Discussions

0:00 And you do get these monsters. Yeah, the monsters will come out of the... There's an... And, you know, assuming, as I said, it's absolutely basic that we have consciousness. Well, there's an interesting thing... If you deny that, then maybe... There... Robert Penrose wrote a book called... It was on the bestseller list, the Times bestseller list for non-fiction for a long time, and he says in the introduction, anybody that knows any high school algebra can read this, well of course that's a complete illusion, right? But the interesting thing is that when he comes to talk about the calculus and analytic function theory and so on, his basic question is, what would Euler have said about this, what would Euler have accepted as a function? I mean, basically, I guess this idea of thinking of this infinitesimal approach to reals is really what's maybe behind Euler's intuitions. It's just, I mean, it's certainly, calculus is never taught that way, you know, past the, it's taught either these are the functions you use and here are their derivatives and so on, or if it's taught as a beginning analysis, you get this full-blown geometrical dedicated version where the monsters are all there, but there's an Eulerian synthesis of this or something. Now, the Sierpinski snowflake, that's a clear example of something that couldn't possibly exist, and yet it flows out of any sort of analysis of the continuum that's based on natural numbers, which I think, I mean, I don't think you can...

2:30 No, I'm sympathetic with your point of view, but I don't see how that's sufficient to explain all this stuff. Well, it's not sufficient, but... It's certainly part of the same kind of... It's certainly the simplest example, maybe, of the same kind of idealization. You see, it's sort of, if you think of Dedekind's proof that the natural numbers exist, tried, carried it out for any topos, right, that if you have a Dedekind infinite set. I.e. if you have an endo-map that's epic but not much, then you can construct something that satisfies the recurrence of the mathematics object. And you do that by intersecting all the parts that are closed under this. So you have this infinite inter... in other words, you have a class of all these subsets somehow representable so that you can take this intersection. So in some sense, with that little bit about... and clearly there exists static and infinite objects. I mean, this is the continuum of adding one on a continuum set, whatever. So that's not part of the general axiom for topos, but certainly something that you want an animal to have, to dedicate. So it's already the idealization of omega itself. Because omega to the x is the arena in which you take this infinite intersection. So beyond the basic idea of function space, the second idealization... But you have truth values, which is the second being unique, the axiom of kilproses. That already is a big idealization. Clearly, it's something subjective, right? In French, it's a judgment. The truth or false is a judgment about some statement, you see, so you've objectified this aspect of the subjective. And that implicitly gives you the natural numbers, and hence Szybrenki-Stoltenberg, or whatever non-milliple sets of all that. So somehow, but again, my version of this, and my hope that one can more precisely show, you see, is that,

5:00 okay, this consists of idealizations. We already idealized motion itself. And the natural numbers. But the continuum, the D is a more modest idealization than the natural numbers. So there's a part of this category that's clothed under things like finite inverse numbers and hopefully even exponentiation, but that's the key problem, which does not contain the natural numbers, but it does contain the reals. The natural numbers exist only in the larger context. In other words, it's more of an idealization. And the fact that the natural numbers and the real numbers seem to be so closely related in any way is because everybody knows you can additively include the one. The idealization of truth, the equalizer of maps into omega, matching the constant map true and something else. So the thing is that in the subcategory of, you know, so to speak, actual geometrical objects, r is there, arbitrary equalizers are there, but omega is not, and neither is n. And this is precisely the kind of thing that wrote the came topology. And which the logicians, under the guise of O-minimality, are on the way toward achieving. The O-minimal world, such a world, contains a good model of the real. I mean, I know there are many O-minimals in the world. It's a good model of the real that does not contain the natural. In fact, in these terms, there's a very crude but decisive thing that pi zero of any object here is finite. None of these, no matter how complicated a pair of equations that are real value to consider, will construct things with lots of pieces but never an infinite number of pieces. Always an infinite number of pieces.

7:30 So already at this level, one can make this distinction that there are plenty of objects in which pi zero is finite. Closed under, closed under, it's a big contradiction. See, closed under equalizers. And, at the same time, Paisley or Feynman. But that's what Ominimality achieves as well. Well, there's a kind of abyss of mathematical difficulties on this. Because, I mean, these Ominimal people are concentrating on the... What's behind all that, at least, this generalized Karski problem and all that stuff? Yeah, well, they put it in subjective terms of, you know, decidability. But I say the finiteness of phi zero is an objective criteria which is very similar, very close to, may not be identical, but the finiteness of the basis of components of the objects that you construct, in other words the natural numbers, i.e. precisely the infinite arithmas, this is the real evil, in that sense, that it creates for us. Yes, I remember the last time I had a long talk with you, when you left, I had that phrase, the natural numbers are the ultimate... Thank you for your attention. But that doesn't mean one shouldn't use them. In other words, you should use the whole topos for its purpose and at the same time you should recognize this is something, of course, you can never do if you think everything is taking place inside of us. You can't make that distinction. But once you have a more coupled topos, you can see easily that... You may have infinite discrete objects, but in this subcategory of geometrically reasonable objects, you do not have infinite discrete ones. You've got some infinite ones, and there's a very complicated sub going on, but you don't have this in Princeton's new framework.

