Mathematical conversations post-ENS seminar: FW Lawvere

0:00 If you have this induced map on the function space of x, well you could ask, as I did here, you could have the section and the discomposite here, that's the prolongation, for example, the section of the equation or anything like that. But you could also ask for the fraction, so that you have composition equal on the other end. So you have composition equal on the other end. Well, I guess going in the opposite direction. Either composite that has to be the identity. Yes, I'm sorry. Well, if you ask the other one to be the identity. If you find this sort of a formal case, we call it an averaging math. Basically, in this case, variable to constant. But in other special cases, we just consider the math from t to 1. Yeah. Now that induces, for any x, that induces... And so on and so forth, and so forth, and so forth, and so forth, and so forth, and so forth, Ah, right. So that's why you're averaging. Now I understand why. It's just what I was wondering. I understood the concept, but I was wondering what the term was about. Now I understand perfectly. It's very simple. It's algebra. It's these retractions and sections that apply to the context. Yes, I'd love to explain that. I mean, you've got to get the sense of mathematics. And learn. Thank you for watching this video.

2:30 No, no, no. Oh, no, actually, we already changed. No, this is the one we want. It's okay, we're back up here on the stairs, which is what we wanted. We wanted the slide going back the other direction. Did we change that already? We did, we did. No, it's okay. Anyway, we already changed. We did not change it. If x were a linear space... Hang on, have I got that right? Christ, no, I've lost it. So we do change it. I'm sorry. I'm sorry, when I get involved in your explanation, we're back again. No, but that's where we want to get back. It is where we want to get back. We now want to get it online, too. I'm sorry. I'd better concentrate on getting up now. Campanella said something like that. He defended him, his right to have this opinion and all this stuff. One act. And this comes to true, you know, Galileo and Hitchin came up having this from Campanella. And this is not from Romano. Exactly. So rather than studying Campanella, you know, rather than studying Galileo, I just said brothers and daughters. I did come to know that I was saying it was probably a convention. And it finally talks about the book of nature. That's what the thesis is. We knew that Galileo said that. No, no, it probably has some context. Well, there's some context.

5:00 Well, you could also refer to this as linear math. Yeah. That's the usual idea of averaging. Yeah. Because not only is it a refraction, you are signing. The same values are constant. That's the normalization. And it's linear as well. But you see, so, somehow the idea of averaging is more fundamental than insurance. It could be required in the cruiser or something else, you know. But, well, for example, the geometric mean isn't averaging, although it's not linear in the usual sense. That's not correct. But it's clearly an averaging. It's an averaging, yeah. And this point about thinking of it is... Reducing variables to constants, in a way, in a way, you see, that's what we're doing thousands of times a day, we have to think about the world, we reduce variables to constants in a way, think about it, and statistics, in that sense, is already part of basic human thinking, if you say the temperature is 30 degrees today, or in the... You might be measuring the motion of these. You can't actually measure the mean limit. This is our stop. I'm sure 9 minutes has got a great deal on us. Here's an example from the universe.

7:30 There are actually partitions between the huge distribution of mass and the fact that partitions is a 9 or a 12 depending on whether you're going to count all the asteroids in it. Partitions is the actual distribution of mass relative to these... Key terms may include mathematical physics. Speakers include mathematics, geometry, algebra, analysis, quantum mechanics, algebra, algebra theory, quantum mechanics, algebra theory, quantum mechanics, algebra theory, quantum mechanics, algebra theory, quantum mechanics, algebra theory, quantum mechanics, algebra theory, quantum mechanics, algebra theory, quantum mechanics, algebra theory, The other one is the value of mass values. Yes, yes, I see that. The geometric mean is an even better example in some ways. You do make the point very, very clearly. The average doesn't have to be linear. You're going to need a general map, not just a unique map to one. This is shown by the apples right here, the solar system. The solar system is the finest of the extended bodies of the solar system in the simplified theory of the point line. I'd like to remark at this point that the very-centric calculus is invented by an astronomer named Monubius. What it actually invented in connection with celestial mechanics? I don't know. I'd like to put in as a suggestion for further investigation. Sure. Now that's where detailed historical scholarship would be used.

