Discussions & exposition FW Lawvere (contd.)

0:00 They can't do. Well, in the case of non-trivial, they can't do. But you've got this action of a group or a monoi, which... Intuitively speaking, it mixes stuff up, so to speak, in such a way that the domain has got so much internal variation that it can't, unlike in the case of a set, be expressed in the condition that... In general, pullbacks of pairs of maps do not preserve co-equalizers, in the case of a non-trivial action, they can't do this. Is that right? 29 euro 50, please. Gosh. Are you sure? Because I was going to try and do this on my card. Well, I don't want to do it, but... I'll tell you, this is only on condition that I get you dinner when we get back, because otherwise I'm really going to be... And the essence of them is, as I understand it, is this existence of a kind of non-trivial... the action of a group, or perhaps more generally of a monoid, some kind of monoidal group action that... So there may not be co-equalizers. I'm sorry, the preservation under pullback of co-equalizers is a condition which fails in general when you've got a non-trivial action in the... You mean in the two forms of action? Yes, yes. Isn't that an étendue?

2:30 Okay, I misunderstood in that case because I thought in the case of non-trivial etendu, there were cases where it didn't preserve it. I thought I must have been completely misunderstood in that case. I must have misunderstood something in that case. Okay, I'm completely misunderstood because I thought they were examples and I thought this connected with the thesis of your student, I'm sorry, whose name I've forgotten, but the guy who wrote about quantaloids. That's right, Kimo Rosenthal, yeah, yeah, yes, yeah, yeah, and it's a long time since I looked at that, but I should obviously try to make a more careful study, but I thought that there were cases where the thesis was wrong in certain crucial details, which was later proved by Hawking, but he just failed to prove it in one direction. This, in fact, has a site to be presented by the site. So in other words, for you, we're talking about monoid action. For you, not for an arbitrary monoid, but for a special kind of monoid. Yeah. Any monoid, it has a cancellation law from one side. Yeah. AX equals BX. No, the other way around. AX equals AY implies X equals Y, just in the monoid itself. Yeah. Then the actions will form an étendue. Otherwise not, basically. Okay, but I remember reading in your... Well, let's remember to define the properties of an étendue locally. Locally, there exists a covering on the whole topos, so you're not getting that epically to one.

5:00 Yeah. If you take the slice and add another one to that, that achieves an topological space. In other words, it's... well, I mean, in any time period, if you confuse bases and the corresponding topos with achieves, it's a portion of a topological space which is no longer a cosmological space. That could be the... I mean, no longer any trope of, according to a more general weaker theorem, is some kind of potion of a post-set. The point about a space is that the site is just a post-set of regions. But in order to get a good, a very good potion of the type just to slice over... Yeah, yeah. I mean, the simplest, almost the simplest example is to take this topological space with three points, which we've only got three points, but there's one way of thinking of it. Sometimes I draw it as a face. Never mind. Think of the real line with the point called zero. Now, what you do is you say everything's typically greater than zero. So you have three-point space, but it's not a discrete space because this other point is still an open set, and this one is still an open set, but the third one is not an open set. So it's really this division. So the sheaves on that, so this three-point space, we've only got five open sets, namely the two, I mean this is one, this is one, the union of the two, of course, is another one, and the whole space, and the input.

7:30 But, you see, modulo the covering conditions to be achieved with some kind of compatibility with genius. And so you can, you can, you can prove that. ...and just keep the basic ones and get the presentation of the sheaves on that space as mere pre-sheaves, because the sheave condition is wiped out by just taking a smaller site, you see, pre-sheaves on this kind of a post-set. Imagine the big hole space and then these two sub-spaces and then the act of restricting to each of those. That's the sheaves. There's some sat attached to the big space. Some, some, some set attached to strictly greater than zero, another set, a kind of restriction map. Whatever these magic elements are globally, they can be restricted to live here, locally, and some, and, but the nature of these two restricted sets and the restriction maps, completely arbitrary. Yeah, any, any diagram of that form is equivalent to a sheaf on this three-point space. All right, so that's, that's, that's an actual space now. The quotient of it, which is going to be an H undue, in other words a local homeomorphic quotient, a special quotient. In these terms, in this combinatorial term, is this. You make these two sets the same set, but you still have two restriction maps that are distinct, you see? That's a funny way to do it. Well, it's no longer a mere process of regions in any sense, not a process, because you have two arrows. So it's a quotient in the world of categories between a process and something that's not a process. Literally, it's just a parallel, parallel arrows. I mean, if you back up now to the sites, which are just little abstract finite categories, it's clear. ...that this thing has as a quotient this thing. You still have, you have two points, they're parallel arrows, and you've just identified the two points, but you haven't identified the arrows themselves, and so what you get, what you get now, you see, is the topos of irreflexive graphs, you think of, you know, you think of the set attached to this end as being a set of nodes, the set attached to this as being a set of arrows, and the two structural arrows...

10:00 I'm going to show you for which arrow, which is at the beginning, which is the end. So an arbitrary, irreflexive graph is just an object of that. The reptile pose is an 8x2. But now notice, almost tautologically, every map in this category is a monomorphism, because there's no non-trivial things to test it with. So each one, you know, if you took two things and suppose that this map became the same, then they were the same. Well, that's right, because they're both the identity anyway. So in a tautological sense, even this category has this non-trivial... Okay, that's very helpful. I guess I ought to have understood that already, but that doesn't make it clearer. But you made a remark, I recall, in the Eilenberg Freshrift paper about Etan Du, I think in the last section. In connection, I think, precisely, pulling back, yeah, yeah, yeah, yeah, that's some fragment of some older abbreviated argument. Yes, I think that must be where I picked it up. But you make particularly the point that it's precisely this kind of, this characteristic of the domains, in the case of domains of variation, and Etan does as domains of variation, this kind of topological characteristic that they have that involves this kind of two-fold aspect of the topology, which I guess is the point about the way that the math... There are a lot of maps, actually, like restriction maps, that makes it impossible in general for them to be just sets. I mean, they are varying internally in a way that is richer than you can have in the case where... I mean, sets don't vary internally at all. Yeah, exactly. But that somehow doesn't...

12:30 You might say that HMU is sort of the simplest form, but just to make the point that the points don't vary per se, but they do vary per se, you don't have to go to the case of HMU for that. No, I guess not. But maybe there was some point there about HMU. Well, it's... Oh, no, I know, you're right, you're right, of course. Well, assuming that... No, but the idea that... This... OK, this isn't... This isn't... This is the third... The earliest of three failed attempts to characterize pure variation. Ah. That's right. OK. Well, now... Now it's interesting. You must tell me of the other two, but tell me of this first one first. At least it's reasonable, you see. The idea is that... OK. Sure, some toposes represent pure cohesion, and others represent pure variation, and the general topos is some kind of combination of the two, but then to this clear, this contrast is fairly intuitively clear in the examples, but to give an axiomatic definition. What is really this? This is still going on. My paper, Axiomatic Cohesion, is about trying to kind of grasp that class of categories which, in some sense, are relative to their own internal notion of discrete, so they're just sort of a relativized canterism. Cantor's absolutely constant things, things that are at least qualitatively constant relative to the spaces involved. These objects, what makes them different from the spheres is just that they are cohesions, it's like being, pure being that's ready to move but isn't moving yet, you see, this is, in other words, cohesion is like potential being, becoming, being is like potential being. And of course, in general, some of it will be potentially becoming, some of it will be actually becoming. And the other pure case is where there's no cohesion there except that which is a trivial byproduct of becoming.

