the truth values correspond to sub-objects. Obviously, that meshes with this. In other words, the sub-objects can be identified with a very special case of intensive quantity. By the way, what people should emphasize is that This fact that there is this correspondence between sub-objects and computational concepts already implies all of the logic. You don't impose implications, you don't impose Heiting's rules. It just follows that if you have, so this very strong dialectic, the fact that there is this contradiction between sub-objects, it forces the whole the whole of logic to This is a point which I think would be very, pedagogically very, very helpful to get across stronger. The first of your talks to the Bristol people, to Bristol logicians. I'm also connecting on what we were saying earlier, which is just emphasizing how much we have to learn from Grassman and the general characterization of the relationship between intensive and extensive quantity, which has just been so neglected by philosophers for 150 years and still haven't even heard of it. Thank you for your attention.

20:00 I'd like to say the roots of intensive and the supports of extensive quantities is the characterization that I meant to say. For the latter, it's completely unstudied as logic. The branch of mathematics that I prefer is that nobody's worked it out. The so-called theory of objective logic. Certainly does. Objective logic in general, in this case, is predictive. I'm going to sing. It's very striking to me how the point in fact that you were underlining conversation with here this afternoon. Which is the astonishing way that even back in the early 50s, before the kind of category theoretic machinery had been explicitly developed, intuited very powerfully these relationships between. Limits and co-limits. Inverses. The role of projectives and objects and injective objects. This characterization of logic says the study of the roots of intensive and the as yet relatively undeveloped study of the supports of extensive quantities. Excuse me, I'm sorry, presumably has important implications for the understanding from the deeper understanding of In fact, the deeper understanding of things like the axiom of choice, the condition that all the objects should be projected, I mean, one should see it in this algebraic way rather than in terms of the way that logicians see it in terms of these, you know, kind of, you know, Rossellian notions to do with that.

22:30 Childish stories about socks and shoes. It's to do with whether the domains of variation really are varying organically or in a more simple overall way. I remember one thing you said a long time ago now. You actually said it in Bristol, I seem to recall, in the Bristol Colloquium paper. Or maybe it was in the Eilenberg Pressure paper, I can't quite remember now. But it was one of those two. You drew attention to the distinction between two different ways of thinking of the structure of domains of variation in general, the one in terms of the lattice homomorphisms between parts of a quantity varying with respect to the domain and parts of the domain itself, which is, it was the more general frame, more powerful, more general framework. And the case where... You can think of the values simply as assuming quantities of points, which is the more restricted case, which is this kind of special instance of the first, but which the logicians in Spreger, for the most part, have taken to be the general framework to the great kind of crippling, dare I say it, philosophy of mathematics. I think that's another point which it might be worth developing and bringing out strongly in the discussions in Bristol. I'm just offering a kind of suggested, you know... What the French call grand lean of an agenda. But I'm sure you've got your own ideas. I'd like to push those as suggestions anyway. By the way, I wrote to Pettigrew. I don't know if Pettigrew knew about the Bristol paper in 1973. Probably not, but he will probably by now, because he's making a careful study of your published work, so that I do know. He's a good guy, Richard. Remember, he was John's research student and he's worked mainly on this program of arithmetic in non-Euclidean set theory, which allows there to be number systems of different lengths. Some closed under exponentiation, some closed under product, and so on. Which is an interesting idea, and which I think may possibly connect up with the additional right of jointness in some compasses. I'm not sure about that now, but understanding...

25:00 How one should really think of non-standard analysis from the point of view of a categorical, from a functorial viewpoint is quite an interesting question. We pretty much have the line on that from a long time ago, I mean, I proposed, and actually Anders worked out, Anders Krauss, actually a meeting in Victoria, British Columbia, which was attended also by Robinson himself. Really? Yeah. Rather simple, specific constructions. Yes. Reduced products. That was an incredibly fertile period, wasn't it, in Thomas' theory, in those years in the early 70s when you and Miles and Annas and others were 69, 70, 71, 72, 73, which makes a look-a-loosies. Remarks even more scandalous than they already are. It's just ridiculous. I mean, quite pitiful, really. I didn't know that period at all. Oh, it's just ridiculous, I agree. With a kind of half-hearted admission, of course I really ought to, but, you know, life is short and I was being asked to produce a very short survey note. But that really was the period when the key ideas were all being worked out, because Robinson died in 74, I think, didn't he? 74, 73, 74, I think. So it must have been around. So this meeting must have been, was it 72 or 73? Was it when you were still at Dalhousie or was it after you had left there?

27:30 After Dalhousie. So it must have been about 73. Yeah. I'm quite sure he died in 74. I'm quite sure he did because he and Moshtovsky both died almost within literally a few days of each other. And I attended a logic meeting in Kiel in Germany. Where Moshtovsky spoke. It was the last meeting he ever attended. And Dana Scott was there as well. In fact, it was where Dana Scott gave his first talk on what subsequently grew into domain theory. Topological models of intuition and trig logic. And the lambda calculus. I'm coming back again to the lambda calculus. Well, it's rather different. Yes, but, but, but... These exponentials are equal to themselves. Yeah. It's a brilliant idea. I immediately translated it into category theory. This was when I was still in Dalhousie. I remember it very well. Yes, even though they didn't realize it at the time. Yes, I mean, the objects are co-sets that are on certain limits, but the morphism preserves filter co-limits. And this is critical because what it means is that the function of two variables, a times b into c, very rare property, that it's a morphism, for each element of a, it's a morphism from b to c. And the other way around, if and only if it's a work as an S on the product, you see, but as if in linear algebra, a bilinear function is the same thing as a linear function, you see.

30:00 Precisely this choice of filtered co-limits, which is, you know, basically finite that peters out or something, sort of, that makes this possible. Special ingredients and then he has... The chain of approximations, and it means also that certain kinds of sequences, the inverse limit, which is computed in sort of a naive way as part of an infinite product, actually serves as the co-limit of the same sequence, you see. Again, because, of course, the kind of sequence that it is, it's a sequence of maps, each of which has an adjoin. Yes, yes. I mean, all these ingredients, and it's incredibly... Functorial. Functorial. There's an interesting thing about Dana's work. It's always been... That's how he constructs the thing. You see, the object y, after which y to the y equals y, is a limit. What kind of limit? Well, it's an inverse limit, which works well for the base of the... So it works also to transform the top, you know, co-limits at the top into limits at the bottom, so in the end you still get... Well, I remember, of course, seeing him drawing on the board for the first time all these lattice topologies, which illustrated how this was... Yeah. It really has not much to do with topology, I think. No, no. He, of course, motivated it in a very simple way to do it. I think I'd like to think of it that way. Special object of the category, then we run with the initial element, because the initial element is itself an adjoint between the terminal space and the space of Hawking. Sorry, the terminal space and the... I just didn't catch what you said.

