Discussions & exposition FW Lawvere

0:00 I look forward very much to hearing about that when we get, you know, possibly when we sit down over in lunch at Mechelen. I noticed I have two copies of the paper, so I can give you... Oh yes, please, because I noticed this is a slightly revised version of the one you sent me. Yes, I'd very much like to study that. Oh yes, without a secret. Yeah, of course, of course. I never circulate those into anybody at all, not without instructions, of course. One of the things I hope to get a chance to talk about today is, in fact, this archive project. You know, my catalogue of all the recordings that I have over the years now, actually since 1989, since the Cambridge conference. ...of your talks, but also, of course, of colloquia and discussions and interventions, which I think we've got to sit down and decide on the priorities for transcription and for you, as it were, to have things that you might want to use as a text, a textual basis for writing stuff up in more permanent form. So in what form did you do this? Well, I haven't finished the catalogue yet, I have to admit, unfortunately, because I've been so busy with others. At the moment, it's just written down. It's on a disc, but I mean, I can just print out everything. You just found this by looking through your boxes and looking at it. But essentially, yes, simple as that, simply by going through and photocopying all the labels. You're actually listening to it. Oh, no, no, I listen to a lot of it as well. Well, I'm sure it's interesting. I'm not sure it corresponds to the title. Correct, but also to ensure that there hadn't been any severe deterioration in the recording. So some of them are now almost 20 years old.

2:30 Unfortunately, I'm sorry to say, in the case of the Cambridge talks, not your lectures, I'm glad to say, but some of the talks that were given later in the workshop. Like Gonzalo's talk, for instance, for some reason it's just completely, you know, just deterioration of the tape. I must have used a rather cheap tape when I recorded it and it's been sitting there for almost 20 years and it's now almost inaudible. It's a great pity because I remember it was an extremely good talk. But it's just possible that with this... Thank you very much for your time, and I look forward to seeing you again soon. Two objects being more common with the function space is equal to the first one. Yeah. I.D.O. maps are constant, except in one direction. It gives rise to these two great subcategories. Discrete, quote-unquote. Yeah. And co-discrete. Yeah. And connected. And connected. Connected. Connected. Oh, yeah. The Hodescreen distance is a different thing. It's a subcategory which is isomorphic to the screen. The Hodescreen object is an object. The object is an object. The object is an object. The object is an object. The object is an object. The object is an object. The object is an object. The object is an object. The object is an object. The object is an object. The object is an object. The object is an object. The object is an object. The object is an object. The object is an object. The object is an object. The object is an object. The object is an object. The object is an object. The object is an object. The object is an object. But now we can go further with the power of the function space, which nobody seems to fully realize yet for some reason, and go further because we can ask for those connected objects for which every power c to the x or n to the x is still connected, and that's going to be a Cartesian closed, a smaller Cartesian closed subcategory. My name for these objects is contractible, because they're terminal for objects in the homotomy category, so more or less that's it.

5:00 So the idea is that that little subcategory should be both adequate and co-adequate relative to the screen. Yes, relative to the screen. Yes, relative to some other category, as you say, to some environing category or something. Well, the category for which these, the objects in this subcategory act as in some sense a kind of probe to the structure, hence the special position obviously of, you know, the condition that points should be adequate and co-adequate, the peculiarity of that situation, specificity I should say of that situation. We want to cross over here actually... In case every space is discrete. Yeah, except in the case of category sets, of course, of things which are sufficiently set-like. Yeah, yeah. Yes, yes, yes, exactly. Hence, of course, what a tremendously important discovery the, you know, John Isbell's isolated discovery of the notion of adequate and co-adequate subcategories was. Oh yeah. This is the way we want to go. Just cut down the steps here. That's where we would have gotten, that's where the meeting was of course, I know why you were heading there. Yes, whose line was that? Yes, yes, yes, thank you. No, but the point is we shall have penetrated through the correct use of dialectical... You said, well, precisely the recognition of adjoint situations in that chapter, the dialectical aspects of these positions, we should have made further progress. Yeah, who was it whose line who said that Bartow, mathematicians going off and living in Nirvana, was that, who was that? Was that Jan Alidze or was it some other guy?

7:30 Could have been him, could have been him, inconsistent with his rules. I have to say, well, it's just my ignorance, perhaps, that makes me think this, but I did think he gave a very good and clear account of the history of that particular chapter. It's so clear, definitively, that it's all totally without interest. Ah, okay. He hadn't made it so clear. I would like to be confused about that. Okay, well, I shall take that on board. So maybe he did shoot himself in the foot, I don't know. No, no, no. No, I realize you were teasing him. I'd like to explain this to you. It's a striking phenomenon I have here with this group. I just want to say, though, that there's another phenomenon of having some idea and waking up while traveling. And two, that would be nirvana, no more than that, it's so many more. Yeah, yeah, I know, I know, but yes, using it as an expression, yes. It could be wrong, because I did a calculation yesterday, which I somehow know, and I'm still not quite right. If you have a category, there's both subcategories, small ones and small ones. And it could be the same, but in fact there's no real loss, you can assume. It's interesting too, we find that just at the beginning. You can assume for general purposes it's the same subcategory as both, but the thing is, if that's true, remember it's very difficult, I mean, very difficult according to the ordinary point of view to find a co-addictive subcategory because it means you have to have a set theoretic axiom, there are no measurables. Measurable cardinals are not small. Yes, yes. It's the definition of small. Yes, the Euler measure has to... If they exist at all, they're outside my universe. Yeah. Yeah, yeah, yeah. Okay. So that's not completely... Although that's easy to state and show, it's not unusual. Excellent. Never mind. Let's suppose it's true.

10:00 Well, one of the striking things is you have this adequate and co-adequate means what? But your category, your interest in category, you're trying to analyze the X that receives one adjoint pair from pre-sheaves and another adjoint pair from the opposite of the covariant pre-sheaves on the same small category, right? Now, but these are, so you have these two pre-sheave categories with a kind of duality between them, and it's just the image. In other words, if it were a topos, for example, the Gruden-Dick topology, which extracts it from the pre-sheet, is precisely given by the algebra functions. What? It's not an additional datum. It's not an additional datum. It's already there. Try it again. It must be something very special. It's no longer a parameter, it's determined by... Sorry, you said... You know, so I'm saying, any Groton-Deke topos has a small adequate subcategory, namely the choice of a site. Yeah. But in order to extract your original topos from the pre-sheaves on the site, you have to have the Groton-Deke topology to impose these existential destructive conditions. But somehow, that is automatically definable, uniquely, if you have also the co-adequate subcategory. If this is true, it's amazing. I never noticed it before. Maybe it's wrong, but that seems to be... That's right. That's interesting. Oh, and it's an image in a very... ...inspectable sense, namely, going all the way across... Be careful on the steps, aren't you?

12:30 Yeah, I'm going. Between three sheets and three algebras, so it's going to have a complex set of algebras. You're fracturing them, you're fracturing them, so that you can get a full inclusion. On one side, full inclusion. An inescapable notion of image. Yeah, this is very striking, isn't it? This rocks, I'm sorry, go on. This is... Find a dualization of... The thing is... Same issue. So if you thought what was not going to be true? I just didn't catch what you said. Every pre-algebra, that is every covariance set value, it has a spectrum, which is always a sheaf. It's not just a pre-sheaf. It's automatically something that the in-category people didn't catch, you know. Somebody in there would say, what did LeVere mean by these dotted arrows? And dotted arrows are a damned actual theorem about an inclusion. But a trivial one, if you only think about what it means, because the spectrum of an algebra... It involves, you know, mapping into all possible things. Therefore, it obviously transforms co-limits into limits. Yes, again. Therefore, everything that you get is not only a sheaf, it's even better than a sheaf. It's an inverse limit of representable pre-sheaves. These inverse limits are always, you know, sheaves are always sheaves, right? Yeah, yeah, yes. So you're getting sheaves, some very strong, whatever topology.

15:00 And likewise, on the other side, if you take the function algebra of an arbitrary pre-sheet, never mind whether it's an actual space or not, but any arbitrary pre-sheet, the function algebra will always be an algebra, that is to say, a product-preserving functor, not just a covariant functor. And so, and moreover then, since these are adjoined processes, there's an adjunction map in any space x maps to the spectrum of the function algebra. Well, with the usual idea of algebra, this is basically never an equivalence because it's sort of the affine shadow of the space. For projective space, x, it will just be trivial points, but I collapse it all together. Whereas with this, the difference is here that we're using a much richer notion of algebra, because we're considering the functions which take values in all of these connected, you know, contractible spaces, not just, you know, r and r squared and so on. So maybe that's the secret to this. Now this is absolutely fascinating. In this richer notion of algebra, you might achieve this. Is this additional enrichment of the notion of algebra, do you think, connected in any way with the point that you made the other day, that one sees algebraic? ...operations as one side of, again, I don't want to use the term duality, but the guiding idea that it's figures and their incidence relations on one side, which are really the generators of the correct notion of defining notions of space in general, and one sees the algebraic aspect, the kind of algebraic operations as a kind of shadow of that, as a dual of those operations, and this is a further illustration of that principle. Incident relations are algebraic operations? Yes, yes, exactly. In the epsilon of Peano and Frege, we have the perfectly natural dual concept of determined by. So one random variable is determined by another one. If there exists a process for transforming one into the other, well that process is exactly the dual notion of mince-mince relation that's taking place.