10:00 Yes, because you don't have the completed discrete infinity. So, Matthias, now this was all supposed to be a preface. There's one more thing that you should do. Which is the axiom of the body zero of R being one, connectedness of R. Connectedness of R. Because it's related to this. Oh, that's right, that's right, that's right. All of these have one component. But the fact that d to the d, it seems to be a further axiom. I mean, it may be possible to prove it in some other way, but at the moment, I mean, what it really, I mean, in the general way, it's concentrated there, but in the general way, it's just the fact that the natural map from phi zero of any x to phi zero of x itself is going to be less than one. Right? Of course, R being a retract of a connected object will be connected and so on. And R is a retract of D to the D. So if X has one component, then so does X to the D. But again, this is something which people who have worked in books will say, of course this is true for microlinear objects. But the best thing to believe is it's true for all objects in the appropriate topos. Because of this general fact that a contractible object, by contractible I'm going to be objective to this and say don't you mean acyclic or something, but just to have one word that roughly corresponds to what topologists usually mean, which fits very well into the setup, I just mean an X to the Y.

12:30 All of this is connected through all y. That means x is contractible, which is simply the statement that in this homotopy category that you construct by defining the new mass, that it becomes a terminal object, so what could be more natural as a condition of contractibility type than to say that the object becomes terminal and you look at it in the homotopy? Of course, this is not exactly the homotopic category that homotopies want. It's the modification of the In Gabriel and Eastman's book, it's explicitly a modification of this. Well, they start with some special sets, as E, which of course is an example of that. Cocos, which is a model of axiomatic cohesion in the sense of my paper, although axiomatic cohesion is not generated by a D in that case. In fact, any presheet topos, in any presheet topos, you can reconstruct the so-called basis of it, the sets, by just requiring it for every representable object. You get the same equation. That is, you cut yourself away completely from any idea that the pre-existing foundation, you reconstruct the foundation inside the category that the category is ample enough, you can do that. So this will single out the constant factors, the constant pre-sheets, i.e. the sets, among all pre-sheets, if you let D be arbitrary. Now, of course, if you're talking about pre-sheets on a category that has finite products, then... The representable functions are atoms, and so it's a special case. On the other hand, the fact that you're singling out the constant pre-sheet by that equation remains true, even if the d's are not the best representable, so there's a little bit of excess particularity from the point of view of that particular construction.

15:00 So, contractible, yes. So, there's a following lemma, you see. What if you have a connected, pointed monoid? You have a monoid that no longer has, no longer has a zero, you know, in other words, an element that's a constant, zero times, in fact you don't even really need a monoid, you just need a zero and a one, but if it's connected, something, and let this given object act on it, your constant acts as a constant, it's automatically not just connected. You assume the monoid was connected, that was... So it's like the unit interval of 0 and 1, except it doesn't have to be that one. It can be any monoid, as long as it's connected and has a 1, which acts as the identity, and a 0, which acts as a genuine constant. The two ingredients of the components automatically have that action. So you have that action on the base and you have it on the exponent and then you can use that to construct a homotopy in the sense that, you know, so this is, we're not thinking about weak equivalences. A homotopy equivalence is when you have maps in both directions

17:30 so that both, each composite is in the component of the quantum space where the identity is. So to prove that something is in such a component, it suffices to have a parametrized path, but the path isn't really a path, it's just a parametrized, linearly connected one. So there's a couple of basic lemmas about that which are so basic that one never seems to have written them down. It doesn't have anything to do with the actual human general. D to the D is the best example of a bi-pointed thing, because in any, in fact you have any pointed object, then in the self-function space you have the name of the constant and the name of the identity. So those are two points. And so if this is connected, in other words if D is contractible, then Grotendieck's proof, Grotendieck proved then that You can take the characteristic map of the false, the zero one, and that will be a proof that omega itself is connected. So my axiom there is that for sufficient, the truth value object should be connected and transform Q into false. In a way that you can't include in the logic, it follows from the existence, somewhere, of a connected object with two distinct points. That's quite a brilliant, brilliant example. And so if this is connected, you will then have the whole topos. Every object in the topos is embeddable into a contractible object. So the conclusion from what you're talking about is that this is connected. As the huge conclusion that every object in the whole topos is monomorphically mapped into something which is not only connected but actually contractible.