32:30 Terminal space, you have 1 and x, and the map from x to 1 has a left adjoint. Yes, yes. So what you do is when you're iterating, you're iterating that. Yeah. There is a special kind of object in the category. So it's a slightly bigger category than the one to which the construction applies. And we start with those that have this extra little address. Let's start actually with the hiding algebra at the beginning, for example. Well, the limit's a hiding algebra, too. It is actually a hiding algebra, so you get this hiding algebra, which is equal to its own limit, blah, blah, blah, blah, blah. I don't think we ever remarked that, but the thing is that these, since the trunks are going up, is a right adjoint that preserves every kind of algebraic structure. So, having a structure of a lattice, or a hiding lattice, or a colliding lattice, many of those things are preserved. Well, the same is true for categories. You start, for example, well, for example, you start with two. Specific example, you miss construction, you get a hiding algebra, which has a space that's equal to its own exponential, the big hiding algebra. Well, instead, let's apply it to categories. Start with the category of sets. You get a topos, which means you fill it into its own. It's a topos with the property that every endo-funker would have. Yes, another example of how one gets logical notions through thinking directly about limits and co-limits. Yes, I think so, yes, yes. Gets clarification of logical notions in that way, which I think is a... ...without perhaps being aware of it, was in fact doing, even back then. Well, yes, but the thing is I remember this was the last meeting Moshtovsky attended and he died about a month afterwards and Robinson died almost exactly the same time, so he was 74.

35:00 Actually, Robinson might have died before, but I'm quite clear that he was in the same year as Meshtovsky, because there was a joint memorial meeting for them both in London that autumn. So, yeah, and I knew a little bit about Robinson because he was the supervisor of our friend, oh, yes, indeed, I'd forgotten that. Actually, I saw Angus, he was at the IHS meeting, I was talking to him the other day, of course he would have given his eye teeth to have known that you were down here, he would have tried to stay in France to come down to see you, he had to go back to London for some, you know, committee meeting or that, he's firing old guns, in fact he actually asked me when I thought I'd be seeing you again, of course I said not till Bristol, he was even wondering in fact, I don't know what you think, if he might... I'll be able to get down to Bristol for a couple of days to take part in these discussions. Oh, that would be wonderful. Yeah, I was quite certain you'd approve that, yeah. Well, unlike our friend Simon Butterfield, he's a real, you know, he really is a man who adds value wherever he goes. Well, I told him about it, and he's very keen, if he can, to get down. So I'll mention that I saw you, and he'll be very interested to know what you're working on. It's fascinating. He was talking to, he was also particularly impressed by this talk by this guy, Gizier, the guy that Gary was talking about this evening, who gave the talk about Grothendieck's work on functional analysis on the work on nuclear spaces and the Grothendieckian ecology and its possible relevance to understanding issues in physics. He was also saying to me, as Pierre was, that he thought that would probably have been the best talk of this whole IHS meeting, and it was the one which they tried to suppress because it wasn't on Algebra Geometry. I mean, there were very good talks on algebraic geometry. Who is this on today? Who is this today? Well, I'm not sure. The committee was Konsevich and Bourguignon and Cartier and some other guy who I'm not quite sure of his name. Not Kohn? No, Kohn wasn't involved. At one point, Cartier had mentioned something to me about Kohn, but Kohn wasn't even there. Kohn didn't attend any of the talks. He didn't come at all. Well, actually, there was one day I wasn't there, so he might, but to my knowledge, he didn't come.

37:30 And, of course, there are lots and lots of talks about motivic cohomology and the Hodge duality and all of this machinery which they do these very, to me, completely dazzling, but, as I say, quite impossible for me to follow things in the mother of all cohomology theories. I didn't really get much sense of the grand lean of the landscape in the way that I do whenever I listen to you and Pierre talking. I just got a lot of very, you know, technically very refined and recherché. No real sense of the underlying guiding ideas. Too bad you missed the first talk today. What, was Pierre? No, no, no. Oh, this other guy, you mean. He gave an introduction. No, he gave an introduction. I came in halfway through that. I had a little bit. Well, anyway, the point is not that you might have learned anything from it. I didn't. Oh, good. But the thing is, you see, he started off as saying, let A and B be two non-commuting variables. We're not sure. It was absolutely, totally formal, working with these problems of formal powers. Absolutely, without any idea of what kind of domain. In the second talk, I immediately saw why all that might be interesting. This is amazing. The second talk is incredibly clear. Interesting. I'm getting the impression these people have been muddying about for 15 years trying to define motifs and they haven't. In fact, they did very well, but they got so much better off than in categories. Well, they're much better off than in categories. Without proper category theory. You know, really what... Well... Peter mentioned something about the Hodge conjecture. That featured in at least three or four talks in Boer. It seems to be a big area of investigation right now.

40:00 Cartier explains things rather well sometimes. The point is that... The Hodge conjectures, these were very difficult. He thought of something still more complicated, okay, and people weren't heamed to. I interpret it this way, that he had some vague image of a structure, of a concept, but instead of being able to define it yet, He had some other conjecture about how an example might be constructed, and this example was Much too complicated because it depended on untruth-based. So perhaps now, recently, according to here again, they have been able to come up with a concept, which may be or may approximate at least the concept that Schrodinger intuitively had. This sometimes happens, you see, in other words. By the way, speaking of Steve, I was struck also by some of the things which this guy Pezzio said in his talk, which seemed to be very relevant too. The work that you were telling me about five years ago when you were in England of Steve's work on triangular matrices and on this, this very interesting...

42:30 Naturality implies smoothness. Exactly, naturality implies smoothness paper. What was going on in the background of... It seemed to me that what he was saying about Grodenig's work on functional analysis seemed to strongly connect with that, and I would love to have understood that better. Again, is Steve still working on that? Well, no, I mean, why on earth is it that none of the people in the foundations of physics ever take this up, ever look at it? Because it seems to hold the key to understanding, well, at least it's a, could a place of any, actually a far better place than most of the places they've been looking for the last 50 years, to understanding, you know, what's going on in quantum theory, so it seems to me. Yeah. Oh, well, actually you think so too. But nobody looks at it at all. They just go off on this, they just do all this stuff like Kirke, who's promoting this, oh, you know, Cartesian closeness is absolutely a dead issue, you have to work in symmetric, bimonoidal categories, and that is, oh, absolutely... No, it's such a pity that Steve hasn't been able to get those ideas across. I'm sure it's not his fault, he's not the one to try, because he doesn't go to conferences much, that's the problem, isn't it, Chet? As I say, it really bugs me that hardly anybody in the Foundations of Physics community even knows that paper and realizes how important it is for any understanding of Foundations of Physics.

45:00 Steve's advisor was Serge Lange. Serge Lange died recently, but there's a statement which I've come to regard as typical of Serge Lange, because, you know, they didn't talk much in later years, but in earlier times, Steve would sometimes talk with Serge Lange on the telephone. At one time, he described this fact that non-communicative naturalness implies . And Lange probably was, I'm not impressed by that. This becomes kind of the canonical answer of Serge Lange, probably with conceptual content, even highly non-trivial in this case. It's certainly not just a formal trick. Now, if you want to just look at what are just formal tricks, go to the, you know, what the non-standard analysis do, which are just, you know, kind of, but, no, on the contrary, this relationship between naturalness and analyticity seems extraordinarily deep, isn't it? Now, I keep trying to get the foundations of physics people to take an interest in this. In the paper it is explicitly stating that stress per function through domain is the line or complex vector.

