Discussions, incl. FW Lawvere, G Khatcherian & M Wright

0:00 Which I think he may have thought of as grids or pebbles, given the materialist world picture, which probably wasn't. I don't think the way that he thought about structure was... I think he really did think about that there was something as pure structure, in the sense that... Well, I think you'll be terribly disappointed, and I should warn you. I mean, I was very disappointed. It's a letter that he wrote to John Mabry, which he sent me a copy, and it came the day after I got back before, and well, I've already scribbled all over it in my opinions of it, but it's very strange for me, indeed, I, well, you've been to Sweden, so I'm very surprised because I learned a lot from going down an analysis, I learned a lot of mathematical logic from my show. I'm surprised that he, well, I think, will misinterpret. It's, to me, as if he's never really listened to a word you're saying. But, uh, anyway, leave it. It's all right. For example, I'll make an important example of C, of the, what are called probabilities. Right, in other words, instead of having the, uh, usual, uh, additivity, if you have two disjoint subsets of events and you take their union, the probability of a union should...

2:30 And so on and so forth. But not one of, not something that I'm probably, these are POX in the sense of, roughly speaking, used in the sense of additive set functions, you see, that they'll be the kind of thing, there may be small technical differences, but conceptually it's the same kind of thing as an additive set function, but the things that are additive on subspaces of Hilbert space, with respect to the orthogonality relation... There is a convex set, so it's an object in this category that is very much not of the form p of x. It's a quotient of p of x, which is the general fact, but Riesens' theorem is sort of going in the other way, you know, it's a property that this particular example has, which shows how much, how really far different it is from a pre-convex set. Yeah. Well, I've got a... Actually, Helzberg, in that survey of quantum logic, does have a fairly full statement of this. Well, in fact, the original paper is reprinted in that book on the... Well, just for the purpose of summarizing this, so therefore we have a contradiction of why didn't Brown persist?

5:00 Well, Walter Point, yes, why didn't Heiley work on this road instead of the way... Well, I think because he was already going down algebraic geometry, which I, the thing I was saying, I think might be an instance of... This is, this is somewhat... That I want to talk to you about. Not quite so simple and abstract, but still it's clear that his accents are crucial for some kind of convex set of states defined on this convex set. This convex set could never be P without some relation. But you see, this says it's P modulo. Let's make this more precise even. Statistical systems, modulo, and other ones, where you're using code relations. And of course, these maps could be thought of as two random maps, and that's a direct, and you co-equalize those with the concept sets, and that's the arbitrary concept set, including this example, which is considered kind of a fun kind, by the Hansi and the Seminari. I mean, abstractly, this is just like, say, group theory, you know, the three groups, which are very, very special, but an arbitrary group is a quotient of a three-group module and other three groups. Now, due to the influence of Lebesgue and Kronecker, the countable additivity is considered very important. Yeah, I was going to ask you, because it's not in fact going off the subject as much as all that, about your remark about Lebesgue integration.

7:30 Sorry, my sleep deficiency is catching up with me a little bit. Kroniker, Lebesgue. Yeah, Kroniker, Lebesgue, as opposed to Piano, who's the other guy? Grassmann? No. Well, you didn't mention Grassmann. Oh, yeah, yeah, Kurzweil and his stuff. That's right. Can you explain that to me, the difference between your practice? That was one of my sort of six top questions I wanted to ask you about. The role of industrial processes, their rise to differential equations. Autonomous means the T occurs there, or F is not Lipschitz, let's say. I mean, the usual elementary theory of differential equations is based on the assumption that F is Lipschitz, so that one can apply various simple-minded methods of fixed-point support to get existence of the differential equation. In actual practice, F may not be Lipschitz, so one needs a more subtle kind of integration theory in order to... We developed this notion of integration, which then end-stocked. I think it's British, actually, perhaps Scottish. I forget the dates. A little bit later, I guess Scottish included in British mentality. And then later on, even the great American integration, like Shane, there's a Charis monograph.

10:00 There are systematic methods. It's 1893. But anyway, it does have... This is called the generalized Riemann integral. The basic point is this. Suppose you have a measure mu and a function f. And the integral of fd mu is base x. They're all epsilon greater than zero. There exists a delta. So that they're all partitions of x. Size, less than delta. The elementary energy, less than epsilon. The whole Riemann's early notes, but this is a story like Lakotas and John Fleet's, about Cauchy's. So I don't know if it's valid or not. But I should tell you who told me about this.

12:30 He was an American who lived and worked in Florence. He says in the marvelous 12th century, power is what enjoys the best of girls. He collects Picassos in a way for a hard time. But he's the one who first explained it, and I think he's the first one to have proved the fundamental theorem of calculus in this context. I'll come to that in a moment. Please do. It's extremely natural for the fundamental experiment to calculate. But anyway, the historical problem, if you did, is maybe Riemann meant the right thing all along. It's just that he was misinterpreted later, or maybe he made a mistake himself later when he wrote it up more precisely. It has to do with what do we mean by delta? We mean by partition. So the vague idea was two things. Delta is a constant. And a partition is countable. The AI, the family of sets, is summed through the whole space. Of course, there are measurable sets, whatever that means. I mean, in a simple case, you could just take intervals, but it's a countable number of them. Of course, in either case, a partition involves a family of sets and a family of chosen points in each of the sets. You evaluate the function at the chosen point. You multiply that by the measure of the piece that it's in, and you just have this sort of disjoint, I mean, I suppose the natural thing is to say that the measure of the intersection, they're two different ones, so you could say that they're literally disjoint, but again, that's not the important thing, but the thing is, delta is just a constant number, partition is countable, and size, oh, size, means that the soup is right, let me go back.