20:00 There are these lemmas there and the two or three things I mentioned about actions of monoids and expectations. These little lemmas together imply that every object can be embedded into a contractible object if this is connected. And in particular, same sort of argument, if that's connected, that's a monoid that acts on x and the means of x and the means of x without the same number of components in the same Makoki theory of the other one, actually. But I think this is the, this sort of expresses the intuition of it, right? In other words, that no matter what you mean by being, it's becoming, it's precisely the kind of germ of connection that you want. The space of the comings, of elementary comings, should be just as connected as the space to be in its happening. And so that intuition then concentrates itself into the fact that this is connected, but that in turn, by these limits, implies the difference. Why it should be true? Well, why it should actually be true, well that's another matter, but you see... Any sort of intuitive proof does seem to involve the idea that maybe D does have at least the existence of some binary operations that you can connect things together. I mean, if you imagine that we are working in a topos of M sets where D is M, something like that, then to connect things together and sort of appreciate by actions. There may be a sequence of different actions that basically connect by actions, so you have to sort of map D into D into D to do that connecting, but that's a binary operation of D. So, in other words, that gives you an idea. In certain kinds of pre-sheet categories, you can explicitly compute what components mean.

22:30 In fact, Pi g was also called direct limit, by the way. Direct element is the same thing, so you can see how this connecting to the other works and so you can compute the exponential of it and so you can see that something is taking place among the representatives and so they have to exist in operations, you know, A times B to see that all three are representative. So again, it would be nice to have a precise number of that sort. It's not something that's, you go to these concrete examples, and of course, the general toposes are contained in pre-sheath toposes, and this computation is not going to change, and so basically the pre-sheath case will explain when this is true and when it's not, that D to the D is, oh yeah, part of this circle of lemmas about contractibility is that this is the natural definition of contractibility, arbitrary Y. However, it's equivalent of putting y equal x. In other words, the connectivity of the end-of-organism space itself implies the connectivity of all the function spaces valued in x. Again, it's sort of the intuition in terms of quantum mechanics. Any more questions before you talk? Listen, I'm going to have to get back to my brother. I've got to get him out. What's going to happen if you're going to do something this afternoon? Well, I may or may not be able to make it. If I'm not here, then it cannot be as absolute.

25:00 I admire your bladder control, Bill. That's really impressive. Oh, bladder control, yeah. I'm going to stop off at the gents before I head off for home. Well, that's not a bad idea. Is there a long line? No, no, it's completely free and right next door. But before you do go, John, do we have any plans for this evening? Okay. All right, so anything? I'll get in touch anyway. I think I'm out of it this evening. Oh, of course, because of brain... OK, that's fine. OK, take care. Would this be a suitable place to... What is the time, by the way? I'm sorry. One o'clock. How do you feel? Might it not be an idea to break for lunch and then for Mathis to talk to us in the afternoon? No problem at all. A good idea. OK, right. And then we can... It is interesting that this program of the low-finance sub-analytic sets, that this has come through the consideration by McIntyre and the other logicians, the model theorists, of the Tarski problem, which you say is focusing on the decidability aspect. Does so naturally tie across to the separability aspect, which is the more objective, well, to the more geometric aspect of what's underlying, yes, yes, yes, yes, definability is what I should say, but yes, but there is a geometric aspect, obviously, to do with the connect, there's no behaviour of coverings. I saw Angus McIntyre the other day, he's going to be at this, he's one of the speakers at this meeting for Atiyah in Edinburgh, and he asked of course to be reminded, remembered to you, so he would very much like to meet up with you again if at all possible, actually he said he really wanted to try and get down here to this workshop, but it's absolutely impossible because he's got...

27:30 He's stuck in Edinburgh with the administration. He would very much like to have come. I told him about it. He said, I wish I'd known sooner because I would really have to try to move things around to get there. I was just trying to seize the opportunity that seemed to exist because Richard was able to do this. Oh, no, it's great. But I'll certainly... We have to do this more often. I'll certainly spread the... It's a long trip for you, isn't it? Yeah. Maybe we should come to Argentina. You could come to Argentina. That would be very interesting. I would love to see that. If it's not burning down right now, then you should come back. Oh, I see. I can get that. Yes, I'd like to. We were talking a lot about what it was. Now that's done. Is it? Okay. Yeah. Someone could pull strings. Something like that. Yeah. I'd be glad that you did that. Thank you. Thank you. Thank you. Thank you. Thank you. Thank you. Thank you. Thank you. Thank you. I think I seriously have got a pretty good clear set of notes, you know, in sequence as you wrote it down, but I may need a little bit of tweaking. Well, if you want to transcribe it... Yeah, well, this is exactly what I intended to do. I mean, I won't be able to do that in real time, as it were, but I'm still transcribing some of your discussions from Montpellier, but... It was good that John raised the question at the point that he did that because it led very naturally into the whole discussion of continuous and discrete from Grassman to the program of tape topology and still in a very natural, very very natural, conceptually natural development.