17:30 Exactly. This is the relation between the figures. Yes, yes. Yeah, the dual to that. That's how refined it is, you see. Well, at a certain level you can talk about figures and incidence relations when the figures have arbitrary objects as shapes and functions where the types of functions are arbitrary, you see. But I'm thinking you're having a limited notion of shape, a limited notion of type of function. And what I seem to be saying is that if I, so for example, if you have a reasonable notion of shape of figures, i.e. a site for a topos, then you might say, well, I'll take as types of functions the same objects. And indeed those are interesting functions, the functions most people are worried about, but it leaves out things like truth values, which are also functions for a larger choice of other categories. But would you still fit into this way of thinking of domains of variation for quantity? That one doesn't have to think of, I mean the very point you made about precisely in terms of this notion of figure and incidence relation is generating the notion of an Atiyah. You don't have to take the logician's notion of the value of a variable as involving commitment to some all-encompassing ontology given by set theory or by some other weird... In other words, it's the origin of Frege's kind of absolutely disastrous, it's always seemed to me, disastrous notion of object, and his refusal, which I think came largely from his neglect of co-domains of maths, to... To see that, you know, the correct analysis of the structural domain of variation for quantity is precisely one that does involve these Lachis homomorphisms from parts of the quantity into parts of the domain. But you see, even in a topos, even you could think of three levels in a topos.

20:00 Never mind what we choose as figures, you could think, well, the obvious kind of functions are truth functions, so you can take omega as a type and omega squared, blah, blah, and you'll have a contravariant representation of the objects as the opposite of the category of hiding algorithms, which is never full, so in other words, this truth is never co-adequate. Sometimes it's faithful. In fact, it was Francie Bourceau who many, many years ago first pointed out that, well, for some toposes, this is actually faithful, although it can never be full. For example, for graphs, it just means that whereas in any arbitrary elementary topos, there's the basic result that... Or say you have a family that shapes A, then omega to the power A in the sense of function space, these are not co-adequate but co-generators in the sense that it's faithful. But the stronger thing is that not omega to the A, but omega to the absolute value A. In other words, the actual just discrete product of copies of omega is already... Co-generating, so that's a special property. Some go deep, some don't, but you'll never arrive at co-adequacy this way. Now, Isabel's observation was, and he proved it, it wasn't just for sets, you see. The basic thing was for sets, countable subsets, countable sets are co-adequate if and only if there are no rule-out numbers. But he immediately, in another paper, extended that to arbitrary. Basically, any category that has a small adequate subcategory will also have a small co-adequate one if there are no Ulam numbers. There's a direct construction of just lifting the... So it means that if you consider, let's say if you consider... Countable products of omega instead of just finite functions. Then, oddly enough, this would actually be co-adequate for sets or it would be a cohomology. It would be an adequate notion. This would be the enlargement of the notion of function from just plain old propositional function or pair of propositional functions into sequence of propositional functions.

22:30 There are enough sequences of propositional functions not only to distinguish between figures. But to recapture figures, if you insist, on naturality in comparing with respect to algebraic operations, which, notice, are now infinitary operations, omega to the n into omega to the n, arbitrary. Of course, again, if you analyze it more closely, you don't need to take all these operations, but at least they have to be infinitary in some of them. So from kind of a set theoretic point of view, this is what I'm talking about. But it should have also, from a point of view of functional analysis, a similar interpretation. You just have to enlarge the notion of... See, I mean, it's implicit even in ordinary analysis. If we think the basic notion of function is real value function, but trivially we therefore know what is a triple, what is a map into our cube, but even also a map into the circle, because the circle is part of R squared defined by an equation, so a map that satisfies that equation will be, so you have a large number of types of functions implicit even if you didn't make them explicit. But then there are others, going to be others, but even take that, you see, if you consider the line and the circle, including geometry, you consider those alone, let's say, as function types. In algebra, by algebra, it's now a two-sorted algebra. Functions on any space are a two-sorted algebra. So like the exponential map is one of the crucial operations, as well as the other obvious ones. Well, in algebra, on a multi-sorted theory, it's still just a product-preserving function. So the mere fact that you consider an arbitrary algebra on that two-sorted theory...

25:00 It won't necessarily mean that the circle-valued part of it is the equalizer of the plane-valued part. No, I see. This will be true for the functions on an actual space, but now you need to know a little bit about the arbitrary algebra in order to go back, because there's a little gap there, because you have products as opposed to products and equalizers, and the doctrine of them. Yeah, again, as you say. By the way, we're at the station now. This is where we were aiming for. This is where we get the train. In fact, the whole question of when pullbacks preserve the maps is quite an interesting one because it introduces the whole. The issue of Etendu and their characteristics. Something else I wanted to ask you about, about this two topological character. But let me just focus for a moment on the timetable and see when the next move on to Mechelen is. Actually, I suspect we don't see... No, no, because Mechelen's not important enough to have a separate destination. Lots of these trains will stop at Mechelen. We just have to find out which one. Ah, well, it's OK. There's a travel centre over there. I'll get the information from there. I should be interested in your reaction, but I don't do it. Don't hold your... Well, she has got this weird theory that... Oh, yes. Oh, yes, yes, yes. Sure, sure, sure. Yes, that's always about... I don't want to apply to Karen's solution to Zeno's paradox, but no, she's got this rather strange view that makes some sense to me. She thinks that Aristotle's discussion of the infinite, his distinction between the potential and the actual infinite, actually derives from Zeno. And that Zeno already recognized this distinction between the two kinds of infinity. The paradox had absolutely nothing to do with any kind of constructive procedure there was it was entirely kind of static notion that the line was already in some way divided and contained the dyad and therefore the principle potentially of infinite division within it but she also has this argument which I must say I don't think stands up to serious scrutiny that

27:30 That Zeno must have understood the notion of cardinal number in something like the same way that Cantor did. Obviously they understood the notion of incommensurable magnitudes. That's the whole reason the Pythagoras' program broke down. They might have extended the notion of incommensurability into trans-finite constructions, although they wouldn't of course have thought of them in terms of the notion of the apiron. D-Limited is yet somehow internally founded in terms of the potential infinite that you can reach through division, but I'm a little bit suspicious of her analysis for several reasons, one is I don't tend to know about the philology and the... The textual scholarship, she does know a lot about that, but no, no, no, she said not till 11, 9 past 11, so we still have about 20 minutes. It should actually say up there on the... There is actually a timetable, I'll check it out, just to be on the safe side. Thank you. But she also, which I think is very dangerous, I'll tell you where she's got some of her ideas from. First of all, her supervisor was Bob Cook, who persuaded her that she should read, which I think was good advice, read some of Scott and Foreman's papers where she started to learn about domain theory. But the other person who's exercised, I'm sorry to say, a considerable influence on her is Graham Priest.

30:00 It's well disposed of the idea of inconsistent logic, and I'm pretty convinced that that is a nonsense, and for various reasons to do with the nature of their awareness and the fact it was an oral culture, they didn't have hang-ups about consistency. Well, I think this is just nonsense on stilts. Complete nonsense on stills. Oh, Granbury's whole thing is absolutely pernicious nonsense. And the other thing which I was very shocked to find last night, I shouldn't, I mean, I do like her, she's a very spirited girl. She's a spirited girl with a big, large personality and lots of, you know, quite fun to be around. But she does believe some, sure, she has been led to believe some really very ridiculous things. One of the, first of all, I was shocked to discover... You get last night that not only is she a big, not only is she a big picker, Dillema Randall a fan, Oh really? Oh yes, and you're all completely confused about that. The project to restore the Platonic Academy, they know Platonic Academy, that wasn't a reactionary project at all, that was nothing to do with buttressing the power of the Medici, that was a terribly progressive project, because it was to do with undermining Aristotelianism. Generally, it's loosening up the mindset of the culture. People like Kepler would never have been guided to the deep mathematical insights that they had into nature if it hadn't been for these developments. And the other thing, which is much more horrific, is that she's recently... Well, even if you were a fluent Flemish speaker, I don't think they would have understood a word of that. They wouldn't have understood why they bother to make these announcements on stage, because nobody can ever hear what they're saying. Here we are, here we are... 11. Yes, there it is. It must be this Nivelle train. Except that's number 4.

32:30 No more, do none. Yes, but don't start stopping there. It could well be it. It probably is it. Correct. But that goes from four, not from three, so I'm thinking that... Printed timetable.