15:00 In other words, if I move from here to there, then that means that here and there I have some cohesion just by virtue of the fact that I don't have to explain how it was that I moved. This is sort of one side of the argument. I don't have to explain why I couldn't move. If I did, then I can declare that there is cohesion. It's not that the cohesion will necessarily be null, but it's in a context where everything is variation, pure motion. It's a motion without matter, if you like, because matter is... Cohesion is really about matter and how it is potentially. Substance is different from... The division is about substance. Substance is different from matter, I think, in the sense that this division in mathematics... Modeling between the domain of variation and the quantities that vary, so the substance can be the domain, but then the particular material, in the sense of null, is the substance equipped with the further structure, which is a response to stress, to strain, a stress response to strain, a particular law. That makes it a kind of matter, a kind of material, so that the substance is a substratum for that. I'm sure all the philosophers go crazy on that. Yes, but don't worry about them for now. I'm much more interested in that. One has to pay more attention to the actual practice of mathematics. Of course, of course. Well, you made the point yourself in Calais, you've got to... Treat it, treat it in its entirety, not necessarily. Taking this nonsense about other modular categories and typical mathematics that needs to be summed up, but certainly there's some totals of physics and engineering as mathematical practice, one has to take that into account, and in that there is this idea, implicitly there is this idea that space, you see, ether and space... And substance. They're all the same kind of thing. I mean, they're different topos as maybe precise models, but they're the same kind of thing, and they can carry additional structure like stress-strain response, which makes them then matter in the sense that they're fully... Which would, of course, really make them richer, as it were, as domains of variation than the kind of domain in which...

17:30 As you have, obviously, in the extreme case, one of the two aspects of Cantor's notion, namely the co-discrete space, which is the case where one point can become any other in a completely trivial way without having to pay any attention at all to any kind of parametrizing, any parametrization of the variation coming from, as it were, the medium of variation itself, which is the typical case, as we say, for spaces with cohesion. It's also possibly connected in some ways with the reason why one sees the, well, it's precisely because you impose things like the axiom of choice that you get. Things like the Banach-Tarski paradox, which is clearly not true of actual variation in the real world, because we know perfectly well that things don't move around in this completely unobstructed way without any attention at all to what is actually parametrizing them, so that in fact it is not the case that one point can become any other in a completely unrestricted or unparametrized way. In fact, obstructions to inverses of maths. The fact that there are all sorts of geometrically reasonable properties on the space itself which have resulted, which is what homology is, amongst other things, about. You can't do that, but neither can actual substance. See, that's an even more something in between. Yeah, yeah, yeah, yes. In fact, really nothing can do that except the pure abstract constancy sets where the axiom of choice holds, Cut the moon up into pieces and reassemble it into spheres. Maybe you could sell that to the Cray Foundation or the Pentagon. That's one of the crazier things that they're trying. I think the difference between the two is that the Templeton Foundation might well accept that. Yes, the Pentagon would still hopefully be too small.

20:00 Yes, exactly. But you take your point. But actually, conceptually, as a way of analyzing the issues that are involved in Zena, this point about the Banach-Tarski paradox I think is a very useful one, and understanding how it originates with the imposition on the structure of the domain as precisely as understood in terms of... The lattice homomorphisms from the parts of quantity into the parts of domain over which quantity is varying is only by imposing the condition that you have in the axiom of choice, which ensures that everything is ground down to completely structuralist points. It allows you to have this condition. That is the point where in fact the analysis on the basis of lattice homomorphisms from parts of front view, parts of space, does actually become the same as what you get from the idea that the domain just consists entirely of, is just determined entirely by its points, that somehow it's what Frege and Piano and Plato indirectly all said that it was, but it's determined by... Some notion of a pure transcan. Trans-categorial notion of objecthood, just the things which have to be there, the same or different absolutely, to be the values and the variables. So the Etan Du is coming back to that. Yes, and also to the others. To describe the opposite of the special class, where we completely ignore any delusion. This is the case of what I often advocate, that the use of negation in ordinary discourse is different than that. Not A means, not A implies false, as the intuitionists say. It means A implies the trivial case of A. Yeah, yeah. So you can say, I mean I like this example, you can say that the definition of a brother is a male. These parents are the same as mine, and so therefore I'm my own brother by ordinary logic.

22:30 You know people normally say I have no brothers. Which means that the only person who satisfies the definition of brotherhood is me and myself, which is the trivial case. Yes, exactly, which is the trivial case. You don't have to constantly chop things up, you just notice that the co-domain of the implications is trivial case, not total case. So when I say, you know, variations... Cohesion, it means, well, only the trivial cohesion that would be there in this kind of situation. It was an attempt to capture that. I think certainly whatever pure variation means, Etan Du is the case of it. It should be broadened, however, because there are these other things that Johnstone studied. You mentioned that there were two more successes, later attempts, to characterize pure variation. Can you tell me a bit more about that? In the Como 90, it's this SUD. Ah, yes, of course. I was going to ask you about that, of course. Let's come back on to SCU, the separable, unramified, and decidable. So this is again a very good notion, which is a second attempt at capturing the pure variation as opposed to the pure confusion. There's a lot to it. I mean, each of these steps has a lot of content, but then the thing is that when that one in particular captures, this is crucial because contrary to the idea that toposes are... There really is only one actual thing that deserves the name of generalized space, and that's Grotendieck's etal. See, the thing that everybody knows about, but again... Like these so-called algebraists we've been visiting, refuse to look at actual algebra in the same way the so-called topos theory refuse to look at the actual application of topos theory. Etal is the one that every algebraic geometer has heard of, but it's never been approached by topos theorists. So in that sense, the Parisian reactionaries are right. Well, anyway, so the etalte, as an analog to the sheaves on a topological space, wrote an etalte attached to every algebraic scheme, a topos, called the etalte, the petit etalte, which is not an etalte, do we?

25:00 But it's very, very, very special as a topo. It's certainly a generalized, you think a generalized topological space, not generalized space in some of those other places. No, no, no. It's a generalization of the topological space. And it is, but it is one of these SUD things. But that was a big step that we could actually capture the main example. I'm afraid nobody understands it. I don't know why, because I thought I wrote it rather clearly. But anyway, nobody seems to do that. That's something which the philosophers should have taken up, too, because it's so clearly, as I say, crucial to understand. They're afraid to look at the actual category of punitive rings. Which is the next most important algebraic category after groups, in the same way this is the next most important, or probably even more important example of the two topos beyond ordinary classical spaces is the Hilton New one, that nobody's looked at anymore. I think, I think that even, I can't say for sure, but I think even when Johnstone invented this class, it says that... Decidables, yes. Equation of decidables. I don't think he looked at the separability and unramified aspects. No. In particular, I don't think he realized that Godin-Dix-Petit-Héton-Doux was being captured. It was just an abstract exercise in logic from other aspects. I don't know for sure. I mean, he's a very smart guy. Well, and it was being captured, even if he didn't... So they made no mention of it. Can you explain to me again about the unity of the conditions of the separability? The decidable objects, the separability, and the one which I don't really understand, which is this business of the unramified. Well, S and D are the same thing, simply as the classical algebraic idea of a separable polynomial is that it's one that doesn't have any repeated roots, and so that the space that it defines, all the space of roots, is sort of well separated, you know, you don't have double, you don't have the function being sort of confused.