30:00 Yes, yes, yes. I haven't really followed what they do, but I have a feeling they're going into some very tactical things. That's my impression. Yes, yes. And not so much, precisely my point. Yes, yes. As you say, because they've become too hung up on the subjective aspect, on the decidability, obviously it was a motivation, a very strong motivation, but one has to see the decidability as just one aspect of this more general connection. Certain people like Angus have very strong geometric ideas. Oh, very strong, very, very strong indeed, yes. And they know, as it were, what Grotendieck's vision of tamed topology was and how he saw it as relating to the whole issue of arithmetic, geometry and the source of the pathological functions. In fact, Angus McIntyre wanted to come to this workshop. I talked to him. I saw him in Paris and when was it? Last month, he was at a workshop at the IHS, not the Grotendieck thing, although he was there as well, but a later one, and I told him about this, and he said, oh, and Bill is coming, and who's it organized? I said, well, that's all right, what are your countrymen? Very impressive young Scott called, Richard Pattinger, but unfortunately, because he's committed to, he's one of the speakers at the Satir thing, the Satir Fest next month. And I think they've got some sort of congress, I think it's actually to do with model theory, going on in Edinburgh at the moment at this IC, whatever it's called, International Centre of Mathematical Sciences. But he said, I wish I'd known a month ago, if I would have moved things around to get down there, I really would like to have come. Well, it's a real shame. He, of course, took part for about a week in these discussions with Bill and Cartier and Fougere two years ago. Right. No, more than that, three years ago. He's a very impressive guy. He's talking at a nice event, actually. It's the 14 year anniversary of the philosophy, the maths and philosophy undergrad course at Oxford this year.

32:30 Ah, yes, because of course he was there, wasn't he, for many years. And he and Scott are giving talks on Saturday. Oh, I must try to get to that if at all possible. Oh, hang on. I will be able. Oh, is it June or is it July? Our trustees meeting in Oxford, I think, is the 2nd of July, so I might be able to get that. And then the bunch of us who were undergraduates on that group. Oh, that would be fantastic. I'd really like to try to get to that, if there's any possible way. Yeah, it should be interesting. Yeah, very interesting. And David Scott's going to be there as well, because he's some... That's something I wanted to talk to you about, Bill, actually, the possibility of our doing... You know the exercise that Angus and... Pierre and obviously Colin were all involved with Foucher three years back when we did the debrief for you. The debrief we called it, whatever. I'm not sure how you want to, rencontre, whatever you call you wanted. Anyway, the extensive discussions with you, the intensive discussions with you, what I call the grill bill exercise, I think it was Nick codenamed actually. It's alright, you came out of it very well, don't worry. No, we were thinking of trying to do a similar thing with, well in fact there are at least two kind of candidates in the frame. One of him is André Joël, two possible subjects for a similar exercise. One of him is André Joël and one of him is Dana Scott. And in both cases, of course, we very, very much want you to be one of the people on the panel and one of the people who are playing the same role that Cartier and Colin and co. played vis-a-vis you. That's one of the things I was hoping we could talk about. And I was also thinking it would be a rather good idea if we got Christian Huzel. ...along, at least further. Because then, of course, you and he could do some more talking, which would itself be a marvellous contribution to verification of a lot of things, particularly in why the whole thrust of Grotendieck's direction of research in this and other areas appears to have been diverted since the 80s in her...

35:00 One other thing. He was talking to me a bit the other day about Grotendieck's work in functional analysis, Grusel, I mean just in nursing. There was this big science book fair in Paris and I ran into him there but we went off to dinner afterwards and he was very much echoing the things that you were saying. Montpellier and elsewhere about how, why it is that, you know, this incredibly rich, but also these such profound connecting ideas and insights that Grotendieck had in algebraic geometry have, I wouldn't say it's petered out, but certainly it's nothing, made nothing like the progress that might have been anticipated, say, in the early 80s. What was the craze of the IHS, the ongoing research? It's pretty thin, actually. Well, exactly what Husserl was saying. He said he's considering just how tremendous the early progress was, particularly with topos theory. Theory from, say, from the late 60s to around 1980 and the 70s. It's depressing how thin it has been since around the early mid-80s. Can I just try and use the lid? Oh, sorry, somebody in there, obviously. Who's that? Well, the door seems to be locked. Oh, it's okay. Well, is there somebody in there? Sorry. Is there somebody in there or are we... Well, nobody's answering. Is it just stuck? Is the door just? Oh, no, there is somebody in there. I apologize. I wasn't trying to force the... Yeah, it's okay. Yeah. There's another toilet. Oh, is there? Okay. From the darkened pub. Oh, that's fine. I managed to get to the pub. Oh, maybe half past two, maybe an hour, an hour and a bit for lunch, does that give us enough time, yeah, and then we'll be back here at half past two, is that something you'll go do here? And the lunch would be at about 1.30 or? We'll just go for lunch just now, just at 1.20.

37:30 Or we can make it later. Where, where, where? Just going to meet the actors. No, for the lunch, where are we going to? Just up the road, just up the road. I would go to Tom. I'll be back in one hour here. Right, just a little bit over, so it's 5050, that's the only code or anything you shouldn't use. Okay, goodbye. Love to meet you. Yeah, he said we are, that's why. Do you need it? Excuse me a minute. Even if the thing you're starting with wasn't a movie, and that's basically Galois theory, not this outrageous thing that Rhode Island Tierney called the generalized Galois theory. It just doesn't seem like that at all. It's an actual concrete Galois theory. Olivia is what is given here. Thank you for your attention. Well, the Greeks of course, I think the Greeks would have loved this if they had got their minds around it, because the idea that algebra really comes out of geometry in the end is, of course, right, when one digs deep enough, that's right, well, of course, the whole line.