35:00 That was absolutely fascinating, what you were just telling me. No, the other thing which I came as a shock was that she is... This is a recent development. I mean, this is certainly not something which she was positioned, which she was attracted to at all when I spoke to her a year or so ago, had the last serious discussion with her about philosophy, is that she's now become a great fan of Heidegger. Very worrying development. She wants to wind up with no-photon infinitesimals in this paper. That's what she wants to wind up with. From what she told me, I didn't know that, but in the paper this is clear. But then she says that these were introduced by John Bell. She first heard about them from John Bell. I think that's her understanding of the notion of introduced by. That's the problem. My God, she not only hasn't read, but she doesn't even know they exist. Well, she does, because I've told her about it for at least 20 times for the last five years. Well, she does, if she's been listening to anything at all that I've said to her in three years. Well, she doesn't listen. That's a problem with her. I mean, John Bell, bless his soul, I love his book. Yeah, so do I. It's an excellent little primer. It's just to say that he introduced them to Karen. That's what it amounts to, exactly. She being the primary information repository of all knowledge, especially on the subject of the pre-Socratics. So I thought, well, the country where he is must be, he must be having a real effect on students somewhere. Sure, sure, because his popular science books, popular anti-pseudo-science books, I should say, were immensely influential. And I still come across many, many people who were seduced into... I suspect that she may have this from Kirke, right? I mean, Kirke... Cooker? I don't know. I really don't know about that. She may have come to Prigogine quite independently. I've never heard Cooker talk about Prigogine and Heidegger and Picot. It's very disturbing.

37:30 There's not much hope. Well, no, I agree. It's not looking too good. The only thing you can do is overthrow the whole ensemble, or it's hopeless. Well, I'm afraid that's what I tried to do last night, but it all ended in tears with her kind of marching off in a high dungeon. Really? Well, I didn't see you bringing her to that point. I'm sorry. I feel a bit bad about it because I should probably have exercised a little bit more patience and less aggression, but I'm afraid when I got, I didn't get Prigogine, but I got Heidegger, Picard, Marandola and various defensive Nazis as being... You shouldn't really judge them until you've actually sort of entered into their souls and been part of them and, you know, you're just, you know, you can only judge their actions as I'm afraid it became very sarcastic and said, oh, yeah, you know, I'm sure Himmler had many really nice facets to his character. And we all know that he looked after his dear old mother and, you know, always intended for them gasoline every time he took her for a drive. I mean, Jesus, you've got a guy, you've got to judge these guys and the whole kind of just pick on little things like they murdered, you know, millions of human beings. I mean, and she said something about, oh, but that was only one incident in Heidegger's life. I pointed out, you know, Heidegger's dismissal of Zamella for refusing to give the Hitler salute. Yes, I like his talk. He had this, he had this. Thank you for watching. Between him and Rousseau at the beginning, I mean, it was in English, but I didn't understand the context. By chance, I saw both through this morning and I got a better idea of it. Namely, I must say I have mixed feelings about this. There's something analogous to this incident eight years ago. Cartier came here, came to Irvine. Oh, that's right, too. As the IHS, or the International Committee for Judging Program.

40:00 Yes, that's right, yes, yes, yes. And the conclusion had been that there wasn't sufficient international, and I had been conveniently invited by Berceau. Cartier had read this, and then looked up and saw me and said, The country was momentarily defeated, but by the fact that I was there, I apologized to him after many years later. It's entirely possible that's why I was invited since they weren't discussing it. I'm not saying Cartier himself could also be infected by this French disease in this day and age. Well, I think he's much less affected by it than any other French mathematician I've come across. There may be less so after that. It's also possible. Yes, could be. He's become much more... Certainly when I was invited, even by him, he never even spoke to me. Well, I think, partly thanks, I hope, to some extent, to Fougere. He certainly made an impression on him. I had this sharing the dormitory for the sake of mathematics. That's quite right, too. It's very well planned. Anyway, so that end goes to something totally analogous to Dutch. I remember him saying at the beginning of his talk about this, I didn't quite understand what was going on, except that they have some sort of commission that sits to decide the quality of research, and it's once or twice every ten years, and they have to be Flemish speakers, which of course rather narrows the possible range of participation, although there can be some purely French speakers, but they're invited as honorary, but they have to be, and the trouble is that... He was saying that they tend to be elderly Dutch mathematicians and half of them are constructivists and have absolutely no understanding of what's going on in areas like sheaf theory or anything. Well, in any case, you see, the point is, okay, that's what Sir explained it. As he explained it, of course, your friend Freddie is not in this. The group that was being almost condemned. It was categorical topology. So, which is, you know, vestige of this German school. By now, hardly exist, but which includes many of these people.

42:30 And of course, Bursu explained it that way, that they don't like, these Dutch don't like categories, and they don't like topology, so they like category topology less. And so he himself, as a Belgian, although... He moved in as it were to compo them, to provide compo to them. He came into this and argued eloquently before they... You are the commissioner that they have the right to love anybody they want. It just occurred to me. Well, I think in this context... I can't imagine what else you could have been. That's my point. Well, as you said yourself, the French committee was almost right, actually, and this one is probably totally right, and yet they were reversed, coming from one of the clever organizational villains, Franci, who knows who he called in as the... As the laurvier of the second, you know, the second instance, yes, to fill your role in the script of the second time around. So I'm glad I met him this morning because I hadn't understood at all that that was the... In the context of the earlier remarks, when Canigo was thanking Franski, of course, but on the other hand, of course, Canigo clearly doesn't belong to that group as such because, I don't think so, it seems like the category of topology was dead from the beginning. I think that categorical universal algebra is due to Le Verre and followers, but that was plagued by a long list of actions and incomprehensibility and unfortunately there is something, finally, finally there is a reasonable categorical universal algebra which is due to Bern. So I thought this was some kind of leftover hostility toward myself from Bell and Johns, but at least I understand now what he means, because he's not talking about universal algebra, he's talking about provincial universal algebra, and for that, you see, in other words, it's a review so the sentences are wrapped together, it's not really about me and my gang at all, it's about the people who try to capture, generalize the notion. In other words, a special class within the universe of algebra in which we have something like boot-crawl properties, which I think is a stupid idea, but it doesn't serve any purpose.

45:00 It doesn't serve any purpose. Anyway, so the attempts to do that in the 50s are by algebraic apologists. So the attempts at some kind of universal algebra in that direction indeed did have long lists of axioms in the names of insecure relationships. So according to the outline given by, so, by Danilitsa, you see he had the two blackboards. Yes, that's right. Old and the new. This guy was talking about, no, it was certainly a copy, that's what was in the text, at least as he wrote it, but he had divided it up into three parts, which could be, and indeed, Boren's actions then are a little more streamlined somehow. Well, the protomodularities. Protomodularities, and yet equivalent to the same. So, in that sense, there is a progress. Within that field, it has no reason to exist. As I say, within the idea that you put everything inside some kind of group, groupification. Yeah, yeah, Thelma gives an expression to this idea of group life and the rest of all the rest of it. But you see, it has no known application. As this was clear to me, unfortunately, you've seen this interview with me and Coimbra. Oh yes, of course, you sent it to me. You've seen it? Yeah, you sent it to me, yes, yes, I've read it very carefully, both halves of it. I sent you both? Yes, oh yes, absolutely, yes, yes, you did. Anyway, it happens that in that same, there's an article... This lecture is brought to you by Clementino, where she is extolling the virtues of the Bourne-Dirac versions about homological construction, chain complexes, homology, exact, strong exact sequence, just so you can see. The motivations, so-called motivations, in this field seem to be almost always, well, the real mathematicians did something like that. We need to be able to display a kind of imitation of that in order to justify ourselves.

47:30 Not that we'll ever go back and apply our imitation to an actual case, but at least we can. So the induced long exact sequence in homology, completely abstract words, what does that mean? Well, we've got something like that. Now, now, now, now, I can immediately imagine... Applications for something like this, for example, in the talk with Giselle Chargois, Shapira, these people are doing partial differential equations in which they construct same complexes of differential operators. Something like a long exact sequence would be useful. But, if they went and seriously tried to apply Ben and Joanna's paper, that is, Boone, Genoisa, and so forth, I don't think it would work because they need additional hypotheses. So, in any case, these analysts know perfectly well they need additional hypotheses because of the failure of normal, ordinary exactness when topology, when cohesion is present, we have to add additional... I think that these people are not putting it in terms of additional cohesion, they're saying, well, in our sort of semi-abelian, maybe call it that, semi-abelian... Anyway. Well. Now, allegedly, subsumed in homological, whatever that means. Yeah, that's again, the thing is they have this abstract homological theory, but to get the long exact sequence, they need an additional hypothesis, even there, on the chain complex. So whether this additional hypothesis is interpretable in ordinary analysis or not is another matter to be considered. The worst one, okay, I've noticed that there is a mood of mathematical reason, which is not fully mathematical. I isolated this.

50:00 Thank you for your attention. Well, it all goes together, doesn't it? But anyway, it was he and Sheldrake who wrote some books about, you know, they have horses in California. The yam-yam tree, the bang-yam tree, you know, and stare at the navel of the universe, or rather Gaia, yes. Or Ralph Abraham. I knew his brother, his brother was a psychologist, he wants another story, but then I met him as well, he was crazy, and chaos, he was one of those. Yes, I know, all those people who preached, you know, to chaos theory, I was one of the first really extreme examples, I mean, much more extreme than similar things which had gone before it, of just complete pseudoscience being invoked. They require a very superficial level of understanding and they were no longer even interested in trying to bamboozle people. They just came out with what anybody could see was complete fiction. And just resting on appeals to authority. You were saying that the two-year-olds of today... Before I got the interruption from the two-year-olds. Oh, before that, you were talking about the program of Van Poser and of... Well, there are many schools in Berlin, but you see, what they call graph transformation, they try to... I learned a little more, somewhat categorically. But the point is, I noticed that the way they reasoned, you see, that there would be certain actions, certain constructions and calculations, and then at a certain stage, they would say, and then they'd go find a source, and go on. It was a process, it was like categorification.