47:30 But he agreed with my generalization of this. In the same topos, you know, there are objects corresponding to open subsets of n-dimensional complex space and so on and so on. And on these things you get also the same, the correct answer that the natural transformations from that are standard in the analytic functions or the smooth functions in these other cases. So it's quite, not just one example, it's a whole setting. It's a whole setting for everybody to challenge themselves. Yes, very good, thank you. It's not entirely unconnected, well this is a separate topic I know, but it's still not totally unconnected with what you were talking to me about in Calais six months ago. And on the last day, which was the issue of the natural numbers and how they, as in the piano characterisation, let in all of the so-called pathological functions. Well, coverings involved, and obviously there's some notion of growth, there's some role being played by growth in the topology here, and by the coverings, the stability of coverings in the growth in the topology, but why should they be stable under pullback along absolutely arbitrary maps?

50:00 Something like finite, piecewise, linear, or analytic. Well, I understand that was the whole motivation of the program for tamed topology, that just finite sub-analytic functions would actually be enough to allow for that. Does anyone have a kind of characterization? By the way, our Japanese friend forgot to mention that in the Keystone program also. That's also where it comes in, yeah, yeah. Sorry, go ahead. No, no, it's all right. Actually, I believe the words change apology were actually your coinage, weren't they? No, they're golden digs. I thought you were. I told it to Vandengrist. Vandengrist didn't have any idea about them before. It was Vandengrist, right? Yeah. Vandengrist published papers and a book. With that title, which you wouldn't have known about if I hadn't told you, it came originally from... From Gregory. Well, if anything, it's even more scandalous that he didn't acknowledge that, but we won't go there. Well, no, I was just interested in understanding better how this... I'm interested in trying to understand historically why it was that this characterisation of the natural numbers, the Pianet characterisation, and also the Dedekind characterisation of the real line, which as it were completely washes out all the cohesion, became so firmly established as the default option definitions. It would have seemed more natural to re-examine the definitions involved in understanding our concept of continuity rather than going to this huge generalization of the notion of function in the direction of just completely arbitrary ordered pairs and therefore to the idea that the Coverings of the must-be-stable-under-absolutely-arbitrary, you know, pull-back-along-even-completely-arbitrary maps, why everybody, as it were, went in that direction, and nobody, at least until very recently, appears to have looked at the alternative strategy, which seems to be much more natural from the point of view of retaining connection between our understanding of the physical world and, you know...

52:30 The ultimate ingredients of definition of our mathematical concepts are rethinking the definition of continuity and until the topology program came along and the work on O-minimality that Angus and his colleagues work on Nobody seemed to look at, nobody seemed to consider grasping the other horn of the dilemma and looking at the redefinite, looking, rethinking continuity. That's very strange. I mean, strange to me as a kind of historical, cultural development, not just as a... Mathematical one. Specifically, Piano constructed the space-telling curve. I always meant to find some genuine expert on history. It's hard to imagine that there wasn't somebody who raised the... Yeah. Descending voice. Yes, yes, yes, exactly. That was the real crisis in founding. Oh yes, absolutely. The real crisis was the crisis of geometry, the crisis that was posed. The incredible consequence of the conflict between discrete and continuous in this sort of uncontrolled fashion. In a sufficiently controlled fashion, the definition of continuity which was used at Sturridge was based on general topology, on locale, so locale theory, you know, no way solves this. No, obviously the solution to the problem and to the rethinking of continuity involves also rethinking topology, and taking some concept other than that of open set as the defining...

55:00 Defining notion for a notion of space in general and the same the same dilemma. We have to define a general notion of structure in terms of functions in nice spaces and in terms of figures and nice shapes because, of course, continuity in open sets, these are called sets, but really they're sophisticated functions, really an intensive type of definition that never gave us... No, which is a very strong indicator that it's not the correct general notion. So perhaps, yeah. I mean, here we're talking about the basic thing itself, but still the idea that the interval is, I mean the interval, but whether the interval is, whether its structure is defined by Well, we're back again, of course, with the role of understanding the distinction between intensive and extensive quantity, aren't we? Because that typically involves that distinction, then. Again, just pedagogically, I really think you should hammer Grassman into the heads of those guys in Bristol for at least the first three days, until they're reeling. You know, there's another Grossmann converse coming up. Yes, I do. You told me about it. But unfortunately, you were saying to me you thought it might not be as productive as it could be because it's just going to be one of these huge, great jamborees with about 500 people there. Yeah, it's huge, you know. Yeah, absolutely. And they'll all just be, well, three-quarters of them will be putting crudely up their own back sides and just wanting to stand up and give their little talk and then disappear.

57:30 Well, that's good. Well, David Rowe, I told you that David Rowe was absolutely, you know, firing old cylinders when he learned that you were going to be coming to that Grasman meeting later this year. Yeah, well, of course, he was at the other one. Yeah, he's at CETE. Yes, CETE. Yes, thank you. Very good. Very nice. What's the alternative? Ice cream or fruit? What type of fruit? Light cheese or ice cream? It's an either or. I like cheese. Me too. I like cheese. Sounds good to me. Somebody was telling me the other day that light cheese apparently are supposed to be even more chock-full of antioxidants than pomegranates, which are the new health food fads. I'm still here in spite of all these health food fads. Ignoring them is probably... Do you want some more wine? No, no, I really don't. You finish it. I'll carry you up the stairs if I have to, but you'll wake up fine. Remember, we have a little bit of a line somewhere. The first talk isn't until 11. Not until 11, that's good. Tomorrow should be extremely interesting because all four talks will be interesting for various reasons. Yes, yes. It's tomorrow's the day, the reason I came down, because if I'd known you were going to be here, I would have come down under any circumstances, but as it is, it was because of the talks tomorrow that I definitely wanted to come, because they did look very interesting. Well, I'm not so sure about that. It's interesting because we have another opportunity to verify if he really is a menace or not. Yes, well, my impression is that he may not. He's not a kind of, he doesn't set out to be an ideological menace. He's not, as it were, a conscious peddler of mysticism, but he is somebody who is...

1:00:00 Very willing to spend, to give far too much time and attention to the mysticism and also to the anarchism and to the, you know, to the, you know, to the, you know, to the, you know, to the, you know, to the, you know, to the, you know, to the, you know, to the, you know, to the, you know, to the, you know, to the, you know, to the, you know, to the, you know, to the, you know, to the, you know, to the, you know, to the, you know, to the, you know, to the, you know, to the, you know, to the, you know, to the, you know, to the, you know, to the, you know, to the, you know, to the, you know, to the, you know, He has reams in the third volume of his projected biography of Grotendieck about this kind of crazy thing that Grotendieck wrote where he discovered God through dreams. He has several volumes? The book is actually going to be in three volumes. The first is... and the second is supposed to be... no, it's awful. or less sweet anarchy, which is really mainly about Grotendieck's father and his childhood. The second is about his life in mathematics. That's going to be like I have no idea because Charles is not a mathematician. On the other hand... No, he is a mathematician. Actually, no, he is a mathematician, but he's... He worked in K-theory. Okay, actually, I stand corrected on that because somebody had told me when they first encountered him, he was actually a psychologist. But no, I understand. You're quite right, actually. That had already been corrected. Pierre corrected me on that. He may not have done any mathematics for a long time. No, but he is a trained mathematician, so I'm sorry. I apologize. That was wrong information. Well, that's my worry. I remember this randomly comes in my mind when I was young. Thank you for your attention. So, in that way, even though we're just describing what he said, at the same time we're avoiding what he said.