40:00 Yes, yes, the synthetic geometry and the idea that algebra and, in fact, even arithmetic eventually emerge from it via, possibly via, no, understanding the homotopy theory, which, after all, is all about the deformability of mapping. So, anyway... Well we shall see. This is going to be an awfully interesting discussion. Bill I think obviously sees arithmoids themselves having some kind of geometrical basis but I wouldn't be so fast to say anything is obvious but that's my very strong impression from everything that I've heard him say both here and before. But this is precisely exactly which is why I would really like to get... I think this would be an extremely fruitful discussion. I'd like him to say more about Grassman's views about continuous and discrete. That's a very interesting topic as well. Anyway, it's going to be a great few days, I think. Yeah, I really get a good feeling about what Sandor is going to say. Yes, yes. And you say you think he's coming in this afternoon or early this evening? Probably. I think so. Well, maybe you would be able to take them into the vaults. Absolutely, absolutely. I was going to say, if I could do something useful. And I'll phone just to make sure that everything's all right. Okay. No, I'll go up to the vaults now with them and we'll see you there a little bit later on, yeah? Yeah. Okay. Andrew Roden arrives today. Yes, he's going in the evening though, isn't he? That's right. Yeah. I thought I might meet up with him for a drink. Where is he staying? Also... Oh, right, in that case I'll certainly see you there. You, Anders, Matthias. Oh, OK. Well, we could all go off in that case. Well, when he contacts you, let him know that that is where we are as well. Yeah, he's got my mobile number. Oh, OK. I'm not quite sure when he said he was getting in. I think he said... There's a dark extension. We're using that... This is contrary to the so-called Mishnevich topology. I don't know why they want that. There's reasons for that, too, but it precisely does not have this property. But anyway, the traditional would. All right. And, yeah. So, you see, you get the... Now, to talk about the infinite Galois group and the zero-dimensional topology and all that stuff...

42:30 That's just a spurious analysis of this topos. It's one way of analyzing, but you don't have to analyze it that way. You get an infinite Galois group because you choose extensions. You see, you make some choice. So you can take a limit of the individual automorphism groups. The automorphism groups, of course, are finite groups because of the finite dimensionality of the thing individually. But you can pass to a limit and get this infinite group. But it's a little bit unstable you see, it's only defined up to conjugation. So to think of Galois theory as the representations of that group is already twisting, because this topos is precisely the continuous representation of that group, you can always say that too, but it's not the most basic way to approach it. Yeah, it's taking the representation for the real... Substance of the theory, isn't it, once again. This is what you're explaining, why you don't like the so-called generalized Galois theory approach. Well, no, that's another... Oh, it's a separate issue, okay. But I think they're more concrete. By the way, this man is giving away, they're giving away free pizza in that cafe bar over there, if you want to go for that rather than go to the Highbury vaults. But it's up to you, what would you prefer? I'm happy with free pizza. You're happy with free pizza? Okay, well, let's go for it. Okay, you've got three customers. Okay, we'll just go inside and help ourselves. Be careful how you cross. It's a superb story. Good. I haven't heard it quite enough times before. That's great. That's a great story. Yeah. Which Bob is this? Is this Bob... Walters. This is Bob Walters, yeah. But, well, actually it was really Basil Hiley's people that placed up the road. Well, yes, I arranged it. Yeah, Basil Hiley, yes. What about, what was I going to say? The Leicester meeting was on foundations of computation, wasn't it? Yeah, yeah. The theory of computation. The theory of science, yes. My audience was completely... Oh, dear. Disappointing.

45:00 Computer scientists who are prepared to think about the foundations of the theory of computability and computable functions ought to be more interesting category than almost any other audience that, you know, first time they're exposed to it, I would have thought. No, not sure it's... Yeah, you were, obviously. Perhaps there's much more of a tradition in Argentina of people thinking about what I think, you know, thinking about real foundations, real foundations of real fields. Subtitles by the Amara.org community A bunch of computer scientists I've met have an appreciation for category theory. I don't know if they got it right in their fields, precisely. They seem to respect it. These domain theorists take very seriously the idea that they have categories of domains. I don't know if they're doing it the right thing. No, no. Anybody who's interested in fundamental semantics for... Programming and higher programming languages has got to be interested in domain theory and therefore got to be interested in category theory but you about a year before that you've been at a There was a summer conference on computing in Galway, hadn't you, in the west of Ireland, with that change. That, I had the impression, they were much more interested in fundamental issues. Yeah, there were a number of other talks. General topologists who got into computer science. What was the name of the guy who organised that? He was a professor in Galway, almost unpronounceable name, McConnachie. Ah, no, McAnachie. That's right, McAnachie. I won't even try and say it. No, he's actually in Dublin. Oh, he's in Dublin, in Trinity. Okay, right, okay. He's the one who invited me. Yeah, right, okay. I gathered from that. No, but I've heard from people who have met him, and they all say the same thing as you, that he is quite a character.