52:30 We find this stage, and then we go on, but how it came about, what it's got to do, I don't know, so it's kind of a discontinuity and a change, an ordinary reason of construction and proof. Yes. It's what I call the, and then a little, and then a miracle occurs. These are all presented as a mathematical theory, which you could easily make it into by taking that thing out and putting it in front as an apostrophe, you know, it's not that it's before the way of contact, but simply because you sort of, it's like the propagation of the philosophical concept of the universe theory, where the theorem has an incredibly important spot, so you just pretend that the hypothesis is automatic. And then you can use it as a prop for objective idealism. Exactly. I only thought of that as an analogy, you see, because... Also, of course, notably, you completely ignore the real algebraic content of the theorem, which is a fixed point theorem. That's the one thing that all these mystery-mongers, you know, people who dance around with dithyrambic hymns around the figure of Gödel, never mention, that it's a fixed point theorem. They're just coming round for the tickets, you should be able to get them, please. Oh, that was another thing, actually that was one of the reasons that we ended up having a real blow-up, which I really must bring up and apologise to her because I was a bit aggressive. Again, Karen was coming out with this very anti-materialist diatribe, saying that, of course, the ideas of these pre-Socratics, the first principle, had absolutely nothing to do with materialism.

55:00 They were completely different. But they were, again, it's all to do with this. It's so completely the ultra-left is right in essence, you know, syndrome, that it was all to do with this business about what she called deictic utterance, you know, because they were in an oral culture and they didn't have written language, subjective experience was somehow different, and it was terribly important to be able to point to things. Now, so far... I'm so confused and I have no idea what she's saying, but somehow this explains the way in which the pre-Socratics, all of them apparently, even though they're clearly extremely different from each other, and also they could write indeed, just unfortunately we don't have much of their take. Matter of inconvenience, they all in fact wrote, except we don't have much of their take. For this reason, their conception of the arche or the first principle was something which we wouldn't recognize post. Plato and Aristotle as a first principle, as a kind of candidate for first being or some kind of limiting, source of limiting structure on whatever it is in any way. In other words, it's not the kind of thing which would be doing as a category, I mean in the loose. Metaphysician sense of character, the kind of job that the notion of matter in motion or any other fundamental first attempt to characterize that being as such, being and becoming, being in relation to becoming, would be doing for us or for any Western philosophers since, say, Aristotle, it's something quite different, which is connected with the fact that they were still an oral culture. I was just saying, well, this is all nonsense on skills. It's what they have got here is... There's an anticipation of materialism, a materialist understanding of reality. That's precisely what makes them so interesting. Oh, no, that's terrible, that's nonsense. No, no, it was all about pointing at things. So I said, excuse me, but how would Anaximenes or Anaximander point at the apiron, you know? How is it still in order? Yes, well, and besides, how do you point to the apiron? And how do you point, for that matter, to the... How do you how do you point to Heraclitus's everlasting fire. Obviously, he's using that as a metaphor, but I mean, it's just crazy, you know, the idea that...

57:30 No, I couldn't understand what she was saying, but it all sounded very subjective, idealist, and very... Subjective, idealist, and rather relative. Yeah, exactly. Whilst also claiming that, you know, it's all a great liberation. But I mean, I got a strong whiff of people like Marcuse and other, you know, other purveyors of this kind of ultra-left. Because she is an ultra-leftist, she does, of course, occasionally say things which are entirely correctly justified, such as the nature of what happened in Baghdad after the American occupation, the deliberate policy of destroying the National Library and raping the National Museums, and, you know, there are things just as a matter of concrete historical record that I can completely agree with her about. But if you say anything, even to try and set this in the context... We would actually try to provide some understanding of the broader agenda of imperialism, rather than just this subjective, oh, they're demons, they're just completely insane, mad demons who want to destroy everything. Just, well, no, actually, they are. A class with class interests and a system of power which provides them with an agenda but then you know if you try to provide even that much kind of serious historical context for understanding what is happening you're immediately accused as Colleen was accused by her a couple of years ago of being an apologist for American imperialism and I'm accused of being an apologist for British imperialism because I say that the British ruling class was actually Actually, despite its, in fact it was decided, acting against its own interest in the longer term, actually had no serious alternative. Oh great, that's a good idea. Party of the Italian Communists. Ah, oh so it's a much older... So even among the Maoist Party, there were two Maoists. There were two, I see, I see. There were two who took this, took over this distinction and took it seriously, yes. I was unfortunately for a time mixed up with the other one, which was more thoroughly Trotskyite in some way, which I appreciate.

1:00:00 Born's work and that of the so-called homological categories. No, not a group setting. The groups have this causal property that quotients are described by also by subgroups, namely normal subgroups, which are the kernels of normal morphisms. Well, that's nice. It's also a Maltsev category, so that means that they're a kind of movable difference operation. And indeed this does tend to give the category algebras of exactness properties. But now the idea is that you should try to force these exactness properties on some other kind of category. It doesn't naturally have it. For example, the key example, the reason I brought up the Berlin computer scientist, there's nothing to do with it except for this mode of reasoning that in the chain of... Yes, there was a little miracle happened. A certain little statement which carries the argument forward but doesn't really... A little statement is this. Let's assume for convenience that it's a pointed category. I think all of them said that. So the hint is, well, we do have the theory of non-pointed categories in our back pocket, but let's, because it'll be easier, let's talk about the pointed case. You see, the pointed case is to the right. Groups happen to be a pointed category. Because there's a one-point group that's contained in a terminal object, an initial object, and a category are the same. That's what makes a point. And so, you do have this idea of a kernel. You can then clearly define something called the kernel of a map, namely the inverse image of the trivial subalgebra, which consists only of a point. There is something about the kernel that doesn't mean anything. Well, for pointed sets, for example, it obviously doesn't mean anything.

1:02:30 It doesn't permit you to reconstruct anything just knowing that. But the trick is, they talk about groups and rings. So they're smuggling, left for convenience, let's assume it's a pointed category. So they are smuggling in this old idea. That rings don't have to have a unit which of course was the analogy he was coming out with precisely in his talk yeah in response to my certain demands that they should have a unit for them to say something about the case of the category of ordinary category of commutative rings yes which algebraic geometry is totally based on it's not based on commutative rings where the maps don't present one or the rings don't even have one you see we need the consequent I mean every category should at least understand that much Even if they do have ones, and they're not required to be preserved in a uniquely pointed category, maps have to be preserved, you have the field of constants, that you can have points of a space, these values give you field values and all this, that is excluded by this innocent remark, to make things easier let's assume the category is pointed. And so all of their paddling about exactness properties and the neat comparison of different lists of axioms with some class of algebraic theories that they doesn't in any sense touch the example of ordinary basic, the most basic example beyond. Groups of any algebraic theory is, of course, commutative rings with units. Yes, yes. Which, as you say, the whole of algebraic geometry rests on. Yes, algebraic geometry and linear algebra and so forth and so on. But even non-commutative rings have numbers. But they deny that, too, and so forth. And so there's an elaborate side operation that Boursier and some of his students went in saying, oh, well, maybe after all, weight of the unit are a good thing. So let's have even categories without identities. So there's a whole garbage thing that you see about this. When they talk about multiplicative graphs, that's nothing but a category without identity maps.

1:05:00 But again, it's a sort of natural move that the ultra-left come up against all the time, is to deny the fact that... Because we must get rid of status, comrade. Let's get rid of that, yes. Yeah, sure. Because all this business about units and one is terribly... It's freezing everything, which should of course always... Everything is fluid. Everything is fluid. Everything is fluid. Process is the only fundamental reality. But there's no dialectic. By the way, also, with the idea of the study of monads and comonads... That's why it appears that it's clear there that the unit is just as important as the multiplication, even though in a particular case it looks like a trivial thing alongside the multiplication. In the formal calculation, it plays a completely dual role, rather than a subsidiary one. But they go back to the philosophy of the subsidiary and then further back to the philosophy of we should leave it out in order that we can include ourselves in this sub-world which is, quote, group-like. So it's largely square. So I, you know, pointed this out. See, they never tell you this. They say, this is what I mean by commuted reasoning. They say, let us for convenience assume. But they never actually point out that their theorems only apply in the case. Yes, yes, yes. So it's a good deal more than convenience at stake, it's actually, you know. So, I mean, certainly the general project is going beyond bar exactness. In other words, finding special, further exactness properties in special kinds of other great categories. Yeah, of course. And so I thought, well, that's what they claim they're doing. It should, sure as hell, apply to community of ranks. And so I told Aurelio that, and so he and General Issa came up with this, which was no answer at all. They think there was no answer there. There was a diversion in fact. They say, well, we could measure the deviation from the median by this thing and we could call it the tangent. What the hell's that got to do with it? I wanted to know, I wanted to know what are, in fact, the unique exact properties of the category of community graph. So even this discussion is saying, well, we could approach, we could construct that category.