15:00 For the size, the largest one of these should be less than this constant value, whereas the Kirchweil-Henstock, McShane generalizes this, the point, the sample point might sort of just be on the edge of the set and not really in it and stuff like this, but again, I don't think that's, at least Goodman and I agree that that didn't seem to be a good thing to make that generalization, Kirchweil-Henstock. So delta is an arbitrary rather than a constant. On the other hand, the partitions are all finite. There's a finite number of sets, so the countable, of course, is arbitrary, so you make the epsilon very small, you need to take very many small pieces, but finite, no mysterious... The definition of size, well, no problem, the size of the partition, you have a finite family of sample points contained in a finite family of Well, I mean, to say that the size of f is less than delta, which is the actual clause that occurs in the definition, is simply to say that the measure of the i-th set is smaller than the sum of delta applied to the sample point.

17:30 Now notice that the original thing has a trick here, because it's by definition, if not F, then F is the value of F. Yeah, yeah. Now you sense that they do not require that. Mm-hmm. So that implies that F is, in fact, not conversable. So the finite partitions, such as the functional delta, The fundamental theorem of calculus is actually more general than Lebesgue, more functional than Lebesgue, and the other thing is the fundamental theorem of calculus. Yes, I was going to ask you that. You see, again, Lebesgue and his followers have some version of the fundamental theorem, of course, but it has hypotheses. The true fundamental theorem of calculus has no hypotheses. Yeah, but if big F has a derivative, that's the only assumption, one which is necessary to even make sense out of the formula. Sure, I was going to say, I was going to get into it. The derivative of F implies that F is a little less, or has a little less, but the vague version has additional strange hypotheses due to the unnatural assumption that delta is constant and due to the business about it. ...requiring absolute instability to make theory work.

20:00 And you see this is, why do I say this is very natural? Because the definition of derivative is local in nature. There's no uniformity between the delta and the definition of derivative. The function of derivative, say f has derivative little f, means at each point it has a derivative. So you have some story where f belongs in delta, but at each point independently, more or less relatively independently. So naturally your delta is a function. So you just sort of substitute that back in. And if you read that formula backwards, you have this theorem. There's no... it's because this is divine to me, and it's the result of... It's the usual... I mean, as a special, special example of this implication, your famous assumption would be zero and one according to whether the point is rational or irrational. Classical Riemann, or classical Riemann, or constant delta. Not that anyone would ever want to use this function in real life, but it's the one that they put a great store in. It's easily seen to be, in a sense, it's just a point that the function delta has to be chosen in a very clever way, so it's behaved one way in a rational way and the other way in an irrational way. Partitions can be finite. You see, intuitively, the idea is that if you're going to integrate, if you're going to achieve this approximation, if f is very, very highly variable in a certain region, then you're going to have to take very small partition pieces there, and the function delta is going to have to be quite variable there as well, whereas in a part where f is more slowly varying, you can afford to use quite coarse partitions and quite relatively constant delta. But all these possibilities... All of these are allowed, and so even this monstrosity can be easily accounted for. Near a rational, you add a rational, you do one thing with the delta. Near a rational, you do another thing, likewise for the size. I mean, that's all explained in MacLeod's book. Right. Whereas, of course, in Riemann's version, you really couldn't handle this.

22:30 That's right. That's what I'm saying. That's the example that's always mentioned, first of all. How hopelessly backward this remodeling is, it can't deal with these important functions that come up in everyday life. Well, it's actually, yeah, a happy irony. Of course, it's exactly the sort of thing that Lakatosch lent his support to by all this stuff about monster-barring, the way that one stretched concepts. The monster-barring was the most important way in which concepts got stretched in its reputation. The big integration, which is really a lot of it is seen when you look at it really carefully, to be very hasty. Right, right. I mean, just something I could mention. Well, I mean, in other words, all of these things that are produced by bad infinity, or the examples that are constantly provided with me, you must have these. You must have this function integrable. You must have the Piano curve still in space. You must have the... Well, no, I don't think that, you know, see, the trouble is monster barring is supposed to be an ad hoc procedure. You, you know, once you stretch your concepts, you no longer, you know, the monsters no longer need to be barred because they fall under the concept once they, to accommodate the new examples. But I, this is very, this is very helpful to me now. It's a lot more about, I always thought that the Riemann theory sort of intuitively seemed to matter. Yeah, well, the finite partitions, as I say, it's a thing which anyone would use in actual mathematical analysis. So, easy-peasy definitions of each point, that gives non-constants, delta, whatever it is. Well, thanks. That's helped me a lot, that. Now, the next thing I really wanted to talk about, I guess, is to go back to the intensive-extensive quantities, pulling forward into the CCRs, but I think my sleep deficit has come up. I'd like to go and just score some crash for about an hour. Can I do that? And then, because Jerry Kacherin's coming at 6, between 6 and 6.30, which is, it's now 4.30. So if I can just crank out for an hour. And if you want to, incidentally, Peter will have plenty more cigarettes, because he's the only man in the world I know. So, if you want to get some. Is that okay?

