Morning Discussions, incl. FW Lawvere, Pierre Cartier, Angus MacIntyre, John L Bell

0:00 Profoundly exciting and important topic of objective number theory has come up. I mean, I envisage that this was something that we would discuss in the course of what we've loosely called, labelled, the Law of Irrational Days, but if you want to continue to pursue it with Pierre while he's here, put it this way, I shall quite understand. On the other hand, I have rather hoped that while Pierre was here, we might... We have one final discussion with him available to input on Grotendieck and specifically on this remarkable Grotendieck of points and other merely points, which Pierre took as the leitmotif of his expository paper about Grotendieck and those aspects of Grotendieck's legacy that are now pursued or allegedly pursued. Or possibly not conceded by people like Korn and Korn-Singert. So it's a question of which we do. If you feel that while Pierre is here, you'd rather... If you think that's a more valuable scientific use of the guys or so that we still have left with him around. Especially since he seems to have grasped the idea and wants to run with it. And Mad Day's Night is on paper, isn't it? Yeah, I can read that paper. Oh, sorry, which paper were you talking about? Chartier's paper in the book. I mean, it would be nice to discuss it, but it is also on paper, right? Well, yes, exactly.

2:30 While you're still here, I'm here this morning to discuss... I'm going to try and keep some time in reserve before you leave to touch on the Grotendieck on points. Okay, the recording's run. The recording night must go, I'm afraid, and we'll bring Leo Coric to you as soon as we can. Okay. Okay. Good luck. Thanks. You said you had a friend, John Bell, didn't you? Marcello Fiori, I suppose you know him, do you? I mean, he's in Cambridge. I'm in Cambridge at the moment. We're always trying to get together to talk about the high school algebra. He's saying we will keep stuff on the high school algebra homework. What's the metric theory of the category of finance? Yes, yes. Mathematics is what the program also mentioned in that.

5:00 Yeah, temporarily lost. You see, I mean, there could be mimics in piano with elements in an exponential set by using the exponential operation and all kinds of, you know, encoding and... That's right, but you think they're natural. But one should be able to, you know, from the point of view of the category of finite sets, it's much more direct, at least what these basic operations mean. Right, so, so what is the, it's a context, okay, I mean, in other words, you could say, what's, how the hell are the category of finite sets different from the category of sets? An obvious thing to me is that data can finite this. So in other words, any two-ended workings of an object which compose to the identity one way also compose to the identity of the other way. That's just the crucial thing now. This as a property of a single object is very difficult to do anything with in the category. So finite is for each object. You formulate it, but the condition is formulated to be one object for every object. You forget the other one. You formulate it just in the whole. And by the way, there's a little trick. This very algebraic way of putting things, composing, implies a more usual formulation by just passing through the power set. In other words, if you have a self-monomorphism, you take the power set, that will split. If it splits, it implies it's a nice morphism. Drop back down, that was a nice morphism too. So the nice algebraic formulation implies the more topos-theoretic one, or the more traditional way of saying either a mono-endo or a mono-epi must be an identity. Now, how, see the question is, and in Piano with Dick, I don't know if one ever formulated a precise, there must be some analog of this statement, but I don't know.

7:30 Well, it's part of the lifting statement. Justify it that it can, everything is that it can finite. The interesting thing about Dedic and Finite is that it has nothing to do with induction, with building up. It's an objective property from the outside. So you say epic and mono are the same? No, endomorphism. Endomorphism. Oh, I see. Mono implies identity. It's equivalent to epi implies. Epi equals identity for every object. Not identity. I don't think so. Mono is identity. I mean it's like fidget, it's like fidget hole principle in some sense. Oh no, you're right. Yeah. Well, no, he's, I don't think he's invisible. Right. I think in terms of, and we won't know he's invisible. Yeah, right. I mean, what you say about piano arithmetic, I mean this is a, this has been a very important issue. At the level of, say, first order piano arithmetic, in first order logic, there is no difference whatsoever between having induction... And having this pigeonhole principle for all formulas, pigeonhole principle being the following thing, that if you have, let's say, two initial segments of him or her, with endpoints, and you have a definable map of one initial segment and a definable map, you can't have a one-one map from the segment into itself, or a one-one map is subject. Now, at the level of all formulas, there is no distinction, well... Locally, there's a distinction. I mean, to have the induction principle for one class of formulas is not equivalent to having the pigeonhole principle for exactly the same class. And it's a very, very difficult issue. I raised the problem 25 years ago. It's never been solved. But if you work in some kind of restricted, complex theory, the kind of thing that Nelson and people propose, Bound and quantifier formulas, essentially the kind of thing that Nelson wants to have in Earth today. We don't know whether this pigeonhole principle for maps defined by bound and quantifiers is true or not. And in fact, there's a very delicate issue. Of course, we're interested in the language, just the ring language, minus also if you wish, but plus times that.