1:07:30 If we could first construct the category of rings without them, and then do this, this, and this, then we could sort of construct it, but that doesn't show what properties it has. As far as I can follow it, this seems to be very much the same kind of project that Van Oystone's non-committal anthropology is all about, taking precisely that kind of... Partial determination of the great and deep topology and then saying oh but we can now generalize this then we can providing all that's to the non-commutative case and then we've got something which allows well in his case he's got some he has got some weird philosophical motivation for it which as far as I can make it out is some kind of Heraclitian all this flux. And so he's very big on this as the explanation of non-locality and quantum theory and many other things besides. But he's a very interesting guy, though, and I think you would find, apart from Engels, I think you might be able to administer some much-needed correction to him. He's a very fine algebrist, and as I say, he's done some good work. But there seems to be a lot of this in the air in Belgium. I don't know. At least it seems to be a slightly different wavelength from... This is not the first time they talk about these commutator operations, which the universal algebraists, so-called, have worked on. It's one of their only contributions, really, that there is this notion of commutator, because I insisted. Maria Cristina Pedicchio worked out at least one kind of categorical formulation, and then these people have learned that and probably simplified it too, and so they use this freely, this community. So that's it. But again, you see, again, they never talk about the real mathematics. You see, the whole business in algebraic geometry, intersection theory. The fact that when Africa intersects Europe, you get the Alps, that's my example. That's a good one. Plate tectonics is actually quite a good example of intersection. No, no, but you see, even if, that somehow this is not so, the mere fact of intersecting produces something a little bit more than your...

1:10:00 Contact, yeah, yeah, yeah. The vast part of algebraic geometry is about calculating things that measure that sort of phenomenon. Now, in algebraic terms, it comes from the fact that in the lattice of ideals of a community of rank, you have the product of two ideals, which is an important operation, and of course you have the intersection and the... The soup of two ideals is meant to give it a lattice structure, but it has also this multiplication, which is not a soup or an end. It's another operation. The fact is, it is the same thing as this so-called commutative case of the category of community of rings. And therefore... All sorts of benefits for algebraic geometry, intersection theory in particular, and arithmetic, because all of classical arithmetic and number theory is based on this multiplication of ideals, which is not itself, it's not a categorical operation. It's not a suple or an ish, but it is a categorical operation. After all, somehow... By a suitable, not completely trivial combination of co-limits and limits, which are categorical. You get, you can get this other, this other operation. And that's what Maria Cristina showed us, and other people have shown us. Marino Grand, for example, knows all about this. I still don't understand it though, because they've got all these upscranted ways of formulating it. But in any case, there is no doubt this concept, but they refuse to apply it. Marino Grand wanted to write a chapter for a book, which is now coming out. I said, well, certainly about commutators. So I said, certainly an important part of this will be a paragraph or a section which explains the connection with algebraic, with arithmetic, algebraic geometry, ring theory and so forth. And he never did it. He said, well, I don't know about it. In other words, he has this abstract theory so well in hand and yet not well enough that it can be applied to the most obvious case. The disaster situation. So not only intersection theory, of course, the word commutator itself, which is due to universal algebra, it's not to these people, that word itself is an incredible misleading thing because in the group case, that's what it is, if you have two normal subgroups, then the commutator, you know, you take A, B inverse, no, A.

1:12:30 A, B inverse, A, B, this type of thing where A comes from one normal subgroup and B comes from the other normal subgroup and you generate a new group out of that. So it involves these commutator symbols. Therefore this is called the commutator and justly so in the group case. However, there's nothing to do really with commuting in general because it is just the product of the ideal in the case of the ideal. Yes, yes, yes, I see. Somehow they wanted to just formally, the old algebra, they wanted to somehow say that ideals were also rings, which they're not. They're the kernels of your homomorphisms, they're modules, but they are not themselves, because they don't contain the unit, at least, ever. Hence, of course, your insistence on the rig. That's the right basis for formulating. Yeah, the rig should have one, even if they don't have negation. Exactly, even if they don't have negation, they should still have the unit, yeah. Yeah, yeah. That's right, that's right. So, in other words, both in attempts to apply this theory, both by the connotators to intersection theory and arithmetic algebra and intonation, as well as in case just of characterizing what are the exactness properties of community events, which of course are needed. It would be needed to make an explicit application of that construction to see why is it that this particular slightly peculiar combination of limits and co-limits happens to come out with this enormous significance. Yeah, clearly enormous significance for the commutative subgroup case, which... But it has the enormous, it turns out, since it is the product of ideals, when I say is, I'm talking about, I mean, basically, given A and two surjective maps from A to B and B, B1 and B2, say. Then, besides the obvious push-out, besides the slightly less obvious but universal thing going up against it, this third thing, it ain't the Tercevia, no, it's not Tony Blair, no, no, no, no, no, not the third way, no, no, no, no, in terms of elements, it's the ordinary product of ideals, small sums of products, I think, it's the thing.

1:15:00 You see, for the nilpotency is that the nilpotent ideal is one whose square is zero, in this sense, so that's in some sense why this comes into intersection, I mean, when you intersect Africa and Europe, the nilpotents pop up, you see, some extra motion there. Exactly, some extra motion, just as in the case of nilpotency infinitesimals, of course. The sort of contrast between the naive intersection or the naive direct sum of ideals, between that and the product, is the difference between the sort of very flat interpretation of that intersection and the one where you have given some voice to the alps. Yeah, yeah. I don't mean... No, no, no, not yodeling, but just... Not gargling, but yodeling. No, but just to the, as you say, to the trace of motion that's involved, to the trace of motion that this product contains, or records. The product of ideals can be zero, even though they're not zero. It can still, as it were, register this trace of motion. Yeah, well, and one sees this obviously particularly in the case of, well, synthetic differential geometry was to a large extent. ...constructed to clarify this and express it. There's another thing I wanted to... Sir, there are three things... Sorry, go on. There is the rhetoric for simplicity to just consider the point of case. Yeah. There are several publications trying to show that, well, after all, things without unit are important, too. And that says, well, I don't know about this. And so this is, you know, this whole theory doesn't apply. You want to accept groups. Except the group case, yeah. In other words, the group case. And certain small modifications of it, namely groups with given automorphisms and stuff like this, is the only real application, so if you want, you know, that's a serious, but this commutator theory does apply more generally, and there's more kinds of algebraic theories that this commutator applies, so.

1:17:30 What is needed is really a different study of special exactness properties of certain algebraic categories, special exactness properties which will be like community of rings, with unit not pointed, very inconvenient for you but not for me, instead of groups. You see, even groups, of course. Again, even the simplest thing, from a categorical point of view, and also known long before category theory, is that really groups should be replaced by groupoids. And as soon as you do that, you lose this uniqueness of the point because it's the objects in the groupoid that are, you know, a group is a groupoid with one point. That's why the category of groups is pointed. So even there, in a way, the real content of group theory is being obscured by focusing on this more or less accidental property of the restriction of the category. The crucial point is pi zero. You just have to mention that in the category of all groupoids there is also a separation of pi zero and in terms of that you can express exactness properties with the same content, the same force, but without restricting to the case of... So, anyway, this tells you, this gives you some hint of some of the things I was thinking into my talk. Yeah. Pi zero is very important. And also, in fact, into your... Rings with functions of compact support. Compact support, yeah. You could even have one, Francie. You notice now, rubbing it into his face. I hope. I think one or two of them might have got that. One or two of them might notice it. One or two might notice it. I really would like to. Actually, if you want to give it to me now, so that way I won't forget to ask for it when we get back.

1:20:00 Because then I can take my haranguing to the food. Yes, give it my chance to look at it. There were also a couple of things I wanted to ask you about your Calais talk, which I, of course, haven't had a chance to look at properly since then. I've got that here, too. Ah, yeah, okay. Well, that would be good to have another copy of that if you've got a spare one, although I do have one. You don't have a spare one. Oh, no, it's okay. I have a copy of that anyway. I have it at home that's fine. And again I wanted to ask you a little bit more about this whole issue about the three distinct ways that you explained to me in Calais, well you explained two of them, I think the third one we got distracted by some other development, in which one can see set theory as fitting into place as a fragment of algebraic geometry. One you explained to me in Calais. Some detail was, was obviously, I wonder if I could, yes, why don't I get some more water while you go to the loo? I'll do that. Just hot water? Hot water only? Just hot water? Yeah, that'd be great, thanks. That'd be great, actually. Can I get some as well? I'm ready for anything. And if I seem particularly sluggish, and more so than usual even today, then I'm afraid I have to blame my weakness of will in allowing Karen to press at least four beers too many on me last night. I was going to ask you a little bit more about the discussion we had in Calais about the manner in which one should consider... Set theory, here obviously one's thinking of sets in the setting of. Cantor is seen in the light of Galois, the Galois construction, as fitting into algebraic geometry, and you mentioned, I seem to recall, that there were at least three distinct aspects to this, one of which was seeing the case sets as effectively the booleanization of various constructions in algebraic geometry, the other of which was, which is not unconnected, was to do with the...