25:00 Can I take those? Oh, yeah, okay. Yeah, and then if I can just, as I say, get a bit of rest, and then I'll be fit for talking this evening. And I've got the papers, the notes of the intensive and extensive modules, the CCRs, which are to go through. And I've got the list of the other things I wanted to ask you. Oh, and I'll try and dig the tape out, too, before we go out tonight, just so I've got it there to copy in the morning. Okay, anywhere else? If you want to get some more tea or anything, I'll just watch a bit of television. Hi, great. Hi. I'm very nice having you both. Have you got any cigarettes? Yeah, yeah, he stayed here last night. He stayed here tonight. He stayed in the back. I'm just going to go and crash out for now, because we didn't get any cigarettes last night. Well, he did, but I didn't. Because we came down from the... on this little train from Bangor. So, I'm just going to crash out for an hour or so, because Gerard is turning around about 6 o'clock. Oh, well, he might be here a bit early, actually, if I can give you a lift back in. Yeah, sure, that's one of the things I wanted to do. You really can't sort it out with this little bit, because it sounded very bad. What happened? Well, the situation is that it's okay, but there is a problem with this guy on the stage because he hasn't produced the tickets. Yeah. I mean, have they got the tickets down? No, they haven't. Well, that means they haven't split it down the two, doesn't it? Well, goodbye Witten. They won't give me anything at all. They're all the same.

27:30 Maybe at one time. Anyway, let's talk about that now. Yeah, there are too many to be... A convex polyhedron is finally generated as a convex set, but you just take the obvious extreme points and choose on that, we'll map onto mapping as a convex set. But as soon as you have some round part in it, then even if the object itself is a finite dimension. Of course, the thing in quantum mechanics is itself a dimension. Yes, of course. The number of generators being necessary already occurs for a finite dimension.

30:00 So this approach, the sort of metallic, matty approach, is really much more general than, say, the lattice approach. Anyway, the next thing I wanted to ask you about, actually, is Molde and the quantal stuff, which I know you think is bad math, but I know... I'll give you a few... Anyway, I'm just going to crash out for an hour. There's a version of this. Right, that's what I want to understand, the difference between those two different versions. So let me... Oh, do you want to see this paper of Adrian's? Oh, it's only a... No, I haven't got that. At least I have, but I can't find it. I was looking for it last night. No, this is just the, this is not, so this is just an abstract. Oh, yeah, well, I have this too. Oh, you have that, yeah, too. But that's where he makes the point. This is the stuff about algebra and geometry. You see, this is his, I'm putting, this is where he's coming from. And this is a very sort of generated topic. Yeah, well, I have this. Oh, you have it, okay, fine. Okay, well, I understand it. We must come up with some... Well, I actually sent him tapes of them. He's one of the messages that was left on the floor, but I promised to do some for him when he gave talks he hadn't got notes of, and now he's, I mean, you know, simply because, you know, he's saved people. It's not actually part of the defense, but Peter has to be involved with that in his fellowship. God, that's why he's been invited to Berkeley for years. But mainly because he doesn't have a universe left to show. So anything you can bet even the level of shows he's had were published in another field, like science. I mean, my basic strategy, I think, for having done so much work and point out that he should do about twice as much, is to find the opposite conclusion. Uh, yeah, well, anyway, I'll dig, I, I, actually, we'll try and dig it out. Oh, you're going to stay here, are you?

32:30 No, I'm going to get a bit out of bed, I think. I, I'll just try to get to bed before I go. Okay, so I'll leave my stuff. Yeah, I'll leave your sleep stuff. Uh, I just need to get a piece of some of these satellites in. Oh, that's okay. No, no, we'll help. Well, it's okay, I'll take this. Uh, yeah. Uh, well, you could, uh, if you want to use the washing machine, hang it in the yard. Well, yeah, you can put it in the dryer. You can put it in the drying cabinet. Except there's a whole lot of stuff in there at the moment, but we can do it overnight. I can just put it out in the yard. Well, you can also, I mean, it'll get it dried here quicker than anywhere else. Why don't you put it in here? We'll just, um... If you just stick it in here, Bill, just sling it down here. I mean, I'll clear that lot out of there, but all I have to do is just put it in there and turn it on and that will dry it. That's the dryer, isn't it? Yeah, that's the drying cabinet. The argument seems to be, given the equation of definition, how do we...