1:02:30 We all admire Aristotle, these guys certainly admire Aristotle, I take him more seriously than Jonathan Barnes appears to, but Aristotle believed in some form of a god, or you know, the prime mover, but then Aristotle was 4th century Greek, he was thinking in terms of the, what was the kind of limiting intelligible structure of the universe from the point of view? No, I mean it's just completely unprincipled. ...claims to not be promoting something when it in fact is promoting something. Again, I have this terrific... All the temple images, you see, they say, oh, temple is not forcing me to grow the religion. There's nothing wrong with that. It's interesting because Lenin says Mach is tending toward Buddhism in his analysis. Yes, in materialism. But in fact, he was... I didn't know that. No, you know, he was part of the Monas gang. Keres was one of the very active promoters of Buddhism. I don't know precisely how it came about, but in any case, Bach in his later years was very much an open Buddhist. Unfortunately, Einstein was very much influenced by Schopenhauer, who of course also embraced Buddhism. There's a Berkeley thesis. I'll Google it and do some reading. I should have known that. But I can understand, you know, Markham Buddhism is a kind of slightly easier, well, pretty straightforward. I can see how he would have made the transition. But Mark and Platonism, I find, is a little bit more. Because Plato's ontology, although it's objective idealism, obviously, in its purest form, is Lenin. There is no wall between subjective idealism and objective idealism. On the contrary, subjective idealism as a piece of propaganda or whatever, it is precisely a road toward objective idealism.

1:05:00 We have the similar perceptions, therefore, we must be perceptions in the eye of God. Yeah, it is. It's very easy. No, I understand. There's a road, you see, there's a road. No, I understand there have been many roads from subjective to objective idealism and indeed vice versa. But it's inevitable that this is actually the purpose which it serves, you see. I mean, I consider this question of maintaining ignorance of the maths as a form of subjective idealism, too, or the philosophy of doubt as Lord Balfour. Yes, yes, yes. It exists as a road to objectification. Very explicitly, Malfoy, of course, who was encouraged in it by his father-in-law, by Salisbury. By Salisbury. Exactly. No, sorry, not his father-in-law, because Malfoy never married. His cousin, I should say. His cousin, I'm sorry. Oh, I didn't know that, Salisbury. Oh, yes. Oh, yes, yes. And Salisbury had also gone through... It's consistent with all the pragmas I have. Oh, no... No, actually, there's a very interesting essay about this by, actually, an extreme reactionary historian of ideas, Maurice Cowley, who was a fellow of my old college at Cambridge. There's a very interesting essay on Salisbury's religious views as a young man, because he wrote very extensively on religious subjects. This was well before his political career took off. What he was called then, whatever the, whatever, yes, this is why, this is why, no, it wasn't that, it's just that these usual nonsense with the aristocracy that the eldest son who is going to inherit actually has a courtesy title until his father dies, he then actually inherits the main title, so he's known as something else until whatever age it is at which his dad pops off, so it's going to be quite confusing, you realise that you're reading, you know, books by Guys with two quite different names, but they are in fact the same name, they are in fact the same person. They underwent a kind of name change when they inherited the title. So whatever is the courtesy title of the Marquis de Salisbury, I can't remember what it is now, but anyway, that was the name under which Salisbury wrote all of these religious tracts in the 1860s, 1850s, 60s, even a little bit earlier than that actually.

1:07:30 He actually makes a point quite specifically about the need for Christian apologetics to assume a more radically skeptical point of departure, because that's the way in which you're going to be able to disarm the materialists. And I can't help thinking that... Salsbury was a very powerful intelligent something was through reactionary of course but Within, you know, the spectrum of 19th century politicians, there's no doubt at all that he had one of the most powerful intellects, that he might even argue he's possibly the most powerful intellect to ever occupy the premiership. I mean, as I say, arch reactionary, but a highly intelligent man. He wrote voluminously. He didn't publish anything after he became Marquess, which was already about 20 years before he became Prime Minister. So these early writings are particularly interesting. And just wish I could think of his, well, of course, Cecil, the family name is Cecil. The family name is Cecil. The family name is Cecil. Which goes right back to Elizabeth. Of course. The family was raised, you know, to fame and wealth and power by, by Elizabeth I. In fact, two, you know, the first two generations of the family were her respective, well, they weren't called prime ministers then, but effectively her chief, you know, chancellor, councillor, you know, most important person in the country after the queen. And he was, went on into, uh, James. We just had a sweet. We didn't break it. We, we just had the lachis. Lachis? No coffee? Uh, no coffee. Ah, no, I'm okay for coffee. I will just... No coffee? Maybe a tea? Yes, actually, could we get some tea? Two? Yes, two teas is a good idea. Good for the throat as well. And I was in LA. Everything was going smoothly, wonderfully, wonderfully. And then in the evening, someone suggested an espresso. And I was in conversation de la compagnia, as they say. Yes, what marvelous. It ruined my night. Oh dear, yes, it does tend to do that. No coffee in the evening. No, no, no, no, no, me too. Sorry. No, not at all. I actually...

1:10:00 There was this people who was actually, let's see, part of the formation of the United States... Oh yes, Lord Hugh Cecil. Tremendously important figure. He was a very important figure in the Conservative Party in the 1920s. Again, very much identified as a kind of Christian apologist. He wrote a lot of semi-popular works of Christian apologetics. He was a great friend of Dean Ng, who was the Dean of St Paul's. ...from the 1920s and 30s, who was a very accomplished Plato scholar and had been a contemporary of people like Curzon at Oxford, but again an interesting figure, very influential, very, very reactionary, I mean much more so than by the 1920s and 30s, the bulk of the episcopacy, they had become, you know, more quite progressive, inverted commas, you know, loosely by that time, like, you know, it was a good thing for Christians to have a social conscience but not in... Ingersoll absolutely died in the world reactionary. He published pamphlets denouncing the trade unions and all these greedy, greedy workers who were ruining the country. But he was a very close associate of Cecil's who was, again, very influential in foreign policy in the 20s. He was, again, very close to Austin Chamberlain, who was the foreign secretary, and indeed to Neville Chamberlain. And he had a major influence over what you might call the more... Given to uplift and high-sounding ideals wing of the conservative foreign policy establishment in the in the 20s and 30s. We use all this high-minded stuff about the British family of nations, never call it the Empire, only these terribly provocative dangerous people like Churchill go around calling it the Empire. No, no, no, no, we call it the British family of nations. I really believe that it was all part of this wonderful, uplifting, civilizing, ameliorating mission of the British people and all this kind of sentimental nonsense, which of course just concealed the white man's burden, we wouldn't have taken up, we do it because we know it's the duty which God has laid on us, you know, I think with half of their brain they actually believe there's garbage, and the other half of course they knew perfectly well it was just an excellent pretext for conning people into believing.