1:22:30 The way that they fit into the treatment of varieties, the whole treatment of the notion of algebraic variety is a kind of special case. And then the third one was the one which unfortunately we didn't really, for what reason I can't remember, didn't actually get round to developing. And I was wondering if there was any chance you could run through really all of these aspects together because I was really getting some very, I thought for the first time, some really good understanding at least of the scenery, of the landscape. What was the third one? That's what I can't remember, because we didn't get around to it. I can't remember the name of the second one. No, no, I can't even, I'm afraid. I think yesterday, I think yesterday I could have done, but right now, that's why I'm struggling to remember. I don't know what that means either. Okay, alright. Well, let's forget what it was. I was struggling to remember and just you tell me what you think I should know about the whole issue of this conception of set theory as a, in the setting of, as a kind of, in some sense a fragment of. I have a feeling that I must have reached at that time the height of understanding it that I don't have at the moment. Okay. Well, in that case, we'll postpone it for another occasion. But that is certainly a phrase that you have used, and not just to me, but in many lectures. So it's more than one lecture. So I don't think I'm misremembering the slogan, the guiding idea that one should reconceive set theory and... As I say, as naturally fitting into place within the 17th Algebraic Geometry. That I'm quite clear. But yes, I'm afraid I can't now remember the details of the... I think I'll probably remember some of the details of the Booleanization. Obviously related to some of the things that you said in your talk in Calais about rigs, about extensivity and the role of rigs in algebraic geometry.

1:25:00 In fact, amongst this now pretty substantial archive of recordings of you and your lectures and discussions, the first one I have, of course, is Cambridge 1989, when you were telling us all about rigs, about the Burnside rig and its centrality. Already there. We're quite remiss on that. Objective number theory. Well, actually, in all seriousness, I think one useful thing I could do is if we could complete the catalogue of these archive recordings and then go through them with you, obviously, listening to extracts and then telling me which you think are the ones which would be most valuable to have a complete transcript of, it might serve as quite a useful... Do you have a name for Steve Shaniel? Unfortunately, no. The only thing I have of Steve Shaniel is when you and he were both in Cuomo in 1990. I think it was 1990. Yes, it was. Yes, it was 1990. And some of the discussions that we had there with you, him, and Alberto, and I have maybe took a couple of hours. Oh, so is that where he gave his early conclusion? Yes, I have, I have, yeah, I have his talk. Oh, I have the recordings of all the talks at... Well, hang on, I must be careful, probably not all the talks, because there were some parallel sessions, but I certainly have the recording of Steve's talk at the Como, and yours, of course. And actually also, do you remember there was an open forum, a discussion on the last day, it was Sammy Eilemberg. That's right, that's right, that's right. No, no, no, that was extraordinary, of course that was a really visionary glimpse. Where Freud conjectured that the converse in my theorem was true and it was later proved by a student of Ray's. And Freud, of course, played quite a major role in those discussions in the... Do you have any recordings of that? Of the discussion in the open forum at Como, yes, I do. I was particularly interested in that because it was the last time I think Sammy came to a category theory meeting. The last one he attended before he had his stroke. I think he had his stroke in 1990. Yeah, but for some reason, I don't think he, for whatever reason, I don't think he came to another technical category. We expected him in big time for a tourist meeting, which was in 1994. But that was when he found out that it wasn't...

1:27:30 It wasn't Abel any longer to travel. But whether this was the last meeting he attended, I have the impression it was, it was certainly, he was certainly firing on all cylinders, he made a very powerful intervention in the foundational discussion together both with Peter Johnston and Pride. And in fact he also took Benabu apart at one point in the discussion. Penelope got very upset about this, not surprisingly, just pointing out to him that he was talking arid nonsense at about five of the categories, and that he had, well, I say completely, but he had certainly misunderstood some of what it was that he was doing with steam rod in his book, which is an interesting discussion. But I remember him making this analogy, which... I think Peter also drew on in the discussion that mathematics is kind of many-fingered activity and it grows out and as the dendrites form it's a little bit like the human brain across the web and one dendrite connects with another and reinforces the way that that neuron, how partly that neuron is firing in terms of the idea that you can reduce the whole thing to... A tree-like diagram all resting on the universe of sets or an ultimate ingredient of definition, an abstract kind of hierarchy is completely crazy. It distorts the real nature of mathematical activity. It's a very general philosophical point, but still one that I remember very clearly that he made in those discussions. But Steve's, I was probably not quite, I know I wasn't capable intellectually of appreciating the importance of what Steve was saying in those talks, but I do recall, I do particularly recall the remark about the possibility that this subjective number theory program might lead to a proof of Fermat via a quite different route, obviously, from the one that Riles took. And by making an objective instead of more abstract, making the ring theory more abstract is what they actually do. Yes, exactly, going off from the ring of the complete. But make it more concrete. Yes, make the ring theory more concrete, but still. What it really says is something about one concrete function phase of the big choice sum of two odd concrete function phases.

1:30:00 Clearly a very rare sort of occurrence. But in principle quite an intelligible one. Yes, I can't help thinking that that is the route that might have appealed far more to Grotendieck than the going in the direction of the super abstraction. Has Greg indeed ever made any comment in private about Weyl's proof, about the proof of Fermat? I mean, I know he has retained contact with a handful of people. I haven't seen him at all. Well, 1980s or something. That might just hit me when you said this. That might be one of the only conceivable ways of bringing him back from the brink of madness that he's been in for so long. That might conceivably be a route. I don't think I ever discussed it with him, because we didn't have the idea before, in 1990, when I saw him in 89. I'm sure we didn't have time to discuss it all, but that's the sort of thing that would really, really, really appeal to him, more than all this crap that's going on. Certainly more than Konsevich and Kahn and all this crap. No, and as you say, it might bring him back from the brink and back into mathematics, which I'm sure that even at his age he's more than capable of still making tremendous contributions to. Yeah. It's such a tragedy that a mind of such great human, of such greatness should end in... In the opposite direction, you see, I think, I'm afraid that it... End in religious murder. ...affects upon him, it would have an effect on him too, as well as the desired effect on the masses. Students, this vicious propaganda of Sherlock. To the effect that Rotary's whole embracing of dreams as a message could be more important than his previous, could be more important. This is absolutely disgusting. This is so disgusting. Did you read this article? No, I didn't, but it now explains to me what the illusion was in your email to me. You said, I hope I'm not being too polemical or too... You made some reference to something which I caught an illusion, that there was somebody who had been mongering the idea that Grotendieck's madness,

1:32:30 the things he now dreams about God and the devil, were... More important or as important as his scientific work, but I didn't know if it was, and I got an allusion to it, but no, and who is this person and is he part of this? You see, this kind of growth in the industry is actually quite sinister, and some of the people involved in it, and Colin. It's involved with the Grozendieck circle, and obviously he's a very serious person to be, but at least one of the people in Paris who regularly descends to give lectures about Grozendieck is a complete charlatan, a dreadful man called Loeschak, if you've come across him. Pierre Loeschak, he's just a... I made the mistake on Rodin's urging before I saw through Rodin. I mean, he's a nice enough guy, don't get me wrong, I'm not personal animus against him, but he is intellectually a menace. And I'm amazed they've given him a job now at Paris CETIEM in this unit of this, they have this project called Ideals of Proof, which is funded by Notre Dame. And they have this guy, well, it's not all funded by Notre Dame, some of the money comes from the French, some of the money comes from the CNRS, and it's divided. The US, sir? Yes, yes, Notre Dame, the Catholic University of Notre Dame in Indiana, who, of course, have Potsdam, one of the most richly endowed universities in the world. And they, together with the CNRS, have sponsored this three-year program. Okay, yes, yes, well, you know, you've got to go, you know, yeah, yeah, well, okay, I seem to be permanently. No, no, I think he's good enough to see through them really quickly, but he's maybe he's not here that you can discuss with them and bring things around to a more rational level. Yes, well, I still labor under this delusion, though I must admit I get more and more disabused of it. We've both just had two nights of, yes, yes, yes, yes, exactly, exactly, but anyway, um, the, the, the, the, the, the, the, the, the, the, the, the, the, the, the, the, the, the, the, the, the, the, the, the, the, the,

1:35:00 Certainly not in terms of so-called philosophy of math to contribute, although he's a nice enough guy. He's got this big obsession with something he calls the purity of methods in proofs, whatever that means. Anyway, it's largely a pot here. But the point is that they've got all this funding, something like $8 million, I'll see to set up this thing for three years and they created eight appointments in Paris at the end One of which is rota, although he's really just the cook and bottle-washer I'm to be I don't want to say this to someone who would upset him He's basically just more or less used as the go from the office boy to run around doing some relatively minor administrative task He's not actually really Really doing any serious research. The two people who are... The kind of head honchos, as far as the research is concerned, is this guy, Jean-Jacques Chachignac, who you may have come across, Polish, he gave, actually, I thought, quite a competent talk on the Galois theory, they had a conference on the history of Galois theory about a month ago, and he seems to be a reasonably serious person. He's a big bête noire of Aluni's, which speaks well for him, but Aluni won't even allow him within the halls of his outfit, because he shows up Aluni's ignorance and pretentiousness rather too readily, so I think that's a bug, but anyway, Shishiniach is one of these guys, and I've completely forgotten why we got onto these people, apart from the fact they were having a conference on diagrams. Sorry, it's being zonked with all that there. I can't even remember what it was that was the reason of this. Yeah, we should probably go up and get some fresh air. Yes, I don't know if you noticed the... Oh, yes, this terrible guy who propagandizes for Grosvendieck's session with dreams. That's what you were telling me about. No, that's how we got on to this, because I was saying there's also this dreadful character called Loschek who comes and talks dribble about Grosvendieck.