35:00 Alright, A and A is A. I call this the three wheel, because there are a lot of operations that are associated with this. You can go ahead and call them that. And then there are axioms like distributivity, which are mysteria. What? Mysteria. Mysteria. This is an attack on Birkhoff and von Neumann. If you look at them, you see it's a very formalist sort of comparison of two lattices, hence quantum logic is borned painlessly. You can't axe distributivity. And then you maintain all the other operations, maintain their meaning, and how do they, how can we read the equation of the definition of the meaning at face value as the logic, unless we go to the core of the equation, we can start before and after. So I'm not doing that. It's a pity you're leaving too long. The non-distributivity is equivalent to the absence of implication. And you obviously know John's paper on this, don't you, that it's equivalent to... Well, they make the point that it's not the eye of the Trinity, so... Well, who makes that point? Ben and Herod. Ben and Halit, what they're saying... No, I'm not talking about Ben and Halit, I'm talking about John's paper on the semantics of ortho-logic, the failure of persistence, the frame semantics... Yeah, you gave me that paper with all the ortho-pages without even... Oh, I've got the copy here, so look at it. This is what I want... But that's interesting, because I think they were biased... Maybe I'll let you read this, because it's been eight years since I read it. But the argument is that distributivity is equivalent to failure of persistence of statements in ortho-phrase, that is, say, once a statement is forced in a Boolean phrase or a Heiting algorithm.

37:30 Statements, once forced, stay forced, but they don't in an ortho-phrase. And that is also equivalent to the breakdown of the covering, localization of coverings. The condition that you use in topology, one of the three conditions on a sheet that the sieve restricts to only a smaller open set, fails for an ordinary frame. It's equivalent to covers don't localize. So you can have a disjunction of R and B. Say R is red. Let's say R is red, and think of it in terms of manifestation of properties or attributes over parts of the space. You can have a manifestation of a disjunction of properties over the whole space, but no subpart of that space will manifest a disjunction of those properties. And that gives you a model for superposition. It also gives you a model for incompatibility, because you can also have... Well, what I said is, let's talk about one thing at a time, we're moving from subsets to subsets, if we look at the definition of implication, the right adjoint, it doesn't exist, however, If we say, instead of looking at intersection, let's shift our attention to projection, orthogonal projection, it doesn't help. That was the diagram. What I said is, if you know projection, the axiom for homogeneity, see what I'm doing here is, these expressions are known, they don't present the subject, you have to look for them.

40:00 Yes, that's right. Sasaki, Wilkins. From that point of view, these are derived operations. Right. So say A is the subspace of the plane in E3. Yeah. And A star is the octagon. Yeah. And say X is simply a vector. How do I obtain the projection? If I take the subspace generated by X and the octagon with A, I get this. That's all. So I get this plane. As I intersect it with one, I obtain my projection of x. That is the one that has the right edge. In fact, this very statement is the technical axiom of the modern architecture. These are just technical things. They're the definition of adjoining elements. You see, distributivity is the statement that intersection has the right edge. Now, in this quantum logic, not in the sense of Jung, but with the automodularity, because already it's weak in other structures, what we're saying, automodularity is projection, hence the notation. Yeah, exactly what Helmsworth is trying to do with his quantum telephones. Well, I think we gathered that he was trying to use... Well, that's one of the main points of the way that he did that. He was trying to introduce you to our admiration academy. That's... That's the same statement as this statement of automobility, as this statement. I mean, there is an industry of producing equivalents of the automobiles, which none of them say anything really geometric. What if you compose two of these, you take A and B? I don't need to. No, you take two, A and B. I don't know. Maybe we can find a few more. Let's proceed on. I can give you a copy of this one. That's the last copy. So let's proceed because you come in.

42:30 Inherent, but not expected. Multi-layered. No, I don't mean to. That's your definition. So you have to. And this is what I say is the categorical. Here we have the noun. So this is the equivalent to the orthomodularity. Yeah, that's equivalent to what they call contralogic. If by contralogic they mean automodular orthologics. Yeah. Although there is a sense in that just to work with the orthomodularity. Yeah, but this is the modularity, isn't it? Slightly weaker than modularity. I don't quite pinpoint how... Well, they do distinguish between orthologics and contralogics by that very property, but the monomod is the one that the total modularity holds. Well, if you want to ax the ax in one, it's going to go the whole way, and the ax will go the other way, and the ax will go the other way, and the ax will go the other way, and the ax will go the other way, and the ax will go the other way, and the ax will go the other way, and the ax will go the other way, and the ax will go the other way, and the ax will go the other way, and the ax will go the other way, and the ax will go the other way, and the ax will go the other way, and the ax will go the other way, and the ax will go the other way, and the ax will go the other way, and the ax will go the other way, and the ax will go the other way, and the ax will go the other way, and the ax will go the other way, and the ax will go the other way, and the ax will go the other way. I'm very exhausted and this isn't because somebody else is here. There are points about, I've done it for that thing. It's an N. Okay, and let's say you were again adjuncts. They have to be. Are they of that form? In other words, just a binary operation of A and B there? Uh, I can't answer you on this topic. No, I think that's just crazy.