1:12:30 You know, there's something good about imperialism, but, you know, that was Cecil, and he again was a, quite interestingly, he never inherited the title, he never became Marcus of Salisbury, because I think he was a younger son, the Marcus of Salisbury who was, would have been his nephew. In the 1950s, Harold Macmillan's government was probably the most reactionary member of that government at the time that the British were decolonizing in Africa, i.e. when they were moving over to the neo-colonialist model and giving nominal independence to all these places where, of course, making sure that British finance capital continued to... He was the big opponent of that. He was the one who was the kind of die-hard imperialist. He made a furious attack upon MacLeod, who was Macmillan's colonial secretary, who was actually over-forcing the agenda and giving fairly rapid independence to all these countries because the British didn't want to get trapped in a long, hopeless colonial war like the French had done in Algeria. But it's absolutely clear that was the motive. They just saw what was happening to the French in Algeria. And they decided they must scuttle. Originally the plan had been to hold on to the African colonies for another 50 years at least, and even places like South Africa were not going to be kind of given independence until well into the 21st century, but then when they saw the obviously enormous boost to the self-confidence of the colonial peoples that had been given by the experience of the Second World War, and when they saw the Disastrous things which happened when the French tried to hang on to their empire in India, China and Algeria, they very quickly changed their mind and of course there was the debacle of Suez which also helped to accelerate things and but there's a speech which Macmillan made, well he didn't make it publicly but he made it to his cabinet when they were having this argument in about 1957 or 1958 and he said do you not realize that you know we've got to get out of Africa, we've got to basically make sure that the British flag is...

1:15:00 I don't give a damn about the consequences as long as our economic interest is protected. But the alternative is obvious. The French have 500,000 men under arms in Algeria. They have, you know, basically sucking in the whole of their conscript class for the next 20 years in an absolutely unwinnable war. If we do that, we shall be bankrupted and it will be a complete social disaster. It's unthinkable. We cannot try and hold the empire by force, so we must get out quickly now. So he accelerated the agenda. And MacLeod, who was the kind of instrument of this policy, He came under very savage attack from Salisbury, who once famously declared he was the trouble with Mr. McLeod is that he was too clever by half. I mean, a curious jive, but somehow it really stuck. It was the label which hung around McLeod's neck for the rest of his political career. Because at that time the Conservative Party did pride itself on being the stupid party. Brains were not regarded as an asset in the Conservative Party. That's why Salisbury of course kept so quiet about Balfour kept so quiet about having them. Anyway, that's the background. But no, the Salisbury's and the Cecil's have been around for a very long time, say over 400 years, as a major player in British ruling class politics. It's a pity that Lenin had only a few months in Swiss libraries. Oh yes, he would have done an amazing job of writing them. Because he doesn't mention Salisbury or Talfour in this connection at all. What he did do in the short time he had was simply incredible. The real point is he got to the essence of things, even without the major examples, he had enough examples too. I think it's absolutely astonishing that you say, I never knew that about Mark, that's very interesting, about Mark ending as a Buddhist. Oh no, yes, that's right. And the interesting thing of course is that this connection, I think I describe it more as a kind of permeable membrane between subjective and objective idealism. It's certainly very permeable most of the time.

1:17:30 From the point of view just of using it as an instrument for anti-science and obfuscation and of ruling class politics, policy, then the fact that Buddhism around that time did become very influential with a certain section of the British ruling class is itself, I think, extraordinarily significant. We talked about this before, but I think it's extraordinary that these people like... Besson, Annie Besson, turned young husband, and the people who created this whole, well, really, the theosophy and the whole of this kind of pseudo-religious, what's now subsequently become this whole... And the funny thing is that people on the left have some kind of admiration for these figures. It's worse than I ever guessed, because in Charlot, in his book... Which I, of course, skimmed through quickly. You may well have already known this, but apparently Grotendieck circulated, this was some years after he published his A list of the people who would be regarded as avatars or figures who symbolize the coming of a new age and these include Krishnamurti, who of course was cultivated, invented, raised up entirely by Besant and those people. He became a greater posse of Krishnamurti, who, as with whatever his faults, sort of saw through right away as being an obvious scoundrel, a complete charlatan, a very nasty piece of work, actually, apparently used to sort of seduce young girls and all sorts of... But, you know, it's astonishing, and it's exactly... Krishnamurti, Tihar Deshadar is another of this list that wrote and drew up. Just don't understand how the mind could have created this. This is in the AMS version as well. No, I haven't seen this. It's just very, very depressing. I can't tell you how depressed I was after I read that. ...a mind so great and had done so much for science could have ended in this. So Godendieck is crazy, let's say, but Bohm, was he crazy too? He flirted with this mysticism in the last 20 years or so. No, I mean crazy in the sense of, you know, something really mental wrong.

1:20:00 No, no, I don't think so. He was very naive. He was very, very naive and almost completely childish. You say it's astonishing, but it really is astonishing how this power can influence intelligent people in some way. I don't know, switching off some particular defense mechanism and just opening them up to complete surrender. Well, it does seem to work with some minds, but I don't know. It doesn't work with all kinds of intelligent minds. It obviously doesn't work with you, it doesn't work with Basil, it doesn't work with a lot of people. A significant number of people with whom it works only too well. I mean, Bohm actually began as a dialectical materialist. I mean, his first book on causality and chance was actually really written from a pretty soundly dialectical materialist viewpoint. He ended up leaving in all of this woozy Buddhist business of the hollow movement. I don't think he particularly... he'd done all of his good work in physics by then anyway. He was, Basil says, a very gentle, spirited man, but very, very naive and very easily bamboozled by people. He's always been taken in by people who want to take advantage of him. He was such a generous and unsuspicious character, he was just rather easily taken for a sucker. I mean, the classic example of that which Basil managed to prevent. By literally stamping on him and getting him in his office and almost literally pinning him against the wall for half an hour until he talked some sense into it, not to be so bloody silly, was his involvement with Geller, with the ridiculous Uri Geller, because he turned up on the scene, as it was in the 1970s, and he persuaded the man who was the head of the experimental physics group in Birkbeck. And several other people in London to take him seriously. Halstead, I think it was. I think that was the name of the man, Halstead, Beverly Halstead. He was the experimental physicist at Birkbeck at that time and he was completely suckered in by Geller. His reputation was completely destroyed of course as a result of this. Another man he happened to was the professor of physics at King's College, John Taylor. In his case