1:37:30 Yes, yes, no, that's how I came about it. No, but this American Mass Society notices a long article by Charval. Are there two letters missing from that name? I think there's an A.T. in there somewhere, isn't there? That should be... Well, you said his name is Charlot. Charlot. Oh, Charlot. I miss hearing it. No, it's... there's a reference... He described all these, you know, girlfriends and that sort of thing. As if this had anything to do with Grokendijk's scientific work. This exclusively says that he's not going to talk about any mathematical things. I'm dead, it's a long article in the AMS. Well, I was going to say, yes. But anyway, but that's worse. So one of the incidents... Worse than gossip. One of the gossips... Yeah. ...is that Grodendeep, when he lived, you know, in the country... I don't know. ...this is later... Windmill, yeah. ...whatever, that he would collect his own extra bucket and peddle it to the farmers. It's fertilizer, you see, and why I'm doing this very, very... Because of his belief in deep ecology. Deep ecology, yeah. At the same time, believing in the special values of his... His excrement. Excement, okay. So, Schallau just mentions this, you know, it's one of these many rumors that may or may not be true, okay? But he mentions this along with other doctors. So that's what I was giving you, this analysis. The scientific fact is that dreams are waste products of the same intellect, the waking intellect. Yes, yes. I was going to say, absolutely. It's unbelievable to me that even though I know what the standards of the political of the AMS now are, I still am amazed they would publish such an article. That's just dreadful. Well, clearly, such an article belongs in the bucket in which Gurdendieck is alleged to have peddled his poop. Yeah, yeah. Oh, I just can't believe that. What started me on this, to remind me about this, is that probably that article, that kind of activity by this group, he's part of the circle. Vile garbage, yeah. It could even influence Gurdendieck himself. I'm even more convinced that what you had just suggested was actually a prospect, admittedly probably not to work, but an actual idea to bring him back to reality.

1:40:00 Which of course is what we would all deeply wish would happen, could happen. Well I don't think there's anybody apart possibly from you that is capable of doing it. I don't know if you've ever considered making a direct attempt, even though it probably would. I think the route that the INGS people have taken has just turned out to be quite disastrous, because you know that Kahn, because Cartier had told me that at the end of last year, in fact around the time he came over to Boston for the meeting where you met Daniel Kahn later the first time, I think he was in fact, because he told me about him on the plane on the way over, so there was a couple of... Maybe a couple of months before that, that Grodendieck had made contact with the IHS. Yes, apparently about September of last year. That must have been about the time that you were in Paris talking to... Charles Fussell, to Christian Fussell I should say, Grothendieck, this was the first time he had made any kind of contact with the IGS for, well I think, from what Cartier said, I think certainly for at least 20 years. Suddenly out of the blue, they received a letter from him. And Cartier was a little secretive about this, and the letter in the first place was simply asked if he might have permission to borrow some books from their library. One of the books he particularly asked was for the third edition of Newton's Principia. So, naturally, they wrote back saying, well, yes, of course, you know, it's an honor that you should ask. This is the kind of work that only French functionnaires could manage to achieve. Bourguignon didn't give the letter straight to the librarian. It got lost somewhere in the toilet in the memory system, and the librarian didn't get it until about two or three weeks later, which I refused to believe. I didn't realise it was important. I don't believe that part of the story. I don't believe in the librarian. They would not know who Grosvenor was. But anyway, five weeks later Grosvenor still hadn't had a reply or received the book. So he wrote another really rather injured, angry letter saying, you know, I know that, I know you all think I'm crazy, but, you know, I was, I actually must have something to do with your institution, and even if you're going to refuse me, you know, library borrowing rights, I think you might at least have the decency to reply to my... So at that point, Bourguignon kind of went into panic mode, and he kind of called in Partridge and said that we, I'm going to, you know, we write a groveling letter of apology, saying we're very, very sorry about what happened, and we courier the books down to him, you know, together with the letter of apology.

1:42:30 And, you know, I want you to help me draft it, so that's what they did. And then when I saw Katya in, when was it, in April, to talk about this meeting in January, which has now been taken over by bloody Conan Koncevich, and obviously his scientific value of which has been very largely rendered nugatory, where you were intended, it was intended that you should give these two talks, he said that he had had two more communications with Gertrude since then, and he was... The last one seemed, although very, there's an awful lot about, so I'm going to go very soon now from this world of kind of corruption and evil and into the world of light, but, you know, I'd still like to, he had the impression that he did still want some contact, and he was wondering whether he kind of should take it on himself, you know, to actually go down directly to try and, but the problem is that... At the time that he did his disappearing act, Grotenbeek had kind of laid a very solemn injunction on Cartier and other people they should not. ...and try and seek him out or visit him without, unless he made some express invitation to them to do so, and since he did indeed give to this undertaking, he thought that it might actually be counterproductive and disastrous, and he did. On the other hand, he thought that since this is the only contract in almost 20 years, directly with his colleagues, although there has been some private contract, I understand... And it was so important because of the anniversary coming up and maybe the very last chance of bringing him back into it. So bringing them back from the brink and back into mathematics, that they ought to take the risk. I have no idea what happened beyond that, but of course if Kohn and Konsevich and co. got into the act and started writing letters to him, then it's almost certainly sabotaged any chance that there might have been. So the tentative plan was that Kote himself would make... That was what Cartier told me he was seriously wondering if he should do, but that was back in April, I have no idea what has happened since, so of course he was hoping that Kurt Grotenbeek himself might actually come to this thing in January, but that would have been too much, and besides, I don't think he would have wanted to be part of any kind of a circus, and those for certain he would not have wanted to be.

1:45:00 But that he would be prepared to meet again with people with whom he feels comfortable and who he respects and whose seriousness he is, of whom, I mean, this is not a personal thing at all, I'm not saying this because, obviously, you know, for my admiration of you, but I think you're quite clearly the founding candidate for that role. That's not just. You've certainly got along very well. And quite honestly, you're the only mathematician in the world I can think of that he's, I think he's pretty certain that he would, he wouldn't suspect some kind of hidden agenda. He really cares about his mathematics. Exactly. See these other characters, they want to use his name. We are the cutting edge, we don't want to go back to past achievements. Just using his name as a mantra. Yes, yes, I know, I know. And the thing which I do find very worrying is that, although she's never actually come out and said this, given the papers, I think she's written two papers on Newton, given what I've heard her say in Newton conferences, I think she probably would endorse the statement that Newton's theology is more important than his contributions to mathematics, which is certainly an echo of exactly what we're... On Grosendieck, exactly. Yeah, yeah, I mean, of course they've been on there. Oh, exactly, yeah. You were really quite rarely a magician, were you not? Yes, of course, I was an alchemist. It's far more important than this. Admittedly important, but still secondary contributions, like creating mechanics. Yeah, creating mechanics, exactly. I don't know, but really it's... I don't know, exactly. I mean, I know this kind of thing only too well. I know this kind of rhetoric. It's absolutely horrific. And how he was the last Magus. And that's just so much more exciting somehow than the idea that he was actually what the 19th century recognized him to be. It was, of course, together with Galileo, the founder of mechanics and of the scientific world here.

1:47:30 It's all part of the general cultural project, too. We can attack science from various directions. One is just the straightforward anti-science and glorification of... Obscurantist and religious ideology. And the other, of course, is trying to hollow science out from the inside by pretending that it really doesn't contradict religion at all, i.e. that one wing is controlled by Templeton and the other wing is, I think, well, not sure that the other wing is under one single unified direction in the way that the Templeton effort is to try and get inside the citadel and destroy science from within, but it's definitely the wing of... Science anyway is a really bad thing. We don't really care whether it can be reconciled with religion and show us the way to God because all these people who believe in dull old mechanics, gee, I tried to take that in. I did math 101 and I just couldn't get my head around that stuff, so I just went off and did drugs instead. I'm afraid there's a lot of it out there. And I got enlightenment through... The doors of perception through William Blake and the higher mysticism and Ilya Prigogine. Now, I still think that his wife, an otherwise intelligent woman, she said she had seen Otto on TV and thought that he was good presidential material. If he comes back, he should be president of Hungary. Gosh, that's very scary. Historical memory wiped away. Yeah, yeah, that's exactly. Who was it who persecuted and imprisoned and crushed the Hungarian Revolution? Just like the memory of the German philosopher in Frankfurt, for instance. Well, Heidegger. Heidegger, the fact that Katrin, for example, can't remember that he was a real... He was a bloody Nazi, yes. Well, she can remember, she just somehow doesn't want to pretend that it's not important.