45:00 There may be easier ways of understanding it and being able to experience it. Yeah, yeah. It's very strange because also my colleague, Shanuel, has the same experience. We know this for years. Every time we have to think about it, it takes hours to find the proof again. Everybody refers it to some other place. It's easily proven that it takes a lot of time. It's one of those easy ways. But I've forgotten what you were going to say. Before you embark on that, can I... There is, here, the possibility of an answer beyond that. You know what I'm saying? By the way, Ralph, you know what... It's actually quite unrelated. You know the part you were asking me about this morning, this modeling, this statistic? I mean, the sort of thing you would work on. Yeah, yeah. Now, what I did next, I said, okay, forget about all that. For real about these expressions, let's think in terms of projection what they're going to be, the a-projections, or what's the simplest things, I mean you mentioned something which you know, you start with 18 x-rings and you came down here, so it's roughly the same here, so what do I know of projection, if x is contained in y, then the a-projection is the functor, now if... Well, the projection, the A projection of X will lie within A, and the third axiom, for all X, I tried to simplify this axiom, but I couldn't. If X is within A, then its projection is within A. It's reasonable, but I tried to...

47:30 In fact, this one can be relaxed a little. If you're already in A, then your projection in A is yourself. If X is in A, then the X is in A. Maybe I should also, instead of saying A projection is always in A. Then, having known that, you can prove that's the A projection of A, the X projection of A. These are all reasonably plausible lectures. This is a projectile. What the hell do I call it? I always call it P.R. Lattice. Oh, he's proud. Projective, I always call it. Projective lattice, so it's a tree. And then I say, give me a light to draw. If you give me a light to draw, then I can start moving. Because then, as you develop a heading under that, you have this... The projection becomes intersection, and that becomes intersection, since I have done already the work.

50:00 Now the interesting thing is always that I call it opposition, I don't define it exactly as you define negation. Yeah, what I call it opposition. And an axiom for one of the axioms. But of course you can't, you know, in fact, hate it. All you can do is prove the equivalence, as in hating algebra, and then if you define, if you further ask, you can prove different things, the equivalent of not A or B is less intense than A or B, and this would be the equivalent. Remember these are the expressions we're using. And so on, and then ultimately you prove that. Your auto is the one where A is opposite of A. You're okay? Yeah. So, given that, you can then show that it is just out of the category. We can furthermore, in the case of an auto projectile, furthermore derive the idea that we there assumed, having got projection in the aspects. Having looked at projections in the abstract and asked to write the terms in the abstract, in the case of ortho, we can regard the different expressions as the expressions to follow in that case.

52:30 So the whole thing parallels. I have... So you reproved the... Well, here I say... Okay. You captured the other part. Here it's really a revision, so that once... May we sound impressive if I pop that out? Let's go inside. Oh, right. I think Peter has cigarettes. He's just finished, actually, I think. I've still got a packet of tea. Wait a minute. We want to get there very soon, don't we? Well, where lunch do you want to eat? I know Pete does. What's this? It was a bandit. Did you want the light on? No. You sure? OK. Thank you very much for your time.

55:00 More generally, if you have both subtractions and implications, so you have two negations, A implies false or true minus A, which is just, so these things are equal if and only if it's Boolean, so there's a nice Thank you for your attention. So all I was looking at is if you want a new post-category, you could, the metric space is very good. No, no, it's been really neat. Oh, I see. Oh, yeah, well, that's what I've been doing. Well, that part sounds so good. I don't know, you can't be... Yeah, yeah, this is tough for the Guinness. The metric space is very good. It's very helpful. It's very formal. It's very formal. The metric space is very good. It's very helpful. If you go through the first two series, then you can start to see some examples. You know, you already know vaguely what it should be, so look at the examples and then you can understand the actions. So they have several interesting examples, but one example is this that they attribute to me. The lattice with both simplification and subtraction is only at the two negation degree. So I was very proud about this, you see, for many years, after all our kinds of discovery and modeling. Then last year, when studying Grassmann, I looked through Mathematica Nauen, and just by chance, I found the paper Schroeder, who refers also to Grassmann, so that was interesting for our research. He's got exactly the same calculation that I did in 1963. But also, history of logic footnote, how revealing that this work of Schroeder is so neglected because, you know, Frege stands in for the historians of logic as this great figure blocking, completely distorting their view of the history of logic,