47:30 Occasionally he pops up, you know, and I might post something on the Category Network that particularly appeals to him. And I get this little piece of poetry, all sorts of Irish allusions, but obviously heartfelt. I certainly would like to meet him. He sounds a really splendid day. Oh, he follows in Saunders' footsteps, in that case, expressing himself in poetry. Oh yes, very much so. He sounded like quite a character. I remember you actually sent me an email before the conference asking me on advice as to how to pronounce his name. Why on earth do you think I was qualified to do that? I looked it up. Well, no, I wasn't born there. My mother was Irish. I wasn't actually born there. But it doesn't help you. But she was. Anyway, I think I managed to find out the correct pronunciation for you. But the thing in Leicester was much more of a disappointment, was it? Oh, I'm sorry to hear that. Even the person who invited me didn't really seem to be into what I was doing. Neil Gotti, do you know him? No, I'm afraid not. No, that doesn't mean anything. What did you do? Well, you know the paper I wrote in honor of Aurelio's 60th birthday? It was more or less that subject. Aurelio was very mad at me, by the way. The concrete example I have there is wrong, and because it was intact, I was able to correct this after the fact, so there's a new footnote if you look at it. It's just that the specific, you know, I tried to explain the logical structure of three. I made the completely spurious idea of identifying it with the field of energy of modulo three, and therefore talking about polynomial functions. And I got the polynomials wrong. I mean, it's just that with a little more care, you can get them right. I mean, that is one way of quantitizing it, of course, but I should have totally misplaced con-venus and worn over wrong.

50:00 But it still works. Oh, the idea works, but it's just that the particular way of... I don't need all 27 operations, of course. I think about 10. Even those could be presented by, a monoid of 10 hours could be presented by things that have more of a logical meaning than that. I don't know why I did that. I guess I thought maybe some people would view this as more complicated, so I quickly made some computations and I made a mistake. I'm sorry. I'm really sorry. But it's still three. Yeah, still three. Still three. The thing is, it made Aurelio very mad, because he actually tried to compute with that notation, but it didn't come out, so he was wasted. He was very mad, so I'm sorry. I'm a teacher, I should be more self-concreted. I should be more careful. Well, actually, Andreas Blasch wrote to me recently about this, because he was interested in this idea of unary operations on a tree. He's writing some survey of different approaches to logic. I referred him to that paper, plenty of them. I don't think the idea is wrong. I still believe in the idea. But it still works the same way, right? You have the automata, and it's sufficient to make it a natural transformation. It still works at that level. You don't have to calculate on the powers of three. Right. Just on three. Just on three, yeah. It's a distinction, you know, sort of like, yes, sort of like in C-infinity geometry. A ring homeomorph is already a C-infinity homeomorph.

52:30 Or an ordinary ideal is actually the C-infinity ideal, because of certain basic calculations of the Taylor series and so on. So it's something similar going on there, namely, of course you can't define outright and say the Boolean operations, but that you have them. Then to say that something is a homomorphism requires you to mention less. So I know that a homomorphism of Boolean algebra is really the same thing as a lattice homomorphism between things that happen between Boolean and algebra. You don't have to be able to define not in the theory that you're being naturalized. So there's some of that involved in this, it's not the... Right. You could say that negation is implicit. It's a simple thing, but I think there's still a lot to be explained by it. I gave a talk actually, that's right, in Canada, to Toronto and Ottawa a few years back. About this three giving a very, you know, huge, huge telescope to look at it, a tiny little thing on the view, you see, but it identifies three. With the order-reversing maps from 2 to 2, in other words, pre-shears onto values of 2, and 2 is a closed category. It's actually a total of closed categories, and you may even bet it has a left edge on it, et cetera, I say et cetera, there are con extensions, all flowing out of the idea that you start with the closed category V equals 2.

55:00 And then you look at V to the V-Op, but you see in that way you get rational names for the operations. That's the idea. I'm saying this business of using field of energy as a module of theory is completely irrational. We're trying to do it as logic. There are much more rational names for the operations because they're all functions. First of all, that's part of the thing I was saying before. Instead of 27 abstract operations, the 10 order-preserving operations are sufficient. If you have a whole morphism with respect to those, and you already know that you've got the concrete object, then you have an action. So they're all functions. They're all under-functions of B to the power of B-up, that flow out of Agiornis in one way or another. So that's what I say, putting all this machinery... Totally, totally, totally complete categories of time extension, enrichment, appreciation, majorities, etc. All those things focusing on two. But specifically on V and Vi. So then all these ten endofunctors of interest have rational names so they can reinterpret the project. I've always said this is something that computer science should actually carry out in some way, see, because it actually means something. I tried to relate this also to this idea that the... Category theories that take interesting properties are statements that have already existed in math as invertebrates, so that in logic, you can use sort of the same idea, namely, instead of just bare propositions, you have propositions which are of the form A implies B, that are sort of trivially true, and the interesting thing is that they might be invertebrate.