27:30 So it's separable, but separable is the same as decidable. You can decide whether two parts are equal or not. So categorically it's just expressed by either of these words, it's just that the diagonal map has an actual simple-minded complement. Intersection, empty, union, the whole square. Yeah, yeah, sure, I understand. That's how it was understood, just in terms of the diagonal map being a separator for the whole. Exactly. So the logistic slogan is decidable and the algebraic slogan is separable. Steve, Samuel and I decided we're algebraic and we're going to call it separable. Those are basically the same, but U is the same property, but applied, you know, if you're talking about a certain totos, a certain object, you apply instead to the category of objects over, okay? So it's under-amplified, and again, it's no mysterious thing, and that's the well-known idea from algebraic geometry, again, having no branches, in other words, that it's huge. But it's not just about a map rather than just about a simple space, so it means that each fiber is separable or decidable or less or larger than each other. So that has the byproduct then that once you have a partial section, once it gets started it can't go anywhere else because of... Ramified solutions, the ones where you do have the branching. Yeah, where you wouldn't have branching. So, crudely speaking, you're just saying the fibers are SD over a certain graph. But in a way, I think that expresses better the idea. You think these objects you're talking about, that they are actually defined over something else, and that's why. That's why the fibers are, would you say, separable or decidable? You can imagine sort of moving along, attempting to decide things.

30:00 Yeah, so that's all it is. These are just three, the same conditions, except in the middle one you... And this again is a further attempt to characterize the case of pure variation, where the pure variation is acting as the negation of constancy rather than cohesion, which of course... He's going to negate constancy in a quite different way. Constancy, but in quite different ways. So constancy, you need another word, maybe. Well, because it's a relativized notion of constancy. You call it constancy in one kind of case, and you call it discreteness in another case. Yes, pure discreteness, or in other cases, pure lack of... Pure constancy. That's very interesting. And what was the third attempt at the general characterisation of the opioid variation? You said there have been three attempts, really. The SUD, the AITANTU... Oh, of course, that most recently. Yeah, sure. Well, I'm emphasising the opposite situation. In passing, I refer also to... The SUD, yeah. You refer to the SUD, isn't it? Well, no, wait a minute. I'm proposing a still more general. Well, first of all, I mean, obviously, you could make a general, simultaneous generalization of HATONDU and SUD in the following way, at least in terms of sites. What Johnstone fundamentally showed was that QD, we call it QD, we call it locally separable, not just to be different, but to emphasize the algebraic as against the logical aspect, but it's clearly the same condition. But locally, just in the sense that everyone is at least covered by, and when you say covered by, you don't need to take the family because the sum is the same idea as the whole individual piece, so it's just a single map, a subjective image of the side of the order.

32:30 And this must have implications for the kind of topology and the behavior of coverings. Well, not as such for the site as a category. Yeah. So Johnstone's theorem is that you can describe very fast kind of category. Any topology on it will give rise. If the category is, it has the property that every map is an epimorphism. Almost, just like, almost the opposite, you see, of that. In other words, if the cancellation law of AX equals B, X implies A equals B, holds in a monoid, for example, then the actions will form BD, be locally separable. So that's, again, a further characterization of the case of D'Aurelius. You can see that in terms of this idea of fibers that are decidable. Why the epimorphism condition plays into that. I'm a little too tired to draw the diagram right now. But that's in Johnstone's paper. But now you can have it. ...versus every map is a monomorphism in a site now. In a site. Not in the old topos. Well, actually, the two are different, I would subtly, in that to say every map is a monomorphism in the site is the same as to say every map in that site is a monomorphism when considered in the old topos. Mono is a universal property, but even if you enlarge the universe to the old topos, or look only at the site, mono means the same thing, four things that are in the site. In contrast, for the epic, you see, the fact that every map in the site is epic, as seen from the site, has nothing to do with whether it's epic in the whole tofu. In fact, it usually is not. It's only a cancellation law. It may hold in the Monoi, but it certainly doesn't hold even for the same map when it's considered as living in the tofu. Sure, sure. In the case where you've got the, well, the case where you've got choice and then the epi-mono-factorization, then obviously that is a special case within this. Ah, epi-mono-factorization. Well, if you've got that at the site, then already you're in a way familiar with this case.

35:00 It may not be either constant or variable or coincidental. No, I was going to say that that's going to typically be the case where you have to enforce constancy to the... No, no, no, I misunderstood that, I'm sorry. Because I thought the choice tended to be a fairly strong indicator of constancy. Oh, oh, oh, split-happy. Yeah. I'm sorry, I didn't hear you. No, I'm... Of course, that's a lot. Yeah, yeah, that's what I meant. Yes, but now sort of just saying that's just yes, you know, the obvious generalization would include both would be in terms of sites. It's not so obvious for the total terms of sites is the following sort of more subtle cancellation law. Those who have two arrows in the site, you want to conclude that they're equal. You can conclude they're equal if you have two things. There exists some other map here which has equal composites, and there exists some other map here which has equal composites, so if this alone implies they're equal, then everything is monic. If this alone implies they're equal, then everything is epic. The more general hypothesis in the cancellation law means it's a weaker condition than the whole thing, so this will include everything epic and epic mnemonic as special cases. Of course there is a sub-case where every individual map is both epic and mnemonic, and that comes up pretty often too, both kinds of cancellation law, but it's a special case. So here you have to be careful about the proposition of operators and all that kind of stuff. One thing is like an intersection, and there's a sort of union between classes. There's double cancellation. However, this is getting complicated, and it occurred to me, well, there's something even much more general, which is much simpler. Which is the business of the neural potence. Idympotence. Idympotence is what I meant to say, sorry. No, Idympotence. Where again, no is in this colloquial sense of negation, not some super logical thing. Namely, the only hidden focus is identity. Again, obviously, for the site, for a site. So my present proposal, which is maybe too general, concludes the previous one.

37:30 And I hope it can be investigated more. The topos is for which there is a site that has no identity. Yes, and that ought to be the most general characterization available of pure variation. This is the idea. I can't imagine anything would exclude that. And by the way, if you look at it the other way around, these two things are not logically complementary either. A lot of topos are mixed together, the two, we hope. We hope it's not all being as permanently static, and we hope it's not all motion that's really random, but anyway. No, no, well, we've... I think we have very good evidence for this, comrade, including the fact that we're sitting here having this conversation. They're not really complementary, but if you sort of imagine it in that way, what does it mean that there is no site with no idempotence? It means that every site has to have some non-trivial idempotence. And non-trivial idempotence, by the way, I advocate degeneracies. Not themselves, but again, this is one of those cases where certain terminology, like intensive and extensive quantities, are only talked about in thermodynamics, but why the hell do they apply everywhere? In the same sense, this is only talked about in seclusal sets, not one category. It's well known there to generate, but it applies everywhere. Namely, you see, the idea in terms of figures, you have... In the epic category, you have a notion of figure shape, and so maps is domain of figure shape, called figure. In general, figures have to be singular. Singular means it's not monomorphic. As a map, it's not monomorphic. A figure whose shape is an integral might be a circle, because you identified the ends. That has to be admitted as a figure of integral shape, but it's a singular one. This is the word singular, but singular is a different concept from degenerate. Because one way that he would arrive at this is that there's an interpotence on the figure shape which transforms one figure into another.