10:00 But I like this sort of thing because I feel I'm getting into what I like to do, namely things that are ultimately very simple, which are understandable almost immediately, and which are not simple in the sense of only having trivial examples, but in fact having the most general example ever considered. Electro thermomechanics, plasma physics, and all these things are special cases. I was going to ask you. Once again, the fact that you can put all these things away on the same footing is only possible with the categorical review. You don't have to figure out the detailed presentation of what the large emotion of the plasma is. In order to know what you have, you have to have such and such. I think, well, you've always said that you wanted to go back to physics. Well, that's absolutely, obviously, the perimeter of the galaxy. Mathematical physics would be a very important, very, very important point at the juncture. There's a lot of people here even though it's not a holiday. Yes, because the being Montmartre, it's being the kind of village of Paris, it is where people come. You do make fun of Jim Edgerton. He is very, very bad fun.

12:30 There is a very nice little book about these qualities of the children, which way he's approaching them. Yes, it would be very interesting. Yes, indeed. In the Catholic Church, I have completely forgotten what I was going to see, and then it suddenly triggered my attention, because of course when I do these religious talks, what I have to do, I say I'm next month in Italy, and that's when I'm going to go and see Alberto, and so it's been a while. I'm going to go absolutely practically to the case of the very, very openly, politically, civically adapted version of it, so to move to counter the, you know, to counter the course of the last, I guess, the Royal Court, because it was, I think, at that stage, they'd lost the open area. But we must, you know, cross a broader section of the 20 masses as possible for being effective in this area by actually taking, by taking steps here in the real world, you know, the political organization and the revolutionaries. James had been involved in that novel since the old wrapping. I haven't heard of him, not at all.

15:00 Balfour is underestimating all of his political crimes, obviously, but these two books that he has about skepticism and the ways that he does it, it's interesting because... And another man, of course, who gave the Gifford Lectures in his time, of course, yes, of course. It's remarkable, you see, that Brennan only had a few months in the middle. That's a very great deal. I always learn a great deal. But, you know, I learnt quite a bit about revolutionary parents, too, because I'd never been in some kind of a church before. Yeah, well, if I hadn't even realised, it was named after them, still. And that, and named after reactionaries, too. Still, at least a number of things are known as after mathematicians. And even after, in some cases, after revolutionaries. I'm sure people who were revolutionaries were safely a long time dead, but even so... Even this bit, which is obviously terrible, terrible, terrible. Only you can decide on that.

17:30 I think she might just want it as a souvenir. What I'd like to do is, if we have time on the way back, is try and find that book about the cartoons, but it's very well done, it's a cartoon history of the Commonwealth, which I think I obviously found it now, but in a few years' time, I think everyone might enjoy walking down the hill from here to the ground floor. Well, I'm going to have another beer, and then if you want to wander over there, we're actually not too bad at the time, we still have an hour before we have dinner. We can give ourselves another 10-15 minutes here before we walk down the hill, or do you want to... Oh, we're allowed in there, before I have to... Oh, okay, before you have to confront, okay, well, the next... By the way, I've also just been called from Italy, so I'm going to... Oh, that'd be yesterday then, it's down my number, I don't know how many... Well, whatever it was, that's extremely well jaded, I do know it's one of the... Outlying members of the Fusdell Society for Mathematics and Udinet. Ah yes sir, you told me about them, you talked to me about Udinet certainly. If they have any plans for you, do let me know, because I would with luck be able to drop Helen in for November 2003. I do want to ask you, in that case I guess we have to make a move, we already paid for these didn't we? Yes we did. Well okay, just if I can just give you, if it's not too rude, just for five minutes. You can do more in five minutes than most people I can know can do in five years. Um, the, okay, well it's going to, I'm going to have to take my pick. All right, all right, okay, I'm sorry. Uh, okay, I've got a choice here. I have to ask you about Cusey's Prophecies or I have to ask you about Compton. Okay, I'm going to save the Cusey's Prophecies for later, there's still a couple of things I don't understand in the Boulder paper. Now, I don't think I'm going to be able to ask you to put that in for five minutes, but you were describing an outline to Steve and myself the other evening about the exchanges between the NRO, and I also looked at his blog when I got back to England.