10:00 And we don't know. People have tried without success. If you analyze how you prove pigeonhole from induction, you typically have to add that. ...a function quantifier out front. You're doing induction on a formula with a function quantifier, so you can get... Now, Itai, who's a complexity theorist, did prove something interesting, which may conceivably be... He showed that there is no possibility of a formal proof of this principle, that if you have induction for bounded formulas, you will get pigeonholed for bounded formulas, because he added a generic symbol for one new function... And he was able to show that there are interpretations in which you have induction of above-the-formance and you don't have pigeon-formance. So, in fact, it's an issue that we've, Wilkie and I and others, have known about for a long time. And it's important for complexity theory because many arguments in elementary number theory use, rather than induction, a pigeon-formance. And they're not really logically equivalent down that big. I'm just buying my guestbook. And there, you know, the axiomatic, the first-order theory, the category of finite sets should... But this again is an interesting idea, which I've encountered over the years. Again, for a logician, if you go back to Gödel, there's not any difference between the two things. This is not a growth. From time to time, I remember doing something about the salt. There are results much weaker than that.

12:30 It's not clear that this phenomenon is equivalent to the corresponding five sets. Can you take the right primitives? So there are, they're probably not known to be slightly out of fashion, but I almost thought they were significant things. So, yeah, I mean this business of exponentiation, as you say, exponentiation of the category of five sets, well it's just... So I came up thinking along the lines, whatever, precisely, I mean, you know, I come across Ramsey's theorem, the BC dimension, so they must have a direct interpretation in the category of finite sets, so the question is, what is exactly in this mean box, and then you go even to So this results, this is kind of a test case at this point in time, because the way that it's formulated, it seems that you have sort of two parameters, one of which could be interpreted as a finite set, but the other one still is a natural number. But, not true. I think it's odd, because it seems to be really a category, a property. Just of the morphism category of finite sets, you know, in other words, it's another finite topos, because the effect of this natural number variable is really to say we're quantifying, we're finite families of finite sets, and so we want to be able to make this kind of Ramsey-like reduction at the same time, so I think it can be formative just using maps of finite sets as objects as opposed to... Which raises an interesting question, though. Why should the first-order theory of the category of maths be significantly different from that of the category itself?

15:00 I have the impression that some of the early people did attempt to look at this piracy student. In fact, Kerbe and Haas really got the first results, and then Paris and Haas got the natural approach to it. I think it's a vaguer question. I once saw people perhaps not doing very much. I'm not sure, but at the time I found it an interesting thing, but they didn't do much. Yeah, this is it. I'm not sure. Anyway, there's another reference. So, that paper was in Theoretical Computer Science in honor of Lena Scott's 70th birthday, but now Aurelio Carboni's 60th birthday, so there's a special volume of the electronic journal. And there I have another... I suppose it's in the same sort of spirit, in relation to some idea that they had after that, so you might be interested. I want to see that. I'd love to have your comments on that. Yes, certainly. No, no, this is very nice. And then you should bring measurable cardinals, you see. Let's see where you find out a couple of comments on that. Yeah. Yeah, yeah, sure. Well, of course, in the end, I mean... Maybe there are fewer specific ideas there than anything. I mean, I just think that's too expensive. Well, this has disappeared over the years. We could see it now. I'm sure you could just print it out right away. It's not too long either. I saw this paper by Chuck. I haven't studied it yet. What again is your perspective on this?

17:30 Because at the time I've been working on this stuff for years, so precisely what this thing's got meaning. A few of the crucial points, you know, just in conversation, and the next thing I know he's writing it out as his own paper, and so what can I say? Can I really object to this or not? I'm just, I mean, of course, when Shannon Wells writes quite unhappy with it all, I mean, there are people who are unhappy with it all. Really? I've never met him yet. I mean, he's in Boston though, isn't he? How does he be doing with it? Thank you for watching this video. He was definitely employed. I didn't see him. They were evading Peruzzi's final coming up. I missed. I don't know where. Peruzzi knows. He was saying categories are very important. What about any category? And he gave a response, which he was trying to make a nice little response, but he was like, I prefer to keep N as low as possible. So they said, ha, ha, ha, N equals zero. So there's this derisive presumption that they've got some kind of superior insight just because they've let N increase. Because in fact they have no theorems and no applications. They go around talking about how these 10-day things will have applications. But they have ten different... that... Leinster's book is just about the fact that... I took a look at it the other day, but I didn't pursue it. Leinster's book on... You know him, he was at... he worked with... Leinster, he's... Oh, Leinster. Yeah, he's now in Scotland, in Glasgow, but apparently he was at IHS. I think he may have been a Hodge fellow or something like that at IHS.

20:00 And he asked you to look on it. And categorize it? Yeah. But this may, well, I mean, there's a whole field, and then there are the individuals, and neither of them are very different. It sounds like a great idea. Uncategorized must be abused and want things really uncategorical and so forth. Fine. But then when you come to the way in which they are, the community, that community is attacking us, they're just doing propaganda. They're not getting anywhere, mathematically. This may get sorted out at this meeting in Ross Street in Sydney, which unfortunately I turned out I couldn't go to, because there are a number of people out there who think about it in categories from different perspectives, and not only the perspectives of their own community, but I expect it might get sorted out. Because the one alleged application is the one that Rodenby talked about in pursuing SAC, infinite dimensional reports, as models for... But there are no other applications. No, no, no, even that would not be a certain application. At least, again, what they've actually done with it. I confess I do not care for players. First of all, could you sign my guestbook? No, I'd be delighted to have one. This is my book of secrets. Nothing is over once you've had, you know, all this. Right. Well, I heard him... There was a summer school in Portugal, which didn't have any more hints than that, but that one hour is long. But anyway, so Peoria and Meisner, right? First Peoria and then Peoria and Meisner. They've written this stuff, but you know, from Shadywell point of view. I don't understand where it's all supposed to be going. I don't understand the... So what exactly do they do? Well, there's this one remarkable, just very remarkable, it sounds almost trivial, but it sounds impossible,