57:30 completely destroying and distorting the history of logic. Michael, I'm overshadowing Schroeder. I hope you're overshadowing Schroeder regarding Schroeder as just a kind of Pali-German imitation of Bohle and people working on it completely wrong. Because they were trying to get logic from mathematics, and that wasn't the way to go. It's Frege's set of problematics defining everything in... My God. No, I mean, this is the activity. I mean, the fact that people like... The fact that people like Crispin Wright still hung up on this Neo-Fregean program, the people who think they're doing philosophy and mathematics today still hung up on this perspective. I mean, really, this is why I got you to Cambridge, Bill, to get out a book that would destroy this rotten way of looking at the history of philosophy of mathematics, apart from anything else. This was precisely in the Bismarck period, right? Uh, yeah, yeah, yeah. Anti-socialist logic. Oh, well, Frege was a great, I guess you probably know, I mean, Frege was an extreme, uh, um, rabid anti-semite, francophobe, and a great supporter of, uh, petty-minded Prussians. Yeah, no, I know, I know, but I'm just saying it was really the same period, wasn't it? So have a good laugh, man. The hating algebra is the boolean algebra. Hating algebra is also a co-hating algebra. And the notions of negation of all metals, which is something, they may coincide. Great! Okay, well... Let it be known as the Schroeder-Lover-Kutcherian. Yes, yes, yes. Forget about Kutcherian. No, you had that statement. Why didn't you call it the Sierpinski-Bach-Altdolski paradox? Why can't it be called the Sierpinski-Bach-Altdolski paradox? No, but I don't know of anybody else who thought of this except we three. No, but you mentioned it. I should have read it. You mentioned the notion of the 1800s. Yeah, this is the thing about this, the modulus, the fraction, operator, the joint, the indication. Then you must have known that. Well, it's the essence of the boundary operators and the pure logical operators. But there's one remark you make in introducing that. You say that this comes out of thinking of more general, the kind of generalization of domains of variations that are not determined by a single topos of variable quantities.

1:00:00 Now, what was behind that? And you actually make a comparison. They say rather like Grassmannian manifolds that are not determined by a single affine line of... Sure. Have you already been looking at the outstanding of that at that time? No, no, I was just... You see, I mean, the status of Grassmann is that geometers use... Thousands of times a day, Grossmann algebras, Grossmann manifolds, without ever knowing where it came from. So it's just a common, it's the basic example of a manifold which is not affine. There are many such examples, but it's the most basic. Projective space is a special case of it. In other words, it's simply this. A dimension k, or a fixed k. So the set of all subspaces is itself a space, because you have a notion of smoothly varying. One k-dimensional space can be smoothly varied into another. So if you have a notion of smooth variation, you have a space. It isn't just an abstract set. It's another example of the fact that if you give something... A so-called set theory of cognition, those things that satisfy a certain property, such as k-dimensional linear subspace of a given linear space, it's automatically not an abstract set, but again a space. There's a notion of tangent vector, there's a notion of motion, and so on. So that's the Grassmannian metaphor. It is just the... Yeah, but it's the fact that that prompted the... So for k equals one, you get projective space. The space of all lines in a given n-dimensional space is what is an n-minus-1-dimensional. If you want an example of a non-affine space, then simple-spawn is projective space, but Gauss-spawn is slightly more general because the kind of subspaces that you take could be k-dimensions instead of 1-dimensions, always inside a given n-minus-1-dimensional space. So in algebraic geometry, this is the first example of a non-affine space.

1:02:30 I wrote this down somewhere lately. In my notes about logic, no, she's large and small. Did you have that? No, no. Oh, God, I want to get to that. The lectures that I gave for topology. No, no, no, you never, I never heard of that. In April. So please send me, but go on. Well, but go on with what you're going to say. In April. I mean, the point is that the topic of education in topology is now so fragmented, you see, that they're brought up to work only on hard problems. They've become great, you see, but no general theory at all. So we have very good topologists, young ones at Buffalo, who work on three-dimensional topology, funcary, conjecture, very difficult things, but they don't even know what a sheaf is, a basic tool of algebraic topology. Well, they heard about it, but they never learned it systematically or so. So they were having a seminar in which they found they needed this. So they invited me to give the information I wanted to receive. So that was the origin of it. I wrote it up then, and so it's a sort of, I gave a copy to Tierney and he didn't give it back, but I spoke to Connes. But the point is, it starts out by saying that a topos, what is a topos? A Rodenby topos. It's an algebra of continuous functions, continuous set value functions, on a kind of situation. It's called a topos. In other words, the situation is the topos. But by some confusion, we call this algebra of functions also topos, which it really shouldn't be. The only thing is we don't know what is the topos, so we work only with the algebra. But this is exactly the situation in algebraic geometry. You have the ring of complex value functions, and that is the way that you get at the space. And looking at the sheaves on that, enlarging it, do the same with toposes. Take the category of all toposes, reverse it, sheaves on that, so you get some generalized toposes, which will, but the ordinary toposes are a full subcategory.

1:05:00 I'd very much like to see that. So there's, there's this general speculation, but then there's also the idea that there may be some examples of theory. The ring of an algebra is not necessarily the spirit, yeah. I assume that you were in India when I came. No, I'm not saying that. What I'm saying is, do you have any details? I don't have a statement. It was a, I mean, see, it was a statement. Yeah, I know, it's not a statement. It's not a statement. It's not a statement. By this time he's moved on. So I wanted that one, just to get some sort of description. Oh yes, this is one thing we must talk about while we're with Jerry tonight. Look, sorry, I have to say this because while you're here you can tell us much more about this. While we're having dinner, I don't know about you, but I would quite like to go to dinner with you. The derivation of the CCR of this whole intensive-extensive quantity is pulling forward an intensive-extensive quantity, and the various instances of that construction. Canonical commutation relations, just like the Heisenberg relation, is the standard. Well anyway this is just Bill is saying in his Cambridge lectures and again he explained it to me and very carefully the other night in Bangor but I was too pissed but anyway and so he very kindly promised to go over it again with me and I've got the notes. So we'll go over them. But it occurred to me, and this may be, it's probably just an idea that's based on insufficient understanding, that what Bertel-Heilig was trying to do a few years ago with Frescura, this taking co-gredient and contra-gredient algebra as one.