57:30 They might also imply B, in a sense, so that the ingredients are not individual compositions but pairs of compositions, and that gives you three again, so it's another way of interpreting three. You know, it's sort of ordinary practice in discourse is when you say something... It's because it's relevant to say it, and it's based on certain prepositions that everybody already agrees to. It's like, I don't know, I live in Bristol, but I don't speak English. It's like everybody who lives in Bristol should speak English, but I'm making this a highly non-trivial, exceptional thing to a perversing application. That sort of thing. I think you can tie this together with this other thing about the d to the q off. Delivered ingredients themselves are already functions. It's like d to the d. They are already themselves operations. So, you know what? Achilles is 1.3 times as fast as him. It isn't just the ones in the abstract, but relationally. Why do they like the effective topo so much better than the recursive one? It seems like they've gone off on the road of idealism. Thank you for your attention.

1:00:00 Well, sort of one of those things, sort of the illusion of being extremely practical is that it is super idealistic. It's good to be very, very pure about this whole idea of being constructive and doing constructivism itself into a, as you say, into an ideological... Well, it is already, but if you view it as an idealization, well, yes, it's useful to have the natural number out there in the topos. Of course. Central geometric category. I couldn't possibly have it. There's no such thing as a suppressed snowflake, but still we could... Yes, that's completely inverted a picture. Yeah, yeah, exactly. They didn't realize that they were wrong. Yes, I don't understand the sociology there. I think it's something to do with what they're told when they first learn the effective topos, that this is all you need for computer science because it's the embodiment of this idea of constructivity, of constructible functions. It's of course not his fault, but perhaps Bill Mowry, I don't know, he didn't press it enough. That's right, he's a bit modest and doesn't put himself forward. Yeah, that could be part of it. On the other hand, I don't think that... Martin Hyland is particularly, may make huge claims for the... Is he really the one behind the... Effective topos? Not behind the widespread belief that it's... No, no, just on the opposite. I'm saying just the opposite. I don't think he is. I don't think he is responsible for that himself.

1:02:30 But there is an awful lot of propaganda on those lines made, which I think perhaps he, precisely because he is such a rather modest and retiring guy, he hasn't done as much to counter as he might have done. Well, I mean, as I was saying before, the idea of the possible math class requires a rise to the old. Topos is simply added on to a function space which is obviously basic, you know, that's the whole thing, and so in Golry's topos, again, he himself is sort of modest about it, doesn't fully exploit the use of the partial math class matter, which is extremely powerful there, as you see here. Because you have all these sub-objects of omega, you therefore have all these different graduations of degrees of implicitness and partial maps and a much more objective and systematic way of expressing all these things than the use of logics or something. Directly in terms of limits and co-limits and in more objective ways than by the presentation via traditional narrow sense logic. I hope one of the things we can talk about in at least one of our sessions is the ramifications of those ideas of Grotendieck that he talked about in 73 in Buffalo, which you discussed with me on several occasions in that talk at the Buffalo Colloquium, that do seem to carry this extraordinary vision of that. As you say, which you christened bypassing logic, which I appreciate was not a phrase that Grotting himself used, but it does seem to be implicit in his program. You can have much more widely known in the course of time. Yeah, well, I hope we can discuss this with Olivia. Yeah. There you go. The plate is also quite warm. Okay. Mine are not too firm. Not too, not obviously, well, you have asbestos fingers. Okay, I've got the picture. Thank you very much. Enjoy your pizza. Thank you very much.

1:05:00 Well, the problem is, the trouble is in land security. Because of that. You see, because when she got her residence permit 30 years ago, she had certain fingerprints. But 40 years ago, she had certain fingerprints. He talked a good deal about the category of categories in some of the discussion sessions, and of course I'd read his paper on the category of categories some years back. I never knew he'd actually gotten around to writing a book. Well, I mean, it's a rather strange thing. He could never have consulted with me at all about it. That's utterly weird. You'd think that something like that he'd... Well, that of all topics. He's asked me if there were any recent developments of it. If anybody else had worked on it, which I don't think they have, and so on. Well, that's utterly weird because... He just dropped it there. Utterly weird. A conversation about something else. The same question immediately, what is going on there? Because we were there with him for a week and he never mentioned this to anybody. Well certainly not to me, I'm sure Cartier or one of the others would have mentioned it. And the subject of the category of categories of course came up on the historical side several times. In fact Marquis spoke about it at some length. Jean Pierre Marquis. Marquis made a presentation on his book, which came out as you probably know earlier this year, but he made a presentation on that. And he did consult with you about that? Yes, yes, and in fact he was very concerned to, you know, he wasn't too sure whether you would be satisfied with his account. He had very much hoped that you would be there, but he seems to me to be very... He's a careful and thoughtful guy, but he went into some lengths, obviously, to give an exposition of the historical, the circumstances under which your paper on the category of categories appeared, and he was giving a kind of potted history of the main developments from the adjoint funda theorem and Kahn's work, obviously, and Kahn extensions, and the... You need a dilemma from 1958 until the mid-1960s, really, until your work on, obviously, your thesis and then the Category of Categories paper.