22:30 that if you, you know, so an adjective monolingual is a natural order, and a top element is something that's greater than or equal to, You completely don't have antisymmetry. If there's any top elements, there's a whole cloud of top elements, but it's not a cloud at all. That's the remarkable thing, especially if you're in a rig. If there exists a top element, then the set of all top elements is a ring with a new zero and one. It's the same old addition and multiplication. But a new 0 and 1, so there's a uniquely determined infinite 0 or top 0, there's a uniquely determined minus 1, just like, I always think of the rock C of electrons, you see, you have essentially infinite C, but if you add 1, it's just really different from the other, it's not isomorphic to it or something. So it's an infinity where you can add and subtract and make a cancellation. So what kind of rings can appear here? Any ring. Any commuter you bring can appear this way, but the way that they came up originally was, as I say, what I've just said is very elementary. I told this to Pior, he went up and wrote down, it's very elementary. It just sounds the fact that you're in this extremely loose situation. And suddenly you've got a total uniqueness. It's just very surprising. I still can't quite believe it. But the way that it arises concretely, if you have one of these fixed point equations, x times f of x, where f is a fairly general polynomial, a degree higher than 1. You don't get it with degree 1. You get it completely different. You take the bird-side rig of the category that this generates, and the dimension rig, entering with two, is just in three elements, as I said before. So the top elements are identified with those things whose expression genuinely contains the object X, like the finite sum of them.

25:00 And so actually there is an object there which behaves like zero and added to many other things. And minus one. So we have a whole deal of this kind of, this kind of rigs. You see, it's even, oh, in fact, in fact, what rig is it? I mean, every rig has an associated ring by the well-known high school construct. You have this left, right, right. So of course the junction map is not monic, as it doesn't start with something that's cancellation. So in fact, it's that same rig. The ring, you take the ring, you take the associated This is one of those situations where you have, so to speak, negative sets, where the negative sets would have sort of, normally minus one is totally virtual in the case of it, but in Chanuil's case, it wasn't quite virtual because there was an actual object that gave rise to it, even though, of course, it couldn't possibly satisfy the equation x plus one equals zero as an object, but it played that role. It satisfied instead the equation, what is it, x equals one plus two x. But if you have a thing of a higher degree, this dimensionary sort of collapses, except so then this homomorphism from the original grid to the ring is onto, because the infinite minus one becomes the actual minus one. 0 and 1 go to 0 and 1, but so do the 0 and 1, the infinite 0 and 1, but there's a section, you see. In other words, the ring injects back into the ring by a map that preserves the position of multiplication, but of course not 0 and 1, the right, the 0 and 1 over here, but it's, so he calls this, for short, the Euler section, because the projection, he was calling the Euler characteristic, and this is a section for it. It's almost a ring homework, but actually it doesn't because they're either zero or one. Every ring can arise that way from a suitable category.

27:30 Well, every ring can arise that way by construction, and many of them can arise as Burnside. You know, as the Euler ring or the Burnside rig of a category. I think this may have something to do with an old, old paper of McLean that I never read, which he's associating to combinatorial objects, namely machoids. The association to matroids, algebraic numbers, as in variance. How is that possible? I mean, it's just very, it might have something to do with, in many cases, in many cases, some of these sort of infinite combinatorial categories generated by following the fixed point. You get, you get the associated grades and rates, which have this infinite view or top view or top minus line. So I have mixed feelings about the theory therefore. On the one hand, I'd like to tell him, well, go ahead and find out more stuff about this, but on the other hand, publishing it without being part of the group that is discussing it, he has no idea what it really means, because he knows it must mean something, and it's a very unusual calculation to show this in the case of a community of monoids, that there's one top element. Then, there's a meaning of U invented there, even just added to that. Basic computation is something that's just added to it, and the multiplication will run for the ride, almost. So then there's a unique element which plays a role, an infinite zero, and together with the given addition, is actually the meaning of U at the top of this, at the top of this order set. So, somehow that whole idea about carbon numbers is... Not typical. The abstract cardinals, or even the isols. Well, actually, isols, they go out in isols.

30:00 Oh, yeah, no, I mean, there almost has been. I mean, the ruler had some pretty heroic efforts at the beginning of these things. I mean, he did want to get something visible amongst other models, and he could have done it, but he couldn't do it. I think it has something to do with Schindler, but I'll explain it a little bit better. There's something, it's not exactly the same, certainly, but... Yeah, this connection with Nelson, this thing, it was pretty striking when he discovered it in connection with his philosophical ideas on quantification and so on, but Wilkie also discovered it on some of it, and I think maybe Kudlack and a few others and people... The new non-standard models, you know the system of R. N. Robinson, Q, the final name of the class system, yeah, the thing is there's no, everything is, everything, nothing is not at all cancelated, etc. There's all these funny things with it, but the addition and multiplication of community, etc., and the British model, as I said, is the most understandable. Nevertheless, they all discovered that you can... In the technical sense of mathematical logic, you can interpret, in any model of Q, you can interpret the model of bounded and bounded. I mean, there are formulas, but you need to interpret the model of bounded and bounded. You need formulas in point of fact, so philosophically, I think it's less important. Nelson claims, but nevertheless, as I said, simply as I said, there's something better than that. They've restricted some of this warning in the others to this original slash, you know, it's a restricted exponentiation, you take an integer and you exponentiate it to the binary length of another integer. It looks at all, but in fact it has, it really isn't. First of all, this function is... Did he answer that call? Well, I don't think it's the one downstairs. Yeah, it went down, I think, so presumably it's not the one downstairs.