1:07:30 Well, that's what you're betting, then? Oh, he's your supervisor. Oh, well, I didn't know that. He's a scientist, anyway. He goes to South Africa and works for, you know, he works for the South African government, I think. Sure. Yeah. But I, I, anyway, um... He proved the theorem for all M. One, because two for all M. Well, okay, so the guys are unscrupulous fashions. Well, he's not any fashions, but unscrupulous fashions. Okay, but I wish I hadn't even mentioned his name, and he just happens to appear as the Kerr-Waltham with Basil, so, you know, so I can... To get to the point, was he? Okay, I believe you. But it seems to me that what they're on to is exactly... I want to have a look at that and I want to understand it but they did it in terms of algebra of differential forms well idea deals with it also in terms of left and right sided ideas And they suggested that it was a model for, well, actually, I think the whole point was that they thought they could get CCRs out of it, because the creation and annihilation operators actually come out, you know, pop out of this as an anti-communication. Yeah, yeah, yeah, exactly. And the supersymmetry comes out of it as well. I mean, it is itself another version of the supersymmetry algebra, but the thing is it's done in terms of differential forms rather than in group theoretic terms. And they related, I say they, I mean, first of all, Haley, I think, who was very interested in the connection of Grassman, because he published that historical paper about the spinner and the algebras, and, you know, the clipper, well, and the Grassman algebra, too, and saying that, which, well, it was interesting, because he was making the point that, you know, everybody thinks, because of the late 19th century, because of Klein and...

1:10:00 And Gibbs, everybody thinks of Grasman algebras in terms of vector and the underlying metric space, but Grasman had, well, he says in the paper, there are much deeper ways, which, of course, he thinks ties up with Boehm's ideas about infinite order, but I don't think you necessarily have to buy that software package to be interested in this construction. At least I hope to God if you do, then I'm backing off. I thought of another one. Computer virus. I think Implicit Order is definitely a computer virus. It automatically appears and it starts reproducing itself in short circuits and other things. Somebody said the difference, you know, as between the two MacLeans, Saunders and Shirley. This is definitely more likely to appeal to Shirley than to Saunders. Well, you know that Shirley MacLaine actually turned up at that department, apparently. Was it you who told me that? I told you because I knew another guy. Apparently Shirley MacLaine actually turned up at Birkbeck to talk to Byrne. I mean, I had, you know, because she'd, she'd have, she'd have read about, I think, the Order of Holism, and here is this, well, the Kohlbeck Newsweek, the New World Bank, who was, I don't know, Einstein. He has these ideas which are just like those good things he tells us. Are you a guru? Well, so is his speech, and being a very, very, you know, charming old, meagre gentleman. Took her out, took her out for dinner, apparently. Gourmet is a good thing. He's a lovely man. He's a good person. Well, that, in a sense, he may damn sure, he may, he may go right out of check for half a million dollars for the physics department at Birkbeck while he was about. Well, we didn't get a penny.

1:12:30 Oh, I heard after our last birthday. Yeah, I know, I've never met Francis Presker in my life, I know nothing about him, except what Kylie's told me, which is all bad, but he now works with me. Yeah, you don't need to come to work with us, I'm happy in government. He was a Christian bishop. Christian Scientologist, of course. Christian Scientologist. Well, I never knew him. I don't know how much he could have told me. Yeah. That's what I thought. They're all planted. What is this, man? You're trying to... Do you ask about the cupboard? We are going to eat. No, no. Well, the thing is, the cherries... So the only way you're going to live out with cherries is if you don't eat with us. We're going to have to eat at 10. Oh, that's right. You actually get the pizza night. You'll have all this, and you'll never forget this. Oh, I told you. It's going to be great. I want some more. Yeah. Yeah. Yeah. Yeah. Yeah. Yeah. Yeah. Yeah. Yeah, I said that's what he actually needs. What he needs is a sound chair. There are a couple of guys with kind of sound mics who follow him around every minute of the day. Well, for that paper... Come to Italy. Stenographers. Come to Italy, we come and support you. For that paper, it was just a passing example. Yes, yes, I know. But you're right. I should have, because there are several things connected with linear algebras and so forth, and I thought it should have been. Very good. There are other people who write, who have one idea and write eight papers. He has eight ideas, the others don't. There are people who have one-eighth of an idea and publish eighty papers on it. And there are people who have eighty ideas and he publishes about one-eighth of a paper on it. Yeah, yeah. So, can I get this card? Well, okay, leave this stuff. I would like to talk to you a bit about this tomorrow, though, if you've got any questions.