40:00 So it would be kind of exclusively degenerate already because of something that's happening on the pure figure shape itself. So that's a special way, special reason why a map might not be a monomorphism. This would be a kind of expression of cohesion, as effectively kind of negating the situation of pure variation. Yes, and therefore the negation of the case of pure variation. I'm very interested to see how it connects with this very powerful underlying idea of the role of figures and their incidence relations there. This is a particular kind of incidence relation between two figures, normally of the same shape, but there's literally a map which is hidden from you, so it takes one into the other. If it's a non-trivial, clearly the result of applying it is more degenerate, is more singular than the original, but it's more singular in a very sort of, pardon the expression, constructive way. So this is well-known in simplicial topology. That's why in the simplicial objects of Funker, Genoa, Delphi, there are ephes and monos in Delphi, but the ephes are split. And so the figures are called symphysis. So degenerate symphysis. A degenerate N-symptom is one, sorry, I mean, idempotent always means you can split the idempotent, so you can say that, well, I went down but I went back up, I got the identity, I got the identity down here, I got the idempotent down here, so, if you have a solid track, collapse it all onto one side, well, that's idempotent.

42:30 When you collapse, if you leave that part fixed, if you collapse and collapse again, it's the same as if you collapsed just once, i.e. hit him, okay, so as a figure in some space, you could have a degenerate triangle, which means, well, really, there's an interval, there's a figure interval, and then you just blow it up artificially in that way. It has to then count as a triangle, but it's a degenerate triangle. DeGeneres, it's a clear idea. When you learn, you said in geometry, you normally get this idea that among the triangles there are those that are collapsed. Yes, there are the DeGeneres case, which is, you know, they satisfy the axioms being a triangle, but we recognize them as, in terms of figure shapes, as having, you know, triangular, as being triangular figures. Yeah. Yeah, yeah, yeah. It has to be, it has to be considered, doesn't it? But an interval itself might collapse to a point, in which case there is a point which we just blew up artificially, but with the constant map, it has something to do with constancy, you see. Now the degenerate figure is itself partly constant. Maybe that's why it has something to do with constancy. Geometry, blah, blah, blah, blah. They're always there. They're always these non-trivial info. And then it's sort of, I mean, I don't know how they do it. It seems like any site has to contain those. You can't find, in other words, every site must have, if there is a site that has the property, none of the objects in it have any info. Then it seems like it must be pure variation that comes out of that mixture of intuition and precise definition.

45:00 Which, as you were saying back in Calais, is the... especially when one's... As we're seeking to put these experience of all these centuries of mathematical development of concepts into a form which can be communicated to philosophers is a very subtle art to choose the examples that will put the idea across. You chose the example of Leibniz's cosmonaut. This is a very, very good illustration of the problem, because clearly there is a great deal of mathematical structure in this conceptual determination in Leibniz, and Leibniz himself is arguably aware of the considerable detail of that mathematical content, but you don't, of course, want to… Thank you for your attention. Presented oversimplification of the mathematical concept, or to say, this idea is just simply the mathematical codification of some. The audio is telling me that there is this French expert on Leibniz who had this idea that Leibniz really had a hidden mathematical theory that he was talking about. Yes, yes. I'm wondering if I should go and read that book. I often tell my comrades in the Communist Party of Canada. Don't read too many books. No, no, it's quite a reason. Many a wise man has read himself stupid. This came up completely rather strongly a few years back. You know, these scholars, you see, these hordes of scholars who are studying. So the relations between the British crown and the aboriginal people, in this particular period, the crown was a little more sympathetic and therefore made these and these concessions and this and this same, we know all that have the same concession, blah, blah, blah, 600 page book. Yeah, yeah, yeah, yeah. Yes, yes, I know, it's like Talmudic scholarship in a way, you just become this, the rabbi. On a large scale, that's the reason why these scholars are paid. They'll clog our minds with our lessons. It was precisely the point I was making a while back about the Trotskyites being, you know, whatever you do, don't think about the big picture. You know, whatever you do, don't stick with the essentials. Whatever you do, don't stick with the principle determination. That's why I hesitate to read Karin's book. The principal contradiction is always obscured or left out of sight in these writings.

47:30 Anyway, you see why this condition is related to the SUD. Yes, I do. Because these cancellation conditions would probably immediately kill off any idempotence. Because to say that x squared equals x is an equation in which you can cancel one of the x's on either side. You can cancel on left, or on the right, or on both, or just in any of those cases, you deduce that x equals 1. I like this. I hope this works, you see, because an equation of x squared equals x implies x equals 1. So the idea of monomorphicity itself, you have to quantify over the whole category, right? f of x equals g of x and phi is f of x equals f of y and phi is f of y, where x and y have arbitrary domains. So it's a much more complicated thing to handle logically, so it would be nice. Yes, I think it's an absolutely fascinating idea, but I think I've really understood it fully now for the first time. Particularly how it relates and fills out, generalizes the earlier characterization of the case of pure variation in terms of this SUD, unity of the conditions to be separable, undecidable, and unramified. That's very, very interesting. Thank you very much indeed for that. This is the, incidentally, obviously this is, this is, this is, this is pro, I mean... Precisely because of the way it clarifies the role of physics and their instance relations, it seems to me that this has a very important bearing on physics as well, not just on pure maths. One of the things I wish I could have got you to... It's always my secret agenda. Yes, I know, of course. Well, one of the things I always... I regret that I... I didn't get you talking too highly about when you came for that visit. It was precisely this business of the SUD characterization of pure variation and the role of idempotence because he for a long time worked with Bohm on this idea of the central role of idempotence and particularly left and right ideals mainly in Clifford and Weyl algebras as a way of...

50:00 ...getting at what was really going on in the quantum formulas, and in his case it was all tied up with this unfortunate, again, typical, sort of ultra-leftist thing that processes somehow absolutely fundamental, we've got to get rid of stasis and have everything in terms of Heraclitian fluxes. But there were some interesting ideas about the role of idempotence in the, as I said, particularly in the violent Clifford algebras, it was a much... Explanation for the mathematical computational relations that didn't just fix the problem. Mathematics has shown that an operator is an observation. It's very strange. It's again one of those verbal things that goes in between the equations and makes the reason desired easy. Well, that in fairness is exactly... Who has ever proved to me that it has anything to do with observation? Well, which is why I was always attracted to Basil's work, because he was one of the few people who was saying exactly this. Why should we... what is this... Operator, observable correspondence. What is this dogma that sits in the center of physics that has made everybody assume that the Hilbert space is the default space for modeling mechanics, which is quite illogical. Exactly. No, no, no, but he's been saying this for years. But I think, unfortunately, he's never been able to figure out completely in a thoroughly worked out rigorous way the alternative approach, except that he believes it's... It's going to come from some deeper understanding of the algebraic structures in the phase space of the system, but the Hilbert space, the whole business of the division between, well, this whole business that ties the states, the operators to, and observables to operators is just... Slate of hands, yes. And of course, all depends on the assumption that quantum theory is an exact theory and that it will always be stuck with a linear theory as the only fundamental exact theory we have, which is there's no reason to suppose this at all.

52:30 Especially since the other great theory of 20th century physics is non-linear, which is, of course, so much hand-waving speculation goes on about how to reconcile. All the traditional extension quantities are linear. Even though they are defined, they consist of non-linear things that live on non-linear spaces. So that argument is fairly... Okay, that's a confusion then. It's a confusion which is very commonly thrown about by these people. It shows they're not serious if they think the linearity is only that. I mean, there should be non-linears available. That's true, maybe, but the actual thing, once you've passed your intentional spaces of variable quantities, these are typically linear, even though they're non-linear variables. But I don't think you're going to understand it better by taking templates of non-linears. Who's he? Hailey. Hasn't he? He hasn't taken any Templeton money, has he? Are you sure? Well, he's certainly never told me, and I hope he hasn't, because I should be very ashamed. You have to look in the list of Templeton fellows. I'd be absolutely amazed if he'd taken Teflon. He certainly kept it very quiet from me. F2X. Well, Bob Kirke didn't mention it to his friends either. Well, I know Bob Kirke has encouraged Basil Hiley to come down to give a couple of talks in Oxford, but I never heard him. Basil hasn't even got an official academic position any longer because he was retired from Birkbeck about five or six years ago, but if he has applied for Templeton I shall be very shocked indeed because he's actually heard me come up with exactly the point about the agenda of Templeton. In connection with discussions about Roger Penrose, we'd better make a move if we're going to go and see this place before we close.