20:00 Well, if you can't see where he explains what the conditions are that you would have to put on, I certainly am not going to. But there's something going on in his blog. I mean, the man is. Man, he does talk about biomodules, biomodules and a couple of things. Uh, remember I didn't have the chance to study it, so I only had a couple of hours. I was busy downloading this stuff. Uh, but what's the sequence, I mean, what exactly were, you asked him, obviously, what is, what is, yeah, any of these spaces, and what, I think you said you've asked him about three or four stages, a succession of answers, each of which is unsatisfactory. Okay, the first answer was, you know, they talked to restrict you to pivot less, but how much are lots of math between others? Very few, because they're simple, like the algebra of all operators on Hilbert space. It's not simple, so it has hardly any... So anyway, so I came back and told him not to restrict it. So he didn't defend it. He said he would switch to the other opposite extreme, so to speak. He said it's any bimodule between... Electric space in which they both act and so on. Which is indeed a generalization. If you have a homomorphism, then you can define a bimodule which is just a second algebra which is acting on itself on one hand and the other one acting on it via the homomorphism.

22:30 So a bimodule is definitely, in all sorts of situations, do constitute a kind of generalization. This is, you know, the one thing you'll then have is the... If you can add these continuous maps, then that's not the way continuous maps behave, it's the way random continuous maps might behave, but not actually behave, and so on. So I said, well, this seems to be too general, for various half-baked reasons, I said. So that's when he came up with the third answer. Well, just go read the book. No, no, the third answer was that they are line-by-modules which are something like homomorphism. Fine. Yeah, and then finally, you know, each time I went away and I talked to my friend and I went back and asked him again, well, what specifically are the conditions you have to put on me? The first answer, you know, floored me. Homomorphism. Oh, okay, I'll go home and think about that. But I didn't go home. I just went in the corner and thought, you know, I think, so I came back. It was like that. It wasn't just standing there confronting him. No, no, no. So then finally the fourth answer, you don't read my book. I know. It's actually adorable to read from somebody, you know, of his... The position is probably the highest paid. We got this half-million-dollar chai from the Abdel Foundation, which was just formed in Norway, their famous son, Abdel, and we gave him this chai, again, for founding the subject, and I've known nothing about what that is, or use it might have, but just for founding the subject, kind of the refrain that you find in all sorts of... Citations of Connes, nothing about them. Except that he hasn't founded it at all, that's the whole issue. He produced the subject, which has been called non-conventional geometry. Well, it's called the subject. Yes, but founded, I assume, means that you have actually provided a clear understanding of what the foundations of the subject are. Ah, well. And what is the issue about the meritoriousness at the time of that? Well, certain bimodules induce isomorphisms of categories of modules. So there's an invertible bimodular, semi-invertible bimodular, right? That's the abstract general correspondent of the concrete general idea that two categories of modules should be equivalent as categories.