1:22:30 it couldn't have happened to a nicer man. He thoroughly deserved what he got. I mean, I'm rather pleased that he fell for it because, you know, he was doing so much damage to the department anyway, that the fact that they used it as a pretext for getting rid of him, that's the only thing I can find positive to say about Uri Geller, that he helped to rid them. Taylor is the one who published this theory of ice room. Satan was really a pilot in a spaceship powered by a black hole. It's unbelievable. Unbelievable garbage. How this guy ever got to be the head of a serious, I mean, mainstream physics department, in fact the physics department which probably had more really good specialists in general relativity than anywhere else in England at that in the 50s 60s and 70s. They had to appoint that guy as the head of it. I don't know what was going on there. Felix Birani, I mean there were lots of very good people there. But anyway, Bohm, Holstead went to see Bohm and told him all about Geller and said, you've got to come and see this, you've got to come and see this guy. And Basil, who had actually had some training as an amateur magician, magic circle stuff, who had seen all these tricks, he knew exactly what the bloody tricks were. I don't say he identified all the tricks that Geller was using straight away, but he had enough common sense. ...sense to know what was going on. This guy was obviously smelled with his complete charlatan from start to finish. Went down there and watched and actually he didn't watch Geller. It was interesting he said he didn't want to watch Geller because he knew that Geller would be far too good to slate at hand. But he did watch Halstead and he could see that Halstead was not watching carefully enough. He said it's a classic kind of sociological trism that people like James Randi have often pointed out that in fact scientists are probably easier to fool than almost any other section of the community because they do tend to be rather trusting of what they appear to be seeing. This was a classic instance. Anyway, Basil managed to head David Bohm off from opening his mouth and making some public endorsement of Geller, thank God.

1:25:00 But it was too late to save Halstead. He had already sunk by it. Extraordinary. But how on earth... I mean, I wouldn't... after reading this thing, this list that Grotendieck published, I was so horrified. The list probably starts off with people that Grotendieck admired at age 20 or 30. Yeah, well, those, yes, might have been, you know, still a scientist. It's because Newton is also on the list, I think. Yes, Newton is on the list. Uh, but I just, as you say, I do not understand what the trick is that these people... Of course, now, people have been, again, amazingly large numbers of people have picked up on the slogan that, the main thing about Newton is that he was an astrologer, you know? I mean, it's such a flimsy story, and yet, by being massively propagated and repeated... Many people just, again, repeat it automatically. Coming back to Salisbury. Yes, yes, by all means. I need to know more about that. Well, I also want to come back to what you were saying about Grotendieck, when he was doing his great scientific work on the point about logic, but anyway, about Salisbury. Anyway, the third, yeah, the third thing, which again is sort of a mistake, It's not a, it's not. ...ringed spaces as had been previously defined, but now it was just a sub-category of appreciated categories. Sorry, this is the... I'm sorry, you're talking now about... This is the May 73 Colloquium talk. Right, right, good. Which I seemingly don't have a copy of. No, alas, but... But I remember very well. This was his main point. Right, the construction of the category of schemes. Yes. It should not be, by the way, previously stated... Yes, so people take it still as being wrong, but he refused it and said no, you should start with everything. Well, I mean, you can describe it in various words. You see, it's a matter of taking figures as basic, or you could say it's a matter of taking pre-shapes as the enveloping category, or there are many ways of describing it, but it's a…

1:27:30 And he made it clear how the old definition, any ingredient of the old definition that you might actually need, is easily derivable by natural transformations, but that the basic thing, it should not be that, you know, it was fortunately something I had always been convinced of by cogitating about what Gabriel had taught me. Again, Gabriel himself didn't say this, nor are they afraid to say this. Since 1960, there has been this construction in the analytic case. They never say this. They always think of the Grotto post as something associated to a given space, rather than the Grotto post associated to a point being a branch of geometry, as Collins liked to say. There was a very clear statement that you should reject and throw away the old definition and use this new definition, and nobody seems to have taken it up. A few people have half-heartedly taken it up, which is even worse, because they start with the old one and then say, well, we could use this new one, and then it becomes a mixture, a mix-up. So those are three distinct points which arose, and these were all made in the course of this one talk to the Buffalo Colloquium. No, no, no, this is what I'm saying. No, no, no. The business is about the subtoposis of the ring classifier being recognizable as such without use of... Yes, yes, yes. Oh, I see. That was never in that lecture. I understand. Okay. I mean, you could say it was kind of implicit, but it wasn't at all touched upon. Okay, okay. I'm sorry. I haven't understood that point properly. It's just a basic framework is all he wants to be good at. And what was the third thing now? The third one you just said was about the redefinition of schemes. No, no. That was the first. Okay. So what was the first one? What was that? I was telling you as we came. Well, I thought the first one was what you just said, the kind of read, well, except you tell me that wasn't in fact in the lecture, that it was implicit in what he said about...

1:30:00 The classifying topos that it carried, that one could think directly in terms of limits and co-limits in the structure without having to think at the level of objects in... Okay, well, there was a third one. Gosh, in that case, what was it? I've just been listening to you. I can only retain those two. I retain those two very clearly. Well, there was another point, you're right. Gosh. Well, no, except, no, no, the point we made earlier about intensive and extensive was something which came up because of questions I asked you. It wasn't something that you said had been in the Grothendieck lecture. Sorry, sorry, I can only recall those two things. So there were three? Yeah, I think so. Okay. Which you have managed to wonderfully combine into one. Yes, yes, okay, they are related, but they are in fact three distinct points, obviously, I see that. And one of them, as you say, is made in fact only in a kind of very implicit, not in a direct way. I have to say, this is absolutely crazy because... No, I recall you telling me earlier about his, you know, he offered this redefinition of some completely new way of thinking about theorem schemes. No, because all the other things, the other things we've been discussing tonight, you were discussing domain theory and the functoriality that was implicit in Dana's ideas for the construction of, but that's obviously nothing to do with what Grotendieck was talking about in 73, talking about the... ...the role of the classifying topos and the fact that he thought so directly in terms of limits and co-limits, and I'm afraid apart from that and the point you've just made about his rethinking of scheme theory, I can't recall a third point...

1:32:30 ...that was directly related to what he said in Buffalo in 1973, that we talked about tonight. I'm sure there is a third one, but I don't think we've discussed it this evening. It'll come. Don't worry, we've gone off in so many different directions. About Salisbury, okay, fair enough. I'll come back to scheme theory when you... Okay, what do you need to know about Salisbury? No, I don't know. I think they need to know a lot of things. Well, as you know, he was a... Because I didn't know I needed to know. Well, just that he was a... He was a see-saw. He was a very... in his earlier years, he was a very... he was the uncle... No, he was the uncle. He was the uncle of both. Yes, he was the uncle of... Yes, he was the uncle of Balfour. He was Prime Minister. He was Foreign Minister on Disraeli. Yes, he was Foreign Minister when he was the guy who negotiated the treaty at the Congress of Berlin with Bismarck. Disraeli was there, but it was actually Salisbury who did almost all the negotiations. And he was the... I mean, he was the... Excuse me, I'm so sorry. He was clearly going to be... Oh, you've just thought of what the third thing was? No, no, I thought it was something else. And he was, you know, he was the major power behind the throne in Tory politics in the 1870s, and he succeeded Disraeli as Prime Minister when he died in 1881, I think. Yes, 1881. And... no. No, he didn't. He didn't immediately... Disraeli is the leader of the party. In fact, they lost the election in 1880. Disraeli was out of office when he died. I'd forgotten that. Disraeli lost office in 1880 to Gladstone. Gladstone came in. Gladstone was prime minister from 1880 to 1884. So Disraeli died the following year after he left office. Salisbury became the leader of the party. Disraeli came in and was actually a long period in the opposite side. Disraeli had an extremely checkered career. He had been a key player in the Conservative Party even back in the 1840s, but never actually expected to become leader. He was always kind of more a position of kingmaker. Oh, he wrote many novels. He wrote several novels. He wrote, yes, of course, mainly about it.