55:00 Yeah, sure, sure. And he's denied, he's agreed with every word, so he's, I hate to say it, if this is true, then he's a terrible humbug, and I shall be very sad to learn it. People need money these days. Well, me too, but I wouldn't take it from Templeton, however desperate I was. I look you in the face and say that, not that I think they're ever likely to offer it to me to begin with. I think I'm a pretty safe wicket. But anyway, go ahead and do that. A very clear and sharp argument to show objectively that it is right-wing politics and it is religion without any redefinition. It needs to be demonstrated very clearly. So that these people, at least if their minds have not been completely destroyed, they will at least take some deep determination of pure variation, mathematically speaking, and in wanting to dissolve ignorance wherever you encounter it. But I think you've just got to accept we ain't going to get into this place tonight. Since this crude outline is presumably true, it would be somewhere... I mean, presumably true, but it's not known even to the people in the tourist office, blah, blah, blah, blah, which is strange, actually, as you pointed out to them. But there must exist somewhere the notes. Erasmus was taking notes. Moore was taking notes from Erasmus' private lectures. They were responding, as we've seen, as Leiden himself, in revising the universities, responding to the present revolutions, and so on, in some way. All of it. Well, all of it from both. The talk and everything you said this morning, but especially what you explained to me about the successive clarifications of the notion of pure variation, particularly this point about the role of the impotent sins. Signalling when one has the case where cohesion, as it were, is involved in a way that disrupts the case of pure variation, that's very, very interesting.

57:30 Yes, I think talking about the Banach-Sarsky paradox is a very good way of getting across to intelligent people. People who do not have a mathematical training, what the really key conceptual issues are and why it is the category theory and the topos theory is so important, because it's something that people can understand, they can get a fairly good feel intuitively for what is involved in the considerations of constancy and variation. And particularly seeing why this thing, the axiom of choice, which most of them, of course, are probably hearing about for the first time, imposes this kind of constancy in the topos in the case of... It would be nice if one could actually have an argument, you see, that's showing that, you know, if you have this solid triangle, you want to move it, but then if you could actually degenerate it... The movement is only a natural transformation at the level of dynamics, which is the next level above the cohesion. Dynamics relies on constituted relations which live on cohesive phases and in turn themselves form a topos, another but a higher topos. Which is by the way itself neither probably a pure cohesion or pure variation. I don't know what kind of sense it might have yet, actually, but, uh... Well, let's just get on with it. The idea that the possibility of degenerating a figure is crucial to cohesion, and then so therefore there couldn't be somehow natural fear variation. I don't see what that argument would be, but if there is indeed a mathematical theorem... It's conceptually a very intuitively appealing idea. Yes, yes.

1:00:00 Sorry. Yes, sure. Is a pencil alright, or do you need a... Yeah, sure. That's fair. Oh, you're good. Good man. Cheers. Ah, good man. Do you mind if I ask you, are you a student here in Belgium? No, I'm here on vacation. You're here on vacation, okay, my friend says. Whereabouts are you from? In New York, okay. My friend here is from New York State. He's from Buffalo. Yes, it would be absolutely fascinating to see what such an argument would be and to actually be able to recast dynamics in that form in terms of a kind of topos-theoretic setting as the next. The stage of construction over the case of cohesion, various relativized instances of cohesion would be even more interesting. Well, I know you've always wanted to go back into physics in the end, and that seems to me to be profound, very profound ideas which obviously have implications for physics. You're obviously younger than I am, so you could write a story, I mean. Well, I could write a commentary. He said, here's the equation. Well, it might be a better use of my time than these various... I remember seeing a character who was saying about the use and various other people have also used, including, of course, the people who... There are two mathematicians sitting in an office and... One of them is going through this whole set of equations and it looks like a very complicated mathematical argument on the board and there's like a gap in the thing and he's just written in the words, you know, and here a little miracle occurs.

1:02:30 And he said, Fred, I feel you just need to be a little bit more precise at this stage. Fred, you're going to need to be more precise at this stage. It's actually not unfair commentary on some of the things which happen in math. I was just, oh, of course, of course. And here, and here a little mirror, and here a deeply beautiful, and here something really rather deeply beautiful occurs. It's like, it's like in the old days, you know, when, you know, everybody was, well, my friend. Christine in Fougere calls it back in the days of the great pussy famine of the 1950s when people were rather repressed about talking to their kids about reproductive matters and childbirth and things before there was universal sex education, the way that people allegedly used to talk to young kids about the facts of life. And here something really rather wonderful happens. So, present generations, or just the ones before, assumed that they even used the term Kelly spaces, presuming that it gave itself a name. Actually, K is for German compact, and it was invented by Rewitz. So Rewitz was... nobody knew until I... Well, actually, I go with one other person who didn't go all the way to the source like I did, but in any case, the person who actually did put this forward has only been made known because of my propagizing, but there's nothing wrong with Kelly, it's just that he's a very good expositor of this particular line of thought.

1:05:00 And he happened to share the initial with the... I quite agree, excellent. The idea itself. On the contrary, he gives full credit. He mentions the Coppola-Weir axiom, I think, in his second or third chapter. Well, he mentions it in his preface, and then again, he mentions it several times. Well, of course, how can it not? How can it not, given its subject? It mentions it and explains it very clearly, which I hate to say, does seem to indicate that Karen hasn't studied it very carefully, to put it mildly. But, uh, okay. I don't quite know why we're sitting here. Must be a red light or something. I'll do some digging on what happened here in Mecklenburg, just some googling and checking especially on Moore, whether he might have been on a mission of some kind. I'd be much like to find that out myself. And whether Fattah and Hussain were actually in correspondence about this question of reform in the university in response to the present revolutions in both countries, yes, because they wouldn't have written that, but that's a very clear fact. Any more than the Metaphysical Club in Harvard actually mentioned the Camparis Commune when they drew up. They did, you're right. But they certainly didn't extensively, yes, clearly. It was probably rather more in the forefront of their minds. It was the biggest threat that the class had faced since the French Revolution, at least. There's a very splendid photograph which I have in a book about the Paris Commune of the last, I think the last three survivors of the committee of the Paris Commune, of the actual commune itself, of the Hotel de Ville, standing with Lenin on the platform of the...

1:07:30 This is the second Congress of the Communist Party of the Soviet Union, because it would have been the 1920 Party Congress of Lenin and the, as I say, the three surviving senior members of the Communist Party in their eighties. It's a magnificent lecture. And they still have with them the banner that flew over the Hotel de Ville during the commune, which in fact was being presented to the Soviet Union for safekeeping. Where of course it remained in the Museum of Marxism in Moscow until God knows what's happened to it now because the museum has been broken up. Part of it though, this was in the Brezhnev era, Brezhnev, yes it was Brezhnev, they did actually put on board The first Soviet moonshot, you know, when they landed a soft lander on the moon, that's not a manned one, but the first soft lander that they landed on the moon did actually, was taken out of the flag freight, given what we know of Brezhnev and the Soviet Union by the time of the 1970s was a pretty empty gesture, but still shows that they, hopes not, is still there and on the moon, you know, as a country, which I'm sure Marx would have felt was where he should have been. The first flag that had flown over the Paris Commune should be the first flag of humanity to be landed on a body outside... Humanité. Humanité, yes. All in favour of that idea. I think it's a rather fine idea. It does.