25:00 Being equivalent as categories does not imply such a way to preserve underlying sets or underlying definitions. Anyway, but typically, for example, if you keep classical Morita theory, if you have a, consider a 3x3 matrices. I was going to ask you about matrices. That's very interesting you said that. Well, there are modules, that's a ring. Modules over that ring form a category. And in fact, that's equivalent to the ordinary category vector spaces. Right. In other words, the real... By any means, the ring K is equivalent to the ring of n by n matrices, and what the functor does, you just take any module over K and you just take n copies of it, and then, of course, the matrices act on that. So there's a clear function going that way, but actually it's invertible because the matrix ring has enough idempotence in it that if you have an action of it, you can take these fixed spaces and you'll recover a family of modules. The matrix ring has to have enough idempotence, doesn't it? No, it doesn't. Any matrix ring would have enough idempotence. So idempotence by themselves would suggest that you have... The division into several parts of the module. But then other matrix, standard matrix elements prove that these different subspaces are actually isomorphic. So there's really only one little space which has been expanded, you see. Now you wouldn't know that at the first glance if you looked at modules over the matrix thing, but you know, it's a little analysis. And that's what a Morita equivalence is, it's keeping track of that, the weight of that small... Yeah, yeah, so this is sort of a trivial case in a way, but certainly any ring is equivalent to n by n matrices over it. Then there are more subtle equivalences where you get more into the structure of the particular ring, but at least that suggests what sort of thing it is.

27:30 You can change the underlying set if you take the end power of the given state, but there's more structure on it, namely all these idempotents and little isomorphisms, so that you can actually recover the smaller. That's when we read the equivalent. Right. And the abstract, then, is sort of cognizant of the biomarkers, and sort of all concentrated in, I guess it's the n-dimensional, I mean the n-fold product of the given. As an abstract general, it's got to be independent of a particular vector space, which these big functions call it. Indimensional space seen as a module over the matrices in the obvious way, and over the yields in the obvious way. Something simple like that would be the bimodule, which concentrates or affects a particular marine equivalent. So again, in general, if you have two rings in a bimodule... It's going to give rise to an additive function in one direction, which has an adjoint, and sometimes that adjoint is actually equivalent, which means that the bimodule itself and another bimodule will be the same. Right, okay. And that's the condition which is breaking down. The idea of organic equivalence is the objective one. We want to understand what it means that two logic categories are equivalent independent of the underlying set. Of course, you know, as in my thesis and many other places, the equivalence which preserves the underlying set is just an isomorphism of the range themselves. So if we're considering C-star algebras, these are arranged also. There is meridian equivalence between them. That'll sort of be factored into, you know, if one accepted the general bimodule as the notion of continuous math, there would be this sudden, strange multiplicity of how these things are represented.

30:00 Right, right. I don't think John could use that. I suspect they simply don't know how to talk math. No, no, no. I did get to think about one point, though, you see. It would be rather difficult. I mean, I wanted to know, what is a curve? ...a map from an interval in one of these monstrous spaces... Yes, absolutely, I'm glad about that. ...for example, a curve, or even a projection map, or, you know, very simple maps that are coming up all the time, and how they, how it is actually represented in terms of... I was thinking now that if the, if you think of the interval as a classical communicative space, then, then... Then mapping that space into a general non-commutative space by a homomorphism would involve a homomorphism from a non-commutative algebra to a commutative one, and depending on the nature of the non-commutative algebra, these can be rather rare, actually, because there's lots of non-commutative algebras still, and many of them will... It's basically dividing the algebra by its communicators. It's the communicator ideal, and the algebra along that. That would have to actually map into functions on the integral. Of course, they probably have a non-communicative model of the energy as well, but I'm not sure. It just has one variable. The non-communicative algebra generated by one variable happens to be commuted, if you understand what I'm talking about. Even if it's a formal power series, or a monosat, or whatever it is exactly, it's only got one generator. If you have two generators, you just have one generator, so I don't know how they explain one generator. No, I agree. I don't see whether it's anything at all. I did try to look at it a couple of times. Well, I think it's deplorable that the poem is not more patient of exposition. Actually, I haven't seen any publications on this so-called subject for some time now. It may have already died out.

32:30 I have an oyster that I call the polyg. Now, this guy's had no math in his head. Well, he claims that he has a way of generalizing the natural growth of the cohomology. Well, I thought it was Gabriel who thought that. The non-communicative geometry, if anything, is basically linear algebra. Algebras have additions, so that's, that's, you know... This is Rob Gabriel and his thesis, 1952, which was developed in a so-called Grotendie category, which is a linear category, a viewing category, particularly on the infinite directed co-levels of communication. Such categories can be seen as exactly the same as subcategories. So-called localization is about modular categories or more, somewhat more naturally, categories of additive functions on small additive categories.