1:35:00 All the foregoing passages on Side A were recorded on the evening of the 21st of January 2009 in conversation with Bill Lovier in Montpellier and it's now the morning of the 22nd of January 2009. I'm about to go downstairs for breakfast and then we'll be going to the second day of the Mediterranean. Algebra and topology conference around the work of Grotendieck. We were mentioning in passing yesterday, and to, let this poor lady have something. Yes, that's right, he's the Belgian algebraist, who I think has done some interesting stuff. He sent me an email asking me about the Grothendieck meeting at IHES, because he knew that I had been going to that, asking me all sorts of questions about what had come up and who had spoken and what I thought had been the most interesting talks, so I simply sent him a short message, a kind of holding message, but mentioning that it was actually the talk by the guy on functional analysis, which I thought was by a long way the most interesting. Not in this email he sent me yesterday, but in previous occasions. He has expressed a very keen desire to meet with you for discussions. He's now saying that he thinks that the... The con-construction of non-commutative geometry is completely the wrong way of going about things and that he sees more and more that the problem is that it's lacking in functoriality and the proper use of functoriality so I thought well this will be music to your ears and at one point he said to me in the last six months or so I realize now more and more that I should be I should really be learning to think, as it were, much more instinctively in a functorial way. That's the key to clearing up a lot of the... Oh, yes, oh, crikey, same problem. Well, they seem to come so frequently. Let's... Yeah, let's... Yes, exactly.

1:37:30 Can you not do it for two? No, you have to do it one at a time. That's unusual. I'm so sorry, we just gave our... Bonne année à vous. Bonne année à vous. Merci, merci. Yeah, no, we're not going to be able to get it this time. We'll get the next one. No, no, not another time. Sorry, I haven't got anything more for you. Can we still get it? Quick! How about that for bad timing? No, no, we can still do it. That was a nice drive. Oh, no, I did the same problem we had yesterday. I can never work out. Yeah, so how are you supposed to do it? Oh, c'est ça. Ah, merci, merci, madame. Yesterday I got it right, more luck than judgment. That's OK, now we know. OK. I noticed in Greece in March last year, you know, when there was the conference in honor of Anders Petras, just how terrible now the problem of, you know, this obviously the crisis of capitalism is becoming. The numbers of beggars on the displaced people on the streets in Greece is just, I mean, an order of magnitude more than any other country I've seen it in Europe, even in Italy. And of course, many of them are refugees from, you know, from, but a very large number of them are clearly, you know... ...native Europeans, and a lot of them from the former Soviet bloc countries, and they're just living in complete destitution, these kind of shanty towns on the outskirts of places like Petras and Salonika, just in utter destitution. Obviously Berlusconi and the right in Italy have already sort of more of a scapegoat look for the crisis, don't they? It's a classic fascist technique of trying to stir up the racist element of the lump of proletariat and pretend that it's all the fault of the immigrants taking your jobs.

1:40:00 The very people who, of course, want constantly putting pressure on the workers to undercut their own hourly wage rates and turn around and say, but these people are undercutting you and taking your jobs. Well, you know, it's so cynical. Now that you've had time to study it, what was your overall verdict on the inaugural address? I didn't study it. Oh, no, okay. Well, I don't blame you. I'll do that later, maybe. ...presents the... You know, representative society is an organic growth. It's so important to understand that, if you're a true conservative, that society is essentially an organism and not a mechanism. That's what the liberals and, of course, the Marxists get wrong. But, in fact, in some ways the liberals get it even more wrong than the Marxists, from our point of view, as I... Yes, yes, you see, that's the whole point. Everything grew so naturally and organically. That's why you can't change anything. Except, of course, in absolute extremists, where it's obvious that unless you do change it, the whole thing is going to break down. But you only change under the pressure of the most urgent necessity. The presumption should always be to retain the existing arrangements. This, as it were, is the classic apologia for high Tory reactionary ideology, as propounded by people like Burke. There can never be a planned society. But of course, this all conceals the fact that, yes, there was an original, there was exactly, precisely, originally there was a plan. It was probably devised by somebody like the, well, we probably don't even know their names, but Menes or whoever was the first pharaoh, or Zosia or one of those people that you read about in the... The first galleries of the Museum of Antiquities in Cairo, when they explain about how the first pharaoh consolidated his power and formed a unitary state. In fact, it was actually much earlier than that, because it was when the first tribal kingdoms emerged.

1:42:30 I'm sure there was some chamanry and priestcraft well before. In their world before, the foundation of civilization is based on written records, like Egypt, but it seems to have taken a qualitative leap in terms of the centralization of ideological projects like the creation of authoritarian religion. And Phidiasm, in fact, you know, the only road to salvation lies in submission and side to an old purple deity and, of course, more importantly, to his appointed representatives on Earth. Marx says capitalism, at a certain stage, assisted the development of the productive forces. This is simply a historical fact rather than something which was a necessary... Yes, that's a fine point. I mean, I'm... I'm going to have to go and think about this. I'm not sure, but... Yeah, I'm saying that it's also, not only is it a fine point about the interpretation of history, but it's also on your donor. It's a constant ideological backbone, this idea that's somehow inevitable. It's certainly at the moment... The best you can do is ameliorate it and... Yeah, yeah, sure. Well, it's certainly the line that all reformists take, but... Right now, actually, what I should have said the day before yesterday, the day just before this crisis broke, just before the scale of this crisis of finance capitalism became apparent a year or two ago, you were getting progressive chatterboxes like the dreadful Jack Attali, the classic example here in this. He had some account of why n categories would provide, because they are really the correct foundation for mathematics, the correct general foundation, and he's compiling all this Mackay line about, you know, identity is old hat, we have to have some more general notion on that of identity, and how this was meant to connect with the requirements of... Financial mathematics, I didn't quite understand, but he explained to me it was something to do with stochastic differential equations, that you needed some notion of mapping that was more general than anything, even than that of an isomorphism. How it was supposed to connect with differential equations, I'm afraid I didn't...