32:30 Yeah, he answered them, I don't know which one that is and where it is. Or it's somebody else's question. Excuse me. Yeah, no, no, no, but... I mean, it looks like a very ad hoc function, but it's a moral growth and also an associativity growth. It's a funny thing, but it's technically important coding of substitution, you know, sticking one sequence of strings inside another is much better than using exponential, it's much more economical, much more algebraic. And, in addition, it's connected with these pigeonhole principles that we... I think it's very close to the idea of John Bell, actually. He had this paper, you see, where he was talking about a general phenomenon of well-ordering and sets. Which I interpreted in the following way. That you, you know, if you take the power... But if you take instead the symmetric power, x to the n modulating factorial, then this typically can be comparable with x itself. What you call type reducing. It's also cardinality reducing. You know, so, I mean, for example, even in the finite case, right, like, 2 to the n is pretty big, but if you, if you, if you modulate, if you divide that by the group of automorphisms of n, you get n plus 1. You always get, typically get this plus, right, you get this additional element. You get the empty set. Yeah, that's right. You see what I'm saying? Yeah. You take, take automorphisms of n. And two to the n, modulo of that, this is a kind of definition of cardinality, you know. Well, that's how Fraker used it. I mean, he actually gets omega plus one in Fraker's construction. He gets omega plus, you know, the natural number is plus one, right?

35:00 You all joint work with him. My Greek friend, right? I've never read papers with a Greek friend. At London, Ontario. Oh, Bill, yes. Bill. Bill Dimopoulos. Dimopoulos. Yeah, yeah, sure. I haven't talked to many other Londoners. No, no, no. I mean, I see. Yes, yes. He was referring back, he claimed that this was Hume's principle or something. Well, that's the term that they use. But this, even Barry Smith. Much as I don't like him. But anyway, he immediately caught that there's something strange about this. Because, or at least different, at least different. The idea was if you want to study the science of a class, you look at anamorphisms of the, well, endomorphisms of the universe which are used to isomorphism on that subclass. Now if the universe is finite, of course that's okay, but... But basically, it means that you're getting the negatives or the complements at the same time, so if you actually divide out by the automorphisms of the universe, then the equivalence classes of subsets are sort of when you have the finite ones, then you have the co-finite ones, and you have the... Yes, that's right. So it's just sort of, it's always going to have two ends because of complementation. You know, who knows what in the middle, but it's quite different from the usual notion of cardinality and fact. Well, that's exactly what I've mentioned. There is a three-dimensional calculus of variation. I mean, I've mentioned that I have two complex, one which is finite and the other one which is co-finite. But they never draw, not at all, not at all. In finite dimension, in finite dimensional spaces, they do something. I mean, they reflect each other, but... I mean, they've done it so that they don't do it. That's the way to buy some Barclay. Shannonwell says that this was his reputation of Descartes, you see, because he had some idea to the effect that if you can see something very, very clearly, then it's true, right? Yes, basically.

37:30 But Shangerwell says that he could see very, very, very clearly this ordered set, which was well-ordered from both ends, you see. Yes, I could see it, you know. He saw it so clearly. So he's just convinced that it doesn't exist, by logic. No, it's really something like, I mean, you're describing, it's something like a cardinal, you know, then you have something like co-cardinals, some kind of dual cardinal notion that's emerging. Because, of course, that's the thing with cardinal algebra, I mean, well, with Cantor's cardinals, that there really isn't any, you have the positive end, and there really isn't any corresponding, well, formally, I suppose, it doesn't play any kind of role, this, you know, this, and, of course, that's the real difference, well, at least, between Cantor's cardinal and Z. I don't think there's an analogy at all with negative numbers. I mean, you've got an analogy with positive numbers, which of course was very important for him, but of course negative numbers just really don't appear at all in any serious sense. So, the fact that the exponential doesn't grow as fast there is just a consequence of the fact that the exponent itself doesn't grow as fast. He asked me to try to steer the discussion. Who wrote Antigone? That was his parting shot as he left, so maybe we should do that. He's not going to pick you, Leo Curry. Yes, that's right. Well, he's going to pick you up. He's on his way. So if you just wait. Yeah, if you if you wait, you'll be there shortly. Okay. Just think all this is being recorded for posterity.