1:10:00 I'm afraid it does. You're quite right. There must have been many genuine and sincere French communists who felt pride in that moment. I wouldn't have wanted to take it from them. Now why doesn't somebody make a film about the... You know, this business about lionising these aristos like Stauffenberg and Vasilievich and all these other guys, and von Trott, especially, as I say, pampered aristos who just turned against Hitler at the very last they saw. He failed to be the effective communist in France who died in many recorded instances where German soldiers in those firing squads were so unnerved afterwards. ...or were certainly forced to think for the first time in their lives about what it was they were doing. There were many instances of Germans who deserted to the Red Army or who said that the moment at which scales had fallen from their eyes at least had begun with the shock of hearing French communists dying with those words on their lips. And if that isn't something that should be told about what human bravery is, I don't know what is. I'm kind of carried away, but... But of course, if anybody made a film like that, it would just be described as a sort of utterly naive and sentimental defense of Stalinism.

1:12:30 No, no, no. You're singing to the choir, Conrad, you're singing to the choir. That was a phrase I think I learned from you, it's one I've used quite often. You're singing to the choir. No, hang on, this is Brussels North, the first one, we want Brussels BD for you, so we still have another one after this. Oh, you're going to a different station? No, I'm coming... well, wouldn't you want... do you want me to walk back to the hotel with you? Well, it's not that far from your hotel back to my place anyway. It's only another five, ten minutes walk. It wouldn't be taking me out of my way. I mean, if you're perfectly comfortable, then we can say goodbye now and I'll get off at the south station, but it really doesn't make any difference, so to me I can... No, let's go back to the hotel. It won't take long. Not a bit. It is genuinely... I'm sure there are things we haven't covered yet. I'm sure there are things we haven't covered yet. A million things like the most of math, most of the things like what did you ask about about. But that's just a great excuse for having more teaching, no more lessons in conversations in future. The one thing we should have... Eric Warburg, who killed revolutionaries after the First World War. He's the one that... And helped... Yes, that's the one. He's the one who was Dulles's... Yeah, that's right, the guy you're talking about, the Freikorps, the guy who murdered the... Yeah. Or helped murder Liebknecht in Luxembourg and those people at Hauser. Which again, very interesting, Heisenberg suppressed his participation in that. Freikorner, ultra-rights, counter-revolutionary coup against the Munich Soviets, as the Bavarian Soviet Republic, in which, of course, Hitler also, of course, played a part as a political agitator on behalf of the German general staff before his demobilization, in fact, it was in the immediate aftermath of that that he attended the first meeting of the...

1:15:00 The first meeting, it already existed, but the first meeting that he attended, as they pronounce it, aren't. Yeah, I mean, different things are suppressed and then sometimes they come out. For example, Heisenberg suppressed the fact that he don't know what he's constructing. All of the above. The centrifuges, the pile, the atomic pile, which they constructed. And it was still going strong. Not at all true what he was trying to say. The last 50 junctures in the history of IHES. You know that. Yes, yes. Because this has nothing to do with me. No, no. I didn't know about it before, just a few months ago. Right. I knew nothing about it. So people get confused about it because, you know, I'm the Buffalo man, right? No, but I was not at Buffalo yet. No, no, but you were not at Buffalo when Lord Vicar, sorry, when Grosvenor, that was a Freudian slip, but one that I hope you'll take as a compliment. Yes, I do take it. But when... You were not there when Grotendieck came in 1973. No, I know that, I well understand that. So I've told everybody about this. So a number of people are confused on that point, but I have always corrected them whenever... Colloquium talk is only the tip of the iceberg. We don't seem to have a record... Fantastic if there were recordings of any of Grotendieck's seminars, that's something I... But I never dreamt that... Sorry, I'm listening. I'm sorry. No, the thing is that in 72, 72... Brodenbeek needed money to finance his crazy communes. Oh yeah, Deep Ecology, yeah, Sovivra, yeah, Sovivra. Got it? Yeah, yeah. And it happened that Ismael was able to find, to give him a lot of money in return for three courses, which total about 112 cassettes.

1:17:30 Where are they? Where are they, Bill? In my garage. You... And you've been keeping that under your hat for how many months? Oh, well, I forget now. Okay, two very important questions. Have you actually listened to any of them to check what the actual content of the quality is, I mean, technically? I'm listening, don't worry. Oh, shut up. Shut up for a few minutes, at least. I'll give you the facts that I know, which are limited. First of all, I'm innocent. I knew nothing about this. Duskin had recorded, there are three courses, one is on algebraic geometry, about 50 cassettes, one is on algebraic groups, about maybe 30 cassettes, one is on topos theory, about 40 cassettes. So Duskin had recorded all these. And he's been planning all along, you know, I'm sure you have these plans as well. Well, you have even these specific plans. Well, I have certain plans that I never had carried out, but I'm still meaning to sometime in the future. So he was hanging on to these things for 33 years. Yeah, 33 years. And then, suddenly, there was this tragedy. He had a stroke, you see. Yes, I know. So Duskin is semi-immobilized. He's not really, wouldn't really be capable to carry out such a, well, let alone transcribe. Even to go to somebody's office and have them transcribe would be a major... No, no, he can't. He has a constant companion to help him and he has a cane and so on. I mean, he can do email with one hand and so, this kind of, so he's... Not absolutely disabled. No, no. Somebody disabled himself. Yeah, no, I understand, exactly. So he made the decision. Michael Ray made the decision. That was not just with respect to this. Actually, by coincidence, you know, the young man we saw on the station here, just as we were leaving, he waved. Oh, yeah, the chap just waved at us. Yes, he was at the conference. Yeah, he's from, anyway, his ambition is to, you know, to process and publish, finally, Jack Dustin's great life work, which is a...

1:20:00 A book which is known, I mean all the people in the field know that he had this manuscript and would only have required a few more months of processing to be ready for publication and then now it is in limbo I mean if Jack doesn't get some major help and he might even have not had the spirit to do it because it involves so much well anyway so he has that major project and one or two other things So he's now after the stroke was already maybe three years ago or something but he's he and he tried rehabilitation and it dealt a lot but didn't manage to bring him out of this okay so he's made a conscious decision that now these these projects that he won't be able to complete them on his own so therefore at that point coming back to the one of specific interest he simply gave me this box full of tapes and said do whatever you want Subtitles by the Amara.org community Okay. Right, well we really do have to... If you see what I mean. Yes. Yes. So besides you, I told one or two other people. For example, Huzel. You see, because I had this idea that it turned out to be maybe not so good, but since Huzel has experience in history of math... That he must have had some experience in sort of organizing the extraction of information from strange sources and that he might have sourced, especially, access to grants. ...grants of my finance such an operation. And he said, oh yes, anything of Brogdick is definitely important, and he supports the project in that way, but I think my impression is that his initial attempts to find either funding or willing slaves to do the translation has not... And the other thing I'd say very quickly on that is that my experience, which is now five, six years standing, is that, frankly, the worst people... To try and involve in anything like this is any kind of French official agency because what will happen it'll disappear into the more of somebody like Lillian Beaulieu