1:45:00 Did he cite Baez or anybody? He did indeed cite, he spoke with great enthusiasm about categorification. He was using the term, he must have used the term at least 18 times in the space of 20 minutes, and he was saying, so I said to him at the end, I'm just trying to keep a straight face, so what you're saying is that you actually want to head the project for categorifying financial economics. Exactly. And I said, well, do you think you'll make a better job of it this time round than you did the last? It is extraordinary. So, again, there he is. You might even want to look him up on the net. I don't know what he's published, but. He started as a pure mathematician before he went into the city, but apparently they, and of course Templeton, who, as you already mentioned, are giving quite a lot of support. This N categories meeting that they had at Imperial was funded by FQX, or partly by FQX, through the grant that they've given to Kirker's group in Oxford. I must admit I had no Better understanding at the end of this 20 minutes than I had at the beginning of what it was that n-categories were supposed to contribute to the understanding of either stochastic differential equations or financial math, but he would doubtless be able to explain. That's the line. It's incredible, isn't it? Yeah. Well, actually, one of the leaders... It's interesting, you see, one of my colleagues... You know about the Perimeter Institute? Of course, I know a good deal about the Perimeter Institute. You probably know more than I do. I know quite a bit about them, actually, yeah. Smolin, for example. Yeah, he's their permanent kind of ideological spokesman. Byers, of course, is also very popular there. Apparently the Institute was funded first by Rogers. Thank you very much for your time.

1:47:30 Our friend Lou Crane, of course, is also an enthusiastic endorser and has been there several times. Yeah, but anyway, so anyway, he doesn't realize all of Smolin's activities, but I looked him up and I found out his close collaborator in physics had been a certain Hurd, Tom Hurd, H-U-R-D. Oh, yes, I think I remember you mentioning this guy. Master University. Oh, I mentioned it to you, hardly. Yeah, no, but I can't remember anything more. Well, anyway, this guy gave up physics to devote his life to financial math. Oh, that's right. He's the guy who was giving a lecture at Buffalo, isn't he? Or am I confusing him with somebody else? No, no, he was calling himself a student. No, but there was a guy who was giving a lecture about financial economics. In fact, I think it was his inaugural, but didn't you tell me he sent out this abstract about six months ago before the crisis broke, explaining how derivatives are far from being a destabilizing factor. No, that's Heard himself. That is Heard himself, that's what I thought. That's the website of Heard. Right, that's Heard himself. But Smolin is speaking in Buffalo. I have mixed feelings, I have mixed views about Lee Smolin, because I don't think that everything he does is bad. I mean, he has attacked string theory. A very strong critique of string theory, which I think is thoroughly deserved and soundly, but it may not, of course, be based in... No, no, not just the tact of writers, sort of, is that... Well, yes, okay, fair enough, fair, fair, fair comment. Well, I'm trying to understand what the ideological affiliations and background of this loop quantum gravity program are, which is the thing which he and the people in the Perimeter Institute particularly push very strongly. And Baez. And Baez, yes. But also people like Abhay Ashtekar, who I think have to be taken rather more seriously. Okay, let's see. Hello, how are you? Hello, haha. As soon as it would be published, I will send you a copy of it. Thank you, thank you. Professor Huzel, very nice to meet you. Michael Wright, very nice to meet you.

1:50:00 Topos theory in the other day at my movie, which I went to, about linguistics, the application to linguistics. It was a very nice talk. That was a very good discussion afterwards, too, I thought, with Benabou and the others about it. Very good, very good. Interesting for me. Yes, yes. Yes, it was very good. It was very nice to have the clarification about how the monoidal structure actually connects with the topology in that construction. But I'd never seen it. It occurred to me after listening to you, I'm afraid I didn't follow it completely because my French is not sufficiently fluid, but I think I followed the gist, Pedagogically, one could turn this whole talk on its head and make the theory of generative grammar into a way of introducing people to chief theory. Not just chief theory as a tool for clarifying what's going on in generative grammar. It's quite helpful also for teaching chief theory to non-mathematicians, perhaps. Maybe, maybe. There's hope for Chomsky after all. Even though I explain the sweeps to the non-mathematicals, I'm a little afraid with that because... I'm afraid that non-mathematicians understand what she's... I'm afraid it's not very... Did you read the paper by François Nicollat? A bit of it, no, not completely. I've read enough of it to be a little... Well, that was exactly why I was thinking that this might be a better way of putting it across to them. Because... My point of view was completely different from that of François-Nicolas about music. You know, when mathematicians discuss together, they know, they may use an informal language, but they know what is behind. But if you speak the same language to people who are not mathematicians, they think that mathematics and mathematics is something purely informal. And they think they could reach for analogies which may be, in some contexts, be partly helpful.

1:52:30 I don't know, that's a big problem. Oh, you have my permission. No, I don't think that they are. I will follow that. I will speak French, but it will be totally elementary. It will be neither historical, nor mathematical. I was not in the IETS last week, but Michael tells me that there was a very interesting talk about cultural analysis. Yes, I think so. Yes, yes. I thought so. It was very good. I went only one day. Yes. And it was contrasted, the talks. Ah, yeah. Yeah. Was that the Wednesday? The Tuesday. Ah, the Tuesday. No, I went on the Tuesday morning. That was some great food. The morning was okay. At the end. Yeah. No, Pessia gave, I thought, a very interesting talk about the Groton-Deacon equality and functional analysis. And that's what I did. Did you do a language lecture I heard? Yes, on Saturday morning. What was it? Well, it was a bit of the same kind of thing, but a bit more progressive. Where did you go? In a seminar called Mathematics, Music and Philosophy. At École Normale. Because it was at École Normale. I don't know why, but I have a few kilos, okay. Every month at 11 p.m. I heard you at the exhibition 30 years ago. Yes, yes, yes. You had promised notes for the lecture. Yes, but they are still not there. But now, I'm going to put them there. Ah, you put them there. Thank you very much for your attention. You were here yesterday? Yes, yesterday afternoon. I was not able to come if you want for some reasons.

1:55:00 No, no, no. I will come today or tomorrow. And me too. And you stay until tomorrow? Excellent. I will introduce you to my friend here. The great mathematician. The point is, you see, what is it? It's called... The reduced form was isomorphism, so it seems to be some kind of generalized Galois theory, but in fact it's very close to category theory, in the same year as Heidelberg and MacLean, but independently, because the Nazi occupation of Rome prevented any kind of communication. In the early years, maybe it's coming partly from there, you see, because he studied Silva, and cited Silva, and Silva actually reviewed some of his own work, so I think the study of this original thesis, which is now published in Lisbon, is very interesting, extremely interesting, historically, as well as to understand the functional analysis to... Complex geometry, all these connections, evolutions. It's very interesting. And just on a historical point, would I be right in thinking that Grotendieck spoke Portuguese? Because he was teaching in Brazil in the early 50s, 52, 53. I don't know, maybe he was teaching in French. He was teaching in French, but he was in Brazil for a time, it's quite likely. Well, of course, Hilbert published in Portuguese and also in French. Oh, he did, okay. But his thesis is in Portuguese, yes? Yes, unfortunately. So I have to, I have very difficult, very difficult, but I have a... For us French it is not difficult to read Portuguese. No, no, no. So I have a collaborator in this bond who helps me to... Yeah, yeah. But at least some fundamental points, you see, for example, at one time homomorphisms were surjective. But for him not. No, homomorphisms are general homomorphisms. They're not, so this is already, I think, one of the fundamental insights of categories. I'd really like to know more about this, of course, for the contribution I'm hoping to make in Oberwolfach on the early history of category theory next month, because at the moment, of course,

1:57:30 Of course, it's going to be the gospel according to our pragmatist friend, Krömer, who of course has never heard of the silver. I think this would be a useful work to know more about.