1:07:30 And Colin obviously said a number of things in the discussion about that, but he never mentioned at all that he'd written something about it, other than the article which he published some years ago about this. Strange business about. There's a consistency proof using Quine's NF. Well, the category of categories has been... there's some sort of consistency theorem. Relative to new foundations. Why that should excite anybody, I don't know, but it might excite some of the magicians. I mean, I think one of the reasons he said he thought it might be interesting is that it's the only thing that NF has ever been shown to be good for. He came out in JSL about maybe six, seven years ago now. It's a very short paper, it's only about three pages, so I knew he'd published a book, but I never had no idea he was writing a book, that's extraordinary, he wouldn't say anything to you, very, very strange. Who published the book? It's to be published by Oxford, oh sorry, Colin McLaren, sorry, so I thought you meant which publisher, yeah, yeah, yeah, thanks. And it's not even clear that it's actually come out. He actually did write me, a few weeks before that, and said, how were these comma categories defined, or something like that. Okay, alright. It just seemed like an isolated question. He didn't mention anything about this getting into his book or something. Oh well. And that's the only thing I meant to say. The category of categories is not about large categories, it's about small categories. Yes, yes. Being able to have a thing called the category of small categories is crucial. It's just a crucial point. It makes a difference. Which in fairness is something he always underlines whenever he's explaining it. Subtitles by the Amara.org community

1:10:00 Pierre Martin Lurps, retirement, 70th birthday, I think, and Colin is speaking at that, he and Steve Audi, and then they're having a mini-meeting in Paris on the 13th of May, at which Steve Audi and Colin are both speaking, at the research, at their history, math unit, a little kind of mini-meeting on. Category theory, which might be quite a useful exercise. So I'll see him then. I'll ask him. Maybe it will have appeared by then. I wonder why he's played it so close to his chest. The publisher has not sent it to me for refereeing. Something is really going wrong. Something is really strange. I don't think it will have appeared by then because it's only been sent to the publisher. It won't appear at least until next year. Probably not even then. No, it hasn't even been submitted for refereeing then. Well, well, we'll soon find out. I'll soon find out. Okay, shall we make a move? Um, so we don't keep Davidian. See, Valdi actually switched from Cambridge to Oxford because of the referee who was very insistent on him. I heard something about that actually. I heard something about that a little indirectly. I rather, reading between the lines, I rather guessed that that was what had happened. Steve was moaning a little bit about that in his 1.1. No, he's basically a good guy. He did say one thing at, what was it? Oh, no, it was actually something much earlier. He was, he was, he was, he was, he was, he was, he was, he was, he was, he was, he was, he was, He was still hurting from the fact that he produced this translation of the introduction, I think, of the Ars Danum's Lyra. Or was it even an attempt to try... No, no, no, it was an introductory essay. There was a translation of the Ars Danum's Lyra, which Open Court produced. Yeah. And the story he told me...

1:12:30 I can't go and repeat this to him, as I don't want to be forced his telling tales out of school, but that he had asked you, he had solicited you to write an introduction for this translation of the Ars Dei in the museum. Not me, but one of the editors. Well, the way he told it, he had actually solicited you himself. But anyway, you were solicited. Probably he told the editors. You were solicited. Maybe he told the editors to solicit you. Anyway, at his instigation, he says, you were solicited to write an introduction. And because the translation was so god-am-awful, which I've looked at and I think you did, you did not wish to write an induction for it. Well, I don't know. Well, exactly, that's the point, the choice of publisher is as much to do with it. But no, no, certainly not Grassman, obviously. My version of the story is that actually Steve himself, although in a relatively lower position, made some major improvements in the German, so the public approved of that. So your objection was to the publisher rather than to the translation? Well, it was to the translation. Definitely the translation. No, no. I saw the preliminary translation. He, after the fact, after... My complaint established in the eyes of the publisher that indeed the translation needed to be improved and he, sort of as a typesetter as opposed to a copywriter on a certain level, he changed various things because he's very good in German. Oh, his German is absolutely fluent. In fact, he is absolutely fluent. That I knew. His German is absolutely fluent. He brought it up to the more nearly readable stage, which it is. But the thing is, what's clear to me is that he just doesn't take on board the issues that you have identified with open court. Well, no, he says that he agrees with me. Well, why does he? Well, okay. Anybody can say he agrees with anybody else, I know, but, I mean, no, I think he does have doubts about it. I mean, he worked with him because he needed a job. And then the grandson, the great-grandson is his friend, but he says, he claims that his great-grandson agrees with me.

1:15:00 Now, that probably means... This is all interpretation. That probably means that both Andre Karras and Bill LeVere agree that the main thrust of the Open Public Company is to promote religion by testing science. The difference is that he agrees with that. He thinks that's a really good project. Maybe that's the sense in which we agree. I don't know, I've never even met Andre. Chiara, sort of. Certainly. Could be, could well be. No, that background I hadn't heard of. That would be a very plausible interpretation of this idea. Yes, it would be.