1:22:30 and it'll be put into the sort of Warbaki archive and it will never see the light of day. That's the problem there. Whatever you do I'd say don't entrust it to the you know to any official I think, no, we've obviously, you know, my mind is racing like, what have I done now, obviously. I hope it's not sounding selfish. I really would want to be part of this because it would, for me, be the fulfillment of my life. Yeah, yeah, yeah. As much as the project of, you know, transcribing your stuff and not getting it all into the public domain is, as you already know. Quick question. Did you also talk to, I know this is a relatively minor point, but I can pass them. Did you actually talk to Jack about the fate of this? ...kind of chart or list that Grotendieck drew up... Well, in fact, I went to his house in order to ask him about that. But then, this was so jaw-dropping. Well, this obviously was so much huger than anything like that. Yeah, that's a pretty minor thing, but it would still be extremely valuable to have as a document, extremely valuable. He said, yeah, I have it, it's in that drawer over there, but I can't get to it. There's a physical problem. But it does still, the thing is, it still exists. Yeah. Okay, that's the thing. Physical problem. He's not going to throw it out or anything like that. Okay. He's aware of its importance. Okay. Well, that answers one question. Okay. Well, the obvious thing is, I, okay, first of all, I obviously trust, I will keep what I mean, my strong wish, of course, was to tell you and Pierre Cartier about it, but I thought because, I mean, this was just my imagination, because of his particular involvement with history. That he might have, and he has a different character than most of the French anyway. Oh, totally different character. But anyway, this one, he was the one I told first and that's it. Oh, you said Pierre Nozopredito. No, no, no, no, no, no, no, I meant Christian Hazel. Oh, Christian Hazel, yes, yes, of course. But certainly Pierre should know about that. No, no, well, I'm quite happy, okay, but I certainly... I doubt if he will have any, actually, any practical suggestions. Probably not, because although he's a wonderful human being, a delightful person isn't the most practical person in the world, to put it mildly. I don't want to get too many people involved. This was my instinct, was not to get too many people involved. Your instinct is extremely sound, Comrade. I really mean that. There's one other aspect about that. I absolutely do admire and respect and revere Pierre Carter intensely, both as a mathematician and a person. But I'm not quite sure I'd tell him about this, because he is one of the world's biggest...

1:25:00 If you tell him this, it'll go all around the houses. I mean, half the people in France will know about it. I'm going to tell him at a certain point. Yeah, at a certain point, but at the point when we've already got it exactly sorted out, what we're going to do with the things. Yes, no, I certainly would tell him, but at the right time, because otherwise... You may recall that a couple of months ago on the phone, I quizzed you, or I wrote you in a letter. Yeah, yeah, about... Detailed questions about how difficult is transcription and so on. Well, I think I mentioned it the other day. I had this extremely useful and very helpful discussion with David Rowe in Mainz last month. About the possibility of my archive coming under the wing of the German Historic Mathematics Society and they're providing funding indirectly because it would actually be provided to David Rowe and his colleagues but they would then pay me to carry the project out. For its transcription. And I've done a very detailed costing on this, very detailed. Oh, yes, I did this already a couple of years ago when I was trying, forming the thing into a trust, into a charitable trust. And trying to get various people to agree to be trustees, which I did succeed in doing. Is this the way to ask? Yes, it is. Yes, we just go up here and to the left. The reason I didn't ask you was the very simple question, the very simple reason that the... It does video-audio work as well, you know, and so you told me the name of the program you should use, was it Audacity or... Yes, yes, but is that not the same one that I mentioned to you that has come out in the last year? There are two or three now that you can use, but the only problem is that it depends entirely on the voice that you're listening to on the tapes. For instance, your material, quite honestly, wouldn't work with that program. What you'd have to do with your material, because you do have a fairly soft voice... Would be for somebody to speak your words into the program. I've tested it. I've actually tested it. Not on your recordings, I have to say, but on about half a dozen recordings. And I find you can transcribe material at about three times the speed that it could possibly be transcribed manually, just by listening to it, which is very good. Obviously, it's a huge improvement. You speak into a word recognition program and you'll then of course still have to correct the text but if provided that you've got a clear speaking voice and with the latest generation of this software it only has to be pretty lightly edited and then approved at the end it's certainly by far the quickest way of getting it done and that would probably be the effective way of transcribing these of course.

1:27:30 I've never heard a recording of Grotendieck's voice, so I have no idea how clear a speaker he is. I'm just going to tell you a little bit, if I may. I'm sorry, I've already said too much. Something I had never realized in meeting, hearing, and speaking with Grotendieck is that he has a rather thick German accent. It's not so surprising given his background. Well, no, of course, but if you hear him on the tape, it's clearly... ...German accent. Not one that's incomprehensible to an English speaker at all. I don't mean that, but just that it is. No, on the contrary. Actually, on the whole, German accents tend to be a lot more comprehensible to most English speakers than strong French accents. So it's English with German accents. Okay, so it's accented English. It almost certainly will require being transcribed by being re-spoken, which is what I've done, as I say, with the selection of tapes that I already tested this for. It's pretty effective, but of course it requires a massive investment of nerve. So you're probably looking at about three or four years of work there. So it's roughly that many, well, I guess maybe 150 hours or something. Yes, yes. They could do things like that. It would obviously take, first of all, the point is it's, well, first, I'm talking about an initial process, changing a hundred actual cassettes into DVDs. Digitizing, well, that's pretty straightforward. You just, well, I've done that. Wait, wait. I'm sorry. I'm sorry, Neil. Sorry. Wait, you want to estimate your own problems. In a way, I mean. Straightforward, but it has to be done in real time. The audio cassette cannot be run at a faster speed, apparently, therefore 100 hours, 100 tapes means 100 hours of somebody sitting there feeding it in. And he said, well, that's the first thing, and then the second thing is, well, we have students who do that for $15 an hour.

1:30:00 It becomes very expensive. Look, I'm sure this guy knows what he's talking about if he's in the computer sciences department, but it's not my information, the guy who gives me information, who is very good, who's a guy called Benoit Daval, whom I'll be seeing in fact tomorrow in Paris, assures me that the programs have existed now for at least three years, whereby you can both compress and re-record audio to digital. It doesn't have to be done in real time. I could be wrong. Audio tape. We're talking about... Yes, we're talking about standard audio... 35 years old audio tapes. Well, if we're talking about standard audio cassettes, we're talking about this sort of thing. Yeah, exactly. Yes, those, unless... I see no reason, I could be wrong, I will check with Benoit, but I see no reason why those would have to be done in real time. Okay, well in that case I'm open to correction on this, certainly, because I don't, by the way, we want to go this way I think to get back to your hotel, in fact let me get my bearings so I don't want to wander off, it should be just going behind the church here, it's the Avenue Namur isn't it? Yeah, just let me get the map out because otherwise we're going to wander around in circles and I'm not quite sure, I think... That it's up there behind that church and then up to the left. Yes, yes, which is why I didn't want to go down that way because it's obviously going down. Let me just check. Okay, well look, I'm completely open to correction on that. There may be some technical reason that I'm not aware of whereby it has to be done in real time, but all the same, real time, 125 hours, this is so important that you just make the time to do it. Also, you don't actually have to have somebody sitting there, provided the machines are running properly, you should just be able to run it on an audio tape deck, a double audio tape deck, you know, which you use to re-record, which I've got a couple, really, all you ought to need is a lead, which you put into a computer with a USB port and with the right program to transcribe the stuff, to burn it into the...