1:15:00 It helps me clarify my ideas by arguing against Mabry, and I just happen to find that weakness in me a good way of clarifying my own ideas, to argue against him. Because I think he is a clear writer, and that you can't take away from him. He's a clear writer, and he pins down exactly where the commitments are, the metaphysical commitments are, particularly what he says are... Let's go and then we can come back and talk about this. Come on this morning. Come on this morning in France. Oh, he brings some conclusions. Yeah, actually the conclusions seem to be pretty well that... He's... well, there's a big concession at the beginning. I don't know if it's... It's, um... I've reached my original conclusions and where is it? Norbert is not a structuralist. It has, as I am now beginning to see, the most revolutionary implications for mathematical practice. This is your favourite? This is your favourite? Yeah, you wrote it yesterday. It's a letter. Yeah, well, at least the man is not black. No, I don't know. It's Darlene. Jose. Nobody else. Vissarionovic. Yeah, you've got three of them. Jagat Ili, Koba. Yeah, I'll give him up on that. I've just been re-reading Thebes. You like Stalin, don't you? He was a top man. You don't take on defeat after something, do you?

1:17:30 Do you think the condemnation of Stalin is made up, or...? Largely. You think it's just a hannibaloo? It's a way to, it's a road toward attacking Lenin and dismantling socialism entirely. But are you aware that Lenin had a bit of fiction? Thank you for your attention. What is that idea? The distinction you make between formalized set theory and ordinary mathematics as it's found by Bohr, Barkey is absolutely crucial. At this fundamental level, the distinctions between set, collection, domain, etc. are, as you point out, superfluous. There is one intuitive idea and one intuitive idea alone here. Whatever you call it, what is that idea? It is, I'm convinced, the ancient notion of arithmos, as analyzed by Aristotle and employed by Euclid, and arithmos is a plurality of determinants size composed of units, monads, and a unit is, according to Euclid, just that, in comparison with which each of the entities of the kind under consideration is called one. In my example, a mena, in Lord Beer's sense, may contain or give rise to arrhythmoi sets in several ways, depending on how the unit is chosen. Thus, a heap of shoes contains both the set of six pairs and the set of twelve shoes. In the former case, the unit is a pair of shoes. In the latter, a shoe. Well, I think he's, you know, he has understood an important part of that, but he hasn't, I think, understood the point about motion, has he, about cohesion or domain of space. Well, that's because if you take the actual pairs of shoes, you see, that means that they cohere, whereas something from two disparate pairs doesn't cohere. Ah, right, and that's, of course, the point about the failure of the excellent choice. Thank you very much for your time, and I hope to see you again soon.

1:20:00 It's a choice of units, don't you see? A generic figure is a choice of units. So it gives rise to, for every object, an arithmos of one case of shoes and the other case of pairs of shoes that are things together. Now, actually, this is a good one. There is a rival to this arithmetical notion of set as a plurality of determinate size, namely the logical notion of set as a plurality of determinate size, namely the logical notion of set as the extension of a concept. Only too well, that notion is fraught with difficulty. It is the notion employed by Dedekind in and by Frege in the Wundersetter, and it was in his review of Frege's Grundlagen that Kant offered his finger on the difficulty only when the objects falling under a concept have a determinant power, cardinality, finite or trans-finite, can the concept even be said to have an extension. I'm not sure that's right, because I think there are ways of thinking about extension. I mean, certainly when you look at the work of Jones, they have to have an extension in that sense for a concept, or you can recover set intersections, a special case where you have that kind of compatibility between... There should be an extension of fragility, which... Yeah, this is it. He's saying there should be an extension of fragility. If the extension can't be quantified at all, then there's some doubt if it is an extension. That quantifying would be an extension quantity. Hmm. Wow. Well, you're cutting a pretty good bit in here.

1:22:30 Well, you know, he's learned a lot from that week. He's decided there's more to it than just sonorous words now. Well, a lot more. He obviously seems to mean it. You see, I'm here and I have thought I would be a philosopher. I can only think about problems at the level of somebody like Mabry thinks about them. I can't follow you sawing eagles into the empire. I don't know. It's very important. We will. We'll come back. Ross Street is using this for a systematic definition of international categories. I'm guessing what one it is. What one is it? We've been combated. I would be sorry if you didn't tell us what you saw. You wouldn't look out of the station, would you? Well, hang on, wait a minute, wait a minute. It's only 7 o'clock. What I have got is a capital card. Oh, you have?

1:25:00 Yeah, these are the ones I did actually put out. We'll ring as soon as I drop Bill off. Yeah, well, his plane is at six, so it'll be before that.

1:27:30 You can distinguish them ultimately by their position. And now you see why you need a category, because which element is which? Once you lose track, you don't know which one went where. Therefore, you don't know the number of matters. Therefore, you have no explanation. Those are Einstein's theories. And I'm throwing sketchy remarks here, but it might trigger something in you. The Bose-Einstein situation gives you very funny probabilities. Now, if you think in terms of categories, then the correct numbers come out if you restrict yourself to the order. The question of identical, non-identical wasn't the important question. The important question is that this thing is not happening in sort of classical space. This is where I wanted to... I will read it. Let's see. Well, that's what I want to tell you, yeah. Because I came to the conclusion... Pete! Can you just hang on a second? I'll just get an umbrella. Yeah, of course there is. There's one in there. You know where it is. Oh, I closed the door. Yeah, start again, Jerry, because I want to hear all of this.