35:00 For example, a ring is a small ring. Just glopping together the objects is not very natural. The kind of typical reflective sub-category, you know, it's like you're preserved from it, but it'll still be a good category because the definition has to do with filtered tone, I must say, to be preserved by any reflection, and finally, pretty much, it's a hypothesis, a hypothesis. Sorry, say that laugh again. It would be preserved by me. Any reflection preserves any co-limits, particularly the filtered co-limits that are involved in the mathematics machine. And the other congenial in mathematics machine is left-exactness, which by half-hypothesis, which is left-exact, will give you another Grund B category. And conversely, any Grund B category is in fact an exact reflection of... But now the technology of explaining how to single out the subcategory, just as in the case of Rodenby's topos, you know, it's in terms of covering, that these are the objects which think that certainly they're actually isomorphisms even though they aren't in the larger. Well, that's the same idea as you would in deep topology, except it's probably required in the additive linear situation, so those are called Gabriel topologies. The axioms are similar to those. And I see how they involve this notion of localization. Yeah, the localization simply means concretely what it meant was, take an open subset of a bigger space, then the ring of functions on the big space maps into the ring of functions on the open set, and that's what's called a localization because The essential point is to make invertible those functions whose restrictions in the open set is, quote, non-zero.

37:30 You take the range of functions and you make, for example, one function invertible, and you define the open subset where that function does not vanish. But that kind of epimorphic map is a good example of what introduces an inclusion to the module. Over the open set and the margins over the big set, and you have it as an adjoin and so on, so it's a typical example, or almost typical, of this abstract notion of localization, of the concrete localization. It seems to me a little bit strange, actually, in many contexts, between terminology and... Well, certainly in his problem of non-convection as a pathologist, he gave in the Imperial College, that he gave his point, and Einstein did make several allusions to that within this. So he must be the god of all pathologies. Although I don't recall his... But that really must be... Well, I hate to take the attitude of Algebraic and Geometry, but I'm not saying... Yeah, yeah, I think there is... There may be cases where you're doing this kind of pure Algebraic technology and losing sight of just what the fundamental relation between Algebraic and Geometric concepts is. I mean, you see, this is the problem with Malleus and Raptus for always saying a kind of general mode of order. You must understand that in geometry, there is no notion, there must be no notion of geometry, except when you go back down to Stanford, it comes out louder. It's just too early. My whole talk last night was a denial of that. Exactly, exactly. I understood, even I understood that very clearly, and I think that is completely the wrong position. But that's why I have this resurgence in the non-communicative definition of cohomology as the image of the map between the two different adlerites of the including.

40:00 Nothing to do with propriety, necessarily. Yes, this is, I know, I understand much more clearly where it is that they've taken the wrong theory. I would say, in defense of that, that he and Amalek, particularly, they are serious people, and I think will benefit enormously from being exposed to these. I, Bill, thank you again a million times. Another point is that, you see, that even in algebraic geometry, where this idea... The space coming from algebras is most used. Everybody working in that field knows, but it's only certain spaces that turn plot exactly to algebras. They're the more general ones that can be perhaps covered by several... It's something like the picture I was drawing, you see, that you can use this subjective tool to grasp some object, but you may need to use the tool in several different interrelated ways. So that is the image which distinguishes the site of the affine schemes from the topos, or even the category of schemes, which is somewhere in between the affine schemes. You need to pass there and then it's inherent in algebraic math. But in any case, most of these spaces are merely obtained by gluing together spaces which individually are entirely determined by it. Especially in C-infinity, especially the classical C-infinity manifold is one I can mention. By a theorem of Miller, they are in certain terms determined by their function, just the ring, but still within this realm of manifolds, you could get the idea that everything is just a ring. Even there, I mean, it would take a certain kind of code, subcategories, sub-spaces of manifolds, and certainly the function spaces are not going to be function spaces,