1:50:00 You've got it filed away in a digital way. Yeah. All right. That's very interesting. Science is great, but I say renaissance of what? And the fact that it was not one rebirth of... Beauty and light, it was a vicious struggle between two rebirths of everything that was most reactionary under the Roman Empire and so forth. ... came to fight with the upcoming Enlightenment, I mean, the scientific viewpoint was already, you see, even before there was Galileo, even before the basis for this was growing, right, and in fact, what we're just talking about, why do we have to reform the university? Well, because we have to counter a certain threat. Yes, a certain threat, which is already clearly there in the, well, look at the Oxford. The Merton, you know, the Merton schoolmen, the people who were trying to wrestle. ...with ideas about dynamics already in the 14th century. They were trying to analyze motion, and they didn't obviously have the appropriate mathematical equipment, but nonetheless they were trying to build on what Aristotle had achieved and to push forward and to try to create new formal tools for the understanding of motion, and this clearly is a very progressive and enlightened scientific project, pushing in the direction of mechanics. Therefore, let's go back to Plato and make everything completely frozen. Here are these ideas and, you know, hang on, we've got to move out of the roadway. Reality of matter in motion. Oh, no, that doesn't sound like a good idea at all. That might give people the idea that it's part of materialism. Because the problems that we're facing are going to be more serious even than those we faced before. But the peasant revolution, of course, you know there was peasant revolution here too, of course. Yes, of course, of course, as there was all over the Germanic world. Less famously in Germany, but of course there was here as well. Even in Italy, northern Italy, all over, of course. And as there were, it was in England too, not just the great, much earlier, great peasants' revolt in 1381, but there were also... Major peasant rural uprisings in the late 1400s as well.

1:52:30 The whole thing, I mean, the trouble is they're taught to people in history classes in a completely misleading way because they're represented as just basically the fag end of the dynastic struggles of the Wars of the Roses, the so-called Perkin-Warbeck's Rebellion, but in fact it had almost nothing to do with the, it had almost nothing to do with the net existence of a pretender to the throne. That was just the form of expression that it took because they had to have some obviously kind of focus of, but in fact it was caused, it was actually a revolt against the... ...against the huge additional taxation, in fact, which Henry had started clamping on, actually more the fact that they were, because the barons had been bought so firmly under control, the king was actually now able to collect taxes much more effectively over the whole country, so people who... Dick Bluntly had been pretty well able to dodge their tax when the previous generation was suddenly having to pay it. And this actually, of course, and tithes of course. There was also a revolt against the church, against tithes. And you see this sort of thing again later in the, under Henry VIII in the 16th century. Obviously dressed up and taking the form of a... Expressed through religious ideology of those kinds, but there's quite clearly a material economic motive behind it. And I was well aware that they had had the revolts here in the Netherlands too. In fact, I told you about being locked in a room with the paintings. Incredible experience, it's still on the line now. In Nervin. When we had a meeting in Louvain. Yeah, I don't think you did tell me that. No, this was 1980. I think meeting in Levin organized by but somehow in that particular case he had to make probably to get the funding he had to prove that he was going to support the Flemish as well you know so one day So, we all went by bus. It was very impressive in a way, the absolutely crazy division between the languages and so forth. So, we got to Leuven and I gave the first talk. I commented on this. I worked it into my paper called Unity and Identity of Opposites. All the more reason for transcribing your talks and writing them up in good time.

1:55:00 The manuscript, first manuscripts, again, you might be even amused by them, but the unexpurgated version, you know, sort of thing. They're always the most interesting. I've got you all three of those of yours, of course, including the whole parts one from 1993. This one had the short title, Belgian Unity. The theory of unity and identity. The theory of opposites, yes. Based on the example of the so-called unity of Remmisch. Yes, yes, yes. Yes, well, there certainly are. The Belgian Unity. That was the original title of the paper. By 1994, this paper would no longer have been... The title would no longer have been... The joke. People would have got the joke. But the unity identity of adjoint opposites. Good question as to which is the left and right adjoint respectively in the case of the Flemish and the Walloons. Anyway, let's get back to this point about the painting in the room. That's why we were in Leuven. We went on an excursion and there is a tower in which there is the... The town council, or parliament or something, Scott Rock, is what they call it, so we were looking at this, I'm not sure it's a meeting, it's a parliament, but in the top of this tower, but on the wall is a giant painting of the peasant revolution, you see the peasants, you know, who fire and smoke and... And so forth, you see. Interesting. In other words, the lawmakers must constantly keep in mind what happens if you get too oppressive. We have to be careful to avoid this. So it's very, very, very striking. Well, okay. So I was with three or four other people. I remember Joanne Palajé, for example. I forget who they were, but three or four.

1:57:30 Same light. And then suddenly I saw the guide. There was an actual guide who was showing us through there. Even though he saw us, closed the door and locked it. So we were imprisoned, you see, consciously. It's not that he made a mistake because he saw us. My comrades told me explicitly that he was looking at us when he did that. Why did he do that? Who knows? I mean, maybe he's just crazy or maybe he overheard the remarks and there's some kind of some kind of virulent anti-communist. Yeah, yeah. That's the only explanation I can even think of. You wanted to make sure nobody else came in to listen to what you were telling people. I have no idea. I wasn't there. It's certainly obviously a conscious action, so there must have been some kind of action. But the objective fact, whatever the intent was, it was that we were locked in. We had no way of escaping from the tower. Shades of a Browning poem or something. I found the way out, you see, because there was no way out, but I climbed up to a window. I had to climb up to get the window and look down in the plots below, far, far below. Kassir was also a very wealthy man, apparently. I was hearing there was a talk, there was a series of talks last year in the Collège de France about, curiously, about Heidegger, Kassirer and Karner. It was quite an interesting series of talks. It was a series of three talks that were given by a guy called Michael Friedman, who is a Kant scholar at the, I think he's at the Stanford. He's published a couple of major books on Kant's philosophy of science. And he is, I think, a very serious scholar, mainly, as I said, Kantian. It was a whole very influential neo-Kantian school in Germany just before the First World War, not all. It's very likely that they were given large amounts of stock or something like that. At the same time they were quite subservient, so it wasn't that they were independently rich. I'm just trying to make up a consistent story about that. No, it's perfectly reasonable speculation. They were rich and also superfluous. Anyway, well... Yeah, I say they're probably rather in a position of Harvey Friedman. They deluded themselves into thinking that they were members of the ruling class. That's right. Because they had a couple of million dollars' worth of stock options. Believe me, that doesn't make you a member of the ruling class.

2:00:00 Yeah, yeah, yeah, right. It's like John Mabry's great line about Nelson Bunker Hunt, sort of mentioning how much he worked. Honey, if you know how much you're worth, you ain't worth much. That's exactly what she thinks. She used the very phrase to me last night. Well, in fact, no, let me be precise. She said what you and Bill fail to realize is that materialism is just another form of objective idealism. Yeah. So it's the complete lack of the role of concepts. The human scientific sense of general concepts is important speech to Platonism as a civil state. But, at the same time, amidst the confusion... Well, that might have been the gist of what she was trying to drive at with what she was saying to me about her account of the theory of the forms, but you certainly expressed it much more clearly in the way it came across from... No, I'm sorry, I didn't express it very well at all. No, it's... It's a simple idea. No, it is. It's a perfectly valid idea. So, probably people will never come upon this idea. I don't know why. But, of course, general concepts are an absolutely indispensable tool for the... But to reify them and deify them is... Also, they have incredible permanence. I think this is underestimated because they're never mentioned at all. They've been placed by a productive idea. The role of concepts... I mean, there is a certain miraculous quality to it. I mean, not really miraculous. No, no, but... Apparently, because they are so precise. They are all consistent, you see, and they are communicable. If I phone up Australia and talk about a category of categories, it's going to be about the same thing. The thing which, of course, impressed Frege so deeply. This phenomenon which has never been adequately explained precisely because no one has realized it, it too is a materialist phenomenon and therefore must be explained in real time. Yes, I completely agree. I passionately agree.

2:02:30 There's very, very little understanding about it scientifically. But it was Frege's great argument for Platonism that the objectivity and transmissibility of the content of concepts, he just thought that this was an open and shut. Proof that they must live in some eternal realm. The cutting edge of this kind of leftism is Rubin Hirsch. No, wait a minute. I mean, he actually says, against all the others, concepts are a social phenomenon. He doesn't say social. He doesn't say concept. Anyway, that idea that the concepts are a social phenomenon and therefore they're not ideal in that sense, which is a way of opposing objective ideals. In order to re-establish it, blah, blah, blah, he's the only one who even says that. That's the thing. All the other subjective ideas ridicule him for saying these things, which is not well-established scientifically. That's what I'm saying. It's not well-explained or well-established. No, no, I agree. Yes, and I think we're obviously going to have to develop our science. I saw a recent article by him. Not least our understanding of science. Brain sites are much greater in depth before we... Oh, in fact, in the European, in the regular journal of the European Maths Society, a member, if I perceive these things, there is precisely this debate about Platonism. Ah, no. Comes up again and again. Obviously this journal is considerably more serious than the American one. Who I remember also had such a debate about 25 years ago, which McLean actually contributed to. Anyway, I think you're right. He was in opposition to somebody else. Both were Americans. I don't know why. They were publishing it. Can't the Europeans find anybody else to talk about this? Can't the Americans publish it anywhere else? All sorts of questions like that. But one point which I noted.

2:05:00 In order to try to illustrate his point that concepts are social phenomena, he said, well, these concepts, they change or they may die also.