40:00 Viewed as a reflection of the apparent state of the historic interest. When archaeologists, when archaeologists, when archaeologists, when archaeologists, when archaeologists, when archaeologists, when archaeologists, when archaeologists, when archaeologists, when archaeologists, when archaeologists, I confess, it's been a long time since I've thought about matrix theory. Especially the Paris-Warrington, the stronger version of it. Yeah. It seems to be a property of the morphisms of finite steps. Yes, I mean, this is so. I mean, you start with something that's relatively innocent, like boundary conduction, and then you assume you have these in the terminals, which you would get for free. You're assuming a slightly stronger version of the misanthropy we should get, but it's this version of Baran's graph, from the infinite Baran's easily, and then purely formally, somehow with the stretching of the misanthropy, which has clearly got something to do with the categorical considerations that Girard uses and so on, and you've just suddenly got a model of something much, much bigger.

42:30 Of course, but that's the same game as they're playing in large jargons. I guess one should have looked, or should still be looking much, because most of the set there is thought. To admit the existence of category theory is... I don't know. There are not a great number of set theorists left, actually. By comparison, certainly in the 1960s. No, but I think it's in steady state. It has become very much a priesthood. I mean, it's a small, small number. Exactly. But they do still get... there are enough of both. They get enough people to maintain a very, very high level. Yeah, yeah. Of course, it's gone in Britain. And quite possibly more, but it won't go in France because they've got the Serbs, the two Serbs in Paris, but it's in California, there's two schools in LA and Berkeley, but I really do think they're not yet under threat. Yeah, Woodin is a pretty representative of that school of Berkeley, I mean, he has his own, whether he caved, I mean, to the AMS, to the draconian powers. What's the subject of the lecture? I think my impression was that he really, well, his presence, which is very smooth, you know, I have to say, he looks very much like the actor Zachary Scott. He used to play, you know, he used to play Dylan, you see, in movies. Very, very smooth. It's very smooth. It is, it is. But you see, it's smooth in the way, in some sense, it may be like, perhaps the program isn't smooth, but it's something like...

45:00 These are rather physicist presentations, you know. You don't really get down to anything. It's sort of suggestive somewhat, you know, and you get the illusion, well, maybe the illusion of understanding something. A bit like some of the physics presentations I've seen. And also the, of course, as I say, he does it, you know, he presents it so smoothly, it's a little, well, you know, it's a little lulled into, it's a sense that you sort of understand the topic. However, I do think that there's a rather more serious, I only meant that analogy with physics in terms of the sort of popular presentation that he gives, but actually I think there's a stronger analogy with physics in some sense. The way he's describing it, it's a bit like casting around for axioms. He's a Platonist of some kind, a realist in some sense, and, you know, you're looking at something that's over there. I did tell him I thought he was looking for axioms that further, for static sets, making the whole universe even more static. And he agreed. He understood what he knew. He did know that description of things. And then I got suggested, well, in that case, what you're really doing is a kind of description, rather than the way that physicists are looking for the right for plausible evidence, let's put it that way, for properties, in this case, of a really static universe. Something like, you know, what physicists do. I mean, they haven't got any, you know, the evidence may be indirect. Of course, the difference in physics goes on to the, well, one presumes you'll be able to determine by experiment, although God knows how long that will continue. It may be that you just can't do it because the energy is required are too high. It's just got to be plausible evidence again. That's all you have. All of these terms are very important.

47:30 Thank you for watching. Now, that's the problem, of course. I'm not saying that plausible evidence may be somebody else's dubious claim, but I do think that's... You know, that's the kind of, and I think perhaps that's why, you know, what enables him to present these, partly, you know, this sort of idea. Of course, it's complicated also. One has to say that the, you know, the actual detail, I don't understand. Oh, no, no, not at all, of course. But I do think that's partly why it comes out, and maybe the work of his own type of presentation, which is over smooth, perhaps, smoothed out far too much. But I do think that there's some... The whole sort of program has this, you know, like some enormously complicated, you know, trying to do it in quantum gravity or something like that, you know, you really can't, the details are very, perhaps very obscure, and you kind of present it as if, well, we're not quite, we have some evidence for this, and this is plausible, and it all comes out as an extremely smooth kind of package, which conceals really the, you know, the actual intricacies of it. I think that there's actual photos of theories. Yeah, I never had sugar in my face. Well, Conan Doyle, right? That's right. Well, of course, he also had ectoplasmic color. That's right. We should wrap this up here. Otherwise, Michael will be very... It's the tape. It's not the problem, it's financing care. It seems to have a very great resonance to the stuff on the, on the, on the very Shanyuil rig, as it corresponds to things he'd been working on in the commentatorial situation. So Michael was torn between allowing free reign to this and having... He left, he left me, he didn't want me to carry him, he didn't want me to carry him. We should find a reason, because it is a pity, I mean, we don't need it. Yeah, I agree, I agree.

50:00 Michael will be back in five minutes or something like that. I mean, who else is going to believe it? You know, it's not really like the... it would be a very hard sell. ...understanding... It's only a very good relationalism. ...mathematical... Oh, yeah. ...any other... There's no question about that. You know the actor, I mean, you must have seen him. Yes, yes, of course, absolutely. And Mildred Pearson in Mask of Demetrius. Yes, yes, yes. He's very smooth and always, always played villains. Well, I understood some of it of his approach, you know, from this talk he gave. Of course, it involves, I mean, it involves 30 years of hard, hard work. I know, I know. I mean, these are hard. I suppose every experience actually takes them further away from other things. It doesn't inspire a philosophical discussion by set you as much further down the...