42:30 It's certainly not to be determined just by its ring. I don't know, I thought of a slogan and I'm not sure. This thing is bigger than both of us. So if you have two affine spaces, they may cover a projected space. So this projected space, you know, one says to the other, this thing is bigger than both of us. Then we have it, you see, it's really entirely ours, but, you know, but it's not an algebraic geometry as well as an arithmetic geometry. Projective space is not equivalent to the center. It's not equivalent to the center, right. Again, it's smooth geometry it is. Because it has many more functions. You can actually embed projective space in a higher dimensional map of space. But you can't do it with analytic or algebraic functions. It's a distinction of... I've never seen this point of doubt in the book, but it's very fundamental. Is it not perhaps the whole clue to where this non-chromative, geometry program has taken the wrong turn from the beginning? Because they do seem, so far as I'm able to follow the idea, they do seem to think that everything is determined by... No, oh yeah, oh yeah. The idea is that there is nothing more than space except elements in the ring. Yeah, yeah. Because they constantly cite Gell-Pan. We're doing the commutative case. And then we've just taken the non-conversative version of it. Yeah, yeah. It's fairly interesting. Well, of course, Galpin had more general theory. But it's funny, the mathematicians who claim to work on this stuff, they only mention the commutative. Well, there's all kinds of things wrong with this non-community job.

45:00 Sure. Another is the very fact that it's based on the sense of product, rather than the truth of it. So that, for example, a so-called quantum group is not a group at all. It's only a couple of poor relatives of mine. Genevieve agrees with me on a totally different political position, obviously, but he agrees with me on the most likely thing in the wording of the Fields Medal to Piotr, the one who's now in Chicago. Grenfell talks to me on non-TVL, on quantum groups. There's no doubt something to be engineered by the CIA and the national security. Part of the drive to recruit several Soviet mathematicians and U.S. universities. So there, as soon as something is given the notoriety of being awarded the field level at the International Congress, then there are several people who jump on the bandwagon. So there was a flurry of papers on Pomp and Groot, which again has pretty much died down in the West. I attended one of his seminars. So the slogan is like pocket groups. Yeah, he agrees with it. He refers to these people as mafia, hysteria, a certain section of Jewish intellectuals, who are completely without roots, quote-unquote. He calls it the Mafia, for short, and this is his elaboration of it, because they're without roots. Not all the Jewish people are without roots. They don't own land.

47:30 If they only owned land and castles, they'd understand what the nature of functional spaces are, they wouldn't do. They'd understand the need for confusing closeness. He attributes the term mafia also to Manin. Manin, I was going to ask. Manin, he recently received the Cantor Prize. What is the analytical line on Manning? Did you notice that Manning had been put in as, there were two or three bibliographic references to Manning in Rooney's reading list? I noticed. I thought you might have done that. I didn't. That's why I didn't bother to mention it until now, because I guess you would already have picked up on that. There is this 12-page address that Manning gave on the Cancer Prize. It has some things that are true, but some wild stories as well. And have you read his very general book on the nature of mathematics and physics? Ah, no, I think he hasn't written the same title as well as what we've talked about yesterday. Again, some of the things he says seem to be perfectly correct and unexceptional, but I don't think there's any correct depth there. He's also, I understand, prone to make very sweeping generalizations about national character, kind of well-known in reaction, as well. Well, Bill, that has been absolutely fascinating, but I am afraid I've been very naughty. I'm afraid it's going to be late before Conrad will leave me. I don't think you're going to lose too much sleep over that, but I'm afraid you are going to lose your ten minutes of sleep. Merci. Merci. Thank you. Well, you sound, your English accent is so perfect, aren't you? I don't think you know all of that stuff. Oh, excuse me, well, I've got a good one in my mouth. That's all right, chum. Must be all the coffee you guessed that you get.