1:32:30 Well, obviously to re-record it digitally and then burn it to a CD. It may have to be done in real time. So again, about the CD, I mean, the guy pointed out that if you do it on a CD, it's more or less one-to-one, one cassette tape per CD disc. Where as the DVD did, he said you could get the whole hundred things, I think. Anyway, the ideal would be to have three DVDs, one for each of the three courses. Yes, yes, yes, I see from the point of view of actually making it available. And then even though the DVDs are not, you can't play them on an ordinary home. Sound system, I think. You should be able to play a DVD in any computer. But you can do it in a computer, yeah. Of course, yes, anybody. I mean, a CD, you can play it in a home audio system. Well, true, but most people have PCs these days. It should be, yes, a DVD would be much better. I have to say, I think... Sorry I'm thinking so slowly. No, no, you're not thinking slowly at all, but I don't think, I think, I think he's probably being over-optimistic if he thinks you could get, if he thinks you could get up to 30 or 40, which is what it would be, assuming they're one-hour tapes, up to 30 or 40 tapes on a single DVD. That sounds to me, even with the... The very best current DVD technology, being a bit optimistic, I would think they would probably fill at least three DVDs each, but still that would make a complete set of maybe nine or ten DVDs. Well, DVDs are as cheap as anything, obviously, you can buy a pack of 50 or 100 of the things for about 20 bucks, that's not the problem. The problem is having the thing in a form which does make it accessible, but I think you're probably looking at a set of about nine or ten DVDs at the end. You're quite right, DVDs would be the way to go. Okay, well, look, I think this is so important that... The next step, then, since we didn't have, we didn't find a solution yet, I had another brainstorm, and I proposed it to Vito, but he hasn't answered yet. Maybe it's too stupid or crazy, but I thought, well, my guess is that instead of transcribing it at all, we could make this set, when you say nine, small... Whatever number, yes, whatever number, whatever number. And simply make ten copies of this and give it to various students. Well, that would be very easily done, yes. Who could then have their own students transcribe their part of it? That would be one route to go, certainly. One route to go. So, I mean, there are several people, like the more responsible people of this rodent-deep circle, like this lady Lila herself.

1:35:00 Yeah, be very careful, obviously, with them. But you see, the point is... If ten or so people have it, one can't do too much damage in terms of falsifying it. I see that's a very important consideration. This is why I think that actually having a transcript would be rather valuable backup to make sure nothing like that can happen. Well, yes, I mean, in the course of that, then we could also get the money. You know how to make our own full transcript. Yes, yes. Well, now I think this is, I think it's very important for many of you to know that one is that it shows that Topo's theory didn't stop 40 years ago because he was giving a major course on it. Yes, only 35 years ago. Exactly, exactly, yes. But worse than that, you see, I have this intuition that... He himself sort of rediscovered topos theory in the process of doing it, because after that, but not before that, well, to my knowledge, at least, well, to me, he said, topos theory is the most important of the many things I've done. So I have the feeling maybe he realized that during this course. During this course, yes. I mean, he knew it was an important tool for this, that, and the other thing, but realized the significance and perhaps, you know, I mean. It could be very interesting in that way, that there are new ideas about toposes that arose, classifying toposes in particular, that would be relevant for algebraic geometry and algebraic groups, and which could finally Take the scales away from some of the people who listen to those people in Atiyah. Not that they themselves could ever have the scales, but you see, so for that one, that course in particular. No, this would be far more than purely historically important. Yeah, exactly. Oh yes, I understand that. This would be much, much more than just historically important. Even purely historically it's very insignificant. Of course. But it could be even more than that in the sense of, to some extent, participating in the present. Mathematical discourse. Absolutely. The other thing is that this colloquium talk which I did here was clearly the product of his teaching the algebraic geometry course and so the details, I mean this was just a sketch really of the fact we need to overthrow the Groton-Dudonnet notion of scheme. Yes, yes. To go to a completely different notion of scheme based on the... He described this. So he said that there but he must have developed it in detail again.

1:37:30 Well, I mean, the likelihood is that we either knew or have discovered like maybe 80% of what, but the remaining 20% would be, could be quite shocking to those people or even to us, you know, I mean, anyway, I'm just aligning the speculative facts. Hardly anybody seems to know about his redefinition of the notion of scheme. People just don't listen to it. They just don't listen to it at all. I mean, all of this non-commutative topology stuff that von Oesterden and people do is entirely based on the old, you know, previous Grotenbeek definition of scheme. Oh, yeah. And, uh... Really, it's completely undercut by his later rethinking of it. No, I completely see that this would not only be of great historical importance, but could also be of quite transformative significance to parts of mathematics itself. So it's obviously a project of huge importance. It has to be taken very seriously. And of course the algebraic groups is one field where they use the nilpotent things the most, systematically for differentiating problems. So this could... This could demonstrate to her, you see, that it wasn't John Bell, or even me. Yeah, yeah, yes. Well, I hate to say it, but letting the scales fall from... Karen's eyes is almost the least of my concerns right now after what you've just told me. I mean, no, no disrespect to her. I think this is really the most important thing you've told me today, and I've told me lots of very important things today, which I've learned. The other thing, I think the final thing, and I think this is all I know, and it's very sketchy. I have tried myself to listen a little bit to these tapes. And apart from the fact that the accent is German, there is this observation. They're incredibly boring! You know, because he's going on, he says, well, let's take a space X over Y, and then let's look at a section over the empty set. Let's look at a section at a single point. You know, he's sort of pedagogically going through very simple moves on everything that's introduced. And so it's incredibly boring. Even if one didn't already know this stuff, it would be boring. So the toll on the... What the transcriber would have to be paid is not just for...

1:40:00 I'm sure practically any transcription job is like that. Yes, that's no different from any other job. Your transcripts are probably boring, too. They're much more boring than anything that Grotting's going to have to say, believe me. Not really in terms of the content, but the content per minute is much less than you imagine when you listen to a talk. Yes, of course. That always is when you listen to it. When you listen to it on a tape, it's going to appear boring. And it's also, of course, which is added to by the fact that unless you actually... If you see somebody writing on the board or it's a chalk and talk performance, just having to reconstruct it from the words is always much more difficult. That's even the disastrous part of course, we don't have that. No, but you don't have from any, I mean that's, I mean it's enough that such an incredible treasure trove of recordings exist, to have them to complete with the working notes. ...of what he wrote would just be too good to be possible, too much of a miracle to hope for, but even so, what you have is of incredible value, clearly, so we've got to do something about this and I will... And so another conjectural point is that because of this, because of the total lack of equations and diagrams, it may not even be the most practical thing. Transcribing, to transcribe word-for-word the actual... No, no, that is always a problem. Rather to, for somebody who knows the subject in a general way, to just listen to the whole thing and write a detailed synopsis. And write up a synopsis. Detailed synopsis. But that would of course require a very competent mathematician. A very confident, trustworthy mathematician is willing to make a very considerable investment of their time. And one who has a real resistance to boredom. Yes, yes. I keep trying, I keep underlining that. I think that the raw transcript probably is indispensable, however boring the project would be. I would be quite willing to take it on and give, you know, four or five years of my life to it if somebody could find the means of, you know, keeping me, you know, body and soul together. But, hang on, let's go across here. But, okay, look, I mean, obviously, we've got to stay in touch on this. Well, stay in touch anyway, for all sorts of reasons. Yes, you've given me enough to keep me awake all night. Leave me, no, with excitement.