52:30 I haven't seen articles on the philosophy of sector. I've seen all the stuff over and over again. You know? Yeah. I mean, and it's... You don't know it. I should also know all about it. But you don't know. It is history. You know what it is, don't you? I was glad to see in Philosophia Mathematica that there really is a reasonable spectrum of papers now on different topics. It wasn't the same old thing. It wasn't neologism or the philosophy of set theory. And there was a danger, of course, you see, that that's what it would become. And I'm glad to say that it's not confined to that at all now. It's a pretty good spectrum. Because I assumed it was so set in these two different kinds of components. No, it's actually after the time. There are different inputs there, you know, phenomenology, working for Ray, you know, and Hilbert. It's quite a good range, not that same old stuff, you know. I mean, give me a break. I'm delegating myself. This number theory has sort of come to three. There are three levels, rational, algebraic, and passive-denominational. Because it's obviously combinatorial. Elaborations, as I said, there's a tremendous range of these. There are direct interests in all kinds of combinatorial problems. So there we notice sometimes that objects satisfy algebraic equations.

55:00 So if an object satisfies any equation, then it satisfies the following power equation, and in turn, that's equivalent to being separable. Yes, yes, that is, that's separable. You see, this decidable, this property, you can detach the... A third way of saying it is that you find a spectrum in terms of punctures to the category of actually finite sets. Can't find it! No, that's the realist downstairs, he said... He was sleeping in the front there? Yeah, yeah, yeah, yeah. Well, he's not here. He's not here. Maybe he's having a shave or something downstairs. He's not in my bathroom. No, I was just there. Well, is his bag still in there? It might be loading his car. It's not leaving for 12. It was 12. So the separability. In another way, it just means that the equations, it's equivalent to some kind of finiteness as well. Objects are very finite. Yes, yes, yes. Then, of course, the only solutions that you could get by mapping into sets, or finite sets, so that the roots of the polynomial are natural numbers, which is, in this infinite case, we get, you know, one of these fixed-point phrases, and if it's a suitable polynomial, then you will find that you have no potent objects, in the sense that… Relative to their top objects, and their square is the top zero, you see. Ah, yes, yes, okay. It's literally an associated drain, you get no quantum elements, but only by going through these infinite, these higher degree equations. And then apart from that, then there's a sort of infinite side starting with these negative sets.

57:30 The linear equation is tremendously interesting. And then the higher degree equations, which unfortunately we don't have real models. There is a negative set, which is a geometrical, a known topological or some other interpretation. Dimensional space, which is so twisted around that it can satisfy a polynomial equation without collapsing everything. Yes, yes, yes, I understand. It would be nice to have such a more geometrical interpretation of this. I think I was saying the other day, the older product formula is an isomorphism of monolite objects, and that's not only the tip of an isopressure tube, but that's also on a third level, where you actually have to use the existence of the free monolite functor, and also the direct limits of all crimes. Yes, yes, yes. See, the free monolite functor is a much stronger thing. I think something that satisfies the list equation that's what that's part of that I figured that in a positive way in that paper but it means that uh it's not clear how strong it would be to postulate a category that has an anomaly addition to multiplication with one over one minus x as a function because if you just if you just take two things that satisfy the equation there's no okay you map x into y and you take the three more than x

1:00:00 Not the free one, but the one something that satisfies the same equation. Well, there's no, there's no reason to have a map between them, and postulating it, therefore, as a functor would be a much, a functor satisfying the equation for one over one minus x is a much stronger thing than just having the existence of an object. So that's probably somewhere in between the algebraic and the analytic. I can't find Pierre. I don't know where he's gone. It's a standard of resolving the fact that there are enough points to basically...

1:02:30 Right, right, right. So he had this general theorem that was coherent to the one for which the Gruden big topology was finite. Yeah, yeah, yeah. Then, Denny showed that you have an environment. Now later, you can do something similar for arbitrary group decodals. You have a human value model instead of sets. So there are enough boolean value models, and that's good enough because the idea is you apply the forward and backward geometric morphism and you do something over here that depends on booleanness of choice and so on.

1:05:00 But that's fine, you can do that in a boolean value model and construe it as a good and big topos. So after we got the boolean value models, we did that. But meanwhile, of course, I recognized very early on that Deligny had basically reproved Gerlitz completely because of this business of actually anything boolean can be coded in positive, but more primitives as negative. I think there was a connection. I mean, there's a sort of... Some kind of connection also, although it was a different, quite independent program, between the notion of points as introduced in this highly elaborated setting in algebraic geometry and also the program that was emerging of pointless topology from, you know, in topology using lattices and then trying to replace, what was the name, a dowker. Which was sort of a different somewhat different level technically but there of course all of this is now really you know you could see the kind of idea of trying to get rid of points. You have in some sense gotten rid of points you know when you when you work with the topos and then or at least when you go from a topological space right to sheaves and so on and then You've essentially got spaces which may or may not have points, and the problem is then somehow to reconstruct the idea of a point. But it's a petitopos instead of a point. Yeah, okay, but right, right. In some sense, yeah, the obvious fact that the terminal object no longer separates is true in that sense. That's an internal sense of points, which is rather different from the external one. It was possible, really, to make these sorts of distinctions. There was already this proto, you know, sort of program of trying to do topology without points, which was a, well, it goes back, that idea, I think, goes back in some sense quite a long way. I mean, you know, the idea of replacing the topological space by its lattice of open sets, and that goes back all the way, back to stone, at least.