50:00 Nah, mate, I mean, you can't drink, can't you? Well, I'm sorry, no, I couldn't. Actually, so is the other one that you looked at, right? It's an impediment of that. So, if you have a morphism between abstract and an interpretation of one theory and another... That will induce a structure between the corresponding concrete generals, or categories, or models, or whatever you call it. As you explained at the opening of your talk. But this will, well, it depends on what amorphism the feedback is, but you can ask more generally, what kind of adjoint structures or equivalences are there between these large categories of concrete generals, and how could that be reflected as a relation between the abstracts? The first thing that comes up is that the generalization of something like a bimodule, something like a bimodule, will not be exactly like a map, but a more general thing of a bimodule. And of course, a few bimodules are available. That's what the standard is here. Or in fact, they're usually there. Again, you can have adjoint pairs of bimodules, which will induce adjoint pairs of functions between the big categories. Thank you for your attention. But you can usually find some way to reflect on this objective thing, this objective level, but it will typically take the form of something like a bible. Right. So this applies to many different situations. D in rich categories for arbitrary D. For example, metric spaces. Yes, indeed. Going back to the metric spaces, for example.

52:30 Nipchits, maps, and so on. Or just plain categories and set value pre-sheets and pre-sheets on small categories. You see actually with these, and the connection is fairly, because considering the categories of all realizations, you can usually find the subjecting within the object. In other words, these, for example, in geometrical pictures, again, the figure forms. ...which are going to play the role of abstract generals. They're found among the spaces, so they're a tiny part of the concrete general also. Now, of course, if you impose strong properties on your realizations, then it may no longer be the case that you can realize straightforwardly the abstract general within the concrete general. But for things like modules and metric spaces and figure forms. Cartier was saying something about how Groton did very recently. We recently distributed an old paper about shapes, this idea of figure forms. But in fact, this was the paper that you were discussing with Steve when you were in Newton-Helmsfield, right? I'm not sure. I'm not sure if it's the same now. It's okay, we're in Pleiades Island, okay, we're in the right place. It's still quite a way, I'm sure. No, it is. Okay, we're all right. Oh, only two? There aren't so many stops on this transit system. It's good. Of course, it's a very nice example. I can't really put the half an hour of your time that he wasted, you would have put it to much better use.

55:00 Still, I did get to get the private tutorial, so I can't really leave. Please put the blame on me, as I say. That's what I'm seeing. It's coming together quite beautifully. As I say, I really hope we can attempt a synoptic presentation of all of this next year and get something out that will be a book, not just a conference proceedings. We'll see what we can get on here. Yeah. So he talks about my paper on the fact that he points out that it's a minor error. Yeah, it is though. But he is saying nobody followed this up. You see, well, it's true in the sense that the formal development of such a theory was not carried out, except by these few... Except by a few people like Colin McLaughlin, who has developed it. Yeah, but the point is that, in fact, everyone works in that framework. Every category appears, and they're not quite so exquisite. Basically, the world is punctured, you see. Categories are punctured. You see, if you want to have one framework, that's it. So it's not really true of the same power. The reason I didn't publish a new version of this might have been a conceptual way to express some of the simple actions, but it's going to be... Yeah, we don't have to put in all the great big universes. The reason I delayed because I thought it was really going to be a second publication and show four and stage, well, this one has exactly ten maps, stuff like that.

57:30 Well, you should be able to prove that. You need to have something like one, two, three, four objects. Those are the figure type. Yes, put your finger here. That's it. This is your stop. It is? Oh my god. I think. Yes, this is it. This is the last of them. Go ahead, right guy. Yeah? Uh, hey, welcome to Le Mans. I'm Michel. Oh, we won't have to sign that. No, I'm saying that, of course, at that time, at the time we wrote that thing, we signed it. Yeah. Of course, it's time that we introduced it. There's a vocabulary about physics, quantum physics, although it's really interesting to me.