1:07:30 And working with that and those theorems that he proved on representing distributive lattices back in the 30s, and there is a kind of, well Peter Johnston makes these, you know, these sort of points in his book on stone spaces. Well, anyway, I mean, there is... This is all, this is all Eilenberg could say in his last days. So you could talk to him and say, if you like, you could say, yeah, yeah, yeah, you could say that. But the other answer was... Distributive lattices are great, right? There was also the work of, I'm not sure if you can characterize the so-called stages underlying the spectrum as being really just equivalent to distributive lattices. These coherent spaces are just distributive lattices, you know. I had run into that quite early on. That was also, I think it was Hopper. What was the purpose? By the way, there's a, you know, there's a sort of, okay, how do you construct a Grotendieck Topa? So, you have a site, small category, notion of cover, fine. And then, of course, there's the dual thing that I think Terry and I were the first to point it out, but essentially a quotient is described by a left-exact homonet. Yeah. Okay. So that's, you know, left-exact homonet. But there's a third method as well. Namely, you could take any category which has filtered co-limits, with some set theoretical, and maybe there's a small set of objects such as everything is a, and now you simply look at the filtered co-limit preserving functions into sets. That's going to be a good indeed topos. I mean, at least it obviously satisfies Giroux's axiom, because the filtered co-limits commute with both arbitrary co-limits and finite limits, and that's what it's all about, you see, so.

1:10:00 So this is kind of a Fourier transform of the topos. You take the category of all points, as a special pick, take the category of all points of the topos, that's the category of the filtered co-limits. So you can then go back and take the punctures to sets from it, from the category point. And that's sort of another topos that's connected with the original one by a geometric morphism. If you have enough points, then that morphism is conservative in one direction or the other. In other words, every, for example, the Shannon-Rowe-Myhill topos is one that we, it's usually thought of as, it represents the notion of infinite decidable. So decidable, separable is clear, but then infinite in the sense, which is probably the most restrictive notion of, I mean, the broadest notion of finite available, namely that given any map from a concrete finite thing, It can be extended a bit to a bigger finite thing as a definition of infinite. So infinite, decidable objects are classified by the Chandrila-Mahill Tokos. There's an idea of Mahill, it's just a Chandrila, it's actually an atomic Boolean Tokos. The site being the opposite of the category of finite sets and monomorphisms to take care of the separability. Take the category of all infinite, decidable sets, so to speak, in other words the models of that thing, points of that total, and take the hunkers back into the category of sets as equivalent. So there's a kind of balance between, if you think of it within the category of all sets, on the one hand there's the filtered co-limits, the infinite things.

1:12:30 And they're finite things with monomorphisms and some kind of Fourier duality with each other. So it's never been explored systematically, this construction. It's almost like completeness and definability and everything all in one. When you recapture, you can sort of identify the topos which are classifying topos of a certain theory. But only with colons, not with ultrapowers, then you can sometimes recover it. I have no idea in what generality. Points of Copa are points of locales. That's sort of, even though it's a much smaller thing, it's part of the same general thing. Whereas the points in algebraic geometry are something very different. The idea was going to wind up inside a big topos rather than in a category of topos. I thought the notions must have, well, some deeper connection then. Well, they're both glorifications of the idea of a map from a terminal object. Yes, but also I mean, well, perhaps the purposes are different. No, what I mean is if you look at the sorts of use of points or areas... In an earlier setting, the notion of point of locale or point of a distributive line, that was really some way of introducing, reconstructing, or characterizing those distributive frames or complete hiding algebra, which do arise in open sets in a topological space.

1:15:00 In other words, it was a way, of course, it was an early, very early thing, quite a natural question. In the usual pattern, I mean, when you have an abstract struggle, when does it actually arise from one that you regarded as concrete? That was the program there. Now, it's difficult to see how that relates to... But it holds in the algebraic geometry. Model at this point. That's right. Well, maybe if it does, then you do have a parallel. The last case was very misleading. In fact, they misled Grotenbeek, I think, in the first instance. Raya Gelfand, of course. Because, you know, among the distributive lattices, there's only two is the only one where you would reasonably, you know, speak about points. But, and the complex numbers are the only one that, you know, so Galvan could identify maximal ideal points. That's right. But this is basically wrong. I mean, atypical, because in this slightly more general... In other situations, it's not a matter of maximal ideals, or even prime ideals, because, again, it was Gabriel who explained this to me, and I haven't seen it in print anywhere else, but basically the points, let's say the affine scheme, this sort of simplicity, they're actually a functor from the category of fields, namely, given the ring A, you map it into every field K, those are the K-valued points. And so on and so forth, it's not just a single object, really, in a Boolean atomic topos, actually. So it's the underlying set, you see, even though it's not actually actually a Boolean atomic topos. Now, if you want to get prime ideals, you just take the direct limit of that functor, you see, because given the map to a field, The kernel is a prime ideal, it isn't onto, the kernel is prime, and you know, if you have the cumulus relation, there's lots of points, and there's all the morphisms of points, but those get washed out when you take the direct limit, and that way you get just a set of prime ideals.

1:17:30 But you see, notice what an unnatural thing this is, because this particular direct limit is not filtered. It doesn't even preserve products, let alone left exact, you see. So it kind of destroys the elementary set theoretical operations that you might like to apply to. So it's again like the slogan of Verdi and Grote, don't take cohomology, don't take this direct limit. It's much better if you don't take this direct limit. So you are, I mean, and you consider the k value, and you consider this as a factor from the category of field to the category of set. Right. Let's just say it's a finite field extension. It's a small category. So along with any small category, you have the notion of a co-limit of any set value function. Any diagram of that shape averages out into a single set. So if you take the direct limit of that particular kind of function, it turns out to be isomorphic to the set of prime ideals of the original ring. Because taking a kernel of a given map as a prime ideal, and then the equivalence relation identifies these, but on the other hand, given any prime ideal, you can mod out, you can enter the domain, take the field of fractions, it's the kernel of that, so as long as your category of fields was large enough to include the field of fractions of any trained in that way, then it's going to be isomorphic, instead of prime ideals. So there was, as I said, Gabriel explained certain things to me and so I already had pretty much developed an ideology different from the standard one.

1:20:00 But then Grubendieck himself came to Buffalo in 1973 and gave a lecture which was exactly, guys, the old definition of scheme was wrong, throw it away and use this new simple one. He was very emphatic. I use this, I use this. Use this new definition. That a scheme is a set-valued functor on a ring, which is actually due to you anyway, at least in the group case, where you emphasize that an algebraic group is not only not a topological group, it's not even a group. Precisely, if you look at this, if you think of it as living on this direct limit set, taking this direct limit destroys products, and therefore destroys any algebraics. This is the approach that they take from the beginning in Gabriel and Gabriel drives, right, which they take from the beginning but then unfortunately abandoned to a certain extent, I always thought. It wasn't really a completely consistent execution at this point of view, but they certainly start with it. Well, after all the prime ideals, there's this non-exact direct limit of the perfect limit. Well, the trouble is, it was late and I didn't realize we had to go. I thought it was kind of this evening. Well, I see that now. I didn't realize that. He said 5.30, it was 5.30 in the afternoon. Yeah. No, he arrived at 5.30. Well, I had to go. I had to go. We have to save our honor. Something intended. In particular, after his lecture in Buffalo, I wasn't at Buffalo at the time, I was just...

1:22:30 It happened by tremendous good fortune to be visiting and he gave us, and after the lecture he went on to say, oh, points have automorphisms, this is a main point, you have to realize themselves, have automorphisms. There are objects in the topos, functors on steels alone. That's another, I think, he didn't say this, but it's just a matter of terminology, the idea of speaking about the b-valued points for any ring b. I think it's not quite right, you see, to call it points, because points are, those are B-shaped figures, but among the figures, there are special figures that deserve the name of points, and they may be quite a bit more general than just maps from one, you see, but still. Well, I like, and it maps to simple, if you have an algebra, you can map to the simple objects, or the sub-directly irreducible objects. You get better. To atoms in some kind of sense. Some kind of sense, maybe atoms with little clouds around them, but still something much more special than the whole curve. Irreducible minima of some sort. Yeah, yeah, yeah. So, but, I mean, of course... Speaking about the b-valued points, in a way it makes sense if you pass to the category of spaces over b, because then you have this old Greek idea of a locus of a moving point. So the figure as defined over a base is the locus of a moving point. But if you say in the same topos, the meaning of points should be more restrictive. If you consider for a given scheme, the joint union of properties, I could say that precisely these counters on fields rather than abstract sets, and so it's very discreet, and then it, yes, there is, there is the inclusion, which has a left adjoint, which is to take components, to take the base of components.

1:30:00 Or in the algebraic geometry case, it's the space of components equipped with the Galois action, may or may not be there. And that function preserves products but not equalizers. That's the only exactness property you have to mention because all the others follow from the joints. So the inclusion has a right of joint, which is the Cantorian idea of extracting from a cohesive space just the discrete space in its points. The points without structure, except this very, very small structure, roughly speaking, just the Galois group, so that if you were working over an algebraically closed field, then this is abstract set, so it's a set of components and a set of points. Rational points. But in the general case, it's not just rational points. So we have these two, actually. So the inclusion has a left and a right. Left and a right. That makes all the discrete factors. But then there's one more, which is the co-discrete space. The inclusion is like... the inclusion is you start from a discrete thing and you just view it as a very special extreme case of a cohesive space. But you can also start with a discrete thing and make a... it's like the co-skeleton in the Semplitian case. Which is sort of at least filling out all the holes, all the things that look like they might have been holes, getting an object which really intrinsically has exactly the same information as the discrete space, but it's plugged into the cohesive world in a very different way, so that, you know, it's the maps to it that are maps from it, so that there's no... It's the furthest right. So if you want to look at it in a row, then there's the components, discrete inclusion, points, co-discrete inclusion.

1:32:30 And this is a full subcategory, so various composites are actually the identity here, the adjuvant retraction. And I said I want the leftmost one to preserve products, that I have to put. It's automatically true that you'll find a lot of commonality, but I'll leave that at that. Now, another action which makes the following construction...